LMFDB - Embedded newform 4160.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.a (trivial)

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:	yes
Analytic conductor:	$33.2177672409$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{9} +4.24264 q^{11} -1.00000 q^{13} -1.41421 q^{15} -2.00000 q^{17} -4.24264 q^{19} +1.41421 q^{23} +1.00000 q^{25} +5.65685 q^{27} -1.41421 q^{31} -6.00000 q^{33} -4.00000 q^{37} +1.41421 q^{39} -10.0000 q^{41} +9.89949 q^{43} -1.00000 q^{45} -11.3137 q^{47} -7.00000 q^{49} +2.82843 q^{51} +10.0000 q^{53} +4.24264 q^{55} +6.00000 q^{57} -1.41421 q^{59} +4.00000 q^{61} -1.00000 q^{65} -2.82843 q^{67} -2.00000 q^{69} +7.07107 q^{71} -4.00000 q^{73} -1.41421 q^{75} +8.48528 q^{79} -5.00000 q^{81} +5.65685 q^{83} -2.00000 q^{85} -18.0000 q^{89} +2.00000 q^{93} -4.24264 q^{95} -18.0000 q^{97} -4.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9} - 2 q^{13} - 4 q^{17} + 2 q^{25} - 12 q^{33} - 8 q^{37} - 20 q^{41} - 2 q^{45} - 14 q^{49} + 20 q^{53} + 12 q^{57} + 8 q^{61} - 2 q^{65} - 4 q^{69} - 8 q^{73} - 10 q^{81} - 4 q^{85}+ \cdots - 36 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^13 - 4 * q^17 + 2 * q^25 - 12 * q^33 - 8 * q^37 - 20 * q^41 - 2 * q^45 - 14 * q^49 + 20 * q^53 + 12 * q^57 + 8 * q^61 - 2 * q^65 - 4 * q^69 - 8 * q^73 - 10 * q^81 - 4 * q^85 - 36 * q^89 + 4 * q^93 - 36 * q^97

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
$2$	0	0
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.41421	−0.254000	−0.127000	−	0.991903i	$-0.540535\pi$
$31$	−1.41421	−0.254000	−0.127000	+	0.991903i	$0.540535\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	9.89949	1.50966	0.754829	−	0.655921i	$-0.227720\pi$
$43$	9.89949	1.50966	0.754829	+	0.655921i	$0.227720\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−11.3137	−1.65027	−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137	−1.65027	−0.825137	+	0.564933i	$0.808902\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	−1.41421	−0.184115	−0.0920575	−	0.995754i	$-0.529344\pi$
$59$	−1.41421	−0.184115	−0.0920575	+	0.995754i	$0.529344\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−2.82843	−0.345547	−0.172774	−	0.984962i	$-0.555273\pi$
$67$	−2.82843	−0.345547	−0.172774	+	0.984962i	$0.555273\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	7.07107	0.839181	0.419591	−	0.907713i	$-0.362174\pi$
$71$	7.07107	0.839181	0.419591	+	0.907713i	$0.362174\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−1.41421	−0.163299
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	5.65685	0.620920	0.310460	−	0.950586i	$-0.399517\pi$
$83$	5.65685	0.620920	0.310460	+	0.950586i	$0.399517\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	0	0
$95$	−4.24264	−0.435286
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	15.5563	1.53281	0.766406	−	0.642356i	$-0.222043\pi$
$103$	15.5563	1.53281	0.766406	+	0.642356i	$0.222043\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	5.65685	0.536925
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	1.41421	0.131876
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	14.1421	1.27515
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	18.3848	1.63139	0.815693	−	0.578486i	$-0.196356\pi$
$127$	18.3848	1.63139	0.815693	+	0.578486i	$0.196356\pi$
$128$	0	0
$129$	−14.0000	−1.23263
$130$	0	0
$131$	−11.3137	−0.988483	−0.494242	−	0.869325i	$-0.664554\pi$
$131$	−11.3137	−0.988483	−0.494242	+	0.869325i	$0.664554\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−8.48528	−0.719712	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	−8.48528	−0.719712	−0.359856	+	0.933008i	$0.617174\pi$
$140$	0	0
$141$	16.0000	1.34744
$142$	0	0
$143$	−4.24264	−0.354787
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.89949	0.816497
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−18.3848	−1.49613	−0.748066	−	0.663624i	$-0.769017\pi$
$151$	−18.3848	−1.49613	−0.748066	+	0.663624i	$0.769017\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−1.41421	−0.113592
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	−14.1421	−1.12154
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.48528	0.664619	0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528	0.664619	0.332309	+	0.943170i	$0.392172\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.24264	0.324443
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.00000	0.150329
$178$	0	0
$179$	−16.9706	−1.26844	−0.634220	−	0.773153i	$-0.718679\pi$
$179$	−16.9706	−1.26844	−0.634220	+	0.773153i	$0.718679\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	−5.65685	−0.418167
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−8.48528	−0.620505
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−11.3137	−0.818631	−0.409316	−	0.912393i	$-0.634232\pi$
$191$	−11.3137	−0.818631	−0.409316	+	0.912393i	$0.634232\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	1.41421	0.101274
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	−18.0000	−1.24509
$210$	0	0
$211$	−5.65685	−0.389434	−0.194717	−	0.980859i	$-0.562379\pi$
$211$	−5.65685	−0.389434	−0.194717	+	0.980859i	$0.562379\pi$
$212$	0	0
$213$	−10.0000	−0.685189
$214$	0	0
$215$	9.89949	0.675140
$216$	0	0
$217$	0	0
$218$	0	0
$219$	5.65685	0.382255
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−25.4558	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$227$	−25.4558	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−11.3137	−0.738025
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	7.07107	0.457389	0.228695	−	0.973498i	$-0.426554\pi$
$239$	7.07107	0.457389	0.228695	+	0.973498i	$0.426554\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−7.00000	−0.447214
$246$	0	0
$247$	4.24264	0.269953
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−25.4558	−1.60676	−0.803379	−	0.595468i	$-0.796967\pi$
$251$	−25.4558	−1.60676	−0.803379	+	0.595468i	$0.796967\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.8701	−1.65688	−0.828439	−	0.560079i	$-0.810771\pi$
$263$	−26.8701	−1.65688	−0.828439	+	0.560079i	$0.810771\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	25.4558	1.55787
$268$	0	0
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−29.6985	−1.80405	−0.902027	−	0.431679i	$-0.857921\pi$
$271$	−29.6985	−1.80405	−0.902027	+	0.431679i	$0.857921\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.24264	0.255841
$276$	0	0
$277$	−26.0000	−1.56219	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000	−1.56219	−0.781094	+	0.624413i	$0.785338\pi$
$278$	0	0
$279$	1.41421	0.0846668
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−1.41421	−0.0840663	−0.0420331	−	0.999116i	$-0.513384\pi$
$283$	−1.41421	−0.0840663	−0.0420331	+	0.999116i	$0.513384\pi$
$284$	0	0
$285$	6.00000	0.355409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	25.4558	1.49225
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−1.41421	−0.0823387
$296$	0	0
$297$	24.0000	1.39262
$298$	0	0
$299$	−1.41421	−0.0817861
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.1421	0.812444
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−22.0000	−1.25154
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.00000	−0.449325	−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000	−0.449325	−0.224662	+	0.974437i	$0.572128\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	10.0000	0.558146
$322$	0	0
$323$	8.48528	0.472134
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	2.82843	0.156412
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.24264	−0.233197	−0.116598	−	0.993179i	$-0.537199\pi$
$331$	−4.24264	−0.233197	−0.116598	+	0.993179i	$0.537199\pi$
$332$	0	0
$333$	4.00000	0.219199
$334$	0	0
$335$	−2.82843	−0.154533
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	8.48528	0.460857
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	1.41421	0.0759190	0.0379595	−	0.999279i	$-0.487914\pi$
$347$	1.41421	0.0759190	0.0379595	+	0.999279i	$0.487914\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	7.07107	0.375293
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.3848	0.970311	0.485156	−	0.874428i	$-0.338763\pi$
$359$	18.3848	0.970311	0.485156	+	0.874428i	$0.338763\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	−9.89949	−0.519589
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−29.6985	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$367$	−29.6985	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	0	0
$378$	0	0
$379$	38.1838	1.96137	0.980684	−	0.195598i	$-0.0626648\pi$
$379$	38.1838	1.96137	0.980684	+	0.195598i	$0.0626648\pi$
$380$	0	0
$381$	−26.0000	−1.33202
$382$	0	0
$383$	22.6274	1.15621	0.578103	−	0.815963i	$-0.303793\pi$
$383$	22.6274	1.15621	0.578103	+	0.815963i	$0.303793\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.89949	−0.503220
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	−2.82843	−0.143040
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	8.48528	0.426941
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	1.41421	0.0704470
$404$	0	0
$405$	−5.00000	−0.248452
$406$	0	0
$407$	−16.9706	−0.841200
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	−14.1421	−0.697580
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.65685	0.277684
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	31.1127	1.51995	0.759977	−	0.649950i	$-0.225210\pi$
$419$	31.1127	1.51995	0.759977	+	0.649950i	$0.225210\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	11.3137	0.550091
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−12.7279	−0.613082	−0.306541	−	0.951857i	$-0.599172\pi$
$431$	−12.7279	−0.613082	−0.306541	+	0.951857i	$0.599172\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−16.9706	−0.809961	−0.404980	−	0.914325i	$-0.632722\pi$
$439$	−16.9706	−0.809961	−0.404980	+	0.914325i	$0.632722\pi$
$440$	0	0
$441$	7.00000	0.333333
$442$	0	0
$443$	21.2132	1.00787	0.503935	−	0.863742i	$-0.331885\pi$
$443$	21.2132	1.00787	0.503935	+	0.863742i	$0.331885\pi$
$444$	0	0
$445$	−18.0000	−0.853282
$446$	0	0
$447$	−14.1421	−0.668900
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	−42.4264	−1.99778
$452$	0	0
$453$	26.0000	1.22159
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	0	0
$459$	−11.3137	−0.528079
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	−14.1421	−0.657241	−0.328620	−	0.944462i	$-0.606584\pi$
$463$	−14.1421	−0.657241	−0.328620	+	0.944462i	$0.606584\pi$
$464$	0	0
$465$	2.00000	0.0927478
$466$	0	0
$467$	−7.07107	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$467$	−7.07107	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	19.7990	0.912289
$472$	0	0
$473$	42.0000	1.93116
$474$	0	0
$475$	−4.24264	−0.194666
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	41.0122	1.87389	0.936947	−	0.349470i	$-0.113638\pi$
$479$	41.0122	1.87389	0.936947	+	0.349470i	$0.113638\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.0000	−0.817338
$486$	0	0
$487$	−25.4558	−1.15351	−0.576757	−	0.816916i	$-0.695682\pi$
$487$	−25.4558	−1.15351	−0.576757	+	0.816916i	$0.695682\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	19.7990	0.893516	0.446758	−	0.894655i	$-0.352579\pi$
$491$	19.7990	0.893516	0.446758	+	0.894655i	$0.352579\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.24264	−0.190693
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.5269	−1.45610	−0.728052	−	0.685522i	$-0.759574\pi$
$499$	−32.5269	−1.45610	−0.728052	+	0.685522i	$0.759574\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−18.3848	−0.819737	−0.409868	−	0.912145i	$-0.634425\pi$
$503$	−18.3848	−0.819737	−0.409868	+	0.912145i	$0.634425\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−1.41421	−0.0628074
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−24.0000	−1.05963
$514$	0	0
$515$	15.5563	0.685495
$516$	0	0
$517$	−48.0000	−2.11104
$518$	0	0
$519$	−8.48528	−0.372463
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	0	0
$523$	−7.07107	−0.309196	−0.154598	−	0.987977i	$-0.549408\pi$
$523$	−7.07107	−0.309196	−0.154598	+	0.987977i	$0.549408\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.82843	0.123208
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	1.41421	0.0613716
$532$	0	0
$533$	10.0000	0.433148
$534$	0	0
$535$	−7.07107	−0.305709
$536$	0	0
$537$	24.0000	1.03568
$538$	0	0
$539$	−29.6985	−1.27920
$540$	0	0
$541$	−14.0000	−0.601907	−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000	−0.601907	−0.300954	+	0.953639i	$0.597305\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−7.07107	−0.302337	−0.151169	−	0.988508i	$-0.548304\pi$
$547$	−7.07107	−0.302337	−0.151169	+	0.988508i	$0.548304\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	5.65685	0.240120
$556$	0	0
$557$	20.0000	0.847427	0.423714	−	0.905796i	$-0.360726\pi$
$557$	20.0000	0.847427	0.423714	+	0.905796i	$0.360726\pi$
$558$	0	0
$559$	−9.89949	−0.418704
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	−18.3848	−0.774826	−0.387413	−	0.921906i	$-0.626631\pi$
$563$	−18.3848	−0.774826	−0.387413	+	0.921906i	$0.626631\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	−2.82843	−0.118366	−0.0591830	−	0.998247i	$-0.518850\pi$
$571$	−2.82843	−0.118366	−0.0591830	+	0.998247i	$0.518850\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	1.41421	0.0589768
$576$	0	0
$577$	44.0000	1.83174	0.915872	−	0.401470i	$-0.131501\pi$
$577$	44.0000	1.83174	0.915872	+	0.401470i	$0.131501\pi$
$578$	0	0
$579$	−2.82843	−0.117545
$580$	0	0
$581$	0	0
$582$	0	0
$583$	42.4264	1.75712
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−14.1421	−0.583708	−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421	−0.583708	−0.291854	+	0.956463i	$0.594272\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	8.48528	0.349038
$592$	0	0
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−2.82843	−0.115566	−0.0577832	−	0.998329i	$-0.518403\pi$
$599$	−2.82843	−0.115566	−0.0577832	+	0.998329i	$0.518403\pi$
$600$	0	0
$601$	14.0000	0.571072	0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000	0.571072	0.285536	+	0.958368i	$0.407828\pi$
$602$	0	0
$603$	2.82843	0.115182
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	29.6985	1.20542	0.602712	−	0.797959i	$-0.294087\pi$
$607$	29.6985	1.20542	0.602712	+	0.797959i	$0.294087\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.3137	0.457704
$612$	0	0
$613$	−26.0000	−1.05013	−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000	−1.05013	−0.525065	+	0.851062i	$0.675959\pi$
$614$	0	0
$615$	14.1421	0.570266
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	0	0
$619$	15.5563	0.625262	0.312631	−	0.949875i	$-0.398790\pi$
$619$	15.5563	0.625262	0.312631	+	0.949875i	$0.398790\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.4558	1.01661
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	29.6985	1.18228	0.591139	−	0.806570i	$-0.298678\pi$
$631$	29.6985	1.18228	0.591139	+	0.806570i	$0.298678\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	18.3848	0.729578
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	−7.07107	−0.279727
$640$	0	0
$641$	14.0000	0.552967	0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000	0.552967	0.276483	+	0.961019i	$0.410831\pi$
$642$	0	0
$643$	−39.5980	−1.56159	−0.780796	−	0.624786i	$-0.785186\pi$
$643$	−39.5980	−1.56159	−0.780796	+	0.624786i	$0.785186\pi$
$644$	0	0
$645$	−14.0000	−0.551249
$646$	0	0
$647$	35.3553	1.38996	0.694981	−	0.719028i	$-0.255413\pi$
$647$	35.3553	1.38996	0.694981	+	0.719028i	$0.255413\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	2.82843	0.110180	0.0550899	−	0.998481i	$-0.482455\pi$
$659$	2.82843	0.110180	0.0550899	+	0.998481i	$0.482455\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	−2.82843	−0.109847
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.9706	0.655141
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	0	0
$675$	5.65685	0.217732
$676$	0	0
$677$	2.00000	0.0768662	0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000	0.0768662	0.0384331	+	0.999261i	$0.487763\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	36.0000	1.37952
$682$	0	0
$683$	−2.82843	−0.108227	−0.0541134	−	0.998535i	$-0.517233\pi$
$683$	−2.82843	−0.108227	−0.0541134	+	0.998535i	$0.517233\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	0	0
$687$	8.48528	0.323734
$688$	0	0
$689$	−10.0000	−0.380970
$690$	0	0
$691$	41.0122	1.56018	0.780089	−	0.625669i	$-0.215174\pi$
$691$	41.0122	1.56018	0.780089	+	0.625669i	$0.215174\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.48528	−0.321865
$696$	0	0
$697$	20.0000	0.757554
$698$	0	0
$699$	8.48528	0.320943
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	16.9706	0.640057
$704$	0	0
$705$	16.0000	0.602595
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−30.0000	−1.12667	−0.563337	−	0.826227i	$-0.690483\pi$
$709$	−30.0000	−1.12667	−0.563337	+	0.826227i	$0.690483\pi$
$710$	0	0
$711$	−8.48528	−0.318223
$712$	0	0
$713$	−2.00000	−0.0749006
$714$	0	0
$715$	−4.24264	−0.158666
$716$	0	0
$717$	−10.0000	−0.373457
$718$	0	0
$719$	−39.5980	−1.47676	−0.738378	−	0.674387i	$-0.764408\pi$
$719$	−39.5980	−1.47676	−0.738378	+	0.674387i	$0.764408\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	14.1421	0.525952
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.8701	0.996555	0.498278	−	0.867018i	$-0.333966\pi$
$727$	26.8701	0.996555	0.498278	+	0.867018i	$0.333966\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−19.7990	−0.732292
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	9.89949	0.365148
$736$	0	0
$737$	−12.0000	−0.442026
$738$	0	0
$739$	−4.24264	−0.156068	−0.0780340	−	0.996951i	$-0.524864\pi$
$739$	−4.24264	−0.156068	−0.0780340	+	0.996951i	$0.524864\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−5.65685	−0.207530	−0.103765	−	0.994602i	$-0.533089\pi$
$743$	−5.65685	−0.207530	−0.103765	+	0.994602i	$0.533089\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	0	0
$747$	−5.65685	−0.206973
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25.4558	−0.928897	−0.464448	−	0.885600i	$-0.653747\pi$
$751$	−25.4558	−0.928897	−0.464448	+	0.885600i	$0.653747\pi$
$752$	0	0
$753$	36.0000	1.31191
$754$	0	0
$755$	−18.3848	−0.669091
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	−8.48528	−0.307996
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	1.41421	0.0510643
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−8.48528	−0.305590
$772$	0	0
$773$	28.0000	1.00709	0.503545	−	0.863969i	$-0.332029\pi$
$773$	28.0000	1.00709	0.503545	+	0.863969i	$0.332029\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	0	0
$777$	0	0
$778$	0	0
$779$	42.4264	1.52008
$780$	0	0
$781$	30.0000	1.07348
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14.0000	−0.499681
$786$	0	0
$787$	39.5980	1.41152	0.705758	−	0.708453i	$-0.250607\pi$
$787$	39.5980	1.41152	0.705758	+	0.708453i	$0.250607\pi$
$788$	0	0
$789$	38.0000	1.35284
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−4.00000	−0.142044
$794$	0	0
$795$	−14.1421	−0.501570
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	22.6274	0.800500
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	−16.9706	−0.598878
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−25.4558	−0.896088
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	−4.24264	−0.148979	−0.0744896	−	0.997222i	$-0.523733\pi$
$811$	−4.24264	−0.148979	−0.0744896	+	0.997222i	$0.523733\pi$
$812$	0	0
$813$	42.0000	1.47300
$814$	0	0
$815$	8.48528	0.297226
$816$	0	0
$817$	−42.0000	−1.46939
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	21.2132	0.739446	0.369723	−	0.929142i	$-0.379453\pi$
$823$	21.2132	0.739446	0.369723	+	0.929142i	$0.379453\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	0	0
$827$	−16.9706	−0.590124	−0.295062	−	0.955478i	$-0.595340\pi$
$827$	−16.9706	−0.590124	−0.295062	+	0.955478i	$0.595340\pi$
$828$	0	0
$829$	−28.0000	−0.972480	−0.486240	−	0.873825i	$-0.661632\pi$
$829$	−28.0000	−0.972480	−0.486240	+	0.873825i	$0.661632\pi$
$830$	0	0
$831$	36.7696	1.27552
$832$	0	0
$833$	14.0000	0.485071
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.00000	−0.276520
$838$	0	0
$839$	−52.3259	−1.80649	−0.903245	−	0.429124i	$-0.858822\pi$
$839$	−52.3259	−1.80649	−0.903245	+	0.429124i	$0.858822\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	14.1421	0.487081
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.00000	0.0686398
$850$	0	0
$851$	−5.65685	−0.193914
$852$	0	0
$853$	16.0000	0.547830	0.273915	−	0.961754i	$-0.411681\pi$
$853$	16.0000	0.547830	0.273915	+	0.961754i	$0.411681\pi$
$854$	0	0
$855$	4.24264	0.145095
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	53.7401	1.83359	0.916795	−	0.399359i	$-0.130767\pi$
$859$	53.7401	1.83359	0.916795	+	0.399359i	$0.130767\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.2843	0.962808	0.481404	−	0.876499i	$-0.340127\pi$
$863$	28.2843	0.962808	0.481404	+	0.876499i	$0.340127\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	18.3848	0.624380
$868$	0	0
$869$	36.0000	1.22122
$870$	0	0
$871$	2.82843	0.0958376
$872$	0	0
$873$	18.0000	0.609208
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.0000	0.943344	0.471672	−	0.881774i	$-0.343651\pi$
$881$	28.0000	0.943344	0.471672	+	0.881774i	$0.343651\pi$
$882$	0	0
$883$	26.8701	0.904249	0.452125	−	0.891955i	$-0.350666\pi$
$883$	26.8701	0.904249	0.452125	+	0.891955i	$0.350666\pi$
$884$	0	0
$885$	2.00000	0.0672293
$886$	0	0
$887$	32.5269	1.09215	0.546073	−	0.837737i	$-0.316122\pi$
$887$	32.5269	1.09215	0.546073	+	0.837737i	$0.316122\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−21.2132	−0.710669
$892$	0	0
$893$	48.0000	1.60626
$894$	0	0
$895$	−16.9706	−0.567263
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−20.0000	−0.666297
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29.6985	−0.986122	−0.493061	−	0.869995i	$-0.664122\pi$
$907$	−29.6985	−0.986122	−0.493061	+	0.869995i	$0.664122\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	16.9706	0.562260	0.281130	−	0.959670i	$-0.409291\pi$
$911$	16.9706	0.562260	0.281130	+	0.959670i	$0.409291\pi$
$912$	0	0
$913$	24.0000	0.794284
$914$	0	0
$915$	−5.65685	−0.187010
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.7990	−0.653108	−0.326554	−	0.945178i	$-0.605888\pi$
$919$	−19.7990	−0.653108	−0.326554	+	0.945178i	$0.605888\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.07107	−0.232747
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	−15.5563	−0.510938
$928$	0	0
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	29.6985	0.973329
$932$	0	0
$933$	−12.0000	−0.392862
$934$	0	0
$935$	−8.48528	−0.277498
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	−19.7990	−0.646116
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	−14.1421	−0.460531
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45.2548	1.47058	0.735292	−	0.677750i	$-0.237045\pi$
$947$	45.2548	1.47058	0.735292	+	0.677750i	$0.237045\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	11.3137	0.366872
$952$	0	0
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	0	0
$955$	−11.3137	−0.366103
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	7.07107	0.227862
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	48.0833	1.54625	0.773127	−	0.634252i	$-0.218692\pi$
$967$	48.0833	1.54625	0.773127	+	0.634252i	$0.218692\pi$
$968$	0	0
$969$	−12.0000	−0.385496
$970$	0	0
$971$	45.2548	1.45230	0.726148	−	0.687538i	$-0.241309\pi$
$971$	45.2548	1.45230	0.726148	+	0.687538i	$0.241309\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.41421	0.0452911
$976$	0	0
$977$	−20.0000	−0.639857	−0.319928	−	0.947442i	$-0.603659\pi$
$977$	−20.0000	−0.639857	−0.319928	+	0.947442i	$0.603659\pi$
$978$	0	0
$979$	−76.3675	−2.44072
$980$	0	0
$981$	2.00000	0.0638551
$982$	0	0
$983$	31.1127	0.992341	0.496170	−	0.868225i	$-0.334739\pi$
$983$	31.1127	0.992341	0.496170	+	0.868225i	$0.334739\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	14.0000	0.445174
$990$	0	0
$991$	−11.3137	−0.359392	−0.179696	−	0.983722i	$-0.557511\pi$
$991$	−11.3137	−0.359392	−0.179696	+	0.983722i	$0.557511\pi$
$992$	0	0
$993$	6.00000	0.190404
$994$	0	0
$995$	0	0
$996$	0	0
$997$	58.0000	1.83688	0.918439	−	0.395562i	$-0.129450\pi$
$997$	58.0000	1.83688	0.918439	+	0.395562i	$0.129450\pi$
$998$	0	0
$999$	−22.6274	−0.715900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bi.1.1		2
4.3	odd	2	inner	4160.2.a.bi.1.2		2
8.3	odd	2		2080.2.a.g.1.1	✓	2
8.5	even	2		2080.2.a.g.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.g.1.1	✓	2	8.3	odd	2
2080.2.a.g.1.2	yes	2	8.5	even	2
4160.2.a.bi.1.1		2	1.1	even	1	trivial
4160.2.a.bi.1.2		2	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} +1.00000 q^{5} -1.00000 q^{9} +4.24264 q^{11} -1.00000 q^{13} -1.41421 q^{15} -2.00000 q^{17} -4.24264 q^{19} +1.41421 q^{23} +1.00000 q^{25} +5.65685 q^{27} -1.41421 q^{31} -6.00000 q^{33} -4.00000 q^{37} +1.41421 q^{39} -10.0000 q^{41} +9.89949 q^{43} -1.00000 q^{45} -11.3137 q^{47} -7.00000 q^{49} +2.82843 q^{51} +10.0000 q^{53} +4.24264 q^{55} +6.00000 q^{57} -1.41421 q^{59} +4.00000 q^{61} -1.00000 q^{65} -2.82843 q^{67} -2.00000 q^{69} +7.07107 q^{71} -4.00000 q^{73} -1.41421 q^{75} +8.48528 q^{79} -5.00000 q^{81} +5.65685 q^{83} -2.00000 q^{85} -18.0000 q^{89} +2.00000 q^{93} -4.24264 q^{95} -18.0000 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9} - 2 q^{13} - 4 q^{17} + 2 q^{25} - 12 q^{33} - 8 q^{37} - 20 q^{41} - 2 q^{45} - 14 q^{49} + 20 q^{53} + 12 q^{57} + 8 q^{61} - 2 q^{65} - 4 q^{69} - 8 q^{73} - 10 q^{81} - 4 q^{85}+ \cdots - 36 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^13 - 4 * q^17 + 2 * q^25 - 12 * q^33 - 8 * q^37 - 20 * q^41 - 2 * q^45 - 14 * q^49 + 20 * q^53 + 12 * q^57 + 8 * q^61 - 2 * q^65 - 4 * q^69 - 8 * q^73 - 10 * q^81 - 4 * q^85 - 36 * q^89 + 4 * q^93 - 36 * q^97

Introduction

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