LMFDB - Embedded newform 1617.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,2,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1617.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:	$12.9118100068$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} -1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +1.00000 q^{19} -1.00000 q^{22} -1.00000 q^{23} +3.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} -1.00000 q^{27} -5.00000 q^{29} -10.0000 q^{31} +5.00000 q^{32} +1.00000 q^{33} +3.00000 q^{34} -1.00000 q^{36} -11.0000 q^{37} +1.00000 q^{38} -4.00000 q^{39} +10.0000 q^{41} -3.00000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -9.00000 q^{47} +1.00000 q^{48} -5.00000 q^{50} -3.00000 q^{51} -4.00000 q^{52} -8.00000 q^{53} -1.00000 q^{54} -1.00000 q^{57} -5.00000 q^{58} -9.00000 q^{59} -2.00000 q^{61} -10.0000 q^{62} +7.00000 q^{64} +1.00000 q^{66} +4.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} -7.00000 q^{71} -3.00000 q^{72} +4.00000 q^{73} -11.0000 q^{74} +5.00000 q^{75} -1.00000 q^{76} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +8.00000 q^{83} -3.00000 q^{86} +5.00000 q^{87} +3.00000 q^{88} +1.00000 q^{92} +10.0000 q^{93} -9.00000 q^{94} -5.00000 q^{96} -1.00000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−1.00000	−0.208514	−0.104257	−	0.994550i	$-0.533247\pi$
$23$	−1.00000	−0.208514	−0.104257	+	0.994550i	$0.533247\pi$
$24$	3.00000	0.612372
$25$	−5.00000	−1.00000
$26$	4.00000	0.784465
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000	0.883883
$33$	1.00000	0.174078
$34$	3.00000	0.514496
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	1.00000	0.162221
$39$	−4.00000	−0.640513
$40$	0	0
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	1.00000	0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	−3.00000	−0.420084
$52$	−4.00000	−0.554700
$53$	−8.00000	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	−5.00000	−0.656532
$59$	−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
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	1.00000	0.123091
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−3.00000	−0.363803
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.00000	−0.830747	−0.415374	−	0.909651i	$-0.636349\pi$
$71$	−7.00000	−0.830747	−0.415374	+	0.909651i	$0.636349\pi$
$72$	−3.00000	−0.353553
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	−11.0000	−1.27872
$75$	5.00000	0.577350
$76$	−1.00000	−0.114708
$77$	0	0
$78$	−4.00000	−0.452911
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	10.0000	1.10432
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	0	0
$86$	−3.00000	−0.323498
$87$	5.00000	0.536056
$88$	3.00000	0.319801
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	10.0000	1.03695
$94$	−9.00000	−0.928279
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	5.00000	0.500000
$101$	−5.00000	−0.497519	−0.248759	−	0.968565i	$-0.580023\pi$
$101$	−5.00000	−0.497519	−0.248759	+	0.968565i	$0.580023\pi$
$102$	−3.00000	−0.297044
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−12.0000	−1.17670
$105$	0	0
$106$	−8.00000	−0.777029
$107$	10.0000	0.966736	0.483368	−	0.875417i	$-0.339413\pi$
$107$	10.0000	0.966736	0.483368	+	0.875417i	$0.339413\pi$
$108$	1.00000	0.0962250
$109$	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
$111$	11.0000	1.04407
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	−1.00000	−0.0936586
$115$	0	0
$116$	5.00000	0.464238
$117$	4.00000	0.369800
$118$	−9.00000	−0.828517
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	−10.0000	−0.901670
$124$	10.0000	0.898027
$125$	0	0
$126$	0	0
$127$	−19.0000	−1.68598	−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000	−1.68598	−0.842989	+	0.537931i	$0.819206\pi$
$128$	−3.00000	−0.265165
$129$	3.00000	0.264135
$130$	0	0
$131$	14.0000	1.22319	0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000	1.22319	0.611593	+	0.791173i	$0.290529\pi$
$132$	−1.00000	−0.0870388
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	−9.00000	−0.771744
$137$	20.0000	1.70872	0.854358	−	0.519685i	$-0.173951\pi$
$137$	20.0000	1.70872	0.854358	+	0.519685i	$0.173951\pi$
$138$	1.00000	0.0851257
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−7.00000	−0.587427
$143$	−4.00000	−0.334497
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	11.0000	0.904194
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	5.00000	0.408248
$151$	−1.00000	−0.0813788	−0.0406894	−	0.999172i	$-0.512955\pi$
$151$	−1.00000	−0.0813788	−0.0406894	+	0.999172i	$0.512955\pi$
$152$	−3.00000	−0.243332
$153$	3.00000	0.242536
$154$	0	0
$155$	0	0
$156$	4.00000	0.320256
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−8.00000	−0.636446
$159$	8.00000	0.634441
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	−2.00000	−0.156652	−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000	−0.156652	−0.0783260	+	0.996928i	$0.524958\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	8.00000	0.620920
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	3.00000	0.228748
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	1.00000	0.0753778
$177$	9.00000	0.676481
$178$	0	0
$179$	−1.00000	−0.0747435	−0.0373718	−	0.999301i	$-0.511899\pi$
$179$	−1.00000	−0.0747435	−0.0373718	+	0.999301i	$0.511899\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	3.00000	0.221163
$185$	0	0
$186$	10.0000	0.733236
$187$	−3.00000	−0.219382
$188$	9.00000	0.656392
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	−7.00000	−0.505181
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−1.00000	−0.0717958
$195$	0	0
$196$	0	0
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	−1.00000	−0.0710669
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	15.0000	1.06066
$201$	−4.00000	−0.282138
$202$	−5.00000	−0.351799
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	−4.00000	−0.278693
$207$	−1.00000	−0.0695048
$208$	−4.00000	−0.277350
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	8.00000	0.549442
$213$	7.00000	0.479632
$214$	10.0000	0.683586
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	14.0000	0.948200
$219$	−4.00000	−0.270295
$220$	0	0
$221$	12.0000	0.807207
$222$	11.0000	0.738272
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−12.0000	−0.798228
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	1.00000	0.0662266
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	0	0
$232$	15.0000	0.984798
$233$	25.0000	1.63780	0.818902	−	0.573933i	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	4.00000	0.261488
$235$	0	0
$236$	9.00000	0.585850
$237$	8.00000	0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−10.0000	−0.637577
$247$	4.00000	0.254514
$248$	30.0000	1.90500
$249$	−8.00000	−0.506979
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	1.00000	0.0628695
$254$	−19.0000	−1.19217
$255$	0	0
$256$	−17.0000	−1.06250
$257$	28.0000	1.74659	0.873296	−	0.487190i	$-0.161978\pi$
$257$	28.0000	1.74659	0.873296	+	0.487190i	$0.161978\pi$
$258$	3.00000	0.186772
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	14.0000	0.864923
$263$	−30.0000	−1.84988	−0.924940	−	0.380114i	$-0.875885\pi$
$263$	−30.0000	−1.84988	−0.924940	+	0.380114i	$0.875885\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	20.0000	1.20824
$275$	5.00000	0.301511
$276$	−1.00000	−0.0601929
$277$	32.0000	1.92269	0.961347	−	0.275340i	$-0.0887905\pi$
$277$	32.0000	1.92269	0.961347	+	0.275340i	$0.0887905\pi$
$278$	−13.0000	−0.779688
$279$	−10.0000	−0.598684
$280$	0	0
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	9.00000	0.535942
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	−4.00000	−0.236525
$287$	0	0
$288$	5.00000	0.294628
$289$	−8.00000	−0.470588
$290$	0	0
$291$	1.00000	0.0586210
$292$	−4.00000	−0.234082
$293$	21.0000	1.22683	0.613417	−	0.789760i	$-0.289795\pi$
$293$	21.0000	1.22683	0.613417	+	0.789760i	$0.289795\pi$
$294$	0	0
$295$	0	0
$296$	33.0000	1.91809
$297$	1.00000	0.0580259
$298$	−3.00000	−0.173785
$299$	−4.00000	−0.231326
$300$	−5.00000	−0.288675
$301$	0	0
$302$	−1.00000	−0.0575435
$303$	5.00000	0.287242
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	3.00000	0.171499
$307$	−12.0000	−0.684876	−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000	−0.684876	−0.342438	+	0.939540i	$0.611253\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	17.0000	0.963982	0.481991	−	0.876176i	$-0.339914\pi$
$311$	17.0000	0.963982	0.481991	+	0.876176i	$0.339914\pi$
$312$	12.0000	0.679366
$313$	−9.00000	−0.508710	−0.254355	−	0.967111i	$-0.581863\pi$
$313$	−9.00000	−0.508710	−0.254355	+	0.967111i	$0.581863\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	8.00000	0.450035
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	8.00000	0.448618
$319$	5.00000	0.279946
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	3.00000	0.166924
$324$	−1.00000	−0.0555556
$325$	−20.0000	−1.10940
$326$	−2.00000	−0.110770
$327$	−14.0000	−0.774202
$328$	−30.0000	−1.65647
$329$	0	0
$330$	0	0
$331$	18.0000	0.989369	0.494685	−	0.869072i	$-0.335284\pi$
$331$	18.0000	0.989369	0.494685	+	0.869072i	$0.335284\pi$
$332$	−8.00000	−0.439057
$333$	−11.0000	−0.602796
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	3.00000	0.163178
$339$	12.0000	0.651751
$340$	0	0
$341$	10.0000	0.541530
$342$	1.00000	0.0540738
$343$	0	0
$344$	9.00000	0.485247
$345$	0	0
$346$	−22.0000	−1.18273
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	−5.00000	−0.268028
$349$	24.0000	1.28469	0.642345	−	0.766415i	$-0.277962\pi$
$349$	24.0000	1.28469	0.642345	+	0.766415i	$0.277962\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	−5.00000	−0.266501
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	9.00000	0.478345
$355$	0	0
$356$	0	0
$357$	0	0
$358$	−1.00000	−0.0528516
$359$	−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
$361$	−18.0000	−0.947368
$362$	−10.0000	−0.525588
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	1.00000	0.0521286
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	−10.0000	−0.518476
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−3.00000	−0.155126
$375$	0	0
$376$	27.0000	1.39242
$377$	−20.0000	−1.03005
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	19.0000	0.973399
$382$	−8.00000	−0.409316
$383$	13.0000	0.664269	0.332134	−	0.943232i	$-0.392231\pi$
$383$	13.0000	0.664269	0.332134	+	0.943232i	$0.392231\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−3.00000	−0.152499
$388$	1.00000	0.0507673
$389$	−20.0000	−1.01404	−0.507020	−	0.861934i	$-0.669253\pi$
$389$	−20.0000	−1.01404	−0.507020	+	0.861934i	$0.669253\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	−14.0000	−0.706207
$394$	−1.00000	−0.0503793
$395$	0	0
$396$	1.00000	0.0502519
$397$	−33.0000	−1.65622	−0.828111	−	0.560564i	$-0.810584\pi$
$397$	−33.0000	−1.65622	−0.828111	+	0.560564i	$0.810584\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	5.00000	0.250000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	−4.00000	−0.199502
$403$	−40.0000	−1.99254
$404$	5.00000	0.248759
$405$	0	0
$406$	0	0
$407$	11.0000	0.545250
$408$	9.00000	0.445566
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	4.00000	0.197066
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	20.0000	0.980581
$417$	13.0000	0.636613
$418$	−1.00000	−0.0489116
$419$	7.00000	0.341972	0.170986	−	0.985273i	$-0.445305\pi$
$419$	7.00000	0.341972	0.170986	+	0.985273i	$0.445305\pi$
$420$	0	0
$421$	−3.00000	−0.146211	−0.0731055	−	0.997324i	$-0.523291\pi$
$421$	−3.00000	−0.146211	−0.0731055	+	0.997324i	$0.523291\pi$
$422$	20.0000	0.973585
$423$	−9.00000	−0.437595
$424$	24.0000	1.16554
$425$	−15.0000	−0.727607
$426$	7.00000	0.339151
$427$	0	0
$428$	−10.0000	−0.483368
$429$	4.00000	0.193122
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	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	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−1.00000	−0.0478365
$438$	−4.00000	−0.191127
$439$	1.00000	0.0477274	0.0238637	−	0.999715i	$-0.492403\pi$
$439$	1.00000	0.0477274	0.0238637	+	0.999715i	$0.492403\pi$
$440$	0	0
$441$	0	0
$442$	12.0000	0.570782
$443$	−35.0000	−1.66290	−0.831450	−	0.555599i	$-0.812489\pi$
$443$	−35.0000	−1.66290	−0.831450	+	0.555599i	$0.812489\pi$
$444$	−11.0000	−0.522037
$445$	0	0
$446$	0	0
$447$	3.00000	0.141895
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	−5.00000	−0.235702
$451$	−10.0000	−0.470882
$452$	12.0000	0.564433
$453$	1.00000	0.0469841
$454$	14.0000	0.657053
$455$	0	0
$456$	3.00000	0.140488
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	−22.0000	−1.02799
$459$	−3.00000	−0.140028
$460$	0	0
$461$	33.0000	1.53696	0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000	1.53696	0.768482	+	0.639872i	$0.221013\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	25.0000	1.15810
$467$	−7.00000	−0.323921	−0.161961	−	0.986797i	$-0.551782\pi$
$467$	−7.00000	−0.323921	−0.161961	+	0.986797i	$0.551782\pi$
$468$	−4.00000	−0.184900
$469$	0	0
$470$	0	0
$471$	−7.00000	−0.322543
$472$	27.0000	1.24278
$473$	3.00000	0.137940
$474$	8.00000	0.367452
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−8.00000	−0.366295
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−44.0000	−2.00623
$482$	−4.00000	−0.182195
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	4.00000	0.181257	0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000	0.181257	0.0906287	+	0.995885i	$0.471112\pi$
$488$	6.00000	0.271607
$489$	2.00000	0.0904431
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	10.0000	0.450835
$493$	−15.0000	−0.675566
$494$	4.00000	0.179969
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	−8.00000	−0.358489
$499$	−14.0000	−0.626726	−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000	−0.626726	−0.313363	+	0.949633i	$0.601456\pi$
$500$	0	0
$501$	−12.0000	−0.536120
$502$	−15.0000	−0.669483
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	0	0
$505$	0	0
$506$	1.00000	0.0444554
$507$	−3.00000	−0.133235
$508$	19.0000	0.842989
$509$	−12.0000	−0.531891	−0.265945	−	0.963988i	$-0.585684\pi$
$509$	−12.0000	−0.531891	−0.265945	+	0.963988i	$0.585684\pi$
$510$	0	0
$511$	0	0
$512$	−11.0000	−0.486136
$513$	−1.00000	−0.0441511
$514$	28.0000	1.23503
$515$	0	0
$516$	−3.00000	−0.132068
$517$	9.00000	0.395820
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	−5.00000	−0.218844
$523$	−28.0000	−1.22435	−0.612177	−	0.790721i	$-0.709706\pi$
$523$	−28.0000	−1.22435	−0.612177	+	0.790721i	$0.709706\pi$
$524$	−14.0000	−0.611593
$525$	0	0
$526$	−30.0000	−1.30806
$527$	−30.0000	−1.30682
$528$	−1.00000	−0.0435194
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−9.00000	−0.390567
$532$	0	0
$533$	40.0000	1.73259
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	1.00000	0.0431532
$538$	22.0000	0.948487
$539$	0	0
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	−8.00000	−0.343629
$543$	10.0000	0.429141
$544$	15.0000	0.643120
$545$	0	0
$546$	0	0
$547$	−13.0000	−0.555840	−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000	−0.555840	−0.277920	+	0.960604i	$0.589645\pi$
$548$	−20.0000	−0.854358
$549$	−2.00000	−0.0853579
$550$	5.00000	0.213201
$551$	−5.00000	−0.213007
$552$	−3.00000	−0.127688
$553$	0	0
$554$	32.0000	1.35955
$555$	0	0
$556$	13.0000	0.551323
$557$	−39.0000	−1.65248	−0.826242	−	0.563316i	$-0.809525\pi$
$557$	−39.0000	−1.65248	−0.826242	+	0.563316i	$0.809525\pi$
$558$	−10.0000	−0.423334
$559$	−12.0000	−0.507546
$560$	0	0
$561$	3.00000	0.126660
$562$	9.00000	0.379642
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−9.00000	−0.378968
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	21.0000	0.881140
$569$	43.0000	1.80265	0.901327	−	0.433140i	$-0.142594\pi$
$569$	43.0000	1.80265	0.901327	+	0.433140i	$0.142594\pi$
$570$	0	0
$571$	−13.0000	−0.544033	−0.272017	−	0.962293i	$-0.587691\pi$
$571$	−13.0000	−0.544033	−0.272017	+	0.962293i	$0.587691\pi$
$572$	4.00000	0.167248
$573$	8.00000	0.334205
$574$	0	0
$575$	5.00000	0.208514
$576$	7.00000	0.291667
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	−8.00000	−0.332756
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	1.00000	0.0414513
$583$	8.00000	0.331326
$584$	−12.0000	−0.496564
$585$	0	0
$586$	21.0000	0.867502
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	0	0
$589$	−10.0000	−0.412043
$590$	0	0
$591$	1.00000	0.0411345
$592$	11.0000	0.452097
$593$	1.00000	0.0410651	0.0205325	−	0.999789i	$-0.493464\pi$
$593$	1.00000	0.0410651	0.0205325	+	0.999789i	$0.493464\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	3.00000	0.122885
$597$	20.0000	0.818546
$598$	−4.00000	−0.163572
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	−15.0000	−0.612372
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	1.00000	0.0406894
$605$	0	0
$606$	5.00000	0.203111
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	5.00000	0.202777
$609$	0	0
$610$	0	0
$611$	−36.0000	−1.45640
$612$	−3.00000	−0.121268
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	4.00000	0.160904
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	17.0000	0.681638
$623$	0	0
$624$	4.00000	0.160128
$625$	25.0000	1.00000
$626$	−9.00000	−0.359712
$627$	1.00000	0.0399362
$628$	−7.00000	−0.279330
$629$	−33.0000	−1.31580
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	24.0000	0.954669
$633$	−20.0000	−0.794929
$634$	12.0000	0.476581
$635$	0	0
$636$	−8.00000	−0.317221
$637$	0	0
$638$	5.00000	0.197952
$639$	−7.00000	−0.276916
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−10.0000	−0.394669
$643$	−42.0000	−1.65632	−0.828159	−	0.560493i	$-0.810612\pi$
$643$	−42.0000	−1.65632	−0.828159	+	0.560493i	$0.810612\pi$
$644$	0	0
$645$	0	0
$646$	3.00000	0.118033
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	−3.00000	−0.117851
$649$	9.00000	0.353281
$650$	−20.0000	−0.784465
$651$	0	0
$652$	2.00000	0.0783260
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−14.0000	−0.547443
$655$	0	0
$656$	−10.0000	−0.390434
$657$	4.00000	0.156055
$658$	0	0
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	23.0000	0.894596	0.447298	−	0.894385i	$-0.352386\pi$
$661$	23.0000	0.894596	0.447298	+	0.894385i	$0.352386\pi$
$662$	18.0000	0.699590
$663$	−12.0000	−0.466041
$664$	−24.0000	−0.931381
$665$	0	0
$666$	−11.0000	−0.426241
$667$	5.00000	0.193601
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	−44.0000	−1.69608	−0.848038	−	0.529936i	$-0.822216\pi$
$673$	−44.0000	−1.69608	−0.848038	+	0.529936i	$0.822216\pi$
$674$	−2.00000	−0.0770371
$675$	5.00000	0.192450
$676$	−3.00000	−0.115385
$677$	27.0000	1.03769	0.518847	−	0.854867i	$-0.326361\pi$
$677$	27.0000	1.03769	0.518847	+	0.854867i	$0.326361\pi$
$678$	12.0000	0.460857
$679$	0	0
$680$	0	0
$681$	−14.0000	−0.536481
$682$	10.0000	0.382920
$683$	−1.00000	−0.0382639	−0.0191320	−	0.999817i	$-0.506090\pi$
$683$	−1.00000	−0.0382639	−0.0191320	+	0.999817i	$0.506090\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	0	0
$687$	22.0000	0.839352
$688$	3.00000	0.114374
$689$	−32.0000	−1.21910
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	−15.0000	−0.568574
$697$	30.0000	1.13633
$698$	24.0000	0.908413
$699$	−25.0000	−0.945587
$700$	0	0
$701$	−23.0000	−0.868698	−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000	−0.868698	−0.434349	+	0.900745i	$0.643022\pi$
$702$	−4.00000	−0.150970
$703$	−11.0000	−0.414873
$704$	−7.00000	−0.263822
$705$	0	0
$706$	24.0000	0.903252
$707$	0	0
$708$	−9.00000	−0.338241
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	0	0
$713$	10.0000	0.374503
$714$	0	0
$715$	0	0
$716$	1.00000	0.0373718
$717$	0	0
$718$	−6.00000	−0.223918
$719$	51.0000	1.90198	0.950990	−	0.309223i	$-0.100069\pi$
$719$	51.0000	1.90198	0.950990	+	0.309223i	$0.100069\pi$
$720$	0	0
$721$	0	0
$722$	−18.0000	−0.669891
$723$	4.00000	0.148762
$724$	10.0000	0.371647
$725$	25.0000	0.928477
$726$	−1.00000	−0.0371135
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.00000	−0.332877
$732$	−2.00000	−0.0739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−5.00000	−0.184302
$737$	−4.00000	−0.147342
$738$	10.0000	0.368105
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	18.0000	0.660356	0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000	0.660356	0.330178	+	0.943919i	$0.392891\pi$
$744$	−30.0000	−1.09985
$745$	0	0
$746$	−6.00000	−0.219676
$747$	8.00000	0.292705
$748$	3.00000	0.109691
$749$	0	0
$750$	0	0
$751$	−34.0000	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$751$	−34.0000	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$752$	9.00000	0.328196
$753$	15.0000	0.546630
$754$	−20.0000	−0.728357
$755$	0	0
$756$	0	0
$757$	45.0000	1.63555	0.817776	−	0.575536i	$-0.195207\pi$
$757$	45.0000	1.63555	0.817776	+	0.575536i	$0.195207\pi$
$758$	34.0000	1.23494
$759$	−1.00000	−0.0362977
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	19.0000	0.688297
$763$	0	0
$764$	8.00000	0.289430
$765$	0	0
$766$	13.0000	0.469709
$767$	−36.0000	−1.29988
$768$	17.0000	0.613435
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−28.0000	−1.00840
$772$	10.0000	0.359908
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	−3.00000	−0.107833
$775$	50.0000	1.79605
$776$	3.00000	0.107694
$777$	0	0
$778$	−20.0000	−0.717035
$779$	10.0000	0.358287
$780$	0	0
$781$	7.00000	0.250480
$782$	−3.00000	−0.107280
$783$	5.00000	0.178685
$784$	0	0
$785$	0	0
$786$	−14.0000	−0.499363
$787$	1.00000	0.0356462	0.0178231	−	0.999841i	$-0.494326\pi$
$787$	1.00000	0.0356462	0.0178231	+	0.999841i	$0.494326\pi$
$788$	1.00000	0.0356235
$789$	30.0000	1.06803
$790$	0	0
$791$	0	0
$792$	3.00000	0.106600
$793$	−8.00000	−0.284088
$794$	−33.0000	−1.17113
$795$	0	0
$796$	20.0000	0.708881
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	−25.0000	−0.883883
$801$	0	0
$802$	−6.00000	−0.211867
$803$	−4.00000	−0.141157
$804$	4.00000	0.141069
$805$	0	0
$806$	−40.0000	−1.40894
$807$	−22.0000	−0.774437
$808$	15.0000	0.527698
$809$	−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
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	8.00000	0.280572
$814$	11.0000	0.385550
$815$	0	0
$816$	3.00000	0.105021
$817$	−3.00000	−0.104957
$818$	−10.0000	−0.349642
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−20.0000	−0.697580
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	12.0000	0.418040
$825$	−5.00000	−0.174078
$826$	0	0
$827$	−2.00000	−0.0695468	−0.0347734	−	0.999395i	$-0.511071\pi$
$827$	−2.00000	−0.0695468	−0.0347734	+	0.999395i	$0.511071\pi$
$828$	1.00000	0.0347524
$829$	−31.0000	−1.07667	−0.538337	−	0.842729i	$-0.680947\pi$
$829$	−31.0000	−1.07667	−0.538337	+	0.842729i	$0.680947\pi$
$830$	0	0
$831$	−32.0000	−1.11007
$832$	28.0000	0.970725
$833$	0	0
$834$	13.0000	0.450153
$835$	0	0
$836$	1.00000	0.0345857
$837$	10.0000	0.345651
$838$	7.00000	0.241811
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−3.00000	−0.103387
$843$	−9.00000	−0.309976
$844$	−20.0000	−0.688428
$845$	0	0
$846$	−9.00000	−0.309426
$847$	0	0
$848$	8.00000	0.274721
$849$	4.00000	0.137280
$850$	−15.0000	−0.514496
$851$	11.0000	0.377075
$852$	−7.00000	−0.239816
$853$	20.0000	0.684787	0.342393	−	0.939557i	$-0.388762\pi$
$853$	20.0000	0.684787	0.342393	+	0.939557i	$0.388762\pi$
$854$	0	0
$855$	0	0
$856$	−30.0000	−1.02538
$857$	−17.0000	−0.580709	−0.290354	−	0.956919i	$-0.593773\pi$
$857$	−17.0000	−0.580709	−0.290354	+	0.956919i	$0.593773\pi$
$858$	4.00000	0.136558
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	28.0000	0.953684
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−11.0000	−0.373795
$867$	8.00000	0.271694
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	16.0000	0.542139
$872$	−42.0000	−1.42230
$873$	−1.00000	−0.0338449
$874$	−1.00000	−0.0338255
$875$	0	0
$876$	4.00000	0.135147
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	1.00000	0.0337484
$879$	−21.0000	−0.708312
$880$	0	0
$881$	20.0000	0.673817	0.336909	−	0.941537i	$-0.390619\pi$
$881$	20.0000	0.673817	0.336909	+	0.941537i	$0.390619\pi$
$882$	0	0
$883$	−18.0000	−0.605748	−0.302874	−	0.953031i	$-0.597946\pi$
$883$	−18.0000	−0.605748	−0.302874	+	0.953031i	$0.597946\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−35.0000	−1.17585
$887$	−32.0000	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$888$	−33.0000	−1.10741
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−9.00000	−0.301174
$894$	3.00000	0.100335
$895$	0	0
$896$	0	0
$897$	4.00000	0.133556
$898$	−24.0000	−0.800890
$899$	50.0000	1.66759
$900$	5.00000	0.166667
$901$	−24.0000	−0.799556
$902$	−10.0000	−0.332964
$903$	0	0
$904$	36.0000	1.19734
$905$	0	0
$906$	1.00000	0.0332228
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	−14.0000	−0.464606
$909$	−5.00000	−0.165840
$910$	0	0
$911$	−45.0000	−1.49092	−0.745458	−	0.666552i	$-0.767769\pi$
$911$	−45.0000	−1.49092	−0.745458	+	0.666552i	$0.767769\pi$
$912$	1.00000	0.0331133
$913$	−8.00000	−0.264761
$914$	18.0000	0.595387
$915$	0	0
$916$	22.0000	0.726900
$917$	0	0
$918$	−3.00000	−0.0990148
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	33.0000	1.08680
$923$	−28.0000	−0.921631
$924$	0	0
$925$	55.0000	1.80839
$926$	−32.0000	−1.05159
$927$	−4.00000	−0.131377
$928$	−25.0000	−0.820665
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	−25.0000	−0.818902
$933$	−17.0000	−0.556555
$934$	−7.00000	−0.229047
$935$	0	0
$936$	−12.0000	−0.392232
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	0	0
$939$	9.00000	0.293704
$940$	0	0
$941$	47.0000	1.53216	0.766078	−	0.642747i	$-0.222206\pi$
$941$	47.0000	1.53216	0.766078	+	0.642747i	$0.222206\pi$
$942$	−7.00000	−0.228072
$943$	−10.0000	−0.325645
$944$	9.00000	0.292925
$945$	0	0
$946$	3.00000	0.0975384
$947$	21.0000	0.682408	0.341204	−	0.939989i	$-0.389165\pi$
$947$	21.0000	0.682408	0.341204	+	0.939989i	$0.389165\pi$
$948$	−8.00000	−0.259828
$949$	16.0000	0.519382
$950$	−5.00000	−0.162221
$951$	−12.0000	−0.389127
$952$	0	0
$953$	−34.0000	−1.10137	−0.550684	−	0.834714i	$-0.685633\pi$
$953$	−34.0000	−1.10137	−0.550684	+	0.834714i	$0.685633\pi$
$954$	−8.00000	−0.259010
$955$	0	0
$956$	0	0
$957$	−5.00000	−0.161627
$958$	10.0000	0.323085
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	−44.0000	−1.41862
$963$	10.0000	0.322245
$964$	4.00000	0.128831
$965$	0	0
$966$	0	0
$967$	−31.0000	−0.996893	−0.498446	−	0.866921i	$-0.666096\pi$
$967$	−31.0000	−0.996893	−0.498446	+	0.866921i	$0.666096\pi$
$968$	−3.00000	−0.0964237
$969$	−3.00000	−0.0963739
$970$	0	0
$971$	−32.0000	−1.02693	−0.513464	−	0.858111i	$-0.671638\pi$
$971$	−32.0000	−1.02693	−0.513464	+	0.858111i	$0.671638\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	4.00000	0.128168
$975$	20.0000	0.640513
$976$	2.00000	0.0640184
$977$	52.0000	1.66363	0.831814	−	0.555055i	$-0.187303\pi$
$977$	52.0000	1.66363	0.831814	+	0.555055i	$0.187303\pi$
$978$	2.00000	0.0639529
$979$	0	0
$980$	0	0
$981$	14.0000	0.446986
$982$	12.0000	0.382935
$983$	19.0000	0.606006	0.303003	−	0.952990i	$-0.402011\pi$
$983$	19.0000	0.606006	0.303003	+	0.952990i	$0.402011\pi$
$984$	30.0000	0.956365
$985$	0	0
$986$	−15.0000	−0.477697
$987$	0	0
$988$	−4.00000	−0.127257
$989$	3.00000	0.0953945
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	−50.0000	−1.58750
$993$	−18.0000	−0.571213
$994$	0	0
$995$	0	0
$996$	8.00000	0.253490
$997$	28.0000	0.886769	0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000	0.886769	0.443384	+	0.896332i	$0.353778\pi$
$998$	−14.0000	−0.443162
$999$	11.0000	0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.2.a.g.1.1		1
3.2	odd	2		4851.2.a.e.1.1		1
7.2	even	3		231.2.i.a.67.1	✓	2
7.4	even	3		231.2.i.a.100.1	yes	2
7.6	odd	2		1617.2.a.h.1.1		1
21.2	odd	6		693.2.i.e.298.1		2
21.11	odd	6		693.2.i.e.100.1		2
21.20	even	2		4851.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.a.67.1	✓	2	7.2	even	3
231.2.i.a.100.1	yes	2	7.4	even	3
693.2.i.e.100.1		2	21.11	odd	6
693.2.i.e.298.1		2	21.2	odd	6
1617.2.a.g.1.1		1	1.1	even	1	trivial
1617.2.a.h.1.1		1	7.6	odd	2
4851.2.a.d.1.1		1	21.20	even	2
4851.2.a.e.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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