LMFDB - Embedded newform 4650.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, 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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{18} -5.00000 q^{19} +1.00000 q^{21} -5.00000 q^{22} -9.00000 q^{23} +1.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} +1.00000 q^{31} +1.00000 q^{32} -5.00000 q^{33} +1.00000 q^{36} +8.00000 q^{37} -5.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} +1.00000 q^{42} +1.00000 q^{43} -5.00000 q^{44} -9.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -4.00000 q^{52} -13.0000 q^{53} +1.00000 q^{54} +1.00000 q^{56} -5.00000 q^{57} -2.00000 q^{58} +10.0000 q^{59} -14.0000 q^{61} +1.00000 q^{62} +1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +14.0000 q^{67} -9.00000 q^{69} -9.00000 q^{71} +1.00000 q^{72} -9.00000 q^{73} +8.00000 q^{74} -5.00000 q^{76} -5.00000 q^{77} -4.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +6.00000 q^{83} +1.00000 q^{84} +1.00000 q^{86} -2.00000 q^{87} -5.00000 q^{88} +3.00000 q^{89} -4.00000 q^{91} -9.00000 q^{92} +1.00000 q^{93} -12.0000 q^{94} +1.00000 q^{96} -18.0000 q^{97} -6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−5.00000	−1.06600
$23$	−9.00000	−1.87663	−0.938315	−	0.345782i	$-0.887614\pi$
$23$	−9.00000	−1.87663	−0.938315	+	0.345782i	$0.887614\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	0	0
$35$	0	0
$36$	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	$0.271573\pi$
$38$	−5.00000	−0.811107
$39$	−4.00000	−0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−9.00000	−1.32698
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−13.0000	−1.78569	−0.892844	−	0.450367i	$-0.851293\pi$
$53$	−13.0000	−1.78569	−0.892844	+	0.450367i	$0.851293\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.00000	0.133631
$57$	−5.00000	−0.662266
$58$	−2.00000	−0.262613
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−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
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	14.0000	1.71037	0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000	1.71037	0.855186	+	0.518321i	$0.173443\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	1.00000	0.117851
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−5.00000	−0.573539
$77$	−5.00000	−0.569803
$78$	−4.00000	−0.452911
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	1.00000	0.107833
$87$	−2.00000	−0.214423
$88$	−5.00000	−0.533002
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	−9.00000	−0.938315
$93$	1.00000	0.103695
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	−6.00000	−0.606092
$99$	−5.00000	−0.502519
$100$	0	0
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−13.0000	−1.26267
$107$	11.0000	1.06341	0.531705	−	0.846930i	$-0.321551\pi$
$107$	11.0000	1.06341	0.531705	+	0.846930i	$0.321551\pi$
$108$	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	$0.530535\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	1.00000	0.0944911
$113$	13.0000	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$113$	13.0000	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$114$	−5.00000	−0.468293
$115$	0	0
$116$	−2.00000	−0.185695
$117$	−4.00000	−0.369800
$118$	10.0000	0.920575
$119$	0	0
$120$	0	0
$121$	14.0000	1.27273
$122$	−14.0000	−1.26750
$123$	6.00000	0.541002
$124$	1.00000	0.0898027
$125$	0	0
$126$	1.00000	0.0890871
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	1.00000	0.0880451
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	−5.00000	−0.435194
$133$	−5.00000	−0.433555
$134$	14.0000	1.20942
$135$	0	0
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	−9.00000	−0.766131
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	−9.00000	−0.755263
$143$	20.0000	1.67248
$144$	1.00000	0.0833333
$145$	0	0
$146$	−9.00000	−0.744845
$147$	−6.00000	−0.494872
$148$	8.00000	0.657596
$149$	7.00000	0.573462	0.286731	−	0.958011i	$-0.407431\pi$
$149$	7.00000	0.573462	0.286731	+	0.958011i	$0.407431\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	−5.00000	−0.402911
$155$	0	0
$156$	−4.00000	−0.320256
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	5.00000	0.397779
$159$	−13.0000	−1.03097
$160$	0	0
$161$	−9.00000	−0.709299
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	6.00000	0.465690
$167$	−9.00000	−0.696441	−0.348220	−	0.937413i	$-0.613214\pi$
$167$	−9.00000	−0.696441	−0.348220	+	0.937413i	$0.613214\pi$
$168$	1.00000	0.0771517
$169$	3.00000	0.230769
$170$	0	0
$171$	−5.00000	−0.382360
$172$	1.00000	0.0762493
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	−5.00000	−0.376889
$177$	10.0000	0.751646
$178$	3.00000	0.224860
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−4.00000	−0.296500
$183$	−14.0000	−1.03491
$184$	−9.00000	−0.663489
$185$	0	0
$186$	1.00000	0.0733236
$187$	0	0
$188$	−12.0000	−0.875190
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	1.00000	0.0721688
$193$	−22.0000	−1.58359	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000	−1.58359	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	−5.00000	−0.355335
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	14.0000	0.987484
$202$	−7.00000	−0.492518
$203$	−2.00000	−0.140372
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	−9.00000	−0.625543
$208$	−4.00000	−0.277350
$209$	25.0000	1.72929
$210$	0	0
$211$	−13.0000	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$212$	−13.0000	−0.892844
$213$	−9.00000	−0.616670
$214$	11.0000	0.751945
$215$	0	0
$216$	1.00000	0.0680414
$217$	1.00000	0.0678844
$218$	−2.00000	−0.135457
$219$	−9.00000	−0.608164
$220$	0	0
$221$	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	$0.320040\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	13.0000	0.864747
$227$	15.0000	0.995585	0.497792	−	0.867296i	$-0.334144\pi$
$227$	15.0000	0.995585	0.497792	+	0.867296i	$0.334144\pi$
$228$	−5.00000	−0.331133
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	−5.00000	−0.328976
$232$	−2.00000	−0.131306
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	10.0000	0.650945
$237$	5.00000	0.324785
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	24.0000	1.54598	0.772988	−	0.634421i	$-0.218761\pi$
$241$	24.0000	1.54598	0.772988	+	0.634421i	$0.218761\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	−14.0000	−0.896258
$245$	0	0
$246$	6.00000	0.382546
$247$	20.0000	1.27257
$248$	1.00000	0.0635001
$249$	6.00000	0.380235
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	1.00000	0.0629941
$253$	45.0000	2.82913
$254$	20.0000	1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.00000	−0.436648	−0.218324	−	0.975876i	$-0.570059\pi$
$257$	−7.00000	−0.436648	−0.218324	+	0.975876i	$0.570059\pi$
$258$	1.00000	0.0622573
$259$	8.00000	0.497096
$260$	0	0
$261$	−2.00000	−0.123797
$262$	−6.00000	−0.370681
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	−5.00000	−0.306570
$267$	3.00000	0.183597
$268$	14.0000	0.855186
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−25.0000	−1.51864	−0.759321	−	0.650716i	$-0.774469\pi$
$271$	−25.0000	−1.51864	−0.759321	+	0.650716i	$0.774469\pi$
$272$	0	0
$273$	−4.00000	−0.242091
$274$	2.00000	0.120824
$275$	0	0
$276$	−9.00000	−0.541736
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	16.0000	0.959616
$279$	1.00000	0.0598684
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	−12.0000	−0.714590
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	20.0000	1.18262
$287$	6.00000	0.354169
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−18.0000	−1.05518
$292$	−9.00000	−0.526685
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	8.00000	0.464991
$297$	−5.00000	−0.290129
$298$	7.00000	0.405499
$299$	36.0000	2.08193
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	−7.00000	−0.402139
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	−5.00000	−0.284901
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	−4.00000	−0.226455
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	5.00000	0.281272
$317$	34.0000	1.90963	0.954815	−	0.297200i	$-0.0960529\pi$
$317$	34.0000	1.90963	0.954815	+	0.297200i	$0.0960529\pi$
$318$	−13.0000	−0.729004
$319$	10.0000	0.559893
$320$	0	0
$321$	11.0000	0.613960
$322$	−9.00000	−0.501550
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−2.00000	−0.110600
$328$	6.00000	0.331295
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	6.00000	0.329293
$333$	8.00000	0.438397
$334$	−9.00000	−0.492458
$335$	0	0
$336$	1.00000	0.0545545
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	3.00000	0.163178
$339$	13.0000	0.706063
$340$	0	0
$341$	−5.00000	−0.270765
$342$	−5.00000	−0.270369
$343$	−13.0000	−0.701934
$344$	1.00000	0.0539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−2.00000	−0.107211
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−5.00000	−0.266501
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	3.00000	0.159000
$357$	0	0
$358$	−16.0000	−0.845626
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	17.0000	0.893500
$363$	14.0000	0.734809
$364$	−4.00000	−0.209657
$365$	0	0
$366$	−14.0000	−0.731792
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−9.00000	−0.469157
$369$	6.00000	0.312348
$370$	0	0
$371$	−13.0000	−0.674926
$372$	1.00000	0.0518476
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	8.00000	0.412021
$378$	1.00000	0.0514344
$379$	−11.0000	−0.565032	−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000	−0.565032	−0.282516	+	0.959263i	$0.591169\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	8.00000	0.409316
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−22.0000	−1.11977
$387$	1.00000	0.0508329
$388$	−18.0000	−0.913812
$389$	−16.0000	−0.811232	−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000	−0.811232	−0.405616	+	0.914044i	$0.632943\pi$
$390$	0	0
$391$	0	0
$392$	−6.00000	−0.303046
$393$	−6.00000	−0.302660
$394$	−10.0000	−0.503793
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	−11.0000	−0.551380
$399$	−5.00000	−0.250313
$400$	0	0
$401$	−25.0000	−1.24844	−0.624220	−	0.781248i	$-0.714583\pi$
$401$	−25.0000	−1.24844	−0.624220	+	0.781248i	$0.714583\pi$
$402$	14.0000	0.698257
$403$	−4.00000	−0.199254
$404$	−7.00000	−0.348263
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−28.0000	−1.38451	−0.692255	−	0.721653i	$-0.743383\pi$
$409$	−28.0000	−1.38451	−0.692255	+	0.721653i	$0.743383\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	8.00000	0.394132
$413$	10.0000	0.492068
$414$	−9.00000	−0.442326
$415$	0	0
$416$	−4.00000	−0.196116
$417$	16.0000	0.783523
$418$	25.0000	1.22279
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−13.0000	−0.632830
$423$	−12.0000	−0.583460
$424$	−13.0000	−0.631336
$425$	0	0
$426$	−9.00000	−0.436051
$427$	−14.0000	−0.677507
$428$	11.0000	0.531705
$429$	20.0000	0.965609
$430$	0	0
$431$	−32.0000	−1.54139	−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000	−1.54139	−0.770693	+	0.637207i	$0.780090\pi$
$432$	1.00000	0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	1.00000	0.0480015
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	45.0000	2.15264
$438$	−9.00000	−0.430037
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	13.0000	0.617649	0.308824	−	0.951119i	$-0.400064\pi$
$443$	13.0000	0.617649	0.308824	+	0.951119i	$0.400064\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	16.0000	0.757622
$447$	7.00000	0.331089
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	−30.0000	−1.41264
$452$	13.0000	0.611469
$453$	0	0
$454$	15.0000	0.703985
$455$	0	0
$456$	−5.00000	−0.234146
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−19.0000	−0.887812
$459$	0	0
$460$	0	0
$461$	36.0000	1.67669	0.838344	−	0.545142i	$-0.183524\pi$
$461$	36.0000	1.67669	0.838344	+	0.545142i	$0.183524\pi$
$462$	−5.00000	−0.232621
$463$	30.0000	1.39422	0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000	1.39422	0.697109	+	0.716965i	$0.254469\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−1.00000	−0.0463241
$467$	20.0000	0.925490	0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000	0.925490	0.462745	+	0.886492i	$0.346865\pi$
$468$	−4.00000	−0.184900
$469$	14.0000	0.646460
$470$	0	0
$471$	13.0000	0.599008
$472$	10.0000	0.460287
$473$	−5.00000	−0.229900
$474$	5.00000	0.229658
$475$	0	0
$476$	0	0
$477$	−13.0000	−0.595229
$478$	−6.00000	−0.274434
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	−32.0000	−1.45907
$482$	24.0000	1.09317
$483$	−9.00000	−0.409514
$484$	14.0000	0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−14.0000	−0.633750
$489$	4.00000	0.180886
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	20.0000	0.899843
$495$	0	0
$496$	1.00000	0.0449013
$497$	−9.00000	−0.403705
$498$	6.00000	0.268866
$499$	6.00000	0.268597	0.134298	−	0.990941i	$-0.457122\pi$
$499$	6.00000	0.268597	0.134298	+	0.990941i	$0.457122\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	−28.0000	−1.24970
$503$	22.0000	0.980932	0.490466	−	0.871460i	$-0.336827\pi$
$503$	22.0000	0.980932	0.490466	+	0.871460i	$0.336827\pi$
$504$	1.00000	0.0445435
$505$	0	0
$506$	45.0000	2.00049
$507$	3.00000	0.133235
$508$	20.0000	0.887357
$509$	−20.0000	−0.886484	−0.443242	−	0.896402i	$-0.646172\pi$
$509$	−20.0000	−0.886484	−0.443242	+	0.896402i	$0.646172\pi$
$510$	0	0
$511$	−9.00000	−0.398137
$512$	1.00000	0.0441942
$513$	−5.00000	−0.220755
$514$	−7.00000	−0.308757
$515$	0	0
$516$	1.00000	0.0440225
$517$	60.0000	2.63880
$518$	8.00000	0.351500
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−36.0000	−1.57719	−0.788594	−	0.614914i	$-0.789191\pi$
$521$	−36.0000	−1.57719	−0.788594	+	0.614914i	$0.789191\pi$
$522$	−2.00000	−0.0875376
$523$	17.0000	0.743358	0.371679	−	0.928361i	$-0.378782\pi$
$523$	17.0000	0.743358	0.371679	+	0.928361i	$0.378782\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	−5.00000	−0.217597
$529$	58.0000	2.52174
$530$	0	0
$531$	10.0000	0.433963
$532$	−5.00000	−0.216777
$533$	−24.0000	−1.03956
$534$	3.00000	0.129823
$535$	0	0
$536$	14.0000	0.604708
$537$	−16.0000	−0.690451
$538$	−22.0000	−0.948487
$539$	30.0000	1.29219
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	−25.0000	−1.07384
$543$	17.0000	0.729540
$544$	0	0
$545$	0	0
$546$	−4.00000	−0.171184
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	2.00000	0.0854358
$549$	−14.0000	−0.597505
$550$	0	0
$551$	10.0000	0.426014
$552$	−9.00000	−0.383065
$553$	5.00000	0.212622
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	16.0000	0.678551
$557$	7.00000	0.296600	0.148300	−	0.988942i	$-0.452620\pi$
$557$	7.00000	0.296600	0.148300	+	0.988942i	$0.452620\pi$
$558$	1.00000	0.0423334
$559$	−4.00000	−0.169182
$560$	0	0
$561$	0	0
$562$	12.0000	0.506189
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	26.0000	1.09286
$567$	1.00000	0.0419961
$568$	−9.00000	−0.377632
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	20.0000	0.836242
$573$	8.00000	0.334205
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−17.0000	−0.707107
$579$	−22.0000	−0.914289
$580$	0	0
$581$	6.00000	0.248922
$582$	−18.0000	−0.746124
$583$	65.0000	2.69202
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−6.00000	−0.247647	−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000	−0.247647	−0.123823	+	0.992304i	$0.539516\pi$
$588$	−6.00000	−0.247436
$589$	−5.00000	−0.206021
$590$	0	0
$591$	−10.0000	−0.411345
$592$	8.00000	0.328798
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	7.00000	0.286731
$597$	−11.0000	−0.450200
$598$	36.0000	1.47215
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	1.00000	0.0407570
$603$	14.0000	0.570124
$604$	0	0
$605$	0	0
$606$	−7.00000	−0.284356
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	−5.00000	−0.202777
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−5.00000	−0.201456
$617$	9.00000	0.362326	0.181163	−	0.983453i	$-0.442014\pi$
$617$	9.00000	0.362326	0.181163	+	0.983453i	$0.442014\pi$
$618$	8.00000	0.321807
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	−8.00000	−0.320771
$623$	3.00000	0.120192
$624$	−4.00000	−0.160128
$625$	0	0
$626$	6.00000	0.239808
$627$	25.0000	0.998404
$628$	13.0000	0.518756
$629$	0	0
$630$	0	0
$631$	31.0000	1.23409	0.617045	−	0.786928i	$-0.288330\pi$
$631$	31.0000	1.23409	0.617045	+	0.786928i	$0.288330\pi$
$632$	5.00000	0.198889
$633$	−13.0000	−0.516704
$634$	34.0000	1.35031
$635$	0	0
$636$	−13.0000	−0.515484
$637$	24.0000	0.950915
$638$	10.0000	0.395904
$639$	−9.00000	−0.356034
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	11.0000	0.434135
$643$	−5.00000	−0.197181	−0.0985904	−	0.995128i	$-0.531433\pi$
$643$	−5.00000	−0.197181	−0.0985904	+	0.995128i	$0.531433\pi$
$644$	−9.00000	−0.354650
$645$	0	0
$646$	0	0
$647$	5.00000	0.196570	0.0982851	−	0.995158i	$-0.468664\pi$
$647$	5.00000	0.196570	0.0982851	+	0.995158i	$0.468664\pi$
$648$	1.00000	0.0392837
$649$	−50.0000	−1.96267
$650$	0	0
$651$	1.00000	0.0391931
$652$	4.00000	0.156652
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	−2.00000	−0.0782062
$655$	0	0
$656$	6.00000	0.234261
$657$	−9.00000	−0.351123
$658$	−12.0000	−0.467809
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	−24.0000	−0.932786
$663$	0	0
$664$	6.00000	0.232845
$665$	0	0
$666$	8.00000	0.309994
$667$	18.0000	0.696963
$668$	−9.00000	−0.348220
$669$	16.0000	0.618596
$670$	0	0
$671$	70.0000	2.70232
$672$	1.00000	0.0385758
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	3.00000	0.115385
$677$	51.0000	1.96009	0.980045	−	0.198778i	$-0.0636972\pi$
$677$	51.0000	1.96009	0.980045	+	0.198778i	$0.0636972\pi$
$678$	13.0000	0.499262
$679$	−18.0000	−0.690777
$680$	0	0
$681$	15.0000	0.574801
$682$	−5.00000	−0.191460
$683$	3.00000	0.114792	0.0573959	−	0.998351i	$-0.481720\pi$
$683$	3.00000	0.114792	0.0573959	+	0.998351i	$0.481720\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	−13.0000	−0.496342
$687$	−19.0000	−0.724895
$688$	1.00000	0.0381246
$689$	52.0000	1.98104
$690$	0	0
$691$	27.0000	1.02713	0.513564	−	0.858051i	$-0.328325\pi$
$691$	27.0000	1.02713	0.513564	+	0.858051i	$0.328325\pi$
$692$	−6.00000	−0.228086
$693$	−5.00000	−0.189934
$694$	−18.0000	−0.683271
$695$	0	0
$696$	−2.00000	−0.0758098
$697$	0	0
$698$	4.00000	0.151402
$699$	−1.00000	−0.0378235
$700$	0	0
$701$	29.0000	1.09531	0.547657	−	0.836703i	$-0.315520\pi$
$701$	29.0000	1.09531	0.547657	+	0.836703i	$0.315520\pi$
$702$	−4.00000	−0.150970
$703$	−40.0000	−1.50863
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−4.00000	−0.150542
$707$	−7.00000	−0.263262
$708$	10.0000	0.375823
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	5.00000	0.187515
$712$	3.00000	0.112430
$713$	−9.00000	−0.337053
$714$	0	0
$715$	0	0
$716$	−16.0000	−0.597948
$717$	−6.00000	−0.224074
$718$	−9.00000	−0.335877
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	6.00000	0.223297
$723$	24.0000	0.892570
$724$	17.0000	0.631800
$725$	0	0
$726$	14.0000	0.519589
$727$	1.00000	0.0370879	0.0185440	−	0.999828i	$-0.494097\pi$
$727$	1.00000	0.0370879	0.0185440	+	0.999828i	$0.494097\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	−14.0000	−0.517455
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	18.0000	0.664392
$735$	0	0
$736$	−9.00000	−0.331744
$737$	−70.0000	−2.57848
$738$	6.00000	0.220863
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	20.0000	0.734718
$742$	−13.0000	−0.477245
$743$	−43.0000	−1.57752	−0.788759	−	0.614703i	$-0.789276\pi$
$743$	−43.0000	−1.57752	−0.788759	+	0.614703i	$0.789276\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	1.00000	0.0366126
$747$	6.00000	0.219529
$748$	0	0
$749$	11.0000	0.401931
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	−12.0000	−0.437595
$753$	−28.0000	−1.02038
$754$	8.00000	0.291343
$755$	0	0
$756$	1.00000	0.0363696
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	−11.0000	−0.399538
$759$	45.0000	1.63340
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	20.0000	0.724524
$763$	−2.00000	−0.0724049
$764$	8.00000	0.289430
$765$	0	0
$766$	12.0000	0.433578
$767$	−40.0000	−1.44432
$768$	1.00000	0.0360844
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	−7.00000	−0.252099
$772$	−22.0000	−0.791797
$773$	−27.0000	−0.971123	−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000	−0.971123	−0.485561	+	0.874203i	$0.661385\pi$
$774$	1.00000	0.0359443
$775$	0	0
$776$	−18.0000	−0.646162
$777$	8.00000	0.286998
$778$	−16.0000	−0.573628
$779$	−30.0000	−1.07486
$780$	0	0
$781$	45.0000	1.61023
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	−6.00000	−0.214286
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	−10.0000	−0.356235
$789$	−8.00000	−0.284808
$790$	0	0
$791$	13.0000	0.462227
$792$	−5.00000	−0.177667
$793$	56.0000	1.98862
$794$	−3.00000	−0.106466
$795$	0	0
$796$	−11.0000	−0.389885
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	−5.00000	−0.176998
$799$	0	0
$800$	0	0
$801$	3.00000	0.106000
$802$	−25.0000	−0.882781
$803$	45.0000	1.58802
$804$	14.0000	0.493742
$805$	0	0
$806$	−4.00000	−0.140894
$807$	−22.0000	−0.774437
$808$	−7.00000	−0.246259
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	55.0000	1.93131	0.965656	−	0.259825i	$-0.0836650\pi$
$811$	55.0000	1.93131	0.965656	+	0.259825i	$0.0836650\pi$
$812$	−2.00000	−0.0701862
$813$	−25.0000	−0.876788
$814$	−40.0000	−1.40200
$815$	0	0
$816$	0	0
$817$	−5.00000	−0.174928
$818$	−28.0000	−0.978997
$819$	−4.00000	−0.139771
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	2.00000	0.0697580
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	10.0000	0.347945
$827$	−46.0000	−1.59958	−0.799788	−	0.600282i	$-0.795055\pi$
$827$	−46.0000	−1.59958	−0.799788	+	0.600282i	$0.795055\pi$
$828$	−9.00000	−0.312772
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	−4.00000	−0.138675
$833$	0	0
$834$	16.0000	0.554035
$835$	0	0
$836$	25.0000	0.864643
$837$	1.00000	0.0345651
$838$	30.0000	1.03633
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−26.0000	−0.896019
$843$	12.0000	0.413302
$844$	−13.0000	−0.447478
$845$	0	0
$846$	−12.0000	−0.412568
$847$	14.0000	0.481046
$848$	−13.0000	−0.446422
$849$	26.0000	0.892318
$850$	0	0
$851$	−72.0000	−2.46813
$852$	−9.00000	−0.308335
$853$	21.0000	0.719026	0.359513	−	0.933140i	$-0.382943\pi$
$853$	21.0000	0.719026	0.359513	+	0.933140i	$0.382943\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	11.0000	0.375972
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	20.0000	0.682789
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	−32.0000	−1.08992
$863$	−31.0000	−1.05525	−0.527626	−	0.849477i	$-0.676918\pi$
$863$	−31.0000	−1.05525	−0.527626	+	0.849477i	$0.676918\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−11.0000	−0.373795
$867$	−17.0000	−0.577350
$868$	1.00000	0.0339422
$869$	−25.0000	−0.848067
$870$	0	0
$871$	−56.0000	−1.89749
$872$	−2.00000	−0.0677285
$873$	−18.0000	−0.609208
$874$	45.0000	1.52215
$875$	0	0
$876$	−9.00000	−0.304082
$877$	−6.00000	−0.202606	−0.101303	−	0.994856i	$-0.532301\pi$
$877$	−6.00000	−0.202606	−0.101303	+	0.994856i	$0.532301\pi$
$878$	20.0000	0.674967
$879$	−14.0000	−0.472208
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	−6.00000	−0.202031
$883$	23.0000	0.774012	0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000	0.774012	0.387006	+	0.922077i	$0.373509\pi$
$884$	0	0
$885$	0	0
$886$	13.0000	0.436744
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	8.00000	0.268462
$889$	20.0000	0.670778
$890$	0	0
$891$	−5.00000	−0.167506
$892$	16.0000	0.535720
$893$	60.0000	2.00782
$894$	7.00000	0.234115
$895$	0	0
$896$	1.00000	0.0334077
$897$	36.0000	1.20201
$898$	6.00000	0.200223
$899$	−2.00000	−0.0667037
$900$	0	0
$901$	0	0
$902$	−30.0000	−0.998891
$903$	1.00000	0.0332779
$904$	13.0000	0.432374
$905$	0	0
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	15.0000	0.497792
$909$	−7.00000	−0.232175
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	−5.00000	−0.165567
$913$	−30.0000	−0.992855
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−19.0000	−0.627778
$917$	−6.00000	−0.198137
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	−8.00000	−0.263609
$922$	36.0000	1.18560
$923$	36.0000	1.18495
$924$	−5.00000	−0.164488
$925$	0	0
$926$	30.0000	0.985861
$927$	8.00000	0.262754
$928$	−2.00000	−0.0656532
$929$	−11.0000	−0.360898	−0.180449	−	0.983584i	$-0.557755\pi$
$929$	−11.0000	−0.360898	−0.180449	+	0.983584i	$0.557755\pi$
$930$	0	0
$931$	30.0000	0.983210
$932$	−1.00000	−0.0327561
$933$	−8.00000	−0.261908
$934$	20.0000	0.654420
$935$	0	0
$936$	−4.00000	−0.130744
$937$	−54.0000	−1.76410	−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000	−1.76410	−0.882052	+	0.471153i	$0.843838\pi$
$938$	14.0000	0.457116
$939$	6.00000	0.195803
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	13.0000	0.423563
$943$	−54.0000	−1.75848
$944$	10.0000	0.325472
$945$	0	0
$946$	−5.00000	−0.162564
$947$	−10.0000	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$947$	−10.0000	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$948$	5.00000	0.162392
$949$	36.0000	1.16861
$950$	0	0
$951$	34.0000	1.10253
$952$	0	0
$953$	−52.0000	−1.68445	−0.842223	−	0.539130i	$-0.818753\pi$
$953$	−52.0000	−1.68445	−0.842223	+	0.539130i	$0.818753\pi$
$954$	−13.0000	−0.420891
$955$	0	0
$956$	−6.00000	−0.194054
$957$	10.0000	0.323254
$958$	−21.0000	−0.678479
$959$	2.00000	0.0645834
$960$	0	0
$961$	1.00000	0.0322581
$962$	−32.0000	−1.03172
$963$	11.0000	0.354470
$964$	24.0000	0.772988
$965$	0	0
$966$	−9.00000	−0.289570
$967$	−20.0000	−0.643157	−0.321578	−	0.946883i	$-0.604213\pi$
$967$	−20.0000	−0.643157	−0.321578	+	0.946883i	$0.604213\pi$
$968$	14.0000	0.449977
$969$	0	0
$970$	0	0
$971$	2.00000	0.0641831	0.0320915	−	0.999485i	$-0.489783\pi$
$971$	2.00000	0.0641831	0.0320915	+	0.999485i	$0.489783\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	24.0000	0.769010
$975$	0	0
$976$	−14.0000	−0.448129
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	4.00000	0.127906
$979$	−15.0000	−0.479402
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	15.0000	0.478669
$983$	−52.0000	−1.65854	−0.829271	−	0.558846i	$-0.811244\pi$
$983$	−52.0000	−1.65854	−0.829271	+	0.558846i	$0.811244\pi$
$984$	6.00000	0.191273
$985$	0	0
$986$	0	0
$987$	−12.0000	−0.381964
$988$	20.0000	0.636285
$989$	−9.00000	−0.286183
$990$	0	0
$991$	−3.00000	−0.0952981	−0.0476491	−	0.998864i	$-0.515173\pi$
$991$	−3.00000	−0.0952981	−0.0476491	+	0.998864i	$0.515173\pi$
$992$	1.00000	0.0317500
$993$	−24.0000	−0.761617
$994$	−9.00000	−0.285463
$995$	0	0
$996$	6.00000	0.190117
$997$	−18.0000	−0.570066	−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000	−0.570066	−0.285033	+	0.958518i	$0.592005\pi$
$998$	6.00000	0.189927
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.bq.1.1		1
5.2	odd	4		930.2.d.b.559.2	yes	2
5.3	odd	4		930.2.d.b.559.1	✓	2
5.4	even	2		4650.2.a.f.1.1		1
15.2	even	4		2790.2.d.g.559.1		2
15.8	even	4		2790.2.d.g.559.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.d.b.559.1	✓	2	5.3	odd	4
930.2.d.b.559.2	yes	2	5.2	odd	4
2790.2.d.g.559.1		2	15.2	even	4
2790.2.d.g.559.2		2	15.8	even	4
4650.2.a.f.1.1		1	5.4	even	2
4650.2.a.bq.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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