LMFDB - Embedded newform 3150.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(1,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.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:	$25.1528766367$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 350)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} -1.00000 q^{8} +5.00000 q^{11} -6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} -3.00000 q^{19} -5.00000 q^{22} +6.00000 q^{26} -1.00000 q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.00000 q^{34} +8.00000 q^{37} +3.00000 q^{38} -11.0000 q^{41} -8.00000 q^{43} +5.00000 q^{44} -2.00000 q^{47} +1.00000 q^{49} -6.00000 q^{52} -4.00000 q^{53} +1.00000 q^{56} -6.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +9.00000 q^{67} +1.00000 q^{68} +10.0000 q^{71} -7.00000 q^{73} -8.00000 q^{74} -3.00000 q^{76} -5.00000 q^{77} -2.00000 q^{79} +11.0000 q^{82} -11.0000 q^{83} +8.00000 q^{86} -5.00000 q^{88} +11.0000 q^{89} +6.00000 q^{91} +2.00000 q^{94} -10.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	−5.00000	−1.06600
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	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
$33$	0	0
$34$	−1.00000	−0.171499
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	0	0
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	5.00000	0.753778
$45$	0	0
$46$	0	0
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−6.00000	−0.832050
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	−6.00000	−0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	9.00000	1.09952	0.549762	−	0.835321i	$-0.314718\pi$
$67$	9.00000	1.09952	0.549762	+	0.835321i	$0.314718\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	−3.00000	−0.344124
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	11.0000	1.21475
$83$	−11.0000	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$83$	−11.0000	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−5.00000	−0.533002
$89$	11.0000	1.16600	0.582999	−	0.812473i	$-0.301879\pi$
$89$	11.0000	1.16600	0.582999	+	0.812473i	$0.301879\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	2.00000	0.206284
$95$	0	0
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	4.00000	0.388514
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000	0.368230
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	−4.00000	−0.359211
$125$	0	0
$126$	0	0
$127$	−14.0000	−1.24230	−0.621150	−	0.783692i	$-0.713334\pi$
$127$	−14.0000	−1.24230	−0.621150	+	0.783692i	$0.713334\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−9.00000	−0.777482
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	−10.0000	−0.839181
$143$	−30.0000	−2.50873
$144$	0	0
$145$	0	0
$146$	7.00000	0.579324
$147$	0	0
$148$	8.00000	0.657596
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	5.00000	0.402911
$155$	0	0
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	2.00000	0.159111
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.0000	−1.48819	−0.744097	−	0.668071i	$-0.767120\pi$
$163$	−19.0000	−1.48819	−0.744097	+	0.668071i	$0.767120\pi$
$164$	−11.0000	−0.858956
$165$	0	0
$166$	11.0000	0.853766
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−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
$175$	0	0
$176$	5.00000	0.376889
$177$	0	0
$178$	−11.0000	−0.824485
$179$	−3.00000	−0.224231	−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000	−0.224231	−0.112115	+	0.993695i	$0.535763\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	−6.00000	−0.444750
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.00000	0.365636
$188$	−2.00000	−0.145865
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.00000	−0.421117
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−6.00000	−0.416025
$209$	−15.0000	−1.03757
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	−4.00000	−0.274721
$213$	0	0
$214$	3.00000	0.205076
$215$	0	0
$216$	0	0
$217$	4.00000	0.271538
$218$	18.0000	1.21911
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	−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	$0.562969\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	1.00000	0.0648204
$239$	−4.00000	−0.258738	−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000	−0.258738	−0.129369	+	0.991596i	$0.541295\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	18.0000	1.14531
$248$	4.00000	0.254000
$249$	0	0
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	0	0
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	8.00000	0.494242
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	0	0
$265$	0	0
$266$	−3.00000	−0.183942
$267$	0	0
$268$	9.00000	0.549762
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	11.0000	0.659736
$279$	0	0
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	−13.0000	−0.772770	−0.386385	−	0.922338i	$-0.626276\pi$
$283$	−13.0000	−0.772770	−0.386385	+	0.922338i	$0.626276\pi$
$284$	10.0000	0.593391
$285$	0	0
$286$	30.0000	1.77394
$287$	11.0000	0.649309
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	−7.00000	−0.409644
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	12.0000	0.695141
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	−8.00000	−0.460348
$303$	0	0
$304$	−3.00000	−0.172062
$305$	0	0
$306$	0	0
$307$	13.0000	0.741949	0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000	0.741949	0.370975	+	0.928643i	$0.379024\pi$
$308$	−5.00000	−0.284901
$309$	0	0
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−2.00000	−0.112509
$317$	−4.00000	−0.224662	−0.112331	−	0.993671i	$-0.535832\pi$
$317$	−4.00000	−0.224662	−0.112331	+	0.993671i	$0.535832\pi$
$318$	0	0
$319$	30.0000	1.67968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	0	0
$326$	19.0000	1.05231
$327$	0	0
$328$	11.0000	0.607373
$329$	2.00000	0.110264
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	−11.0000	−0.603703
$333$	0	0
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	−29.0000	−1.57973	−0.789865	−	0.613280i	$-0.789850\pi$
$337$	−29.0000	−1.57973	−0.789865	+	0.613280i	$0.789850\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.00000	0.431331
$345$	0	0
$346$	2.00000	0.107521
$347$	−19.0000	−1.01997	−0.509987	−	0.860182i	$-0.670350\pi$
$347$	−19.0000	−1.01997	−0.509987	+	0.860182i	$0.670350\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	−5.00000	−0.266501
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	11.0000	0.582999
$357$	0	0
$358$	3.00000	0.158555
$359$	−26.0000	−1.37223	−0.686114	−	0.727494i	$-0.740685\pi$
$359$	−26.0000	−1.37223	−0.686114	+	0.727494i	$0.740685\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−10.0000	−0.525588
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	−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
$369$	0	0
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	−4.00000	−0.207112	−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000	−0.207112	−0.103556	+	0.994624i	$0.533022\pi$
$374$	−5.00000	−0.258544
$375$	0	0
$376$	2.00000	0.103142
$377$	−36.0000	−1.85409
$378$	0	0
$379$	9.00000	0.462299	0.231149	−	0.972918i	$-0.425751\pi$
$379$	9.00000	0.462299	0.231149	+	0.972918i	$0.425751\pi$
$380$	0	0
$381$	0	0
$382$	−6.00000	−0.306987
$383$	6.00000	0.306586	0.153293	−	0.988181i	$-0.451012\pi$
$383$	6.00000	0.306586	0.153293	+	0.988181i	$0.451012\pi$
$384$	0	0
$385$	0	0
$386$	19.0000	0.967075
$387$	0	0
$388$	−10.0000	−0.507673
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	0	0
$401$	−37.0000	−1.84769	−0.923846	−	0.382765i	$-0.874972\pi$
$401$	−37.0000	−1.84769	−0.923846	+	0.382765i	$0.874972\pi$
$402$	0	0
$403$	24.0000	1.19553
$404$	0	0
$405$	0	0
$406$	6.00000	0.297775
$407$	40.0000	1.98273
$408$	0	0
$409$	−21.0000	−1.03838	−0.519192	−	0.854658i	$-0.673767\pi$
$409$	−21.0000	−1.03838	−0.519192	+	0.854658i	$0.673767\pi$
$410$	0	0
$411$	0	0
$412$	4.00000	0.197066
$413$	4.00000	0.196827
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	15.0000	0.733674
$419$	39.0000	1.90527	0.952637	−	0.304109i	$-0.0983586\pi$
$419$	39.0000	1.90527	0.952637	+	0.304109i	$0.0983586\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−1.00000	−0.0486792
$423$	0	0
$424$	4.00000	0.194257
$425$	0	0
$426$	0	0
$427$	2.00000	0.0967868
$428$	−3.00000	−0.145010
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	−1.00000	−0.0480569	−0.0240285	−	0.999711i	$-0.507649\pi$
$433$	−1.00000	−0.0480569	−0.0240285	+	0.999711i	$0.507649\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	6.00000	0.285391
$443$	−37.0000	−1.75792	−0.878962	−	0.476893i	$-0.841763\pi$
$443$	−37.0000	−1.75792	−0.878962	+	0.476893i	$0.841763\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	−55.0000	−2.58985
$452$	1.00000	0.0470360
$453$	0	0
$454$	−28.0000	−1.31411
$455$	0	0
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	0	0
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	−9.00000	−0.415581
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−40.0000	−1.83920
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	4.00000	0.182956
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	5.00000	0.227744
$483$	0	0
$484$	14.0000	0.636364
$485$	0	0
$486$	0	0
$487$	−34.0000	−1.54069	−0.770344	−	0.637629i	$-0.779915\pi$
$487$	−34.0000	−1.54069	−0.770344	+	0.637629i	$0.779915\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	−18.0000	−0.809858
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−10.0000	−0.448561
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	0	0
$502$	27.0000	1.20507
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−14.0000	−0.621150
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	−10.0000	−0.439799
$518$	8.00000	0.351500
$519$	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
$523$	13.0000	0.568450	0.284225	−	0.958758i	$-0.408264\pi$
$523$	13.0000	0.568450	0.284225	+	0.958758i	$0.408264\pi$
$524$	−8.00000	−0.349482
$525$	0	0
$526$	−10.0000	−0.436021
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	3.00000	0.130066
$533$	66.0000	2.85878
$534$	0	0
$535$	0	0
$536$	−9.00000	−0.388741
$537$	0	0
$538$	18.0000	0.776035
$539$	5.00000	0.215365
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	6.00000	0.257722
$543$	0	0
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	0	0
$547$	27.0000	1.15444	0.577218	−	0.816590i	$-0.304138\pi$
$547$	27.0000	1.15444	0.577218	+	0.816590i	$0.304138\pi$
$548$	3.00000	0.128154
$549$	0	0
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	2.00000	0.0850487
$554$	30.0000	1.27458
$555$	0	0
$556$	−11.0000	−0.466504
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	0	0
$559$	48.0000	2.03018
$560$	0	0
$561$	0	0
$562$	14.0000	0.590554
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	0	0
$566$	13.0000	0.546431
$567$	0	0
$568$	−10.0000	−0.419591
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−30.0000	−1.25436
$573$	0	0
$574$	−11.0000	−0.459131
$575$	0	0
$576$	0	0
$577$	13.0000	0.541197	0.270599	−	0.962692i	$-0.412778\pi$
$577$	13.0000	0.541197	0.270599	+	0.962692i	$0.412778\pi$
$578$	16.0000	0.665512
$579$	0	0
$580$	0	0
$581$	11.0000	0.456357
$582$	0	0
$583$	−20.0000	−0.828315
$584$	7.00000	0.289662
$585$	0	0
$586$	−14.0000	−0.578335
$587$	13.0000	0.536567	0.268284	−	0.963340i	$-0.413544\pi$
$587$	13.0000	0.536567	0.268284	+	0.963340i	$0.413544\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	0	0
$591$	0	0
$592$	8.00000	0.328798
$593$	39.0000	1.60154	0.800769	−	0.598973i	$-0.204424\pi$
$593$	39.0000	1.60154	0.800769	+	0.598973i	$0.204424\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	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
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	18.0000	0.727013	0.363507	−	0.931592i	$-0.381579\pi$
$613$	18.0000	0.727013	0.363507	+	0.931592i	$0.381579\pi$
$614$	−13.0000	−0.524637
$615$	0	0
$616$	5.00000	0.201456
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	6.00000	0.240578
$623$	−11.0000	−0.440706
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	4.00000	0.159617
$629$	8.00000	0.318981
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	2.00000	0.0795557
$633$	0	0
$634$	4.00000	0.158860
$635$	0	0
$636$	0	0
$637$	−6.00000	−0.237729
$638$	−30.0000	−1.18771
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	0	0
$645$	0	0
$646$	3.00000	0.118033
$647$	8.00000	0.314512	0.157256	−	0.987558i	$-0.449735\pi$
$647$	8.00000	0.314512	0.157256	+	0.987558i	$0.449735\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	0	0
$651$	0	0
$652$	−19.0000	−0.744097
$653$	28.0000	1.09572	0.547862	−	0.836569i	$-0.315442\pi$
$653$	28.0000	1.09572	0.547862	+	0.836569i	$0.315442\pi$
$654$	0	0
$655$	0	0
$656$	−11.0000	−0.429478
$657$	0	0
$658$	−2.00000	−0.0779681
$659$	−1.00000	−0.0389545	−0.0194772	−	0.999810i	$-0.506200\pi$
$659$	−1.00000	−0.0389545	−0.0194772	+	0.999810i	$0.506200\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	17.0000	0.660724
$663$	0	0
$664$	11.0000	0.426883
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	29.0000	1.11704
$675$	0	0
$676$	23.0000	0.884615
$677$	48.0000	1.84479	0.922395	−	0.386248i	$-0.126229\pi$
$677$	48.0000	1.84479	0.922395	+	0.386248i	$0.126229\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	0	0
$682$	20.0000	0.765840
$683$	13.0000	0.497431	0.248716	−	0.968577i	$-0.419992\pi$
$683$	13.0000	0.497431	0.248716	+	0.968577i	$0.419992\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	−8.00000	−0.304997
$689$	24.0000	0.914327
$690$	0	0
$691$	49.0000	1.86405	0.932024	−	0.362397i	$-0.118041\pi$
$691$	49.0000	1.86405	0.932024	+	0.362397i	$0.118041\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	19.0000	0.721230
$695$	0	0
$696$	0	0
$697$	−11.0000	−0.416655
$698$	8.00000	0.302804
$699$	0	0
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	5.00000	0.188445
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	4.00000	0.150223	0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000	0.150223	0.0751116	+	0.997175i	$0.476069\pi$
$710$	0	0
$711$	0	0
$712$	−11.0000	−0.412242
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	26.0000	0.970311
$719$	48.0000	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	10.0000	0.372161
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	6.00000	0.222528	0.111264	−	0.993791i	$-0.464510\pi$
$727$	6.00000	0.222528	0.111264	+	0.993791i	$0.464510\pi$
$728$	−6.00000	−0.222375
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	0	0
$733$	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
$737$	45.0000	1.65760
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	−4.00000	−0.146845
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	5.00000	0.182818
$749$	3.00000	0.109618
$750$	0	0
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	−2.00000	−0.0729325
$753$	0	0
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	−9.00000	−0.326895
$759$	0	0
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	18.0000	0.651644
$764$	6.00000	0.217072
$765$	0	0
$766$	−6.00000	−0.216789
$767$	24.0000	0.866590
$768$	0	0
$769$	19.0000	0.685158	0.342579	−	0.939489i	$-0.388700\pi$
$769$	19.0000	0.685158	0.342579	+	0.939489i	$0.388700\pi$
$770$	0	0
$771$	0	0
$772$	−19.0000	−0.683825
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	8.00000	0.286814
$779$	33.0000	1.18235
$780$	0	0
$781$	50.0000	1.78914
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	52.0000	1.85360	0.926800	−	0.375555i	$-0.122548\pi$
$787$	52.0000	1.85360	0.926800	+	0.375555i	$0.122548\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	0	0
$791$	−1.00000	−0.0355559
$792$	0	0
$793$	12.0000	0.426132
$794$	−10.0000	−0.354887
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−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
$799$	−2.00000	−0.0707549
$800$	0	0
$801$	0	0
$802$	37.0000	1.30652
$803$	−35.0000	−1.23512
$804$	0	0
$805$	0	0
$806$	−24.0000	−0.845364
$807$	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
$814$	−40.0000	−1.40200
$815$	0	0
$816$	0	0
$817$	24.0000	0.839654
$818$	21.0000	0.734248
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	10.0000	0.348578	0.174289	−	0.984695i	$-0.444237\pi$
$823$	10.0000	0.348578	0.174289	+	0.984695i	$0.444237\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	−4.00000	−0.139178
$827$	−41.0000	−1.42571	−0.712855	−	0.701312i	$-0.752598\pi$
$827$	−41.0000	−1.42571	−0.712855	+	0.701312i	$0.752598\pi$
$828$	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
$831$	0	0
$832$	−6.00000	−0.208013
$833$	1.00000	0.0346479
$834$	0	0
$835$	0	0
$836$	−15.0000	−0.518786
$837$	0	0
$838$	−39.0000	−1.34723
$839$	−2.00000	−0.0690477	−0.0345238	−	0.999404i	$-0.510991\pi$
$839$	−2.00000	−0.0690477	−0.0345238	+	0.999404i	$0.510991\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−20.0000	−0.689246
$843$	0	0
$844$	1.00000	0.0344214
$845$	0	0
$846$	0	0
$847$	−14.0000	−0.481046
$848$	−4.00000	−0.137361
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	3.00000	0.102538
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−51.0000	−1.74010	−0.870049	−	0.492966i	$-0.835913\pi$
$859$	−51.0000	−1.74010	−0.870049	+	0.492966i	$0.835913\pi$
$860$	0	0
$861$	0	0
$862$	−36.0000	−1.22616
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	0	0
$866$	1.00000	0.0339814
$867$	0	0
$868$	4.00000	0.135769
$869$	−10.0000	−0.339227
$870$	0	0
$871$	−54.0000	−1.82972
$872$	18.0000	0.609557
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−28.0000	−0.944954
$879$	0	0
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	15.0000	0.504790	0.252395	−	0.967624i	$-0.418782\pi$
$883$	15.0000	0.504790	0.252395	+	0.967624i	$0.418782\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	37.0000	1.24304
$887$	34.0000	1.14161	0.570804	−	0.821086i	$-0.306632\pi$
$887$	34.0000	1.14161	0.570804	+	0.821086i	$0.306632\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	0	0
$892$	22.0000	0.736614
$893$	6.00000	0.200782
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	33.0000	1.10122
$899$	−24.0000	−0.800445
$900$	0	0
$901$	−4.00000	−0.133259
$902$	55.0000	1.83130
$903$	0	0
$904$	−1.00000	−0.0332595
$905$	0	0
$906$	0	0
$907$	−4.00000	−0.132818	−0.0664089	−	0.997792i	$-0.521154\pi$
$907$	−4.00000	−0.132818	−0.0664089	+	0.997792i	$0.521154\pi$
$908$	28.0000	0.929213
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	−55.0000	−1.82023
$914$	−25.0000	−0.826927
$915$	0	0
$916$	−14.0000	−0.462573
$917$	8.00000	0.264183
$918$	0	0
$919$	−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
$921$	0	0
$922$	−38.0000	−1.25146
$923$	−60.0000	−1.97492
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	−6.00000	−0.196960
$929$	−46.0000	−1.50921	−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000	−1.50921	−0.754606	+	0.656179i	$0.772172\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	−6.00000	−0.196537
$933$	0	0
$934$	−4.00000	−0.130884
$935$	0	0
$936$	0	0
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	9.00000	0.293860
$939$	0	0
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	0	0
$943$	0	0
$944$	−4.00000	−0.130189
$945$	0	0
$946$	40.0000	1.30051
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	42.0000	1.36338
$950$	0	0
$951$	0	0
$952$	1.00000	0.0324102
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	0	0
$956$	−4.00000	−0.129369
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	48.0000	1.54758
$963$	0	0
$964$	−5.00000	−0.161039
$965$	0	0
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	−14.0000	−0.449977
$969$	0	0
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	0	0
$973$	11.0000	0.352644
$974$	34.0000	1.08943
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−21.0000	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$977$	−21.0000	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$978$	0	0
$979$	55.0000	1.75781
$980$	0	0
$981$	0	0
$982$	−12.0000	−0.382935
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	18.0000	0.572656
$989$	0	0
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	4.00000	0.127000
$993$	0	0
$994$	10.0000	0.317181
$995$	0	0
$996$	0	0
$997$	−58.0000	−1.83688	−0.918439	−	0.395562i	$-0.870550\pi$
$997$	−58.0000	−1.83688	−0.918439	+	0.395562i	$0.870550\pi$
$998$	−36.0000	−1.13956
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.a.j.1.1		1
3.2	odd	2		350.2.a.d.1.1	yes	1
5.2	odd	4		3150.2.g.v.2899.1		2
5.3	odd	4		3150.2.g.v.2899.2		2
5.4	even	2		3150.2.a.bq.1.1		1
12.11	even	2		2800.2.a.bg.1.1		1
15.2	even	4		350.2.c.a.99.2		2
15.8	even	4		350.2.c.a.99.1		2
15.14	odd	2		350.2.a.c.1.1	✓	1
21.20	even	2		2450.2.a.bg.1.1		1
60.23	odd	4		2800.2.g.a.449.2		2
60.47	odd	4		2800.2.g.a.449.1		2
60.59	even	2		2800.2.a.b.1.1		1
105.62	odd	4		2450.2.c.r.99.2		2
105.83	odd	4		2450.2.c.r.99.1		2
105.104	even	2		2450.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.a.c.1.1	✓	1	15.14	odd	2
350.2.a.d.1.1	yes	1	3.2	odd	2
350.2.c.a.99.1		2	15.8	even	4
350.2.c.a.99.2		2	15.2	even	4
2450.2.a.a.1.1		1	105.104	even	2
2450.2.a.bg.1.1		1	21.20	even	2
2450.2.c.r.99.1		2	105.83	odd	4
2450.2.c.r.99.2		2	105.62	odd	4
2800.2.a.b.1.1		1	60.59	even	2
2800.2.a.bg.1.1		1	12.11	even	2
2800.2.g.a.449.1		2	60.47	odd	4
2800.2.g.a.449.2		2	60.23	odd	4
3150.2.a.j.1.1		1	1.1	even	1	trivial
3150.2.a.bq.1.1		1	5.4	even	2
3150.2.g.v.2899.1		2	5.2	odd	4
3150.2.g.v.2899.2		2	5.3	odd	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more