LMFDB - Embedded newform 3136.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,2,Mod(1,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, 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("3136.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3136.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:	$25.0410860739$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} +O(q^{10})$

$q-3.00000 q^{3} -1.00000 q^{5} +6.00000 q^{9} -1.00000 q^{11} +2.00000 q^{13} +3.00000 q^{15} -3.00000 q^{17} -5.00000 q^{19} +3.00000 q^{23} -4.00000 q^{25} -9.00000 q^{27} +6.00000 q^{29} -1.00000 q^{31} +3.00000 q^{33} +5.00000 q^{37} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -6.00000 q^{45} +1.00000 q^{47} +9.00000 q^{51} +9.00000 q^{53} +1.00000 q^{55} +15.0000 q^{57} -3.00000 q^{59} +3.00000 q^{61} -2.00000 q^{65} +11.0000 q^{67} -9.00000 q^{69} -16.0000 q^{71} -7.00000 q^{73} +12.0000 q^{75} +11.0000 q^{79} +9.00000 q^{81} +4.00000 q^{83} +3.00000 q^{85} -18.0000 q^{87} +9.00000 q^{89} +3.00000 q^{93} +5.00000 q^{95} -6.00000 q^{97} -6.00000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.00000	2.00000
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−9.00000	−1.73205
$28$	0	0
$29$	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
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	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
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−6.00000	−0.894427
$46$	0	0
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	9.00000	1.26025
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	15.0000	1.98680
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	11.0000	1.34386	0.671932	−	0.740613i	$-0.265465\pi$
$67$	11.0000	1.34386	0.671932	+	0.740613i	$0.265465\pi$
$68$	0	0
$69$	−9.00000	−1.08347
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	0	0
$87$	−18.0000	−1.92980
$88$	0	0
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	5.00000	0.512989
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−13.0000	−1.29355	−0.646774	−	0.762682i	$-0.723882\pi$
$101$	−13.0000	−1.29355	−0.646774	+	0.762682i	$0.723882\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−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
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	−15.0000	−1.42374
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	12.0000	1.10940
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	−30.0000	−2.70501
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	−17.0000	−1.48530	−0.742648	−	0.669681i	$-0.766431\pi$
$131$	−17.0000	−1.48530	−0.742648	+	0.669681i	$0.766431\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	9.00000	0.774597
$136$	0	0
$137$	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
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	0	0
$151$	−15.0000	−1.22068	−0.610341	−	0.792139i	$-0.708968\pi$
$151$	−15.0000	−1.22068	−0.610341	+	0.792139i	$0.708968\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	15.0000	1.19713	0.598565	−	0.801074i	$-0.295738\pi$
$157$	15.0000	1.19713	0.598565	+	0.801074i	$0.295738\pi$
$158$	0	0
$159$	−27.0000	−2.14124
$160$	0	0
$161$	0	0
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	0	0
$165$	−3.00000	−0.233550
$166$	0	0
$167$	−20.0000	−1.54765	−0.773823	−	0.633402i	$-0.781658\pi$
$167$	−20.0000	−1.54765	−0.773823	+	0.633402i	$0.781658\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−30.0000	−2.29416
$172$	0	0
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	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
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−9.00000	−0.665299
$184$	0	0
$185$	−5.00000	−0.367607
$186$	0	0
$187$	3.00000	0.219382
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.0000	−1.23008	−0.615038	−	0.788497i	$-0.710860\pi$
$191$	−17.0000	−1.23008	−0.615038	+	0.788497i	$0.710860\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	0	0
$201$	−33.0000	−2.32764
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−10.0000	−0.698430
$206$	0	0
$207$	18.0000	1.25109
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	0	0
$213$	48.0000	3.28891
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	21.0000	1.41905
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	−7.00000	−0.464606	−0.232303	−	0.972643i	$-0.574626\pi$
$227$	−7.00000	−0.464606	−0.232303	+	0.972643i	$0.574626\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.0000	−0.851658	−0.425829	−	0.904804i	$-0.640018\pi$
$233$	−13.0000	−0.851658	−0.425829	+	0.904804i	$0.640018\pi$
$234$	0	0
$235$	−1.00000	−0.0652328
$236$	0	0
$237$	−33.0000	−2.14358
$238$	0	0
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.0000	−0.636285
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−9.00000	−0.563602
$256$	0	0
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	36.0000	2.22834
$262$	0	0
$263$	−3.00000	−0.184988	−0.0924940	−	0.995713i	$-0.529484\pi$
$263$	−3.00000	−0.184988	−0.0924940	+	0.995713i	$0.529484\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	−27.0000	−1.65237
$268$	0	0
$269$	−17.0000	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$270$	0	0
$271$	−3.00000	−0.182237	−0.0911185	−	0.995840i	$-0.529044\pi$
$271$	−3.00000	−0.182237	−0.0911185	+	0.995840i	$0.529044\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−7.00000	−0.420589	−0.210295	−	0.977638i	$-0.567442\pi$
$277$	−7.00000	−0.420589	−0.210295	+	0.977638i	$0.567442\pi$
$278$	0	0
$279$	−6.00000	−0.359211
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	17.0000	1.01055	0.505273	−	0.862960i	$-0.331392\pi$
$283$	17.0000	1.01055	0.505273	+	0.862960i	$0.331392\pi$
$284$	0	0
$285$	−15.0000	−0.888523
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	18.0000	1.05518
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	3.00000	0.174667
$296$	0	0
$297$	9.00000	0.522233
$298$	0	0
$299$	6.00000	0.346989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	39.0000	2.24049
$304$	0	0
$305$	−3.00000	−0.171780
$306$	0	0
$307$	−4.00000	−0.228292	−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000	−0.228292	−0.114146	+	0.993464i	$0.536413\pi$
$308$	0	0
$309$	−15.0000	−0.853320
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.0000	−1.51647	−0.758236	−	0.651981i	$-0.773938\pi$
$317$	−27.0000	−1.51647	−0.758236	+	0.651981i	$0.773938\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	9.00000	0.502331
$322$	0	0
$323$	15.0000	0.834622
$324$	0	0
$325$	−8.00000	−0.443760
$326$	0	0
$327$	33.0000	1.82490
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.00000	−0.384755	−0.192377	−	0.981321i	$-0.561620\pi$
$331$	−7.00000	−0.384755	−0.192377	+	0.981321i	$0.561620\pi$
$332$	0	0
$333$	30.0000	1.64399
$334$	0	0
$335$	−11.0000	−0.600994
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	0	0
$339$	30.0000	1.62938
$340$	0	0
$341$	1.00000	0.0541530
$342$	0	0
$343$	0	0
$344$	0	0
$345$	9.00000	0.484544
$346$	0	0
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	−18.0000	−0.960769
$352$	0	0
$353$	5.00000	0.266123	0.133062	−	0.991108i	$-0.457519\pi$
$353$	5.00000	0.266123	0.133062	+	0.991108i	$0.457519\pi$
$354$	0	0
$355$	16.0000	0.849192
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	30.0000	1.57459
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	19.0000	0.991792	0.495896	−	0.868382i	$-0.334840\pi$
$367$	19.0000	0.991792	0.495896	+	0.868382i	$0.334840\pi$
$368$	0	0
$369$	60.0000	3.12348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−19.0000	−0.983783	−0.491891	−	0.870657i	$-0.663694\pi$
$373$	−19.0000	−0.983783	−0.491891	+	0.870657i	$0.663694\pi$
$374$	0	0
$375$	−27.0000	−1.39427
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−24.0000	−1.22956
$382$	0	0
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.0000	−1.21999
$388$	0	0
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	−9.00000	−0.455150
$392$	0	0
$393$	51.0000	2.57261
$394$	0	0
$395$	−11.0000	−0.553470
$396$	0	0
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−5.00000	−0.247841
$408$	0	0
$409$	−19.0000	−0.939490	−0.469745	−	0.882802i	$-0.655654\pi$
$409$	−19.0000	−0.939490	−0.469745	+	0.882802i	$0.655654\pi$
$410$	0	0
$411$	−9.00000	−0.443937
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	6.00000	0.291730
$424$	0	0
$425$	12.0000	0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	41.0000	1.97490	0.987450	−	0.157930i	$-0.0504821\pi$
$431$	41.0000	1.97490	0.987450	+	0.157930i	$0.0504821\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	18.0000	0.863034
$436$	0	0
$437$	−15.0000	−0.717547
$438$	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
$442$	0	0
$443$	−27.0000	−1.28281	−0.641404	−	0.767203i	$-0.721648\pi$
$443$	−27.0000	−1.28281	−0.641404	+	0.767203i	$0.721648\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	0	0
$447$	45.0000	2.12843
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	45.0000	2.11428
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.0000	−0.795226	−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000	−0.795226	−0.397613	+	0.917553i	$0.630161\pi$
$458$	0	0
$459$	27.0000	1.26025
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	0	0
$465$	−3.00000	−0.139122
$466$	0	0
$467$	−25.0000	−1.15686	−0.578431	−	0.815731i	$-0.696335\pi$
$467$	−25.0000	−1.15686	−0.578431	+	0.815731i	$0.696335\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−45.0000	−2.07349
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	20.0000	0.917663
$476$	0	0
$477$	54.0000	2.47249
$478$	0	0
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	13.0000	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$487$	13.0000	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$488$	0	0
$489$	−27.0000	−1.22098
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	0	0
$495$	6.00000	0.269680
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.00000	−0.313363	−0.156682	−	0.987649i	$-0.550080\pi$
$499$	−7.00000	−0.313363	−0.156682	+	0.987649i	$0.550080\pi$
$500$	0	0
$501$	60.0000	2.68060
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	13.0000	0.578492
$506$	0	0
$507$	27.0000	1.19911
$508$	0	0
$509$	7.00000	0.310270	0.155135	−	0.987893i	$-0.450419\pi$
$509$	7.00000	0.310270	0.155135	+	0.987893i	$0.450419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	45.0000	1.98680
$514$	0	0
$515$	−5.00000	−0.220326
$516$	0	0
$517$	−1.00000	−0.0439799
$518$	0	0
$519$	63.0000	2.76539
$520$	0	0
$521$	−15.0000	−0.657162	−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000	−0.657162	−0.328581	+	0.944476i	$0.606570\pi$
$522$	0	0
$523$	−13.0000	−0.568450	−0.284225	−	0.958758i	$-0.591736\pi$
$523$	−13.0000	−0.568450	−0.284225	+	0.958758i	$0.591736\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.00000	0.130682
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−18.0000	−0.781133
$532$	0	0
$533$	20.0000	0.866296
$534$	0	0
$535$	3.00000	0.129701
$536$	0	0
$537$	3.00000	0.129460
$538$	0	0
$539$	0	0
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	0	0
$543$	−66.0000	−2.83233
$544$	0	0
$545$	11.0000	0.471188
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	18.0000	0.768221
$550$	0	0
$551$	−30.0000	−1.27804
$552$	0	0
$553$	0	0
$554$	0	0
$555$	15.0000	0.636715
$556$	0	0
$557$	−11.0000	−0.466085	−0.233042	−	0.972467i	$-0.574868\pi$
$557$	−11.0000	−0.466085	−0.233042	+	0.972467i	$0.574868\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−9.00000	−0.379980
$562$	0	0
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	10.0000	0.420703
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.00000	−0.0419222	−0.0209611	−	0.999780i	$-0.506673\pi$
$569$	−1.00000	−0.0419222	−0.0209611	+	0.999780i	$0.506673\pi$
$570$	0	0
$571$	−17.0000	−0.711428	−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000	−0.711428	−0.355714	+	0.934595i	$0.615762\pi$
$572$	0	0
$573$	51.0000	2.13056
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	−31.0000	−1.29055	−0.645273	−	0.763952i	$-0.723257\pi$
$577$	−31.0000	−1.29055	−0.645273	+	0.763952i	$0.723257\pi$
$578$	0	0
$579$	15.0000	0.623379
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.00000	−0.372742
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−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
$589$	5.00000	0.206021
$590$	0	0
$591$	54.0000	2.22126
$592$	0	0
$593$	−43.0000	−1.76580	−0.882899	−	0.469563i	$-0.844412\pi$
$593$	−43.0000	−1.76580	−0.882899	+	0.469563i	$0.844412\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.0000	1.10504
$598$	0	0
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	66.0000	2.68773
$604$	0	0
$605$	10.0000	0.406558
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.00000	0.0809113
$612$	0	0
$613$	21.0000	0.848182	0.424091	−	0.905620i	$-0.360594\pi$
$613$	21.0000	0.848182	0.424091	+	0.905620i	$0.360594\pi$
$614$	0	0
$615$	30.0000	1.20972
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	5.00000	0.200967	0.100483	−	0.994939i	$-0.467961\pi$
$619$	5.00000	0.200967	0.100483	+	0.994939i	$0.467961\pi$
$620$	0	0
$621$	−27.0000	−1.08347
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−15.0000	−0.599042
$628$	0	0
$629$	−15.0000	−0.598089
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	−36.0000	−1.43087
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−96.0000	−3.79770
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	44.0000	1.73519	0.867595	−	0.497271i	$-0.165665\pi$
$643$	44.0000	1.73519	0.867595	+	0.497271i	$0.165665\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	43.0000	1.69050	0.845252	−	0.534368i	$-0.179450\pi$
$647$	43.0000	1.69050	0.845252	+	0.534368i	$0.179450\pi$
$648$	0	0
$649$	3.00000	0.117760
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.00000	0.195665	0.0978326	−	0.995203i	$-0.468809\pi$
$653$	5.00000	0.195665	0.0978326	+	0.995203i	$0.468809\pi$
$654$	0	0
$655$	17.0000	0.664245
$656$	0	0
$657$	−42.0000	−1.63858
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−1.00000	−0.0388955	−0.0194477	−	0.999811i	$-0.506191\pi$
$661$	−1.00000	−0.0388955	−0.0194477	+	0.999811i	$0.506191\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	0	0
$666$	0	0
$667$	18.0000	0.696963
$668$	0	0
$669$	−72.0000	−2.78368
$670$	0	0
$671$	−3.00000	−0.115814
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	36.0000	1.38564
$676$	0	0
$677$	−9.00000	−0.345898	−0.172949	−	0.984931i	$-0.555330\pi$
$677$	−9.00000	−0.345898	−0.172949	+	0.984931i	$0.555330\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	21.0000	0.804722
$682$	0	0
$683$	39.0000	1.49229	0.746147	−	0.665782i	$-0.231902\pi$
$683$	39.0000	1.49229	0.746147	+	0.665782i	$0.231902\pi$
$684$	0	0
$685$	−3.00000	−0.114624
$686$	0	0
$687$	−21.0000	−0.801200
$688$	0	0
$689$	18.0000	0.685745
$690$	0	0
$691$	47.0000	1.78796	0.893982	−	0.448103i	$-0.147900\pi$
$691$	47.0000	1.78796	0.893982	+	0.448103i	$0.147900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.00000	0.151729
$696$	0	0
$697$	−30.0000	−1.13633
$698$	0	0
$699$	39.0000	1.47512
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−25.0000	−0.942893
$704$	0	0
$705$	3.00000	0.112987
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−27.0000	−1.01401	−0.507003	−	0.861944i	$-0.669247\pi$
$709$	−27.0000	−1.01401	−0.507003	+	0.861944i	$0.669247\pi$
$710$	0	0
$711$	66.0000	2.47519
$712$	0	0
$713$	−3.00000	−0.112351
$714$	0	0
$715$	2.00000	0.0747958
$716$	0	0
$717$	−12.0000	−0.448148
$718$	0	0
$719$	29.0000	1.08152	0.540759	−	0.841178i	$-0.318137\pi$
$719$	29.0000	1.08152	0.540759	+	0.841178i	$0.318137\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−51.0000	−1.89671
$724$	0	0
$725$	−24.0000	−0.891338
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	11.0000	0.406294	0.203147	−	0.979148i	$-0.434883\pi$
$733$	11.0000	0.406294	0.203147	+	0.979148i	$0.434883\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.0000	−0.405190
$738$	0	0
$739$	−41.0000	−1.50821	−0.754105	−	0.656754i	$-0.771929\pi$
$739$	−41.0000	−1.50821	−0.754105	+	0.656754i	$0.771929\pi$
$740$	0	0
$741$	30.0000	1.10208
$742$	0	0
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	24.0000	0.878114
$748$	0	0
$749$	0	0
$750$	0	0
$751$	47.0000	1.71505	0.857527	−	0.514439i	$-0.172000\pi$
$751$	47.0000	1.71505	0.857527	+	0.514439i	$0.172000\pi$
$752$	0	0
$753$	72.0000	2.62383
$754$	0	0
$755$	15.0000	0.545906
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	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
$763$	0	0
$764$	0	0
$765$	18.0000	0.650791
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	−39.0000	−1.40455
$772$	0	0
$773$	35.0000	1.25886	0.629431	−	0.777056i	$-0.283288\pi$
$773$	35.0000	1.25886	0.629431	+	0.777056i	$0.283288\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−50.0000	−1.79144
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	−54.0000	−1.92980
$784$	0	0
$785$	−15.0000	−0.535373
$786$	0	0
$787$	13.0000	0.463400	0.231700	−	0.972787i	$-0.425571\pi$
$787$	13.0000	0.463400	0.231700	+	0.972787i	$0.425571\pi$
$788$	0	0
$789$	9.00000	0.320408
$790$	0	0
$791$	0	0
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	27.0000	0.957591
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	54.0000	1.90800
$802$	0	0
$803$	7.00000	0.247025
$804$	0	0
$805$	0	0
$806$	0	0
$807$	51.0000	1.79529
$808$	0	0
$809$	−25.0000	−0.878953	−0.439477	−	0.898254i	$-0.644836\pi$
$809$	−25.0000	−0.878953	−0.439477	+	0.898254i	$0.644836\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	9.00000	0.315644
$814$	0	0
$815$	−9.00000	−0.315256
$816$	0	0
$817$	20.0000	0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.0000	0.872506	0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000	0.872506	0.436253	+	0.899824i	$0.356305\pi$
$822$	0	0
$823$	21.0000	0.732014	0.366007	−	0.930612i	$-0.380725\pi$
$823$	21.0000	0.732014	0.366007	+	0.930612i	$0.380725\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−4.00000	−0.139094	−0.0695468	−	0.997579i	$-0.522155\pi$
$827$	−4.00000	−0.139094	−0.0695468	+	0.997579i	$0.522155\pi$
$828$	0	0
$829$	−37.0000	−1.28506	−0.642532	−	0.766259i	$-0.722116\pi$
$829$	−37.0000	−1.28506	−0.642532	+	0.766259i	$0.722116\pi$
$830$	0	0
$831$	21.0000	0.728482
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.0000	0.692129
$836$	0	0
$837$	9.00000	0.311086
$838$	0	0
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	54.0000	1.85986
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−51.0000	−1.75032
$850$	0	0
$851$	15.0000	0.514193
$852$	0	0
$853$	−22.0000	−0.753266	−0.376633	−	0.926363i	$-0.622918\pi$
$853$	−22.0000	−0.753266	−0.376633	+	0.926363i	$0.622918\pi$
$854$	0	0
$855$	30.0000	1.02598
$856$	0	0
$857$	57.0000	1.94708	0.973541	−	0.228510i	$-0.0733855\pi$
$857$	57.0000	1.94708	0.973541	+	0.228510i	$0.0733855\pi$
$858$	0	0
$859$	−5.00000	−0.170598	−0.0852989	−	0.996355i	$-0.527185\pi$
$859$	−5.00000	−0.170598	−0.0852989	+	0.996355i	$0.527185\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−37.0000	−1.25949	−0.629747	−	0.776800i	$-0.716842\pi$
$863$	−37.0000	−1.25949	−0.629747	+	0.776800i	$0.716842\pi$
$864$	0	0
$865$	21.0000	0.714021
$866$	0	0
$867$	24.0000	0.815083
$868$	0	0
$869$	−11.0000	−0.373149
$870$	0	0
$871$	22.0000	0.745442
$872$	0	0
$873$	−36.0000	−1.21842
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−39.0000	−1.31694	−0.658468	−	0.752609i	$-0.728795\pi$
$877$	−39.0000	−1.31694	−0.658468	+	0.752609i	$0.728795\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	28.0000	0.942275	0.471138	−	0.882060i	$-0.343844\pi$
$883$	28.0000	0.942275	0.471138	+	0.882060i	$0.343844\pi$
$884$	0	0
$885$	−9.00000	−0.302532
$886$	0	0
$887$	−55.0000	−1.84672	−0.923360	−	0.383936i	$-0.874568\pi$
$887$	−55.0000	−1.84672	−0.923360	+	0.383936i	$0.874568\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	−5.00000	−0.167319
$894$	0	0
$895$	1.00000	0.0334263
$896$	0	0
$897$	−18.0000	−0.601003
$898$	0	0
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−27.0000	−0.899500
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−22.0000	−0.731305
$906$	0	0
$907$	−13.0000	−0.431658	−0.215829	−	0.976431i	$-0.569245\pi$
$907$	−13.0000	−0.431658	−0.215829	+	0.976431i	$0.569245\pi$
$908$	0	0
$909$	−78.0000	−2.58710
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	9.00000	0.297531
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−17.0000	−0.560778	−0.280389	−	0.959886i	$-0.590464\pi$
$919$	−17.0000	−0.560778	−0.280389	+	0.959886i	$0.590464\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	−32.0000	−1.05329
$924$	0	0
$925$	−20.0000	−0.657596
$926$	0	0
$927$	30.0000	0.985329
$928$	0	0
$929$	41.0000	1.34517	0.672583	−	0.740022i	$-0.265185\pi$
$929$	41.0000	1.34517	0.672583	+	0.740022i	$0.265185\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−33.0000	−1.08037
$934$	0	0
$935$	−3.00000	−0.0981105
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	93.0000	3.03494
$940$	0	0
$941$	39.0000	1.27136	0.635682	−	0.771951i	$-0.280719\pi$
$941$	39.0000	1.27136	0.635682	+	0.771951i	$0.280719\pi$
$942$	0	0
$943$	30.0000	0.976934
$944$	0	0
$945$	0	0
$946$	0	0
$947$	49.0000	1.59229	0.796143	−	0.605108i	$-0.206870\pi$
$947$	49.0000	1.59229	0.796143	+	0.605108i	$0.206870\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	81.0000	2.62660
$952$	0	0
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	17.0000	0.550107
$956$	0	0
$957$	18.0000	0.581857
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−18.0000	−0.580042
$964$	0	0
$965$	5.00000	0.160956
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	0	0
$969$	−45.0000	−1.44561
$970$	0	0
$971$	−57.0000	−1.82922	−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000	−1.82922	−0.914609	+	0.404341i	$0.867501\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	24.0000	0.768615
$976$	0	0
$977$	−37.0000	−1.18373	−0.591867	−	0.806035i	$-0.701609\pi$
$977$	−37.0000	−1.18373	−0.591867	+	0.806035i	$0.701609\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	−66.0000	−2.10722
$982$	0	0
$983$	−33.0000	−1.05254	−0.526268	−	0.850319i	$-0.676409\pi$
$983$	−33.0000	−1.05254	−0.526268	+	0.850319i	$0.676409\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−59.0000	−1.87420	−0.937098	−	0.349065i	$-0.886499\pi$
$991$	−59.0000	−1.87420	−0.937098	+	0.349065i	$0.886499\pi$
$992$	0	0
$993$	21.0000	0.666415
$994$	0	0
$995$	9.00000	0.285319
$996$	0	0
$997$	31.0000	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$997$	31.0000	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$998$	0	0
$999$	−45.0000	−1.42374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.2.a.a.1.1		1
4.3	odd	2		3136.2.a.bb.1.1		1
7.3	odd	6		448.2.i.a.65.1		2
7.5	odd	6		448.2.i.a.193.1		2
7.6	odd	2		3136.2.a.bc.1.1		1
8.3	odd	2		392.2.a.a.1.1		1
8.5	even	2		784.2.a.j.1.1		1
24.5	odd	2		7056.2.a.s.1.1		1
24.11	even	2		3528.2.a.k.1.1		1
28.3	even	6		448.2.i.f.65.1		2
28.19	even	6		448.2.i.f.193.1		2
28.27	even	2		3136.2.a.b.1.1		1
40.19	odd	2		9800.2.a.bp.1.1		1
56.3	even	6		56.2.i.a.9.1	✓	2
56.5	odd	6		112.2.i.c.81.1		2
56.11	odd	6		392.2.i.f.177.1		2
56.13	odd	2		784.2.a.a.1.1		1
56.19	even	6		56.2.i.a.25.1	yes	2
56.27	even	2		392.2.a.f.1.1		1
56.37	even	6		784.2.i.a.753.1		2
56.45	odd	6		112.2.i.c.65.1		2
56.51	odd	6		392.2.i.f.361.1		2
56.53	even	6		784.2.i.a.177.1		2
168.5	even	6		1008.2.s.e.865.1		2
168.11	even	6		3528.2.s.o.3313.1		2
168.59	odd	6		504.2.s.e.289.1		2
168.83	odd	2		3528.2.a.r.1.1		1
168.101	even	6		1008.2.s.e.289.1		2
168.107	even	6		3528.2.s.o.361.1		2
168.125	even	2		7056.2.a.bi.1.1		1
168.131	odd	6		504.2.s.e.361.1		2
280.3	odd	12		1400.2.bh.f.849.2		4
280.19	even	6		1400.2.q.g.1201.1		2
280.59	even	6		1400.2.q.g.401.1		2
280.139	even	2		9800.2.a.b.1.1		1
280.187	odd	12		1400.2.bh.f.249.2		4
280.227	odd	12		1400.2.bh.f.849.1		4
280.243	odd	12		1400.2.bh.f.249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.i.a.9.1	✓	2	56.3	even	6
56.2.i.a.25.1	yes	2	56.19	even	6
112.2.i.c.65.1		2	56.45	odd	6
112.2.i.c.81.1		2	56.5	odd	6
392.2.a.a.1.1		1	8.3	odd	2
392.2.a.f.1.1		1	56.27	even	2
392.2.i.f.177.1		2	56.11	odd	6
392.2.i.f.361.1		2	56.51	odd	6
448.2.i.a.65.1		2	7.3	odd	6
448.2.i.a.193.1		2	7.5	odd	6
448.2.i.f.65.1		2	28.3	even	6
448.2.i.f.193.1		2	28.19	even	6
504.2.s.e.289.1		2	168.59	odd	6
504.2.s.e.361.1		2	168.131	odd	6
784.2.a.a.1.1		1	56.13	odd	2
784.2.a.j.1.1		1	8.5	even	2
784.2.i.a.177.1		2	56.53	even	6
784.2.i.a.753.1		2	56.37	even	6
1008.2.s.e.289.1		2	168.101	even	6
1008.2.s.e.865.1		2	168.5	even	6
1400.2.q.g.401.1		2	280.59	even	6
1400.2.q.g.1201.1		2	280.19	even	6
1400.2.bh.f.249.1		4	280.243	odd	12
1400.2.bh.f.249.2		4	280.187	odd	12
1400.2.bh.f.849.1		4	280.227	odd	12
1400.2.bh.f.849.2		4	280.3	odd	12
3136.2.a.a.1.1		1	1.1	even	1	trivial
3136.2.a.b.1.1		1	28.27	even	2
3136.2.a.bb.1.1		1	4.3	odd	2
3136.2.a.bc.1.1		1	7.6	odd	2
3528.2.a.k.1.1		1	24.11	even	2
3528.2.a.r.1.1		1	168.83	odd	2
3528.2.s.o.361.1		2	168.107	even	6
3528.2.s.o.3313.1		2	168.11	even	6
7056.2.a.s.1.1		1	24.5	odd	2
7056.2.a.bi.1.1		1	168.125	even	2
9800.2.a.b.1.1		1	280.139	even	2
9800.2.a.bp.1.1		1	40.19	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:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3136.a (trivial)

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L-functions

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