LMFDB - Embedded newform 465.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [465,2,Mod(1,465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.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:	$3.71304369399$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{14} -1.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -4.00000 q^{22} -8.00000 q^{23} +3.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} +5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} +8.00000 q^{37} -8.00000 q^{38} -3.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +4.00000 q^{44} +1.00000 q^{45} -8.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} +6.00000 q^{56} +8.00000 q^{57} +10.0000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +1.00000 q^{62} -2.00000 q^{63} +7.00000 q^{64} +4.00000 q^{66} +2.00000 q^{67} -2.00000 q^{68} +8.00000 q^{69} -2.00000 q^{70} +6.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +8.00000 q^{74} -1.00000 q^{75} +8.00000 q^{76} +8.00000 q^{77} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -2.00000 q^{84} +2.00000 q^{85} +12.0000 q^{88} +4.00000 q^{89} +1.00000 q^{90} +8.00000 q^{92} -1.00000 q^{93} +4.00000 q^{94} -8.00000 q^{95} -5.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

Coefficient data

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

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

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

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−2.00000	−0.534522
$15$	−1.00000	−0.258199
$16$	−1.00000	−0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	1.00000	0.235702
$19$	−8.00000	−1.83533	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000	−1.83533	−0.917663	+	0.397360i	$0.869927\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.00000	0.883883
$33$	4.00000	0.696311
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	−1.00000	−0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−8.00000	−1.17954
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	6.00000	0.801784
$57$	8.00000	1.05963
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	1.00000	0.127000
$63$	−2.00000	−0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−2.00000	−0.242536
$69$	8.00000	0.963087
$70$	−2.00000	−0.239046
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−3.00000	−0.353553
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	8.00000	0.929981
$75$	−1.00000	−0.115470
$76$	8.00000	0.917663
$77$	8.00000	0.911685
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	−2.00000	−0.218218
$85$	2.00000	0.216930
$86$	0	0
$87$	0	0
$88$	12.0000	1.27920
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	8.00000	0.834058
$93$	−1.00000	−0.103695
$94$	4.00000	0.412568
$95$	−8.00000	−0.820783
$96$	−5.00000	−0.510310
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	−3.00000	−0.303046
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	−2.00000	−0.198030
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	6.00000	0.582772
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	1.00000	0.0962250
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−4.00000	−0.381385
$111$	−8.00000	−0.759326
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	8.00000	0.749269
$115$	−8.00000	−0.746004
$116$	0	0
$117$	0	0
$118$	10.0000	0.920575
$119$	−4.00000	−0.366679
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	−14.0000	−1.26750
$123$	6.00000	0.541002
$124$	−1.00000	−0.0898027
$125$	1.00000	0.0894427
$126$	−2.00000	−0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0	0
$131$	−14.0000	−1.22319	−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000	−1.22319	−0.611593	+	0.791173i	$0.709471\pi$
$132$	−4.00000	−0.348155
$133$	16.0000	1.38738
$134$	2.00000	0.172774
$135$	−1.00000	−0.0860663
$136$	−6.00000	−0.514496
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	8.00000	0.681005
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	2.00000	0.169031
$141$	−4.00000	−0.336861
$142$	6.00000	0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	3.00000	0.247436
$148$	−8.00000	−0.657596
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	−1.00000	−0.0816497
$151$	−24.0000	−1.95309	−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000	−1.95309	−0.976546	+	0.215308i	$0.930924\pi$
$152$	24.0000	1.94666
$153$	2.00000	0.161690
$154$	8.00000	0.644658
$155$	1.00000	0.0803219
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	5.00000	0.395285
$161$	16.0000	1.26098
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	6.00000	0.468521
$165$	4.00000	0.311400
$166$	4.00000	0.310460
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	−6.00000	−0.462910
$169$	−13.0000	−1.00000
$170$	2.00000	0.153393
$171$	−8.00000	−0.611775
$172$	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
$175$	−2.00000	−0.151186
$176$	4.00000	0.301511
$177$	−10.0000	−0.751646
$178$	4.00000	0.299813
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	24.0000	1.76930
$185$	8.00000	0.588172
$186$	−1.00000	−0.0733236
$187$	−8.00000	−0.585018
$188$	−4.00000	−0.291730
$189$	2.00000	0.145479
$190$	−8.00000	−0.580381
$191$	−14.0000	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$191$	−14.0000	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$192$	−7.00000	−0.505181
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	3.00000	0.214286
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−4.00000	−0.284268
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	−3.00000	−0.212132
$201$	−2.00000	−0.141069
$202$	14.0000	0.985037
$203$	0	0
$204$	2.00000	0.140028
$205$	−6.00000	−0.419058
$206$	2.00000	0.139347
$207$	−8.00000	−0.556038
$208$	0	0
$209$	32.0000	2.21349
$210$	2.00000	0.138013
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	16.0000	1.09374
$215$	0	0
$216$	3.00000	0.204124
$217$	−2.00000	−0.135769
$218$	−14.0000	−0.948200
$219$	16.0000	1.08118
$220$	4.00000	0.269680
$221$	0	0
$222$	−8.00000	−0.536925
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−10.0000	−0.668153
$225$	1.00000	0.0666667
$226$	−6.00000	−0.399114
$227$	−20.0000	−1.32745	−0.663723	−	0.747978i	$-0.731025\pi$
$227$	−20.0000	−1.32745	−0.663723	+	0.747978i	$0.731025\pi$
$228$	−8.00000	−0.529813
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	−8.00000	−0.527504
$231$	−8.00000	−0.526361
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	−10.0000	−0.650945
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	5.00000	0.321412
$243$	−1.00000	−0.0641500
$244$	14.0000	0.896258
$245$	−3.00000	−0.191663
$246$	6.00000	0.382546
$247$	0	0
$248$	−3.00000	−0.190500
$249$	−4.00000	−0.253490
$250$	1.00000	0.0632456
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	2.00000	0.125988
$253$	32.0000	2.01182
$254$	−4.00000	−0.250982
$255$	−2.00000	−0.125245
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	−14.0000	−0.864923
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	−12.0000	−0.738549
$265$	6.00000	0.368577
$266$	16.0000	0.981023
$267$	−4.00000	−0.244796
$268$	−2.00000	−0.122169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−4.00000	−0.241209
$276$	−8.00000	−0.481543
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−12.0000	−0.719712
$279$	1.00000	0.0598684
$280$	6.00000	0.358569
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−4.00000	−0.238197
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−6.00000	−0.356034
$285$	8.00000	0.473879
$286$	0	0
$287$	12.0000	0.708338
$288$	5.00000	0.294628
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−6.00000	−0.351726
$292$	16.0000	0.936329
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	3.00000	0.174964
$295$	10.0000	0.582223
$296$	−24.0000	−1.39497
$297$	4.00000	0.232104
$298$	−18.0000	−1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−24.0000	−1.38104
$303$	−14.0000	−0.804279
$304$	8.00000	0.458831
$305$	−14.0000	−0.801638
$306$	2.00000	0.114332
$307$	22.0000	1.25561	0.627803	−	0.778372i	$-0.283954\pi$
$307$	22.0000	1.25561	0.627803	+	0.778372i	$0.283954\pi$
$308$	−8.00000	−0.455842
$309$	−2.00000	−0.113776
$310$	1.00000	0.0567962
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	2.00000	0.112867
$315$	−2.00000	−0.112687
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	7.00000	0.391312
$321$	−16.0000	−0.893033
$322$	16.0000	0.891645
$323$	−16.0000	−0.890264
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	2.00000	0.110770
$327$	14.0000	0.774202
$328$	18.0000	0.993884
$329$	−8.00000	−0.441054
$330$	4.00000	0.220193
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−4.00000	−0.219529
$333$	8.00000	0.438397
$334$	16.0000	0.875481
$335$	2.00000	0.109272
$336$	−2.00000	−0.109109
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	−13.0000	−0.707107
$339$	6.00000	0.325875
$340$	−2.00000	−0.108465
$341$	−4.00000	−0.216612
$342$	−8.00000	−0.432590
$343$	20.0000	1.07990
$344$	0	0
$345$	8.00000	0.430706
$346$	6.00000	0.322562
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	−20.0000	−1.06600
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	−10.0000	−0.531494
$355$	6.00000	0.318447
$356$	−4.00000	−0.212000
$357$	4.00000	0.211702
$358$	−4.00000	−0.211407
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	−3.00000	−0.158114
$361$	45.0000	2.36842
$362$	−6.00000	−0.315353
$363$	−5.00000	−0.262432
$364$	0	0
$365$	−16.0000	−0.837478
$366$	14.0000	0.731792
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	8.00000	0.417029
$369$	−6.00000	−0.312348
$370$	8.00000	0.415900
$371$	−12.0000	−0.623009
$372$	1.00000	0.0518476
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	−8.00000	−0.413670
$375$	−1.00000	−0.0516398
$376$	−12.0000	−0.618853
$377$	0	0
$378$	2.00000	0.102869
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	8.00000	0.410391
$381$	4.00000	0.204926
$382$	−14.0000	−0.716302
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	3.00000	0.153093
$385$	8.00000	0.407718
$386$	6.00000	0.305392
$387$	0	0
$388$	−6.00000	−0.304604
$389$	16.0000	0.811232	0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000	0.811232	0.405616	+	0.914044i	$0.367057\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	9.00000	0.454569
$393$	14.0000	0.706207
$394$	2.00000	0.100759
$395$	0	0
$396$	4.00000	0.201008
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−8.00000	−0.401004
$399$	−16.0000	−0.801002
$400$	−1.00000	−0.0500000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	−14.0000	−0.696526
$405$	1.00000	0.0496904
$406$	0	0
$407$	−32.0000	−1.58618
$408$	6.00000	0.297044
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	−6.00000	−0.296319
$411$	18.0000	0.887875
$412$	−2.00000	−0.0985329
$413$	−20.0000	−0.984136
$414$	−8.00000	−0.393179
$415$	4.00000	0.196352
$416$	0	0
$417$	12.0000	0.587643
$418$	32.0000	1.56517
$419$	14.0000	0.683945	0.341972	−	0.939710i	$-0.388905\pi$
$419$	14.0000	0.683945	0.341972	+	0.939710i	$0.388905\pi$
$420$	−2.00000	−0.0975900
$421$	18.0000	0.877266	0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000	0.877266	0.438633	+	0.898666i	$0.355463\pi$
$422$	8.00000	0.389434
$423$	4.00000	0.194487
$424$	−18.0000	−0.874157
$425$	2.00000	0.0970143
$426$	−6.00000	−0.290701
$427$	28.0000	1.35501
$428$	−16.0000	−0.773389
$429$	0	0
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	1.00000	0.0481125
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	14.0000	0.670478
$437$	64.0000	3.06154
$438$	16.0000	0.764510
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	12.0000	0.572078
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	8.00000	0.379663
$445$	4.00000	0.189618
$446$	16.0000	0.757622
$447$	18.0000	0.851371
$448$	−14.0000	−0.661438
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	1.00000	0.0471405
$451$	24.0000	1.13012
$452$	6.00000	0.282216
$453$	24.0000	1.12762
$454$	−20.0000	−0.938647
$455$	0	0
$456$	−24.0000	−1.12390
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	6.00000	0.280362
$459$	−2.00000	−0.0933520
$460$	8.00000	0.373002
$461$	12.0000	0.558896	0.279448	−	0.960161i	$-0.409849\pi$
$461$	12.0000	0.558896	0.279448	+	0.960161i	$0.409849\pi$
$462$	−8.00000	−0.372194
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	6.00000	0.277945
$467$	40.0000	1.85098	0.925490	−	0.378773i	$-0.123654\pi$
$467$	40.0000	1.85098	0.925490	+	0.378773i	$0.123654\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	4.00000	0.184506
$471$	−2.00000	−0.0921551
$472$	−30.0000	−1.38086
$473$	0	0
$474$	0	0
$475$	−8.00000	−0.367065
$476$	4.00000	0.183340
$477$	6.00000	0.274721
$478$	−20.0000	−0.914779
$479$	−10.0000	−0.456912	−0.228456	−	0.973554i	$-0.573368\pi$
$479$	−10.0000	−0.456912	−0.228456	+	0.973554i	$0.573368\pi$
$480$	−5.00000	−0.228218
$481$	0	0
$482$	−22.0000	−1.00207
$483$	−16.0000	−0.728025
$484$	−5.00000	−0.227273
$485$	6.00000	0.272446
$486$	−1.00000	−0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	42.0000	1.90125
$489$	−2.00000	−0.0904431
$490$	−3.00000	−0.135526
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	0	0
$495$	−4.00000	−0.179787
$496$	−1.00000	−0.0449013
$497$	−12.0000	−0.538274
$498$	−4.00000	−0.179244
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	−1.00000	−0.0447214
$501$	−16.0000	−0.714827
$502$	20.0000	0.892644
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	6.00000	0.267261
$505$	14.0000	0.622992
$506$	32.0000	1.42257
$507$	13.0000	0.577350
$508$	4.00000	0.177471
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	−2.00000	−0.0885615
$511$	32.0000	1.41560
$512$	−11.0000	−0.486136
$513$	8.00000	0.353209
$514$	−18.0000	−0.793946
$515$	2.00000	0.0881305
$516$	0	0
$517$	−16.0000	−0.703679
$518$	−16.0000	−0.703000
$519$	−6.00000	−0.263371
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	14.0000	0.611593
$525$	2.00000	0.0872872
$526$	−24.0000	−1.04645
$527$	2.00000	0.0871214
$528$	−4.00000	−0.174078
$529$	41.0000	1.78261
$530$	6.00000	0.260623
$531$	10.0000	0.433963
$532$	−16.0000	−0.693688
$533$	0	0
$534$	−4.00000	−0.173097
$535$	16.0000	0.691740
$536$	−6.00000	−0.259161
$537$	4.00000	0.172613
$538$	0	0
$539$	12.0000	0.516877
$540$	1.00000	0.0430331
$541$	34.0000	1.46177	0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000	1.46177	0.730887	+	0.682498i	$0.239107\pi$
$542$	8.00000	0.343629
$543$	6.00000	0.257485
$544$	10.0000	0.428746
$545$	−14.0000	−0.599694
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	18.0000	0.768922
$549$	−14.0000	−0.597505
$550$	−4.00000	−0.170561
$551$	0	0
$552$	−24.0000	−1.02151
$553$	0	0
$554$	−28.0000	−1.18961
$555$	−8.00000	−0.339581
$556$	12.0000	0.508913
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	1.00000	0.0423334
$559$	0	0
$560$	2.00000	0.0845154
$561$	8.00000	0.337760
$562$	30.0000	1.26547
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	4.00000	0.168430
$565$	−6.00000	−0.252422
$566$	10.0000	0.420331
$567$	−2.00000	−0.0839921
$568$	−18.0000	−0.755263
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	8.00000	0.335083
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	14.0000	0.584858
$574$	12.0000	0.500870
$575$	−8.00000	−0.333623
$576$	7.00000	0.291667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−13.0000	−0.540729
$579$	−6.00000	−0.249351
$580$	0	0
$581$	−8.00000	−0.331896
$582$	−6.00000	−0.248708
$583$	−24.0000	−0.993978
$584$	48.0000	1.98625
$585$	0	0
$586$	26.0000	1.07405
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−3.00000	−0.123718
$589$	−8.00000	−0.329634
$590$	10.0000	0.411693
$591$	−2.00000	−0.0822690
$592$	−8.00000	−0.328798
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	4.00000	0.164122
$595$	−4.00000	−0.163984
$596$	18.0000	0.737309
$597$	8.00000	0.327418
$598$	0	0
$599$	−14.0000	−0.572024	−0.286012	−	0.958226i	$-0.592330\pi$
$599$	−14.0000	−0.572024	−0.286012	+	0.958226i	$0.592330\pi$
$600$	3.00000	0.122474
$601$	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
$603$	2.00000	0.0814463
$604$	24.0000	0.976546
$605$	5.00000	0.203279
$606$	−14.0000	−0.568711
$607$	−46.0000	−1.86708	−0.933541	−	0.358470i	$-0.883298\pi$
$607$	−46.0000	−1.86708	−0.933541	+	0.358470i	$0.883298\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	−14.0000	−0.566843
$611$	0	0
$612$	−2.00000	−0.0808452
$613$	−44.0000	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$614$	22.0000	0.887848
$615$	6.00000	0.241943
$616$	−24.0000	−0.966988
$617$	−38.0000	−1.52982	−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000	−1.52982	−0.764911	+	0.644136i	$0.777217\pi$
$618$	−2.00000	−0.0804518
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	−1.00000	−0.0401610
$621$	8.00000	0.321029
$622$	18.0000	0.721734
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	−8.00000	−0.319744
$627$	−32.0000	−1.27796
$628$	−2.00000	−0.0798087
$629$	16.0000	0.637962
$630$	−2.00000	−0.0796819
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	−6.00000	−0.238290
$635$	−4.00000	−0.158735
$636$	6.00000	0.237915
$637$	0	0
$638$	0	0
$639$	6.00000	0.237356
$640$	−3.00000	−0.118585
$641$	40.0000	1.57991	0.789953	−	0.613168i	$-0.210105\pi$
$641$	40.0000	1.57991	0.789953	+	0.613168i	$0.210105\pi$
$642$	−16.0000	−0.631470
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−16.0000	−0.629512
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	−3.00000	−0.117851
$649$	−40.0000	−1.57014
$650$	0	0
$651$	2.00000	0.0783862
$652$	−2.00000	−0.0783260
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	14.0000	0.547443
$655$	−14.0000	−0.547025
$656$	6.00000	0.234261
$657$	−16.0000	−0.624219
$658$	−8.00000	−0.311872
$659$	−42.0000	−1.63609	−0.818044	−	0.575156i	$-0.804941\pi$
$659$	−42.0000	−1.63609	−0.818044	+	0.575156i	$0.804941\pi$
$660$	−4.00000	−0.155700
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−12.0000	−0.465690
$665$	16.0000	0.620453
$666$	8.00000	0.309994
$667$	0	0
$668$	−16.0000	−0.619059
$669$	−16.0000	−0.618596
$670$	2.00000	0.0772667
$671$	56.0000	2.16186
$672$	10.0000	0.385758
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	28.0000	1.07852
$675$	−1.00000	−0.0384900
$676$	13.0000	0.500000
$677$	−26.0000	−0.999261	−0.499631	−	0.866239i	$-0.666531\pi$
$677$	−26.0000	−0.999261	−0.499631	+	0.866239i	$0.666531\pi$
$678$	6.00000	0.230429
$679$	−12.0000	−0.460518
$680$	−6.00000	−0.230089
$681$	20.0000	0.766402
$682$	−4.00000	−0.153168
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	8.00000	0.305888
$685$	−18.0000	−0.687745
$686$	20.0000	0.763604
$687$	−6.00000	−0.228914
$688$	0	0
$689$	0	0
$690$	8.00000	0.304555
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−6.00000	−0.228086
$693$	8.00000	0.303895
$694$	−12.0000	−0.455514
$695$	−12.0000	−0.455186
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−14.0000	−0.529908
$699$	−6.00000	−0.226941
$700$	2.00000	0.0755929
$701$	−6.00000	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$702$	0	0
$703$	−64.0000	−2.41381
$704$	−28.0000	−1.05529
$705$	−4.00000	−0.150649
$706$	−14.0000	−0.526897
$707$	−28.0000	−1.05305
$708$	10.0000	0.375823
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	6.00000	0.225176
$711$	0	0
$712$	−12.0000	−0.449719
$713$	−8.00000	−0.299602
$714$	4.00000	0.149696
$715$	0	0
$716$	4.00000	0.149487
$717$	20.0000	0.746914
$718$	26.0000	0.970311
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	−1.00000	−0.0372678
$721$	−4.00000	−0.148968
$722$	45.0000	1.67473
$723$	22.0000	0.818189
$724$	6.00000	0.222988
$725$	0	0
$726$	−5.00000	−0.185567
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−16.0000	−0.592187
$731$	0	0
$732$	−14.0000	−0.517455
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	−32.0000	−1.18114
$735$	3.00000	0.110657
$736$	−40.0000	−1.47442
$737$	−8.00000	−0.294684
$738$	−6.00000	−0.220863
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	3.00000	0.109985
$745$	−18.0000	−0.659469
$746$	−22.0000	−0.805477
$747$	4.00000	0.146352
$748$	8.00000	0.292509
$749$	−32.0000	−1.16925
$750$	−1.00000	−0.0365148
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−4.00000	−0.145865
$753$	−20.0000	−0.728841
$754$	0	0
$755$	−24.0000	−0.873449
$756$	−2.00000	−0.0727393
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	16.0000	0.581146
$759$	−32.0000	−1.16153
$760$	24.0000	0.870572
$761$	−24.0000	−0.869999	−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000	−0.869999	−0.435000	+	0.900431i	$0.643252\pi$
$762$	4.00000	0.144905
$763$	28.0000	1.01367
$764$	14.0000	0.506502
$765$	2.00000	0.0723102
$766$	0	0
$767$	0	0
$768$	17.0000	0.613435
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	8.00000	0.288300
$771$	18.0000	0.648254
$772$	−6.00000	−0.215945
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	−18.0000	−0.646162
$777$	16.0000	0.573997
$778$	16.0000	0.573628
$779$	48.0000	1.71978
$780$	0	0
$781$	−24.0000	−0.858788
$782$	−16.0000	−0.572159
$783$	0	0
$784$	3.00000	0.107143
$785$	2.00000	0.0713831
$786$	14.0000	0.499363
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	−2.00000	−0.0712470
$789$	24.0000	0.854423
$790$	0	0
$791$	12.0000	0.426671
$792$	12.0000	0.426401
$793$	0	0
$794$	14.0000	0.496841
$795$	−6.00000	−0.212798
$796$	8.00000	0.283552
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	−16.0000	−0.566394
$799$	8.00000	0.283020
$800$	5.00000	0.176777
$801$	4.00000	0.141333
$802$	16.0000	0.564980
$803$	64.0000	2.25851
$804$	2.00000	0.0705346
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	−42.0000	−1.47755
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	1.00000	0.0351364
$811$	32.0000	1.12367	0.561836	−	0.827249i	$-0.310095\pi$
$811$	32.0000	1.12367	0.561836	+	0.827249i	$0.310095\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−32.0000	−1.12160
$815$	2.00000	0.0700569
$816$	2.00000	0.0700140
$817$	0	0
$818$	−18.0000	−0.629355
$819$	0	0
$820$	6.00000	0.209529
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	18.0000	0.627822
$823$	−44.0000	−1.53374	−0.766872	−	0.641800i	$-0.778188\pi$
$823$	−44.0000	−1.53374	−0.766872	+	0.641800i	$0.778188\pi$
$824$	−6.00000	−0.209020
$825$	4.00000	0.139262
$826$	−20.0000	−0.695889
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	8.00000	0.278019
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	4.00000	0.138842
$831$	28.0000	0.971309
$832$	0	0
$833$	−6.00000	−0.207888
$834$	12.0000	0.415526
$835$	16.0000	0.553703
$836$	−32.0000	−1.10674
$837$	−1.00000	−0.0345651
$838$	14.0000	0.483622
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	−6.00000	−0.207020
$841$	−29.0000	−1.00000
$842$	18.0000	0.620321
$843$	−30.0000	−1.03325
$844$	−8.00000	−0.275371
$845$	−13.0000	−0.447214
$846$	4.00000	0.137523
$847$	−10.0000	−0.343604
$848$	−6.00000	−0.206041
$849$	−10.0000	−0.343199
$850$	2.00000	0.0685994
$851$	−64.0000	−2.19389
$852$	6.00000	0.205557
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	28.0000	0.958140
$855$	−8.00000	−0.273594
$856$	−48.0000	−1.64061
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	2.00000	0.0681203
$863$	8.00000	0.272323	0.136162	−	0.990687i	$-0.456523\pi$
$863$	8.00000	0.272323	0.136162	+	0.990687i	$0.456523\pi$
$864$	−5.00000	−0.170103
$865$	6.00000	0.204006
$866$	24.0000	0.815553
$867$	13.0000	0.441503
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	0	0
$872$	42.0000	1.42230
$873$	6.00000	0.203069
$874$	64.0000	2.16483
$875$	−2.00000	−0.0676123
$876$	−16.0000	−0.540590
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	24.0000	0.809961
$879$	−26.0000	−0.876958
$880$	4.00000	0.134840
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	−3.00000	−0.101015
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	−10.0000	−0.336146
$886$	−20.0000	−0.671913
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	24.0000	0.805387
$889$	8.00000	0.268311
$890$	4.00000	0.134080
$891$	−4.00000	−0.134005
$892$	−16.0000	−0.535720
$893$	−32.0000	−1.07084
$894$	18.0000	0.602010
$895$	−4.00000	−0.133705
$896$	6.00000	0.200446
$897$	0	0
$898$	−8.00000	−0.266963
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	12.0000	0.399778
$902$	24.0000	0.799113
$903$	0	0
$904$	18.0000	0.598671
$905$	−6.00000	−0.199447
$906$	24.0000	0.797347
$907$	−18.0000	−0.597680	−0.298840	−	0.954303i	$-0.596600\pi$
$907$	−18.0000	−0.597680	−0.298840	+	0.954303i	$0.596600\pi$
$908$	20.0000	0.663723
$909$	14.0000	0.464351
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	−8.00000	−0.264906
$913$	−16.0000	−0.529523
$914$	−8.00000	−0.264616
$915$	14.0000	0.462826
$916$	−6.00000	−0.198246
$917$	28.0000	0.924641
$918$	−2.00000	−0.0660098
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	24.0000	0.791257
$921$	−22.0000	−0.724925
$922$	12.0000	0.395199
$923$	0	0
$924$	8.00000	0.263181
$925$	8.00000	0.263038
$926$	20.0000	0.657241
$927$	2.00000	0.0656886
$928$	0	0
$929$	−44.0000	−1.44359	−0.721797	−	0.692105i	$-0.756683\pi$
$929$	−44.0000	−1.44359	−0.721797	+	0.692105i	$0.756683\pi$
$930$	−1.00000	−0.0327913
$931$	24.0000	0.786568
$932$	−6.00000	−0.196537
$933$	−18.0000	−0.589294
$934$	40.0000	1.30884
$935$	−8.00000	−0.261628
$936$	0	0
$937$	6.00000	0.196011	0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000	0.196011	0.0980057	+	0.995186i	$0.468754\pi$
$938$	−4.00000	−0.130605
$939$	8.00000	0.261070
$940$	−4.00000	−0.130466
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	−2.00000	−0.0651635
$943$	48.0000	1.56310
$944$	−10.0000	−0.325472
$945$	2.00000	0.0650600
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	0	0
$950$	−8.00000	−0.259554
$951$	6.00000	0.194563
$952$	12.0000	0.388922
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	6.00000	0.194257
$955$	−14.0000	−0.453029
$956$	20.0000	0.646846
$957$	0	0
$958$	−10.0000	−0.323085
$959$	36.0000	1.16250
$960$	−7.00000	−0.225924
$961$	1.00000	0.0322581
$962$	0	0
$963$	16.0000	0.515593
$964$	22.0000	0.708572
$965$	6.00000	0.193147
$966$	−16.0000	−0.514792
$967$	24.0000	0.771788	0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000	0.771788	0.385894	+	0.922543i	$0.373893\pi$
$968$	−15.0000	−0.482118
$969$	16.0000	0.513994
$970$	6.00000	0.192648
$971$	−42.0000	−1.34784	−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000	−1.34784	−0.673922	+	0.738802i	$0.735392\pi$
$972$	1.00000	0.0320750
$973$	24.0000	0.769405
$974$	−4.00000	−0.128168
$975$	0	0
$976$	14.0000	0.448129
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−2.00000	−0.0639529
$979$	−16.0000	−0.511362
$980$	3.00000	0.0958315
$981$	−14.0000	−0.446986
$982$	−24.0000	−0.765871
$983$	−8.00000	−0.255160	−0.127580	−	0.991828i	$-0.540721\pi$
$983$	−8.00000	−0.255160	−0.127580	+	0.991828i	$0.540721\pi$
$984$	−18.0000	−0.573819
$985$	2.00000	0.0637253
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	0	0
$990$	−4.00000	−0.127128
$991$	−56.0000	−1.77890	−0.889449	−	0.457034i	$-0.848912\pi$
$991$	−56.0000	−1.77890	−0.889449	+	0.457034i	$0.848912\pi$
$992$	5.00000	0.158750
$993$	4.00000	0.126936
$994$	−12.0000	−0.380617
$995$	−8.00000	−0.253617
$996$	4.00000	0.126745
$997$	42.0000	1.33015	0.665077	−	0.746775i	$-0.268399\pi$
$997$	42.0000	1.33015	0.665077	+	0.746775i	$0.268399\pi$
$998$	−20.0000	−0.633089
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	465.2.a.b.1.1	✓	1
3.2	odd	2		1395.2.a.a.1.1		1
4.3	odd	2		7440.2.a.ba.1.1		1
5.2	odd	4		2325.2.c.d.1024.2		2
5.3	odd	4		2325.2.c.d.1024.1		2
5.4	even	2		2325.2.a.d.1.1		1
15.14	odd	2		6975.2.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.b.1.1	✓	1	1.1	even	1	trivial
1395.2.a.a.1.1		1	3.2	odd	2
2325.2.a.d.1.1		1	5.4	even	2
2325.2.c.d.1024.1		2	5.3	odd	4
2325.2.c.d.1024.2		2	5.2	odd	4
6975.2.a.q.1.1		1	15.14	odd	2
7440.2.a.ba.1.1		1	4.3	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:	\( N \)	\(=\)	\( 465 = 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	465.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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