LMFDB - Embedded newform 465.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [465,2,Mod(1,465)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 465 = 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	465.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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)

$f(q)$

$=$

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -1.00000 q^{20} -4.00000 q^{21} +4.00000 q^{22} +3.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} -1.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} -4.00000 q^{35} -1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +3.00000 q^{40} -6.00000 q^{41} +4.00000 q^{42} -12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} -4.00000 q^{55} -12.0000 q^{56} -4.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{62} -4.00000 q^{63} +7.00000 q^{64} +2.00000 q^{65} +4.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} -12.0000 q^{71} +3.00000 q^{72} +6.00000 q^{73} -10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +16.0000 q^{77} -2.00000 q^{78} -8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} +4.00000 q^{84} -6.00000 q^{85} +12.0000 q^{86} -6.00000 q^{87} -12.0000 q^{88} +6.00000 q^{89} -1.00000 q^{90} -8.00000 q^{91} -1.00000 q^{93} -4.00000 q^{95} -5.00000 q^{96} -6.00000 q^{97} -9.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	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
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	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
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	4.00000	1.06904
$15$	1.00000	0.258199
$16$	−1.00000	−0.250000
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	3.00000	0.612372
$25$	1.00000	0.200000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	4.00000	0.755929
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	6.00000	1.02899
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	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	$0.655213\pi$
$42$	4.00000	0.617213
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	−1.00000	−0.144338
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−6.00000	−0.840168
$52$	−2.00000	−0.277350
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	−1.00000	−0.136083
$55$	−4.00000	−0.539360
$56$	−12.0000	−1.60357
$57$	−4.00000	−0.529813
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	−1.00000	−0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	1.00000	0.127000
$63$	−4.00000	−0.503953
$64$	7.00000	0.875000
$65$	2.00000	0.248069
$66$	4.00000	0.492366
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	4.00000	0.478091
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	3.00000	0.353553
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−10.0000	−1.16248
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	16.0000	1.82337
$78$	−2.00000	−0.226455
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	4.00000	0.436436
$85$	−6.00000	−0.650791
$86$	12.0000	1.29399
$87$	−6.00000	−0.643268
$88$	−12.0000	−1.27920
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−4.00000	−0.410391
$96$	−5.00000	−0.510310
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	−9.00000	−0.909137
$99$	−4.00000	−0.402015
$100$	−1.00000	−0.100000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	6.00000	0.594089
$103$	20.0000	1.97066	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000	1.97066	0.985329	+	0.170664i	$0.0545913\pi$
$104$	6.00000	0.588348
$105$	−4.00000	−0.390360
$106$	2.00000	0.194257
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	−1.00000	−0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	4.00000	0.381385
$111$	10.0000	0.949158
$112$	4.00000	0.377964
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	2.00000	0.184900
$118$	8.00000	0.736460
$119$	24.0000	2.20008
$120$	3.00000	0.273861
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	−6.00000	−0.541002
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	−12.0000	−1.05654
$130$	−2.00000	−0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000	0.348155
$133$	16.0000	1.38738
$134$	−8.00000	−0.691095
$135$	1.00000	0.0860663
$136$	−18.0000	−1.54349
$137$	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
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	12.0000	1.00702
$143$	−8.00000	−0.668994
$144$	−1.00000	−0.0833333
$145$	−6.00000	−0.498273
$146$	−6.00000	−0.496564
$147$	9.00000	0.742307
$148$	−10.0000	−0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	−1.00000	−0.0816497
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	−12.0000	−0.973329
$153$	−6.00000	−0.485071
$154$	−16.0000	−1.28932
$155$	−1.00000	−0.0803219
$156$	−2.00000	−0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	8.00000	0.636446
$159$	−2.00000	−0.158610
$160$	−5.00000	−0.395285
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	6.00000	0.468521
$165$	−4.00000	−0.311400
$166$	−12.0000	−0.931381
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	−12.0000	−0.925820
$169$	−9.00000	−0.692308
$170$	6.00000	0.460179
$171$	−4.00000	−0.305888
$172$	12.0000	0.914991
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	−4.00000	−0.302372
$176$	4.00000	0.301511
$177$	−8.00000	−0.601317
$178$	−6.00000	−0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	−1.00000	−0.0745356
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	8.00000	0.592999
$183$	6.00000	0.443533
$184$	0	0
$185$	10.0000	0.735215
$186$	1.00000	0.0733236
$187$	24.0000	1.75505
$188$	0	0
$189$	−4.00000	−0.290957
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	7.00000	0.505181
$193$	26.0000	1.87152	0.935760	−	0.352636i	$-0.114715\pi$
$193$	26.0000	1.87152	0.935760	+	0.352636i	$0.114715\pi$
$194$	6.00000	0.430775
$195$	2.00000	0.143223
$196$	−9.00000	−0.642857
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	4.00000	0.284268
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	3.00000	0.212132
$201$	8.00000	0.564276
$202$	10.0000	0.703598
$203$	24.0000	1.68447
$204$	6.00000	0.420084
$205$	−6.00000	−0.419058
$206$	−20.0000	−1.39347
$207$	0	0
$208$	−2.00000	−0.138675
$209$	16.0000	1.10674
$210$	4.00000	0.276026
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	2.00000	0.137361
$213$	−12.0000	−0.822226
$214$	−12.0000	−0.820303
$215$	−12.0000	−0.818393
$216$	3.00000	0.204124
$217$	4.00000	0.271538
$218$	2.00000	0.135457
$219$	6.00000	0.405442
$220$	4.00000	0.269680
$221$	−12.0000	−0.807207
$222$	−10.0000	−0.671156
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	20.0000	1.33631
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	4.00000	0.264906
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	16.0000	1.05272
$232$	−18.0000	−1.18176
$233$	−22.0000	−1.44127	−0.720634	−	0.693316i	$-0.756149\pi$
$233$	−22.0000	−1.44127	−0.720634	+	0.693316i	$0.756149\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	8.00000	0.520756
$237$	−8.00000	−0.519656
$238$	−24.0000	−1.55569
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−1.00000	−0.0645497
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	9.00000	0.574989
$246$	6.00000	0.382546
$247$	−8.00000	−0.509028
$248$	−3.00000	−0.190500
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	4.00000	0.251976
$253$	0	0
$254$	8.00000	0.501965
$255$	−6.00000	−0.375735
$256$	−17.0000	−1.06250
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	12.0000	0.747087
$259$	−40.0000	−2.48548
$260$	−2.00000	−0.124035
$261$	−6.00000	−0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−12.0000	−0.738549
$265$	−2.00000	−0.122859
$266$	−16.0000	−0.981023
$267$	6.00000	0.367194
$268$	−8.00000	−0.488678
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	−1.00000	−0.0608581
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	6.00000	0.363803
$273$	−8.00000	−0.484182
$274$	−10.0000	−0.604122
$275$	−4.00000	−0.241209
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	−12.0000	−0.719712
$279$	−1.00000	−0.0598684
$280$	−12.0000	−0.717137
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	12.0000	0.712069
$285$	−4.00000	−0.236940
$286$	8.00000	0.473050
$287$	24.0000	1.41668
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	6.00000	0.352332
$291$	−6.00000	−0.351726
$292$	−6.00000	−0.351123
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−9.00000	−0.524891
$295$	−8.00000	−0.465778
$296$	30.0000	1.74371
$297$	−4.00000	−0.232104
$298$	−6.00000	−0.347571
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	48.0000	2.76667
$302$	−16.0000	−0.920697
$303$	−10.0000	−0.574485
$304$	4.00000	0.229416
$305$	6.00000	0.343559
$306$	6.00000	0.342997
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−16.0000	−0.911685
$309$	20.0000	1.13776
$310$	1.00000	0.0567962
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	6.00000	0.339683
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	10.0000	0.564333
$315$	−4.00000	−0.225374
$316$	8.00000	0.450035
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	2.00000	0.112154
$319$	24.0000	1.34374
$320$	7.00000	0.391312
$321$	12.0000	0.669775
$322$	0	0
$323$	24.0000	1.33540
$324$	−1.00000	−0.0555556
$325$	2.00000	0.110940
$326$	8.00000	0.443079
$327$	−2.00000	−0.110600
$328$	−18.0000	−0.993884
$329$	0	0
$330$	4.00000	0.220193
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−12.0000	−0.658586
$333$	10.0000	0.547997
$334$	8.00000	0.437741
$335$	8.00000	0.437087
$336$	4.00000	0.218218
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	9.00000	0.489535
$339$	18.0000	0.977626
$340$	6.00000	0.325396
$341$	4.00000	0.216612
$342$	4.00000	0.216295
$343$	−8.00000	−0.431959
$344$	−36.0000	−1.94099
$345$	0	0
$346$	18.0000	0.967686
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	4.00000	0.213809
$351$	2.00000	0.106752
$352$	20.0000	1.06600
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	8.00000	0.425195
$355$	−12.0000	−0.636894
$356$	−6.00000	−0.317999
$357$	24.0000	1.27021
$358$	12.0000	0.634220
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	3.00000	0.158114
$361$	−3.00000	−0.157895
$362$	2.00000	0.105118
$363$	5.00000	0.262432
$364$	8.00000	0.419314
$365$	6.00000	0.314054
$366$	−6.00000	−0.313625
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	−10.0000	−0.519875
$371$	8.00000	0.415339
$372$	1.00000	0.0518476
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	−24.0000	−1.24101
$375$	1.00000	0.0516398
$376$	0	0
$377$	−12.0000	−0.618031
$378$	4.00000	0.205738
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	4.00000	0.205196
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	3.00000	0.153093
$385$	16.0000	0.815436
$386$	−26.0000	−1.32337
$387$	−12.0000	−0.609994
$388$	6.00000	0.304604
$389$	2.00000	0.101404	0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000	0.101404	0.0507020	+	0.998714i	$0.483854\pi$
$390$	−2.00000	−0.101274
$391$	0	0
$392$	27.0000	1.36371
$393$	0	0
$394$	26.0000	1.30986
$395$	−8.00000	−0.402524
$396$	4.00000	0.201008
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	16.0000	0.801002
$400$	−1.00000	−0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	−8.00000	−0.399004
$403$	−2.00000	−0.0996271
$404$	10.0000	0.497519
$405$	1.00000	0.0496904
$406$	−24.0000	−1.19110
$407$	−40.0000	−1.98273
$408$	−18.0000	−0.891133
$409$	34.0000	1.68119	0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000	1.68119	0.840596	+	0.541663i	$0.182205\pi$
$410$	6.00000	0.296319
$411$	10.0000	0.493264
$412$	−20.0000	−0.985329
$413$	32.0000	1.57462
$414$	0	0
$415$	12.0000	0.589057
$416$	−10.0000	−0.490290
$417$	12.0000	0.587643
$418$	−16.0000	−0.782586
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	4.00000	0.195180
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−6.00000	−0.291043
$426$	12.0000	0.581402
$427$	−24.0000	−1.16144
$428$	−12.0000	−0.580042
$429$	−8.00000	−0.386244
$430$	12.0000	0.578691
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	−1.00000	−0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	−4.00000	−0.192006
$435$	−6.00000	−0.287678
$436$	2.00000	0.0957826
$437$	0	0
$438$	−6.00000	−0.286691
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	−12.0000	−0.572078
$441$	9.00000	0.428571
$442$	12.0000	0.570782
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−10.0000	−0.474579
$445$	6.00000	0.284427
$446$	16.0000	0.757622
$447$	6.00000	0.283790
$448$	−28.0000	−1.32288
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	−1.00000	−0.0471405
$451$	24.0000	1.13012
$452$	−18.0000	−0.846649
$453$	16.0000	0.751746
$454$	−12.0000	−0.563188
$455$	−8.00000	−0.375046
$456$	−12.0000	−0.561951
$457$	−42.0000	−1.96468	−0.982339	−	0.187112i	$-0.940087\pi$
$457$	−42.0000	−1.96468	−0.982339	+	0.187112i	$0.940087\pi$
$458$	26.0000	1.21490
$459$	−6.00000	−0.280056
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	−16.0000	−0.744387
$463$	−40.0000	−1.85896	−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000	−1.85896	−0.929479	+	0.368875i	$0.879743\pi$
$464$	6.00000	0.278543
$465$	−1.00000	−0.0463739
$466$	22.0000	1.01913
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−0.0924500
$469$	−32.0000	−1.47762
$470$	0	0
$471$	−10.0000	−0.460776
$472$	−24.0000	−1.10469
$473$	48.0000	2.20704
$474$	8.00000	0.367452
$475$	−4.00000	−0.183533
$476$	−24.0000	−1.10004
$477$	−2.00000	−0.0915737
$478$	24.0000	1.09773
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	−5.00000	−0.228218
$481$	20.0000	0.911922
$482$	−18.0000	−0.819878
$483$	0	0
$484$	−5.00000	−0.227273
$485$	−6.00000	−0.272446
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	18.0000	0.814822
$489$	−8.00000	−0.361773
$490$	−9.00000	−0.406579
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	6.00000	0.270501
$493$	36.0000	1.62136
$494$	8.00000	0.359937
$495$	−4.00000	−0.179787
$496$	1.00000	0.0449013
$497$	48.0000	2.15309
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	−1.00000	−0.0447214
$501$	−8.00000	−0.357414
$502$	20.0000	0.892644
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	−12.0000	−0.534522
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	6.00000	0.265684
$511$	−24.0000	−1.06170
$512$	11.0000	0.486136
$513$	−4.00000	−0.176604
$514$	14.0000	0.617514
$515$	20.0000	0.881305
$516$	12.0000	0.528271
$517$	0	0
$518$	40.0000	1.75750
$519$	−18.0000	−0.790112
$520$	6.00000	0.263117
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	6.00000	0.262613
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	8.00000	0.348817
$527$	6.00000	0.261364
$528$	4.00000	0.174078
$529$	−23.0000	−1.00000
$530$	2.00000	0.0868744
$531$	−8.00000	−0.347170
$532$	−16.0000	−0.693688
$533$	−12.0000	−0.519778
$534$	−6.00000	−0.259645
$535$	12.0000	0.518805
$536$	24.0000	1.03664
$537$	−12.0000	−0.517838
$538$	6.00000	0.258678
$539$	−36.0000	−1.55063
$540$	−1.00000	−0.0430331
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	−16.0000	−0.687259
$543$	−2.00000	−0.0858282
$544$	30.0000	1.28624
$545$	−2.00000	−0.0856706
$546$	8.00000	0.342368
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	−10.0000	−0.427179
$549$	6.00000	0.256074
$550$	4.00000	0.170561
$551$	24.0000	1.02243
$552$	0	0
$553$	32.0000	1.36078
$554$	−2.00000	−0.0849719
$555$	10.0000	0.424476
$556$	−12.0000	−0.508913
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	1.00000	0.0423334
$559$	−24.0000	−1.01509
$560$	4.00000	0.169031
$561$	24.0000	1.01328
$562$	−18.0000	−0.759284
$563$	−28.0000	−1.18006	−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000	−1.18006	−0.590030	+	0.807382i	$0.700884\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	32.0000	1.34506
$567$	−4.00000	−0.167984
$568$	−36.0000	−1.51053
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	4.00000	0.167542
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	8.00000	0.334497
$573$	−12.0000	−0.501307
$574$	−24.0000	−1.00174
$575$	0	0
$576$	7.00000	0.291667
$577$	−30.0000	−1.24892	−0.624458	−	0.781058i	$-0.714680\pi$
$577$	−30.0000	−1.24892	−0.624458	+	0.781058i	$0.714680\pi$
$578$	−19.0000	−0.790296
$579$	26.0000	1.08052
$580$	6.00000	0.249136
$581$	−48.0000	−1.99138
$582$	6.00000	0.248708
$583$	8.00000	0.331326
$584$	18.0000	0.744845
$585$	2.00000	0.0826898
$586$	−6.00000	−0.247858
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−9.00000	−0.371154
$589$	4.00000	0.164817
$590$	8.00000	0.329355
$591$	−26.0000	−1.06950
$592$	−10.0000	−0.410997
$593$	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
$595$	24.0000	0.983904
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	3.00000	0.122474
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	−48.0000	−1.95633
$603$	8.00000	0.325785
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	10.0000	0.406222
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	20.0000	0.811107
$609$	24.0000	0.972529
$610$	−6.00000	−0.242933
$611$	0	0
$612$	6.00000	0.242536
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−6.00000	−0.241943
$616$	48.0000	1.93398
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	−20.0000	−0.804518
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	1.00000	0.0401610
$621$	0	0
$622$	−12.0000	−0.481156
$623$	−24.0000	−0.961540
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	10.0000	0.399680
$627$	16.0000	0.638978
$628$	10.0000	0.399043
$629$	−60.0000	−2.39236
$630$	4.00000	0.159364
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	−24.0000	−0.954669
$633$	−4.00000	−0.158986
$634$	2.00000	0.0794301
$635$	−8.00000	−0.317470
$636$	2.00000	0.0793052
$637$	18.0000	0.713186
$638$	−24.0000	−0.950169
$639$	−12.0000	−0.474713
$640$	3.00000	0.118585
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	−12.0000	−0.473602
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	−24.0000	−0.944267
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	3.00000	0.117851
$649$	32.0000	1.25611
$650$	−2.00000	−0.0784465
$651$	4.00000	0.156772
$652$	8.00000	0.313304
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	0	0
$659$	−40.0000	−1.55818	−0.779089	−	0.626913i	$-0.784318\pi$
$659$	−40.0000	−1.55818	−0.779089	+	0.626913i	$0.784318\pi$
$660$	4.00000	0.155700
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	−4.00000	−0.155464
$663$	−12.0000	−0.466041
$664$	36.0000	1.39707
$665$	16.0000	0.620453
$666$	−10.0000	−0.387492
$667$	0	0
$668$	8.00000	0.309529
$669$	−16.0000	−0.618596
$670$	−8.00000	−0.309067
$671$	−24.0000	−0.926510
$672$	20.0000	0.771517
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	10.0000	0.385186
$675$	1.00000	0.0384900
$676$	9.00000	0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	−18.0000	−0.691286
$679$	24.0000	0.921035
$680$	−18.0000	−0.690268
$681$	12.0000	0.459841
$682$	−4.00000	−0.153168
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	4.00000	0.152944
$685$	10.0000	0.382080
$686$	8.00000	0.305441
$687$	−26.0000	−0.991962
$688$	12.0000	0.457496
$689$	−4.00000	−0.152388
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	18.0000	0.684257
$693$	16.0000	0.607790
$694$	−12.0000	−0.455514
$695$	12.0000	0.455186
$696$	−18.0000	−0.682288
$697$	36.0000	1.36360
$698$	2.00000	0.0757011
$699$	−22.0000	−0.832116
$700$	4.00000	0.151186
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	−2.00000	−0.0754851
$703$	−40.0000	−1.50863
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−10.0000	−0.376355
$707$	40.0000	1.50435
$708$	8.00000	0.300658
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	12.0000	0.450352
$711$	−8.00000	−0.300023
$712$	18.0000	0.674579
$713$	0	0
$714$	−24.0000	−0.898177
$715$	−8.00000	−0.299183
$716$	12.0000	0.448461
$717$	−24.0000	−0.896296
$718$	4.00000	0.149279
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	−1.00000	−0.0372678
$721$	−80.0000	−2.97936
$722$	3.00000	0.111648
$723$	18.0000	0.669427
$724$	2.00000	0.0743294
$725$	−6.00000	−0.222834
$726$	−5.00000	−0.185567
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	−24.0000	−0.889499
$729$	1.00000	0.0370370
$730$	−6.00000	−0.222070
$731$	72.0000	2.66302
$732$	−6.00000	−0.221766
$733$	30.0000	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$734$	16.0000	0.590571
$735$	9.00000	0.331970
$736$	0	0
$737$	−32.0000	−1.17874
$738$	6.00000	0.220863
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	−10.0000	−0.367607
$741$	−8.00000	−0.293887
$742$	−8.00000	−0.293689
$743$	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
$745$	6.00000	0.219823
$746$	−14.0000	−0.512576
$747$	12.0000	0.439057
$748$	−24.0000	−0.877527
$749$	−48.0000	−1.75388
$750$	−1.00000	−0.0365148
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	12.0000	0.437014
$755$	16.0000	0.582300
$756$	4.00000	0.145479
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	−36.0000	−1.30758
$759$	0	0
$760$	−12.0000	−0.435286
$761$	14.0000	0.507500	0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000	0.507500	0.253750	+	0.967270i	$0.418336\pi$
$762$	8.00000	0.289809
$763$	8.00000	0.289619
$764$	12.0000	0.434145
$765$	−6.00000	−0.216930
$766$	8.00000	0.289052
$767$	−16.0000	−0.577727
$768$	−17.0000	−0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	−16.0000	−0.576600
$771$	−14.0000	−0.504198
$772$	−26.0000	−0.935760
$773$	30.0000	1.07903	0.539513	−	0.841978i	$-0.318609\pi$
$773$	30.0000	1.07903	0.539513	+	0.841978i	$0.318609\pi$
$774$	12.0000	0.431331
$775$	−1.00000	−0.0359211
$776$	−18.0000	−0.646162
$777$	−40.0000	−1.43499
$778$	−2.00000	−0.0717035
$779$	24.0000	0.859889
$780$	−2.00000	−0.0716115
$781$	48.0000	1.71758
$782$	0	0
$783$	−6.00000	−0.214423
$784$	−9.00000	−0.321429
$785$	−10.0000	−0.356915
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	26.0000	0.926212
$789$	−8.00000	−0.284808
$790$	8.00000	0.284627
$791$	−72.0000	−2.56003
$792$	−12.0000	−0.426401
$793$	12.0000	0.426132
$794$	18.0000	0.638796
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	46.0000	1.62940	0.814702	−	0.579880i	$-0.196901\pi$
$797$	46.0000	1.62940	0.814702	+	0.579880i	$0.196901\pi$
$798$	−16.0000	−0.566394
$799$	0	0
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	−6.00000	−0.211867
$803$	−24.0000	−0.846942
$804$	−8.00000	−0.282138
$805$	0	0
$806$	2.00000	0.0704470
$807$	−6.00000	−0.211210
$808$	−30.0000	−1.05540
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	−1.00000	−0.0351364
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−24.0000	−0.842235
$813$	16.0000	0.561144
$814$	40.0000	1.40200
$815$	−8.00000	−0.280228
$816$	6.00000	0.210042
$817$	48.0000	1.67931
$818$	−34.0000	−1.18878
$819$	−8.00000	−0.279543
$820$	6.00000	0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	−10.0000	−0.348790
$823$	−48.0000	−1.67317	−0.836587	−	0.547833i	$-0.815453\pi$
$823$	−48.0000	−1.67317	−0.836587	+	0.547833i	$0.815453\pi$
$824$	60.0000	2.09020
$825$	−4.00000	−0.139262
$826$	−32.0000	−1.11342
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	−12.0000	−0.416526
$831$	2.00000	0.0693792
$832$	14.0000	0.485363
$833$	−54.0000	−1.87099
$834$	−12.0000	−0.415526
$835$	−8.00000	−0.276851
$836$	−16.0000	−0.553372
$837$	−1.00000	−0.0345651
$838$	24.0000	0.829066
$839$	28.0000	0.966667	0.483334	−	0.875436i	$-0.339426\pi$
$839$	28.0000	0.966667	0.483334	+	0.875436i	$0.339426\pi$
$840$	−12.0000	−0.414039
$841$	7.00000	0.241379
$842$	−6.00000	−0.206774
$843$	18.0000	0.619953
$844$	4.00000	0.137686
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−20.0000	−0.687208
$848$	2.00000	0.0686803
$849$	−32.0000	−1.09824
$850$	6.00000	0.205798
$851$	0	0
$852$	12.0000	0.411113
$853$	−26.0000	−0.890223	−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000	−0.890223	−0.445112	+	0.895475i	$0.646836\pi$
$854$	24.0000	0.821263
$855$	−4.00000	−0.136797
$856$	36.0000	1.23045
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	8.00000	0.273115
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	12.0000	0.409197
$861$	24.0000	0.817918
$862$	28.0000	0.953684
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	−5.00000	−0.170103
$865$	−18.0000	−0.612018
$866$	−30.0000	−1.01944
$867$	19.0000	0.645274
$868$	−4.00000	−0.135769
$869$	32.0000	1.08553
$870$	6.00000	0.203419
$871$	16.0000	0.542139
$872$	−6.00000	−0.203186
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−6.00000	−0.202721
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	8.00000	0.269987
$879$	6.00000	0.202375
$880$	4.00000	0.134840
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−9.00000	−0.303046
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	12.0000	0.403604
$885$	−8.00000	−0.268917
$886$	−36.0000	−1.20944
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	30.0000	1.00673
$889$	32.0000	1.07325
$890$	−6.00000	−0.201120
$891$	−4.00000	−0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	−6.00000	−0.200670
$895$	−12.0000	−0.401116
$896$	−12.0000	−0.400892
$897$	0	0
$898$	−6.00000	−0.200223
$899$	6.00000	0.200111
$900$	−1.00000	−0.0333333
$901$	12.0000	0.399778
$902$	−24.0000	−0.799113
$903$	48.0000	1.59734
$904$	54.0000	1.79601
$905$	−2.00000	−0.0664822
$906$	−16.0000	−0.531564
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	−12.0000	−0.398234
$909$	−10.0000	−0.331679
$910$	8.00000	0.265197
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	4.00000	0.132453
$913$	−48.0000	−1.58857
$914$	42.0000	1.38924
$915$	6.00000	0.198354
$916$	26.0000	0.859064
$917$	0	0
$918$	6.00000	0.198030
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	−2.00000	−0.0658665
$923$	−24.0000	−0.789970
$924$	−16.0000	−0.526361
$925$	10.0000	0.328798
$926$	40.0000	1.31448
$927$	20.0000	0.656886
$928$	30.0000	0.984798
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	1.00000	0.0327913
$931$	−36.0000	−1.17985
$932$	22.0000	0.720634
$933$	12.0000	0.392862
$934$	12.0000	0.392652
$935$	24.0000	0.784884
$936$	6.00000	0.196116
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	32.0000	1.04484
$939$	−10.0000	−0.326338
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	10.0000	0.325818
$943$	0	0
$944$	8.00000	0.260378
$945$	−4.00000	−0.130120
$946$	−48.0000	−1.56061
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	8.00000	0.259828
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	−2.00000	−0.0648544
$952$	72.0000	2.33353
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	2.00000	0.0647524
$955$	−12.0000	−0.388311
$956$	24.0000	0.776215
$957$	24.0000	0.775810
$958$	4.00000	0.129234
$959$	−40.0000	−1.29167
$960$	7.00000	0.225924
$961$	1.00000	0.0322581
$962$	−20.0000	−0.644826
$963$	12.0000	0.386695
$964$	−18.0000	−0.579741
$965$	26.0000	0.836970
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	15.0000	0.482118
$969$	24.0000	0.770991
$970$	6.00000	0.192648
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	−1.00000	−0.0320750
$973$	−48.0000	−1.53881
$974$	−16.0000	−0.512673
$975$	2.00000	0.0640513
$976$	−6.00000	−0.192055
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	8.00000	0.255812
$979$	−24.0000	−0.767043
$980$	−9.00000	−0.287494
$981$	−2.00000	−0.0638551
$982$	−36.0000	−1.14881
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−18.0000	−0.573819
$985$	−26.0000	−0.828429
$986$	−36.0000	−1.14647
$987$	0	0
$988$	8.00000	0.254514
$989$	0	0
$990$	4.00000	0.127128
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	5.00000	0.158750
$993$	4.00000	0.126936
$994$	−48.0000	−1.52247
$995$	0	0
$996$	−12.0000	−0.380235
$997$	30.0000	0.950110	0.475055	−	0.879956i	$-0.342428\pi$
$997$	30.0000	0.950110	0.475055	+	0.879956i	$0.342428\pi$
$998$	−4.00000	−0.126618
$999$	10.0000	0.316386

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.a.1.1	✓	1	1.1	even	1	trivial
1395.2.a.c.1.1		1	3.2	odd	2
2325.2.a.i.1.1		1	5.4	even	2
2325.2.c.c.1024.1		2	5.2	odd	4
2325.2.c.c.1024.2		2	5.3	odd	4
6975.2.a.g.1.1		1	15.14	odd	2
7440.2.a.o.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 $\F_{q}$
Belyi maps

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

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