LMFDB - Embedded newform 475.4.b.e.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.b (of order $2$, degree $1$, 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:	$28.0259072527$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.4.b.e.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{3} +8.00000 q^{4} +22.0000i q^{7} +11.0000 q^{9} -12.0000 q^{11} +32.0000i q^{12} +8.00000i q^{13} +64.0000 q^{16} +66.0000i q^{17} -19.0000 q^{19} -88.0000 q^{21} -30.0000i q^{23} +152.000i q^{27} +176.000i q^{28} +6.00000 q^{29} -64.0000 q^{31} -48.0000i q^{33} +88.0000 q^{36} +16.0000i q^{37} -32.0000 q^{39} +54.0000 q^{41} +182.000i q^{43} -96.0000 q^{44} -594.000i q^{47} +256.000i q^{48} -141.000 q^{49} -264.000 q^{51} +64.0000i q^{52} +396.000i q^{53} -76.0000i q^{57} +564.000 q^{59} -706.000 q^{61} +242.000i q^{63} +512.000 q^{64} +628.000i q^{67} +528.000i q^{68} +120.000 q^{69} -984.000 q^{71} +14.0000i q^{73} -152.000 q^{76} -264.000i q^{77} +328.000 q^{79} -311.000 q^{81} -294.000i q^{83} -704.000 q^{84} +24.0000i q^{87} -918.000 q^{89} -176.000 q^{91} -240.000i q^{92} -256.000i q^{93} +1564.00i q^{97} -132.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 16 q^{4} + 22 q^{9} - 24 q^{11} + 128 q^{16} - 38 q^{19} - 176 q^{21} + 12 q^{29} - 128 q^{31} + 176 q^{36} - 64 q^{39} + 108 q^{41} - 192 q^{44} - 282 q^{49} - 528 q^{51} + 1128 q^{59} - 1412 q^{61}+ \cdots - 264 q^{99}+O(q^{100}) $ 2 * q + 16 * q^4 + 22 * q^9 - 24 * q^11 + 128 * q^16 - 38 * q^19 - 176 * q^21 + 12 * q^29 - 128 * q^31 + 176 * q^36 - 64 * q^39 + 108 * q^41 - 192 * q^44 - 282 * q^49 - 528 * q^51 + 1128 * q^59 - 1412 * q^61 + 1024 * q^64 + 240 * q^69 - 1968 * q^71 - 304 * q^76 + 656 * q^79 - 622 * q^81 - 1408 * q^84 - 1836 * q^89 - 352 * q^91 - 264 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	4.00000i	0.769800i	0.922958	+	0.384900i	$0.125764\pi$
$3$	4.00000i	0.769800i	−0.922958	+	0.384900i	$0.874236\pi$
$4$	8.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	22.0000i	1.18789i	0.804506	+	0.593944i	$0.202430\pi$
$7$	22.0000i	1.18789i	−0.804506	+	0.593944i	$0.797570\pi$
$8$	0	0
$9$	11.0000	0.407407
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	32.0000i	0.769800i
$13$	8.00000i	0.170677i	0.996352	+	0.0853385i	$0.0271972\pi$
$13$	8.00000i	0.170677i	−0.996352	+	0.0853385i	$0.972803\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	66.0000i	0.941609i	0.882238	+	0.470804i	$0.156036\pi$
$17$	66.0000i	0.941609i	−0.882238	+	0.470804i	$0.843964\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−88.0000	−0.914437
$22$	0	0
$23$	− 30.0000i	− 0.271975i	−0.990711	−	0.135988i	$-0.956579\pi$
$23$	− 30.0000i	− 0.271975i	0.990711	−	0.135988i	$-0.0434208\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.000i	1.08342i
$28$	176.000i	1.18789i
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−64.0000	−0.370798	−0.185399	−	0.982663i	$-0.559358\pi$
$31$	−64.0000	−0.370798	−0.185399	+	0.982663i	$0.559358\pi$
$32$	0	0
$33$	− 48.0000i	− 0.253204i
$34$	0	0
$35$	0	0
$36$	88.0000	0.407407
$37$	16.0000i	0.0710915i	0.999368	+	0.0355457i	$0.0113169\pi$
$37$	16.0000i	0.0710915i	−0.999368	+	0.0355457i	$0.988683\pi$
$38$	0	0
$39$	−32.0000	−0.131387
$40$	0	0
$41$	54.0000	0.205692	0.102846	−	0.994697i	$-0.467205\pi$
$41$	54.0000	0.205692	0.102846	+	0.994697i	$0.467205\pi$
$42$	0	0
$43$	182.000i	0.645459i	0.946491	+	0.322730i	$0.104600\pi$
$43$	182.000i	0.645459i	−0.946491	+	0.322730i	$0.895400\pi$
$44$	−96.0000	−0.328921
$45$	0	0
$46$	0	0
$47$	− 594.000i	− 1.84349i	−0.387802	−	0.921743i	$-0.626766\pi$
$47$	− 594.000i	− 1.84349i	0.387802	−	0.921743i	$-0.373234\pi$
$48$	256.000i	0.769800i
$49$	−141.000	−0.411079
$50$	0	0
$51$	−264.000	−0.724851
$52$	64.0000i	0.170677i
$53$	396.000i	1.02632i	0.858294	+	0.513158i	$0.171525\pi$
$53$	396.000i	1.02632i	−0.858294	+	0.513158i	$0.828475\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 76.0000i	− 0.176604i
$58$	0	0
$59$	564.000	1.24452	0.622259	−	0.782812i	$-0.286215\pi$
$59$	564.000	1.24452	0.622259	+	0.782812i	$0.286215\pi$
$60$	0	0
$61$	−706.000	−1.48187	−0.740935	−	0.671577i	$-0.765617\pi$
$61$	−706.000	−1.48187	−0.740935	+	0.671577i	$0.765617\pi$
$62$	0	0
$63$	242.000i	0.483955i
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	628.000i	1.14511i	0.819866	+	0.572555i	$0.194048\pi$
$67$	628.000i	1.14511i	−0.819866	+	0.572555i	$0.805952\pi$
$68$	528.000i	0.941609i
$69$	120.000	0.209367
$70$	0	0
$71$	−984.000	−1.64478	−0.822390	−	0.568925i	$-0.807360\pi$
$71$	−984.000	−1.64478	−0.822390	+	0.568925i	$0.807360\pi$
$72$	0	0
$73$	14.0000i	0.0224462i	0.999937	+	0.0112231i	$0.00357251\pi$
$73$	14.0000i	0.0224462i	−0.999937	+	0.0112231i	$0.996427\pi$
$74$	0	0
$75$	0	0
$76$	−152.000	−0.229416
$77$	− 264.000i	− 0.390722i
$78$	0	0
$79$	328.000	0.467125	0.233563	−	0.972342i	$-0.424962\pi$
$79$	328.000	0.467125	0.233563	+	0.972342i	$0.424962\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	− 294.000i	− 0.388804i	−0.980922	−	0.194402i	$-0.937723\pi$
$83$	− 294.000i	− 0.388804i	0.980922	−	0.194402i	$-0.0622765\pi$
$84$	−704.000	−0.914437
$85$	0	0
$86$	0	0
$87$	24.0000i	0.0295755i
$88$	0	0
$89$	−918.000	−1.09335	−0.546673	−	0.837346i	$-0.684106\pi$
$89$	−918.000	−1.09335	−0.546673	+	0.837346i	$0.684106\pi$
$90$	0	0
$91$	−176.000	−0.202745
$92$	− 240.000i	− 0.271975i
$93$	− 256.000i	− 0.285440i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1564.00i	1.63711i	0.574425	+	0.818557i	$0.305226\pi$
$97$	1564.00i	1.63711i	−0.574425	+	0.818557i	$0.694774\pi$
$98$	0	0
$99$	−132.000	−0.134005
$100$	0	0
$101$	−294.000	−0.289644	−0.144822	−	0.989458i	$-0.546261\pi$
$101$	−294.000	−0.289644	−0.144822	+	0.989458i	$0.546261\pi$
$102$	0	0
$103$	752.000i	0.719386i	0.933071	+	0.359693i	$0.117119\pi$
$103$	752.000i	0.719386i	−0.933071	+	0.359693i	$0.882881\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	216.000i	0.195154i	0.995228	+	0.0975771i	$0.0311093\pi$
$107$	216.000i	0.195154i	−0.995228	+	0.0975771i	$0.968891\pi$
$108$	1216.00i	1.08342i
$109$	754.000	0.662570	0.331285	−	0.943531i	$-0.392518\pi$
$109$	754.000	0.662570	0.331285	+	0.943531i	$0.392518\pi$
$110$	0	0
$111$	−64.0000	−0.0547262
$112$	1408.00i	1.18789i
$113$	− 12.0000i	− 0.00998996i	−0.999988	−	0.00499498i	$-0.998410\pi$
$113$	− 12.0000i	− 0.00998996i	0.999988	−	0.00499498i	$-0.00158996\pi$
$114$	0	0
$115$	0	0
$116$	48.0000	0.0384197
$117$	88.0000i	0.0695351i
$118$	0	0
$119$	−1452.00	−1.11853
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	216.000i	0.158342i
$124$	−512.000	−0.370798
$125$	0	0
$126$	0	0
$127$	− 344.000i	− 0.240355i	−0.992752	−	0.120177i	$-0.961654\pi$
$127$	− 344.000i	− 0.240355i	0.992752	−	0.120177i	$-0.0383463\pi$
$128$	0	0
$129$	−728.000	−0.496875
$130$	0	0
$131$	2520.00	1.68071	0.840357	−	0.542034i	$-0.182346\pi$
$131$	2520.00	1.68071	0.840357	+	0.542034i	$0.182346\pi$
$132$	− 384.000i	− 0.253204i
$133$	− 418.000i	− 0.272520i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 654.000i	− 0.407847i	−0.978987	−	0.203923i	$-0.934631\pi$
$137$	− 654.000i	− 0.407847i	0.978987	−	0.203923i	$-0.0653693\pi$
$138$	0	0
$139$	2392.00	1.45962	0.729809	−	0.683652i	$-0.239609\pi$
$139$	2392.00	1.45962	0.729809	+	0.683652i	$0.239609\pi$
$140$	0	0
$141$	2376.00	1.41912
$142$	0	0
$143$	− 96.0000i	− 0.0561393i
$144$	704.000	0.407407
$145$	0	0
$146$	0	0
$147$	− 564.000i	− 0.316449i
$148$	128.000i	0.0710915i
$149$	1266.00	0.696072	0.348036	−	0.937481i	$-0.386849\pi$
$149$	1266.00	0.696072	0.348036	+	0.937481i	$0.386849\pi$
$150$	0	0
$151$	3080.00	1.65991	0.829956	−	0.557828i	$-0.188365\pi$
$151$	3080.00	1.65991	0.829956	+	0.557828i	$0.188365\pi$
$152$	0	0
$153$	726.000i	0.383618i
$154$	0	0
$155$	0	0
$156$	−256.000	−0.131387
$157$	− 1838.00i	− 0.934321i	−0.884173	−	0.467160i	$-0.845277\pi$
$157$	− 1838.00i	− 0.934321i	0.884173	−	0.467160i	$-0.154723\pi$
$158$	0	0
$159$	−1584.00	−0.790059
$160$	0	0
$161$	660.000	0.323076
$162$	0	0
$163$	− 850.000i	− 0.408449i	−0.978924	−	0.204224i	$-0.934533\pi$
$163$	− 850.000i	− 0.408449i	0.978924	−	0.204224i	$-0.0654672\pi$
$164$	432.000	0.205692
$165$	0	0
$166$	0	0
$167$	− 3804.00i	− 1.76265i	−0.472512	−	0.881324i	$-0.656653\pi$
$167$	− 3804.00i	− 1.76265i	0.472512	−	0.881324i	$-0.343347\pi$
$168$	0	0
$169$	2133.00	0.970869
$170$	0	0
$171$	−209.000	−0.0934657
$172$	1456.00i	0.645459i
$173$	− 564.000i	− 0.247862i	−0.992291	−	0.123931i	$-0.960450\pi$
$173$	− 564.000i	− 0.247862i	0.992291	−	0.123931i	$-0.0395501\pi$
$174$	0	0
$175$	0	0
$176$	−768.000	−0.328921
$177$	2256.00i	0.958030i
$178$	0	0
$179$	1812.00	0.756621	0.378311	−	0.925679i	$-0.376505\pi$
$179$	1812.00	0.756621	0.378311	+	0.925679i	$0.376505\pi$
$180$	0	0
$181$	−4498.00	−1.84715	−0.923574	−	0.383421i	$-0.874746\pi$
$181$	−4498.00	−1.84715	−0.923574	+	0.383421i	$0.874746\pi$
$182$	0	0
$183$	− 2824.00i	− 1.14074i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 792.000i	− 0.309715i
$188$	− 4752.00i	− 1.84349i
$189$	−3344.00	−1.28699
$190$	0	0
$191$	3588.00	1.35926	0.679630	−	0.733555i	$-0.262140\pi$
$191$	3588.00	1.35926	0.679630	+	0.733555i	$0.262140\pi$
$192$	2048.00i	0.769800i
$193$	− 4492.00i	− 1.67534i	−0.546174	−	0.837672i	$-0.683916\pi$
$193$	− 4492.00i	− 1.67534i	0.546174	−	0.837672i	$-0.316084\pi$
$194$	0	0
$195$	0	0
$196$	−1128.00	−0.411079
$197$	− 2466.00i	− 0.891854i	−0.895069	−	0.445927i	$-0.852874\pi$
$197$	− 2466.00i	− 0.891854i	0.895069	−	0.445927i	$-0.147126\pi$
$198$	0	0
$199$	−824.000	−0.293527	−0.146763	−	0.989172i	$-0.546886\pi$
$199$	−824.000	−0.293527	−0.146763	+	0.989172i	$0.546886\pi$
$200$	0	0
$201$	−2512.00	−0.881507
$202$	0	0
$203$	132.000i	0.0456383i
$204$	−2112.00	−0.724851
$205$	0	0
$206$	0	0
$207$	− 330.000i	− 0.110805i
$208$	512.000i	0.170677i
$209$	228.000	0.0754598
$210$	0	0
$211$	4244.00	1.38469	0.692344	−	0.721568i	$-0.256578\pi$
$211$	4244.00	1.38469	0.692344	+	0.721568i	$0.256578\pi$
$212$	3168.00i	1.02632i
$213$	− 3936.00i	− 1.26615i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1408.00i	− 0.440467i
$218$	0	0
$219$	−56.0000	−0.0172791
$220$	0	0
$221$	−528.000	−0.160711
$222$	0	0
$223$	5480.00i	1.64560i	0.568334	+	0.822798i	$0.307588\pi$
$223$	5480.00i	1.64560i	−0.568334	+	0.822798i	$0.692412\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 5544.00i	− 1.62101i	−0.585735	−	0.810503i	$-0.699194\pi$
$227$	− 5544.00i	− 1.62101i	0.585735	−	0.810503i	$-0.300806\pi$
$228$	− 608.000i	− 0.176604i
$229$	−2930.00	−0.845502	−0.422751	−	0.906246i	$-0.638935\pi$
$229$	−2930.00	−0.845502	−0.422751	+	0.906246i	$0.638935\pi$
$230$	0	0
$231$	1056.00	0.300778
$232$	0	0
$233$	− 4398.00i	− 1.23658i	−0.785951	−	0.618289i	$-0.787826\pi$
$233$	− 4398.00i	− 1.23658i	0.785951	−	0.618289i	$-0.212174\pi$
$234$	0	0
$235$	0	0
$236$	4512.00	1.24452
$237$	1312.00i	0.359593i
$238$	0	0
$239$	6480.00	1.75379	0.876896	−	0.480680i	$-0.159610\pi$
$239$	6480.00	1.75379	0.876896	+	0.480680i	$0.159610\pi$
$240$	0	0
$241$	−5770.00	−1.54223	−0.771117	−	0.636694i	$-0.780302\pi$
$241$	−5770.00	−1.54223	−0.771117	+	0.636694i	$0.780302\pi$
$242$	0	0
$243$	2860.00i	0.755017i
$244$	−5648.00	−1.48187
$245$	0	0
$246$	0	0
$247$	− 152.000i	− 0.0391560i
$248$	0	0
$249$	1176.00	0.299301
$250$	0	0
$251$	−624.000	−0.156918	−0.0784592	−	0.996917i	$-0.525000\pi$
$251$	−624.000	−0.156918	−0.0784592	+	0.996917i	$0.525000\pi$
$252$	1936.00i	0.483955i
$253$	360.000i	0.0894585i
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	− 120.000i	− 0.0291260i	−0.999894	−	0.0145630i	$-0.995364\pi$
$257$	− 120.000i	− 0.0291260i	0.999894	−	0.0145630i	$-0.00463572\pi$
$258$	0	0
$259$	−352.000	−0.0844487
$260$	0	0
$261$	66.0000	0.0156525
$262$	0	0
$263$	1842.00i	0.431873i	0.976407	+	0.215936i	$0.0692804\pi$
$263$	1842.00i	0.431873i	−0.976407	+	0.215936i	$0.930720\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 3672.00i	− 0.841658i
$268$	5024.00i	1.14511i
$269$	3234.00	0.733013	0.366506	−	0.930416i	$-0.380554\pi$
$269$	3234.00	0.733013	0.366506	+	0.930416i	$0.380554\pi$
$270$	0	0
$271$	4664.00	1.04545	0.522727	−	0.852500i	$-0.324915\pi$
$271$	4664.00	1.04545	0.522727	+	0.852500i	$0.324915\pi$
$272$	4224.00i	0.941609i
$273$	− 704.000i	− 0.156073i
$274$	0	0
$275$	0	0
$276$	960.000	0.209367
$277$	− 5222.00i	− 1.13271i	−0.824163	−	0.566353i	$-0.808354\pi$
$277$	− 5222.00i	− 1.13271i	0.824163	−	0.566353i	$-0.191646\pi$
$278$	0	0
$279$	−704.000	−0.151066
$280$	0	0
$281$	7566.00	1.60623	0.803113	−	0.595826i	$-0.203175\pi$
$281$	7566.00	1.60623	0.803113	+	0.595826i	$0.203175\pi$
$282$	0	0
$283$	− 5614.00i	− 1.17921i	−0.807690	−	0.589607i	$-0.799283\pi$
$283$	− 5614.00i	− 1.17921i	0.807690	−	0.589607i	$-0.200717\pi$
$284$	−7872.00	−1.64478
$285$	0	0
$286$	0	0
$287$	1188.00i	0.244339i
$288$	0	0
$289$	557.000	0.113373
$290$	0	0
$291$	−6256.00	−1.26025
$292$	112.000i	0.0224462i
$293$	924.000i	0.184234i	0.995748	+	0.0921172i	$0.0293634\pi$
$293$	924.000i	0.184234i	−0.995748	+	0.0921172i	$0.970637\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1824.00i	− 0.356361i
$298$	0	0
$299$	240.000	0.0464199
$300$	0	0
$301$	−4004.00	−0.766733
$302$	0	0
$303$	− 1176.00i	− 0.222968i
$304$	−1216.00	−0.229416
$305$	0	0
$306$	0	0
$307$	4480.00i	0.832857i	0.909168	+	0.416429i	$0.136718\pi$
$307$	4480.00i	0.832857i	−0.909168	+	0.416429i	$0.863282\pi$
$308$	− 2112.00i	− 0.390722i
$309$	−3008.00	−0.553784
$310$	0	0
$311$	1272.00	0.231924	0.115962	−	0.993254i	$-0.463005\pi$
$311$	1272.00	0.231924	0.115962	+	0.993254i	$0.463005\pi$
$312$	0	0
$313$	1370.00i	0.247402i	0.992320	+	0.123701i	$0.0394764\pi$
$313$	1370.00i	0.247402i	−0.992320	+	0.123701i	$0.960524\pi$
$314$	0	0
$315$	0	0
$316$	2624.00	0.467125
$317$	− 3552.00i	− 0.629338i	−0.949201	−	0.314669i	$-0.898106\pi$
$317$	− 3552.00i	− 0.629338i	0.949201	−	0.314669i	$-0.101894\pi$
$318$	0	0
$319$	−72.0000	−0.0126371
$320$	0	0
$321$	−864.000	−0.150230
$322$	0	0
$323$	− 1254.00i	− 0.216020i
$324$	−2488.00	−0.426612
$325$	0	0
$326$	0	0
$327$	3016.00i	0.510046i
$328$	0	0
$329$	13068.0	2.18985
$330$	0	0
$331$	4532.00	0.752572	0.376286	−	0.926504i	$-0.377201\pi$
$331$	4532.00	0.752572	0.376286	+	0.926504i	$0.377201\pi$
$332$	− 2352.00i	− 0.388804i
$333$	176.000i	0.0289632i
$334$	0	0
$335$	0	0
$336$	−5632.00	−0.914437
$337$	10036.0i	1.62224i	0.584878	+	0.811121i	$0.301142\pi$
$337$	10036.0i	1.62224i	−0.584878	+	0.811121i	$0.698858\pi$
$338$	0	0
$339$	48.0000	0.00769027
$340$	0	0
$341$	768.000	0.121963
$342$	0	0
$343$	4444.00i	0.699573i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2178.00i	0.336949i	0.985706	+	0.168474i	$0.0538840\pi$
$347$	2178.00i	0.336949i	−0.985706	+	0.168474i	$0.946116\pi$
$348$	192.000i	0.0295755i
$349$	10042.0	1.54022	0.770109	−	0.637913i	$-0.220202\pi$
$349$	10042.0	1.54022	0.770109	+	0.637913i	$0.220202\pi$
$350$	0	0
$351$	−1216.00	−0.184915
$352$	0	0
$353$	6102.00i	0.920047i	0.887907	+	0.460024i	$0.152159\pi$
$353$	6102.00i	0.920047i	−0.887907	+	0.460024i	$0.847841\pi$
$354$	0	0
$355$	0	0
$356$	−7344.00	−1.09335
$357$	− 5808.00i	− 0.861042i
$358$	0	0
$359$	1140.00	0.167596	0.0837979	−	0.996483i	$-0.473295\pi$
$359$	1140.00	0.167596	0.0837979	+	0.996483i	$0.473295\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	− 4748.00i	− 0.686516i
$364$	−1408.00	−0.202745
$365$	0	0
$366$	0	0
$367$	5614.00i	0.798497i	0.916843	+	0.399249i	$0.130729\pi$
$367$	5614.00i	0.798497i	−0.916843	+	0.399249i	$0.869271\pi$
$368$	− 1920.00i	− 0.271975i
$369$	594.000	0.0838006
$370$	0	0
$371$	−8712.00	−1.21915
$372$	− 2048.00i	− 0.285440i
$373$	− 3652.00i	− 0.506953i	−0.967342	−	0.253476i	$-0.918426\pi$
$373$	− 3652.00i	− 0.506953i	0.967342	−	0.253476i	$-0.0815740\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	48.0000i	0.00655736i
$378$	0	0
$379$	316.000	0.0428280	0.0214140	−	0.999771i	$-0.493183\pi$
$379$	316.000	0.0428280	0.0214140	+	0.999771i	$0.493183\pi$
$380$	0	0
$381$	1376.00	0.185025
$382$	0	0
$383$	− 8844.00i	− 1.17991i	−0.807434	−	0.589957i	$-0.799145\pi$
$383$	− 8844.00i	− 1.17991i	0.807434	−	0.589957i	$-0.200855\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2002.00i	0.262965i
$388$	12512.0i	1.63711i
$389$	7230.00	0.942354	0.471177	−	0.882039i	$-0.343829\pi$
$389$	7230.00	0.942354	0.471177	+	0.882039i	$0.343829\pi$
$390$	0	0
$391$	1980.00	0.256094
$392$	0	0
$393$	10080.0i	1.29381i
$394$	0	0
$395$	0	0
$396$	−1056.00	−0.134005
$397$	− 3446.00i	− 0.435642i	−0.975989	−	0.217821i	$-0.930105\pi$
$397$	− 3446.00i	− 0.435642i	0.975989	−	0.217821i	$-0.0698949\pi$
$398$	0	0
$399$	1672.00	0.209786
$400$	0	0
$401$	−14478.0	−1.80298	−0.901492	−	0.432795i	$-0.857527\pi$
$401$	−14478.0	−1.80298	−0.901492	+	0.432795i	$0.857527\pi$
$402$	0	0
$403$	− 512.000i	− 0.0632867i
$404$	−2352.00	−0.289644
$405$	0	0
$406$	0	0
$407$	− 192.000i	− 0.0233835i
$408$	0	0
$409$	−9074.00	−1.09702	−0.548509	−	0.836145i	$-0.684804\pi$
$409$	−9074.00	−1.09702	−0.548509	+	0.836145i	$0.684804\pi$
$410$	0	0
$411$	2616.00	0.313960
$412$	6016.00i	0.719386i
$413$	12408.0i	1.47835i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9568.00i	1.12361i
$418$	0	0
$419$	7068.00	0.824092	0.412046	−	0.911163i	$-0.364814\pi$
$419$	7068.00	0.824092	0.412046	+	0.911163i	$0.364814\pi$
$420$	0	0
$421$	−7342.00	−0.849946	−0.424973	−	0.905206i	$-0.639716\pi$
$421$	−7342.00	−0.849946	−0.424973	+	0.905206i	$0.639716\pi$
$422$	0	0
$423$	− 6534.00i	− 0.751050i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 15532.0i	− 1.76030i
$428$	1728.00i	0.195154i
$429$	384.000	0.0432161
$430$	0	0
$431$	2976.00	0.332596	0.166298	−	0.986076i	$-0.446819\pi$
$431$	2976.00	0.332596	0.166298	+	0.986076i	$0.446819\pi$
$432$	9728.00i	1.08342i
$433$	3476.00i	0.385787i	0.981220	+	0.192894i	$0.0617872\pi$
$433$	3476.00i	0.385787i	−0.981220	+	0.192894i	$0.938213\pi$
$434$	0	0
$435$	0	0
$436$	6032.00	0.662570
$437$	570.000i	0.0623954i
$438$	0	0
$439$	−6200.00	−0.674054	−0.337027	−	0.941495i	$-0.609421\pi$
$439$	−6200.00	−0.674054	−0.337027	+	0.941495i	$0.609421\pi$
$440$	0	0
$441$	−1551.00	−0.167477
$442$	0	0
$443$	16026.0i	1.71878i	0.511323	+	0.859389i	$0.329156\pi$
$443$	16026.0i	1.71878i	−0.511323	+	0.859389i	$0.670844\pi$
$444$	−512.000	−0.0547262
$445$	0	0
$446$	0	0
$447$	5064.00i	0.535837i
$448$	11264.0i	1.18789i
$449$	1830.00	0.192345	0.0961726	−	0.995365i	$-0.469340\pi$
$449$	1830.00	0.192345	0.0961726	+	0.995365i	$0.469340\pi$
$450$	0	0
$451$	−648.000	−0.0676566
$452$	− 96.0000i	− 0.00998996i
$453$	12320.0i	1.27780i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 12986.0i	− 1.32923i	−0.747185	−	0.664616i	$-0.768595\pi$
$457$	− 12986.0i	− 1.32923i	0.747185	−	0.664616i	$-0.231405\pi$
$458$	0	0
$459$	−10032.0	−1.02016
$460$	0	0
$461$	−10506.0	−1.06142	−0.530708	−	0.847554i	$-0.678074\pi$
$461$	−10506.0	−1.06142	−0.530708	+	0.847554i	$0.678074\pi$
$462$	0	0
$463$	1562.00i	0.156787i	0.996923	+	0.0783934i	$0.0249790\pi$
$463$	1562.00i	0.156787i	−0.996923	+	0.0783934i	$0.975021\pi$
$464$	384.000	0.0384197
$465$	0	0
$466$	0	0
$467$	− 6.00000i	0 0.000594533i	−1.00000		0.000297266i	$-0.999905\pi$
$467$	− 6.00000i	0 0.000594533i	1.00000		0.000297266i	$-9.46229e-5\pi$
$468$	704.000i	0.0695351i
$469$	−13816.0	−1.36026
$470$	0	0
$471$	7352.00	0.719241
$472$	0	0
$473$	− 2184.00i	− 0.212305i
$474$	0	0
$475$	0	0
$476$	−11616.0	−1.11853
$477$	4356.00i	0.418129i
$478$	0	0
$479$	−3132.00	−0.298757	−0.149379	−	0.988780i	$-0.547727\pi$
$479$	−3132.00	−0.298757	−0.149379	+	0.988780i	$0.547727\pi$
$480$	0	0
$481$	−128.000	−0.0121337
$482$	0	0
$483$	2640.00i	0.248704i
$484$	−9496.00	−0.891811
$485$	0	0
$486$	0	0
$487$	12436.0i	1.15714i	0.815631	+	0.578572i	$0.196390\pi$
$487$	12436.0i	1.15714i	−0.815631	+	0.578572i	$0.803610\pi$
$488$	0	0
$489$	3400.00	0.314424
$490$	0	0
$491$	−7848.00	−0.721335	−0.360667	−	0.932695i	$-0.617451\pi$
$491$	−7848.00	−0.721335	−0.360667	+	0.932695i	$0.617451\pi$
$492$	1728.00i	0.158342i
$493$	396.000i	0.0361764i
$494$	0	0
$495$	0	0
$496$	−4096.00	−0.370798
$497$	− 21648.0i	− 1.95381i
$498$	0	0
$499$	−17720.0	−1.58969	−0.794846	−	0.606811i	$-0.792448\pi$
$499$	−17720.0	−1.58969	−0.794846	+	0.606811i	$0.792448\pi$
$500$	0	0
$501$	15216.0	1.35689
$502$	0	0
$503$	− 5094.00i	− 0.451551i	−0.974179	−	0.225776i	$-0.927508\pi$
$503$	− 5094.00i	− 0.451551i	0.974179	−	0.225776i	$-0.0724916\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8532.00i	0.747376i
$508$	− 2752.00i	− 0.240355i
$509$	5670.00	0.493749	0.246875	−	0.969047i	$-0.420596\pi$
$509$	5670.00	0.493749	0.246875	+	0.969047i	$0.420596\pi$
$510$	0	0
$511$	−308.000	−0.0266636
$512$	0	0
$513$	− 2888.00i	− 0.248554i
$514$	0	0
$515$	0	0
$516$	−5824.00	−0.496875
$517$	7128.00i	0.606362i
$518$	0	0
$519$	2256.00	0.190804
$520$	0	0
$521$	−20670.0	−1.73814	−0.869068	−	0.494692i	$-0.835281\pi$
$521$	−20670.0	−1.73814	−0.869068	+	0.494692i	$0.835281\pi$
$522$	0	0
$523$	− 16816.0i	− 1.40595i	−0.711214	−	0.702975i	$-0.751854\pi$
$523$	− 16816.0i	− 1.40595i	0.711214	−	0.702975i	$-0.248146\pi$
$524$	20160.0	1.68071
$525$	0	0
$526$	0	0
$527$	− 4224.00i	− 0.349147i
$528$	− 3072.00i	− 0.253204i
$529$	11267.0	0.926029
$530$	0	0
$531$	6204.00	0.507026
$532$	− 3344.00i	− 0.272520i
$533$	432.000i	0.0351069i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7248.00i	0.582447i
$538$	0	0
$539$	1692.00	0.135213
$540$	0	0
$541$	9530.00	0.757351	0.378675	−	0.925530i	$-0.376380\pi$
$541$	9530.00	0.757351	0.378675	+	0.925530i	$0.376380\pi$
$542$	0	0
$543$	− 17992.0i	− 1.42193i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5264.00i	− 0.411467i	−0.978608	−	0.205733i	$-0.934042\pi$
$547$	− 5264.00i	− 0.411467i	0.978608	−	0.205733i	$-0.0659580\pi$
$548$	− 5232.00i	− 0.407847i
$549$	−7766.00	−0.603725
$550$	0	0
$551$	−114.000	−0.00881409
$552$	0	0
$553$	7216.00i	0.554892i
$554$	0	0
$555$	0	0
$556$	19136.0	1.45962
$557$	− 16542.0i	− 1.25836i	−0.777259	−	0.629180i	$-0.783391\pi$
$557$	− 16542.0i	− 1.25836i	0.777259	−	0.629180i	$-0.216609\pi$
$558$	0	0
$559$	−1456.00	−0.110165
$560$	0	0
$561$	3168.00	0.238419
$562$	0	0
$563$	− 5232.00i	− 0.391656i	−0.980638	−	0.195828i	$-0.937261\pi$
$563$	− 5232.00i	− 0.391656i	0.980638	−	0.195828i	$-0.0627395\pi$
$564$	19008.0	1.41912
$565$	0	0
$566$	0	0
$567$	− 6842.00i	− 0.506767i
$568$	0	0
$569$	−15114.0	−1.11355	−0.556777	−	0.830662i	$-0.687962\pi$
$569$	−15114.0	−1.11355	−0.556777	+	0.830662i	$0.687962\pi$
$570$	0	0
$571$	−11764.0	−0.862186	−0.431093	−	0.902308i	$-0.641872\pi$
$571$	−11764.0	−0.862186	−0.431093	+	0.902308i	$0.641872\pi$
$572$	− 768.000i	− 0.0561393i
$573$	14352.0i	1.04636i
$574$	0	0
$575$	0	0
$576$	5632.00	0.407407
$577$	25198.0i	1.81804i	0.416757	+	0.909018i	$0.363166\pi$
$577$	25198.0i	1.81804i	−0.416757	+	0.909018i	$0.636834\pi$
$578$	0	0
$579$	17968.0	1.28968
$580$	0	0
$581$	6468.00	0.461855
$582$	0	0
$583$	− 4752.00i	− 0.337578i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 20190.0i	− 1.41964i	−0.704382	−	0.709822i	$-0.748776\pi$
$587$	− 20190.0i	− 1.41964i	0.704382	−	0.709822i	$-0.251224\pi$
$588$	− 4512.00i	− 0.316449i
$589$	1216.00	0.0850669
$590$	0	0
$591$	9864.00	0.686549
$592$	1024.00i	0.0710915i
$593$	− 2886.00i	− 0.199855i	−0.994995	−	0.0999273i	$-0.968139\pi$
$593$	− 2886.00i	− 0.199855i	0.994995	−	0.0999273i	$-0.0318610\pi$
$594$	0	0
$595$	0	0
$596$	10128.0	0.696072
$597$	− 3296.00i	− 0.225957i
$598$	0	0
$599$	4464.00	0.304498	0.152249	−	0.988342i	$-0.451348\pi$
$599$	4464.00	0.304498	0.152249	+	0.988342i	$0.451348\pi$
$600$	0	0
$601$	−15874.0	−1.07739	−0.538697	−	0.842499i	$-0.681083\pi$
$601$	−15874.0	−1.07739	−0.538697	+	0.842499i	$0.681083\pi$
$602$	0	0
$603$	6908.00i	0.466527i
$604$	24640.0	1.65991
$605$	0	0
$606$	0	0
$607$	18916.0i	1.26487i	0.774613	+	0.632436i	$0.217945\pi$
$607$	18916.0i	1.26487i	−0.774613	+	0.632436i	$0.782055\pi$
$608$	0	0
$609$	−528.000	−0.0351324
$610$	0	0
$611$	4752.00	0.314640
$612$	5808.00i	0.383618i
$613$	− 3058.00i	− 0.201487i	−0.994912	−	0.100743i	$-0.967878\pi$
$613$	− 3058.00i	− 0.201487i	0.994912	−	0.100743i	$-0.0321221\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4158.00i	− 0.271304i	−0.990757	−	0.135652i	$-0.956687\pi$
$617$	− 4158.00i	− 0.271304i	0.990757	−	0.135652i	$-0.0433130\pi$
$618$	0	0
$619$	22864.0	1.48462	0.742312	−	0.670055i	$-0.233729\pi$
$619$	22864.0	1.48462	0.742312	+	0.670055i	$0.233729\pi$
$620$	0	0
$621$	4560.00	0.294664
$622$	0	0
$623$	− 20196.0i	− 1.29877i
$624$	−2048.00	−0.131387
$625$	0	0
$626$	0	0
$627$	912.000i	0.0580890i
$628$	− 14704.0i	− 0.934321i
$629$	−1056.00	−0.0669403
$630$	0	0
$631$	18536.0	1.16942	0.584712	−	0.811241i	$-0.301208\pi$
$631$	18536.0	1.16942	0.584712	+	0.811241i	$0.301208\pi$
$632$	0	0
$633$	16976.0i	1.06593i
$634$	0	0
$635$	0	0
$636$	−12672.0	−0.790059
$637$	− 1128.00i	− 0.0701617i
$638$	0	0
$639$	−10824.0	−0.670095
$640$	0	0
$641$	15630.0	0.963101	0.481551	−	0.876418i	$-0.340074\pi$
$641$	15630.0	0.963101	0.481551	+	0.876418i	$0.340074\pi$
$642$	0	0
$643$	− 27574.0i	− 1.69115i	−0.533853	−	0.845577i	$-0.679256\pi$
$643$	− 27574.0i	− 1.69115i	0.533853	−	0.845577i	$-0.320744\pi$
$644$	5280.00	0.323076
$645$	0	0
$646$	0	0
$647$	8826.00i	0.536300i	0.963377	+	0.268150i	$0.0864122\pi$
$647$	8826.00i	0.536300i	−0.963377	+	0.268150i	$0.913588\pi$
$648$	0	0
$649$	−6768.00	−0.409349
$650$	0	0
$651$	5632.00	0.339071
$652$	− 6800.00i	− 0.408449i
$653$	18678.0i	1.11934i	0.828717	+	0.559668i	$0.189071\pi$
$653$	18678.0i	1.11934i	−0.828717	+	0.559668i	$0.810929\pi$
$654$	0	0
$655$	0	0
$656$	3456.00	0.205692
$657$	154.000i	0.00914477i
$658$	0	0
$659$	−16980.0	−1.00371	−0.501857	−	0.864951i	$-0.667349\pi$
$659$	−16980.0	−1.00371	−0.501857	+	0.864951i	$0.667349\pi$
$660$	0	0
$661$	−9358.00	−0.550657	−0.275328	−	0.961350i	$-0.588787\pi$
$661$	−9358.00	−0.550657	−0.275328	+	0.961350i	$0.588787\pi$
$662$	0	0
$663$	− 2112.00i	− 0.123715i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 180.000i	− 0.0104492i
$668$	− 30432.0i	− 1.76265i
$669$	−21920.0	−1.26678
$670$	0	0
$671$	8472.00	0.487419
$672$	0	0
$673$	− 16120.0i	− 0.923299i	−0.887062	−	0.461650i	$-0.847258\pi$
$673$	− 16120.0i	− 0.923299i	0.887062	−	0.461650i	$-0.152742\pi$
$674$	0	0
$675$	0	0
$676$	17064.0	0.970869
$677$	− 27876.0i	− 1.58251i	−0.611484	−	0.791257i	$-0.709427\pi$
$677$	− 27876.0i	− 1.58251i	0.611484	−	0.791257i	$-0.290573\pi$
$678$	0	0
$679$	−34408.0	−1.94471
$680$	0	0
$681$	22176.0	1.24785
$682$	0	0
$683$	− 4872.00i	− 0.272946i	−0.990644	−	0.136473i	$-0.956423\pi$
$683$	− 4872.00i	− 0.272946i	0.990644	−	0.136473i	$-0.0435766\pi$
$684$	−1672.00	−0.0934657
$685$	0	0
$686$	0	0
$687$	− 11720.0i	− 0.650867i
$688$	11648.0i	0.645459i
$689$	−3168.00	−0.175169
$690$	0	0
$691$	13412.0	0.738374	0.369187	−	0.929355i	$-0.379636\pi$
$691$	13412.0	0.738374	0.369187	+	0.929355i	$0.379636\pi$
$692$	− 4512.00i	− 0.247862i
$693$	− 2904.00i	− 0.159183i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3564.00i	0.193682i
$698$	0	0
$699$	17592.0	0.951918
$700$	0	0
$701$	1926.00	0.103772	0.0518859	−	0.998653i	$-0.483477\pi$
$701$	1926.00	0.103772	0.0518859	+	0.998653i	$0.483477\pi$
$702$	0	0
$703$	− 304.000i	− 0.0163095i
$704$	−6144.00	−0.328921
$705$	0	0
$706$	0	0
$707$	− 6468.00i	− 0.344065i
$708$	18048.0i	0.958030i
$709$	−17534.0	−0.928777	−0.464389	−	0.885631i	$-0.653726\pi$
$709$	−17534.0	−0.928777	−0.464389	+	0.885631i	$0.653726\pi$
$710$	0	0
$711$	3608.00	0.190310
$712$	0	0
$713$	1920.00i	0.100848i
$714$	0	0
$715$	0	0
$716$	14496.0	0.756621
$717$	25920.0i	1.35007i
$718$	0	0
$719$	11220.0	0.581969	0.290984	−	0.956728i	$-0.406017\pi$
$719$	11220.0	0.581969	0.290984	+	0.956728i	$0.406017\pi$
$720$	0	0
$721$	−16544.0	−0.854550
$722$	0	0
$723$	− 23080.0i	− 1.18721i
$724$	−35984.0	−1.84715
$725$	0	0
$726$	0	0
$727$	25078.0i	1.27936i	0.768643	+	0.639678i	$0.220932\pi$
$727$	25078.0i	1.27936i	−0.768643	+	0.639678i	$0.779068\pi$
$728$	0	0
$729$	−19837.0	−1.00782
$730$	0	0
$731$	−12012.0	−0.607770
$732$	− 22592.0i	− 1.14074i
$733$	434.000i	0.0218692i	0.999940	+	0.0109346i	$0.00348067\pi$
$733$	434.000i	0.0218692i	−0.999940	+	0.0109346i	$0.996519\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7536.00i	− 0.376651i
$738$	0	0
$739$	−2564.00	−0.127630	−0.0638148	−	0.997962i	$-0.520327\pi$
$739$	−2564.00	−0.127630	−0.0638148	+	0.997962i	$0.520327\pi$
$740$	0	0
$741$	608.000	0.0301423
$742$	0	0
$743$	21948.0i	1.08371i	0.840473	+	0.541853i	$0.182277\pi$
$743$	21948.0i	1.08371i	−0.840473	+	0.541853i	$0.817723\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3234.00i	− 0.158401i
$748$	− 6336.00i	− 0.309715i
$749$	−4752.00	−0.231821
$750$	0	0
$751$	−7648.00	−0.371610	−0.185805	−	0.982587i	$-0.559489\pi$
$751$	−7648.00	−0.371610	−0.185805	+	0.982587i	$0.559489\pi$
$752$	− 38016.0i	− 1.84349i
$753$	− 2496.00i	− 0.120796i
$754$	0	0
$755$	0	0
$756$	−26752.0	−1.28699
$757$	30190.0i	1.44950i	0.689010	+	0.724752i	$0.258046\pi$
$757$	30190.0i	1.44950i	−0.689010	+	0.724752i	$0.741954\pi$
$758$	0	0
$759$	−1440.00	−0.0688652
$760$	0	0
$761$	−1242.00	−0.0591622	−0.0295811	−	0.999562i	$-0.509417\pi$
$761$	−1242.00	−0.0591622	−0.0295811	+	0.999562i	$0.509417\pi$
$762$	0	0
$763$	16588.0i	0.787059i
$764$	28704.0	1.35926
$765$	0	0
$766$	0	0
$767$	4512.00i	0.212411i
$768$	16384.0i	0.769800i
$769$	28738.0	1.34762	0.673809	−	0.738905i	$-0.264657\pi$
$769$	28738.0	1.34762	0.673809	+	0.738905i	$0.264657\pi$
$770$	0	0
$771$	480.000	0.0224212
$772$	− 35936.0i	− 1.67534i
$773$	− 40128.0i	− 1.86715i	−0.358387	−	0.933573i	$-0.616673\pi$
$773$	− 40128.0i	− 1.86715i	0.358387	−	0.933573i	$-0.383327\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 1408.00i	− 0.0650086i
$778$	0	0
$779$	−1026.00	−0.0471890
$780$	0	0
$781$	11808.0	0.541003
$782$	0	0
$783$	912.000i	0.0416248i
$784$	−9024.00	−0.411079
$785$	0	0
$786$	0	0
$787$	15448.0i	0.699697i	0.936806	+	0.349849i	$0.113767\pi$
$787$	15448.0i	0.699697i	−0.936806	+	0.349849i	$0.886233\pi$
$788$	− 19728.0i	− 0.891854i
$789$	−7368.00	−0.332456
$790$	0	0
$791$	264.000	0.0118670
$792$	0	0
$793$	− 5648.00i	− 0.252921i
$794$	0	0
$795$	0	0
$796$	−6592.00	−0.293527
$797$	27324.0i	1.21439i	0.794554	+	0.607193i	$0.207705\pi$
$797$	27324.0i	1.21439i	−0.794554	+	0.607193i	$0.792295\pi$
$798$	0	0
$799$	39204.0	1.73584
$800$	0	0
$801$	−10098.0	−0.445437
$802$	0	0
$803$	− 168.000i	− 0.00738305i
$804$	−20096.0	−0.881507
$805$	0	0
$806$	0	0
$807$	12936.0i	0.564274i
$808$	0	0
$809$	17766.0	0.772088	0.386044	−	0.922480i	$-0.373841\pi$
$809$	17766.0	0.772088	0.386044	+	0.922480i	$0.373841\pi$
$810$	0	0
$811$	−7396.00	−0.320233	−0.160116	−	0.987098i	$-0.551187\pi$
$811$	−7396.00	−0.320233	−0.160116	+	0.987098i	$0.551187\pi$
$812$	1056.00i	0.0456383i
$813$	18656.0i	0.804790i
$814$	0	0
$815$	0	0
$816$	−16896.0	−0.724851
$817$	− 3458.00i	− 0.148078i
$818$	0	0
$819$	−1936.00	−0.0825999
$820$	0	0
$821$	26898.0	1.14342	0.571709	−	0.820456i	$-0.306281\pi$
$821$	26898.0	1.14342	0.571709	+	0.820456i	$0.306281\pi$
$822$	0	0
$823$	− 24442.0i	− 1.03523i	−0.855614	−	0.517615i	$-0.826820\pi$
$823$	− 24442.0i	− 1.03523i	0.855614	−	0.517615i	$-0.173180\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13524.0i	− 0.568652i	−0.958728	−	0.284326i	$-0.908230\pi$
$827$	− 13524.0i	− 0.568652i	0.958728	−	0.284326i	$-0.0917699\pi$
$828$	− 2640.00i	− 0.110805i
$829$	7714.00	0.323183	0.161591	−	0.986858i	$-0.448337\pi$
$829$	7714.00	0.323183	0.161591	+	0.986858i	$0.448337\pi$
$830$	0	0
$831$	20888.0	0.871958
$832$	4096.00i	0.170677i
$833$	− 9306.00i	− 0.387075i
$834$	0	0
$835$	0	0
$836$	1824.00	0.0754598
$837$	− 9728.00i	− 0.401731i
$838$	0	0
$839$	16248.0	0.668586	0.334293	−	0.942469i	$-0.391503\pi$
$839$	16248.0	0.668586	0.334293	+	0.942469i	$0.391503\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	30264.0i	1.23647i
$844$	33952.0	1.38469
$845$	0	0
$846$	0	0
$847$	− 26114.0i	− 1.05937i
$848$	25344.0i	1.02632i
$849$	22456.0	0.907760
$850$	0	0
$851$	480.000	0.0193351
$852$	− 31488.0i	− 1.26615i
$853$	35498.0i	1.42489i	0.701730	+	0.712443i	$0.252411\pi$
$853$	35498.0i	1.42489i	−0.701730	+	0.712443i	$0.747589\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	40344.0i	1.60808i	0.594575	+	0.804040i	$0.297320\pi$
$857$	40344.0i	1.60808i	−0.594575	+	0.804040i	$0.702680\pi$
$858$	0	0
$859$	−31484.0	−1.25055	−0.625274	−	0.780406i	$-0.715013\pi$
$859$	−31484.0	−1.25055	−0.625274	+	0.780406i	$0.715013\pi$
$860$	0	0
$861$	−4752.00	−0.188093
$862$	0	0
$863$	− 28836.0i	− 1.13741i	−0.822540	−	0.568707i	$-0.807444\pi$
$863$	− 28836.0i	− 1.13741i	0.822540	−	0.568707i	$-0.192556\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2228.00i	0.0872743i
$868$	− 11264.0i	− 0.440467i
$869$	−3936.00	−0.153647
$870$	0	0
$871$	−5024.00	−0.195444
$872$	0	0
$873$	17204.0i	0.666973i
$874$	0	0
$875$	0	0
$876$	−448.000	−0.0172791
$877$	− 22796.0i	− 0.877727i	−0.898554	−	0.438863i	$-0.855381\pi$
$877$	− 22796.0i	− 0.877727i	0.898554	−	0.438863i	$-0.144619\pi$
$878$	0	0
$879$	−3696.00	−0.141824
$880$	0	0
$881$	−18822.0	−0.719784	−0.359892	−	0.932994i	$-0.617186\pi$
$881$	−18822.0	−0.719784	−0.359892	+	0.932994i	$0.617186\pi$
$882$	0	0
$883$	7526.00i	0.286829i	0.989663	+	0.143415i	$0.0458082\pi$
$883$	7526.00i	0.286829i	−0.989663	+	0.143415i	$0.954192\pi$
$884$	−4224.00	−0.160711
$885$	0	0
$886$	0	0
$887$	− 33816.0i	− 1.28008i	−0.768342	−	0.640040i	$-0.778918\pi$
$887$	− 33816.0i	− 1.28008i	0.768342	−	0.640040i	$-0.221082\pi$
$888$	0	0
$889$	7568.00	0.285515
$890$	0	0
$891$	3732.00	0.140322
$892$	43840.0i	1.64560i
$893$	11286.0i	0.422925i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	960.000i	0.0357341i
$898$	0	0
$899$	−384.000	−0.0142460
$900$	0	0
$901$	−26136.0	−0.966389
$902$	0	0
$903$	− 16016.0i	− 0.590232i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	33784.0i	1.23680i	0.785863	+	0.618401i	$0.212219\pi$
$907$	33784.0i	1.23680i	−0.785863	+	0.618401i	$0.787781\pi$
$908$	− 44352.0i	− 1.62101i
$909$	−3234.00	−0.118003
$910$	0	0
$911$	−15216.0	−0.553379	−0.276690	−	0.960959i	$-0.589237\pi$
$911$	−15216.0	−0.553379	−0.276690	+	0.960959i	$0.589237\pi$
$912$	− 4864.00i	− 0.176604i
$913$	3528.00i	0.127886i
$914$	0	0
$915$	0	0
$916$	−23440.0	−0.845502
$917$	55440.0i	1.99650i
$918$	0	0
$919$	−19760.0	−0.709273	−0.354637	−	0.935004i	$-0.615395\pi$
$919$	−19760.0	−0.709273	−0.354637	+	0.935004i	$0.615395\pi$
$920$	0	0
$921$	−17920.0	−0.641134
$922$	0	0
$923$	− 7872.00i	− 0.280726i
$924$	8448.00	0.300778
$925$	0	0
$926$	0	0
$927$	8272.00i	0.293083i
$928$	0	0
$929$	−16278.0	−0.574880	−0.287440	−	0.957799i	$-0.592804\pi$
$929$	−16278.0	−0.574880	−0.287440	+	0.957799i	$0.592804\pi$
$930$	0	0
$931$	2679.00	0.0943079
$932$	− 35184.0i	− 1.23658i
$933$	5088.00i	0.178536i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6994.00i	0.243846i	0.992540	+	0.121923i	$0.0389062\pi$
$937$	6994.00i	0.243846i	−0.992540	+	0.121923i	$0.961094\pi$
$938$	0	0
$939$	−5480.00	−0.190451
$940$	0	0
$941$	32502.0	1.12597	0.562983	−	0.826468i	$-0.309653\pi$
$941$	32502.0	1.12597	0.562983	+	0.826468i	$0.309653\pi$
$942$	0	0
$943$	− 1620.00i	− 0.0559432i
$944$	36096.0	1.24452
$945$	0	0
$946$	0	0
$947$	− 50358.0i	− 1.72800i	−0.503493	−	0.864000i	$-0.667952\pi$
$947$	− 50358.0i	− 1.72800i	0.503493	−	0.864000i	$-0.332048\pi$
$948$	10496.0i	0.359593i
$949$	−112.000	−0.00383106
$950$	0	0
$951$	14208.0	0.484465
$952$	0	0
$953$	39816.0i	1.35338i	0.736270	+	0.676688i	$0.236585\pi$
$953$	39816.0i	1.35338i	−0.736270	+	0.676688i	$0.763415\pi$
$954$	0	0
$955$	0	0
$956$	51840.0	1.75379
$957$	− 288.000i	− 0.00972802i
$958$	0	0
$959$	14388.0	0.484476
$960$	0	0
$961$	−25695.0	−0.862509
$962$	0	0
$963$	2376.00i	0.0795073i
$964$	−46160.0	−1.54223
$965$	0	0
$966$	0	0
$967$	− 590.000i	− 0.0196206i	−0.999952	−	0.00981030i	$-0.996877\pi$
$967$	− 590.000i	− 0.0196206i	0.999952	−	0.00981030i	$-0.00312277\pi$
$968$	0	0
$969$	5016.00	0.166292
$970$	0	0
$971$	26820.0	0.886400	0.443200	−	0.896423i	$-0.353843\pi$
$971$	26820.0	0.886400	0.443200	+	0.896423i	$0.353843\pi$
$972$	22880.0i	0.755017i
$973$	52624.0i	1.73386i
$974$	0	0
$975$	0	0
$976$	−45184.0	−1.48187
$977$	33312.0i	1.09083i	0.838165	+	0.545417i	$0.183629\pi$
$977$	33312.0i	1.09083i	−0.838165	+	0.545417i	$0.816371\pi$
$978$	0	0
$979$	11016.0	0.359625
$980$	0	0
$981$	8294.00	0.269936
$982$	0	0
$983$	612.000i	0.0198573i	0.999951	+	0.00992867i	$0.00316045\pi$
$983$	612.000i	0.0198573i	−0.999951	+	0.00992867i	$0.996840\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	52272.0i	1.68575i
$988$	− 1216.00i	− 0.0391560i
$989$	5460.00	0.175549
$990$	0	0
$991$	39416.0	1.26346	0.631731	−	0.775188i	$-0.282345\pi$
$991$	39416.0	1.26346	0.631731	+	0.775188i	$0.282345\pi$
$992$	0	0
$993$	18128.0i	0.579330i
$994$	0	0
$995$	0	0
$996$	9408.00	0.299301
$997$	− 36614.0i	− 1.16307i	−0.813523	−	0.581533i	$-0.802453\pi$
$997$	− 36614.0i	− 1.16307i	0.813523	−	0.581533i	$-0.197547\pi$
$998$	0	0
$999$	−2432.00	−0.0770221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.b.e.324.2		2
5.2	odd	4		95.4.a.a.1.1	✓	1
5.3	odd	4		475.4.a.d.1.1		1
5.4	even	2	inner	475.4.b.e.324.1		2
15.2	even	4		855.4.a.e.1.1		1
20.7	even	4		1520.4.a.b.1.1		1
95.37	even	4		1805.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.4.a.a.1.1	✓	1	5.2	odd	4
475.4.a.d.1.1		1	5.3	odd	4
475.4.b.e.324.1		2	5.4	even	2	inner
475.4.b.e.324.2		2	1.1	even	1	trivial
855.4.a.e.1.1		1	15.2	even	4
1520.4.a.b.1.1		1	20.7	even	4
1805.4.a.f.1.1		1	95.37	even	4

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{3} +8.00000 q^{4} +22.0000i q^{7} +11.0000 q^{9} -12.0000 q^{11} +32.0000i q^{12} +8.00000i q^{13} +64.0000 q^{16} +66.0000i q^{17} -19.0000 q^{19} -88.0000 q^{21} -30.0000i q^{23} +152.000i q^{27} +176.000i q^{28} +6.00000 q^{29} -64.0000 q^{31} -48.0000i q^{33} +88.0000 q^{36} +16.0000i q^{37} -32.0000 q^{39} +54.0000 q^{41} +182.000i q^{43} -96.0000 q^{44} -594.000i q^{47} +256.000i q^{48} -141.000 q^{49} -264.000 q^{51} +64.0000i q^{52} +396.000i q^{53} -76.0000i q^{57} +564.000 q^{59} -706.000 q^{61} +242.000i q^{63} +512.000 q^{64} +628.000i q^{67} +528.000i q^{68} +120.000 q^{69} -984.000 q^{71} +14.0000i q^{73} -152.000 q^{76} -264.000i q^{77} +328.000 q^{79} -311.000 q^{81} -294.000i q^{83} -704.000 q^{84} +24.0000i q^{87} -918.000 q^{89} -176.000 q^{91} -240.000i q^{92} -256.000i q^{93} +1564.00i q^{97} -132.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 16 q^{4} + 22 q^{9} - 24 q^{11} + 128 q^{16} - 38 q^{19} - 176 q^{21} + 12 q^{29} - 128 q^{31} + 176 q^{36} - 64 q^{39} + 108 q^{41} - 192 q^{44} - 282 q^{49} - 528 q^{51} + 1128 q^{59} - 1412 q^{61}+ \cdots - 264 q^{99}+O(q^{100}) \) 2 * q + 16 * q^4 + 22 * q^9 - 24 * q^11 + 128 * q^16 - 38 * q^19 - 176 * q^21 + 12 * q^29 - 128 * q^31 + 176 * q^36 - 64 * q^39 + 108 * q^41 - 192 * q^44 - 282 * q^49 - 528 * q^51 + 1128 * q^59 - 1412 * q^61 + 1024 * q^64 + 240 * q^69 - 1968 * q^71 - 304 * q^76 + 656 * q^79 - 622 * q^81 - 1408 * q^84 - 1836 * q^89 - 352 * q^91 - 264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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