LMFDB - Embedded newform 405.4.e.x.271.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, not 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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} + \cdots + 746496 $ x^12 - 2x^11 + 2x^10 + 32x^9 + 583x^8 - 624x^7 + 594x^6 + 9450x^5 + 90513x^4 - 20304x^3 + 10368x^2 + 124416*x + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.2
Root			$1.25636 + 1.25636i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.4.e.x.136.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.78729 - 3.09567i) q^{2} +(-2.38879 + 4.13751i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-7.07987 - 12.2627i) q^{7} -11.5188 q^{8} +O(q^{10})$	\(q+(-1.78729 - 3.09567i) q^{2} +(-2.38879 + 4.13751i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-7.07987 - 12.2627i) q^{7} -11.5188 q^{8} -17.8729 q^{10} +(31.8319 + 55.1345i) q^{11} +(5.19795 - 9.00312i) q^{13} +(-25.3075 + 43.8339i) q^{14} +(39.6977 + 68.7584i) q^{16} -108.605 q^{17} +18.6145 q^{19} +(11.9440 + 20.6876i) q^{20} +(113.786 - 197.082i) q^{22} +(-64.4120 + 111.565i) q^{23} +(-12.5000 - 21.6506i) q^{25} -37.1609 q^{26} +67.6494 q^{28} +(41.8248 + 72.4426i) q^{29} +(5.23638 - 9.06967i) q^{31} +(95.8273 - 165.978i) q^{32} +(194.109 + 336.206i) q^{34} -70.7987 q^{35} -81.8885 q^{37} +(-33.2695 - 57.6245i) q^{38} +(-28.7969 + 49.8777i) q^{40} +(-153.726 + 266.262i) q^{41} +(111.220 + 192.638i) q^{43} -304.159 q^{44} +460.491 q^{46} +(180.965 + 313.441i) q^{47} +(71.2509 - 123.410i) q^{49} +(-44.6822 + 77.3918i) q^{50} +(24.8337 + 43.0132i) q^{52} -562.215 q^{53} +318.319 q^{55} +(81.5513 + 141.251i) q^{56} +(149.506 - 258.952i) q^{58} +(231.155 - 400.372i) q^{59} +(324.533 + 562.108i) q^{61} -37.4356 q^{62} -49.9208 q^{64} +(-25.9898 - 45.0156i) q^{65} +(-127.090 + 220.126i) q^{67} +(259.435 - 449.355i) q^{68} +(126.538 + 219.170i) q^{70} +1092.93 q^{71} +1034.52 q^{73} +(146.358 + 253.500i) q^{74} +(-44.4663 + 77.0179i) q^{76} +(450.731 - 780.690i) q^{77} +(-575.927 - 997.534i) q^{79} +396.977 q^{80} +1099.01 q^{82} +(262.160 + 454.075i) q^{83} +(-271.513 + 470.274i) q^{85} +(397.563 - 688.599i) q^{86} +(-366.664 - 635.081i) q^{88} -656.485 q^{89} -147.203 q^{91} +(-307.734 - 533.010i) q^{92} +(646.874 - 1120.42i) q^{94} +(46.5363 - 80.6033i) q^{95} +(-562.755 - 974.721i) q^{97} -509.383 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 34 q^{4} + 30 q^{5} - 40 q^{7} - 132 q^{8}+O(q^{10}) $ 12 * q + 4 * q^2 - 34 * q^4 + 30 * q^5 - 40 * q^7 - 132 * q^8	$ 12 q + 4 q^{2} - 34 q^{4} + 30 q^{5} - 40 q^{7} - 132 q^{8} + 40 q^{10} + 88 q^{11} - 20 q^{13} + 180 q^{14} - 58 q^{16} - 248 q^{17} - 92 q^{19} + 170 q^{20} + 74 q^{22} + 210 q^{23} - 150 q^{25} - 8 q^{26} + 704 q^{28} + 296 q^{29} + 104 q^{31} + 722 q^{32} + 428 q^{34} - 400 q^{35} - 408 q^{37} - 20 q^{38} - 330 q^{40} + 344 q^{41} - 512 q^{43} - 1432 q^{44} - 372 q^{46} + 238 q^{47} - 68 q^{49} + 100 q^{50} + 468 q^{52} - 1700 q^{53} + 880 q^{55} + 2316 q^{56} - 890 q^{58} + 1840 q^{59} + 364 q^{61} - 2076 q^{62} - 1980 q^{64} + 100 q^{65} - 88 q^{67} + 236 q^{68} - 900 q^{70} - 2728 q^{71} + 1672 q^{73} + 1316 q^{74} + 2106 q^{76} + 840 q^{77} + 680 q^{79} - 580 q^{80} + 3484 q^{82} + 2148 q^{83} - 620 q^{85} + 2872 q^{86} - 1296 q^{88} - 6000 q^{89} - 6116 q^{91} + 1002 q^{92} + 3662 q^{94} - 230 q^{95} + 612 q^{97} - 3964 q^{98}+O(q^{100}) $ 12 * q + 4 * q^2 - 34 * q^4 + 30 * q^5 - 40 * q^7 - 132 * q^8 + 40 * q^10 + 88 * q^11 - 20 * q^13 + 180 * q^14 - 58 * q^16 - 248 * q^17 - 92 * q^19 + 170 * q^20 + 74 * q^22 + 210 * q^23 - 150 * q^25 - 8 * q^26 + 704 * q^28 + 296 * q^29 + 104 * q^31 + 722 * q^32 + 428 * q^34 - 400 * q^35 - 408 * q^37 - 20 * q^38 - 330 * q^40 + 344 * q^41 - 512 * q^43 - 1432 * q^44 - 372 * q^46 + 238 * q^47 - 68 * q^49 + 100 * q^50 + 468 * q^52 - 1700 * q^53 + 880 * q^55 + 2316 * q^56 - 890 * q^58 + 1840 * q^59 + 364 * q^61 - 2076 * q^62 - 1980 * q^64 + 100 * q^65 - 88 * q^67 + 236 * q^68 - 900 * q^70 - 2728 * q^71 + 1672 * q^73 + 1316 * q^74 + 2106 * q^76 + 840 * q^77 + 680 * q^79 - 580 * q^80 + 3484 * q^82 + 2148 * q^83 - 620 * q^85 + 2872 * q^86 - 1296 * q^88 - 6000 * q^89 - 6116 * q^91 + 1002 * q^92 + 3662 * q^94 - 230 * q^95 + 612 * q^97 - 3964 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.78729	−	3.09567i	−0.631902	−	1.09449i	−0.987163	−	0.159718i	$-0.948941\pi$
$2$	−1.78729	−	3.09567i	−0.631902	−	1.09449i	0.355261	−	0.934767i	$-0.384392\pi$
$3$		0			0
$3$		0			0
$4$	−2.38879	+	4.13751i	−0.298599	+	0.517189i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−7.07987	−	12.2627i	−0.382277	−	0.662123i	0.609110	−	0.793086i	$-0.291527\pi$
$7$	−7.07987	−	12.2627i	−0.382277	−	0.662123i	−0.991387	+	0.130962i	$0.958193\pi$
$8$	−11.5188			−0.509062
$9$		0			0
$10$	−17.8729			−0.565190
$11$	31.8319	+	55.1345i	0.872516	+	1.51124i	0.859385	+	0.511329i	$0.170847\pi$
$11$	31.8319	+	55.1345i	0.872516	+	1.51124i	0.0131312	+	0.999914i	$0.495820\pi$
$12$		0			0
$13$	5.19795	−	9.00312i	0.110896	−	0.192078i	−0.805236	−	0.592955i	$-0.797961\pi$
$13$	5.19795	−	9.00312i	0.110896	−	0.192078i	0.916132	+	0.400877i	$0.131294\pi$
$14$	−25.3075	+	43.8339i	−0.483123	+	0.836794i
$15$		0			0
$16$	39.6977	+	68.7584i	0.620276	+	1.07435i
$17$	−108.605			−1.54945			−0.774724	−	0.632300i	$-0.782111\pi$
$17$	−108.605			−1.54945			−0.774724	+	0.632300i	$0.782111\pi$
$18$		0			0
$19$	18.6145			0.224761			0.112381	−	0.993665i	$-0.464152\pi$
$19$	18.6145			0.224761			0.112381	+	0.993665i	$0.464152\pi$
$20$	11.9440	+	20.6876i	0.133538	+	0.231294i
$21$		0			0
$22$	113.786	−	197.082i	1.10269	−	1.90991i
$23$	−64.4120	+	111.565i	−0.583949	+	1.01143i	0.411057	+	0.911610i	$0.365160\pi$
$23$	−64.4120	+	111.565i	−0.583949	+	1.01143i	−0.995006	+	0.0998192i	$0.968174\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−37.1609			−0.280302
$27$		0			0
$28$	67.6494			0.456590
$29$	41.8248	+	72.4426i	0.267816	+	0.463871i	0.968298	−	0.249799i	$-0.0803647\pi$
$29$	41.8248	+	72.4426i	0.267816	+	0.463871i	−0.700481	+	0.713671i	$0.747031\pi$
$30$		0			0
$31$	5.23638	−	9.06967i	0.0303381	−	0.0525471i	−0.850458	−	0.526043i	$-0.823675\pi$
$31$	5.23638	−	9.06967i	0.0303381	−	0.0525471i	0.880796	+	0.473496i	$0.157008\pi$
$32$	95.8273	−	165.978i	0.529376	−	0.916906i
$33$		0			0
$34$	194.109	+	336.206i	0.979098	+	1.69585i
$35$	−70.7987			−0.341919
$36$		0			0
$37$	−81.8885			−0.363848			−0.181924	−	0.983313i	$-0.558233\pi$
$37$	−81.8885			−0.363848			−0.181924	+	0.983313i	$0.558233\pi$
$38$	−33.2695	−	57.6245i	−0.142027	−	0.245998i
$39$		0			0
$40$	−28.7969	+	49.8777i	−0.113830	+	0.197159i
$41$	−153.726	+	266.262i	−0.585562	+	1.01422i	0.409244	+	0.912425i	$0.365793\pi$
$41$	−153.726	+	266.262i	−0.585562	+	1.01422i	−0.994805	+	0.101797i	$0.967541\pi$
$42$		0			0
$43$	111.220	+	192.638i	0.394438	+	0.683187i	0.993029	−	0.117868i	$-0.0376059\pi$
$43$	111.220	+	192.638i	0.394438	+	0.683187i	−0.598591	+	0.801055i	$0.704273\pi$
$44$	−304.159			−1.04213
$45$		0			0
$46$	460.491			1.47599
$47$	180.965	+	313.441i	0.561628	+	0.972768i	0.997355	+	0.0726889i	$0.0231580\pi$
$47$	180.965	+	313.441i	0.561628	+	0.972768i	−0.435727	+	0.900079i	$0.643509\pi$
$48$		0			0
$49$	71.2509	−	123.410i	0.207728	−	0.359796i
$50$	−44.6822	+	77.3918i	−0.126380	+	0.218897i
$51$		0			0
$52$	24.8337	+	43.0132i	0.0662271	+	0.114709i
$53$	−562.215			−1.45710			−0.728548	−	0.684994i	$-0.759805\pi$
$53$	−562.215			−1.45710			−0.728548	+	0.684994i	$0.759805\pi$
$54$		0			0
$55$	318.319			0.780402
$56$	81.5513	+	141.251i	0.194603	+	0.337062i
$57$		0			0
$58$	149.506	−	258.952i	0.338467	−	0.586242i
$59$	231.155	−	400.372i	0.510065	−	0.883458i	−0.489867	−	0.871797i	$-0.662955\pi$
$59$	231.155	−	400.372i	0.510065	−	0.883458i	0.999932	−	0.0116608i	$-0.00371182\pi$
$60$		0			0
$61$	324.533	+	562.108i	0.681184	+	1.17985i	0.974620	+	0.223867i	$0.0718682\pi$
$61$	324.533	+	562.108i	0.681184	+	1.17985i	−0.293435	+	0.955979i	$0.594799\pi$
$62$	−37.4356			−0.0766827
$63$		0			0
$64$	−49.9208			−0.0975015
$65$	−25.9898	−	45.0156i	−0.0495944	−	0.0859000i
$66$		0			0
$67$	−127.090	+	220.126i	−0.231739	+	0.401384i	−0.958320	−	0.285697i	$-0.907775\pi$
$67$	−127.090	+	220.126i	−0.231739	+	0.401384i	0.726581	+	0.687081i	$0.241108\pi$
$68$	259.435	−	449.355i	0.462664	−	0.801357i
$69$		0			0
$70$	126.538	+	219.170i	0.216059	+	0.374225i
$71$	1092.93			1.82685			0.913426	−	0.407006i	$-0.133427\pi$
$71$	1092.93			1.82685			0.913426	+	0.407006i	$0.133427\pi$
$72$		0			0
$73$	1034.52			1.65865			0.829324	−	0.558769i	$-0.188726\pi$
$73$	1034.52			1.65865			0.829324	+	0.558769i	$0.188726\pi$
$74$	146.358	+	253.500i	0.229916	+	0.398227i
$75$		0			0
$76$	−44.4663	+	77.0179i	−0.0671136	+	0.116244i
$77$	450.731	−	780.690i	0.667086	−	1.15543i
$78$		0			0
$79$	−575.927	−	997.534i	−0.820213	−	1.42065i	−0.905524	−	0.424296i	$-0.860522\pi$
$79$	−575.927	−	997.534i	−0.820213	−	1.42065i	0.0853111	−	0.996354i	$-0.472812\pi$
$80$	396.977			0.554792
$81$		0			0
$82$	1099.01			1.48007
$83$	262.160	+	454.075i	0.346697	+	0.600496i	0.985660	−	0.168741i	$-0.0539700\pi$
$83$	262.160	+	454.075i	0.346697	+	0.600496i	−0.638964	+	0.769237i	$0.720637\pi$
$84$		0			0
$85$	−271.513	+	470.274i	−0.346467	+	0.600099i
$86$	397.563	−	688.599i	0.498492	−	0.863414i
$87$		0			0
$88$	−366.664	−	635.081i	−0.444165	−	0.769316i
$89$	−656.485			−0.781879			−0.390940	−	0.920416i	$-0.627850\pi$
$89$	−656.485			−0.781879			−0.390940	+	0.920416i	$0.627850\pi$
$90$		0			0
$91$	−147.203			−0.169573
$92$	−307.734	−	533.010i	−0.348733	−	0.604024i
$93$		0			0
$94$	646.874	−	1120.42i	0.709787	−	1.22939i
$95$	46.5363	−	80.6033i	0.0502582	−	0.0870497i
$96$		0			0
$97$	−562.755	−	974.721i	−0.589063	−	1.02029i	−0.994355	−	0.106100i	$-0.966163\pi$
$97$	−562.755	−	974.721i	−0.589063	−	1.02029i	0.405292	−	0.914187i	$-0.367170\pi$
$98$	−509.383			−0.525056
$99$		0			0
$100$	119.440			0.119440
$101$	−190.162	−	329.371i	−0.187345	−	0.324491i	0.757019	−	0.653393i	$-0.226655\pi$
$101$	−190.162	−	329.371i	−0.187345	−	0.324491i	−0.944364	+	0.328902i	$0.893322\pi$
$102$		0			0
$103$	−382.139	+	661.884i	−0.365566	+	0.633178i	−0.988867	−	0.148804i	$-0.952458\pi$
$103$	−382.139	+	661.884i	−0.365566	+	0.633178i	0.623301	+	0.781982i	$0.285791\pi$
$104$	−59.8740	+	103.705i	−0.0564531	+	0.0977797i
$105$		0			0
$106$	1004.84	+	1740.43i	0.920742	+	1.59477i
$107$	816.083			0.737325			0.368662	−	0.929563i	$-0.379816\pi$
$107$	816.083			0.737325			0.368662	+	0.929563i	$0.379816\pi$
$108$		0			0
$109$	606.775			0.533198			0.266599	−	0.963808i	$-0.414100\pi$
$109$	606.775			0.533198			0.266599	+	0.963808i	$0.414100\pi$
$110$	−568.928	−	985.412i	−0.493137	−	0.854139i
$111$		0			0
$112$	562.109	−	973.601i	0.474235	−	0.821399i
$113$	−13.6613	+	23.6621i	−0.0113730	+	0.0196986i	−0.871656	−	0.490118i	$-0.836954\pi$
$113$	−13.6613	+	23.6621i	−0.0113730	+	0.0196986i	0.860283	+	0.509817i	$0.170287\pi$
$114$		0			0
$115$	322.060	+	557.824i	0.261150	+	0.452325i
$116$	−399.643			−0.319879
$117$		0			0
$118$	−1652.56			−1.28924
$119$	768.910	+	1331.79i	0.592318	+	1.02593i
$120$		0			0
$121$	−1361.04	+	2357.39i	−1.02257	+	1.77114i
$122$	1160.07	−	2009.30i	0.860883	−	1.49109i
$123$		0			0
$124$	25.0172	+	43.3311i	0.0181179	+	0.0313810i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	−1206.22			−0.842794			−0.421397	−	0.906876i	$-0.638460\pi$
$127$	−1206.22			−0.842794			−0.421397	+	0.906876i	$0.638460\pi$
$128$	−677.396	−	1173.28i	−0.467765	−	0.810192i
$129$		0			0
$130$	−92.9024	+	160.912i	−0.0626775	+	0.108561i
$131$	−766.002	+	1326.75i	−0.510885	+	0.884878i	0.489036	+	0.872264i	$0.337349\pi$
$131$	−766.002	+	1326.75i	−0.510885	+	0.884878i	−0.999920	+	0.0126146i	$0.995985\pi$
$132$		0			0
$133$	−131.789	−	228.264i	−0.0859212	−	0.148820i
$134$	908.585			0.585745
$135$		0			0
$136$	1251.00			0.788765
$137$	1236.06	+	2140.92i	0.770830	+	1.33512i	0.937109	+	0.349038i	$0.113491\pi$
$137$	1236.06	+	2140.92i	0.770830	+	1.33512i	−0.166279	+	0.986079i	$0.553175\pi$
$138$		0			0
$139$	−6.06336	+	10.5020i	−0.00369991	+	0.00640843i	−0.867869	−	0.496792i	$-0.834511\pi$
$139$	−6.06336	+	10.5020i	−0.00369991	+	0.00640843i	0.864170	+	0.503201i	$0.167844\pi$
$140$	169.123	−	292.930i	0.102097	−	0.176837i
$141$		0			0
$142$	−1953.37	−	3383.34i	−1.15439	−	1.99946i
$143$	661.843			0.387036
$144$		0			0
$145$	418.248			0.239542
$146$	−1848.98	−	3202.53i	−1.04810	−	1.81537i
$147$		0			0
$148$	195.615	−	338.815i	0.108645	−	0.188178i
$149$	−1568.26	+	2716.30i	−0.862258	+	1.49348i	0.00748582	+	0.999972i	$0.497617\pi$
$149$	−1568.26	+	2716.30i	−0.862258	+	1.49348i	−0.869744	+	0.493503i	$0.835716\pi$
$150$		0			0
$151$	1350.04	+	2338.33i	0.727580	+	1.26020i	0.957903	+	0.287091i	$0.0926881\pi$
$151$	1350.04	+	2338.33i	0.727580	+	1.26020i	−0.230324	+	0.973114i	$0.573979\pi$
$152$	−214.416			−0.114418
$153$		0			0
$154$	−3222.35			−1.68613
$155$	−26.1819	−	45.3483i	−0.0135676	−	0.0234998i
$156$		0			0
$157$	−221.314	+	383.326i	−0.112502	+	0.194858i	−0.916778	−	0.399397i	$-0.869220\pi$
$157$	−221.314	+	383.326i	−0.112502	+	0.194858i	0.804277	+	0.594255i	$0.202553\pi$
$158$	−2058.69	+	3565.76i	−1.03659	+	1.79542i
$159$		0			0
$160$	−479.136	−	829.889i	−0.236744	−	0.410053i
$161$	1824.11			0.892921
$162$		0			0
$163$	−2116.29			−1.01694			−0.508469	−	0.861080i	$-0.669788\pi$
$163$	−2116.29			−1.01694			−0.508469	+	0.861080i	$0.669788\pi$
$164$	−734.441	−	1272.09i	−0.349696	−	0.605692i
$165$		0			0
$166$	937.111	−	1623.12i	0.438156	−	0.758909i
$167$	647.312	−	1121.18i	0.299943	−	0.519517i	−0.676180	−	0.736737i	$-0.736366\pi$
$167$	647.312	−	1121.18i	0.299943	−	0.519517i	0.976123	+	0.217220i	$0.0696990\pi$
$168$		0			0
$169$	1044.46	+	1809.06i	0.475404	+	0.823424i
$170$	1941.09			0.875732
$171$		0			0
$172$	−1062.72			−0.471116
$173$	1461.09	+	2530.67i	0.642106	+	1.11216i	0.984962	+	0.172772i	$0.0552723\pi$
$173$	1461.09	+	2530.67i	0.642106	+	1.11216i	−0.342856	+	0.939388i	$0.611394\pi$
$174$		0			0
$175$	−176.997	+	306.567i	−0.0764554	+	0.132425i
$176$	−2527.31	+	4377.42i	−1.08240	+	1.87478i
$177$		0			0
$178$	1173.33	+	2032.26i	0.494071	+	0.855756i
$179$	−3955.51			−1.65167			−0.825834	−	0.563913i	$-0.809295\pi$
$179$	−3955.51			−1.65167			−0.825834	+	0.563913i	$0.809295\pi$
$180$		0			0
$181$	−2694.00			−1.10632			−0.553159	−	0.833076i	$-0.686578\pi$
$181$	−2694.00			−1.10632			−0.553159	+	0.833076i	$0.686578\pi$
$182$	263.095	+	455.693i	0.107153	+	0.185595i
$183$		0			0
$184$	741.946	−	1285.09i	0.297266	−	0.514880i
$185$	−204.721	+	354.588i	−0.0813590	+	0.140918i
$186$		0			0
$187$	−3457.11	−	5987.89i	−1.35192	−	2.34159i
$188$	−1729.15			−0.670806
$189$		0			0
$190$	−332.695			−0.127033
$191$	−428.173	−	741.618i	−0.162207	−	0.280951i	0.773453	−	0.633854i	$-0.218528\pi$
$191$	−428.173	−	741.618i	−0.162207	−	0.280951i	−0.935660	+	0.352903i	$0.885195\pi$
$192$		0			0
$193$	867.607	−	1502.74i	0.323584	−	0.560464i	−0.657641	−	0.753332i	$-0.728446\pi$
$193$	867.607	−	1502.74i	0.323584	−	0.560464i	0.981225	+	0.192868i	$0.0617789\pi$
$194$	−2011.61	+	3484.21i	−0.744460	+	1.28944i
$195$		0			0
$196$	340.407	+	589.602i	0.124055	+	0.214870i
$197$	−2943.95			−1.06471			−0.532354	−	0.846522i	$-0.678692\pi$
$197$	−2943.95			−1.06471			−0.532354	+	0.846522i	$0.678692\pi$
$198$		0			0
$199$	−4172.06			−1.48618			−0.743088	−	0.669193i	$-0.766640\pi$
$199$	−4172.06			−1.48618			−0.743088	+	0.669193i	$0.766640\pi$
$200$	143.985	+	249.388i	0.0509062	+	0.0881721i
$201$		0			0
$202$	−679.749	+	1177.36i	−0.236767	+	0.410093i
$203$	592.228	−	1025.77i	0.204760	−	0.354655i
$204$		0			0
$205$	768.632	+	1331.31i	0.261871	+	0.453574i
$206$	2731.97			0.924006
$207$		0			0
$208$	825.387			0.275146
$209$	592.536	+	1026.30i	0.196108	+	0.339669i
$210$		0			0
$211$	−1925.52	+	3335.10i	−0.628239	+	1.08814i	0.359666	+	0.933081i	$0.382891\pi$
$211$	−1925.52	+	3335.10i	−0.628239	+	1.08814i	−0.987905	+	0.155060i	$0.950443\pi$
$212$	1343.01	−	2326.17i	0.435088	−	0.753594i
$213$		0			0
$214$	−1458.58	−	2526.33i	−0.465917	−	0.806991i
$215$	1112.20			0.352796
$216$		0			0
$217$	−148.291			−0.0463902
$218$	−1084.48	−	1878.38i	−0.336928	−	0.583577i
$219$		0			0
$220$	−760.398	+	1317.05i	−0.233027	+	0.403615i
$221$	−564.524	+	977.785i	−0.171828	+	0.297615i
$222$		0			0
$223$	−410.357	−	710.759i	−0.123227	−	0.213435i	0.797812	−	0.602907i	$-0.205991\pi$
$223$	−410.357	−	710.759i	−0.123227	−	0.213435i	−0.921038	+	0.389472i	$0.872658\pi$
$224$	−2713.78			−0.809473
$225$		0			0
$226$	97.6667			0.0287464
$227$	−2630.61	−	4556.36i	−0.769163	−	1.33223i	−0.938018	−	0.346588i	$-0.887340\pi$
$227$	−2630.61	−	4556.36i	−0.769163	−	1.33223i	0.168855	−	0.985641i	$-0.445993\pi$
$228$		0			0
$229$	−9.79945	+	16.9731i	−0.00282780	+	0.00489789i	−0.867436	−	0.497549i	$-0.834233\pi$
$229$	−9.79945	+	16.9731i	−0.00282780	+	0.00489789i	0.864608	+	0.502447i	$0.167567\pi$
$230$	1151.23	−	1993.98i	0.330042	−	0.571649i
$231$		0			0
$232$	−481.770	−	834.449i	−0.136335	−	0.236139i
$233$	3181.79			0.894619			0.447309	−	0.894379i	$-0.352382\pi$
$233$	3181.79			0.894619			0.447309	+	0.894379i	$0.352382\pi$
$234$		0			0
$235$	1809.65			0.502335
$236$	1104.36	+	1912.81i	0.304610	+	0.527599i
$237$		0			0
$238$	2748.53	−	4760.59i	0.748574	−	1.29657i
$239$	817.052	−	1415.18i	0.221133	−	0.383013i	−0.734019	−	0.679128i	$-0.762358\pi$
$239$	817.052	−	1415.18i	0.221133	−	0.383013i	0.955152	+	0.296115i	$0.0956913\pi$
$240$		0			0
$241$	−707.355	−	1225.17i	−0.189065	−	0.327471i	0.755874	−	0.654718i	$-0.227212\pi$
$241$	−707.355	−	1225.17i	−0.189065	−	0.327471i	−0.944939	+	0.327247i	$0.893879\pi$
$242$	9730.28			2.58465
$243$		0			0
$244$	−3100.97			−0.813604
$245$	−356.254	−	617.051i	−0.0928990	−	0.160906i
$246$		0			0
$247$	96.7575	−	167.589i	0.0249252	−	0.0431718i
$248$	−60.3166	+	104.471i	−0.0154440	+	0.0267497i
$249$		0			0
$250$	223.411	+	386.959i	0.0565190	+	0.0978938i
$251$	−5932.53			−1.49186			−0.745932	−	0.666022i	$-0.767995\pi$
$251$	−5932.53			−1.49186			−0.745932	+	0.666022i	$0.767995\pi$
$252$		0			0
$253$	−8201.42			−2.03802
$254$	2155.86	+	3734.07i	0.532563	+	0.922426i
$255$		0			0
$256$	−2621.08	+	4539.85i	−0.639913	+	1.10836i
$257$	3500.65	−	6063.30i	0.849666	−	1.47167i	−0.0318400	−	0.999493i	$-0.510137\pi$
$257$	3500.65	−	6063.30i	0.849666	−	1.47167i	0.881506	−	0.472172i	$-0.156530\pi$
$258$		0			0
$259$	579.760	+	1004.17i	0.139091	+	0.240913i
$260$	248.337			0.0592353
$261$		0			0
$262$	5476.26			1.29132
$263$	1827.27	+	3164.93i	0.428420	+	0.742045i	0.996733	−	0.0807677i	$-0.0257372\pi$
$263$	1827.27	+	3164.93i	0.428420	+	0.742045i	−0.568313	+	0.822812i	$0.692404\pi$
$264$		0			0
$265$	−1405.54	+	2434.46i	−0.325817	+	0.564331i
$266$	−471.088	+	815.948i	−0.108587	+	0.188079i
$267$		0			0
$268$	−607.183	−	1051.67i	−0.138394	−	0.239706i
$269$	−2589.46			−0.586922			−0.293461	−	0.955971i	$-0.594807\pi$
$269$	−2589.46			−0.586922			−0.293461	+	0.955971i	$0.594807\pi$
$270$		0			0
$271$	4854.10			1.08806			0.544032	−	0.839064i	$-0.316897\pi$
$271$	4854.10			1.08806			0.544032	+	0.839064i	$0.316897\pi$
$272$	−4311.37	−	7467.52i	−0.961086	−	1.66465i
$273$		0			0
$274$	4418.39	−	7652.87i	0.974177	−	1.68732i
$275$	795.798	−	1378.36i	0.174503	−	0.302249i
$276$		0			0
$277$	3757.58	+	6508.31i	0.815058	+	1.41172i	0.909287	+	0.416170i	$0.136628\pi$
$277$	3757.58	+	6508.31i	0.815058	+	1.41172i	−0.0942291	+	0.995551i	$0.530039\pi$
$278$	43.3478			0.00935191
$279$		0			0
$280$	815.513			0.174058
$281$	−1806.03	−	3128.14i	−0.383412	−	0.664090i	0.608135	−	0.793834i	$-0.291918\pi$
$281$	−1806.03	−	3128.14i	−0.383412	−	0.664090i	−0.991548	+	0.129744i	$0.958585\pi$
$282$		0			0
$283$	−2583.45	+	4474.67i	−0.542652	+	0.939900i	0.456099	+	0.889929i	$0.349246\pi$
$283$	−2583.45	+	4474.67i	−0.542652	+	0.939900i	−0.998751	+	0.0499712i	$0.984087\pi$
$284$	−2610.77	+	4521.99i	−0.545496	+	0.944827i
$285$		0			0
$286$	−1182.90	−	2048.85i	−0.244568	−	0.423605i
$287$	4353.45			0.895387
$288$		0			0
$289$	6882.07			1.40079
$290$	−747.529	−	1294.76i	−0.151367	−	0.262175i
$291$		0			0
$292$	−2471.25	+	4280.33i	−0.495271	+	0.857834i
$293$	−2113.18	+	3660.13i	−0.421342	+	0.729785i	−0.996071	−	0.0885584i	$-0.971774\pi$
$293$	−2113.18	+	3660.13i	−0.421342	+	0.729785i	0.574729	+	0.818344i	$0.305107\pi$
$294$		0			0
$295$	−1155.77	−	2001.86i	−0.228108	−	0.395094i
$296$	943.255			0.185221
$297$		0			0
$298$	11211.7			2.17945
$299$	669.621	+	1159.82i	0.129516	+	0.224328i
$300$		0			0
$301$	1574.84	−	2727.71i	0.301569	−	0.522333i
$302$	4825.81	−	8358.55i	0.919517	−	1.59265i
$303$		0			0
$304$	738.954	+	1279.91i	0.139414	+	0.241472i
$305$	3245.33			0.609270
$306$		0			0
$307$	5734.25			1.06603			0.533015	−	0.846106i	$-0.321059\pi$
$307$	5734.25			1.06603			0.533015	+	0.846106i	$0.321059\pi$
$308$	2153.41	+	3729.81i	0.398383	+	0.690019i
$309$		0			0
$310$	−93.5891	+	162.101i	−0.0171468	+	0.0296991i
$311$	3310.19	−	5733.42i	0.603549	−	1.04538i	−0.388730	−	0.921352i	$-0.627086\pi$
$311$	3310.19	−	5733.42i	0.603549	−	1.04538i	0.992279	−	0.124026i	$-0.0395807\pi$
$312$		0			0
$313$	−46.7155	−	80.9136i	−0.00843615	−	0.0146118i	0.861777	−	0.507288i	$-0.169352\pi$
$313$	−46.7155	−	80.9136i	−0.00843615	−	0.0146118i	−0.870213	+	0.492676i	$0.836019\pi$
$314$	1582.20			0.284360
$315$		0			0
$316$	5503.08			0.979659
$317$	996.584	+	1726.13i	0.176573	+	0.305834i	0.940705	−	0.339227i	$-0.110165\pi$
$317$	996.584	+	1726.13i	0.176573	+	0.305834i	−0.764131	+	0.645061i	$0.776832\pi$
$318$		0			0
$319$	−2662.72	+	4611.97i	−0.467348	+	0.809470i
$320$	−124.802	+	216.163i	−0.0218020	+	0.0377622i
$321$		0			0
$322$	−3260.21	−	5646.86i	−0.564238	−	0.977289i
$323$	−2021.63			−0.348256
$324$		0			0
$325$	−259.898			−0.0443586
$326$	3782.42	+	6551.35i	0.642605	+	1.11302i
$327$		0			0
$328$	1770.74	−	3067.01i	0.298087	−	0.516302i
$329$	2562.42	−	4438.25i	0.429395	−	0.743734i
$330$		0			0
$331$	1194.59	+	2069.09i	0.198370	+	0.343587i	0.948000	−	0.318270i	$-0.103102\pi$
$331$	1194.59	+	2069.09i	0.198370	+	0.343587i	−0.749630	+	0.661857i	$0.769769\pi$
$332$	−2504.99			−0.414093
$333$		0			0
$334$	−4627.73			−0.758138
$335$	635.450	+	1100.63i	0.103637	+	0.179504i
$336$		0			0
$337$	−5142.75	+	8907.50i	−0.831286	+	1.43983i	0.0657327	+	0.997837i	$0.479062\pi$
$337$	−5142.75	+	8907.50i	−0.831286	+	1.43983i	−0.897019	+	0.441992i	$0.854272\pi$
$338$	3733.51	−	6466.63i	0.600817	−	1.04065i
$339$		0			0
$340$	−1297.18	−	2246.77i	−0.206910	−	0.358378i
$341$	666.735			0.105882
$342$		0			0
$343$	−6874.58			−1.08219
$344$	−1281.11	−	2218.95i	−0.200794	−	0.347785i
$345$		0			0
$346$	5222.76	−	9046.09i	0.811495	−	1.40555i
$347$	2115.10	−	3663.47i	0.327218	−	0.566758i	−0.654741	−	0.755854i	$-0.727222\pi$
$347$	2115.10	−	3663.47i	0.327218	−	0.566758i	0.981959	+	0.189095i	$0.0605555\pi$
$348$		0			0
$349$	5605.46	+	9708.94i	0.859752	+	1.48913i	0.872166	+	0.489211i	$0.162715\pi$
$349$	5605.46	+	9708.94i	0.859752	+	1.48913i	−0.0124138	+	0.999923i	$0.503952\pi$
$350$	1265.38			0.193249
$351$		0			0
$352$	12201.5			1.84756
$353$	−5691.50	−	9857.98i	−0.858154	−	1.48637i	−0.873688	−	0.486486i	$-0.838278\pi$
$353$	−5691.50	−	9857.98i	−0.858154	−	1.48637i	0.0155346	−	0.999879i	$-0.495055\pi$
$354$		0			0
$355$	2732.31	−	4732.51i	0.408496	−	0.707536i
$356$	1568.21	−	2716.21i	0.233468	−	0.404379i
$357$		0			0
$358$	7069.63	+	12245.0i	1.04369	+	1.80773i
$359$	−8942.57			−1.31468			−0.657341	−	0.753593i	$-0.728319\pi$
$359$	−8942.57			−1.31468			−0.657341	+	0.753593i	$0.728319\pi$
$360$		0			0
$361$	−6512.50			−0.949482
$362$	4814.95	+	8339.74i	0.699084	+	1.21085i
$363$		0			0
$364$	351.638	−	609.055i	0.0506342	−	0.0877010i
$365$	2586.30	−	4479.60i	0.370885	−	0.642391i
$366$		0			0
$367$	222.085	+	384.662i	0.0315878	+	0.0547118i	0.881387	−	0.472395i	$-0.156610\pi$
$367$	222.085	+	384.662i	0.0315878	+	0.0547118i	−0.849799	+	0.527106i	$0.823277\pi$
$368$	−10228.0			−1.44884
$369$		0			0
$370$	1463.58			0.205643
$371$	3980.41	+	6894.27i	0.557015	+	0.964778i
$372$		0			0
$373$	−1708.05	+	2958.42i	−0.237103	+	0.410674i	−0.959882	−	0.280405i	$-0.909531\pi$
$373$	−1708.05	+	2958.42i	−0.237103	+	0.410674i	0.722779	+	0.691079i	$0.242864\pi$
$374$	−12357.7	+	21404.1i	−1.70856	+	2.95931i
$375$		0			0
$376$	−2084.50	−	3610.45i	−0.285903	−	0.495199i
$377$	869.613			0.118799
$378$		0			0
$379$	598.277			0.0810855			0.0405428	−	0.999178i	$-0.487091\pi$
$379$	598.277			0.0810855			0.0405428	+	0.999178i	$0.487091\pi$
$380$	222.331	+	385.089i	0.0300141	+	0.0519860i
$381$		0			0
$382$	−1530.54	+	2650.97i	−0.204998	+	0.355066i
$383$	−4528.59	+	7843.74i	−0.604178	+	1.04647i	0.388003	+	0.921658i	$0.373165\pi$
$383$	−4528.59	+	7843.74i	−0.604178	+	1.04647i	−0.992181	+	0.124808i	$0.960168\pi$
$384$		0			0
$385$	−2253.66	−	3903.45i	−0.298330	−	0.516723i
$386$	−6202.65			−0.817893
$387$		0			0
$388$	5377.22			0.703575
$389$	315.703	+	546.814i	0.0411485	+	0.0712714i	0.885866	−	0.463941i	$-0.153565\pi$
$389$	315.703	+	546.814i	0.0411485	+	0.0712714i	−0.844718	+	0.535212i	$0.820232\pi$
$390$		0			0
$391$	6995.47	−	12116.5i	0.904798	−	1.56716i
$392$	−820.722	+	1421.53i	−0.105747	+	0.183159i
$393$		0			0
$394$	5261.68	+	9113.49i	0.672791	+	1.16531i
$395$	−5759.27			−0.733620
$396$		0			0
$397$	−4615.39			−0.583475			−0.291738	−	0.956498i	$-0.594233\pi$
$397$	−4615.39			−0.583475			−0.291738	+	0.956498i	$0.594233\pi$
$398$	7456.66	+	12915.3i	0.939118	+	1.62660i
$399$		0			0
$400$	992.442	−	1718.96i	0.124055	−	0.214870i
$401$	2345.60	−	4062.69i	0.292104	−	0.505938i	−0.682203	−	0.731163i	$-0.738978\pi$
$401$	2345.60	−	4062.69i	0.292104	−	0.505938i	0.974307	+	0.225224i	$0.0723114\pi$
$402$		0			0
$403$	−54.4369	−	94.2874i	−0.00672877	−	0.0116546i
$404$	1817.03			0.223764
$405$		0			0
$406$	−4233.93			−0.517552
$407$	−2606.67	−	4514.88i	−0.317464	−	0.549863i
$408$		0			0
$409$	7781.64	−	13478.2i	0.940776	−	1.62947i	0.176781	−	0.984250i	$-0.443432\pi$
$409$	7781.64	−	13478.2i	0.940776	−	1.62947i	0.763995	−	0.645222i	$-0.223235\pi$
$410$	2747.53	−	4758.86i	0.330953	−	0.573228i
$411$		0			0
$412$	−1825.70	−	3162.21i	−0.218315	−	0.378133i
$413$	−6546.19			−0.779944
$414$		0			0
$415$	2621.60			0.310095
$416$	−996.212	−	1725.49i	−0.117412	−	0.203363i
$417$		0			0
$418$	2118.06	−	3668.60i	0.247842	−	0.429275i
$419$	2139.22	−	3705.23i	0.249422	−	0.432011i	−0.713944	−	0.700203i	$-0.753093\pi$
$419$	2139.22	−	3705.23i	0.249422	−	0.432011i	0.963365	+	0.268192i	$0.0864262\pi$
$420$		0			0
$421$	48.8531	+	84.6161i	0.00565548	+	0.00979558i	0.868839	−	0.495094i	$-0.164866\pi$
$421$	48.8531	+	84.6161i	0.00565548	+	0.00979558i	−0.863184	+	0.504890i	$0.831533\pi$
$422$	13765.8			1.58794
$423$		0			0
$424$	6476.02			0.741753
$425$	1357.56	+	2351.37i	0.154945	+	0.268372i
$426$		0			0
$427$	4595.31	−	7959.31i	0.520802	−	0.902056i
$428$	−1949.45	+	3376.55i	−0.220164	+	0.381336i
$429$		0			0
$430$	−1987.82	−	3443.00i	−0.222932	−	0.386130i
$431$	−11932.9			−1.33361			−0.666806	−	0.745231i	$-0.732339\pi$
$431$	−11932.9			−1.33361			−0.666806	+	0.745231i	$0.732339\pi$
$432$		0			0
$433$	6304.32			0.699691			0.349845	−	0.936807i	$-0.386234\pi$
$433$	6304.32			0.699691			0.349845	+	0.936807i	$0.386234\pi$
$434$	265.039	+	459.062i	0.0293141	+	0.0507734i
$435$		0			0
$436$	−1449.46	+	2510.54i	−0.159212	+	0.275764i
$437$	−1199.00	+	2076.73i	−0.131249	+	0.227330i
$438$		0			0
$439$	6461.34	+	11191.4i	0.702466	+	1.21671i	0.967598	+	0.252495i	$0.0812512\pi$
$439$	6461.34	+	11191.4i	0.702466	+	1.21671i	−0.265132	+	0.964212i	$0.585415\pi$
$440$	−3666.64			−0.397273
$441$		0			0
$442$	4035.87			0.434314
$443$	−1912.21	−	3312.04i	−0.205083	−	0.355214i	0.745076	−	0.666979i	$-0.232413\pi$
$443$	−1912.21	−	3312.04i	−0.205083	−	0.355214i	−0.950159	+	0.311765i	$0.899080\pi$
$444$		0			0
$445$	−1641.21	+	2842.66i	−0.174834	+	0.302821i
$446$	−1466.85	+	2540.66i	−0.155734	+	0.269740i
$447$		0			0
$448$	353.433	+	612.163i	0.0372726	+	0.0645580i
$449$	530.656			0.0557756			0.0278878	−	0.999611i	$-0.491122\pi$
$449$	530.656			0.0557756			0.0278878	+	0.999611i	$0.491122\pi$
$450$		0			0
$451$	−19573.6			−2.04365
$452$	−65.2680	−	113.048i	−0.00679193	−	0.0117640i
$453$		0			0
$454$	−9403.32	+	16287.0i	−0.972070	+	1.68367i
$455$	−368.008	+	637.409i	−0.0379176	+	0.0656752i
$456$		0			0
$457$	7796.80	+	13504.5i	0.798072	+	1.38230i	0.920870	+	0.389870i	$0.127480\pi$
$457$	7796.80	+	13504.5i	0.798072	+	1.38230i	−0.122798	+	0.992432i	$0.539187\pi$
$458$	70.0577			0.00714756
$459$		0			0
$460$	−3077.34			−0.311916
$461$	−3637.30	−	6300.00i	−0.367475	−	0.636486i	0.621695	−	0.783260i	$-0.286444\pi$
$461$	−3637.30	−	6300.00i	−0.367475	−	0.636486i	−0.989170	+	0.146774i	$0.953111\pi$
$462$		0			0
$463$	−1342.57	+	2325.40i	−0.134761	+	0.233413i	−0.925506	−	0.378732i	$-0.876360\pi$
$463$	−1342.57	+	2325.40i	−0.134761	+	0.233413i	0.790745	+	0.612146i	$0.209693\pi$
$464$	−3320.69	+	5751.61i	−0.332240	+	0.575456i
$465$		0			0
$466$	−5686.78	−	9849.79i	−0.565311	−	0.979147i
$467$	8844.77			0.876418			0.438209	−	0.898873i	$-0.355613\pi$
$467$	8844.77			0.876418			0.438209	+	0.898873i	$0.355613\pi$
$468$		0			0
$469$	3599.12			0.354354
$470$	−3234.37	−	5602.09i	−0.317426	−	0.549799i
$471$		0			0
$472$	−2662.62	+	4611.79i	−0.259655	+	0.449735i
$473$	−7080.67	+	12264.1i	−0.688308	+	1.19218i
$474$		0			0
$475$	−232.682	−	403.017i	−0.0224761	−	0.0389298i
$476$	−7347.07			−0.707463
$477$		0			0
$478$	−5841.23			−0.558936
$479$	6167.17	+	10681.9i	0.588278	+	1.01893i	0.994458	+	0.105134i	$0.0335272\pi$
$479$	6167.17	+	10681.9i	0.588278	+	1.01893i	−0.406180	+	0.913793i	$0.633139\pi$
$480$		0			0
$481$	−425.653	+	737.252i	−0.0403495	+	0.0698873i
$482$	−2528.49	+	4379.48i	−0.238941	+	0.413858i
$483$		0			0
$484$	−6502.49	−	11262.6i	−0.610677	−	1.05772i
$485$	−5627.55			−0.526874
$486$		0			0
$487$	3540.31			0.329419			0.164709	−	0.986342i	$-0.447331\pi$
$487$	3540.31			0.329419			0.164709	+	0.986342i	$0.447331\pi$
$488$	−3738.22	−	6474.79i	−0.346765	−	0.600615i
$489$		0			0
$490$	−1273.46	+	2205.69i	−0.117406	+	0.203353i
$491$	3855.63	−	6678.14i	0.354383	−	0.613809i	−0.632629	−	0.774455i	$-0.718024\pi$
$491$	3855.63	−	6678.14i	0.354383	−	0.613809i	0.987012	+	0.160646i	$0.0513577\pi$
$492$		0			0
$493$	−4542.39	−	7867.64i	−0.414967	−	0.718744i
$494$	−691.734			−0.0630012
$495$		0			0
$496$	831.488			0.0752720
$497$	−7737.77	−	13402.2i	−0.698363	−	1.20960i
$498$		0			0
$499$	6081.29	−	10533.1i	0.545563	−	0.944943i	−0.453008	−	0.891506i	$-0.649649\pi$
$499$	6081.29	−	10533.1i	0.545563	−	0.944943i	0.998571	−	0.0534365i	$-0.0170175\pi$
$500$	298.599	−	517.189i	0.0267075	−	0.0462588i
$501$		0			0
$502$	10603.1	+	18365.2i	0.942711	+	1.63282i
$503$	8796.85			0.779786			0.389893	−	0.920860i	$-0.372512\pi$
$503$	8796.85			0.779786			0.389893	+	0.920860i	$0.372512\pi$
$504$		0			0
$505$	−1901.62			−0.167566
$506$	14658.3	+	25388.9i	1.28783	+	2.23058i
$507$		0			0
$508$	2881.41	−	4990.76i	0.251658	−	0.435884i
$509$	−10974.1	+	19007.7i	−0.955634	+	1.65521i	−0.222722	+	0.974882i	$0.571494\pi$
$509$	−10974.1	+	19007.7i	−0.955634	+	1.65521i	−0.732911	+	0.680324i	$0.761839\pi$
$510$		0			0
$511$	−7324.26	−	12686.0i	−0.634063	−	1.09823i
$512$	7900.20			0.681919
$513$		0			0
$514$	−25026.6			−2.14762
$515$	1910.69	+	3309.42i	0.163486	+	0.283166i
$516$		0			0
$517$	−11520.9	+	19954.9i	−0.980059	+	1.69751i
$518$	2072.40	−	3589.50i	0.175784	−	0.304466i
$519$		0			0
$520$	299.370	+	518.524i	0.0252466	+	0.0437284i
$521$	11232.4			0.944531			0.472265	−	0.881456i	$-0.343436\pi$
$521$	11232.4			0.944531			0.472265	+	0.881456i	$0.343436\pi$
$522$		0			0
$523$	−1962.88			−0.164112			−0.0820560	−	0.996628i	$-0.526149\pi$
$523$	−1962.88			−0.164112			−0.0820560	+	0.996628i	$0.526149\pi$
$524$	−3659.64	−	6338.68i	−0.305100	−	0.528448i
$525$		0			0
$526$	6531.72	−	11313.3i	0.541438	−	0.937798i
$527$	−568.697	+	985.013i	−0.0470073	+	0.0814190i
$528$		0			0
$529$	−2214.30	−	3835.28i	−0.181992	−	0.315220i
$530$	10048.4			0.823537
$531$		0			0
$532$	1259.26			0.102624
$533$	1598.12	+	2768.03i	0.129873	+	0.224947i
$534$		0			0
$535$	2040.21	−	3533.74i	0.164871	−	0.285565i
$536$	1463.92	−	2535.58i	0.117970	−	0.204329i
$537$		0			0
$538$	4628.11	+	8016.11i	0.370877	+	0.642378i
$539$	9072.20			0.724986
$540$		0			0
$541$	2816.86			0.223856			0.111928	−	0.993716i	$-0.464297\pi$
$541$	2816.86			0.223856			0.111928	+	0.993716i	$0.464297\pi$
$542$	−8675.67	−	15026.7i	−0.687549	−	1.19087i
$543$		0			0
$544$	−10407.3	+	18026.0i	−0.820240	+	1.42070i
$545$	1516.94	−	2627.41i	0.119227	−	0.206507i
$546$		0			0
$547$	−2264.45	−	3922.15i	−0.177004	−	0.306579i	0.763849	−	0.645395i	$-0.223307\pi$
$547$	−2264.45	−	3922.15i	−0.177004	−	0.306579i	−0.940853	+	0.338815i	$0.889974\pi$
$548$	−11810.8			−0.920676
$549$		0			0
$550$	−5689.28			−0.441076
$551$	778.549	+	1348.49i	0.0601947	+	0.104260i
$552$		0			0
$553$	−8154.97	+	14124.8i	−0.627097	+	1.08616i
$554$	13431.7	−	23264.5i	1.03007	−	1.78414i
$555$		0			0
$556$	−28.9682	−	50.1744i	−0.00220958	−	0.00382710i
$557$	6267.47			0.476771			0.238385	−	0.971171i	$-0.423382\pi$
$557$	6267.47			0.476771			0.238385	+	0.971171i	$0.423382\pi$
$558$		0			0
$559$	2312.46			0.174967
$560$	−2810.54	−	4868.01i	−0.212084	−	0.367341i
$561$		0			0
$562$	−6455.80	+	11181.8i	−0.484558	+	0.839279i
$563$	4897.00	−	8481.86i	0.366579	−	0.634934i	−0.622449	−	0.782660i	$-0.713862\pi$
$563$	4897.00	−	8481.86i	0.366579	−	0.634934i	0.989028	+	0.147727i	$0.0471956\pi$
$564$		0			0
$565$	68.3065	+	118.310i	0.00508615	+	0.00880948i
$566$	18469.5			1.37161
$567$		0			0
$568$	−12589.2			−0.929981
$569$	−3799.94	−	6581.69i	−0.279968	−	0.484919i	0.691409	−	0.722464i	$-0.256990\pi$
$569$	−3799.94	−	6581.69i	−0.279968	−	0.484919i	−0.971376	+	0.237545i	$0.923657\pi$
$570$		0			0
$571$	6047.07	−	10473.8i	0.443191	−	0.767630i	−0.554733	−	0.832028i	$-0.687180\pi$
$571$	6047.07	−	10473.8i	0.443191	−	0.767630i	0.997924	+	0.0643988i	$0.0205130\pi$
$572$	−1581.01	+	2738.38i	−0.115568	+	0.200170i
$573$		0			0
$574$	−7780.87	−	13476.9i	−0.565796	−	0.979988i
$575$	3220.60			0.233580
$576$		0			0
$577$	3113.85			0.224665			0.112332	−	0.993671i	$-0.464168\pi$
$577$	3113.85			0.224665			0.112332	+	0.993671i	$0.464168\pi$
$578$	−12300.2	−	21304.7i	−0.885161	−	1.53314i
$579$		0			0
$580$	−999.107	+	1730.50i	−0.0715270	+	0.123888i
$581$	3712.12	−	6429.58i	0.265068	−	0.459112i
$582$		0			0
$583$	−17896.4	−	30997.4i	−1.27134	−	2.20203i
$584$	−11916.4			−0.844354
$585$		0			0
$586$	15107.4			1.06499
$587$	−2973.96	−	5151.04i	−0.209111	−	0.362191i	0.742324	−	0.670042i	$-0.233724\pi$
$587$	−2973.96	−	5151.04i	−0.209111	−	0.362191i	−0.951435	+	0.307850i	$0.900390\pi$
$588$		0			0
$589$	97.4727	−	168.828i	0.00681883	−	0.0118106i
$590$	−4131.40	+	7155.80i	−0.288283	+	0.499321i
$591$		0			0
$592$	−3250.79	−	5630.53i	−0.225687	−	0.390901i
$593$	22153.0			1.53409			0.767043	−	0.641596i	$-0.221727\pi$
$593$	22153.0			1.53409			0.767043	+	0.641596i	$0.221727\pi$
$594$		0			0
$595$	7689.10			0.529786
$596$	−7492.47	−	12977.3i	−0.514939	−	0.891901i
$597$		0			0
$598$	2393.61	−	4145.85i	0.163682	−	0.283506i
$599$	13438.2	−	23275.6i	0.916643	−	1.58767i	0.112164	−	0.993690i	$-0.464222\pi$
$599$	13438.2	−	23275.6i	0.916643	−	1.58767i	0.804478	−	0.593982i	$-0.202445\pi$
$600$		0			0
$601$	−12920.1	−	22378.2i	−0.876907	−	1.51885i	−0.854717	−	0.519094i	$-0.826269\pi$
$601$	−12920.1	−	22378.2i	−0.876907	−	1.51885i	−0.0221904	−	0.999754i	$-0.507064\pi$
$602$	−11258.8			−0.762249
$603$		0			0
$604$	−12899.8			−0.869019
$605$	6805.20	+	11787.0i	0.457307	+	0.792079i
$606$		0			0
$607$	1384.37	−	2397.80i	0.0925697	−	0.160335i	−0.816022	−	0.578021i	$-0.803825\pi$
$607$	1384.37	−	2397.80i	0.0925697	−	0.160335i	0.908592	+	0.417685i	$0.137159\pi$
$608$	1783.78	−	3089.60i	0.118983	−	0.206085i
$609$		0			0
$610$	−5800.35	−	10046.5i	−0.384999	−	0.666837i
$611$	3762.60			0.249130
$612$		0			0
$613$	10177.3			0.670563			0.335282	−	0.942118i	$-0.391169\pi$
$613$	10177.3			0.670563			0.335282	+	0.942118i	$0.391169\pi$
$614$	−10248.8	−	17751.4i	−0.673626	−	1.16675i
$615$		0			0
$616$	−5191.87	+	8992.58i	−0.339588	+	0.588184i
$617$	−4208.61	+	7289.53i	−0.274607	+	0.475633i	−0.970036	−	0.242962i	$-0.921881\pi$
$617$	−4208.61	+	7289.53i	−0.274607	+	0.475633i	0.695429	+	0.718595i	$0.255214\pi$
$618$		0			0
$619$	6640.13	+	11501.0i	0.431162	+	0.746795i	0.996974	−	0.0777399i	$-0.0247704\pi$
$619$	6640.13	+	11501.0i	0.431162	+	0.746795i	−0.565812	+	0.824535i	$0.691437\pi$
$620$	250.172			0.0162051
$621$		0			0
$622$	−23665.1			−1.52553
$623$	4647.83	+	8050.27i	0.298895	+	0.517700i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	−166.988	+	289.232i	−0.0106616	+	0.0184665i
$627$		0			0
$628$	−1057.34	−	1831.37i	−0.0671857	−	0.116369i
$629$	8893.52			0.563764
$630$		0			0
$631$	−14454.1			−0.911903			−0.455951	−	0.890005i	$-0.650701\pi$
$631$	−14454.1			−0.911903			−0.455951	+	0.890005i	$0.650701\pi$
$632$	6633.96	+	11490.4i	0.417539	+	0.723199i
$633$		0			0
$634$	3562.37	−	6170.20i	0.223154	−	0.386514i
$635$	−3015.55	+	5223.09i	−0.188455	+	0.326413i
$636$		0			0
$637$	−740.717	−	1282.96i	−0.0460727	−	0.0798002i
$638$	19036.2			1.18127
$639$		0			0
$640$	−6773.96			−0.418381
$641$	−10532.5	−	18242.8i	−0.648997	−	1.12410i	−0.983363	−	0.181653i	$-0.941855\pi$
$641$	−10532.5	−	18242.8i	−0.648997	−	1.12410i	0.334366	−	0.942443i	$-0.391478\pi$
$642$		0			0
$643$	3564.41	−	6173.73i	0.218610	−	0.378644i	−0.735773	−	0.677228i	$-0.763181\pi$
$643$	3564.41	−	6173.73i	0.218610	−	0.378644i	0.954383	+	0.298584i	$0.0965143\pi$
$644$	−4357.43	+	7547.29i	−0.266625	+	0.461809i
$645$		0			0
$646$	3613.24	+	6258.32i	0.220064	+	0.381161i
$647$	1364.92			0.0829376			0.0414688	−	0.999140i	$-0.486796\pi$
$647$	1364.92			0.0829376			0.0414688	+	0.999140i	$0.486796\pi$
$648$		0			0
$649$	29432.4			1.78016
$650$	464.512	+	804.558i	0.0280302	+	0.0485498i
$651$		0			0
$652$	5055.39	−	8756.18i	0.303657	−	0.525949i
$653$	−6080.31	+	10531.4i	−0.364381	+	0.631127i	−0.988677	−	0.150061i	$-0.952053\pi$
$653$	−6080.31	+	10531.4i	−0.364381	+	0.631127i	0.624295	+	0.781188i	$0.285386\pi$
$654$		0			0
$655$	3830.01	+	6633.77i	0.228475	+	0.395730i
$656$	−24410.3			−1.45284
$657$		0			0
$658$	−18319.1			−1.08534
$659$	3084.28	+	5342.14i	0.182317	+	0.315782i	0.942669	−	0.333729i	$-0.108307\pi$
$659$	3084.28	+	5342.14i	0.182317	+	0.315782i	−0.760352	+	0.649511i	$0.774974\pi$
$660$		0			0
$661$	−9068.85	+	15707.7i	−0.533642	+	0.924295i	0.465586	+	0.885003i	$0.345844\pi$
$661$	−9068.85	+	15707.7i	−0.533642	+	0.924295i	−0.999228	+	0.0392922i	$0.987490\pi$
$662$	4270.15	−	7396.11i	0.250701	−	0.434226i
$663$		0			0
$664$	−3019.76	−	5230.38i	−0.176490	−	0.305690i
$665$	−1317.89			−0.0768502
$666$		0			0
$667$	−10776.1			−0.625564
$668$	3092.59	+	5356.52i	0.179125	+	0.310254i
$669$		0			0
$670$	2271.46	−	3934.29i	0.130977	−	0.226858i
$671$	−20661.0	+	35786.0i	−1.18869	+	2.05887i
$672$		0			0
$673$	10851.1	+	18794.6i	0.621512	+	1.07649i	0.989204	+	0.146543i	$0.0468148\pi$
$673$	10851.1	+	18794.6i	0.621512	+	1.07649i	−0.367692	+	0.929948i	$0.619852\pi$
$674$	36766.3			2.10116
$675$		0			0
$676$	−9980.02			−0.567821
$677$	5235.29	+	9067.79i	0.297206	+	0.514776i	0.975496	−	0.220019i	$-0.0706118\pi$
$677$	5235.29	+	9067.79i	0.297206	+	0.514776i	−0.678290	+	0.734795i	$0.737279\pi$
$678$		0			0
$679$	−7968.47	+	13801.8i	−0.450371	+	0.780065i
$680$	3127.49	−	5416.97i	0.176373	−	0.305487i
$681$		0			0
$682$	−1191.65	−	2063.99i	−0.0669070	−	0.115886i
$683$	−15386.0			−0.861974			−0.430987	−	0.902358i	$-0.641834\pi$
$683$	−15386.0			−0.861974			−0.430987	+	0.902358i	$0.641834\pi$
$684$		0			0
$685$	12360.6			0.689451
$686$	12286.8	+	21281.4i	0.683840	+	1.18445i
$687$		0			0
$688$	−8830.33	+	15294.6i	−0.489321	+	0.847529i
$689$	−2922.37	+	5061.69i	−0.161587	+	0.279876i
$690$		0			0
$691$	−7595.57	−	13155.9i	−0.418161	−	0.724276i	0.577594	−	0.816324i	$-0.303992\pi$
$691$	−7595.57	−	13155.9i	−0.418161	−	0.724276i	−0.995755	+	0.0920487i	$0.970658\pi$
$692$	−13960.9			−0.766929
$693$		0			0
$694$	−15121.2			−0.827079
$695$	30.3168	+	52.5102i	0.00165465	+	0.00286594i
$696$		0			0
$697$	16695.5	−	28917.4i	0.907297	−	1.57148i
$698$	20037.1	−	34705.3i	1.08656	−	1.88197i
$699$		0			0
$700$	−845.617	−	1464.65i	−0.0456590	−	0.0790838i
$701$	−5181.93			−0.279199			−0.139600	−	0.990208i	$-0.544582\pi$
$701$	−5181.93			−0.279199			−0.139600	+	0.990208i	$0.544582\pi$
$702$		0			0
$703$	−1524.32			−0.0817791
$704$	−1589.07	−	2752.36i	−0.0850717	−	0.147348i
$705$		0			0
$706$	−20344.7	+	35238.1i	−1.08454	+	1.87847i
$707$	−2692.65	+	4663.80i	−0.143235	+	0.248091i
$708$		0			0
$709$	4393.27	+	7609.36i	0.232712	+	0.403069i	0.958605	−	0.284739i	$-0.0919068\pi$
$709$	4393.27	+	7609.36i	0.232712	+	0.403069i	−0.725893	+	0.687807i	$0.758573\pi$
$710$	−19533.7			−1.03252
$711$		0			0
$712$	7561.89			0.398025
$713$	674.570	+	1168.39i	0.0354318	+	0.0613696i
$714$		0			0
$715$	1654.61	−	2865.86i	0.0865438	−	0.149898i
$716$	9448.89	−	16366.0i	0.493187	−	0.854224i
$717$		0			0
$718$	15983.0	+	27683.3i	0.830750	+	1.43890i
$719$	−14433.0			−0.748625			−0.374312	−	0.927303i	$-0.622121\pi$
$719$	−14433.0			−0.748625			−0.374312	+	0.927303i	$0.622121\pi$
$720$		0			0
$721$	10822.0			0.558989
$722$	11639.7	+	20160.6i	0.599979	+	1.03919i
$723$		0			0
$724$	6435.41	−	11146.5i	0.330345	−	0.572175i
$725$	1045.62	−	1811.07i	0.0535632	−	0.0927742i
$726$		0			0
$727$	−13349.5	−	23121.9i	−0.681023	−	1.17957i	−0.974669	−	0.223653i	$-0.928202\pi$
$727$	−13349.5	−	23121.9i	−0.681023	−	1.17957i	0.293646	−	0.955914i	$-0.405131\pi$
$728$	1695.60			0.0863230
$729$		0			0
$730$	−18489.8			−0.937451
$731$	−12079.0	−	20921.5i	−0.611161	−	1.05856i
$732$		0			0
$733$	3866.92	−	6697.71i	0.194854	−	0.337497i	−0.751999	−	0.659165i	$-0.770910\pi$
$733$	3866.92	−	6697.71i	0.194854	−	0.337497i	0.946853	+	0.321668i	$0.104243\pi$
$734$	793.859	−	1375.00i	0.0399208	−	0.0691449i
$735$		0			0
$736$	12344.8	+	21381.9i	0.618257	+	1.07085i
$737$	−16182.1			−0.808784
$738$		0			0
$739$	28142.2			1.40085			0.700426	−	0.713725i	$-0.252994\pi$
$739$	28142.2			1.40085			0.700426	+	0.713725i	$0.252994\pi$
$740$	−978.074	−	1694.07i	−0.0485874	−	0.0841559i
$741$		0			0
$742$	14228.3	−	24644.1i	0.703957	−	1.21929i
$743$	−11384.6	+	19718.6i	−0.562125	+	0.973629i	0.435186	+	0.900341i	$0.356683\pi$
$743$	−11384.6	+	19718.6i	−0.562125	+	0.973629i	−0.997311	+	0.0732884i	$0.976651\pi$
$744$		0			0
$745$	7841.28	+	13581.5i	0.385614	+	0.667902i
$746$	12211.1			0.599302
$747$		0			0
$748$	33033.3			1.61473
$749$	−5777.76	−	10007.4i	−0.281862	−	0.488200i
$750$		0			0
$751$	−254.918	+	441.531i	−0.0123863	+	0.0214537i	−0.872152	−	0.489235i	$-0.837276\pi$
$751$	−254.918	+	441.531i	−0.0123863	+	0.0214537i	0.859766	+	0.510688i	$0.170609\pi$
$752$	−14367.8	+	24885.8i	−0.696729	+	1.20677i
$753$		0			0
$754$	−1554.25	−	2692.04i	−0.0750695	−	0.130024i
$755$	13500.4			0.650767
$756$		0			0
$757$	−28105.6			−1.34943			−0.674713	−	0.738080i	$-0.735733\pi$
$757$	−28105.6			−1.34943			−0.674713	+	0.738080i	$0.735733\pi$
$758$	−1069.29	−	1852.07i	−0.0512381	−	0.0887469i
$759$		0			0
$760$	−536.041	+	928.450i	−0.0255845	+	0.0443137i
$761$	−4347.82	+	7530.64i	−0.207107	+	0.358720i	−0.950802	−	0.309799i	$-0.899738\pi$
$761$	−4347.82	+	7530.64i	−0.207107	+	0.358720i	0.743695	+	0.668519i	$0.233071\pi$
$762$		0			0
$763$	−4295.89	−	7440.70i	−0.203829	−	0.353043i
$764$	4091.27			0.193739
$765$		0			0
$766$	32375.5			1.52712
$767$	−2403.06	−	4162.23i	−0.113129	−	0.195945i
$768$		0			0
$769$	−2730.93	+	4730.12i	−0.128062	+	0.221811i	−0.922926	−	0.384978i	$-0.874209\pi$
$769$	−2730.93	+	4730.12i	−0.128062	+	0.221811i	0.794863	+	0.606788i	$0.207542\pi$
$770$	−8055.87	+	13953.2i	−0.377030	+	0.653036i
$771$		0			0
$772$	4145.06	+	7179.46i	0.193244	+	0.334708i
$773$	37952.9			1.76594			0.882970	−	0.469430i	$-0.155541\pi$
$773$	37952.9			1.76594			0.882970	+	0.469430i	$0.155541\pi$
$774$		0			0
$775$	−261.819			−0.0121352
$776$	6482.24	+	11227.6i	0.299870	+	0.519390i
$777$		0			0
$778$	1128.50	−	1954.63i	0.0520037	−	0.0900730i
$779$	−2861.55	+	4956.34i	−0.131612	+	0.227958i
$780$		0			0
$781$	34789.9	+	60257.9i	1.59396	+	2.76082i
$782$	−50011.7			−2.28697
$783$		0			0
$784$	11314.0			0.515396
$785$	1106.57	+	1916.63i	0.0503122	+	0.0871433i
$786$		0			0
$787$	9107.57	−	15774.8i	0.412516	−	0.714498i	−0.582648	−	0.812724i	$-0.697983\pi$
$787$	9107.57	−	15774.8i	0.412516	−	0.714498i	0.995164	+	0.0982261i	$0.0313168\pi$
$788$	7032.48	−	12180.6i	0.317921	−	0.550655i
$789$		0			0
$790$	10293.5	+	17828.8i	0.463576	+	0.802937i
$791$	386.881			0.0173905
$792$		0			0
$793$	6747.64			0.302164
$794$	8249.03	+	14287.7i	0.368699	+	0.638605i
$795$		0			0
$796$	9966.18	−	17261.9i	0.443771	−	0.768634i
$797$	−14274.8	+	24724.6i	−0.634427	+	1.09886i	0.352209	+	0.935921i	$0.385431\pi$
$797$	−14274.8	+	24724.6i	−0.634427	+	1.09886i	−0.986636	+	0.162938i	$0.947903\pi$
$798$		0			0
$799$	−19653.8	−	34041.3i	−0.870213	−	1.50725i
$800$	−4791.36			−0.211750
$801$		0			0
$802$	−16769.0			−0.738323
$803$	32930.7	+	57037.6i	1.44720	+	2.50662i
$804$		0			0
$805$	4560.28	−	7898.64i	0.199663	−	0.345827i
$806$	−194.589	+	337.037i	−0.00850384	+	0.0147291i
$807$		0			0
$808$	2190.43	+	3793.94i	0.0953703	+	0.165186i
$809$	2685.20			0.116695			0.0583476	−	0.998296i	$-0.481417\pi$
$809$	2685.20			0.116695			0.0583476	+	0.998296i	$0.481417\pi$
$810$		0			0
$811$	−12491.4			−0.540854			−0.270427	−	0.962740i	$-0.587165\pi$
$811$	−12491.4			−0.540854			−0.270427	+	0.962740i	$0.587165\pi$
$812$	2829.42	+	4900.70i	0.122282	+	0.211799i
$813$		0			0
$814$	−9317.73	+	16138.8i	−0.401212	+	0.694919i
$815$	−5290.73	+	9163.82i	−0.227394	+	0.393858i
$816$		0			0
$817$	2070.30	+	3585.87i	0.0886545	+	0.153554i
$818$	−55632.1			−2.37791
$819$		0			0
$820$	−7344.41			−0.312778
$821$	−11758.7	−	20366.7i	−0.499858	−	0.865779i	0.500142	−	0.865943i	$-0.333281\pi$
$821$	−11758.7	−	20366.7i	−0.499858	−	0.865779i	−1.00000		0.000164513i	$0.999948\pi$
$822$		0			0
$823$	9063.09	−	15697.7i	0.383863	−	0.664871i	−0.607747	−	0.794130i	$-0.707927\pi$
$823$	9063.09	−	15697.7i	0.383863	−	0.664871i	0.991611	+	0.129259i	$0.0412600\pi$
$824$	4401.76	−	7624.08i	0.186096	−	0.322327i
$825$		0			0
$826$	11699.9	+	20264.9i	0.492848	+	0.853637i
$827$	−45677.1			−1.92062			−0.960308	−	0.278940i	$-0.910017\pi$
$827$	−45677.1			−1.92062			−0.960308	+	0.278940i	$0.910017\pi$
$828$		0			0
$829$	9130.02			0.382507			0.191254	−	0.981541i	$-0.438745\pi$
$829$	9130.02			0.382507			0.191254	+	0.981541i	$0.438745\pi$
$830$	−4685.56	−	8115.62i	−0.195949	−	0.339394i
$831$		0			0
$832$	−259.486	+	449.443i	−0.0108126	+	0.0187279i
$833$	−7738.21	+	13403.0i	−0.321864	+	0.557486i
$834$		0			0
$835$	−3236.56	−	5605.88i	−0.134139	−	0.232335i
$836$	−5661.78			−0.234231
$837$		0			0
$838$	−15293.6			−0.630440
$839$	9103.20	+	15767.2i	0.374586	+	0.648802i	0.990265	−	0.139195i	$-0.0444516\pi$
$839$	9103.20	+	15767.2i	0.374586	+	0.648802i	−0.615679	+	0.787997i	$0.711118\pi$
$840$		0			0
$841$	8695.88	−	15061.7i	0.356549	−	0.617561i
$842$	174.629	−	302.467i	0.00714741	−	0.0123797i
$843$		0			0
$844$	−9199.35	−	15933.7i	−0.375183	−	0.649836i
$845$	10444.6			0.425214
$846$		0			0
$847$	38543.9			1.56362
$848$	−22318.6	−	38657.0i	−0.903803	−	1.56543i
$849$		0			0
$850$	4852.71	−	8405.15i	0.195820	−	0.339170i
$851$	5274.60	−	9135.88i	0.212469	−	0.368007i
$852$		0			0
$853$	−14896.5	−	25801.4i	−0.597943	−	1.03567i	−0.993124	−	0.117064i	$-0.962652\pi$
$853$	−14896.5	−	25801.4i	−0.597943	−	1.03567i	0.395182	−	0.918603i	$-0.370682\pi$
$854$	−32852.6			−1.31638
$855$		0			0
$856$	−9400.27			−0.375344
$857$	10749.8	+	18619.1i	0.428477	+	0.742144i	0.996738	−	0.0807043i	$-0.0257170\pi$
$857$	10749.8	+	18619.1i	0.428477	+	0.742144i	−0.568261	+	0.822848i	$0.692384\pi$
$858$		0			0
$859$	15206.7	−	26338.9i	0.604013	−	1.04618i	−0.388193	−	0.921578i	$-0.626901\pi$
$859$	15206.7	−	26338.9i	0.604013	−	1.04618i	0.992207	−	0.124604i	$-0.0397660\pi$
$860$	−2656.81	+	4601.73i	−0.105345	+	0.182462i
$861$		0			0
$862$	21327.5	+	36940.3i	0.842711	+	1.45962i
$863$	7021.71			0.276966			0.138483	−	0.990365i	$-0.455777\pi$
$863$	7021.71			0.276966			0.138483	+	0.990365i	$0.455777\pi$
$864$		0			0
$865$	14610.9			0.574317
$866$	−11267.6	−	19516.1i	−0.442136	−	0.765802i
$867$		0			0
$868$	354.238	−	613.557i	0.0138521	−	0.0239925i
$869$	36665.7	−	63506.8i	1.43130	−	2.47908i
$870$		0			0
$871$	1321.22	+	2288.41i	0.0513980	+	0.0890240i
$872$	−6989.30			−0.271431
$873$		0			0
$874$	8571.82			0.331746
$875$	884.984	+	1532.84i	0.0341919	+	0.0592221i
$876$		0			0
$877$	6009.37	−	10408.5i	0.231382	−	0.400765i	−0.726833	−	0.686814i	$-0.759009\pi$
$877$	6009.37	−	10408.5i	0.231382	−	0.400765i	0.958215	+	0.286049i	$0.0923419\pi$
$878$	23096.5	−	40004.4i	0.887779	−	1.53768i
$879$		0			0
$880$	12636.5	+	21887.1i	0.484065	+	0.838425i
$881$	21990.9			0.840968			0.420484	−	0.907300i	$-0.361860\pi$
$881$	21990.9			0.840968			0.420484	+	0.907300i	$0.361860\pi$
$882$		0			0
$883$	−6023.11			−0.229551			−0.114776	−	0.993391i	$-0.536615\pi$
$883$	−6023.11			−0.229551			−0.114776	+	0.993391i	$0.536615\pi$
$884$	−2697.06	−	4671.45i	−0.102615	−	0.177735i
$885$		0			0
$886$	−6835.32	+	11839.1i	−0.259184	+	0.448920i
$887$	4289.05	−	7428.84i	0.162359	−	0.281213i	−0.773356	−	0.633973i	$-0.781423\pi$
$887$	4289.05	−	7428.84i	0.162359	−	0.281213i	0.935714	+	0.352759i	$0.114757\pi$
$888$		0			0
$889$	8539.89	+	14791.5i	0.322181	+	0.558034i
$890$	11733.3			0.441910
$891$		0			0
$892$	3921.03			0.147181
$893$	3368.59	+	5834.56i	0.126232	+	0.218641i
$894$		0			0
$895$	−9888.77	+	17127.9i	−0.369324	+	0.639688i
$896$	−9591.75	+	16613.4i	−0.357631	+	0.619436i
$897$		0			0
$898$	−948.436	−	1642.74i	−0.0352447	−	0.0610455i
$899$	876.041			0.0325001
$900$		0			0
$901$	61059.4			2.25770
$902$	34983.7	+	60593.5i	1.29138	+	2.23674i
$903$		0			0
$904$	157.361	−	272.558i	0.00578956	−	0.0100278i
$905$	−6735.00	+	11665.4i	−0.247380	+	0.428475i
$906$		0			0
$907$	−4670.31	−	8089.21i	−0.170976	−	0.296139i	0.767785	−	0.640707i	$-0.221359\pi$
$907$	−4670.31	−	8089.21i	−0.170976	−	0.296139i	−0.938761	+	0.344568i	$0.888025\pi$
$908$	25136.0			0.918685
$909$		0			0
$910$	2630.95			0.0958407
$911$	−8176.45	−	14162.0i	−0.297363	−	0.515048i	0.678169	−	0.734906i	$-0.262774\pi$
$911$	−8176.45	−	14162.0i	−0.297363	−	0.515048i	−0.975532	+	0.219858i	$0.929441\pi$
$912$		0			0
$913$	−16690.1	+	28908.1i	−0.604997	+	1.04789i
$914$	27870.2	−	48272.7i	1.00861	−	1.74696i
$915$		0			0
$916$	−46.8177	−	81.0906i	−0.00168876	−	0.00292501i
$917$	21692.8			0.781198
$918$		0			0
$919$	1321.13			0.0474211			0.0237105	−	0.999719i	$-0.492452\pi$
$919$	1321.13			0.0474211			0.0237105	+	0.999719i	$0.492452\pi$
$920$	−3709.73	−	6425.44i	−0.132941	−	0.230261i
$921$		0			0
$922$	−13001.8	+	22519.8i	−0.464417	+	0.804393i
$923$	5680.98	−	9839.74i	0.202591	−	0.350898i
$924$		0			0
$925$	1023.61	+	1772.94i	0.0363848	+	0.0630204i
$926$	9598.23			0.340623
$927$		0			0
$928$	16031.8			0.567102
$929$	19159.5	+	33185.2i	0.676643	+	1.17198i	0.975986	+	0.217835i	$0.0698994\pi$
$929$	19159.5	+	33185.2i	0.676643	+	1.17198i	−0.299342	+	0.954146i	$0.596767\pi$
$930$		0			0
$931$	1326.30	−	2297.22i	0.0466894	−	0.0808683i
$932$	−7600.64	+	13164.7i	−0.267132	+	0.462687i
$933$		0			0
$934$	−15808.2	−	27380.5i	−0.553810	−	0.959227i
$935$	−34571.1			−1.20919
$936$		0			0
$937$	−24307.8			−0.847494			−0.423747	−	0.905781i	$-0.639285\pi$
$937$	−24307.8			−0.847494			−0.423747	+	0.905781i	$0.639285\pi$
$938$	−6432.67	−	11141.7i	−0.223917	−	0.387835i
$939$		0			0
$940$	−4322.89	+	7487.46i	−0.149997	+	0.259802i
$941$	11511.2	−	19938.1i	0.398784	−	0.690714i	−0.594792	−	0.803880i	$-0.702766\pi$
$941$	11511.2	−	19938.1i	0.398784	−	0.690714i	0.993576	+	0.113165i	$0.0360990\pi$
$942$		0			0
$943$	−19803.6	−	34300.9i	−0.683876	−	1.18451i
$944$	36705.3			1.26552
$945$		0			0
$946$	50620.8			1.73977
$947$	14162.4	+	24530.0i	0.485972	+	0.841729i	0.999870	−	0.0161227i	$-0.00513224\pi$
$947$	14162.4	+	24530.0i	0.485972	+	0.841729i	−0.513898	+	0.857851i	$0.671799\pi$
$948$		0			0
$949$	5377.38	−	9313.89i	0.183938	−	0.318590i
$950$	−831.738	+	1440.61i	−0.0284054	+	0.0491996i
$951$		0			0
$952$	−8856.89	−	15340.6i	−0.301527	−	0.522260i
$953$	12986.4			0.441417			0.220709	−	0.975340i	$-0.429163\pi$
$953$	12986.4			0.441417			0.220709	+	0.975340i	$0.429163\pi$
$954$		0			0
$955$	−4281.73			−0.145082
$956$	3903.54	+	6761.13i	0.132060	+	0.228735i
$957$		0			0
$958$	22045.0	−	38183.1i	0.743468	−	1.28772i
$959$	17502.3	−	30314.8i	0.589341	−	1.02077i
$960$		0			0
$961$	14840.7	+	25704.8i	0.498159	+	0.862837i
$962$	3043.06			0.101988
$963$		0			0
$964$	6758.90			0.225819
$965$	−4338.03	−	7513.69i	−0.144711	−	0.250647i
$966$		0			0
$967$	19626.0	−	33993.2i	0.652666	−	1.13045i	−0.329807	−	0.944048i	$-0.606984\pi$
$967$	19626.0	−	33993.2i	0.652666	−	1.13045i	0.982473	−	0.186403i	$-0.0596830\pi$
$968$	15677.5	−	27154.2i	0.520551	−	0.901621i
$969$		0			0
$970$	10058.1	+	17421.1i	0.332933	+	0.576656i
$971$	2404.97			0.0794843			0.0397422	−	0.999210i	$-0.487346\pi$
$971$	2404.97			0.0794843			0.0397422	+	0.999210i	$0.487346\pi$
$972$		0			0
$973$	171.711			0.00565756
$974$	−6327.56	−	10959.7i	−0.208160	−	0.360544i
$975$		0			0
$976$	−25766.5	+	44628.8i	−0.845045	+	1.46366i
$977$	−8820.02	+	15276.7i	−0.288820	+	0.500252i	−0.973528	−	0.228566i	$-0.926596\pi$
$977$	−8820.02	+	15276.7i	−0.288820	+	0.500252i	0.684708	+	0.728818i	$0.259930\pi$
$978$		0			0
$979$	−20897.2	−	36194.9i	−0.682202	−	1.18161i
$980$	3404.07			0.110958
$981$		0			0
$982$	−27564.4			−0.895740
$983$	−26611.4	−	46092.3i	−0.863451	−	1.49554i	−0.868577	−	0.495554i	$-0.834965\pi$
$983$	−26611.4	−	46092.3i	−0.863451	−	1.49554i	0.00512618	−	0.999987i	$-0.498368\pi$
$984$		0			0
$985$	−7359.86	+	12747.7i	−0.238076	+	0.412360i
$986$	−16237.1	+	28123.5i	−0.524437	+	0.908351i
$987$		0			0
$988$	462.267	+	800.670i	0.0148853	+	0.0257821i
$989$	−28655.5			−0.921327
$990$		0			0
$991$	−33124.1			−1.06178			−0.530889	−	0.847441i	$-0.678142\pi$
$991$	−33124.1			−1.06178			−0.530889	+	0.847441i	$0.678142\pi$
$992$	−1003.58	−	1738.24i	−0.0321205	−	0.0556344i
$993$		0			0
$994$	−27659.2	+	47907.2i	−0.882594	+	1.52870i
$995$	−10430.1	+	18065.5i	−0.332319	+	0.575594i
$996$		0			0
$997$	5868.45	+	10164.5i	0.186415	+	0.322880i	0.944052	−	0.329795i	$-0.106980\pi$
$997$	5868.45	+	10164.5i	0.186415	+	0.322880i	−0.757637	+	0.652676i	$0.773646\pi$
$998$	−43476.1			−1.37897
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.x.271.2		12
3.2	odd	2		405.4.e.w.271.5		12
9.2	odd	6		405.4.e.w.136.5		12
9.4	even	3		405.4.a.k.1.5	✓	6
9.5	odd	6		405.4.a.l.1.2	yes	6
9.7	even	3	inner	405.4.e.x.136.2		12
45.4	even	6		2025.4.a.z.1.2		6
45.14	odd	6		2025.4.a.y.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.5	✓	6	9.4	even	3
405.4.a.l.1.2	yes	6	9.5	odd	6
405.4.e.w.136.5		12	9.2	odd	6
405.4.e.w.271.5		12	3.2	odd	2
405.4.e.x.136.2		12	9.7	even	3	inner
405.4.e.x.271.2		12	1.1	even	1	trivial
2025.4.a.y.1.5		6	45.14	odd	6
2025.4.a.z.1.2		6	45.4	even	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.78729 - 3.09567i) q^{2} +(-2.38879 + 4.13751i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-7.07987 - 12.2627i) q^{7} -11.5188 q^{8} +O(q^{10})\)	\(q+(-1.78729 - 3.09567i) q^{2} +(-2.38879 + 4.13751i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-7.07987 - 12.2627i) q^{7} -11.5188 q^{8} -17.8729 q^{10} +(31.8319 + 55.1345i) q^{11} +(5.19795 - 9.00312i) q^{13} +(-25.3075 + 43.8339i) q^{14} +(39.6977 + 68.7584i) q^{16} -108.605 q^{17} +18.6145 q^{19} +(11.9440 + 20.6876i) q^{20} +(113.786 - 197.082i) q^{22} +(-64.4120 + 111.565i) q^{23} +(-12.5000 - 21.6506i) q^{25} -37.1609 q^{26} +67.6494 q^{28} +(41.8248 + 72.4426i) q^{29} +(5.23638 - 9.06967i) q^{31} +(95.8273 - 165.978i) q^{32} +(194.109 + 336.206i) q^{34} -70.7987 q^{35} -81.8885 q^{37} +(-33.2695 - 57.6245i) q^{38} +(-28.7969 + 49.8777i) q^{40} +(-153.726 + 266.262i) q^{41} +(111.220 + 192.638i) q^{43} -304.159 q^{44} +460.491 q^{46} +(180.965 + 313.441i) q^{47} +(71.2509 - 123.410i) q^{49} +(-44.6822 + 77.3918i) q^{50} +(24.8337 + 43.0132i) q^{52} -562.215 q^{53} +318.319 q^{55} +(81.5513 + 141.251i) q^{56} +(149.506 - 258.952i) q^{58} +(231.155 - 400.372i) q^{59} +(324.533 + 562.108i) q^{61} -37.4356 q^{62} -49.9208 q^{64} +(-25.9898 - 45.0156i) q^{65} +(-127.090 + 220.126i) q^{67} +(259.435 - 449.355i) q^{68} +(126.538 + 219.170i) q^{70} +1092.93 q^{71} +1034.52 q^{73} +(146.358 + 253.500i) q^{74} +(-44.4663 + 77.0179i) q^{76} +(450.731 - 780.690i) q^{77} +(-575.927 - 997.534i) q^{79} +396.977 q^{80} +1099.01 q^{82} +(262.160 + 454.075i) q^{83} +(-271.513 + 470.274i) q^{85} +(397.563 - 688.599i) q^{86} +(-366.664 - 635.081i) q^{88} -656.485 q^{89} -147.203 q^{91} +(-307.734 - 533.010i) q^{92} +(646.874 - 1120.42i) q^{94} +(46.5363 - 80.6033i) q^{95} +(-562.755 - 974.721i) q^{97} -509.383 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 34 q^{4} + 30 q^{5} - 40 q^{7} - 132 q^{8}+O(q^{10}) \) 12 * q + 4 * q^2 - 34 * q^4 + 30 * q^5 - 40 * q^7 - 132 * q^8	\( 12 q + 4 q^{2} - 34 q^{4} + 30 q^{5} - 40 q^{7} - 132 q^{8} + 40 q^{10} + 88 q^{11} - 20 q^{13} + 180 q^{14} - 58 q^{16} - 248 q^{17} - 92 q^{19} + 170 q^{20} + 74 q^{22} + 210 q^{23} - 150 q^{25} - 8 q^{26} + 704 q^{28} + 296 q^{29} + 104 q^{31} + 722 q^{32} + 428 q^{34} - 400 q^{35} - 408 q^{37} - 20 q^{38} - 330 q^{40} + 344 q^{41} - 512 q^{43} - 1432 q^{44} - 372 q^{46} + 238 q^{47} - 68 q^{49} + 100 q^{50} + 468 q^{52} - 1700 q^{53} + 880 q^{55} + 2316 q^{56} - 890 q^{58} + 1840 q^{59} + 364 q^{61} - 2076 q^{62} - 1980 q^{64} + 100 q^{65} - 88 q^{67} + 236 q^{68} - 900 q^{70} - 2728 q^{71} + 1672 q^{73} + 1316 q^{74} + 2106 q^{76} + 840 q^{77} + 680 q^{79} - 580 q^{80} + 3484 q^{82} + 2148 q^{83} - 620 q^{85} + 2872 q^{86} - 1296 q^{88} - 6000 q^{89} - 6116 q^{91} + 1002 q^{92} + 3662 q^{94} - 230 q^{95} + 612 q^{97} - 3964 q^{98}+O(q^{100}) \) 12 * q + 4 * q^2 - 34 * q^4 + 30 * q^5 - 40 * q^7 - 132 * q^8 + 40 * q^10 + 88 * q^11 - 20 * q^13 + 180 * q^14 - 58 * q^16 - 248 * q^17 - 92 * q^19 + 170 * q^20 + 74 * q^22 + 210 * q^23 - 150 * q^25 - 8 * q^26 + 704 * q^28 + 296 * q^29 + 104 * q^31 + 722 * q^32 + 428 * q^34 - 400 * q^35 - 408 * q^37 - 20 * q^38 - 330 * q^40 + 344 * q^41 - 512 * q^43 - 1432 * q^44 - 372 * q^46 + 238 * q^47 - 68 * q^49 + 100 * q^50 + 468 * q^52 - 1700 * q^53 + 880 * q^55 + 2316 * q^56 - 890 * q^58 + 1840 * q^59 + 364 * q^61 - 2076 * q^62 - 1980 * q^64 + 100 * q^65 - 88 * q^67 + 236 * q^68 - 900 * q^70 - 2728 * q^71 + 1672 * q^73 + 1316 * q^74 + 2106 * q^76 + 840 * q^77 + 680 * q^79 - 580 * q^80 + 3484 * q^82 + 2148 * q^83 - 620 * q^85 + 2872 * q^86 - 1296 * q^88 - 6000 * q^89 - 6116 * q^91 + 1002 * q^92 + 3662 * q^94 - 230 * q^95 + 612 * q^97 - 3964 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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