LMFDB - Embedded newform 350.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,2,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("350.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	350.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:	$2.79476407074$
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	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{9} +3.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{18} -7.00000 q^{19} +1.00000 q^{21} -3.00000 q^{22} +1.00000 q^{24} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +3.00000 q^{34} -2.00000 q^{36} -8.00000 q^{37} +7.00000 q^{38} +2.00000 q^{39} -9.00000 q^{41} -1.00000 q^{42} -8.00000 q^{43} +3.00000 q^{44} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} -2.00000 q^{52} +12.0000 q^{53} -5.00000 q^{54} +1.00000 q^{56} +7.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} +7.00000 q^{67} -3.00000 q^{68} +6.00000 q^{71} +2.00000 q^{72} -5.00000 q^{73} +8.00000 q^{74} -7.00000 q^{76} -3.00000 q^{77} -2.00000 q^{78} +14.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} +9.00000 q^{83} +1.00000 q^{84} +8.00000 q^{86} +6.00000 q^{87} -3.00000 q^{88} -15.0000 q^{89} +2.00000 q^{91} +4.00000 q^{93} -6.00000 q^{94} +1.00000 q^{96} +10.0000 q^{97} -1.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	2.00000	0.471405
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	2.00000	0.392232
$27$	5.00000	0.962250
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	7.00000	1.13555
$39$	2.00000	0.320256
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	−1.00000	−0.154303
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	−2.00000	−0.277350
$53$	12.0000	1.64833	0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000	1.64833	0.824163	+	0.566352i	$0.191646\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	1.00000	0.133631
$57$	7.00000	0.927173
$58$	6.00000	0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	2.00000	0.235702
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−7.00000	−0.802955
$77$	−3.00000	−0.341882
$78$	−2.00000	−0.226455
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	9.00000	0.993884
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	1.00000	0.109109
$85$	0	0
$86$	8.00000	0.862662
$87$	6.00000	0.643268
$88$	−3.00000	−0.319801
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	4.00000	0.414781
$94$	−6.00000	−0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	−1.00000	−0.101015
$99$	−6.00000	−0.603023
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−3.00000	−0.297044
$103$	−20.0000	−1.97066	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−20.0000	−1.97066	−0.985329	+	0.170664i	$0.945409\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−12.0000	−1.16554
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	5.00000	0.481125
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	−1.00000	−0.0944911
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−7.00000	−0.655610
$115$	0	0
$116$	−6.00000	−0.557086
$117$	4.00000	0.369800
$118$	−12.0000	−1.10469
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	10.0000	0.905357
$123$	9.00000	0.811503
$124$	−4.00000	−0.359211
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−1.00000	−0.0883883
$129$	8.00000	0.704361
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	−3.00000	−0.261116
$133$	7.00000	0.606977
$134$	−7.00000	−0.604708
$135$	0	0
$136$	3.00000	0.257248
$137$	−21.0000	−1.79415	−0.897076	−	0.441877i	$-0.854313\pi$
$137$	−21.0000	−1.79415	−0.897076	+	0.441877i	$0.854313\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−6.00000	−0.503509
$143$	−6.00000	−0.501745
$144$	−2.00000	−0.166667
$145$	0	0
$146$	5.00000	0.413803
$147$	−1.00000	−0.0824786
$148$	−8.00000	−0.657596
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	7.00000	0.567775
$153$	6.00000	0.485071
$154$	3.00000	0.241747
$155$	0	0
$156$	2.00000	0.160128
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	−14.0000	−1.11378
$159$	−12.0000	−0.951662
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−5.00000	−0.391630	−0.195815	−	0.980641i	$-0.562735\pi$
$163$	−5.00000	−0.391630	−0.195815	+	0.980641i	$0.562735\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−9.00000	−0.698535
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	−1.00000	−0.0771517
$169$	−9.00000	−0.692308
$170$	0	0
$171$	14.0000	1.07061
$172$	−8.00000	−0.609994
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	3.00000	0.226134
$177$	−12.0000	−0.901975
$178$	15.0000	1.12430
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−2.00000	−0.148250
$183$	10.0000	0.739221
$184$	0	0
$185$	0	0
$186$	−4.00000	−0.293294
$187$	−9.00000	−0.658145
$188$	6.00000	0.437595
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	−1.00000	−0.0721688
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	6.00000	0.426401
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	6.00000	0.421117
$204$	3.00000	0.210042
$205$	0	0
$206$	20.0000	1.39347
$207$	0	0
$208$	−2.00000	−0.138675
$209$	−21.0000	−1.45260
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	12.0000	0.824163
$213$	−6.00000	−0.411113
$214$	3.00000	0.205076
$215$	0	0
$216$	−5.00000	−0.340207
$217$	4.00000	0.271538
$218$	−14.0000	−0.948200
$219$	5.00000	0.337869
$220$	0	0
$221$	6.00000	0.403604
$222$	−8.00000	−0.536925
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	7.00000	0.463586
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−4.00000	−0.261488
$235$	0	0
$236$	12.0000	0.781133
$237$	−14.0000	−0.909398
$238$	−3.00000	−0.194461
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	2.00000	0.128565
$243$	−16.0000	−1.02640
$244$	−10.0000	−0.640184
$245$	0	0
$246$	−9.00000	−0.573819
$247$	14.0000	0.890799
$248$	4.00000	0.254000
$249$	−9.00000	−0.570352
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	−8.00000	−0.498058
$259$	8.00000	0.497096
$260$	0	0
$261$	12.0000	0.742781
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	3.00000	0.184637
$265$	0	0
$266$	−7.00000	−0.429198
$267$	15.0000	0.917985
$268$	7.00000	0.427593
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−3.00000	−0.181902
$273$	−2.00000	−0.121046
$274$	21.0000	1.26866
$275$	0	0
$276$	0	0
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	7.00000	0.419832
$279$	8.00000	0.478947
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	6.00000	0.357295
$283$	1.00000	0.0594438	0.0297219	−	0.999558i	$-0.490538\pi$
$283$	1.00000	0.0594438	0.0297219	+	0.999558i	$0.490538\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	6.00000	0.354787
$287$	9.00000	0.531253
$288$	2.00000	0.117851
$289$	−8.00000	−0.470588
$290$	0	0
$291$	−10.0000	−0.586210
$292$	−5.00000	−0.292603
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	8.00000	0.464991
$297$	15.0000	0.870388
$298$	−12.0000	−0.695141
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−7.00000	−0.401478
$305$	0	0
$306$	−6.00000	−0.342997
$307$	7.00000	0.399511	0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000	0.399511	0.199756	+	0.979846i	$0.435985\pi$
$308$	−3.00000	−0.170941
$309$	20.0000	1.13776
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−2.00000	−0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	14.0000	0.787562
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	12.0000	0.672927
$319$	−18.0000	−1.00781
$320$	0	0
$321$	3.00000	0.167444
$322$	0	0
$323$	21.0000	1.16847
$324$	1.00000	0.0555556
$325$	0	0
$326$	5.00000	0.276924
$327$	−14.0000	−0.774202
$328$	9.00000	0.496942
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	9.00000	0.493939
$333$	16.0000	0.876795
$334$	12.0000	0.656611
$335$	0	0
$336$	1.00000	0.0545545
$337$	13.0000	0.708155	0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000	0.708155	0.354078	+	0.935216i	$0.384795\pi$
$338$	9.00000	0.489535
$339$	−9.00000	−0.488813
$340$	0	0
$341$	−12.0000	−0.649836
$342$	−14.0000	−0.757033
$343$	−1.00000	−0.0539949
$344$	8.00000	0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	21.0000	1.12734	0.563670	−	0.826000i	$-0.309389\pi$
$347$	21.0000	1.12734	0.563670	+	0.826000i	$0.309389\pi$
$348$	6.00000	0.321634
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	−10.0000	−0.533761
$352$	−3.00000	−0.159901
$353$	−30.0000	−1.59674	−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000	−1.59674	−0.798369	+	0.602168i	$0.794304\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−15.0000	−0.794998
$357$	−3.00000	−0.158777
$358$	−3.00000	−0.158555
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−2.00000	−0.105118
$363$	2.00000	0.104973
$364$	2.00000	0.104828
$365$	0	0
$366$	−10.0000	−0.522708
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	18.0000	0.937043
$370$	0	0
$371$	−12.0000	−0.623009
$372$	4.00000	0.207390
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000	0.618031
$378$	5.00000	0.257172
$379$	17.0000	0.873231	0.436616	−	0.899648i	$-0.356177\pi$
$379$	17.0000	0.873231	0.436616	+	0.899648i	$0.356177\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	6.00000	0.306987
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	5.00000	0.254493
$387$	16.0000	0.813326
$388$	10.0000	0.507673
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−6.00000	−0.301511
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−14.0000	−0.701757
$399$	−7.00000	−0.350438
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	7.00000	0.349128
$403$	8.00000	0.398508
$404$	0	0
$405$	0	0
$406$	−6.00000	−0.297775
$407$	−24.0000	−1.18964
$408$	−3.00000	−0.148522
$409$	−25.0000	−1.23617	−0.618085	−	0.786111i	$-0.712091\pi$
$409$	−25.0000	−1.23617	−0.618085	+	0.786111i	$0.712091\pi$
$410$	0	0
$411$	21.0000	1.03585
$412$	−20.0000	−0.985329
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	2.00000	0.0980581
$417$	7.00000	0.342791
$418$	21.0000	1.02714
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−17.0000	−0.827547
$423$	−12.0000	−0.583460
$424$	−12.0000	−0.582772
$425$	0	0
$426$	6.00000	0.290701
$427$	10.0000	0.483934
$428$	−3.00000	−0.145010
$429$	6.00000	0.289683
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	5.00000	0.240563
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	−5.00000	−0.238909
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−2.00000	−0.0952381
$442$	−6.00000	−0.285391
$443$	−21.0000	−0.997740	−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000	−0.997740	−0.498870	+	0.866677i	$0.666252\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	14.0000	0.662919
$447$	−12.0000	−0.567581
$448$	−1.00000	−0.0472456
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	9.00000	0.423324
$453$	−8.00000	−0.375873
$454$	12.0000	0.563188
$455$	0	0
$456$	−7.00000	−0.327805
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	−26.0000	−1.21490
$459$	−15.0000	−0.700140
$460$	0	0
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	−3.00000	−0.139573
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	4.00000	0.184900
$469$	−7.00000	−0.323230
$470$	0	0
$471$	20.0000	0.921551
$472$	−12.0000	−0.552345
$473$	−24.0000	−1.10352
$474$	14.0000	0.643041
$475$	0	0
$476$	3.00000	0.137505
$477$	−24.0000	−1.09888
$478$	12.0000	0.548867
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	25.0000	1.13872
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	0	0
$486$	16.0000	0.725775
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	10.0000	0.452679
$489$	5.00000	0.226108
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	9.00000	0.405751
$493$	18.0000	0.810679
$494$	−14.0000	−0.629890
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−6.00000	−0.269137
$498$	9.00000	0.403300
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	−15.0000	−0.669483
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	−2.00000	−0.0887357
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	−1.00000	−0.0441942
$513$	−35.0000	−1.54529
$514$	−30.0000	−1.32324
$515$	0	0
$516$	8.00000	0.352180
$517$	18.0000	0.791639
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	0	0
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	−12.0000	−0.525226
$523$	7.00000	0.306089	0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000	0.306089	0.153044	+	0.988219i	$0.451092\pi$
$524$	0	0
$525$	0	0
$526$	6.00000	0.261612
$527$	12.0000	0.522728
$528$	−3.00000	−0.130558
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−24.0000	−1.04151
$532$	7.00000	0.303488
$533$	18.0000	0.779667
$534$	−15.0000	−0.649113
$535$	0	0
$536$	−7.00000	−0.302354
$537$	−3.00000	−0.129460
$538$	6.00000	0.258678
$539$	3.00000	0.129219
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	−2.00000	−0.0859074
$543$	−2.00000	−0.0858282
$544$	3.00000	0.128624
$545$	0	0
$546$	2.00000	0.0855921
$547$	−35.0000	−1.49649	−0.748246	−	0.663421i	$-0.769104\pi$
$547$	−35.0000	−1.49649	−0.748246	+	0.663421i	$0.769104\pi$
$548$	−21.0000	−0.897076
$549$	20.0000	0.853579
$550$	0	0
$551$	42.0000	1.78926
$552$	0	0
$553$	−14.0000	−0.595341
$554$	2.00000	0.0849719
$555$	0	0
$556$	−7.00000	−0.296866
$557$	−36.0000	−1.52537	−0.762684	−	0.646771i	$-0.776119\pi$
$557$	−36.0000	−1.52537	−0.762684	+	0.646771i	$0.776119\pi$
$558$	−8.00000	−0.338667
$559$	16.0000	0.676728
$560$	0	0
$561$	9.00000	0.379980
$562$	18.0000	0.759284
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−1.00000	−0.0420331
$567$	−1.00000	−0.0419961
$568$	−6.00000	−0.251754
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	−6.00000	−0.250873
$573$	6.00000	0.250654
$574$	−9.00000	−0.375653
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	7.00000	0.291414	0.145707	−	0.989328i	$-0.453454\pi$
$577$	7.00000	0.291414	0.145707	+	0.989328i	$0.453454\pi$
$578$	8.00000	0.332756
$579$	5.00000	0.207793
$580$	0	0
$581$	−9.00000	−0.373383
$582$	10.0000	0.414513
$583$	36.0000	1.49097
$584$	5.00000	0.206901
$585$	0	0
$586$	−6.00000	−0.247858
$587$	−39.0000	−1.60970	−0.804851	−	0.593477i	$-0.797755\pi$
$587$	−39.0000	−1.60970	−0.804851	+	0.593477i	$0.797755\pi$
$588$	−1.00000	−0.0412393
$589$	28.0000	1.15372
$590$	0	0
$591$	−6.00000	−0.246807
$592$	−8.00000	−0.328798
$593$	27.0000	1.10876	0.554379	−	0.832265i	$-0.312956\pi$
$593$	27.0000	1.10876	0.554379	+	0.832265i	$0.312956\pi$
$594$	−15.0000	−0.615457
$595$	0	0
$596$	12.0000	0.491539
$597$	−14.0000	−0.572982
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	−8.00000	−0.326056
$603$	−14.0000	−0.570124
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	−44.0000	−1.78590	−0.892952	−	0.450151i	$-0.851370\pi$
$607$	−44.0000	−1.78590	−0.892952	+	0.450151i	$0.851370\pi$
$608$	7.00000	0.283887
$609$	−6.00000	−0.243132
$610$	0	0
$611$	−12.0000	−0.485468
$612$	6.00000	0.242536
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	3.00000	0.120873
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	−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	$0.690240\pi$
$620$	0	0
$621$	0	0
$622$	18.0000	0.721734
$623$	15.0000	0.600962
$624$	2.00000	0.0800641
$625$	0	0
$626$	−10.0000	−0.399680
$627$	21.0000	0.838659
$628$	−20.0000	−0.798087
$629$	24.0000	0.956943
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−14.0000	−0.556890
$633$	−17.0000	−0.675689
$634$	−12.0000	−0.476581
$635$	0	0
$636$	−12.0000	−0.475831
$637$	−2.00000	−0.0792429
$638$	18.0000	0.712627
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−3.00000	−0.118401
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	0	0
$645$	0	0
$646$	−21.0000	−0.826234
$647$	24.0000	0.943537	0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000	0.943537	0.471769	+	0.881722i	$0.343616\pi$
$648$	−1.00000	−0.0392837
$649$	36.0000	1.41312
$650$	0	0
$651$	−4.00000	−0.156772
$652$	−5.00000	−0.195815
$653$	−36.0000	−1.40879	−0.704394	−	0.709809i	$-0.748781\pi$
$653$	−36.0000	−1.40879	−0.704394	+	0.709809i	$0.748781\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	−9.00000	−0.351391
$657$	10.0000	0.390137
$658$	6.00000	0.233904
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	25.0000	0.971653
$663$	−6.00000	−0.233021
$664$	−9.00000	−0.349268
$665$	0	0
$666$	−16.0000	−0.619987
$667$	0	0
$668$	−12.0000	−0.464294
$669$	14.0000	0.541271
$670$	0	0
$671$	−30.0000	−1.15814
$672$	−1.00000	−0.0385758
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−9.00000	−0.346154
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	9.00000	0.345643
$679$	−10.0000	−0.383765
$680$	0	0
$681$	12.0000	0.459841
$682$	12.0000	0.459504
$683$	−3.00000	−0.114792	−0.0573959	−	0.998351i	$-0.518280\pi$
$683$	−3.00000	−0.114792	−0.0573959	+	0.998351i	$0.518280\pi$
$684$	14.0000	0.535303
$685$	0	0
$686$	1.00000	0.0381802
$687$	−26.0000	−0.991962
$688$	−8.00000	−0.304997
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	6.00000	0.228086
$693$	6.00000	0.227921
$694$	−21.0000	−0.797149
$695$	0	0
$696$	−6.00000	−0.227429
$697$	27.0000	1.02270
$698$	−8.00000	−0.302804
$699$	6.00000	0.226941
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	10.0000	0.377426
$703$	56.0000	2.11208
$704$	3.00000	0.113067
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	−12.0000	−0.450988
$709$	−28.0000	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$709$	−28.0000	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$710$	0	0
$711$	−28.0000	−1.05008
$712$	15.0000	0.562149
$713$	0	0
$714$	3.00000	0.112272
$715$	0	0
$716$	3.00000	0.112115
$717$	12.0000	0.448148
$718$	6.00000	0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	−30.0000	−1.11648
$723$	25.0000	0.929760
$724$	2.00000	0.0743294
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	34.0000	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$727$	34.0000	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$728$	−2.00000	−0.0741249
$729$	13.0000	0.481481
$730$	0	0
$731$	24.0000	0.887672
$732$	10.0000	0.369611
$733$	40.0000	1.47743	0.738717	−	0.674016i	$-0.235432\pi$
$733$	40.0000	1.47743	0.738717	+	0.674016i	$0.235432\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	21.0000	0.773545
$738$	−18.0000	−0.662589
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	−14.0000	−0.514303
$742$	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	$0.354894\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	−4.00000	−0.146450
$747$	−18.0000	−0.658586
$748$	−9.00000	−0.329073
$749$	3.00000	0.109618
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	6.00000	0.218797
$753$	−15.0000	−0.546630
$754$	−12.0000	−0.437014
$755$	0	0
$756$	−5.00000	−0.181848
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	−17.0000	−0.617468
$759$	0	0
$760$	0	0
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	−2.00000	−0.0724524
$763$	−14.0000	−0.506834
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−30.0000	−1.08394
$767$	−24.0000	−0.866590
$768$	−1.00000	−0.0360844
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	−5.00000	−0.179954
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	−16.0000	−0.575108
$775$	0	0
$776$	−10.0000	−0.358979
$777$	−8.00000	−0.286998
$778$	24.0000	0.860442
$779$	63.0000	2.25721
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	−30.0000	−1.07211
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	6.00000	0.213741
$789$	6.00000	0.213606
$790$	0	0
$791$	−9.00000	−0.320003
$792$	6.00000	0.213201
$793$	20.0000	0.710221
$794$	2.00000	0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	7.00000	0.247797
$799$	−18.0000	−0.636794
$800$	0	0
$801$	30.0000	1.06000
$802$	27.0000	0.953403
$803$	−15.0000	−0.529339
$804$	−7.00000	−0.246871
$805$	0	0
$806$	−8.00000	−0.281788
$807$	6.00000	0.211210
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	6.00000	0.210559
$813$	−2.00000	−0.0701431
$814$	24.0000	0.841200
$815$	0	0
$816$	3.00000	0.105021
$817$	56.0000	1.95919
$818$	25.0000	0.874105
$819$	−4.00000	−0.139771
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	−21.0000	−0.732459
$823$	−26.0000	−0.906303	−0.453152	−	0.891434i	$-0.649700\pi$
$823$	−26.0000	−0.906303	−0.453152	+	0.891434i	$0.649700\pi$
$824$	20.0000	0.696733
$825$	0	0
$826$	12.0000	0.417533
$827$	−9.00000	−0.312961	−0.156480	−	0.987681i	$-0.550015\pi$
$827$	−9.00000	−0.312961	−0.156480	+	0.987681i	$0.550015\pi$
$828$	0	0
$829$	−4.00000	−0.138926	−0.0694629	−	0.997585i	$-0.522129\pi$
$829$	−4.00000	−0.138926	−0.0694629	+	0.997585i	$0.522129\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−2.00000	−0.0693375
$833$	−3.00000	−0.103944
$834$	−7.00000	−0.242390
$835$	0	0
$836$	−21.0000	−0.726300
$837$	−20.0000	−0.691301
$838$	3.00000	0.103633
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−20.0000	−0.689246
$843$	18.0000	0.619953
$844$	17.0000	0.585164
$845$	0	0
$846$	12.0000	0.412568
$847$	2.00000	0.0687208
$848$	12.0000	0.412082
$849$	−1.00000	−0.0343199
$850$	0	0
$851$	0	0
$852$	−6.00000	−0.205557
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	3.00000	0.102538
$857$	−15.0000	−0.512390	−0.256195	−	0.966625i	$-0.582469\pi$
$857$	−15.0000	−0.512390	−0.256195	+	0.966625i	$0.582469\pi$
$858$	−6.00000	−0.204837
$859$	−31.0000	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$859$	−31.0000	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	36.0000	1.22616
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	11.0000	0.373795
$867$	8.00000	0.271694
$868$	4.00000	0.135769
$869$	42.0000	1.42475
$870$	0	0
$871$	−14.0000	−0.474372
$872$	−14.0000	−0.474100
$873$	−20.0000	−0.676897
$874$	0	0
$875$	0	0
$876$	5.00000	0.168934
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	4.00000	0.134993
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−6.00000	−0.202145	−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000	−0.202145	−0.101073	+	0.994879i	$0.532227\pi$
$882$	2.00000	0.0673435
$883$	−47.0000	−1.58168	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000	−1.58168	−0.790838	+	0.612026i	$0.790355\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	21.0000	0.705509
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−8.00000	−0.268462
$889$	2.00000	0.0670778
$890$	0	0
$891$	3.00000	0.100504
$892$	−14.0000	−0.468755
$893$	−42.0000	−1.40548
$894$	12.0000	0.401340
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	15.0000	0.500556
$899$	24.0000	0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	27.0000	0.899002
$903$	−8.00000	−0.266223
$904$	−9.00000	−0.299336
$905$	0	0
$906$	8.00000	0.265782
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	0	0
$911$	36.0000	1.19273	0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000	1.19273	0.596367	+	0.802712i	$0.296610\pi$
$912$	7.00000	0.231793
$913$	27.0000	0.893570
$914$	17.0000	0.562310
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	15.0000	0.495074
$919$	−34.0000	−1.12156	−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000	−1.12156	−0.560778	+	0.827966i	$0.689498\pi$
$920$	0	0
$921$	−7.00000	−0.230658
$922$	−18.0000	−0.592798
$923$	−12.0000	−0.394985
$924$	3.00000	0.0986928
$925$	0	0
$926$	8.00000	0.262896
$927$	40.0000	1.31377
$928$	6.00000	0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−6.00000	−0.196537
$933$	18.0000	0.589294
$934$	36.0000	1.17796
$935$	0	0
$936$	−4.00000	−0.130744
$937$	−29.0000	−0.947389	−0.473694	−	0.880689i	$-0.657080\pi$
$937$	−29.0000	−0.947389	−0.473694	+	0.880689i	$0.657080\pi$
$938$	7.00000	0.228558
$939$	−10.0000	−0.326338
$940$	0	0
$941$	24.0000	0.782378	0.391189	−	0.920310i	$-0.372064\pi$
$941$	24.0000	0.782378	0.391189	+	0.920310i	$0.372064\pi$
$942$	−20.0000	−0.651635
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	24.0000	0.780307
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−14.0000	−0.454699
$949$	10.0000	0.324614
$950$	0	0
$951$	−12.0000	−0.389127
$952$	−3.00000	−0.0972306
$953$	−57.0000	−1.84641	−0.923206	−	0.384307i	$-0.874441\pi$
$953$	−57.0000	−1.84641	−0.923206	+	0.384307i	$0.874441\pi$
$954$	24.0000	0.777029
$955$	0	0
$956$	−12.0000	−0.388108
$957$	18.0000	0.581857
$958$	−18.0000	−0.581554
$959$	21.0000	0.678125
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−16.0000	−0.515861
$963$	6.00000	0.193347
$964$	−25.0000	−0.805196
$965$	0	0
$966$	0	0
$967$	34.0000	1.09337	0.546683	−	0.837340i	$-0.315890\pi$
$967$	34.0000	1.09337	0.546683	+	0.837340i	$0.315890\pi$
$968$	2.00000	0.0642824
$969$	−21.0000	−0.674617
$970$	0	0
$971$	9.00000	0.288824	0.144412	−	0.989518i	$-0.453871\pi$
$971$	9.00000	0.288824	0.144412	+	0.989518i	$0.453871\pi$
$972$	−16.0000	−0.513200
$973$	7.00000	0.224410
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−10.0000	−0.320092
$977$	3.00000	0.0959785	0.0479893	−	0.998848i	$-0.484719\pi$
$977$	3.00000	0.0959785	0.0479893	+	0.998848i	$0.484719\pi$
$978$	−5.00000	−0.159882
$979$	−45.0000	−1.43821
$980$	0	0
$981$	−28.0000	−0.893971
$982$	12.0000	0.382935
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	−18.0000	−0.573237
$987$	6.00000	0.190982
$988$	14.0000	0.445399
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	4.00000	0.127000
$993$	25.0000	0.793351
$994$	6.00000	0.190308
$995$	0	0
$996$	−9.00000	−0.285176
$997$	−62.0000	−1.96356	−0.981780	−	0.190022i	$-0.939144\pi$
$997$	−62.0000	−1.96356	−0.981780	+	0.190022i	$0.939144\pi$
$998$	28.0000	0.886325
$999$	−40.0000	−1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.2.a.a.1.1	✓	1
3.2	odd	2		3150.2.a.x.1.1		1
4.3	odd	2		2800.2.a.x.1.1		1
5.2	odd	4		350.2.c.c.99.1		2
5.3	odd	4		350.2.c.c.99.2		2
5.4	even	2		350.2.a.e.1.1	yes	1
7.6	odd	2		2450.2.a.m.1.1		1
15.2	even	4		3150.2.g.f.2899.2		2
15.8	even	4		3150.2.g.f.2899.1		2
15.14	odd	2		3150.2.a.m.1.1		1
20.3	even	4		2800.2.g.i.449.2		2
20.7	even	4		2800.2.g.i.449.1		2
20.19	odd	2		2800.2.a.h.1.1		1
35.13	even	4		2450.2.c.h.99.2		2
35.27	even	4		2450.2.c.h.99.1		2
35.34	odd	2		2450.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.a.a.1.1	✓	1	1.1	even	1	trivial
350.2.a.e.1.1	yes	1	5.4	even	2
350.2.c.c.99.1		2	5.2	odd	4
350.2.c.c.99.2		2	5.3	odd	4
2450.2.a.m.1.1		1	7.6	odd	2
2450.2.a.x.1.1		1	35.34	odd	2
2450.2.c.h.99.1		2	35.27	even	4
2450.2.c.h.99.2		2	35.13	even	4
2800.2.a.h.1.1		1	20.19	odd	2
2800.2.a.x.1.1		1	4.3	odd	2
2800.2.g.i.449.1		2	20.7	even	4
2800.2.g.i.449.2		2	20.3	even	4
3150.2.a.m.1.1		1	15.14	odd	2
3150.2.a.x.1.1		1	3.2	odd	2
3150.2.g.f.2899.1		2	15.8	even	4
3150.2.g.f.2899.2		2	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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