LMFDB - Embedded newform 3465.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.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:	$27.6681643004$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +O(q^{10})$

$q+2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.00000 q^{10} -1.00000 q^{11} -2.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -1.00000 q^{17} -7.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} -5.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -2.00000 q^{28} -3.00000 q^{29} -4.00000 q^{31} -8.00000 q^{32} -2.00000 q^{34} -1.00000 q^{35} -2.00000 q^{37} -14.0000 q^{38} +12.0000 q^{41} -1.00000 q^{43} -2.00000 q^{44} -10.0000 q^{46} +4.00000 q^{47} +1.00000 q^{49} +2.00000 q^{50} -4.00000 q^{52} +1.00000 q^{53} -1.00000 q^{55} -6.00000 q^{58} -9.00000 q^{59} -11.0000 q^{61} -8.00000 q^{62} -8.00000 q^{64} -2.00000 q^{65} +2.00000 q^{67} -2.00000 q^{68} -2.00000 q^{70} +8.00000 q^{71} +8.00000 q^{73} -4.00000 q^{74} -14.0000 q^{76} +1.00000 q^{77} +2.00000 q^{79} -4.00000 q^{80} +24.0000 q^{82} +9.00000 q^{83} -1.00000 q^{85} -2.00000 q^{86} +3.00000 q^{89} +2.00000 q^{91} -10.0000 q^{92} +8.00000 q^{94} -7.00000 q^{95} -13.0000 q^{97} +2.00000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$2$	2.00000	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	1.00000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	2.00000	0.632456
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−2.00000	−0.426401
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−8.00000	−1.41421
$33$	0	0
$34$	−2.00000	−0.342997
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−14.0000	−2.27110
$39$	0	0
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−10.0000	−1.47442
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.00000	0.282843
$51$	0	0
$52$	−4.00000	−0.554700
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−9.00000	−1.17170	−0.585850	−	0.810419i	$-0.699239\pi$
$59$	−9.00000	−1.17170	−0.585850	+	0.810419i	$0.699239\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−2.00000	−0.239046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−14.0000	−1.60591
$77$	1.00000	0.113961
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	24.0000	2.65036
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−2.00000	−0.215666
$87$	0	0
$88$	0	0
$89$	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
$91$	2.00000	0.209657
$92$	−10.0000	−1.04257
$93$	0	0
$94$	8.00000	0.825137
$95$	−7.00000	−0.718185
$96$	0	0
$97$	−13.0000	−1.31995	−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000	−1.31995	−0.659975	+	0.751288i	$0.729433\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	2.00000	0.200000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−9.00000	−0.886796	−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000	−0.886796	−0.443398	+	0.896325i	$0.646227\pi$
$104$	0	0
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	4.00000	0.377964
$113$	−11.0000	−1.03479	−0.517396	−	0.855746i	$-0.673099\pi$
$113$	−11.0000	−1.03479	−0.517396	+	0.855746i	$0.673099\pi$
$114$	0	0
$115$	−5.00000	−0.466252
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−18.0000	−1.65703
$119$	1.00000	0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	−22.0000	−1.99179
$123$	0	0
$124$	−8.00000	−0.718421
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	−4.00000	−0.350823
$131$	18.0000	1.57267	0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000	1.57267	0.786334	+	0.617802i	$0.211977\pi$
$132$	0	0
$133$	7.00000	0.606977
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	16.0000	1.34269
$143$	2.00000	0.167248
$144$	0	0
$145$	−3.00000	−0.249136
$146$	16.0000	1.32417
$147$	0	0
$148$	−4.00000	−0.328798
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	0	0
$154$	2.00000	0.161165
$155$	−4.00000	−0.321288
$156$	0	0
$157$	23.0000	1.83560	0.917800	−	0.397043i	$-0.129964\pi$
$157$	23.0000	1.83560	0.917800	+	0.397043i	$0.129964\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−8.00000	−0.632456
$161$	5.00000	0.394055
$162$	0	0
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	24.0000	1.87409
$165$	0	0
$166$	18.0000	1.39707
$167$	−16.0000	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$167$	−16.0000	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−2.00000	−0.152499
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	4.00000	0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	−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
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	1.00000	0.0731272
$188$	8.00000	0.583460
$189$	0	0
$190$	−14.0000	−1.01567
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−26.0000	−1.86669
$195$	0	0
$196$	2.00000	0.142857
$197$	16.0000	1.13995	0.569976	−	0.821661i	$-0.306952\pi$
$197$	16.0000	1.13995	0.569976	+	0.821661i	$0.306952\pi$
$198$	0	0
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	0	0
$202$	28.0000	1.97007
$203$	3.00000	0.210559
$204$	0	0
$205$	12.0000	0.838116
$206$	−18.0000	−1.25412
$207$	0	0
$208$	8.00000	0.554700
$209$	7.00000	0.484200
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−8.00000	−0.546869
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	4.00000	0.271538
$218$	−16.0000	−1.08366
$219$	0	0
$220$	−2.00000	−0.134840
$221$	2.00000	0.134535
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	8.00000	0.534522
$225$	0	0
$226$	−22.0000	−1.46342
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	0	0
$229$	4.00000	0.264327	0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000	0.264327	0.132164	+	0.991228i	$0.457808\pi$
$230$	−10.0000	−0.659380
$231$	0	0
$232$	0	0
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	−18.0000	−1.17170
$237$	0	0
$238$	2.00000	0.129641
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	2.00000	0.128565
$243$	0	0
$244$	−22.0000	−1.40841
$245$	1.00000	0.0638877
$246$	0	0
$247$	14.0000	0.890799
$248$	0	0
$249$	0	0
$250$	2.00000	0.126491
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	5.00000	0.314347
$254$	−26.0000	−1.63139
$255$	0	0
$256$	16.0000	1.00000
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	−4.00000	−0.248069
$261$	0	0
$262$	36.0000	2.22409
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	1.00000	0.0614295
$266$	14.0000	0.858395
$267$	0	0
$268$	4.00000	0.244339
$269$	−5.00000	−0.304855	−0.152428	−	0.988315i	$-0.548709\pi$
$269$	−5.00000	−0.304855	−0.152428	+	0.988315i	$0.548709\pi$
$270$	0	0
$271$	31.0000	1.88312	0.941558	−	0.336851i	$-0.109362\pi$
$271$	31.0000	1.88312	0.941558	+	0.336851i	$0.109362\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	20.0000	1.20824
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	22.0000	1.30776	0.653882	−	0.756596i	$-0.273139\pi$
$283$	22.0000	1.30776	0.653882	+	0.756596i	$0.273139\pi$
$284$	16.0000	0.949425
$285$	0	0
$286$	4.00000	0.236525
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−16.0000	−0.941176
$290$	−6.00000	−0.352332
$291$	0	0
$292$	16.0000	0.936329
$293$	−5.00000	−0.292103	−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000	−0.292103	−0.146052	+	0.989277i	$0.546657\pi$
$294$	0	0
$295$	−9.00000	−0.524000
$296$	0	0
$297$	0	0
$298$	20.0000	1.15857
$299$	10.0000	0.578315
$300$	0	0
$301$	1.00000	0.0576390
$302$	−36.0000	−2.07157
$303$	0	0
$304$	28.0000	1.60591
$305$	−11.0000	−0.629858
$306$	0	0
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	−8.00000	−0.454369
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	9.00000	0.508710	0.254355	−	0.967111i	$-0.418137\pi$
$313$	9.00000	0.508710	0.254355	+	0.967111i	$0.418137\pi$
$314$	46.0000	2.59593
$315$	0	0
$316$	4.00000	0.225018
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	−8.00000	−0.447214
$321$	0	0
$322$	10.0000	0.557278
$323$	7.00000	0.389490
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−4.00000	−0.221540
$327$	0	0
$328$	0	0
$329$	−4.00000	−0.220527
$330$	0	0
$331$	−11.0000	−0.604615	−0.302307	−	0.953211i	$-0.597757\pi$
$331$	−11.0000	−0.604615	−0.302307	+	0.953211i	$0.597757\pi$
$332$	18.0000	0.987878
$333$	0	0
$334$	−32.0000	−1.75096
$335$	2.00000	0.109272
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	−18.0000	−0.979071
$339$	0	0
$340$	−2.00000	−0.108465
$341$	4.00000	0.216612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	−4.00000	−0.215041
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	8.00000	0.426401
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	6.00000	0.317999
$357$	0	0
$358$	−32.0000	−1.69125
$359$	−25.0000	−1.31945	−0.659725	−	0.751507i	$-0.729327\pi$
$359$	−25.0000	−1.31945	−0.659725	+	0.751507i	$0.729327\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−20.0000	−1.05118
$363$	0	0
$364$	4.00000	0.209657
$365$	8.00000	0.418739
$366$	0	0
$367$	7.00000	0.365397	0.182699	−	0.983169i	$-0.441517\pi$
$367$	7.00000	0.365397	0.182699	+	0.983169i	$0.441517\pi$
$368$	20.0000	1.04257
$369$	0	0
$370$	−4.00000	−0.207950
$371$	−1.00000	−0.0519174
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	37.0000	1.90056	0.950281	−	0.311393i	$-0.100796\pi$
$379$	37.0000	1.90056	0.950281	+	0.311393i	$0.100796\pi$
$380$	−14.0000	−0.718185
$381$	0	0
$382$	−48.0000	−2.45589
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	12.0000	0.610784
$387$	0	0
$388$	−26.0000	−1.31995
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	0	0
$394$	32.0000	1.61214
$395$	2.00000	0.100631
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−4.00000	−0.200000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	8.00000	0.398508
$404$	28.0000	1.39305
$405$	0	0
$406$	6.00000	0.297775
$407$	2.00000	0.0991363
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	24.0000	1.18528
$411$	0	0
$412$	−18.0000	−0.886796
$413$	9.00000	0.442861
$414$	0	0
$415$	9.00000	0.441793
$416$	16.0000	0.784465
$417$	0	0
$418$	14.0000	0.684762
$419$	−35.0000	−1.70986	−0.854931	−	0.518742i	$-0.826401\pi$
$419$	−35.0000	−1.70986	−0.854931	+	0.518742i	$0.826401\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	11.0000	0.532327
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	35.0000	1.67428
$438$	0	0
$439$	−15.0000	−0.715911	−0.357955	−	0.933739i	$-0.616526\pi$
$439$	−15.0000	−0.715911	−0.357955	+	0.933739i	$0.616526\pi$
$440$	0	0
$441$	0	0
$442$	4.00000	0.190261
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	3.00000	0.142214
$446$	−14.0000	−0.662919
$447$	0	0
$448$	8.00000	0.377964
$449$	−40.0000	−1.88772	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000	−1.88772	−0.943858	+	0.330350i	$0.892833\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	−22.0000	−1.03479
$453$	0	0
$454$	18.0000	0.844782
$455$	2.00000	0.0937614
$456$	0	0
$457$	3.00000	0.140334	0.0701670	−	0.997535i	$-0.477647\pi$
$457$	3.00000	0.140334	0.0701670	+	0.997535i	$0.477647\pi$
$458$	8.00000	0.373815
$459$	0	0
$460$	−10.0000	−0.466252
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	12.0000	0.557086
$465$	0	0
$466$	−28.0000	−1.29707
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	−2.00000	−0.0923514
$470$	8.00000	0.369012
$471$	0	0
$472$	0	0
$473$	1.00000	0.0459800
$474$	0	0
$475$	−7.00000	−0.321182
$476$	2.00000	0.0916698
$477$	0	0
$478$	−10.0000	−0.457389
$479$	38.0000	1.73626	0.868132	−	0.496333i	$-0.165321\pi$
$479$	38.0000	1.73626	0.868132	+	0.496333i	$0.165321\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	20.0000	0.910975
$483$	0	0
$484$	2.00000	0.0909091
$485$	−13.0000	−0.590300
$486$	0	0
$487$	−26.0000	−1.17817	−0.589086	−	0.808070i	$-0.700512\pi$
$487$	−26.0000	−1.17817	−0.589086	+	0.808070i	$0.700512\pi$
$488$	0	0
$489$	0	0
$490$	2.00000	0.0903508
$491$	−43.0000	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$491$	−43.0000	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$492$	0	0
$493$	3.00000	0.135113
$494$	28.0000	1.25978
$495$	0	0
$496$	16.0000	0.718421
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−25.0000	−1.11915	−0.559577	−	0.828778i	$-0.689036\pi$
$499$	−25.0000	−1.11915	−0.559577	+	0.828778i	$0.689036\pi$
$500$	2.00000	0.0894427
$501$	0	0
$502$	24.0000	1.07117
$503$	−5.00000	−0.222939	−0.111469	−	0.993768i	$-0.535556\pi$
$503$	−5.00000	−0.222939	−0.111469	+	0.993768i	$0.535556\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	10.0000	0.444554
$507$	0	0
$508$	−26.0000	−1.15356
$509$	33.0000	1.46270	0.731350	−	0.682003i	$-0.238891\pi$
$509$	33.0000	1.46270	0.731350	+	0.682003i	$0.238891\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	32.0000	1.41421
$513$	0	0
$514$	−28.0000	−1.23503
$515$	−9.00000	−0.396587
$516$	0	0
$517$	−4.00000	−0.175920
$518$	4.00000	0.175750
$519$	0	0
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	36.0000	1.57267
$525$	0	0
$526$	32.0000	1.39527
$527$	4.00000	0.174243
$528$	0	0
$529$	2.00000	0.0869565
$530$	2.00000	0.0868744
$531$	0	0
$532$	14.0000	0.606977
$533$	−24.0000	−1.03956
$534$	0	0
$535$	−4.00000	−0.172935
$536$	0	0
$537$	0	0
$538$	−10.0000	−0.431131
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−32.0000	−1.37579	−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000	−1.37579	−0.687894	+	0.725811i	$0.741464\pi$
$542$	62.0000	2.66313
$543$	0	0
$544$	8.00000	0.342997
$545$	−8.00000	−0.342682
$546$	0	0
$547$	−23.0000	−0.983409	−0.491704	−	0.870762i	$-0.663626\pi$
$547$	−23.0000	−0.983409	−0.491704	+	0.870762i	$0.663626\pi$
$548$	20.0000	0.854358
$549$	0	0
$550$	−2.00000	−0.0852803
$551$	21.0000	0.894630
$552$	0	0
$553$	−2.00000	−0.0850487
$554$	−20.0000	−0.849719
$555$	0	0
$556$	−16.0000	−0.678551
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	4.00000	0.169031
$561$	0	0
$562$	−12.0000	−0.506189
$563$	−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
$565$	−11.0000	−0.462773
$566$	44.0000	1.84946
$567$	0	0
$568$	0	0
$569$	−7.00000	−0.293455	−0.146728	−	0.989177i	$-0.546874\pi$
$569$	−7.00000	−0.293455	−0.146728	+	0.989177i	$0.546874\pi$
$570$	0	0
$571$	34.0000	1.42286	0.711428	−	0.702759i	$-0.248049\pi$
$571$	34.0000	1.42286	0.711428	+	0.702759i	$0.248049\pi$
$572$	4.00000	0.167248
$573$	0	0
$574$	−24.0000	−1.00174
$575$	−5.00000	−0.208514
$576$	0	0
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−9.00000	−0.373383
$582$	0	0
$583$	−1.00000	−0.0414158
$584$	0	0
$585$	0	0
$586$	−10.0000	−0.413096
$587$	10.0000	0.412744	0.206372	−	0.978474i	$-0.433834\pi$
$587$	10.0000	0.412744	0.206372	+	0.978474i	$0.433834\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	−18.0000	−0.741048
$591$	0	0
$592$	8.00000	0.328798
$593$	−26.0000	−1.06769	−0.533846	−	0.845582i	$-0.679254\pi$
$593$	−26.0000	−1.06769	−0.533846	+	0.845582i	$0.679254\pi$
$594$	0	0
$595$	1.00000	0.0409960
$596$	20.0000	0.819232
$597$	0	0
$598$	20.0000	0.817861
$599$	−34.0000	−1.38920	−0.694601	−	0.719395i	$-0.744419\pi$
$599$	−34.0000	−1.38920	−0.694601	+	0.719395i	$0.744419\pi$
$600$	0	0
$601$	47.0000	1.91717	0.958585	−	0.284807i	$-0.0919294\pi$
$601$	47.0000	1.91717	0.958585	+	0.284807i	$0.0919294\pi$
$602$	2.00000	0.0815139
$603$	0	0
$604$	−36.0000	−1.46482
$605$	1.00000	0.0406558
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	56.0000	2.27110
$609$	0	0
$610$	−22.0000	−0.890754
$611$	−8.00000	−0.323645
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	68.0000	2.74426
$615$	0	0
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	0	0
$619$	22.0000	0.884255	0.442127	−	0.896952i	$-0.354224\pi$
$619$	22.0000	0.884255	0.442127	+	0.896952i	$0.354224\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	−24.0000	−0.962312
$623$	−3.00000	−0.120192
$624$	0	0
$625$	1.00000	0.0400000
$626$	18.0000	0.719425
$627$	0	0
$628$	46.0000	1.83560
$629$	2.00000	0.0797452
$630$	0	0
$631$	−19.0000	−0.756378	−0.378189	−	0.925728i	$-0.623453\pi$
$631$	−19.0000	−0.756378	−0.378189	+	0.925728i	$0.623453\pi$
$632$	0	0
$633$	0	0
$634$	−12.0000	−0.476581
$635$	−13.0000	−0.515889
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	6.00000	0.237542
$639$	0	0
$640$	0	0
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	0	0
$643$	−21.0000	−0.828159	−0.414080	−	0.910241i	$-0.635896\pi$
$643$	−21.0000	−0.828159	−0.414080	+	0.910241i	$0.635896\pi$
$644$	10.0000	0.394055
$645$	0	0
$646$	14.0000	0.550823
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	−4.00000	−0.156893
$651$	0	0
$652$	−4.00000	−0.156652
$653$	−13.0000	−0.508729	−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000	−0.508729	−0.254365	+	0.967108i	$0.581866\pi$
$654$	0	0
$655$	18.0000	0.703318
$656$	−48.0000	−1.87409
$657$	0	0
$658$	−8.00000	−0.311872
$659$	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
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	−22.0000	−0.855054
$663$	0	0
$664$	0	0
$665$	7.00000	0.271448
$666$	0	0
$667$	15.0000	0.580802
$668$	−32.0000	−1.23812
$669$	0	0
$670$	4.00000	0.154533
$671$	11.0000	0.424650
$672$	0	0
$673$	−19.0000	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$673$	−19.0000	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$674$	46.0000	1.77185
$675$	0	0
$676$	−18.0000	−0.692308
$677$	35.0000	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$677$	35.0000	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$678$	0	0
$679$	13.0000	0.498894
$680$	0	0
$681$	0	0
$682$	8.00000	0.306336
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	−2.00000	−0.0763604
$687$	0	0
$688$	4.00000	0.152499
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−4.00000	−0.152057
$693$	0	0
$694$	32.0000	1.21470
$695$	−8.00000	−0.303457
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−22.0000	−0.832712
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	15.0000	0.566542	0.283271	−	0.959040i	$-0.408580\pi$
$701$	15.0000	0.566542	0.283271	+	0.959040i	$0.408580\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	8.00000	0.301511
$705$	0	0
$706$	−28.0000	−1.05379
$707$	−14.0000	−0.526524
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	0	0
$713$	20.0000	0.749006
$714$	0	0
$715$	2.00000	0.0747958
$716$	−32.0000	−1.19590
$717$	0	0
$718$	−50.0000	−1.86598
$719$	27.0000	1.00693	0.503465	−	0.864016i	$-0.332058\pi$
$719$	27.0000	1.00693	0.503465	+	0.864016i	$0.332058\pi$
$720$	0	0
$721$	9.00000	0.335178
$722$	60.0000	2.23297
$723$	0	0
$724$	−20.0000	−0.743294
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−23.0000	−0.853023	−0.426511	−	0.904482i	$-0.640258\pi$
$727$	−23.0000	−0.853023	−0.426511	+	0.904482i	$0.640258\pi$
$728$	0	0
$729$	0	0
$730$	16.0000	0.592187
$731$	1.00000	0.0369863
$732$	0	0
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	40.0000	1.47442
$737$	−2.00000	−0.0736709
$738$	0	0
$739$	42.0000	1.54499	0.772497	−	0.635018i	$-0.219007\pi$
$739$	42.0000	1.54499	0.772497	+	0.635018i	$0.219007\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−14.0000	−0.513610	−0.256805	−	0.966463i	$-0.582670\pi$
$743$	−14.0000	−0.513610	−0.256805	+	0.966463i	$0.582670\pi$
$744$	0	0
$745$	10.0000	0.366372
$746$	−50.0000	−1.83063
$747$	0	0
$748$	2.00000	0.0731272
$749$	4.00000	0.146157
$750$	0	0
$751$	−15.0000	−0.547358	−0.273679	−	0.961821i	$-0.588241\pi$
$751$	−15.0000	−0.547358	−0.273679	+	0.961821i	$0.588241\pi$
$752$	−16.0000	−0.583460
$753$	0	0
$754$	12.0000	0.437014
$755$	−18.0000	−0.655087
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	74.0000	2.68780
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	−48.0000	−1.73658
$765$	0	0
$766$	−36.0000	−1.30073
$767$	18.0000	0.649942
$768$	0	0
$769$	9.00000	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$769$	9.00000	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$770$	2.00000	0.0720750
$771$	0	0
$772$	12.0000	0.431889
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0	0
$778$	28.0000	1.00385
$779$	−84.0000	−3.00961
$780$	0	0
$781$	−8.00000	−0.286263
$782$	10.0000	0.357599
$783$	0	0
$784$	−4.00000	−0.142857
$785$	23.0000	0.820905
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	32.0000	1.13995
$789$	0	0
$790$	4.00000	0.142314
$791$	11.0000	0.391115
$792$	0	0
$793$	22.0000	0.781243
$794$	28.0000	0.993683
$795$	0	0
$796$	−8.00000	−0.283552
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	−8.00000	−0.282843
$801$	0	0
$802$	48.0000	1.69494
$803$	−8.00000	−0.282314
$804$	0	0
$805$	5.00000	0.176227
$806$	16.0000	0.563576
$807$	0	0
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	4.00000	0.140200
$815$	−2.00000	−0.0700569
$816$	0	0
$817$	7.00000	0.244899
$818$	20.0000	0.699284
$819$	0	0
$820$	24.0000	0.838116
$821$	−13.0000	−0.453703	−0.226852	−	0.973929i	$-0.572843\pi$
$821$	−13.0000	−0.453703	−0.226852	+	0.973929i	$0.572843\pi$
$822$	0	0
$823$	38.0000	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$823$	38.0000	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$824$	0	0
$825$	0	0
$826$	18.0000	0.626300
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	16.0000	0.554700
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−16.0000	−0.553703
$836$	14.0000	0.484200
$837$	0	0
$838$	−70.0000	−2.41811
$839$	3.00000	0.103572	0.0517858	−	0.998658i	$-0.483509\pi$
$839$	3.00000	0.103572	0.0517858	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	2.00000	0.0689246
$843$	0	0
$844$	−20.0000	−0.688428
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−4.00000	−0.137361
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	10.0000	0.342796
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	22.0000	0.752825
$855$	0	0
$856$	0	0
$857$	50.0000	1.70797	0.853984	−	0.520300i	$-0.174180\pi$
$857$	50.0000	1.70797	0.853984	+	0.520300i	$0.174180\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−48.0000	−1.63489
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	−52.0000	−1.76703
$867$	0	0
$868$	8.00000	0.271538
$869$	−2.00000	−0.0678454
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	0	0
$874$	70.0000	2.36779
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−19.0000	−0.641584	−0.320792	−	0.947150i	$-0.603949\pi$
$877$	−19.0000	−0.641584	−0.320792	+	0.947150i	$0.603949\pi$
$878$	−30.0000	−1.01245
$879$	0	0
$880$	4.00000	0.134840
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	24.0000	0.806296
$887$	−51.0000	−1.71241	−0.856206	−	0.516634i	$-0.827185\pi$
$887$	−51.0000	−1.71241	−0.856206	+	0.516634i	$0.827185\pi$
$888$	0	0
$889$	13.0000	0.436006
$890$	6.00000	0.201120
$891$	0	0
$892$	−14.0000	−0.468755
$893$	−28.0000	−0.936984
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	0	0
$898$	−80.0000	−2.66963
$899$	12.0000	0.400222
$900$	0	0
$901$	−1.00000	−0.0333148
$902$	−24.0000	−0.799113
$903$	0	0
$904$	0	0
$905$	−10.0000	−0.332411
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	4.00000	0.132599
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−9.00000	−0.297857
$914$	6.00000	0.198462
$915$	0	0
$916$	8.00000	0.264327
$917$	−18.0000	−0.594412
$918$	0	0
$919$	−10.0000	−0.329870	−0.164935	−	0.986304i	$-0.552741\pi$
$919$	−10.0000	−0.329870	−0.164935	+	0.986304i	$0.552741\pi$
$920$	0	0
$921$	0	0
$922$	4.00000	0.131733
$923$	−16.0000	−0.526646
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	−28.0000	−0.920137
$927$	0	0
$928$	24.0000	0.787839
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	−28.0000	−0.917170
$933$	0	0
$934$	48.0000	1.57061
$935$	1.00000	0.0327035
$936$	0	0
$937$	4.00000	0.130674	0.0653372	−	0.997863i	$-0.479188\pi$
$937$	4.00000	0.130674	0.0653372	+	0.997863i	$0.479188\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	8.00000	0.260931
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	−60.0000	−1.95387
$944$	36.0000	1.17170
$945$	0	0
$946$	2.00000	0.0650256
$947$	35.0000	1.13735	0.568674	−	0.822563i	$-0.307457\pi$
$947$	35.0000	1.13735	0.568674	+	0.822563i	$0.307457\pi$
$948$	0	0
$949$	−16.0000	−0.519382
$950$	−14.0000	−0.454220
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	−24.0000	−0.776622
$956$	−10.0000	−0.323423
$957$	0	0
$958$	76.0000	2.45545
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−15.0000	−0.483871
$962$	8.00000	0.257930
$963$	0	0
$964$	20.0000	0.644157
$965$	6.00000	0.193147
$966$	0	0
$967$	35.0000	1.12552	0.562762	−	0.826619i	$-0.309739\pi$
$967$	35.0000	1.12552	0.562762	+	0.826619i	$0.309739\pi$
$968$	0	0
$969$	0	0
$970$	−26.0000	−0.834810
$971$	−17.0000	−0.545556	−0.272778	−	0.962077i	$-0.587942\pi$
$971$	−17.0000	−0.545556	−0.272778	+	0.962077i	$0.587942\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	−52.0000	−1.66619
$975$	0	0
$976$	44.0000	1.40841
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	−3.00000	−0.0958804
$980$	2.00000	0.0638877
$981$	0	0
$982$	−86.0000	−2.74437
$983$	18.0000	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$983$	18.0000	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$984$	0	0
$985$	16.0000	0.509802
$986$	6.00000	0.191079
$987$	0	0
$988$	28.0000	0.890799
$989$	5.00000	0.158991
$990$	0	0
$991$	−17.0000	−0.540023	−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000	−0.540023	−0.270011	+	0.962857i	$0.587027\pi$
$992$	32.0000	1.01600
$993$	0	0
$994$	−16.0000	−0.507489
$995$	−4.00000	−0.126809
$996$	0	0
$997$	−14.0000	−0.443384	−0.221692	−	0.975117i	$-0.571158\pi$
$997$	−14.0000	−0.443384	−0.221692	+	0.975117i	$0.571158\pi$
$998$	−50.0000	−1.58272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.t.1.1		1
3.2	odd	2		1155.2.a.b.1.1	✓	1
15.14	odd	2		5775.2.a.z.1.1		1
21.20	even	2		8085.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.b.1.1	✓	1	3.2	odd	2
3465.2.a.t.1.1		1	1.1	even	1	trivial
5775.2.a.z.1.1		1	15.14	odd	2
8085.2.a.c.1.1		1	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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