LMFDB - Embedded newform 50.5.c.b.43.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,5,Mod(7,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("50.7");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 50 = 2 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	50.c (of order $4$, degree $2$, 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:	$5.16849815419$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			43.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	50.43
Dual form			50.5.c.b.7.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.00000 - 2.00000i) q^{2} +(-9.00000 - 9.00000i) q^{3} -8.00000i q^{4} -36.0000 q^{6} +(-29.0000 + 29.0000i) q^{7} +(-16.0000 - 16.0000i) q^{8} +81.0000i q^{9} -118.000 q^{11} +(-72.0000 + 72.0000i) q^{12} +(-69.0000 - 69.0000i) q^{13} +116.000i q^{14} -64.0000 q^{16} +(271.000 - 271.000i) q^{17} +(162.000 + 162.000i) q^{18} -280.000i q^{19} +522.000 q^{21} +(-236.000 + 236.000i) q^{22} +(-269.000 - 269.000i) q^{23} +288.000i q^{24} -276.000 q^{26} +(232.000 + 232.000i) q^{28} -680.000i q^{29} +202.000 q^{31} +(-128.000 + 128.000i) q^{32} +(1062.00 + 1062.00i) q^{33} -1084.00i q^{34} +648.000 q^{36} +(651.000 - 651.000i) q^{37} +(-560.000 - 560.000i) q^{38} +1242.00i q^{39} +1682.00 q^{41} +(1044.00 - 1044.00i) q^{42} +(-1089.00 - 1089.00i) q^{43} +944.000i q^{44} -1076.00 q^{46} +(-1269.00 + 1269.00i) q^{47} +(576.000 + 576.000i) q^{48} +719.000i q^{49} -4878.00 q^{51} +(-552.000 + 552.000i) q^{52} +(611.000 + 611.000i) q^{53} +928.000 q^{56} +(-2520.00 + 2520.00i) q^{57} +(-1360.00 - 1360.00i) q^{58} -1160.00i q^{59} -5598.00 q^{61} +(404.000 - 404.000i) q^{62} +(-2349.00 - 2349.00i) q^{63} +512.000i q^{64} +4248.00 q^{66} +(751.000 - 751.000i) q^{67} +(-2168.00 - 2168.00i) q^{68} +4842.00i q^{69} +6442.00 q^{71} +(1296.00 - 1296.00i) q^{72} +(2951.00 + 2951.00i) q^{73} -2604.00i q^{74} -2240.00 q^{76} +(3422.00 - 3422.00i) q^{77} +(2484.00 + 2484.00i) q^{78} -10560.0i q^{79} +6561.00 q^{81} +(3364.00 - 3364.00i) q^{82} +(6231.00 + 6231.00i) q^{83} -4176.00i q^{84} -4356.00 q^{86} +(-6120.00 + 6120.00i) q^{87} +(1888.00 + 1888.00i) q^{88} +14480.0i q^{89} +4002.00 q^{91} +(-2152.00 + 2152.00i) q^{92} +(-1818.00 - 1818.00i) q^{93} +5076.00i q^{94} +2304.00 q^{96} +(7311.00 - 7311.00i) q^{97} +(1438.00 + 1438.00i) q^{98} -9558.00i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 18 q^{3} - 72 q^{6} - 58 q^{7} - 32 q^{8} - 236 q^{11} - 144 q^{12} - 138 q^{13} - 128 q^{16} + 542 q^{17} + 324 q^{18} + 1044 q^{21} - 472 q^{22} - 538 q^{23} - 552 q^{26} + 464 q^{28}+ \cdots + 2876 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 18 * q^3 - 72 * q^6 - 58 * q^7 - 32 * q^8 - 236 * q^11 - 144 * q^12 - 138 * q^13 - 128 * q^16 + 542 * q^17 + 324 * q^18 + 1044 * q^21 - 472 * q^22 - 538 * q^23 - 552 * q^26 + 464 * q^28 + 404 * q^31 - 256 * q^32 + 2124 * q^33 + 1296 * q^36 + 1302 * q^37 - 1120 * q^38 + 3364 * q^41 + 2088 * q^42 - 2178 * q^43 - 2152 * q^46 - 2538 * q^47 + 1152 * q^48 - 9756 * q^51 - 1104 * q^52 + 1222 * q^53 + 1856 * q^56 - 5040 * q^57 - 2720 * q^58 - 11196 * q^61 + 808 * q^62 - 4698 * q^63 + 8496 * q^66 + 1502 * q^67 - 4336 * q^68 + 12884 * q^71 + 2592 * q^72 + 5902 * q^73 - 4480 * q^76 + 6844 * q^77 + 4968 * q^78 + 13122 * q^81 + 6728 * q^82 + 12462 * q^83 - 8712 * q^86 - 12240 * q^87 + 3776 * q^88 + 8004 * q^91 - 4304 * q^92 - 3636 * q^93 + 4608 * q^96 + 14622 * q^97 + 2876 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$.

$n$	$27$
$\chi(n)$	$e\left(\frac{3}{4}\right)$

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 $

$2$	2.00000	−	2.00000i	0.500000	−	0.500000i
$2$	2.00000	−	2.00000i	0.500000	−	0.500000i
$3$	−9.00000	−	9.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	−9.00000	−	9.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$4$		−	8.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−36.0000			−1.00000
$7$	−29.0000	+	29.0000i	−0.591837	+	0.591837i	−0.938127	−	0.346291i	$-0.887441\pi$
$7$	−29.0000	+	29.0000i	−0.591837	+	0.591837i	0.346291	+	0.938127i	$0.387441\pi$
$8$	−16.0000	−	16.0000i	−0.250000	−	0.250000i
$9$			81.0000i			1.00000i
$10$		0			0
$11$	−118.000			−0.975207			−0.487603	−	0.873065i	$-0.662129\pi$
$11$	−118.000			−0.975207			−0.487603	+	0.873065i	$0.662129\pi$
$12$	−72.0000	+	72.0000i	−0.500000	+	0.500000i
$13$	−69.0000	−	69.0000i	−0.408284	−	0.408284i	0.472856	−	0.881140i	$-0.343223\pi$
$13$	−69.0000	−	69.0000i	−0.408284	−	0.408284i	−0.881140	+	0.472856i	$0.843223\pi$
$14$			116.000i			0.591837i
$15$		0			0
$16$	−64.0000			−0.250000
$17$	271.000	−	271.000i	0.937716	−	0.937716i	−0.0604547	−	0.998171i	$-0.519255\pi$
$17$	271.000	−	271.000i	0.937716	−	0.937716i	0.998171	+	0.0604547i	$0.0192551\pi$
$18$	162.000	+	162.000i	0.500000	+	0.500000i
$19$		−	280.000i		−	0.775623i	−0.921739	−	0.387812i	$-0.873231\pi$
$19$		−	280.000i		−	0.775623i	0.921739	−	0.387812i	$-0.126769\pi$
$20$		0			0
$21$	522.000			1.18367
$22$	−236.000	+	236.000i	−0.487603	+	0.487603i
$23$	−269.000	−	269.000i	−0.508507	−	0.508507i	0.405561	−	0.914068i	$-0.367076\pi$
$23$	−269.000	−	269.000i	−0.508507	−	0.508507i	−0.914068	+	0.405561i	$0.867076\pi$
$24$			288.000i			0.500000i
$25$		0			0
$26$	−276.000			−0.408284
$27$		0			0
$28$	232.000	+	232.000i	0.295918	+	0.295918i
$29$		−	680.000i		−	0.808561i	−0.914635	−	0.404281i	$-0.867522\pi$
$29$		−	680.000i		−	0.808561i	0.914635	−	0.404281i	$-0.132478\pi$
$30$		0			0
$31$	202.000			0.210198			0.105099	−	0.994462i	$-0.466484\pi$
$31$	202.000			0.210198			0.105099	+	0.994462i	$0.466484\pi$
$32$	−128.000	+	128.000i	−0.125000	+	0.125000i
$33$	1062.00	+	1062.00i	0.975207	+	0.975207i
$34$		−	1084.00i		−	0.937716i
$35$		0			0
$36$	648.000			0.500000
$37$	651.000	−	651.000i	0.475530	−	0.475530i	−0.428169	−	0.903699i	$-0.640841\pi$
$37$	651.000	−	651.000i	0.475530	−	0.475530i	0.903699	+	0.428169i	$0.140841\pi$
$38$	−560.000	−	560.000i	−0.387812	−	0.387812i
$39$			1242.00i			0.816568i
$40$		0			0
$41$	1682.00			1.00059			0.500297	−	0.865854i	$-0.333224\pi$
$41$	1682.00			1.00059			0.500297	+	0.865854i	$0.333224\pi$
$42$	1044.00	−	1044.00i	0.591837	−	0.591837i
$43$	−1089.00	−	1089.00i	−0.588967	−	0.588967i	0.348385	−	0.937352i	$-0.386730\pi$
$43$	−1089.00	−	1089.00i	−0.588967	−	0.588967i	−0.937352	+	0.348385i	$0.886730\pi$
$44$			944.000i			0.487603i
$45$		0			0
$46$	−1076.00			−0.508507
$47$	−1269.00	+	1269.00i	−0.574468	+	0.574468i	−0.933374	−	0.358906i	$-0.883150\pi$
$47$	−1269.00	+	1269.00i	−0.574468	+	0.574468i	0.358906	+	0.933374i	$0.383150\pi$
$48$	576.000	+	576.000i	0.250000	+	0.250000i
$49$			719.000i			0.299459i
$50$		0			0
$51$	−4878.00			−1.87543
$52$	−552.000	+	552.000i	−0.204142	+	0.204142i
$53$	611.000	+	611.000i	0.217515	+	0.217515i	0.807450	−	0.589935i	$-0.200847\pi$
$53$	611.000	+	611.000i	0.217515	+	0.217515i	−0.589935	+	0.807450i	$0.700847\pi$
$54$		0			0
$55$		0			0
$56$	928.000			0.295918
$57$	−2520.00	+	2520.00i	−0.775623	+	0.775623i
$58$	−1360.00	−	1360.00i	−0.404281	−	0.404281i
$59$		−	1160.00i		−	0.333238i	−0.986021	−	0.166619i	$-0.946715\pi$
$59$		−	1160.00i		−	0.333238i	0.986021	−	0.166619i	$-0.0532849\pi$
$60$		0			0
$61$	−5598.00			−1.50443			−0.752217	−	0.658915i	$-0.771016\pi$
$61$	−5598.00			−1.50443			−0.752217	+	0.658915i	$0.771016\pi$
$62$	404.000	−	404.000i	0.105099	−	0.105099i
$63$	−2349.00	−	2349.00i	−0.591837	−	0.591837i
$64$			512.000i			0.125000i
$65$		0			0
$66$	4248.00			0.975207
$67$	751.000	−	751.000i	0.167298	−	0.167298i	−0.618493	−	0.785791i	$-0.712256\pi$
$67$	751.000	−	751.000i	0.167298	−	0.167298i	0.785791	+	0.618493i	$0.212256\pi$
$68$	−2168.00	−	2168.00i	−0.468858	−	0.468858i
$69$			4842.00i			1.01701i
$70$		0			0
$71$	6442.00			1.27792			0.638961	−	0.769240i	$-0.279365\pi$
$71$	6442.00			1.27792			0.638961	+	0.769240i	$0.279365\pi$
$72$	1296.00	−	1296.00i	0.250000	−	0.250000i
$73$	2951.00	+	2951.00i	0.553762	+	0.553762i	0.927525	−	0.373762i	$-0.121932\pi$
$73$	2951.00	+	2951.00i	0.553762	+	0.553762i	−0.373762	+	0.927525i	$0.621932\pi$
$74$		−	2604.00i		−	0.475530i
$75$		0			0
$76$	−2240.00			−0.387812
$77$	3422.00	−	3422.00i	0.577163	−	0.577163i
$78$	2484.00	+	2484.00i	0.408284	+	0.408284i
$79$		−	10560.0i		−	1.69204i	−0.533154	−	0.846018i	$-0.678993\pi$
$79$		−	10560.0i		−	1.69204i	0.533154	−	0.846018i	$-0.321007\pi$
$80$		0			0
$81$	6561.00			1.00000
$82$	3364.00	−	3364.00i	0.500297	−	0.500297i
$83$	6231.00	+	6231.00i	0.904485	+	0.904485i	0.995820	−	0.0913348i	$-0.0291134\pi$
$83$	6231.00	+	6231.00i	0.904485	+	0.904485i	−0.0913348	+	0.995820i	$0.529113\pi$
$84$		−	4176.00i		−	0.591837i
$85$		0			0
$86$	−4356.00			−0.588967
$87$	−6120.00	+	6120.00i	−0.808561	+	0.808561i
$88$	1888.00	+	1888.00i	0.243802	+	0.243802i
$89$			14480.0i			1.82805i	0.405656	+	0.914026i	$0.367043\pi$
$89$			14480.0i			1.82805i	−0.405656	+	0.914026i	$0.632957\pi$
$90$		0			0
$91$	4002.00			0.483275
$92$	−2152.00	+	2152.00i	−0.254253	+	0.254253i
$93$	−1818.00	−	1818.00i	−0.210198	−	0.210198i
$94$			5076.00i			0.574468i
$95$		0			0
$96$	2304.00			0.250000
$97$	7311.00	−	7311.00i	0.777022	−	0.777022i	−0.202301	−	0.979323i	$-0.564842\pi$
$97$	7311.00	−	7311.00i	0.777022	−	0.777022i	0.979323	+	0.202301i	$0.0648420\pi$
$98$	1438.00	+	1438.00i	0.149729	+	0.149729i
$99$		−	9558.00i		−	0.975207i
$100$		0			0
$101$	−878.000			−0.0860700			−0.0430350	−	0.999074i	$-0.513703\pi$
$101$	−878.000			−0.0860700			−0.0430350	+	0.999074i	$0.513703\pi$
$102$	−9756.00	+	9756.00i	−0.937716	+	0.937716i
$103$	−10429.0	−	10429.0i	−0.983033	−	0.983033i	0.0168252	−	0.999858i	$-0.494644\pi$
$103$	−10429.0	−	10429.0i	−0.983033	−	0.983033i	−0.999858	+	0.0168252i	$0.994644\pi$
$104$			2208.00i			0.204142i
$105$		0			0
$106$	2444.00			0.217515
$107$	4711.00	−	4711.00i	0.411477	−	0.411477i	−0.470776	−	0.882253i	$-0.656026\pi$
$107$	4711.00	−	4711.00i	0.411477	−	0.411477i	0.882253	+	0.470776i	$0.156026\pi$
$108$		0			0
$109$		−	22040.0i		−	1.85506i	−0.373745	−	0.927531i	$-0.621927\pi$
$109$		−	22040.0i		−	1.85506i	0.373745	−	0.927531i	$-0.378073\pi$
$110$		0			0
$111$	−11718.0			−0.951059
$112$	1856.00	−	1856.00i	0.147959	−	0.147959i
$113$	2111.00	+	2111.00i	0.165322	+	0.165322i	0.784920	−	0.619597i	$-0.212704\pi$
$113$	2111.00	+	2111.00i	0.165322	+	0.165322i	−0.619597	+	0.784920i	$0.712704\pi$
$114$			10080.0i			0.775623i
$115$		0			0
$116$	−5440.00			−0.404281
$117$	5589.00	−	5589.00i	0.408284	−	0.408284i
$118$	−2320.00	−	2320.00i	−0.166619	−	0.166619i
$119$			15718.0i			1.10995i
$120$		0			0
$121$	−717.000			−0.0489721
$122$	−11196.0	+	11196.0i	−0.752217	+	0.752217i
$123$	−15138.0	−	15138.0i	−1.00059	−	1.00059i
$124$		−	1616.00i		−	0.105099i
$125$		0			0
$126$	−9396.00			−0.591837
$127$	−5909.00	+	5909.00i	−0.366359	+	0.366359i	−0.866147	−	0.499789i	$-0.833411\pi$
$127$	−5909.00	+	5909.00i	−0.366359	+	0.366359i	0.499789	+	0.866147i	$0.333411\pi$
$128$	1024.00	+	1024.00i	0.0625000	+	0.0625000i
$129$			19602.0i			1.17793i
$130$		0			0
$131$	−6358.00			−0.370491			−0.185246	−	0.982692i	$-0.559308\pi$
$131$	−6358.00			−0.370491			−0.185246	+	0.982692i	$0.559308\pi$
$132$	8496.00	−	8496.00i	0.487603	−	0.487603i
$133$	8120.00	+	8120.00i	0.459042	+	0.459042i
$134$		−	3004.00i		−	0.167298i
$135$		0			0
$136$	−8672.00			−0.468858
$137$	−20409.0	+	20409.0i	−1.08738	+	1.08738i	−0.0915804	+	0.995798i	$0.529192\pi$
$137$	−20409.0	+	20409.0i	−1.08738	+	1.08738i	−0.995798	+	0.0915804i	$0.970808\pi$
$138$	9684.00	+	9684.00i	0.508507	+	0.508507i
$139$			9400.00i			0.486517i	0.969961	+	0.243259i	$0.0782164\pi$
$139$			9400.00i			0.486517i	−0.969961	+	0.243259i	$0.921784\pi$
$140$		0			0
$141$	22842.0			1.14894
$142$	12884.0	−	12884.0i	0.638961	−	0.638961i
$143$	8142.00	+	8142.00i	0.398161	+	0.398161i
$144$		−	5184.00i		−	0.250000i
$145$		0			0
$146$	11804.0			0.553762
$147$	6471.00	−	6471.00i	0.299459	−	0.299459i
$148$	−5208.00	−	5208.00i	−0.237765	−	0.237765i
$149$		−	13800.0i		−	0.621594i	−0.950476	−	0.310797i	$-0.899404\pi$
$149$		−	13800.0i		−	0.621594i	0.950476	−	0.310797i	$-0.100596\pi$
$150$		0			0
$151$	−18998.0			−0.833209			−0.416605	−	0.909088i	$-0.636780\pi$
$151$	−18998.0			−0.833209			−0.416605	+	0.909088i	$0.636780\pi$
$152$	−4480.00	+	4480.00i	−0.193906	+	0.193906i
$153$	21951.0	+	21951.0i	0.937716	+	0.937716i
$154$		−	13688.0i		−	0.577163i
$155$		0			0
$156$	9936.00			0.408284
$157$	16371.0	−	16371.0i	0.664165	−	0.664165i	−0.292194	−	0.956359i	$-0.594385\pi$
$157$	16371.0	−	16371.0i	0.664165	−	0.664165i	0.956359	+	0.292194i	$0.0943853\pi$
$158$	−21120.0	−	21120.0i	−0.846018	−	0.846018i
$159$		−	10998.0i		−	0.435030i
$160$		0			0
$161$	15602.0			0.601906
$162$	13122.0	−	13122.0i	0.500000	−	0.500000i
$163$	−20009.0	−	20009.0i	−0.753096	−	0.753096i	0.221960	−	0.975056i	$-0.428755\pi$
$163$	−20009.0	−	20009.0i	−0.753096	−	0.753096i	−0.975056	+	0.221960i	$0.928755\pi$
$164$		−	13456.0i		−	0.500297i
$165$		0			0
$166$	24924.0			0.904485
$167$	−1549.00	+	1549.00i	−0.0555416	+	0.0555416i	−0.734332	−	0.678790i	$-0.762504\pi$
$167$	−1549.00	+	1549.00i	−0.0555416	+	0.0555416i	0.678790	+	0.734332i	$0.262504\pi$
$168$	−8352.00	−	8352.00i	−0.295918	−	0.295918i
$169$		−	19039.0i		−	0.666608i
$170$		0			0
$171$	22680.0			0.775623
$172$	−8712.00	+	8712.00i	−0.294484	+	0.294484i
$173$	−2789.00	−	2789.00i	−0.0931872	−	0.0931872i	0.658976	−	0.752164i	$-0.270990\pi$
$173$	−2789.00	−	2789.00i	−0.0931872	−	0.0931872i	−0.752164	+	0.658976i	$0.770990\pi$
$174$			24480.0i			0.808561i
$175$		0			0
$176$	7552.00			0.243802
$177$	−10440.0	+	10440.0i	−0.333238	+	0.333238i
$178$	28960.0	+	28960.0i	0.914026	+	0.914026i
$179$			2600.00i			0.0811460i	0.999177	+	0.0405730i	$0.0129183\pi$
$179$			2600.00i			0.0811460i	−0.999177	+	0.0405730i	$0.987082\pi$
$180$		0			0
$181$	−44398.0			−1.35521			−0.677604	−	0.735427i	$-0.736982\pi$
$181$	−44398.0			−1.35521			−0.677604	+	0.735427i	$0.736982\pi$
$182$	8004.00	−	8004.00i	0.241637	−	0.241637i
$183$	50382.0	+	50382.0i	1.50443	+	1.50443i
$184$			8608.00i			0.254253i
$185$		0			0
$186$	−7272.00			−0.210198
$187$	−31978.0	+	31978.0i	−0.914467	+	0.914467i
$188$	10152.0	+	10152.0i	0.287234	+	0.287234i
$189$		0			0
$190$		0			0
$191$	−14678.0			−0.402346			−0.201173	−	0.979556i	$-0.564475\pi$
$191$	−14678.0			−0.402346			−0.201173	+	0.979556i	$0.564475\pi$
$192$	4608.00	−	4608.00i	0.125000	−	0.125000i
$193$	−42849.0	−	42849.0i	−1.15034	−	1.15034i	−0.986484	−	0.163855i	$-0.947607\pi$
$193$	−42849.0	−	42849.0i	−1.15034	−	1.15034i	−0.163855	−	0.986484i	$-0.552393\pi$
$194$		−	29244.0i		−	0.777022i
$195$		0			0
$196$	5752.00			0.149729
$197$	10971.0	−	10971.0i	0.282692	−	0.282692i	−0.551490	−	0.834182i	$-0.685940\pi$
$197$	10971.0	−	10971.0i	0.282692	−	0.282692i	0.834182	+	0.551490i	$0.185940\pi$
$198$	−19116.0	−	19116.0i	−0.487603	−	0.487603i
$199$			38160.0i			0.963612i	0.876278	+	0.481806i	$0.160019\pi$
$199$			38160.0i			0.963612i	−0.876278	+	0.481806i	$0.839981\pi$
$200$		0			0
$201$	−13518.0			−0.334596
$202$	−1756.00	+	1756.00i	−0.0430350	+	0.0430350i
$203$	19720.0	+	19720.0i	0.478536	+	0.478536i
$204$			39024.0i			0.937716i
$205$		0			0
$206$	−41716.0			−0.983033
$207$	21789.0	−	21789.0i	0.508507	−	0.508507i
$208$	4416.00	+	4416.00i	0.102071	+	0.102071i
$209$			33040.0i			0.756393i
$210$		0			0
$211$	72842.0			1.63613			0.818063	−	0.575128i	$-0.195048\pi$
$211$	72842.0			1.63613			0.818063	+	0.575128i	$0.195048\pi$
$212$	4888.00	−	4888.00i	0.108758	−	0.108758i
$213$	−57978.0	−	57978.0i	−1.27792	−	1.27792i
$214$		−	18844.0i		−	0.411477i
$215$		0			0
$216$		0			0
$217$	−5858.00	+	5858.00i	−0.124403	+	0.124403i
$218$	−44080.0	−	44080.0i	−0.927531	−	0.927531i
$219$		−	53118.0i		−	1.10752i
$220$		0			0
$221$	−37398.0			−0.765709
$222$	−23436.0	+	23436.0i	−0.475530	+	0.475530i
$223$	30891.0	+	30891.0i	0.621187	+	0.621187i	0.945835	−	0.324648i	$-0.105246\pi$
$223$	30891.0	+	30891.0i	0.621187	+	0.621187i	−0.324648	+	0.945835i	$0.605246\pi$
$224$		−	7424.00i		−	0.147959i
$225$		0			0
$226$	8444.00			0.165322
$227$	54911.0	−	54911.0i	1.06563	−	1.06563i	0.0679438	−	0.997689i	$-0.478356\pi$
$227$	54911.0	−	54911.0i	1.06563	−	1.06563i	0.997689	−	0.0679438i	$-0.0216439\pi$
$228$	20160.0	+	20160.0i	0.387812	+	0.387812i
$229$			50280.0i			0.958792i	0.877599	+	0.479396i	$0.159144\pi$
$229$			50280.0i			0.958792i	−0.877599	+	0.479396i	$0.840856\pi$
$230$		0			0
$231$	−61596.0			−1.15433
$232$	−10880.0	+	10880.0i	−0.202140	+	0.202140i
$233$	2391.00	+	2391.00i	0.0440421	+	0.0440421i	0.728785	−	0.684743i	$-0.240085\pi$
$233$	2391.00	+	2391.00i	0.0440421	+	0.0440421i	−0.684743	+	0.728785i	$0.740085\pi$
$234$		−	22356.0i		−	0.408284i
$235$		0			0
$236$	−9280.00			−0.166619
$237$	−95040.0	+	95040.0i	−1.69204	+	1.69204i
$238$	31436.0	+	31436.0i	0.554975	+	0.554975i
$239$			17760.0i			0.310919i	0.987842	+	0.155459i	$0.0496858\pi$
$239$			17760.0i			0.310919i	−0.987842	+	0.155459i	$0.950314\pi$
$240$		0			0
$241$	−28238.0			−0.486183			−0.243092	−	0.970003i	$-0.578162\pi$
$241$	−28238.0			−0.486183			−0.243092	+	0.970003i	$0.578162\pi$
$242$	−1434.00	+	1434.00i	−0.0244860	+	0.0244860i
$243$	−59049.0	−	59049.0i	−1.00000	−	1.00000i
$244$			44784.0i			0.752217i
$245$		0			0
$246$	−60552.0			−1.00059
$247$	−19320.0	+	19320.0i	−0.316675	+	0.316675i
$248$	−3232.00	−	3232.00i	−0.0525494	−	0.0525494i
$249$		−	112158.i		−	1.80897i
$250$		0			0
$251$	121002.			1.92064			0.960318	−	0.278907i	$-0.0899722\pi$
$251$	121002.			1.92064			0.960318	+	0.278907i	$0.0899722\pi$
$252$	−18792.0	+	18792.0i	−0.295918	+	0.295918i
$253$	31742.0	+	31742.0i	0.495899	+	0.495899i
$254$			23636.0i			0.366359i
$255$		0			0
$256$	4096.00			0.0625000
$257$	72431.0	−	72431.0i	1.09663	−	1.09663i	0.101823	−	0.994803i	$-0.467533\pi$
$257$	72431.0	−	72431.0i	1.09663	−	1.09663i	0.994803	−	0.101823i	$-0.0324674\pi$
$258$	39204.0	+	39204.0i	0.588967	+	0.588967i
$259$			37758.0i			0.562872i
$260$		0			0
$261$	55080.0			0.808561
$262$	−12716.0	+	12716.0i	−0.185246	+	0.185246i
$263$	14771.0	+	14771.0i	0.213549	+	0.213549i	0.805773	−	0.592224i	$-0.201750\pi$
$263$	14771.0	+	14771.0i	0.213549	+	0.213549i	−0.592224	+	0.805773i	$0.701750\pi$
$264$		−	33984.0i		−	0.487603i
$265$		0			0
$266$	32480.0			0.459042
$267$	130320.	−	130320.i	1.82805	−	1.82805i
$268$	−6008.00	−	6008.00i	−0.0836489	−	0.0836489i
$269$		−	89720.0i		−	1.23989i	−0.784644	−	0.619947i	$-0.787154\pi$
$269$		−	89720.0i		−	1.23989i	0.784644	−	0.619947i	$-0.212846\pi$
$270$		0			0
$271$	68202.0			0.928664			0.464332	−	0.885661i	$-0.346294\pi$
$271$	68202.0			0.928664			0.464332	+	0.885661i	$0.346294\pi$
$272$	−17344.0	+	17344.0i	−0.234429	+	0.234429i
$273$	−36018.0	−	36018.0i	−0.483275	−	0.483275i
$274$			81636.0i			1.08738i
$275$		0			0
$276$	38736.0			0.508507
$277$	−18549.0	+	18549.0i	−0.241747	+	0.241747i	−0.817573	−	0.575826i	$-0.804681\pi$
$277$	−18549.0	+	18549.0i	−0.241747	+	0.241747i	0.575826	+	0.817573i	$0.304681\pi$
$278$	18800.0	+	18800.0i	0.243259	+	0.243259i
$279$			16362.0i			0.210198i
$280$		0			0
$281$	2322.00			0.0294069			0.0147035	−	0.999892i	$-0.495320\pi$
$281$	2322.00			0.0294069			0.0147035	+	0.999892i	$0.495320\pi$
$282$	45684.0	−	45684.0i	0.574468	−	0.574468i
$283$	91711.0	+	91711.0i	1.14511	+	1.14511i	0.987501	+	0.157613i	$0.0503797\pi$
$283$	91711.0	+	91711.0i	1.14511	+	1.14511i	0.157613	+	0.987501i	$0.449620\pi$
$284$		−	51536.0i		−	0.638961i
$285$		0			0
$286$	32568.0			0.398161
$287$	−48778.0	+	48778.0i	−0.592189	+	0.592189i
$288$	−10368.0	−	10368.0i	−0.125000	−	0.125000i
$289$		−	63361.0i		−	0.758624i
$290$		0			0
$291$	−131598.			−1.55404
$292$	23608.0	−	23608.0i	0.276881	−	0.276881i
$293$	4851.00	+	4851.00i	0.0565062	+	0.0565062i	0.734795	−	0.678289i	$-0.237278\pi$
$293$	4851.00	+	4851.00i	0.0565062	+	0.0565062i	−0.678289	+	0.734795i	$0.737278\pi$
$294$		−	25884.0i		−	0.299459i
$295$		0			0
$296$	−20832.0			−0.237765
$297$		0			0
$298$	−27600.0	−	27600.0i	−0.310797	−	0.310797i
$299$			37122.0i			0.415230i
$300$		0			0
$301$	63162.0			0.697145
$302$	−37996.0	+	37996.0i	−0.416605	+	0.416605i
$303$	7902.00	+	7902.00i	0.0860700	+	0.0860700i
$304$			17920.0i			0.193906i
$305$		0			0
$306$	87804.0			0.937716
$307$	−42849.0	+	42849.0i	−0.454636	+	0.454636i	−0.896890	−	0.442254i	$-0.854179\pi$
$307$	−42849.0	+	42849.0i	−0.454636	+	0.454636i	0.442254	+	0.896890i	$0.354179\pi$
$308$	−27376.0	−	27376.0i	−0.288582	−	0.288582i
$309$			187722.i			1.96607i
$310$		0			0
$311$	−72278.0			−0.747283			−0.373642	−	0.927573i	$-0.621891\pi$
$311$	−72278.0			−0.747283			−0.373642	+	0.927573i	$0.621891\pi$
$312$	19872.0	−	19872.0i	0.204142	−	0.204142i
$313$	−18249.0	−	18249.0i	−0.186273	−	0.186273i	0.607810	−	0.794083i	$-0.292048\pi$
$313$	−18249.0	−	18249.0i	−0.186273	−	0.186273i	−0.794083	+	0.607810i	$0.792048\pi$
$314$		−	65484.0i		−	0.664165i
$315$		0			0
$316$	−84480.0			−0.846018
$317$	−25149.0	+	25149.0i	−0.250266	+	0.250266i	−0.821080	−	0.570814i	$-0.806628\pi$
$317$	−25149.0	+	25149.0i	−0.250266	+	0.250266i	0.570814	+	0.821080i	$0.306628\pi$
$318$	−21996.0	−	21996.0i	−0.217515	−	0.217515i
$319$			80240.0i			0.788514i
$320$		0			0
$321$	−84798.0			−0.822954
$322$	31204.0	−	31204.0i	0.300953	−	0.300953i
$323$	−75880.0	−	75880.0i	−0.727315	−	0.727315i
$324$		−	52488.0i		−	0.500000i
$325$		0			0
$326$	−80036.0			−0.753096
$327$	−198360.	+	198360.i	−1.85506	+	1.85506i
$328$	−26912.0	−	26912.0i	−0.250149	−	0.250149i
$329$		−	73602.0i		−	0.679983i
$330$		0			0
$331$	−54038.0			−0.493223			−0.246611	−	0.969114i	$-0.579317\pi$
$331$	−54038.0			−0.493223			−0.246611	+	0.969114i	$0.579317\pi$
$332$	49848.0	−	49848.0i	0.452243	−	0.452243i
$333$	52731.0	+	52731.0i	0.475530	+	0.475530i
$334$			6196.00i			0.0555416i
$335$		0			0
$336$	−33408.0			−0.295918
$337$	−8529.00	+	8529.00i	−0.0750997	+	0.0750997i	−0.743659	−	0.668559i	$-0.766911\pi$
$337$	−8529.00	+	8529.00i	−0.0750997	+	0.0750997i	0.668559	+	0.743659i	$0.266911\pi$
$338$	−38078.0	−	38078.0i	−0.333304	−	0.333304i
$339$		−	37998.0i		−	0.330645i
$340$		0			0
$341$	−23836.0			−0.204986
$342$	45360.0	−	45360.0i	0.387812	−	0.387812i
$343$	−90480.0	−	90480.0i	−0.769067	−	0.769067i
$344$			34848.0i			0.294484i
$345$		0			0
$346$	−11156.0			−0.0931872
$347$	56551.0	−	56551.0i	0.469658	−	0.469658i	−0.432146	−	0.901804i	$-0.642244\pi$
$347$	56551.0	−	56551.0i	0.469658	−	0.469658i	0.901804	+	0.432146i	$0.142244\pi$
$348$	48960.0	+	48960.0i	0.404281	+	0.404281i
$349$			22520.0i			0.184892i	0.995718	+	0.0924459i	$0.0294685\pi$
$349$			22520.0i			0.184892i	−0.995718	+	0.0924459i	$0.970531\pi$
$350$		0			0
$351$		0			0
$352$	15104.0	−	15104.0i	0.121901	−	0.121901i
$353$	44511.0	+	44511.0i	0.357205	+	0.357205i	0.862782	−	0.505576i	$-0.168720\pi$
$353$	44511.0	+	44511.0i	0.357205	+	0.357205i	−0.505576	+	0.862782i	$0.668720\pi$
$354$			41760.0i			0.333238i
$355$		0			0
$356$	115840.			0.914026
$357$	141462.	−	141462.i	1.10995	−	1.10995i
$358$	5200.00	+	5200.00i	0.0405730	+	0.0405730i
$359$			9680.00i			0.0751080i	0.999295	+	0.0375540i	$0.0119566\pi$
$359$			9680.00i			0.0751080i	−0.999295	+	0.0375540i	$0.988043\pi$
$360$		0			0
$361$	51921.0			0.398409
$362$	−88796.0	+	88796.0i	−0.677604	+	0.677604i
$363$	6453.00	+	6453.00i	0.0489721	+	0.0489721i
$364$		−	32016.0i		−	0.241637i
$365$		0			0
$366$	201528.			1.50443
$367$	14971.0	−	14971.0i	0.111152	−	0.111152i	−0.649343	−	0.760496i	$-0.724956\pi$
$367$	14971.0	−	14971.0i	0.111152	−	0.111152i	0.760496	+	0.649343i	$0.224956\pi$
$368$	17216.0	+	17216.0i	0.127127	+	0.127127i
$369$			136242.i			1.00059i
$370$		0			0
$371$	−35438.0			−0.257467
$372$	−14544.0	+	14544.0i	−0.105099	+	0.105099i
$373$	13811.0	+	13811.0i	0.0992676	+	0.0992676i	0.754996	−	0.655729i	$-0.227639\pi$
$373$	13811.0	+	13811.0i	0.0992676	+	0.0992676i	−0.655729	+	0.754996i	$0.727639\pi$
$374$			127912.i			0.914467i
$375$		0			0
$376$	40608.0			0.287234
$377$	−46920.0	+	46920.0i	−0.330123	+	0.330123i
$378$		0			0
$379$		−	251080.i		−	1.74797i	−0.485954	−	0.873984i	$-0.661528\pi$
$379$		−	251080.i		−	1.74797i	0.485954	−	0.873984i	$-0.338472\pi$
$380$		0			0
$381$	106362.			0.732717
$382$	−29356.0	+	29356.0i	−0.201173	+	0.201173i
$383$	86091.0	+	86091.0i	0.586895	+	0.586895i	0.936789	−	0.349894i	$-0.113783\pi$
$383$	86091.0	+	86091.0i	0.586895	+	0.586895i	−0.349894	+	0.936789i	$0.613783\pi$
$384$		−	18432.0i		−	0.125000i
$385$		0			0
$386$	−171396.			−1.15034
$387$	88209.0	−	88209.0i	0.588967	−	0.588967i
$388$	−58488.0	−	58488.0i	−0.388511	−	0.388511i
$389$			75000.0i			0.495635i	0.968807	+	0.247818i	$0.0797134\pi$
$389$			75000.0i			0.495635i	−0.968807	+	0.247818i	$0.920287\pi$
$390$		0			0
$391$	−145798.			−0.953670
$392$	11504.0	−	11504.0i	0.0748646	−	0.0748646i
$393$	57222.0	+	57222.0i	0.370491	+	0.370491i
$394$		−	43884.0i		−	0.282692i
$395$		0			0
$396$	−76464.0			−0.487603
$397$	−29149.0	+	29149.0i	−0.184945	+	0.184945i	−0.793507	−	0.608562i	$-0.791747\pi$
$397$	−29149.0	+	29149.0i	−0.184945	+	0.184945i	0.608562	+	0.793507i	$0.291747\pi$
$398$	76320.0	+	76320.0i	0.481806	+	0.481806i
$399$		−	146160.i		−	0.918085i
$400$		0			0
$401$	−45918.0			−0.285558			−0.142779	−	0.989755i	$-0.545604\pi$
$401$	−45918.0			−0.285558			−0.142779	+	0.989755i	$0.545604\pi$
$402$	−27036.0	+	27036.0i	−0.167298	+	0.167298i
$403$	−13938.0	−	13938.0i	−0.0858204	−	0.0858204i
$404$			7024.00i			0.0430350i
$405$		0			0
$406$	78880.0			0.478536
$407$	−76818.0	+	76818.0i	−0.463740	+	0.463740i
$408$	78048.0	+	78048.0i	0.468858	+	0.468858i
$409$		−	78720.0i		−	0.470585i	−0.971925	−	0.235293i	$-0.924395\pi$
$409$		−	78720.0i		−	0.470585i	0.971925	−	0.235293i	$-0.0756049\pi$
$410$		0			0
$411$	367362.			2.17476
$412$	−83432.0	+	83432.0i	−0.491517	+	0.491517i
$413$	33640.0	+	33640.0i	0.197222	+	0.197222i
$414$		−	87156.0i		−	0.508507i
$415$		0			0
$416$	17664.0			0.102071
$417$	84600.0	−	84600.0i	0.486517	−	0.486517i
$418$	66080.0	+	66080.0i	0.378196	+	0.378196i
$419$			14760.0i			0.0840733i	0.999116	+	0.0420367i	$0.0133846\pi$
$419$			14760.0i			0.0840733i	−0.999116	+	0.0420367i	$0.986615\pi$
$420$		0			0
$421$	221282.			1.24848			0.624240	−	0.781232i	$-0.285409\pi$
$421$	221282.			1.24848			0.624240	+	0.781232i	$0.285409\pi$
$422$	145684.	−	145684.i	0.818063	−	0.818063i
$423$	−102789.	−	102789.i	−0.574468	−	0.574468i
$424$		−	19552.0i		−	0.108758i
$425$		0			0
$426$	−231912.			−1.27792
$427$	162342.	−	162342.i	0.890379	−	0.890379i
$428$	−37688.0	−	37688.0i	−0.205738	−	0.205738i
$429$		−	146556.i		−	0.796323i
$430$		0			0
$431$	212522.			1.14406			0.572031	−	0.820232i	$-0.306156\pi$
$431$	212522.			1.14406			0.572031	+	0.820232i	$0.306156\pi$
$432$		0			0
$433$	−145409.	−	145409.i	−0.775560	−	0.775560i	0.203512	−	0.979072i	$-0.434764\pi$
$433$	−145409.	−	145409.i	−0.775560	−	0.775560i	−0.979072	+	0.203512i	$0.934764\pi$
$434$			23432.0i			0.124403i
$435$		0			0
$436$	−176320.			−0.927531
$437$	−75320.0	+	75320.0i	−0.394410	+	0.394410i
$438$	−106236.	−	106236.i	−0.553762	−	0.553762i
$439$			299440.i			1.55375i	0.629656	+	0.776874i	$0.283196\pi$
$439$			299440.i			1.55375i	−0.629656	+	0.776874i	$0.716804\pi$
$440$		0			0
$441$	−58239.0			−0.299459
$442$	−74796.0	+	74796.0i	−0.382855	+	0.382855i
$443$	−240609.	−	240609.i	−1.22604	−	1.22604i	−0.965450	−	0.260590i	$-0.916083\pi$
$443$	−240609.	−	240609.i	−1.22604	−	1.22604i	−0.260590	−	0.965450i	$-0.583917\pi$
$444$			93744.0i			0.475530i
$445$		0			0
$446$	123564.			0.621187
$447$	−124200.	+	124200.i	−0.621594	+	0.621594i
$448$	−14848.0	−	14848.0i	−0.0739796	−	0.0739796i
$449$			82480.0i			0.409125i	0.978854	+	0.204562i	$0.0655772\pi$
$449$			82480.0i			0.409125i	−0.978854	+	0.204562i	$0.934423\pi$
$450$		0			0
$451$	−198476.			−0.975787
$452$	16888.0	−	16888.0i	0.0826611	−	0.0826611i
$453$	170982.	+	170982.i	0.833209	+	0.833209i
$454$		−	219644.i		−	1.06563i
$455$		0			0
$456$	80640.0			0.387812
$457$	188151.	−	188151.i	0.900895	−	0.900895i	−0.0946187	−	0.995514i	$-0.530163\pi$
$457$	188151.	−	188151.i	0.900895	−	0.900895i	0.995514	+	0.0946187i	$0.0301632\pi$
$458$	100560.	+	100560.i	0.479396	+	0.479396i
$459$		0			0
$460$		0			0
$461$	−326158.			−1.53471			−0.767355	−	0.641223i	$-0.778427\pi$
$461$	−326158.			−1.53471			−0.767355	+	0.641223i	$0.778427\pi$
$462$	−123192.	+	123192.i	−0.577163	+	0.577163i
$463$	218731.	+	218731.i	1.02035	+	1.02035i	0.999789	+	0.0205595i	$0.00654474\pi$
$463$	218731.	+	218731.i	1.02035	+	1.02035i	0.0205595	+	0.999789i	$0.493455\pi$
$464$			43520.0i			0.202140i
$465$		0			0
$466$	9564.00			0.0440421
$467$	−59249.0	+	59249.0i	−0.271673	+	0.271673i	−0.829774	−	0.558100i	$-0.811530\pi$
$467$	−59249.0	+	59249.0i	−0.271673	+	0.271673i	0.558100	+	0.829774i	$0.311530\pi$
$468$	−44712.0	−	44712.0i	−0.204142	−	0.204142i
$469$			43558.0i			0.198026i
$470$		0			0
$471$	−294678.			−1.32833
$472$	−18560.0	+	18560.0i	−0.0833094	+	0.0833094i
$473$	128502.	+	128502.i	0.574365	+	0.574365i
$474$			380160.i			1.69204i
$475$		0			0
$476$	125744.			0.554975
$477$	−49491.0	+	49491.0i	−0.217515	+	0.217515i
$478$	35520.0	+	35520.0i	0.155459	+	0.155459i
$479$		−	273440.i		−	1.19177i	−0.803071	−	0.595883i	$-0.796802\pi$
$479$		−	273440.i		−	1.19177i	0.803071	−	0.595883i	$-0.203198\pi$
$480$		0			0
$481$	−89838.0			−0.388302
$482$	−56476.0	+	56476.0i	−0.243092	+	0.243092i
$483$	−140418.	−	140418.i	−0.601906	−	0.601906i
$484$			5736.00i			0.0244860i
$485$		0			0
$486$	−236196.			−1.00000
$487$	123651.	−	123651.i	0.521362	−	0.521362i	−0.396620	−	0.917983i	$-0.629817\pi$
$487$	123651.	−	123651.i	0.521362	−	0.521362i	0.917983	+	0.396620i	$0.129817\pi$
$488$	89568.0	+	89568.0i	0.376109	+	0.376109i
$489$			360162.i			1.50619i
$490$		0			0
$491$	198442.			0.823134			0.411567	−	0.911379i	$-0.364982\pi$
$491$	198442.			0.823134			0.411567	+	0.911379i	$0.364982\pi$
$492$	−121104.	+	121104.i	−0.500297	+	0.500297i
$493$	−184280.	−	184280.i	−0.758201	−	0.758201i
$494$			77280.0i			0.316675i
$495$		0			0
$496$	−12928.0			−0.0525494
$497$	−186818.	+	186818.i	−0.756321	+	0.756321i
$498$	−224316.	−	224316.i	−0.904485	−	0.904485i
$499$		−	269240.i		−	1.08128i	−0.841254	−	0.540640i	$-0.818182\pi$
$499$		−	269240.i		−	1.08128i	0.841254	−	0.540640i	$-0.181818\pi$
$500$		0			0
$501$	27882.0			0.111083
$502$	242004.	−	242004.i	0.960318	−	0.960318i
$503$	−109869.	−	109869.i	−0.434249	−	0.434249i	0.455822	−	0.890071i	$-0.349345\pi$
$503$	−109869.	−	109869.i	−0.434249	−	0.434249i	−0.890071	+	0.455822i	$0.849345\pi$
$504$			75168.0i			0.295918i
$505$		0			0
$506$	126968.			0.495899
$507$	−171351.	+	171351.i	−0.666608	+	0.666608i
$508$	47272.0	+	47272.0i	0.183179	+	0.183179i
$509$			211000.i			0.814417i	0.913335	+	0.407209i	$0.133498\pi$
$509$			211000.i			0.814417i	−0.913335	+	0.407209i	$0.866502\pi$
$510$		0			0
$511$	−171158.			−0.655474
$512$	8192.00	−	8192.00i	0.0312500	−	0.0312500i
$513$		0			0
$514$		−	289724.i		−	1.09663i
$515$		0			0
$516$	156816.			0.588967
$517$	149742.	−	149742.i	0.560225	−	0.560225i
$518$	75516.0	+	75516.0i	0.281436	+	0.281436i
$519$			50202.0i			0.186374i
$520$		0			0
$521$	297282.			1.09520			0.547600	−	0.836740i	$-0.315542\pi$
$521$	297282.			1.09520			0.547600	+	0.836740i	$0.315542\pi$
$522$	110160.	−	110160.i	0.404281	−	0.404281i
$523$	25071.0	+	25071.0i	0.0916576	+	0.0916576i	0.751449	−	0.659791i	$-0.229355\pi$
$523$	25071.0	+	25071.0i	0.0916576	+	0.0916576i	−0.659791	+	0.751449i	$0.729355\pi$
$524$			50864.0i			0.185246i
$525$		0			0
$526$	59084.0			0.213549
$527$	54742.0	−	54742.0i	0.197106	−	0.197106i
$528$	−67968.0	−	67968.0i	−0.243802	−	0.243802i
$529$		−	135119.i		−	0.482842i
$530$		0			0
$531$	93960.0			0.333238
$532$	64960.0	−	64960.0i	0.229521	−	0.229521i
$533$	−116058.	−	116058.i	−0.408527	−	0.408527i
$534$		−	521280.i		−	1.82805i
$535$		0			0
$536$	−24032.0			−0.0836489
$537$	23400.0	−	23400.0i	0.0811460	−	0.0811460i
$538$	−179440.	−	179440.i	−0.619947	−	0.619947i
$539$		−	84842.0i		−	0.292034i
$540$		0			0
$541$	−142478.			−0.486803			−0.243402	−	0.969926i	$-0.578263\pi$
$541$	−142478.			−0.486803			−0.243402	+	0.969926i	$0.578263\pi$
$542$	136404.	−	136404.i	0.464332	−	0.464332i
$543$	399582.	+	399582.i	1.35521	+	1.35521i
$544$			69376.0i			0.234429i
$545$		0			0
$546$	−144072.			−0.483275
$547$	−291009.	+	291009.i	−0.972594	+	0.972594i	−0.999634	−	0.0270399i	$-0.991392\pi$
$547$	−291009.	+	291009.i	−0.972594	+	0.972594i	0.0270399	+	0.999634i	$0.491392\pi$
$548$	163272.	+	163272.i	0.543689	+	0.543689i
$549$		−	453438.i		−	1.50443i
$550$		0			0
$551$	−190400.			−0.627139
$552$	77472.0	−	77472.0i	0.254253	−	0.254253i
$553$	306240.	+	306240.i	1.00141	+	1.00141i
$554$			74196.0i			0.241747i
$555$		0			0
$556$	75200.0			0.243259
$557$	83091.0	−	83091.0i	0.267820	−	0.267820i	−0.560401	−	0.828221i	$-0.689353\pi$
$557$	83091.0	−	83091.0i	0.267820	−	0.267820i	0.828221	+	0.560401i	$0.189353\pi$
$558$	32724.0	+	32724.0i	0.105099	+	0.105099i
$559$			150282.i			0.480932i
$560$		0			0
$561$	575604.			1.82893
$562$	4644.00	−	4644.00i	0.0147035	−	0.0147035i
$563$	−43449.0	−	43449.0i	−0.137076	−	0.137076i	0.635239	−	0.772316i	$-0.280902\pi$
$563$	−43449.0	−	43449.0i	−0.137076	−	0.137076i	−0.772316	+	0.635239i	$0.780902\pi$
$564$		−	182736.i		−	0.574468i
$565$		0			0
$566$	366844.			1.14511
$567$	−190269.	+	190269.i	−0.591837	+	0.591837i
$568$	−103072.	−	103072.i	−0.319480	−	0.319480i
$569$			270560.i			0.835678i	0.908521	+	0.417839i	$0.137212\pi$
$569$			270560.i			0.835678i	−0.908521	+	0.417839i	$0.862788\pi$
$570$		0			0
$571$	57482.0			0.176303			0.0881515	−	0.996107i	$-0.471904\pi$
$571$	57482.0			0.176303			0.0881515	+	0.996107i	$0.471904\pi$
$572$	65136.0	−	65136.0i	0.199081	−	0.199081i
$573$	132102.	+	132102.i	0.402346	+	0.402346i
$574$			195112.i			0.592189i
$575$		0			0
$576$	−41472.0			−0.125000
$577$	−195889.	+	195889.i	−0.588381	+	0.588381i	−0.937193	−	0.348812i	$-0.886585\pi$
$577$	−195889.	+	195889.i	−0.588381	+	0.588381i	0.348812	+	0.937193i	$0.386585\pi$
$578$	−126722.	−	126722.i	−0.379312	−	0.379312i
$579$			771282.i			2.30068i
$580$		0			0
$581$	−361398.			−1.07062
$582$	−263196.	+	263196.i	−0.777022	+	0.777022i
$583$	−72098.0	−	72098.0i	−0.212122	−	0.212122i
$584$		−	94432.0i		−	0.276881i
$585$		0			0
$586$	19404.0			0.0565062
$587$	404631.	−	404631.i	1.17431	−	1.17431i	0.193139	−	0.981171i	$-0.438133\pi$
$587$	404631.	−	404631.i	1.17431	−	1.17431i	0.981171	−	0.193139i	$-0.0618669\pi$
$588$	−51768.0	−	51768.0i	−0.149729	−	0.149729i
$589$		−	56560.0i		−	0.163034i
$590$		0			0
$591$	−197478.			−0.565384
$592$	−41664.0	+	41664.0i	−0.118882	+	0.118882i
$593$	210991.	+	210991.i	0.600005	+	0.600005i	0.940314	−	0.340309i	$-0.110532\pi$
$593$	210991.	+	210991.i	0.600005	+	0.600005i	−0.340309	+	0.940314i	$0.610532\pi$
$594$		0			0
$595$		0			0
$596$	−110400.			−0.310797
$597$	343440.	−	343440.i	0.963612	−	0.963612i
$598$	74244.0	+	74244.0i	0.207615	+	0.207615i
$599$		−	300560.i		−	0.837679i	−0.908060	−	0.418839i	$-0.862437\pi$
$599$		−	300560.i		−	0.837679i	0.908060	−	0.418839i	$-0.137563\pi$
$600$		0			0
$601$	367442.			1.01728			0.508639	−	0.860980i	$-0.330149\pi$
$601$	367442.			1.01728			0.508639	+	0.860980i	$0.330149\pi$
$602$	126324.	−	126324.i	0.348572	−	0.348572i
$603$	60831.0	+	60831.0i	0.167298	+	0.167298i
$604$			151984.i			0.416605i
$605$		0			0
$606$	31608.0			0.0860700
$607$	−146469.	+	146469.i	−0.397529	+	0.397529i	−0.877360	−	0.479832i	$-0.840698\pi$
$607$	−146469.	+	146469.i	−0.397529	+	0.397529i	0.479832	+	0.877360i	$0.340698\pi$
$608$	35840.0	+	35840.0i	0.0969529	+	0.0969529i
$609$		−	354960.i		−	0.957072i
$610$		0			0
$611$	175122.			0.469092
$612$	175608.	−	175608.i	0.468858	−	0.468858i
$613$	−160989.	−	160989.i	−0.428425	−	0.428425i	0.459666	−	0.888092i	$-0.347969\pi$
$613$	−160989.	−	160989.i	−0.428425	−	0.428425i	−0.888092	+	0.459666i	$0.847969\pi$
$614$			171396.i			0.454636i
$615$		0			0
$616$	−109504.			−0.288582
$617$	−320409.	+	320409.i	−0.841656	+	0.841656i	−0.989074	−	0.147419i	$-0.952904\pi$
$617$	−320409.	+	320409.i	−0.841656	+	0.841656i	0.147419	+	0.989074i	$0.452904\pi$
$618$	375444.	+	375444.i	0.983033	+	0.983033i
$619$		−	341160.i		−	0.890383i	−0.895435	−	0.445191i	$-0.853136\pi$
$619$		−	341160.i		−	0.890383i	0.895435	−	0.445191i	$-0.146864\pi$
$620$		0			0
$621$		0			0
$622$	−144556.	+	144556.i	−0.373642	+	0.373642i
$623$	−419920.	−	419920.i	−1.08191	−	1.08191i
$624$		−	79488.0i		−	0.204142i
$625$		0			0
$626$	−72996.0			−0.186273
$627$	297360.	−	297360.i	0.756393	−	0.756393i
$628$	−130968.	−	130968.i	−0.332082	−	0.332082i
$629$		−	352842.i		−	0.891824i
$630$		0			0
$631$	−390998.			−0.982010			−0.491005	−	0.871157i	$-0.663370\pi$
$631$	−390998.			−0.982010			−0.491005	+	0.871157i	$0.663370\pi$
$632$	−168960.	+	168960.i	−0.423009	+	0.423009i
$633$	−655578.	−	655578.i	−1.63613	−	1.63613i
$634$			100596.i			0.250266i
$635$		0			0
$636$	−87984.0			−0.217515
$637$	49611.0	−	49611.0i	0.122264	−	0.122264i
$638$	160480.	+	160480.i	0.394257	+	0.394257i
$639$			521802.i			1.27792i
$640$		0			0
$641$	−585038.			−1.42386			−0.711931	−	0.702249i	$-0.752179\pi$
$641$	−585038.			−1.42386			−0.711931	+	0.702249i	$0.752179\pi$
$642$	−169596.	+	169596.i	−0.411477	+	0.411477i
$643$	31911.0	+	31911.0i	0.0771824	+	0.0771824i	0.744644	−	0.667462i	$-0.232619\pi$
$643$	31911.0	+	31911.0i	0.0771824	+	0.0771824i	−0.667462	+	0.744644i	$0.732619\pi$
$644$		−	124816.i		−	0.300953i
$645$		0			0
$646$	−303520.			−0.727315
$647$	280931.	−	280931.i	0.671106	−	0.671106i	−0.286865	−	0.957971i	$-0.592613\pi$
$647$	280931.	−	280931.i	0.671106	−	0.671106i	0.957971	+	0.286865i	$0.0926131\pi$
$648$	−104976.	−	104976.i	−0.250000	−	0.250000i
$649$			136880.i			0.324975i
$650$		0			0
$651$	105444.			0.248805
$652$	−160072.	+	160072.i	−0.376548	+	0.376548i
$653$	−523989.	−	523989.i	−1.22884	−	1.22884i	−0.964402	−	0.264439i	$-0.914813\pi$
$653$	−523989.	−	523989.i	−1.22884	−	1.22884i	−0.264439	−	0.964402i	$-0.585187\pi$
$654$			793440.i			1.85506i
$655$		0			0
$656$	−107648.			−0.250149
$657$	−239031.	+	239031.i	−0.553762	+	0.553762i
$658$	−147204.	−	147204.i	−0.339991	−	0.339991i
$659$			404360.i			0.931102i	0.885021	+	0.465551i	$0.154144\pi$
$659$			404360.i			0.931102i	−0.885021	+	0.465551i	$0.845856\pi$
$660$		0			0
$661$	−5278.00			−0.0120800			−0.00603999	−	0.999982i	$-0.501923\pi$
$661$	−5278.00			−0.0120800			−0.00603999	+	0.999982i	$0.501923\pi$
$662$	−108076.	+	108076.i	−0.246611	+	0.246611i
$663$	336582.	+	336582.i	0.765709	+	0.765709i
$664$		−	199392.i		−	0.452243i
$665$		0			0
$666$	210924.			0.475530
$667$	−182920.	+	182920.i	−0.411159	+	0.411159i
$668$	12392.0	+	12392.0i	0.0277708	+	0.0277708i
$669$		−	556038.i		−	1.24237i
$670$		0			0
$671$	660564.			1.46713
$672$	−66816.0	+	66816.0i	−0.147959	+	0.147959i
$673$	332111.	+	332111.i	0.733252	+	0.733252i	0.971263	−	0.238011i	$-0.0764953\pi$
$673$	332111.	+	332111.i	0.733252	+	0.733252i	−0.238011	+	0.971263i	$0.576495\pi$
$674$			34116.0i			0.0750997i
$675$		0			0
$676$	−152312.			−0.333304
$677$	−578309.	+	578309.i	−1.26178	+	1.26178i	−0.311546	+	0.950231i	$0.600847\pi$
$677$	−578309.	+	578309.i	−1.26178	+	1.26178i	−0.950231	+	0.311546i	$0.899153\pi$
$678$	−75996.0	−	75996.0i	−0.165322	−	0.165322i
$679$			424038.i			0.919740i
$680$		0			0
$681$	−988398.			−2.13127
$682$	−47672.0	+	47672.0i	−0.102493	+	0.102493i
$683$	349311.	+	349311.i	0.748809	+	0.748809i	0.974255	−	0.225447i	$-0.0723842\pi$
$683$	349311.	+	349311.i	0.748809	+	0.748809i	−0.225447	+	0.974255i	$0.572384\pi$
$684$		−	181440.i		−	0.387812i
$685$		0			0
$686$	−361920.			−0.769067
$687$	452520.	−	452520.i	0.958792	−	0.958792i
$688$	69696.0	+	69696.0i	0.147242	+	0.147242i
$689$		−	84318.0i		−	0.177616i
$690$		0			0
$691$	282762.			0.592195			0.296098	−	0.955158i	$-0.404315\pi$
$691$	282762.			0.592195			0.296098	+	0.955158i	$0.404315\pi$
$692$	−22312.0	+	22312.0i	−0.0465936	+	0.0465936i
$693$	277182.	+	277182.i	0.577163	+	0.577163i
$694$		−	226204.i		−	0.469658i
$695$		0			0
$696$	195840.			0.404281
$697$	455822.	−	455822.i	0.938274	−	0.938274i
$698$	45040.0	+	45040.0i	0.0924459	+	0.0924459i
$699$		−	43038.0i		−	0.0880841i
$700$		0			0
$701$	270242.			0.549942			0.274971	−	0.961453i	$-0.411332\pi$
$701$	270242.			0.549942			0.274971	+	0.961453i	$0.411332\pi$
$702$		0			0
$703$	−182280.	−	182280.i	−0.368832	−	0.368832i
$704$		−	60416.0i		−	0.121901i
$705$		0			0
$706$	178044.			0.357205
$707$	25462.0	−	25462.0i	0.0509394	−	0.0509394i
$708$	83520.0	+	83520.0i	0.166619	+	0.166619i
$709$			297800.i			0.592423i	0.955122	+	0.296212i	$0.0957234\pi$
$709$			297800.i			0.592423i	−0.955122	+	0.296212i	$0.904277\pi$
$710$		0			0
$711$	855360.			1.69204
$712$	231680.	−	231680.i	0.457013	−	0.457013i
$713$	−54338.0	−	54338.0i	−0.106887	−	0.106887i
$714$		−	565848.i		−	1.10995i
$715$		0			0
$716$	20800.0			0.0405730
$717$	159840.	−	159840.i	0.310919	−	0.310919i
$718$	19360.0	+	19360.0i	0.0375540	+	0.0375540i
$719$		−	913760.i		−	1.76756i	−0.467902	−	0.883780i	$-0.654990\pi$
$719$		−	913760.i		−	1.76756i	0.467902	−	0.883780i	$-0.345010\pi$
$720$		0			0
$721$	604882.			1.16359
$722$	103842.	−	103842.i	0.199204	−	0.199204i
$723$	254142.	+	254142.i	0.486183	+	0.486183i
$724$			355184.i			0.677604i
$725$		0			0
$726$	25812.0			0.0489721
$727$	417651.	−	417651.i	0.790214	−	0.790214i	−0.191315	−	0.981529i	$-0.561275\pi$
$727$	417651.	−	417651.i	0.790214	−	0.790214i	0.981529	+	0.191315i	$0.0612751\pi$
$728$	−64032.0	−	64032.0i	−0.120819	−	0.120819i
$729$			531441.i			1.00000i
$730$		0			0
$731$	−590238.			−1.10457
$732$	403056.	−	403056.i	0.752217	−	0.752217i
$733$	−394549.	−	394549.i	−0.734333	−	0.734333i	0.237142	−	0.971475i	$-0.423789\pi$
$733$	−394549.	−	394549.i	−0.734333	−	0.734333i	−0.971475	+	0.237142i	$0.923789\pi$
$734$		−	59884.0i		−	0.111152i
$735$		0			0
$736$	68864.0			0.127127
$737$	−88618.0	+	88618.0i	−0.163150	+	0.163150i
$738$	272484.	+	272484.i	0.500297	+	0.500297i
$739$		−	109880.i		−	0.201201i	−0.994927	−	0.100600i	$-0.967924\pi$
$739$		−	109880.i		−	0.201201i	0.994927	−	0.100600i	$-0.0320764\pi$
$740$		0			0
$741$	347760.			0.633349
$742$	−70876.0	+	70876.0i	−0.128733	+	0.128733i
$743$	466451.	+	466451.i	0.844945	+	0.844945i	0.989497	−	0.144552i	$-0.0461742\pi$
$743$	466451.	+	466451.i	0.844945	+	0.844945i	−0.144552	+	0.989497i	$0.546174\pi$
$744$			58176.0i			0.105099i
$745$		0			0
$746$	55244.0			0.0992676
$747$	−504711.	+	504711.i	−0.904485	+	0.904485i
$748$	255824.	+	255824.i	0.457234	+	0.457234i
$749$			273238.i			0.487054i
$750$		0			0
$751$	1.01092e6			1.79241			0.896206	−	0.443638i	$-0.146313\pi$
$751$	1.01092e6			1.79241			0.896206	+	0.443638i	$0.146313\pi$
$752$	81216.0	−	81216.0i	0.143617	−	0.143617i
$753$	−1.08902e6	−	1.08902e6i	−1.92064	−	1.92064i
$754$			187680.i			0.330123i
$755$		0			0
$756$		0			0
$757$	−313269.	+	313269.i	−0.546671	+	0.546671i	−0.925476	−	0.378806i	$-0.876335\pi$
$757$	−313269.	+	313269.i	−0.546671	+	0.546671i	0.378806	+	0.925476i	$0.376335\pi$
$758$	−502160.	−	502160.i	−0.873984	−	0.873984i
$759$		−	571356.i		−	0.991798i
$760$		0			0
$761$	142082.			0.245341			0.122670	−	0.992447i	$-0.460854\pi$
$761$	142082.			0.245341			0.122670	+	0.992447i	$0.460854\pi$
$762$	212724.	−	212724.i	0.366359	−	0.366359i
$763$	639160.	+	639160.i	1.09789	+	1.09789i
$764$			117424.i			0.201173i
$765$		0			0
$766$	344364.			0.586895
$767$	−80040.0	+	80040.0i	−0.136056	+	0.136056i
$768$	−36864.0	−	36864.0i	−0.0625000	−	0.0625000i
$769$		−	13280.0i		−	0.0224567i	−0.999937	−	0.0112283i	$-0.996426\pi$
$769$		−	13280.0i		−	0.0224567i	0.999937	−	0.0112283i	$-0.00357417\pi$
$770$		0			0
$771$	−1.30376e6			−2.19325
$772$	−342792.	+	342792.i	−0.575170	+	0.575170i
$773$	782211.	+	782211.i	1.30908	+	1.30908i	0.922082	+	0.386994i	$0.126487\pi$
$773$	782211.	+	782211.i	1.30908	+	1.30908i	0.386994	+	0.922082i	$0.373513\pi$
$774$		−	352836.i		−	0.588967i
$775$		0			0
$776$	−233952.			−0.388511
$777$	339822.	−	339822.i	0.562872	−	0.562872i
$778$	150000.	+	150000.i	0.247818	+	0.247818i
$779$		−	470960.i		−	0.776085i
$780$		0			0
$781$	−760156.			−1.24624
$782$	−291596.	+	291596.i	−0.476835	+	0.476835i
$783$		0			0
$784$		−	46016.0i		−	0.0748646i
$785$		0			0
$786$	228888.			0.370491
$787$	−201409.	+	201409.i	−0.325184	+	0.325184i	−0.850752	−	0.525568i	$-0.823853\pi$
$787$	−201409.	+	201409.i	−0.325184	+	0.325184i	0.525568	+	0.850752i	$0.323853\pi$
$788$	−87768.0	−	87768.0i	−0.141346	−	0.141346i
$789$		−	265878.i		−	0.427099i
$790$		0			0
$791$	−122438.			−0.195688
$792$	−152928.	+	152928.i	−0.243802	+	0.243802i
$793$	386262.	+	386262.i	0.614236	+	0.614236i
$794$			116596.i			0.184945i
$795$		0			0
$796$	305280.			0.481806
$797$	36291.0	−	36291.0i	0.0571324	−	0.0571324i	−0.677963	−	0.735096i	$-0.737137\pi$
$797$	36291.0	−	36291.0i	0.0571324	−	0.0571324i	0.735096	+	0.677963i	$0.237137\pi$
$798$	−292320.	−	292320.i	−0.459042	−	0.459042i
$799$			687798.i			1.07738i
$800$		0			0
$801$	−1.17288e6			−1.82805
$802$	−91836.0	+	91836.0i	−0.142779	+	0.142779i
$803$	−348218.	−	348218.i	−0.540033	−	0.540033i
$804$			108144.i			0.167298i
$805$		0			0
$806$	−55752.0			−0.0858204
$807$	−807480.	+	807480.i	−1.23989	+	1.23989i
$808$	14048.0	+	14048.0i	0.0215175	+	0.0215175i
$809$			71600.0i			0.109400i	0.998503	+	0.0546998i	$0.0174202\pi$
$809$			71600.0i			0.109400i	−0.998503	+	0.0546998i	$0.982580\pi$
$810$		0			0
$811$	−103318.			−0.157085			−0.0785424	−	0.996911i	$-0.525027\pi$
$811$	−103318.			−0.157085			−0.0785424	+	0.996911i	$0.525027\pi$
$812$	157760.	−	157760.i	0.239268	−	0.239268i
$813$	−613818.	−	613818.i	−0.928664	−	0.928664i
$814$			307272.i			0.463740i
$815$		0			0
$816$	312192.			0.468858
$817$	−304920.	+	304920.i	−0.456817	+	0.456817i
$818$	−157440.	−	157440.i	−0.235293	−	0.235293i
$819$			324162.i			0.483275i
$820$		0			0
$821$	−157438.			−0.233573			−0.116787	−	0.993157i	$-0.537259\pi$
$821$	−157438.			−0.233573			−0.116787	+	0.993157i	$0.537259\pi$
$822$	734724.	−	734724.i	1.08738	−	1.08738i
$823$	−791309.	−	791309.i	−1.16828	−	1.16828i	−0.982613	−	0.185666i	$-0.940556\pi$
$823$	−791309.	−	791309.i	−1.16828	−	1.16828i	−0.185666	−	0.982613i	$-0.559444\pi$
$824$			333728.i			0.491517i
$825$		0			0
$826$	134560.			0.197222
$827$	889671.	−	889671.i	1.30082	−	1.30082i	0.372987	−	0.927837i	$-0.378334\pi$
$827$	889671.	−	889671.i	1.30082	−	1.30082i	0.927837	−	0.372987i	$-0.121666\pi$
$828$	−174312.	−	174312.i	−0.254253	−	0.254253i
$829$			618280.i			0.899655i	0.893115	+	0.449828i	$0.148515\pi$
$829$			618280.i			0.899655i	−0.893115	+	0.449828i	$0.851485\pi$
$830$		0			0
$831$	333882.			0.483494
$832$	35328.0	−	35328.0i	0.0510355	−	0.0510355i
$833$	194849.	+	194849.i	0.280807	+	0.280807i
$834$		−	338400.i		−	0.486517i
$835$		0			0
$836$	264320.			0.378196
$837$		0			0
$838$	29520.0	+	29520.0i	0.0420367	+	0.0420367i
$839$			821360.i			1.16684i	0.812172	+	0.583418i	$0.198285\pi$
$839$			821360.i			1.16684i	−0.812172	+	0.583418i	$0.801715\pi$
$840$		0			0
$841$	244881.			0.346229
$842$	442564.	−	442564.i	0.624240	−	0.624240i
$843$	−20898.0	−	20898.0i	−0.0294069	−	0.0294069i
$844$		−	582736.i		−	0.818063i
$845$		0			0
$846$	−411156.			−0.574468
$847$	20793.0	−	20793.0i	0.0289835	−	0.0289835i
$848$	−39104.0	−	39104.0i	−0.0543788	−	0.0543788i
$849$		−	1.65080e6i		−	2.29023i
$850$		0			0
$851$	−350238.			−0.483620
$852$	−463824.	+	463824.i	−0.638961	+	0.638961i
$853$	698291.	+	698291.i	0.959706	+	0.959706i	0.999219	−	0.0395127i	$-0.0125806\pi$
$853$	698291.	+	698291.i	0.959706	+	0.959706i	−0.0395127	+	0.999219i	$0.512581\pi$
$854$		−	649368.i		−	0.890379i
$855$		0			0
$856$	−150752.			−0.205738
$857$	−144489.	+	144489.i	−0.196731	+	0.196731i	−0.798597	−	0.601866i	$-0.794424\pi$
$857$	−144489.	+	144489.i	−0.196731	+	0.196731i	0.601866	+	0.798597i	$0.294424\pi$
$858$	−293112.	−	293112.i	−0.398161	−	0.398161i
$859$			943480.i			1.27863i	0.768943	+	0.639317i	$0.220783\pi$
$859$			943480.i			1.27863i	−0.768943	+	0.639317i	$0.779217\pi$
$860$		0			0
$861$	878004.			1.18438
$862$	425044.	−	425044.i	0.572031	−	0.572031i
$863$	−438149.	−	438149.i	−0.588302	−	0.588302i	0.348869	−	0.937171i	$-0.386566\pi$
$863$	−438149.	−	438149.i	−0.588302	−	0.588302i	−0.937171	+	0.348869i	$0.886566\pi$
$864$		0			0
$865$		0			0
$866$	−581636.			−0.775560
$867$	−570249.	+	570249.i	−0.758624	+	0.758624i
$868$	46864.0	+	46864.0i	0.0622014	+	0.0622014i
$869$			1.24608e6i			1.65009i
$870$		0			0
$871$	−103638.			−0.136610
$872$	−352640.	+	352640.i	−0.463766	+	0.463766i
$873$	592191.	+	592191.i	0.777022	+	0.777022i
$874$			301280.i			0.394410i
$875$		0			0
$876$	−424944.			−0.553762
$877$	−281469.	+	281469.i	−0.365958	+	0.365958i	−0.866001	−	0.500043i	$-0.833318\pi$
$877$	−281469.	+	281469.i	−0.365958	+	0.365958i	0.500043	+	0.866001i	$0.333318\pi$
$878$	598880.	+	598880.i	0.776874	+	0.776874i
$879$		−	87318.0i		−	0.113012i
$880$		0			0
$881$	876722.			1.12956			0.564781	−	0.825241i	$-0.308961\pi$
$881$	876722.			1.12956			0.564781	+	0.825241i	$0.308961\pi$
$882$	−116478.	+	116478.i	−0.149729	+	0.149729i
$883$	327431.	+	327431.i	0.419951	+	0.419951i	0.885187	−	0.465236i	$-0.154031\pi$
$883$	327431.	+	327431.i	0.419951	+	0.419951i	−0.465236	+	0.885187i	$0.654031\pi$
$884$			299184.i			0.382855i
$885$		0			0
$886$	−962436.			−1.22604
$887$	477171.	−	477171.i	0.606494	−	0.606494i	−0.335534	−	0.942028i	$-0.608917\pi$
$887$	477171.	−	477171.i	0.606494	−	0.606494i	0.942028	+	0.335534i	$0.108917\pi$
$888$	187488.	+	187488.i	0.237765	+	0.237765i
$889$		−	342722.i		−	0.433649i
$890$		0			0
$891$	−774198.			−0.975207
$892$	247128.	−	247128.i	0.310593	−	0.310593i
$893$	355320.	+	355320.i	0.445571	+	0.445571i
$894$			496800.i			0.621594i
$895$		0			0
$896$	−59392.0			−0.0739796
$897$	334098.	−	334098.i	0.415230	−	0.415230i
$898$	164960.	+	164960.i	0.204562	+	0.204562i
$899$		−	137360.i		−	0.169958i
$900$		0			0
$901$	331162.			0.407935
$902$	−396952.	+	396952.i	−0.487893	+	0.487893i
$903$	−568458.	−	568458.i	−0.697145	−	0.697145i
$904$		−	67552.0i		−	0.0826611i
$905$		0			0
$906$	683928.			0.833209
$907$	−1.11209e6	+	1.11209e6i	−1.35184	+	1.35184i	−0.468235	+	0.883604i	$0.655110\pi$
$907$	−1.11209e6	+	1.11209e6i	−1.35184	+	1.35184i	−0.883604	+	0.468235i	$0.844890\pi$
$908$	−439288.	−	439288.i	−0.532816	−	0.532816i
$909$		−	71118.0i		−	0.0860700i
$910$		0			0
$911$	−883958.			−1.06511			−0.532556	−	0.846395i	$-0.678768\pi$
$911$	−883958.			−1.06511			−0.532556	+	0.846395i	$0.678768\pi$
$912$	161280.	−	161280.i	0.193906	−	0.193906i
$913$	−735258.	−	735258.i	−0.882060	−	0.882060i
$914$		−	752604.i		−	0.900895i
$915$		0			0
$916$	402240.			0.479396
$917$	184382.	−	184382.i	0.219270	−	0.219270i
$918$		0			0
$919$		−	1.24040e6i		−	1.46869i	−0.678775	−	0.734346i	$-0.737489\pi$
$919$		−	1.24040e6i		−	1.46869i	0.678775	−	0.734346i	$-0.262511\pi$
$920$		0			0
$921$	771282.			0.909272
$922$	−652316.	+	652316.i	−0.767355	+	0.767355i
$923$	−444498.	−	444498.i	−0.521755	−	0.521755i
$924$			492768.i			0.577163i
$925$		0			0
$926$	874924.			1.02035
$927$	844749.	−	844749.i	0.983033	−	0.983033i
$928$	87040.0	+	87040.0i	0.101070	+	0.101070i
$929$		−	1.22744e6i		−	1.42223i	−0.703077	−	0.711113i	$-0.748191\pi$
$929$		−	1.22744e6i		−	1.42223i	0.703077	−	0.711113i	$-0.251809\pi$
$930$		0			0
$931$	201320.			0.232267
$932$	19128.0	−	19128.0i	0.0220210	−	0.0220210i
$933$	650502.	+	650502.i	0.747283	+	0.747283i
$934$			236996.i			0.271673i
$935$		0			0
$936$	−178848.			−0.204142
$937$	1.07047e6	−	1.07047e6i	1.21926	−	1.21926i	0.251366	−	0.967892i	$-0.419120\pi$
$937$	1.07047e6	−	1.07047e6i	1.21926	−	1.21926i	0.967892	−	0.251366i	$-0.0808799\pi$
$938$	87116.0	+	87116.0i	0.0990130	+	0.0990130i
$939$			328482.i			0.372546i
$940$		0			0
$941$	558642.			0.630891			0.315446	−	0.948944i	$-0.397846\pi$
$941$	558642.			0.630891			0.315446	+	0.948944i	$0.397846\pi$
$942$	−589356.	+	589356.i	−0.664165	+	0.664165i
$943$	−452458.	−	452458.i	−0.508809	−	0.508809i
$944$			74240.0i			0.0833094i
$945$		0			0
$946$	514008.			0.574365
$947$	191711.	−	191711.i	0.213770	−	0.213770i	−0.592097	−	0.805867i	$-0.701700\pi$
$947$	191711.	−	191711.i	0.213770	−	0.213770i	0.805867	+	0.592097i	$0.201700\pi$
$948$	760320.	+	760320.i	0.846018	+	0.846018i
$949$		−	407238.i		−	0.452185i
$950$		0			0
$951$	452682.			0.500532
$952$	251488.	−	251488.i	0.277487	−	0.277487i
$953$	630231.	+	630231.i	0.693927	+	0.693927i	0.963094	−	0.269166i	$-0.0867482\pi$
$953$	630231.	+	630231.i	0.693927	+	0.693927i	−0.269166	+	0.963094i	$0.586748\pi$
$954$			197964.i			0.217515i
$955$		0			0
$956$	142080.			0.155459
$957$	722160.	−	722160.i	0.788514	−	0.788514i
$958$	−546880.	−	546880.i	−0.595883	−	0.595883i
$959$		−	1.18372e6i		−	1.28710i
$960$		0			0
$961$	−882717.			−0.955817
$962$	−179676.	+	179676.i	−0.194151	+	0.194151i
$963$	381591.	+	381591.i	0.411477	+	0.411477i
$964$			225904.i			0.243092i
$965$		0			0
$966$	−561672.			−0.601906
$967$	345491.	−	345491.i	0.369474	−	0.369474i	−0.497811	−	0.867285i	$-0.665863\pi$
$967$	345491.	−	345491.i	0.369474	−	0.369474i	0.867285	+	0.497811i	$0.165863\pi$
$968$	11472.0	+	11472.0i	0.0122430	+	0.0122430i
$969$			1.36584e6i			1.45463i
$970$		0			0
$971$	1.08308e6			1.14874			0.574372	−	0.818595i	$-0.305247\pi$
$971$	1.08308e6			1.14874			0.574372	+	0.818595i	$0.305247\pi$
$972$	−472392.	+	472392.i	−0.500000	+	0.500000i
$973$	−272600.	−	272600.i	−0.287939	−	0.287939i
$974$		−	494604.i		−	0.521362i
$975$		0			0
$976$	358272.			0.376109
$977$	146751.	−	146751.i	0.153742	−	0.153742i	−0.626045	−	0.779787i	$-0.715327\pi$
$977$	146751.	−	146751.i	0.153742	−	0.153742i	0.779787	+	0.626045i	$0.215327\pi$
$978$	720324.	+	720324.i	0.753096	+	0.753096i
$979$		−	1.70864e6i		−	1.78273i
$980$		0			0
$981$	1.78524e6			1.85506
$982$	396884.	−	396884.i	0.411567	−	0.411567i
$983$	−466909.	−	466909.i	−0.483198	−	0.483198i	0.422953	−	0.906151i	$-0.360993\pi$
$983$	−466909.	−	466909.i	−0.483198	−	0.483198i	−0.906151	+	0.422953i	$0.860993\pi$
$984$			484416.i			0.500297i
$985$		0			0
$986$	−737120.			−0.758201
$987$	−662418.	+	662418.i	−0.679983	+	0.679983i
$988$	154560.	+	154560.i	0.158337	+	0.158337i
$989$			585882.i			0.598987i
$990$		0			0
$991$	−901238.			−0.917682			−0.458841	−	0.888518i	$-0.651735\pi$
$991$	−901238.			−0.917682			−0.458841	+	0.888518i	$0.651735\pi$
$992$	−25856.0	+	25856.0i	−0.0262747	+	0.0262747i
$993$	486342.	+	486342.i	0.493223	+	0.493223i
$994$			747272.i			0.756321i
$995$		0			0
$996$	−897264.			−0.904485
$997$	−152149.	+	152149.i	−0.153066	+	0.153066i	−0.779486	−	0.626420i	$-0.784520\pi$
$997$	−152149.	+	152149.i	−0.153066	+	0.153066i	0.626420	+	0.779486i	$0.284520\pi$
$998$	−538480.	−	538480.i	−0.540640	−	0.540640i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	50.5.c.b.43.1		2
3.2	odd	2		450.5.g.a.343.1		2
4.3	odd	2		400.5.p.c.193.1		2
5.2	odd	4	inner	50.5.c.b.7.1		2
5.3	odd	4		10.5.c.a.7.1	yes	2
5.4	even	2		10.5.c.a.3.1	✓	2
15.2	even	4		450.5.g.a.307.1		2
15.8	even	4		90.5.g.b.37.1		2
15.14	odd	2		90.5.g.b.73.1		2
20.3	even	4		80.5.p.b.17.1		2
20.7	even	4		400.5.p.c.257.1		2
20.19	odd	2		80.5.p.b.33.1		2
40.3	even	4		320.5.p.i.257.1		2
40.13	odd	4		320.5.p.b.257.1		2
40.19	odd	2		320.5.p.i.193.1		2
40.29	even	2		320.5.p.b.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.5.c.a.3.1	✓	2	5.4	even	2
10.5.c.a.7.1	yes	2	5.3	odd	4
50.5.c.b.7.1		2	5.2	odd	4	inner
50.5.c.b.43.1		2	1.1	even	1	trivial
80.5.p.b.17.1		2	20.3	even	4
80.5.p.b.33.1		2	20.19	odd	2
90.5.g.b.37.1		2	15.8	even	4
90.5.g.b.73.1		2	15.14	odd	2
320.5.p.b.193.1		2	40.29	even	2
320.5.p.b.257.1		2	40.13	odd	4
320.5.p.i.193.1		2	40.19	odd	2
320.5.p.i.257.1		2	40.3	even	4
400.5.p.c.193.1		2	4.3	odd	2
400.5.p.c.257.1		2	20.7	even	4
450.5.g.a.307.1		2	15.2	even	4
450.5.g.a.343.1		2	3.2	odd	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:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.00000 - 2.00000i) q^{2} +(-9.00000 - 9.00000i) q^{3} -8.00000i q^{4} -36.0000 q^{6} +(-29.0000 + 29.0000i) q^{7} +(-16.0000 - 16.0000i) q^{8} +81.0000i q^{9} -118.000 q^{11} +(-72.0000 + 72.0000i) q^{12} +(-69.0000 - 69.0000i) q^{13} +116.000i q^{14} -64.0000 q^{16} +(271.000 - 271.000i) q^{17} +(162.000 + 162.000i) q^{18} -280.000i q^{19} +522.000 q^{21} +(-236.000 + 236.000i) q^{22} +(-269.000 - 269.000i) q^{23} +288.000i q^{24} -276.000 q^{26} +(232.000 + 232.000i) q^{28} -680.000i q^{29} +202.000 q^{31} +(-128.000 + 128.000i) q^{32} +(1062.00 + 1062.00i) q^{33} -1084.00i q^{34} +648.000 q^{36} +(651.000 - 651.000i) q^{37} +(-560.000 - 560.000i) q^{38} +1242.00i q^{39} +1682.00 q^{41} +(1044.00 - 1044.00i) q^{42} +(-1089.00 - 1089.00i) q^{43} +944.000i q^{44} -1076.00 q^{46} +(-1269.00 + 1269.00i) q^{47} +(576.000 + 576.000i) q^{48} +719.000i q^{49} -4878.00 q^{51} +(-552.000 + 552.000i) q^{52} +(611.000 + 611.000i) q^{53} +928.000 q^{56} +(-2520.00 + 2520.00i) q^{57} +(-1360.00 - 1360.00i) q^{58} -1160.00i q^{59} -5598.00 q^{61} +(404.000 - 404.000i) q^{62} +(-2349.00 - 2349.00i) q^{63} +512.000i q^{64} +4248.00 q^{66} +(751.000 - 751.000i) q^{67} +(-2168.00 - 2168.00i) q^{68} +4842.00i q^{69} +6442.00 q^{71} +(1296.00 - 1296.00i) q^{72} +(2951.00 + 2951.00i) q^{73} -2604.00i q^{74} -2240.00 q^{76} +(3422.00 - 3422.00i) q^{77} +(2484.00 + 2484.00i) q^{78} -10560.0i q^{79} +6561.00 q^{81} +(3364.00 - 3364.00i) q^{82} +(6231.00 + 6231.00i) q^{83} -4176.00i q^{84} -4356.00 q^{86} +(-6120.00 + 6120.00i) q^{87} +(1888.00 + 1888.00i) q^{88} +14480.0i q^{89} +4002.00 q^{91} +(-2152.00 + 2152.00i) q^{92} +(-1818.00 - 1818.00i) q^{93} +5076.00i q^{94} +2304.00 q^{96} +(7311.00 - 7311.00i) q^{97} +(1438.00 + 1438.00i) q^{98} -9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 18 q^{3} - 72 q^{6} - 58 q^{7} - 32 q^{8} - 236 q^{11} - 144 q^{12} - 138 q^{13} - 128 q^{16} + 542 q^{17} + 324 q^{18} + 1044 q^{21} - 472 q^{22} - 538 q^{23} - 552 q^{26} + 464 q^{28}+ \cdots + 2876 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 18 * q^3 - 72 * q^6 - 58 * q^7 - 32 * q^8 - 236 * q^11 - 144 * q^12 - 138 * q^13 - 128 * q^16 + 542 * q^17 + 324 * q^18 + 1044 * q^21 - 472 * q^22 - 538 * q^23 - 552 * q^26 + 464 * q^28 + 404 * q^31 - 256 * q^32 + 2124 * q^33 + 1296 * q^36 + 1302 * q^37 - 1120 * q^38 + 3364 * q^41 + 2088 * q^42 - 2178 * q^43 - 2152 * q^46 - 2538 * q^47 + 1152 * q^48 - 9756 * q^51 - 1104 * q^52 + 1222 * q^53 + 1856 * q^56 - 5040 * q^57 - 2720 * q^58 - 11196 * q^61 + 808 * q^62 - 4698 * q^63 + 8496 * q^66 + 1502 * q^67 - 4336 * q^68 + 12884 * q^71 + 2592 * q^72 + 5902 * q^73 - 4480 * q^76 + 6844 * q^77 + 4968 * q^78 + 13122 * q^81 + 6728 * q^82 + 12462 * q^83 - 8712 * q^86 - 12240 * q^87 + 3776 * q^88 + 8004 * q^91 - 4304 * q^92 - 3636 * q^93 + 4608 * q^96 + 14622 * q^97 + 2876 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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