LMFDB - Embedded newform 78.5.c.a.53.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,5,Mod(53,78)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("78.53");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	78.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.06285712054$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 154 x^{14} - 938 x^{13} + 8635 x^{12} - 39980 x^{11} + 231013 x^{10} + \cdots + 81960012 $ x^16 - 8x^15 + 154x^14 - 938x^13 + 8635x^12 - 39980x^11 + 231013x^10 - 809984x^9 + 3221227x^8 - 8428176x^7 + 23768367x^6 - 44807058x^5 + 89131275x^4 - 111997512x^3 + 150809040x^2 - 101086056*x + 81960012
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{9}\cdot 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.7
Root			$0.500000 + 1.80454i$ of defining polynomial
Character	$\chi$	$=$	78.53
Dual form			78.5.c.a.53.15

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} +(7.25482 + 5.32612i) q^{3} -8.00000 q^{4} -41.0662i q^{5} +(15.0646 - 20.5197i) q^{6} -47.8807 q^{7} +22.6274i q^{8} +(24.2648 + 77.2801i) q^{9} +O(q^{10})$	\(q-2.82843i q^{2} +(7.25482 + 5.32612i) q^{3} -8.00000 q^{4} -41.0662i q^{5} +(15.0646 - 20.5197i) q^{6} -47.8807 q^{7} +22.6274i q^{8} +(24.2648 + 77.2801i) q^{9} -116.153 q^{10} -221.470i q^{11} +(-58.0386 - 42.6090i) q^{12} +46.8722 q^{13} +135.427i q^{14} +(218.724 - 297.928i) q^{15} +64.0000 q^{16} -300.682i q^{17} +(218.581 - 68.6313i) q^{18} -130.401 q^{19} +328.529i q^{20} +(-347.366 - 255.018i) q^{21} -626.413 q^{22} +344.605i q^{23} +(-120.516 + 164.158i) q^{24} -1061.43 q^{25} -132.575i q^{26} +(-235.566 + 689.891i) q^{27} +383.045 q^{28} -298.001i q^{29} +(-842.667 - 618.644i) q^{30} +1559.62 q^{31} -181.019i q^{32} +(1179.58 - 1606.73i) q^{33} -850.458 q^{34} +1966.28i q^{35} +(-194.119 - 618.241i) q^{36} -226.295 q^{37} +368.829i q^{38} +(340.049 + 249.647i) q^{39} +929.222 q^{40} -700.729i q^{41} +(-721.301 + 982.499i) q^{42} +3404.20 q^{43} +1771.76i q^{44} +(3173.60 - 996.464i) q^{45} +974.689 q^{46} +2023.04i q^{47} +(464.308 + 340.872i) q^{48} -108.440 q^{49} +3002.18i q^{50} +(1601.47 - 2181.40i) q^{51} -374.977 q^{52} +3631.22i q^{53} +(1951.31 + 666.283i) q^{54} -9094.95 q^{55} -1083.42i q^{56} +(-946.033 - 694.530i) q^{57} -842.873 q^{58} +218.371i q^{59} +(-1749.79 + 2383.42i) q^{60} +2899.00 q^{61} -4411.27i q^{62} +(-1161.82 - 3700.23i) q^{63} -512.000 q^{64} -1924.86i q^{65} +(-4544.51 - 3336.35i) q^{66} +3907.95 q^{67} +2405.46i q^{68} +(-1835.41 + 2500.05i) q^{69} +5561.47 q^{70} -364.565i q^{71} +(-1748.65 + 549.050i) q^{72} -749.233 q^{73} +640.060i q^{74} +(-7700.50 - 5653.32i) q^{75} +1043.21 q^{76} +10604.2i q^{77} +(706.108 - 961.804i) q^{78} +7073.30 q^{79} -2628.24i q^{80} +(-5383.44 + 3750.38i) q^{81} -1981.96 q^{82} -1378.77i q^{83} +(2778.93 + 2040.15i) q^{84} -12347.9 q^{85} -9628.54i q^{86} +(1587.19 - 2161.94i) q^{87} +5011.30 q^{88} -11357.8i q^{89} +(-2818.43 - 8976.30i) q^{90} -2244.27 q^{91} -2756.84i q^{92} +(11314.8 + 8306.72i) q^{93} +5722.01 q^{94} +5355.06i q^{95} +(964.131 - 1313.26i) q^{96} -9460.80 q^{97} +306.715i q^{98} +(17115.3 - 5373.94i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9}+O(q^{10}) $ 16 * q + 12 * q^3 - 128 * q^4 - 96 * q^6 + 80 * q^7 + 116 * q^9	$ 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9} - 64 q^{10} - 96 q^{12} + 40 q^{15} + 1024 q^{16} + 448 q^{18} + 816 q^{19} - 128 q^{21} - 1344 q^{22} + 768 q^{24} - 3896 q^{25} - 2520 q^{27} - 640 q^{28} - 544 q^{30} + 5472 q^{31} + 2920 q^{33} + 1408 q^{34} - 928 q^{36} - 3456 q^{37} + 512 q^{40} - 3520 q^{42} - 3112 q^{43} + 3632 q^{45} + 4864 q^{46} + 768 q^{48} + 10776 q^{49} + 4132 q^{51} + 10080 q^{54} - 13792 q^{55} - 17264 q^{57} - 10688 q^{58} - 320 q^{60} + 11504 q^{61} - 2904 q^{63} - 8192 q^{64} - 8512 q^{66} - 10080 q^{67} + 16320 q^{69} + 8576 q^{70} - 3584 q^{72} + 19568 q^{73} - 4840 q^{75} - 6528 q^{76} - 7216 q^{79} - 6364 q^{81} + 6592 q^{82} + 1024 q^{84} + 8320 q^{85} + 14664 q^{87} + 10752 q^{88} + 7360 q^{90} - 1352 q^{91} + 49392 q^{93} + 4224 q^{94} - 6144 q^{96} - 45056 q^{97} - 51016 q^{99}+O(q^{100}) $ 16 * q + 12 * q^3 - 128 * q^4 - 96 * q^6 + 80 * q^7 + 116 * q^9 - 64 * q^10 - 96 * q^12 + 40 * q^15 + 1024 * q^16 + 448 * q^18 + 816 * q^19 - 128 * q^21 - 1344 * q^22 + 768 * q^24 - 3896 * q^25 - 2520 * q^27 - 640 * q^28 - 544 * q^30 + 5472 * q^31 + 2920 * q^33 + 1408 * q^34 - 928 * q^36 - 3456 * q^37 + 512 * q^40 - 3520 * q^42 - 3112 * q^43 + 3632 * q^45 + 4864 * q^46 + 768 * q^48 + 10776 * q^49 + 4132 * q^51 + 10080 * q^54 - 13792 * q^55 - 17264 * q^57 - 10688 * q^58 - 320 * q^60 + 11504 * q^61 - 2904 * q^63 - 8192 * q^64 - 8512 * q^66 - 10080 * q^67 + 16320 * q^69 + 8576 * q^70 - 3584 * q^72 + 19568 * q^73 - 4840 * q^75 - 6528 * q^76 - 7216 * q^79 - 6364 * q^81 + 6592 * q^82 + 1024 * q^84 + 8320 * q^85 + 14664 * q^87 + 10752 * q^88 + 7360 * q^90 - 1352 * q^91 + 49392 * q^93 + 4224 * q^94 - 6144 * q^96 - 45056 * q^97 - 51016 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.82843i		−	0.707107i
$2$		−	2.82843i		−	0.707107i
$3$	7.25482	+	5.32612i	0.806091	+	0.591791i
$3$	7.25482	+	5.32612i	0.806091	+	0.591791i
$4$	−8.00000			−0.500000
$5$		−	41.0662i		−	1.64265i	−0.570462	−	0.821324i	$-0.693236\pi$
$5$		−	41.0662i		−	1.64265i	0.570462	−	0.821324i	$-0.306764\pi$
$6$	15.0646	−	20.5197i	0.418460	−	0.569993i
$7$	−47.8807			−0.977157			−0.488578	−	0.872520i	$-0.662484\pi$
$7$	−47.8807			−0.977157			−0.488578	+	0.872520i	$0.662484\pi$
$8$			22.6274i			0.353553i
$9$	24.2648	+	77.2801i	0.299566	+	0.954076i
$10$	−116.153			−1.16153
$11$		−	221.470i		−	1.83033i	−0.403075	−	0.915167i	$-0.632059\pi$
$11$		−	221.470i		−	1.83033i	0.403075	−	0.915167i	$-0.367941\pi$
$12$	−58.0386	−	42.6090i	−0.403046	−	0.295896i
$13$	46.8722			0.277350
$13$	46.8722			0.277350
$14$			135.427i			0.690954i
$15$	218.724	−	297.928i	0.972105	−	1.32412i
$16$	64.0000			0.250000
$17$		−	300.682i		−	1.04042i	−0.854037	−	0.520212i	$-0.825853\pi$
$17$		−	300.682i		−	1.04042i	0.854037	−	0.520212i	$-0.174147\pi$
$18$	218.581	−	68.6313i	0.674633	−	0.211825i
$19$	−130.401			−0.361221			−0.180610	−	0.983555i	$-0.557807\pi$
$19$	−130.401			−0.361221			−0.180610	+	0.983555i	$0.557807\pi$
$20$			328.529i			0.821324i
$21$	−347.366	−	255.018i	−0.787677	−	0.578273i
$22$	−626.413			−1.29424
$23$			344.605i			0.651427i	0.945469	+	0.325713i	$0.105604\pi$
$23$			344.605i			0.651427i	−0.945469	+	0.325713i	$0.894396\pi$
$24$	−120.516	+	164.158i	−0.209230	+	0.284996i
$25$	−1061.43			−1.69829
$26$		−	132.575i		−	0.196116i
$27$	−235.566	+	689.891i	−0.323136	+	0.946352i
$28$	383.045			0.488578
$29$		−	298.001i		−	0.354341i	−0.984180	−	0.177170i	$-0.943306\pi$
$29$		−	298.001i		−	0.354341i	0.984180	−	0.177170i	$-0.0566944\pi$
$30$	−842.667	−	618.644i	−0.936297	−	0.687382i
$31$	1559.62			1.62291			0.811456	−	0.584413i	$-0.198675\pi$
$31$	1559.62			1.62291			0.811456	+	0.584413i	$0.198675\pi$
$32$		−	181.019i		−	0.176777i
$33$	1179.58	−	1606.73i	1.08318	−	1.47542i
$34$	−850.458			−0.735690
$35$			1966.28i			1.60512i
$36$	−194.119	−	618.241i	−0.149783	−	0.477038i
$37$	−226.295			−0.165300			−0.0826498	−	0.996579i	$-0.526338\pi$
$37$	−226.295			−0.165300			−0.0826498	+	0.996579i	$0.526338\pi$
$38$			368.829i			0.255422i
$39$	340.049	+	249.647i	0.223569	+	0.164133i
$40$	929.222			0.580764
$41$		−	700.729i		−	0.416852i	−0.978038	−	0.208426i	$-0.933166\pi$
$41$		−	700.729i		−	0.416852i	0.978038	−	0.208426i	$-0.0668341\pi$
$42$	−721.301	+	982.499i	−0.408901	+	0.556972i
$43$	3404.20			1.84110			0.920552	−	0.390620i	$-0.127739\pi$
$43$	3404.20			1.84110			0.920552	+	0.390620i	$0.127739\pi$
$44$			1771.76i			0.915167i
$45$	3173.60	−	996.464i	1.56721	−	0.492081i
$46$	974.689			0.460628
$47$			2023.04i			0.915816i	0.889000	+	0.457908i	$0.151401\pi$
$47$			2023.04i			0.915816i	−0.889000	+	0.457908i	$0.848599\pi$
$48$	464.308	+	340.872i	0.201523	+	0.147948i
$49$	−108.440			−0.0451645
$50$			3002.18i			1.20087i
$51$	1601.47	−	2181.40i	0.615713	−	0.838676i
$52$	−374.977			−0.138675
$53$			3631.22i			1.29271i	0.763037	+	0.646355i	$0.223707\pi$
$53$			3631.22i			1.29271i	−0.763037	+	0.646355i	$0.776293\pi$
$54$	1951.31	+	666.283i	0.669172	+	0.228492i
$55$	−9094.95			−3.00659
$56$		−	1083.42i		−	0.345477i
$57$	−946.033	−	694.530i	−0.291177	−	0.213767i
$58$	−842.873			−0.250557
$59$			218.371i			0.0627322i	0.999508	+	0.0313661i	$0.00998578\pi$
$59$			218.371i			0.0627322i	−0.999508	+	0.0313661i	$0.990014\pi$
$60$	−1749.79	+	2383.42i	−0.486052	+	0.662062i
$61$	2899.00			0.779092			0.389546	−	0.921007i	$-0.372632\pi$
$61$	2899.00			0.779092			0.389546	+	0.921007i	$0.372632\pi$
$62$		−	4411.27i		−	1.14757i
$63$	−1161.82	−	3700.23i	−0.292723	−	0.932282i
$64$	−512.000			−0.125000
$65$		−	1924.86i		−	0.455588i
$66$	−4544.51	−	3336.35i	−1.04328	−	0.765921i
$67$	3907.95			0.870562			0.435281	−	0.900295i	$-0.356649\pi$
$67$	3907.95			0.870562			0.435281	+	0.900295i	$0.356649\pi$
$68$			2405.46i			0.520212i
$69$	−1835.41	+	2500.05i	−0.385509	+	0.525109i
$70$	5561.47			1.13499
$71$		−	364.565i		−	0.0723200i	−0.999346	−	0.0361600i	$-0.988487\pi$
$71$		−	364.565i		−	0.0723200i	0.999346	−	0.0361600i	$-0.0115126\pi$
$72$	−1748.65	+	549.050i	−0.337317	+	0.105913i
$73$	−749.233			−0.140595			−0.0702977	−	0.997526i	$-0.522395\pi$
$73$	−749.233			−0.140595			−0.0702977	+	0.997526i	$0.522395\pi$
$74$			640.060i			0.116885i
$75$	−7700.50	−	5653.32i	−1.36898	−	1.00503i
$76$	1043.21			0.180610
$77$			10604.2i			1.78852i
$78$	706.108	−	961.804i	0.116060	−	0.158087i
$79$	7073.30			1.13336			0.566680	−	0.823938i	$-0.308228\pi$
$79$	7073.30			1.13336			0.566680	+	0.823938i	$0.308228\pi$
$80$		−	2628.24i		−	0.410662i
$81$	−5383.44	+	3750.38i	−0.820521	+	0.571617i
$82$	−1981.96			−0.294759
$83$		−	1378.77i		−	0.200141i	−0.994980	−	0.100071i	$-0.968093\pi$
$83$		−	1378.77i		−	0.200141i	0.994980	−	0.100071i	$-0.0319069\pi$
$84$	2778.93	+	2040.15i	0.393839	+	0.289137i
$85$	−12347.9			−1.70905
$86$		−	9628.54i		−	1.30186i
$87$	1587.19	−	2161.94i	0.209696	−	0.285631i
$88$	5011.30			0.647121
$89$		−	11357.8i		−	1.43389i	−0.697131	−	0.716944i	$-0.745540\pi$
$89$		−	11357.8i		−	1.43389i	0.697131	−	0.716944i	$-0.254460\pi$
$90$	−2818.43	−	8976.30i	−0.347954	−	1.10818i
$91$	−2244.27			−0.271015
$92$		−	2756.84i		−	0.325713i
$93$	11314.8	+	8306.72i	1.30822	+	0.960426i
$94$	5722.01			0.647579
$95$			5355.06i			0.593358i
$96$	964.131	−	1313.26i	0.104615	−	0.142498i
$97$	−9460.80			−1.00551			−0.502753	−	0.864430i	$-0.667679\pi$
$97$	−9460.80			−1.00551			−0.502753	+	0.864430i	$0.667679\pi$
$98$			306.715i			0.0319362i
$99$	17115.3	−	5373.94i	1.74628	−	0.548305i
$100$	8491.45			0.849145
$101$			1750.64i			0.171614i	0.996312	+	0.0858072i	$0.0273469\pi$
$101$			1750.64i			0.171614i	−0.996312	+	0.0858072i	$0.972653\pi$
$102$	−6169.92	−	4529.64i	−0.593033	−	0.435375i
$103$	−15027.3			−1.41647			−0.708233	−	0.705978i	$-0.750508\pi$
$103$	−15027.3			−1.41647			−0.708233	+	0.705978i	$0.750508\pi$
$104$			1060.60i			0.0980581i
$105$	−10472.6	+	14265.0i	−0.949899	+	1.29388i
$106$	10270.6			0.914084
$107$		−	19723.3i		−	1.72271i	−0.508001	−	0.861357i	$-0.669615\pi$
$107$		−	19723.3i		−	1.72271i	0.508001	−	0.861357i	$-0.330385\pi$
$108$	1884.53	−	5519.13i	0.161568	−	0.473176i
$109$	−5496.69			−0.462646			−0.231323	−	0.972877i	$-0.574305\pi$
$109$	−5496.69			−0.462646			−0.231323	+	0.972877i	$0.574305\pi$
$110$			25724.4i			2.12598i
$111$	−1641.73	−	1205.28i	−0.133247	−	0.0978229i
$112$	−3064.36			−0.244289
$113$		−	24419.0i		−	1.91236i	−0.292772	−	0.956182i	$-0.594578\pi$
$113$		−	24419.0i		−	1.91236i	0.292772	−	0.956182i	$-0.405422\pi$
$114$	−1964.43	+	2675.79i	−0.151156	+	0.205893i
$115$	14151.6			1.07006
$116$			2384.00i			0.177170i
$117$	1137.35	+	3622.29i	0.0830846	+	0.264613i
$118$	617.646			0.0443584
$119$			14396.9i			1.01666i
$120$	6741.34	+	4949.15i	0.468148	+	0.343691i
$121$	−34408.1			−2.35012
$122$		−	8199.61i		−	0.550901i
$123$	3732.17	−	5083.66i	0.246690	−	0.336021i
$124$	−12477.0			−0.811456
$125$			17922.6i			1.14705i
$126$	−10465.8	+	3286.11i	−0.659223	+	0.206986i
$127$	8060.15			0.499730			0.249865	−	0.968281i	$-0.419614\pi$
$127$	8060.15			0.499730			0.249865	+	0.968281i	$0.419614\pi$
$128$			1448.15i			0.0883883i
$129$	24696.9	+	18131.2i	1.48410	+	1.08955i
$130$	−5444.33			−0.322150
$131$			18556.2i			1.08130i	0.841248	+	0.540650i	$0.181822\pi$
$131$			18556.2i			1.08130i	−0.841248	+	0.540650i	$0.818178\pi$
$132$	−9436.63	+	12853.8i	−0.541588	+	0.737708i
$133$	6243.67			0.352969
$134$		−	11053.4i		−	0.615580i
$135$	28331.2	+	9673.82i	1.55452	+	0.530799i
$136$	6803.66			0.367845
$137$			7393.26i			0.393908i	0.980413	+	0.196954i	$0.0631050\pi$
$137$			7393.26i			0.393908i	−0.980413	+	0.196954i	$0.936895\pi$
$138$	7071.20	+	5191.32i	0.371308	+	0.272596i
$139$	25870.2			1.33897			0.669483	−	0.742827i	$-0.266516\pi$
$139$	25870.2			1.33897			0.669483	+	0.742827i	$0.266516\pi$
$140$		−	15730.2i		−	0.802562i
$141$	−10774.9	+	14676.8i	−0.541972	+	0.738231i
$142$	−1031.15			−0.0511380
$143$		−	10380.8i		−	0.507643i
$144$	1552.95	+	4945.93i	0.0748915	+	0.238519i
$145$	−12237.7			−0.582057
$146$			2119.15i			0.0994159i
$147$	−786.713	−	577.565i	−0.0364067	−	0.0267280i
$148$	1810.36			0.0826498
$149$			7609.68i			0.342763i	0.985205	+	0.171381i	$0.0548230\pi$
$149$			7609.68i			0.342763i	−0.985205	+	0.171381i	$0.945177\pi$
$150$	−15990.0	+	21780.3i	−0.710666	+	0.968013i
$151$	15709.3			0.688974			0.344487	−	0.938791i	$-0.388053\pi$
$151$	15709.3			0.688974			0.344487	+	0.938791i	$0.388053\pi$
$152$		−	2950.63i		−	0.127711i
$153$	23236.8	−	7296.00i	0.992642	−	0.311675i
$154$	29993.1			1.26468
$155$		−	64047.6i		−	2.66587i
$156$	−2720.39	−	1997.18i	−0.111785	−	0.0820667i
$157$	30517.0			1.23806			0.619031	−	0.785366i	$-0.287525\pi$
$157$	30517.0			1.23806			0.619031	+	0.785366i	$0.287525\pi$
$158$		−	20006.3i		−	0.801407i
$159$	−19340.3	+	26343.9i	−0.765015	+	1.04204i
$160$	−7433.77			−0.290382
$161$		−	16499.9i		−	0.636546i
$162$	10607.7	+	15226.7i	0.404194	+	0.580196i
$163$	−28377.2			−1.06806			−0.534029	−	0.845466i	$-0.679323\pi$
$163$	−28377.2			−1.06806			−0.534029	+	0.845466i	$0.679323\pi$
$164$			5605.83i			0.208426i
$165$	−65982.2	−	48440.8i	−2.42359	−	1.77928i
$166$	−3899.76			−0.141521
$167$		−	4013.41i		−	0.143907i	−0.997408	−	0.0719534i	$-0.977077\pi$
$167$		−	4013.41i		−	0.143907i	0.997408	−	0.0719534i	$-0.0229233\pi$
$168$	5770.41	−	7859.99i	0.204450	−	0.278486i
$169$	2197.00			0.0769231
$170$			34925.1i			1.20848i
$171$	−3164.15	−	10077.4i	−0.108209	−	0.344632i
$172$	−27233.6			−0.920552
$173$			27816.5i			0.929416i	0.885464	+	0.464708i	$0.153841\pi$
$173$			27816.5i			0.929416i	−0.885464	+	0.464708i	$0.846159\pi$
$174$	−6114.89	−	4489.25i	−0.201972	−	0.148277i
$175$	50822.1			1.65950
$176$		−	14174.1i		−	0.457583i
$177$	−1163.07	+	1584.24i	−0.0371244	+	0.0505679i
$178$	−32124.8			−1.01391
$179$		−	32595.8i		−	1.01731i	−0.860969	−	0.508657i	$-0.830142\pi$
$179$		−	32595.8i		−	1.01731i	0.860969	−	0.508657i	$-0.169858\pi$
$180$	−25388.8	+	7971.71i	−0.783605	+	0.246041i
$181$	−23110.1			−0.705416			−0.352708	−	0.935733i	$-0.614739\pi$
$181$	−23110.1			−0.705416			−0.352708	+	0.935733i	$0.614739\pi$
$182$			6347.76i			0.191636i
$183$	21031.7	+	15440.4i	0.628019	+	0.461060i
$184$	−7797.52			−0.230314
$185$			9293.08i			0.271529i
$186$	23495.0	−	32003.0i	0.679124	−	0.925048i
$187$	−66592.2			−1.90432
$188$		−	16184.3i		−	0.457908i
$189$	11279.1	−	33032.4i	0.315755	−	0.924735i
$190$	15146.4			0.419568
$191$			3686.10i			0.101042i	0.998723	+	0.0505208i	$0.0160881\pi$
$191$			3686.10i			0.101042i	−0.998723	+	0.0505208i	$0.983912\pi$
$192$	−3714.47	−	2726.97i	−0.100761	−	0.0739739i
$193$	52393.7			1.40658			0.703291	−	0.710902i	$-0.251713\pi$
$193$	52393.7			1.40658			0.703291	+	0.710902i	$0.251713\pi$
$194$			26759.2i			0.710999i
$195$	10252.0	−	13964.5i	0.269613	−	0.367246i
$196$	867.521			0.0225823
$197$			9637.14i			0.248322i	0.992262	+	0.124161i	$0.0396240\pi$
$197$			9637.14i			0.248322i	−0.992262	+	0.124161i	$0.960376\pi$
$198$	−15199.8	−	48409.3i	−0.387711	−	1.23480i
$199$	8725.42			0.220333			0.110167	−	0.993913i	$-0.464862\pi$
$199$	8725.42			0.220333			0.110167	+	0.993913i	$0.464862\pi$
$200$		−	24017.5i		−	0.600436i
$201$	28351.5	+	20814.2i	0.701752	+	0.515191i
$202$	4951.56			0.121350
$203$			14268.5i			0.346247i
$204$	−12811.8	+	17451.2i	−0.307857	+	0.419338i
$205$	−28776.3			−0.684742
$206$			42503.6i			1.00159i
$207$	−26631.1	+	8361.78i	−0.621510	+	0.195145i
$208$	2999.82			0.0693375
$209$			28879.9i			0.661154i
$210$	40347.5	+	29621.1i	0.914909	+	0.671680i
$211$	37891.0			0.851081			0.425541	−	0.904939i	$-0.360084\pi$
$211$	37891.0			0.851081			0.425541	+	0.904939i	$0.360084\pi$
$212$		−	29049.8i		−	0.646355i
$213$	1941.72	−	2644.85i	0.0427984	−	0.0582965i
$214$	−55786.0			−1.21814
$215$		−	139798.i		−	3.02428i
$216$	−15610.4	−	5330.26i	−0.334586	−	0.114246i
$217$	−74675.6			−1.58584
$218$			15547.0i			0.327140i
$219$	−5435.55	−	3990.51i	−0.113333	−	0.0832031i
$220$	72759.6			1.50330
$221$		−	14093.6i		−	0.288561i
$222$	−3409.04	+	4643.52i	−0.0691713	+	0.0942196i
$223$	30903.2			0.621433			0.310716	−	0.950503i	$-0.399431\pi$
$223$	30903.2			0.621433			0.310716	+	0.950503i	$0.399431\pi$
$224$			8667.33i			0.172739i
$225$	−25755.5	−	82027.6i	−0.508750	−	1.62030i
$226$	−69067.3			−1.35225
$227$			85286.5i			1.65512i	0.561380	+	0.827558i	$0.310271\pi$
$227$			85286.5i			1.65512i	−0.561380	+	0.827558i	$0.689729\pi$
$228$	7568.27	+	5556.24i	0.145588	+	0.106884i
$229$	−22841.9			−0.435573			−0.217786	−	0.975996i	$-0.569884\pi$
$229$	−22841.9			−0.435573			−0.217786	+	0.975996i	$0.569884\pi$
$230$		−	40026.8i		−	0.756650i
$231$	−56479.0	+	76931.2i	−1.05843	+	1.44171i
$232$	6742.98			0.125278
$233$			86336.0i			1.59030i	0.606411	+	0.795152i	$0.292609\pi$
$233$			86336.0i			1.59030i	−0.606411	+	0.795152i	$0.707391\pi$
$234$	10245.4	−	3216.90i	0.187110	−	0.0587497i
$235$	83078.4			1.50436
$236$		−	1746.97i		−	0.0313661i
$237$	51315.5	+	37673.3i	0.913592	+	0.670713i
$238$	40720.5			0.718885
$239$			50513.3i			0.884320i	0.896936	+	0.442160i	$0.145788\pi$
$239$			50513.3i			0.884320i	−0.896936	+	0.442160i	$0.854212\pi$
$240$	13998.3	−	19067.4i	0.243026	−	0.331031i
$241$	−24564.4			−0.422933			−0.211466	−	0.977385i	$-0.567824\pi$
$241$	−24564.4			−0.422933			−0.211466	+	0.977385i	$0.567824\pi$
$242$			97320.9i			1.66179i
$243$	−59030.8	−	1464.52i	−0.999692	−	0.0248018i
$244$	−23192.0			−0.389546
$245$			4453.22i			0.0741894i
$246$	−14378.8	−	10556.2i	−0.237603	−	0.174436i
$247$	−6112.16			−0.100185
$248$			35290.1i			0.573786i
$249$	7343.52	−	10002.8i	0.118442	−	0.161332i
$250$	50692.7			0.811084
$251$		−	11427.8i		−	0.181390i	−0.995879	−	0.0906952i	$-0.971091\pi$
$251$		−	11427.8i		−	0.181390i	0.995879	−	0.0906952i	$-0.0289089\pi$
$252$	9294.53	+	29601.8i	0.146361	+	0.466141i
$253$	76319.8			1.19233
$254$		−	22797.5i		−	0.353363i
$255$	−89581.6	−	65766.3i	−1.37765	−	1.01140i
$256$	4096.00			0.0625000
$257$		−	41797.5i		−	0.632825i	−0.948622	−	0.316413i	$-0.897522\pi$
$257$		−	41797.5i		−	0.632825i	0.948622	−	0.316413i	$-0.102478\pi$
$258$	51282.8	−	69853.3i	0.770428	−	1.04942i
$259$	10835.2			0.161524
$260$			15398.9i			0.227794i
$261$	23029.5	−	7230.93i	0.338068	−	0.106148i
$262$	52484.8			0.764594
$263$			8629.13i			0.124754i	0.998053	+	0.0623771i	$0.0198682\pi$
$263$			8629.13i			0.124754i	−0.998053	+	0.0623771i	$0.980132\pi$
$264$	36356.1	+	26690.8i	0.521638	+	0.382961i
$265$	149120.			2.12347
$266$		−	17659.8i		−	0.249587i
$267$	60493.2	−	82399.0i	0.848563	−	1.15584i
$268$	−31263.6			−0.435281
$269$		−	31181.1i		−	0.430911i	−0.976514	−	0.215455i	$-0.930876\pi$
$269$		−	31181.1i		−	0.430911i	0.976514	−	0.215455i	$-0.0691236\pi$
$270$	27361.7	−	80132.7i	0.375332	−	1.09921i
$271$	−113077.			−1.53970			−0.769848	−	0.638228i	$-0.779668\pi$
$271$	−113077.			−1.53970			−0.769848	+	0.638228i	$0.779668\pi$
$272$		−	19243.7i		−	0.260106i
$273$	−16281.8	−	11953.3i	−0.218462	−	0.160384i
$274$	20911.3			0.278535
$275$			235076.i			3.10844i
$276$	14683.3	−	20000.4i	0.192754	−	0.262555i
$277$	−82038.4			−1.06920			−0.534599	−	0.845106i	$-0.679537\pi$
$277$	−82038.4			−1.06920			−0.534599	+	0.845106i	$0.679537\pi$
$278$		−	73171.9i		−	0.946792i
$279$	37843.9	+	120528.i	0.486169	+	1.54838i
$280$	−44491.8			−0.567497
$281$			31056.3i			0.393312i	0.980473	+	0.196656i	$0.0630083\pi$
$281$			31056.3i			0.393312i	−0.980473	+	0.196656i	$0.936992\pi$
$282$	41512.2	+	30476.1i	0.522008	+	0.383232i
$283$	25497.7			0.318367			0.159183	−	0.987249i	$-0.449114\pi$
$283$	25497.7			0.318367			0.159183	+	0.987249i	$0.449114\pi$
$284$			2916.52i			0.0361600i
$285$	−28521.7	+	38850.0i	−0.351144	+	0.478301i
$286$	−29361.3			−0.358958
$287$			33551.4i			0.407330i
$288$	13989.2	−	4392.40i	0.168658	−	0.0529563i
$289$	−6888.82			−0.0824801
$290$			34613.6i			0.411576i
$291$	−68636.4	−	50389.4i	−0.810529	−	0.595049i
$292$	5993.86			0.0702977
$293$			93848.5i			1.09318i	0.837400	+	0.546591i	$0.184075\pi$
$293$			93848.5i			1.09318i	−0.837400	+	0.546591i	$0.815925\pi$
$294$	−1633.60	+	2225.16i	−0.0188995	+	0.0257435i
$295$	8967.66			0.103047
$296$		−	5120.48i		−	0.0584423i
$297$	152790.	+	52171.0i	1.73214	+	0.591448i
$298$	21523.4			0.242370
$299$			16152.4i			0.180673i
$300$	61604.0	+	45226.5i	0.684489	+	0.502517i
$301$	−162995.			−1.79905
$302$		−	44432.6i		−	0.487179i
$303$	−9324.12	+	12700.6i	−0.101560	+	0.138337i
$304$	−8345.64			−0.0903052
$305$		−	119051.i		−	1.27977i
$306$	−20636.2	−	65723.5i	−0.220388	−	0.701904i
$307$	170115.			1.80495			0.902476	−	0.430741i	$-0.141748\pi$
$307$	170115.			1.80495			0.902476	+	0.430741i	$0.141748\pi$
$308$		−	84833.2i		−	0.894262i
$309$	−109020.	−	80037.2i	−1.14180	−	0.838253i
$310$	−181154.			−1.88506
$311$			20237.3i			0.209234i	0.994513	+	0.104617i	$0.0333617\pi$
$311$			20237.3i			0.209234i	−0.994513	+	0.104617i	$0.966638\pi$
$312$	−5648.86	+	7694.43i	−0.0580299	+	0.0790437i
$313$	−20507.5			−0.209326			−0.104663	−	0.994508i	$-0.533376\pi$
$313$	−20507.5			−0.209326			−0.104663	+	0.994508i	$0.533376\pi$
$314$		−	86315.1i		−	0.875442i
$315$	−151954.	+	47711.4i	−1.53141	+	0.480840i
$316$	−56586.4			−0.566680
$317$			55302.9i			0.550338i	0.961396	+	0.275169i	$0.0887337\pi$
$317$			55302.9i			0.550338i	−0.961396	+	0.275169i	$0.911266\pi$
$318$	74511.7	+	54702.7i	0.736835	+	0.540947i
$319$	−65998.3			−0.648562
$320$			21025.9i			0.205331i
$321$	105049.	−	143089.i	1.01949	−	1.38866i
$322$	−46668.8			−0.450106
$323$			39209.2i			0.375822i
$324$	43067.5	−	30003.0i	0.410260	−	0.285808i
$325$	−49751.6			−0.471021
$326$			80262.9i			0.755231i
$327$	−39877.5	−	29276.1i	−0.372934	−	0.273790i
$328$	15855.7			0.147380
$329$		−	96864.4i		−	0.894895i
$330$	−137011.	+	186626.i	−1.25814	+	1.71374i
$331$	−93353.9			−0.852072			−0.426036	−	0.904706i	$-0.640090\pi$
$331$	−93353.9			−0.852072			−0.426036	+	0.904706i	$0.640090\pi$
$332$			11030.2i			0.100071i
$333$	−5491.02	−	17488.1i	−0.0495181	−	0.157708i
$334$	−11351.7			−0.101757
$335$		−	160485.i		−	1.43003i
$336$	−22231.4	−	16321.2i	−0.196919	−	0.144568i
$337$	60455.6			0.532325			0.266162	−	0.963928i	$-0.414244\pi$
$337$	60455.6			0.532325			0.266162	+	0.963928i	$0.414244\pi$
$338$		−	6214.05i		−	0.0543928i
$339$	130058.	−	177155.i	1.13172	−	1.54154i
$340$	98783.0			0.854524
$341$		−	345409.i		−	2.97047i
$342$	−28503.1	+	8949.57i	−0.243691	+	0.0765156i
$343$	120154.			1.02129
$344$			77028.3i			0.650929i
$345$	102667.	+	75373.2i	0.862569	+	0.633255i
$346$	78676.9			0.657196
$347$			67971.4i			0.564504i	0.959340	+	0.282252i	$0.0910815\pi$
$347$			67971.4i			0.564504i	−0.959340	+	0.282252i	$0.908919\pi$
$348$	−12697.5	+	17295.5i	−0.104848	+	0.142815i
$349$	−149995.			−1.23148			−0.615739	−	0.787950i	$-0.711143\pi$
$349$	−149995.			−1.23148			−0.615739	+	0.787950i	$0.711143\pi$
$350$		−	143747.i		−	1.17344i
$351$	−11041.5	+	32336.7i	−0.0896219	+	0.262471i
$352$	−40090.4			−0.323560
$353$			59709.0i			0.479171i	0.970875	+	0.239586i	$0.0770116\pi$
$353$			59709.0i			0.479171i	−0.970875	+	0.239586i	$0.922988\pi$
$354$	4480.91	+	3289.66i	0.0357569	+	0.0262509i
$355$	−14971.3			−0.118796
$356$			90862.6i			0.716944i
$357$	−76679.5	+	104447.i	−0.601649	+	0.819518i
$358$	−92194.7			−0.719350
$359$		−	37296.6i		−	0.289388i	−0.989476	−	0.144694i	$-0.953780\pi$
$359$		−	37296.6i		−	0.289388i	0.989476	−	0.144694i	$-0.0462198\pi$
$360$	22547.4	+	71810.4i	0.173977	+	0.554092i
$361$	−113317.			−0.869520
$362$			65365.4i			0.498805i
$363$	−249625.	−	183262.i	−1.89441	−	1.39078i
$364$	17954.2			0.135507
$365$			30768.1i			0.230949i
$366$	43672.1	−	59486.7i	0.326018	−	0.444076i
$367$	197785.			1.46845			0.734227	−	0.678904i	$-0.237545\pi$
$367$	197785.			1.46845			0.734227	+	0.678904i	$0.237545\pi$
$368$			22054.7i			0.162857i
$369$	54152.4	−	17003.1i	0.397709	−	0.124875i
$370$	26284.8			0.192000
$371$		−	173865.i		−	1.26318i
$372$	−90518.0	−	66453.8i	−0.654108	−	0.480213i
$373$	164614.			1.18317			0.591586	−	0.806242i	$-0.298502\pi$
$373$	164614.			1.18317			0.591586	+	0.806242i	$0.298502\pi$
$374$			188351.i			1.34656i
$375$	−95457.9	+	130025.i	−0.678812	+	0.924623i
$376$	−45776.1			−0.323790
$377$		−	13967.9i		−	0.0982765i
$378$	−93429.9	−	31902.1i	−0.653886	−	0.223272i
$379$	−102477.			−0.713426			−0.356713	−	0.934214i	$-0.616103\pi$
$379$	−102477.			−0.713426			−0.356713	+	0.934214i	$0.616103\pi$
$380$		−	42840.5i		−	0.296679i
$381$	58474.9	+	42929.4i	0.402828	+	0.295736i
$382$	10425.9			0.0714473
$383$			23132.1i			0.157695i	0.996887	+	0.0788474i	$0.0251240\pi$
$383$			23132.1i			0.157695i	−0.996887	+	0.0788474i	$0.974876\pi$
$384$	−7713.05	+	10506.1i	−0.0523075	+	0.0712491i
$385$	435472.			2.93791
$386$		−	148192.i		−	0.994603i
$387$	82602.4	+	263077.i	0.551532	+	1.75655i
$388$	75686.4			0.502753
$389$			108954.i			0.720017i	0.932949	+	0.360009i	$0.117226\pi$
$389$			108954.i			0.720017i	−0.932949	+	0.360009i	$0.882774\pi$
$390$	−39497.6	−	28997.2i	−0.259682	−	0.190645i
$391$	103617.			0.677759
$392$		−	2453.72i		−	0.0159681i
$393$	−98832.5	+	134622.i	−0.639904	+	0.871626i
$394$	27258.0			0.175590
$395$		−	290474.i		−	1.86171i
$396$	−136922.	+	42991.5i	−0.873138	+	0.274153i
$397$	236633.			1.50139			0.750697	−	0.660647i	$-0.229718\pi$
$397$	236633.			1.50139			0.750697	+	0.660647i	$0.229718\pi$
$398$		−	24679.2i		−	0.155799i
$399$	45296.7	+	33254.6i	0.284525	+	0.208884i
$400$	−67931.6			−0.424573
$401$		−	219070.i		−	1.36237i	−0.732111	−	0.681185i	$-0.761465\pi$
$401$		−	219070.i		−	1.36237i	0.732111	−	0.681185i	$-0.238535\pi$
$402$	58871.5	−	80190.1i	0.364295	−	0.496214i
$403$	73102.7			0.450115
$404$		−	14005.1i		−	0.0858072i
$405$	154014.	+	221077.i	0.938965	+	1.34783i
$406$	40357.3			0.244833
$407$			50117.7i			0.302554i
$408$	49359.4	+	36237.1i	0.296517	+	0.217688i
$409$	−32186.1			−0.192408			−0.0962038	−	0.995362i	$-0.530670\pi$
$409$	−32186.1			−0.192408			−0.0962038	+	0.995362i	$0.530670\pi$
$410$			81391.6i			0.484185i
$411$	−39377.4	+	53636.8i	−0.233111	+	0.317526i
$412$	120218.			0.708233
$413$		−	10455.7i		−	0.0612992i
$414$	23650.7	+	75324.1i	0.137988	+	0.439474i
$415$	−56621.0			−0.328762
$416$		−	8484.77i		−	0.0490290i
$417$	187683.	+	137788.i	1.07933	+	0.792389i
$418$	81684.6			0.467507
$419$			26496.1i			0.150922i	0.997149	+	0.0754612i	$0.0240429\pi$
$419$			26496.1i			0.150922i	−0.997149	+	0.0754612i	$0.975957\pi$
$420$	83781.1	−	114120.i	0.474949	−	0.646938i
$421$	65281.2			0.368319			0.184159	−	0.982896i	$-0.441044\pi$
$421$	65281.2			0.368319			0.184159	+	0.982896i	$0.441044\pi$
$422$		−	107172.i		−	0.601805i
$423$	−156341.	+	49088.6i	−0.873757	+	0.274347i
$424$	−82165.2			−0.457042
$425$			319154.i			1.76694i
$426$	−7480.78	−	5492.01i	−0.0412219	−	0.0302630i
$427$	−138806.			−0.761295
$428$			157787.i			0.861357i
$429$	55289.4	−	75310.8i	0.300419	−	0.409207i
$430$	−395407.			−2.13849
$431$		−	33379.9i		−	0.179693i	−0.995956	−	0.0898463i	$-0.971362\pi$
$431$		−	33379.9i		−	0.179693i	0.995956	−	0.0898463i	$-0.0286376\pi$
$432$	−15076.3	+	44153.0i	−0.0807841	+	0.236588i
$433$	−271245.			−1.44672			−0.723362	−	0.690469i	$-0.757404\pi$
$433$	−271245.			−1.44672			−0.723362	+	0.690469i	$0.757404\pi$
$434$			211215.i			1.12136i
$435$	−88782.7	−	65179.8i	−0.469191	−	0.344456i
$436$	43973.5			0.231323
$437$		−	44936.7i		−	0.235309i
$438$	−11286.9	+	15374.1i	−0.0588335	+	0.0801383i
$439$	−75908.0			−0.393875			−0.196938	−	0.980416i	$-0.563100\pi$
$439$	−75908.0			−0.393875			−0.196938	+	0.980416i	$0.563100\pi$
$440$		−	205795.i		−	1.06299i
$441$	−2631.28	−	8380.26i	−0.0135298	−	0.0430904i
$442$	−39862.8			−0.204044
$443$			169038.i			0.861345i	0.902508	+	0.430673i	$0.141724\pi$
$443$			169038.i			0.861345i	−0.902508	+	0.430673i	$0.858276\pi$
$444$	13133.9	+	9642.21i	0.0666233	+	0.0489115i
$445$	−466423.			−2.35537
$446$		−	87407.5i		−	0.439419i
$447$	−40530.1	+	55206.8i	−0.202844	+	0.276298i
$448$	24514.9			0.122145
$449$			220525.i			1.09387i	0.837176	+	0.546933i	$0.184205\pi$
$449$			220525.i			1.09387i	−0.837176	+	0.546933i	$0.815795\pi$
$450$	−232009.	+	72847.4i	−1.14572	+	0.359740i
$451$	−155191.			−0.762979
$452$			195352.i			0.956182i
$453$	113968.	+	83669.7i	0.555376	+	0.407729i
$454$	241227.			1.17034
$455$			92163.7i			0.445181i
$456$	15715.4	−	21406.3i	0.0755781	−	0.102947i
$457$	137417.			0.657975			0.328988	−	0.944334i	$-0.393293\pi$
$457$	137417.			0.657975			0.328988	+	0.944334i	$0.393293\pi$
$458$			64606.6i			0.307997i
$459$	207438.	+	70830.7i	0.984607	+	0.336199i
$460$	−113213.			−0.535032
$461$		−	307950.i		−	1.44903i	−0.689257	−	0.724517i	$-0.742063\pi$
$461$		−	307950.i		−	1.44903i	0.689257	−	0.724517i	$-0.257937\pi$
$462$	217594.	+	159747.i	1.01944	+	0.748425i
$463$	−232326.			−1.08377			−0.541884	−	0.840453i	$-0.682289\pi$
$463$	−232326.			−1.08377			−0.541884	+	0.840453i	$0.682289\pi$
$464$		−	19072.0i		−	0.0885852i
$465$	341125.	−	464654.i	1.57764	−	2.14894i
$466$	244195.			1.12451
$467$			160856.i			0.737570i	0.929515	+	0.368785i	$0.120226\pi$
$467$			160856.i			0.737570i	−0.929515	+	0.368785i	$0.879774\pi$
$468$	−9098.76	−	28978.3i	−0.0415423	−	0.132306i
$469$	−187115.			−0.850675
$470$		−	234981.i		−	1.06374i
$471$	221395.	+	162537.i	0.997991	+	0.732675i
$472$	−4941.17			−0.0221792
$473$		−	753930.i		−	3.36984i
$474$	106556.	−	145142.i	0.474266	−	0.646007i
$475$	138411.			0.613458
$476$		−	115175.i		−	0.508328i
$477$	−280621.	+	88111.0i	−1.23334	+	0.387252i
$478$	142873.			0.625309
$479$		−	95588.3i		−	0.416614i	−0.978064	−	0.208307i	$-0.933205\pi$
$479$		−	95588.3i		−	0.416614i	0.978064	−	0.208307i	$-0.0667953\pi$
$480$	−53930.7	−	39593.2i	−0.234074	−	0.171845i
$481$	−10606.9			−0.0458459
$482$			69478.5i			0.299059i
$483$	87880.6	−	119704.i	0.376703	−	0.513114i
$484$	275265.			1.17506
$485$			388519.i			1.65169i
$486$	−4142.29	+	166964.i	−0.0175375	+	0.706889i
$487$	451192.			1.90241			0.951204	−	0.308563i	$-0.0998481\pi$
$487$	451192.			1.90241			0.951204	+	0.308563i	$0.0998481\pi$
$488$			65596.9i			0.275451i
$489$	−205872.	−	151141.i	−0.860951	−	0.632067i
$490$	12595.6			0.0524598
$491$			401370.i			1.66488i	0.554118	+	0.832438i	$0.313056\pi$
$491$			401370.i			1.66488i	−0.554118	+	0.832438i	$0.686944\pi$
$492$	−29857.3	+	40669.3i	−0.123345	+	0.168011i
$493$	−89603.5			−0.368664
$494$			17287.8i			0.0708412i
$495$	−220687.	−	702859.i	−0.900673	−	2.86852i
$496$	99815.6			0.405728
$497$			17455.6i			0.0706680i
$498$	−28292.1	−	20770.6i	−0.114079	−	0.0837511i
$499$	−142476.			−0.572189			−0.286095	−	0.958201i	$-0.592357\pi$
$499$	−142476.			−0.572189			−0.286095	+	0.958201i	$0.592357\pi$
$500$		−	143381.i		−	0.573523i
$501$	21375.9	−	29116.6i	0.0851628	−	0.116002i
$502$	−32322.6			−0.128262
$503$			129435.i			0.511581i	0.966732	+	0.255791i	$0.0823358\pi$
$503$			129435.i			0.511581i	−0.966732	+	0.255791i	$0.917664\pi$
$504$	83726.5	−	26288.9i	0.329611	−	0.103493i
$505$	71892.1			0.281902
$506$		−	215865.i		−	0.843104i
$507$	15938.8	+	11701.5i	0.0620070	+	0.0455224i
$508$	−64481.2			−0.249865
$509$		−	3954.51i		−	0.0152636i	−0.999971	−	0.00763179i	$-0.997571\pi$
$509$		−	3954.51i		−	0.0152636i	0.999971	−	0.00763179i	$-0.00242930\pi$
$510$	−186015.	+	253375.i	−0.715168	+	0.974145i
$511$	35873.8			0.137384
$512$		−	11585.2i		−	0.0441942i
$513$	30718.0	−	89962.2i	0.116724	−	0.341842i
$514$	−118221.			−0.447475
$515$			617114.i			2.32676i
$516$	−197575.	−	145050.i	−0.742049	−	0.544775i
$517$	448043.			1.67625
$518$		−	30646.5i		−	0.114215i
$519$	−148154.	+	201804.i	−0.550020	+	0.749194i
$520$	43554.6			0.161075
$521$		−	384986.i		−	1.41831i	−0.705054	−	0.709153i	$-0.749077\pi$
$521$		−	384986.i		−	1.41831i	0.705054	−	0.709153i	$-0.250923\pi$
$522$	−20452.2	−	65137.3i	−0.0750582	−	0.239050i
$523$	−338703.			−1.23827			−0.619135	−	0.785285i	$-0.712517\pi$
$523$	−338703.			−1.23827			−0.619135	+	0.785285i	$0.712517\pi$
$524$		−	148449.i		−	0.540650i
$525$	368705.	+	270685.i	1.33771	+	0.982076i
$526$	24406.9			0.0882146
$527$		−	468950.i		−	1.68852i
$528$	75493.0	−	102831.i	0.270794	−	0.368854i
$529$	161089.			0.575643
$530$		−	421776.i		−	1.50152i
$531$	−16875.7	+	5298.73i	−0.0598513	+	0.0187924i
$532$	−49949.4			−0.176485
$533$		−	32844.7i		−	0.115614i
$534$	−233060.	−	171101.i	−0.817306	−	0.600024i
$535$	−809963.			−2.82981
$536$			88426.9i			0.307790i
$537$	173609.	−	236476.i	0.602038	−	0.820048i
$538$	−88193.6			−0.304700
$539$			24016.3i			0.0826662i
$540$	−226650.	−	77390.5i	−0.777262	−	0.265400i
$541$	−524534.			−1.79217			−0.896085	−	0.443883i	$-0.853601\pi$
$541$	−524534.			−1.79217			−0.896085	+	0.443883i	$0.853601\pi$
$542$			319829.i			1.08873i
$543$	−167660.	−	123087.i	−0.568630	−	0.417459i
$544$	−54429.3			−0.183923
$545$			225728.i			0.759964i
$546$	−33808.9	+	46051.8i	−0.113409	+	0.154476i
$547$	2563.44			0.00856739			0.00428370	−	0.999991i	$-0.498636\pi$
$547$	2563.44			0.00856739			0.00428370	+	0.999991i	$0.498636\pi$
$548$		−	59146.1i		−	0.196954i
$549$	70343.7	+	224035.i	0.233389	+	0.743312i
$550$	664895.			2.19800
$551$			38859.5i			0.127995i
$552$	−56569.6	−	41530.5i	−0.185654	−	0.136298i
$553$	−338675.			−1.10747
$554$			232040.i			0.756036i
$555$	−49496.1	+	67419.6i	−0.160689	+	0.218877i
$556$	−206961.			−0.669483
$557$		−	100687.i		−	0.324536i	−0.986747	−	0.162268i	$-0.948119\pi$
$557$		−	100687.i		−	0.324536i	0.986747	−	0.162268i	$-0.0518809\pi$
$558$	340903.	−	107039.i	1.09487	−	0.343773i
$559$	159562.			0.510630
$560$			125842.i			0.401281i
$561$	−483115.	−	354678.i	−1.53506	−	1.12696i
$562$	87840.6			0.278114
$563$			362042.i			1.14220i	0.820880	+	0.571101i	$0.193483\pi$
$563$			362042.i			1.14220i	−0.820880	+	0.571101i	$0.806517\pi$
$564$	86199.5	−	117414.i	0.270986	−	0.369115i
$565$	−1.00279e6			−3.14134
$566$		−	72118.3i		−	0.225119i
$567$	257763.	−	179571.i	0.801777	−	0.558559i
$568$	8249.17			0.0255690
$569$			438663.i			1.35490i	0.735570	+	0.677449i	$0.236915\pi$
$569$			438663.i			1.35490i	−0.735570	+	0.677449i	$0.763085\pi$
$570$	109884.	+	80671.5i	0.338210	+	0.248296i
$571$	142313.			0.436490			0.218245	−	0.975894i	$-0.429967\pi$
$571$	142313.			0.436490			0.218245	+	0.975894i	$0.429967\pi$
$572$			83046.4i			0.253822i
$573$	−19632.6	+	26742.0i	−0.0597956	+	0.0814488i
$574$	94897.6			0.288026
$575$		−	365774.i		−	1.10631i
$576$	−12423.6	−	39567.4i	−0.0374457	−	0.119259i
$577$	−159352.			−0.478635			−0.239318	−	0.970941i	$-0.576924\pi$
$577$	−159352.			−0.478635			−0.239318	+	0.970941i	$0.576924\pi$
$578$			19484.5i			0.0583223i
$579$	380107.	+	279056.i	1.13383	+	0.832403i
$580$	97902.0			0.291029
$581$			66016.6i			0.195569i
$582$	−142523.	+	194133.i	−0.420763	+	0.573130i
$583$	804208.			2.36609
$584$		−	16953.2i		−	0.0497080i
$585$	148754.	−	46706.4i	0.434666	−	0.136479i
$586$	265444.			0.772996
$587$		−	413412.i		−	1.19980i	−0.800077	−	0.599898i	$-0.795208\pi$
$587$		−	413412.i		−	1.19980i	0.800077	−	0.599898i	$-0.204792\pi$
$588$	6293.71	+	4620.52i	0.0182034	+	0.0133640i
$589$	−203375.			−0.586229
$590$		−	25364.4i		−	0.0728652i
$591$	−51328.6	+	69915.7i	−0.146955	+	0.200170i
$592$	−14482.9			−0.0413249
$593$		−	394229.i		−	1.12109i	−0.828125	−	0.560544i	$-0.810592\pi$
$593$		−	394229.i		−	1.12109i	0.828125	−	0.560544i	$-0.189408\pi$
$594$	147562.	−	432157.i	0.418217	−	1.22481i
$595$	591225.			1.67001
$596$		−	60877.4i		−	0.171381i
$597$	63301.4	+	46472.7i	0.177609	+	0.130391i
$598$	45685.8			0.127755
$599$			72163.0i			0.201123i	0.994931	+	0.100561i	$0.0320639\pi$
$599$			72163.0i			0.201123i	−0.994931	+	0.100561i	$0.967936\pi$
$600$	127920.	−	174242.i	0.355333	−	0.484007i
$601$	385633.			1.06764			0.533821	−	0.845598i	$-0.320756\pi$
$601$	385633.			1.06764			0.533821	+	0.845598i	$0.320756\pi$
$602$			461021.i			1.27212i
$603$	94825.8	+	302007.i	0.260791	+	0.830582i
$604$	−125674.			−0.344487
$605$			1.41301e6i			3.86042i
$606$	35922.6	+	26372.6i	0.0978190	+	0.0718137i
$607$	101238.			0.274769			0.137385	−	0.990518i	$-0.456130\pi$
$607$	101238.			0.274769			0.137385	+	0.990518i	$0.456130\pi$
$608$			23605.0i			0.0638554i
$609$	−75995.6	+	103515.i	−0.204906	+	0.279106i
$610$	−336727.			−0.904936
$611$			94824.1i			0.254002i
$612$	−185894.	+	58368.0i	−0.496321	+	0.155838i
$613$	−10546.2			−0.0280657			−0.0140328	−	0.999902i	$-0.504467\pi$
$613$	−10546.2			−0.0280657			−0.0140328	+	0.999902i	$0.504467\pi$
$614$		−	481157.i		−	1.27629i
$615$	−208767.	−	153266.i	−0.551964	−	0.405224i
$616$	−239945.			−0.632338
$617$		−	310630.i		−	0.815967i	−0.912989	−	0.407983i	$-0.866232\pi$
$617$		−	310630.i		−	0.815967i	0.912989	−	0.407983i	$-0.133768\pi$
$618$	−226379.	+	308356.i	−0.592734	+	0.807375i
$619$	336842.			0.879113			0.439557	−	0.898215i	$-0.355136\pi$
$619$	336842.			0.879113			0.439557	+	0.898215i	$0.355136\pi$
$620$			512381.i			1.33294i
$621$	−237740.	−	81177.3i	−0.616479	−	0.210500i
$622$	57239.9			0.147951
$623$			543821.i			1.40113i
$624$	21763.1	+	15977.4i	0.0558924	+	0.0410334i
$625$	72617.5			0.185901
$626$			58003.9i			0.148016i
$627$	−153818.	+	209518.i	−0.391265	+	0.532951i
$628$	−244136.			−0.619031
$629$			68043.0i			0.171982i
$630$	134948.	+	429791.i	0.340005	+	1.08287i
$631$	341708.			0.858215			0.429107	−	0.903253i	$-0.358828\pi$
$631$	341708.			0.858215			0.429107	+	0.903253i	$0.358828\pi$
$632$			160051.i			0.400703i
$633$	274892.	+	201812.i	0.686049	+	0.503663i
$634$	156420.			0.389147
$635$		−	331000.i		−	0.820881i
$636$	154723.	−	210751.i	0.382507	−	0.521021i
$637$	−5082.82			−0.0125264
$638$			186671.i			0.458603i
$639$	28173.6	−	8846.11i	0.0689987	−	0.0216646i
$640$	59470.2			0.145191
$641$			331478.i			0.806749i	0.915035	+	0.403375i	$0.132163\pi$
$641$			331478.i			0.806749i	−0.915035	+	0.403375i	$0.867837\pi$
$642$	−404718.	−	297123.i	−0.981934	−	0.720886i
$643$	−195545.			−0.472961			−0.236481	−	0.971636i	$-0.575994\pi$
$643$	−195545.			−0.472961			−0.236481	+	0.971636i	$0.575994\pi$
$644$			131999.i			0.318273i
$645$	744579.	−	1.01421e6i	1.78975	−	2.43785i
$646$	110900.			0.265746
$647$		−	318885.i		−	0.761772i	−0.924622	−	0.380886i	$-0.875619\pi$
$647$		−	318885.i		−	0.761772i	0.924622	−	0.380886i	$-0.124381\pi$
$648$	−84861.4	−	121813.i	−0.202097	−	0.290098i
$649$	48362.7			0.114821
$650$			140719.i			0.333062i
$651$	−541758.	−	397732.i	−1.27833	−	0.938487i
$652$	227018.			0.534029
$653$			780816.i			1.83114i	0.402154	+	0.915572i	$0.368262\pi$
$653$			780816.i			1.83114i	−0.402154	+	0.915572i	$0.631738\pi$
$654$	−82805.2	+	112791.i	−0.193599	+	0.263705i
$655$	762032.			1.77619
$656$		−	44846.6i		−	0.104213i
$657$	−18180.0	−	57900.8i	−0.0421176	−	0.134139i
$658$	−273974.			−0.632787
$659$		−	529393.i		−	1.21901i	−0.792782	−	0.609505i	$-0.791368\pi$
$659$		−	529393.i		−	1.21901i	0.792782	−	0.609505i	$-0.208632\pi$
$660$	527858.	+	387526.i	1.21179	+	0.889638i
$661$	99630.3			0.228028			0.114014	−	0.993479i	$-0.463629\pi$
$661$	99630.3			0.228028			0.114014	+	0.993479i	$0.463629\pi$
$662$			264045.i			0.602506i
$663$	75064.4	−	102247.i	0.170768	−	0.232607i
$664$	31198.1			0.0707607
$665$		−	256404.i		−	0.579804i
$666$	−49463.9	+	15530.9i	−0.111517	+	0.0350146i
$667$	102692.			0.230827
$668$			32107.3i			0.0719534i
$669$	224197.	+	164594.i	0.500931	+	0.367758i
$670$	−453919.			−1.01118
$671$		−	642043.i		−	1.42600i
$672$	−46163.3	+	62879.9i	−0.102225	+	0.139243i
$673$	142775.			0.315225			0.157613	−	0.987501i	$-0.449620\pi$
$673$	142775.			0.315225			0.157613	+	0.987501i	$0.449620\pi$
$674$		−	170994.i		−	0.376411i
$675$	250038.	−	732272.i	0.548780	−	1.60718i
$676$	−17576.0			−0.0384615
$677$			292592.i			0.638388i	0.947689	+	0.319194i	$0.103412\pi$
$677$			292592.i			0.638388i	−0.947689	+	0.319194i	$0.896588\pi$
$678$	−501071.	−	367861.i	−1.09003	−	0.800248i
$679$	452989.			0.982536
$680$		−	279400.i		−	0.604240i
$681$	−454246.	+	618738.i	−0.979483	+	1.33417i
$682$	−976965.			−2.10044
$683$		−	535853.i		−	1.14869i	−0.818612	−	0.574346i	$-0.805256\pi$
$683$		−	535853.i		−	1.14869i	0.818612	−	0.574346i	$-0.194744\pi$
$684$	25313.2	+	80619.0i	0.0541047	+	0.172316i
$685$	303613.			0.647052
$686$		−	339846.i		−	0.722161i
$687$	−165714.	−	121659.i	−0.351111	−	0.257768i
$688$	217869.			0.460276
$689$			170203.i			0.358533i
$690$	213188.	−	290387.i	0.447779	−	0.609929i
$691$	−320322.			−0.670857			−0.335429	−	0.942066i	$-0.608881\pi$
$691$	−320322.			−0.670857			−0.335429	+	0.942066i	$0.608881\pi$
$692$		−	222532.i		−	0.464708i
$693$	−819490.	+	257308.i	−1.70639	+	0.535780i
$694$	192252.			0.399165
$695$		−	1.06239e6i		−	2.19945i
$696$	48919.1	+	35914.0i	0.100986	+	0.0741387i
$697$	−210697.			−0.433703
$698$			424251.i			0.870787i
$699$	−459836.	+	626352.i	−0.941128	+	1.28193i
$700$	−406577.			−0.829748
$701$		−	486338.i		−	0.989697i	−0.868979	−	0.494849i	$-0.835223\pi$
$701$		−	486338.i		−	0.989697i	0.868979	−	0.494849i	$-0.164777\pi$
$702$	91461.9	+	31230.1i	0.185595	+	0.0633723i
$703$	29509.0			0.0597097
$704$			113393.i			0.228792i
$705$	602719.	+	442486.i	1.21265	+	0.890269i
$706$	168883.			0.338825
$707$		−	83821.8i		−	0.167694i
$708$	9304.56	−	12673.9i	0.0185622	−	0.0252839i
$709$	−333144.			−0.662735			−0.331367	−	0.943502i	$-0.607510\pi$
$709$	−333144.			−0.662735			−0.331367	+	0.943502i	$0.607510\pi$
$710$			42345.2i			0.0840016i
$711$	171632.	+	546626.i	0.339516	+	1.08131i
$712$	256998.			0.506956
$713$			537452.i			1.05721i
$714$	295420.	+	216882.i	0.579487	+	0.425430i
$715$	−426300.			−0.833879
$716$			260766.i			0.508657i
$717$	−269040.	+	366465.i	−0.523333	+	0.712843i
$718$	−105491.			−0.204628
$719$			8227.08i			0.0159143i	0.999968	+	0.00795715i	$0.00253287\pi$
$719$			8227.08i			0.0159143i	−0.999968	+	0.00795715i	$0.997467\pi$
$720$	203110.	−	63773.7i	0.391802	−	0.123020i
$721$	719517.			1.38411
$722$			320508.i			0.614843i
$723$	−178210.	−	130833.i	−0.340923	−	0.250288i
$724$	184881.			0.352708
$725$			316307.i			0.601774i
$726$	−518343.	+	706046.i	−0.983432	+	1.33955i
$727$	−427026.			−0.807952			−0.403976	−	0.914770i	$-0.632372\pi$
$727$	−427026.			−0.807952			−0.403976	+	0.914770i	$0.632372\pi$
$728$		−	50782.1i		−	0.0958181i
$729$	−420458.	−	325030.i	−0.791166	−	0.611602i
$730$	87025.4			0.163305
$731$		−	1.02358e6i		−	1.91553i
$732$	−168254.	−	123523.i	−0.314009	−	0.230530i
$733$	11654.5			0.0216913			0.0108457	−	0.999941i	$-0.496548\pi$
$733$	11654.5			0.0216913			0.0108457	+	0.999941i	$0.496548\pi$
$734$		−	559419.i		−	1.03835i
$735$	−23718.4	+	32307.3i	−0.0439047	+	0.0598034i
$736$	62380.1			0.115157
$737$		−	865496.i		−	1.59342i
$738$	−48091.9	−	153166.i	−0.0882998	−	0.281223i
$739$	−110450.			−0.202245			−0.101122	−	0.994874i	$-0.532243\pi$
$739$	−110450.			−0.202245			−0.101122	+	0.994874i	$0.532243\pi$
$740$		−	74344.7i		−	0.135765i
$741$	−44342.6	−	32554.1i	−0.0807579	−	0.0592884i
$742$	−491766.			−0.893204
$743$			830506.i			1.50441i	0.658931	+	0.752203i	$0.271009\pi$
$743$			830506.i			1.50441i	−0.658931	+	0.752203i	$0.728991\pi$
$744$	−187960.	+	256024.i	−0.339562	+	0.462524i
$745$	312500.			0.563039
$746$		−	465597.i		−	0.836629i
$747$	106552.	−	33455.7i	0.190950	−	0.0599555i
$748$	532738.			0.952161
$749$			944367.i			1.68336i
$750$	367767.	+	269996.i	0.653807	+	0.479992i
$751$	−176693.			−0.313285			−0.156642	−	0.987655i	$-0.550067\pi$
$751$	−176693.			−0.313285			−0.156642	+	0.987655i	$0.550067\pi$
$752$			129474.i			0.228954i
$753$	60865.8	−	82906.5i	0.107345	−	0.146217i
$754$	−39507.3			−0.0694919
$755$		−	645121.i		−	1.13174i
$756$	−90232.7	+	264260.i	−0.157877	+	0.462367i
$757$	−841125.			−1.46781			−0.733903	−	0.679254i	$-0.762303\pi$
$757$	−841125.			−1.46781			−0.733903	+	0.679254i	$0.762303\pi$
$758$			289849.i			0.504468i
$759$	553686.	+	406488.i	0.961125	+	0.705610i
$760$	−121171.			−0.209784
$761$		−	331584.i		−	0.572564i	−0.958145	−	0.286282i	$-0.907581\pi$
$761$		−	331584.i		−	0.572564i	0.958145	−	0.286282i	$-0.0924195\pi$
$762$	121423.	−	165392.i	0.209117	−	0.284843i
$763$	263185.			0.452077
$764$		−	29488.8i		−	0.0505208i
$765$	−299619.	−	954245.i	−0.511972	−	1.63056i
$766$	65427.4			0.111507
$767$			10235.5i			0.0173988i
$768$	29715.7	+	21815.8i	0.0503807	+	0.0369870i
$769$	−513803.			−0.868848			−0.434424	−	0.900708i	$-0.643048\pi$
$769$	−513803.			−0.868848			−0.434424	+	0.900708i	$0.643048\pi$
$770$		−	1.23170e6i		−	2.07742i
$771$	222618.	−	303233.i	0.374500	−	0.510115i
$772$	−419150.			−0.703291
$773$		−	711897.i		−	1.19140i	−0.803207	−	0.595700i	$-0.796875\pi$
$773$		−	711897.i		−	1.19140i	0.803207	−	0.595700i	$-0.203125\pi$
$774$	744094.	−	233635.i	1.24207	−	0.389992i
$775$	−1.65543e6			−2.75618
$776$		−	214073.i		−	0.355500i
$777$	78607.2	+	57709.5i	0.130203	+	0.0955883i
$778$	308168.			0.509129
$779$			91375.5i			0.150576i
$780$	−82016.4	+	111716.i	−0.134807	+	0.183623i
$781$	−80740.4			−0.132370
$782$		−	293072.i		−	0.479248i
$783$	205588.	+	70198.9i	0.335331	+	0.114500i
$784$	−6940.16			−0.0112911
$785$		−	1.25322e6i		−	2.03370i
$786$	380768.	+	279541.i	0.616333	+	0.452480i
$787$	1.05256e6			1.69940			0.849701	−	0.527265i	$-0.176782\pi$
$787$	1.05256e6			1.69940			0.849701	+	0.527265i	$0.176782\pi$
$788$		−	77097.1i		−	0.124161i
$789$	−45959.8	+	62602.8i	−0.0738285	+	0.100563i
$790$	−821583.			−1.31643
$791$			1.16920e6i			1.86868i
$792$	121598.	+	387274.i	0.193855	+	0.617402i
$793$	135882.			0.216081
$794$		−	669300.i		−	1.06165i
$795$	1.08184e6	+	794234.i	1.71171	+	1.25665i
$796$	−69803.4			−0.110167
$797$			995008.i			1.56643i	0.621753	+	0.783213i	$0.286421\pi$
$797$			995008.i			1.56643i	−0.621753	+	0.783213i	$0.713579\pi$
$798$	94058.1	−	128118.i	0.147703	−	0.201190i
$799$	608291.			0.952836
$800$			192140.i			0.300218i
$801$	877734.	−	275596.i	1.36804	−	0.429544i
$802$	−619625.			−0.963341
$803$			165933.i			0.257336i
$804$	−226812.	−	166514.i	−0.350876	−	0.257595i
$805$	−677588.			−1.04562
$806$		−	206766.i		−	0.318279i
$807$	166075.	−	226214.i	0.255009	−	0.347353i
$808$	−39612.4			−0.0606749
$809$			486446.i			0.743254i	0.928382	+	0.371627i	$0.121200\pi$
$809$			486446.i			0.743254i	−0.928382	+	0.371627i	$0.878800\pi$
$810$	625301.	−	435617.i	0.953057	−	0.663949i
$811$	−623875.			−0.948540			−0.474270	−	0.880379i	$-0.657288\pi$
$811$	−623875.			−0.948540			−0.474270	+	0.880379i	$0.657288\pi$
$812$		−	114148.i		−	0.173123i
$813$	−820352.	−	602261.i	−1.24113	−	0.911178i
$814$	141754.			0.213938
$815$			1.16534e6i			1.75444i
$816$	102494.	−	139609.i	0.153928	−	0.209669i
$817$	−443910.			−0.665045
$818$			91036.2i			0.136053i
$819$	−54456.9	−	173438.i	−0.0811867	−	0.258568i
$820$	230210.			0.342371
$821$		−	819874.i		−	1.21636i	−0.793801	−	0.608178i	$-0.791901\pi$
$821$		−	819874.i		−	1.21636i	0.793801	−	0.608178i	$-0.208099\pi$
$822$	151708.	+	111376.i	0.224525	+	0.164835i
$823$	951611.			1.40495			0.702473	−	0.711710i	$-0.252079\pi$
$823$	951611.			1.40495			0.702473	+	0.711710i	$0.252079\pi$
$824$		−	340029.i		−	0.500797i
$825$	−1.25204e6	+	1.70543e6i	−1.83955	+	2.50569i
$826$	−29573.3			−0.0433451
$827$			806124.i			1.17867i	0.807890	+	0.589333i	$0.200610\pi$
$827$			806124.i			1.17867i	−0.807890	+	0.589333i	$0.799390\pi$
$828$	213049.	−	66894.2i	0.310755	−	0.0975726i
$829$	−595041.			−0.865841			−0.432920	−	0.901432i	$-0.642517\pi$
$829$	−595041.			−0.865841			−0.432920	+	0.901432i	$0.642517\pi$
$830$			160148.i			0.232470i
$831$	−595174.	−	436947.i	−0.861870	−	0.632742i
$832$	−23998.5			−0.0346688
$833$			32606.0i			0.0469902i
$834$	389723.	−	530849.i	0.560304	−	0.763201i
$835$	−164816.			−0.236388
$836$		−	231039.i		−	0.330577i
$837$	−367394.	+	1.07597e6i	−0.524422	+	1.53585i
$838$	74942.3			0.106718
$839$		−	585198.i		−	0.831341i	−0.909515	−	0.415670i	$-0.863547\pi$
$839$		−	585198.i		−	0.831341i	0.909515	−	0.415670i	$-0.136453\pi$
$840$	−322780.	−	236969.i	−0.457454	−	0.335840i
$841$	618477.			0.874443
$842$		−	184643.i		−	0.260441i
$843$	−165410.	+	225308.i	−0.232759	+	0.317046i
$844$	−303128.			−0.425541
$845$		−	90222.4i		−	0.126357i
$846$	138844.	+	442198.i	0.193993	+	0.617840i
$847$	1.64749e6			2.29644
$848$			232398.i			0.323178i
$849$	184981.	+	135804.i	0.256632	+	0.188407i
$850$	902703.			1.24942
$851$		−	77982.4i		−	0.107681i
$852$	−15533.7	+	21158.8i	−0.0213992	+	0.0291483i
$853$	790603.			1.08658			0.543288	−	0.839546i	$-0.317179\pi$
$853$	790603.			1.08658			0.543288	+	0.839546i	$0.317179\pi$
$854$			392603.i			0.538317i
$855$	−413840.	+	129940.i	−0.566109	+	0.177750i
$856$	446288.			0.609071
$857$		−	372096.i		−	0.506633i	−0.967383	−	0.253317i	$-0.918479\pi$
$857$		−	372096.i		−	0.506633i	0.967383	−	0.253317i	$-0.0815215\pi$
$858$	−213011.	−	156382.i	−0.289353	−	0.212428i
$859$	−338886.			−0.459270			−0.229635	−	0.973277i	$-0.573753\pi$
$859$	−338886.			−0.459270			−0.229635	+	0.973277i	$0.573753\pi$
$860$			1.11838e6i			1.51214i
$861$	−178699.	+	243409.i	−0.241054	+	0.328345i
$862$	−94412.5			−0.127062
$863$			1.11362e6i			1.49526i	0.664116	+	0.747629i	$0.268808\pi$
$863$			1.11362e6i			1.49526i	−0.664116	+	0.747629i	$0.731192\pi$
$864$	124884.	+	42642.1i	0.167293	+	0.0571230i
$865$	1.14232e6			1.52670
$866$			767197.i			1.02299i
$867$	−49977.2	−	36690.7i	−0.0664865	−	0.0488110i
$868$	597405.			0.792920
$869$		−	1.56653e6i		−	2.07443i
$870$	−184356.	+	251115.i	−0.243567	+	0.331768i
$871$	183174.			0.241450
$872$		−	124376.i		−	0.163570i
$873$	−229565.	−	731132.i	−0.301215	−	0.959328i
$874$	−127100.			−0.166388
$875$		−	858146.i		−	1.12084i
$876$	43484.4	+	31924.0i	0.0566663	+	0.0416016i
$877$	835156.			1.08585			0.542923	−	0.839782i	$-0.317317\pi$
$877$	835156.			1.08585			0.542923	+	0.839782i	$0.317317\pi$
$878$			214700.i			0.278512i
$879$	−499849.	+	680854.i	−0.646935	+	0.881204i
$880$	−582076.			−0.751648
$881$		−	62725.3i		−	0.0808148i	−0.999183	−	0.0404074i	$-0.987134\pi$
$881$		−	62725.3i		−	0.0808148i	0.999183	−	0.0404074i	$-0.0128656\pi$
$882$	−23703.0	+	7442.38i	−0.0304695	+	0.00956698i
$883$	145758.			0.186944			0.0934722	−	0.995622i	$-0.470203\pi$
$883$	145758.			0.186944			0.0934722	+	0.995622i	$0.470203\pi$
$884$			112749.i			0.144281i
$885$	65058.7	+	47762.8i	0.0830652	+	0.0609823i
$886$	478112.			0.609063
$887$		−	638598.i		−	0.811672i	−0.913946	−	0.405836i	$-0.866980\pi$
$887$		−	638598.i		−	0.811672i	0.913946	−	0.405836i	$-0.133020\pi$
$888$	27272.3	−	37148.1i	0.0345856	−	0.0471098i
$889$	−385926.			−0.488315
$890$			1.31924e6i			1.66550i
$891$	830598.	+	1.19227e6i	1.04625	+	1.50183i
$892$	−247226.			−0.310716
$893$		−	263805.i		−	0.330811i
$894$	156149.	+	114636.i	0.195372	+	0.143432i
$895$	−1.33858e6			−1.67109
$896$		−	69338.6i		−	0.0863693i
$897$	−86029.5	+	117183.i	−0.106921	+	0.145639i
$898$	623738.			0.773480
$899$		−	464767.i		−	0.575064i
$900$	206044.	+	656221.i	0.254375	+	0.810149i
$901$	1.09184e6			1.34497
$902$			438946.i			0.539508i
$903$	−1.18250e6	−	868134.i	−1.45020	−	1.06466i
$904$	552538.			0.676123
$905$			949045.i			1.15875i
$906$	236654.	−	322351.i	0.288308	−	0.392710i
$907$	−55065.7			−0.0669371			−0.0334685	−	0.999440i	$-0.510655\pi$
$907$	−55065.7			−0.0669371			−0.0334685	+	0.999440i	$0.510655\pi$
$908$		−	682292.i		−	0.827558i
$909$	−135290.	+	42479.0i	−0.163733	+	0.0514098i
$910$	260678.			0.314791
$911$			1.61260e6i			1.94307i	0.236889	+	0.971537i	$0.423872\pi$
$911$			1.61260e6i			1.94307i	−0.236889	+	0.971537i	$0.576128\pi$
$912$	−60546.1	−	44449.9i	−0.0727942	−	0.0534418i
$913$	−305358.			−0.366326
$914$		−	388675.i		−	0.465259i
$915$	634080.	−	863693.i	0.757359	−	1.03161i
$916$	182735.			0.217786
$917$		−	888483.i		−	1.05660i
$918$	200339.	−	586723.i	0.237728	−	0.696222i
$919$	−1.57097e6			−1.86011			−0.930054	−	0.367423i	$-0.880240\pi$
$919$	−1.57097e6			−1.86011			−0.930054	+	0.367423i	$0.880240\pi$
$920$			320214.i			0.378325i
$921$	1.23415e6	+	906053.i	1.45496	+	1.06815i
$922$	−871015.			−1.02462
$923$		−	17088.0i		−	0.0200580i
$924$	451832.	−	615450.i	0.529216	−	0.720856i
$925$	240197.			0.280727
$926$			657118.i			0.766339i
$927$	−364635.	−	1.16131e6i	−0.424325	−	1.35142i
$928$	−53943.9			−0.0626392
$929$			1.27703e6i			1.47969i	0.672778	+	0.739845i	$0.265101\pi$
$929$			1.27703e6i			1.47969i	−0.672778	+	0.739845i	$0.734899\pi$
$930$	−1.31424e6	−	964848.i	−1.51953	−	1.11556i
$931$	14140.7			0.0163144
$932$		−	690688.i		−	0.795152i
$933$	−107787.	+	146818.i	−0.123823	+	0.168662i
$934$	454969.			0.521541
$935$			2.73469e6i			3.12813i
$936$	−81963.0	+	25735.2i	−0.0935548	+	0.0293748i
$937$	1.22260e6			1.39253			0.696267	−	0.717783i	$-0.254843\pi$
$937$	1.22260e6			1.39253			0.696267	+	0.717783i	$0.254843\pi$
$938$			529242.i			0.601518i
$939$	−148778.	−	109225.i	−0.168736	−	0.123878i
$940$	−664627.			−0.752181
$941$		−	209227.i		−	0.236287i	−0.992997	−	0.118143i	$-0.962306\pi$
$941$		−	209227.i		−	0.236287i	0.992997	−	0.118143i	$-0.0376942\pi$
$942$	459725.	−	626201.i	0.518079	−	0.705686i
$943$	241474.			0.271549
$944$			13975.7i			0.0156831i
$945$	−1.35652e6	−	463189.i	−1.51901	−	0.518674i
$946$	−2.13244e6			−2.38283
$947$		−	319767.i		−	0.356561i	−0.983980	−	0.178280i	$-0.942947\pi$
$947$		−	319767.i		−	0.356561i	0.983980	−	0.178280i	$-0.0570534\pi$
$948$	−410524.	−	301386.i	−0.456796	−	0.335356i
$949$	−35118.2			−0.0389941
$950$		−	391487.i		−	0.433780i
$951$	−294550.	+	401212.i	−0.325685	+	0.443622i
$952$	−325764.			−0.359442
$953$			72285.5i			0.0795912i	0.999208	+	0.0397956i	$0.0126707\pi$
$953$			72285.5i			0.0795912i	−0.999208	+	0.0397956i	$0.987329\pi$
$954$	249216.	+	793717.i	0.273828	+	0.872105i
$955$	151374.			0.165976
$956$		−	404106.i		−	0.442160i
$957$	−478806.	−	351515.i	−0.522800	−	0.383813i
$958$	−270365.			−0.294590
$959$		−	353994.i		−	0.384910i
$960$	−111986.	+	152539.i	−0.121513	+	0.165515i
$961$	1.50889e6			1.63384
$962$			30001.0i			0.0324179i
$963$	1.52422e6	−	478584.i	1.64360	−	0.516066i
$964$	196515.			0.211466
$965$		−	2.15161e6i		−	2.31052i
$966$	−338574.	−	248564.i	−0.362826	−	0.266369i
$967$	−1.40209e6			−1.49942			−0.749708	−	0.661769i	$-0.769806\pi$
$967$	−1.40209e6			−1.49942			−0.749708	+	0.661769i	$0.769806\pi$
$968$		−	778567.i		−	0.830894i
$969$	−208833.	+	284455.i	−0.222408	+	0.302947i
$970$	1.09890e6			1.16792
$971$			911523.i			0.966783i	0.875404	+	0.483392i	$0.160595\pi$
$971$			911523.i			0.966783i	−0.875404	+	0.483392i	$0.839405\pi$
$972$	472247.	+	11716.2i	0.499846	+	0.0124009i
$973$	−1.23868e6			−1.30838
$974$		−	1.27616e6i		−	1.34521i
$975$	−360939.	−	264983.i	−0.379686	−	0.278746i
$976$	185536.			0.194773
$977$			512120.i			0.536516i	0.963347	+	0.268258i	$0.0864479\pi$
$977$			512120.i			0.536516i	−0.963347	+	0.268258i	$0.913552\pi$
$978$	−427490.	+	582293.i	−0.446939	+	0.608785i
$979$	−2.51542e6			−2.62449
$980$		−	35625.8i		−	0.0370947i
$981$	−133376.	−	424785.i	−0.138593	−	0.441399i
$982$	1.13525e6			1.17725
$983$		−	160570.i		−	0.166172i	−0.996542	−	0.0830860i	$-0.973522\pi$
$983$		−	160570.i		−	0.166172i	0.996542	−	0.0830860i	$-0.0264776\pi$
$984$	115030.	+	84449.3i	0.118801	+	0.0872180i
$985$	395761.			0.407906
$986$			253437.i			0.260685i
$987$	515912.	−	702734.i	0.529591	−	0.721367i
$988$	48897.3			0.0500923
$989$			1.17310e6i			1.19934i
$990$	−1.98798e6	+	624198.i	−2.02835	+	0.636872i
$991$	−1.68520e6			−1.71594			−0.857972	−	0.513697i	$-0.828275\pi$
$991$	−1.68520e6			−1.71594			−0.857972	+	0.513697i	$0.828275\pi$
$992$		−	282321.i		−	0.286893i
$993$	−677266.	−	497214.i	−0.686848	−	0.504249i
$994$	49372.0			0.0499698
$995$		−	358320.i		−	0.361930i
$996$	−58748.1	+	80022.0i	−0.0592210	+	0.0806661i
$997$	1.45152e6			1.46026			0.730132	−	0.683306i	$-0.239459\pi$
$997$	1.45152e6			1.46026			0.730132	+	0.683306i	$0.239459\pi$
$998$			402982.i			0.404599i
$999$	53307.6	−	156119.i	0.0534143	−	0.156432i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.5.c.a.53.7	✓	16
3.2	odd	2	inner	78.5.c.a.53.15	yes	16
4.3	odd	2		624.5.f.a.209.3		16
12.11	even	2		624.5.f.a.209.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.5.c.a.53.7	✓	16	1.1	even	1	trivial
78.5.c.a.53.15	yes	16	3.2	odd	2	inner
624.5.f.a.209.3		16	4.3	odd	2
624.5.f.a.209.4		16	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	78.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} +(7.25482 + 5.32612i) q^{3} -8.00000 q^{4} -41.0662i q^{5} +(15.0646 - 20.5197i) q^{6} -47.8807 q^{7} +22.6274i q^{8} +(24.2648 + 77.2801i) q^{9} +O(q^{10})\)	\(q-2.82843i q^{2} +(7.25482 + 5.32612i) q^{3} -8.00000 q^{4} -41.0662i q^{5} +(15.0646 - 20.5197i) q^{6} -47.8807 q^{7} +22.6274i q^{8} +(24.2648 + 77.2801i) q^{9} -116.153 q^{10} -221.470i q^{11} +(-58.0386 - 42.6090i) q^{12} +46.8722 q^{13} +135.427i q^{14} +(218.724 - 297.928i) q^{15} +64.0000 q^{16} -300.682i q^{17} +(218.581 - 68.6313i) q^{18} -130.401 q^{19} +328.529i q^{20} +(-347.366 - 255.018i) q^{21} -626.413 q^{22} +344.605i q^{23} +(-120.516 + 164.158i) q^{24} -1061.43 q^{25} -132.575i q^{26} +(-235.566 + 689.891i) q^{27} +383.045 q^{28} -298.001i q^{29} +(-842.667 - 618.644i) q^{30} +1559.62 q^{31} -181.019i q^{32} +(1179.58 - 1606.73i) q^{33} -850.458 q^{34} +1966.28i q^{35} +(-194.119 - 618.241i) q^{36} -226.295 q^{37} +368.829i q^{38} +(340.049 + 249.647i) q^{39} +929.222 q^{40} -700.729i q^{41} +(-721.301 + 982.499i) q^{42} +3404.20 q^{43} +1771.76i q^{44} +(3173.60 - 996.464i) q^{45} +974.689 q^{46} +2023.04i q^{47} +(464.308 + 340.872i) q^{48} -108.440 q^{49} +3002.18i q^{50} +(1601.47 - 2181.40i) q^{51} -374.977 q^{52} +3631.22i q^{53} +(1951.31 + 666.283i) q^{54} -9094.95 q^{55} -1083.42i q^{56} +(-946.033 - 694.530i) q^{57} -842.873 q^{58} +218.371i q^{59} +(-1749.79 + 2383.42i) q^{60} +2899.00 q^{61} -4411.27i q^{62} +(-1161.82 - 3700.23i) q^{63} -512.000 q^{64} -1924.86i q^{65} +(-4544.51 - 3336.35i) q^{66} +3907.95 q^{67} +2405.46i q^{68} +(-1835.41 + 2500.05i) q^{69} +5561.47 q^{70} -364.565i q^{71} +(-1748.65 + 549.050i) q^{72} -749.233 q^{73} +640.060i q^{74} +(-7700.50 - 5653.32i) q^{75} +1043.21 q^{76} +10604.2i q^{77} +(706.108 - 961.804i) q^{78} +7073.30 q^{79} -2628.24i q^{80} +(-5383.44 + 3750.38i) q^{81} -1981.96 q^{82} -1378.77i q^{83} +(2778.93 + 2040.15i) q^{84} -12347.9 q^{85} -9628.54i q^{86} +(1587.19 - 2161.94i) q^{87} +5011.30 q^{88} -11357.8i q^{89} +(-2818.43 - 8976.30i) q^{90} -2244.27 q^{91} -2756.84i q^{92} +(11314.8 + 8306.72i) q^{93} +5722.01 q^{94} +5355.06i q^{95} +(964.131 - 1313.26i) q^{96} -9460.80 q^{97} +306.715i q^{98} +(17115.3 - 5373.94i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9}+O(q^{10}) \) 16 * q + 12 * q^3 - 128 * q^4 - 96 * q^6 + 80 * q^7 + 116 * q^9	\( 16 q + 12 q^{3} - 128 q^{4} - 96 q^{6} + 80 q^{7} + 116 q^{9} - 64 q^{10} - 96 q^{12} + 40 q^{15} + 1024 q^{16} + 448 q^{18} + 816 q^{19} - 128 q^{21} - 1344 q^{22} + 768 q^{24} - 3896 q^{25} - 2520 q^{27} - 640 q^{28} - 544 q^{30} + 5472 q^{31} + 2920 q^{33} + 1408 q^{34} - 928 q^{36} - 3456 q^{37} + 512 q^{40} - 3520 q^{42} - 3112 q^{43} + 3632 q^{45} + 4864 q^{46} + 768 q^{48} + 10776 q^{49} + 4132 q^{51} + 10080 q^{54} - 13792 q^{55} - 17264 q^{57} - 10688 q^{58} - 320 q^{60} + 11504 q^{61} - 2904 q^{63} - 8192 q^{64} - 8512 q^{66} - 10080 q^{67} + 16320 q^{69} + 8576 q^{70} - 3584 q^{72} + 19568 q^{73} - 4840 q^{75} - 6528 q^{76} - 7216 q^{79} - 6364 q^{81} + 6592 q^{82} + 1024 q^{84} + 8320 q^{85} + 14664 q^{87} + 10752 q^{88} + 7360 q^{90} - 1352 q^{91} + 49392 q^{93} + 4224 q^{94} - 6144 q^{96} - 45056 q^{97} - 51016 q^{99}+O(q^{100}) \) 16 * q + 12 * q^3 - 128 * q^4 - 96 * q^6 + 80 * q^7 + 116 * q^9 - 64 * q^10 - 96 * q^12 + 40 * q^15 + 1024 * q^16 + 448 * q^18 + 816 * q^19 - 128 * q^21 - 1344 * q^22 + 768 * q^24 - 3896 * q^25 - 2520 * q^27 - 640 * q^28 - 544 * q^30 + 5472 * q^31 + 2920 * q^33 + 1408 * q^34 - 928 * q^36 - 3456 * q^37 + 512 * q^40 - 3520 * q^42 - 3112 * q^43 + 3632 * q^45 + 4864 * q^46 + 768 * q^48 + 10776 * q^49 + 4132 * q^51 + 10080 * q^54 - 13792 * q^55 - 17264 * q^57 - 10688 * q^58 - 320 * q^60 + 11504 * q^61 - 2904 * q^63 - 8192 * q^64 - 8512 * q^66 - 10080 * q^67 + 16320 * q^69 + 8576 * q^70 - 3584 * q^72 + 19568 * q^73 - 4840 * q^75 - 6528 * q^76 - 7216 * q^79 - 6364 * q^81 + 6592 * q^82 + 1024 * q^84 + 8320 * q^85 + 14664 * q^87 + 10752 * q^88 + 7360 * q^90 - 1352 * q^91 + 49392 * q^93 + 4224 * q^94 - 6144 * q^96 - 45056 * q^97 - 51016 * q^99

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