LMFDB - Embedded newform 3249.2.a.bc.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,4,0,-8,0,0,0,10,0,0,0,0,-14,8,0,0,12,0,0,6,0,4,-10,0, 6,0,0,0,0,0,0,12,0,0,0,0,0,0,0,6,0,0,0,28,0,8,0,0,0,0,0,0,0,0,10,0,0,-20, 10,0,8,0,0,0,24,0,0,0,0,36,-10,0,0,-30,0,0,-24,0,30,4,0,48,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 361)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.90211$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90211 q^{2} +1.61803 q^{4} +3.23607 q^{5} +0.236068 q^{7} -0.726543 q^{8} +6.15537 q^{10} +1.38197 q^{11} -0.726543 q^{13} +0.449028 q^{14} -4.61803 q^{16} +6.47214 q^{17} +5.23607 q^{20} +2.62866 q^{22} +0.381966 q^{23} +5.47214 q^{25} -1.38197 q^{26} +0.381966 q^{28} +4.25325 q^{29} +1.90211 q^{31} -7.33094 q^{32} +12.3107 q^{34} +0.763932 q^{35} -6.60440 q^{37} -2.35114 q^{40} +2.17963 q^{41} +9.32624 q^{43} +2.23607 q^{44} +0.726543 q^{46} +11.4721 q^{47} -6.94427 q^{49} +10.4086 q^{50} -1.17557 q^{52} +4.97980 q^{53} +4.47214 q^{55} -0.171513 q^{56} +8.09017 q^{58} -8.78402 q^{59} -9.47214 q^{61} +3.61803 q^{62} -4.70820 q^{64} -2.35114 q^{65} +13.0373 q^{67} +10.4721 q^{68} +1.45309 q^{70} -10.1311 q^{71} +9.00000 q^{73} -12.5623 q^{74} +0.326238 q^{77} -4.70228 q^{79} -14.9443 q^{80} +4.14590 q^{82} +3.23607 q^{83} +20.9443 q^{85} +17.7396 q^{86} -1.00406 q^{88} -7.43694 q^{89} -0.171513 q^{91} +0.618034 q^{92} +21.8213 q^{94} -4.42477 q^{97} -13.2088 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 8 q^{7} + 10 q^{11} - 14 q^{16} + 8 q^{17} + 12 q^{20} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{35} + 6 q^{43} + 28 q^{47} + 8 q^{49} + 10 q^{58} - 20 q^{61} + 10 q^{62}+ \cdots - 2 q^{92}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 8 * q^7 + 10 * q^11 - 14 * q^16 + 8 * q^17 + 12 * q^20 + 6 * q^23 + 4 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^35 + 6 * q^43 + 28 * q^47 + 8 * q^49 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 8 * q^64 + 24 * q^68 + 36 * q^73 - 10 * q^74 - 30 * q^77 - 24 * q^80 + 30 * q^82 + 4 * q^83 + 48 * q^85 - 2 * q^92

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 $

$2$	1.90211	1.34500	0.672499	−	0.740098i	$-0.265221\pi$
$2$	1.90211	1.34500	0.672499	+	0.740098i	$0.265221\pi$
$3$	0	0
$3$	0	0
$4$	1.61803	0.809017
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	−0.726543	−0.256872
$9$	0	0
$10$	6.15537	1.94650
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	−0.726543	−0.201507	−0.100753	−	0.994911i	$-0.532125\pi$
$13$	−0.726543	−0.201507	−0.100753	+	0.994911i	$0.532125\pi$
$14$	0.449028	0.120008
$15$	0	0
$16$	−4.61803	−1.15451
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	5.23607	1.17082
$21$	0	0
$22$	2.62866	0.560431
$23$	0.381966	0.0796454	0.0398227	−	0.999207i	$-0.487321\pi$
$23$	0.381966	0.0796454	0.0398227	+	0.999207i	$0.487321\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	−1.38197	−0.271026
$27$	0	0
$28$	0.381966	0.0721848
$29$	4.25325	0.789809	0.394905	−	0.918722i	$-0.370778\pi$
$29$	4.25325	0.789809	0.394905	+	0.918722i	$0.370778\pi$
$30$	0	0
$31$	1.90211	0.341630	0.170815	−	0.985303i	$-0.445360\pi$
$31$	1.90211	0.341630	0.170815	+	0.985303i	$0.445360\pi$
$32$	−7.33094	−1.29594
$33$	0	0
$34$	12.3107	2.11127
$35$	0.763932	0.129128
$36$	0	0
$37$	−6.60440	−1.08576	−0.542878	−	0.839812i	$-0.682665\pi$
$37$	−6.60440	−1.08576	−0.542878	+	0.839812i	$0.682665\pi$
$38$	0	0
$39$	0	0
$40$	−2.35114	−0.371748
$41$	2.17963	0.340401	0.170200	−	0.985409i	$-0.445559\pi$
$41$	2.17963	0.340401	0.170200	+	0.985409i	$0.445559\pi$
$42$	0	0
$43$	9.32624	1.42224	0.711119	−	0.703072i	$-0.248189\pi$
$43$	9.32624	1.42224	0.711119	+	0.703072i	$0.248189\pi$
$44$	2.23607	0.337100
$45$	0	0
$46$	0.726543	0.107123
$47$	11.4721	1.67338	0.836692	−	0.547674i	$-0.184487\pi$
$47$	11.4721	1.67338	0.836692	+	0.547674i	$0.184487\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	10.4086	1.47200
$51$	0	0
$52$	−1.17557	−0.163022
$53$	4.97980	0.684028	0.342014	−	0.939695i	$-0.388891\pi$
$53$	4.97980	0.684028	0.342014	+	0.939695i	$0.388891\pi$
$54$	0	0
$55$	4.47214	0.603023
$56$	−0.171513	−0.0229194
$57$	0	0
$58$	8.09017	1.06229
$59$	−8.78402	−1.14358	−0.571791	−	0.820399i	$-0.693751\pi$
$59$	−8.78402	−1.14358	−0.571791	+	0.820399i	$0.693751\pi$
$60$	0	0
$61$	−9.47214	−1.21278	−0.606391	−	0.795166i	$-0.707384\pi$
$61$	−9.47214	−1.21278	−0.606391	+	0.795166i	$0.707384\pi$
$62$	3.61803	0.459491
$63$	0	0
$64$	−4.70820	−0.588525
$65$	−2.35114	−0.291623
$66$	0	0
$67$	13.0373	1.59276	0.796378	−	0.604799i	$-0.206746\pi$
$67$	13.0373	1.59276	0.796378	+	0.604799i	$0.206746\pi$
$68$	10.4721	1.26993
$69$	0	0
$70$	1.45309	0.173677
$71$	−10.1311	−1.20234	−0.601171	−	0.799121i	$-0.705299\pi$
$71$	−10.1311	−1.20234	−0.601171	+	0.799121i	$0.705299\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	−12.5623	−1.46034
$75$	0	0
$76$	0	0
$77$	0.326238	0.0371783
$78$	0	0
$79$	−4.70228	−0.529048	−0.264524	−	0.964379i	$-0.585215\pi$
$79$	−4.70228	−0.529048	−0.264524	+	0.964379i	$0.585215\pi$
$80$	−14.9443	−1.67082
$81$	0	0
$82$	4.14590	0.457838
$83$	3.23607	0.355205	0.177602	−	0.984102i	$-0.443166\pi$
$83$	3.23607	0.355205	0.177602	+	0.984102i	$0.443166\pi$
$84$	0	0
$85$	20.9443	2.27173
$86$	17.7396	1.91291
$87$	0	0
$88$	−1.00406	−0.107033
$89$	−7.43694	−0.788314	−0.394157	−	0.919043i	$-0.628963\pi$
$89$	−7.43694	−0.788314	−0.394157	+	0.919043i	$0.628963\pi$
$90$	0	0
$91$	−0.171513	−0.0179795
$92$	0.618034	0.0644345
$93$	0	0
$94$	21.8213	2.25070
$95$	0	0
$96$	0	0
$97$	−4.42477	−0.449267	−0.224634	−	0.974443i	$-0.572118\pi$
$97$	−4.42477	−0.449267	−0.224634	+	0.974443i	$0.572118\pi$
$98$	−13.2088	−1.33429
$99$	0	0
$100$	8.85410	0.885410
$101$	−3.94427	−0.392470	−0.196235	−	0.980557i	$-0.562871\pi$
$101$	−3.94427	−0.392470	−0.196235	+	0.980557i	$0.562871\pi$
$102$	0	0
$103$	11.1352	1.09718	0.548590	−	0.836091i	$-0.315165\pi$
$103$	11.1352	1.09718	0.548590	+	0.836091i	$0.315165\pi$
$104$	0.527864	0.0517613
$105$	0	0
$106$	9.47214	0.920015
$107$	−5.98385	−0.578481	−0.289240	−	0.957256i	$-0.593403\pi$
$107$	−5.98385	−0.578481	−0.289240	+	0.957256i	$0.593403\pi$
$108$	0	0
$109$	−3.07768	−0.294789	−0.147394	−	0.989078i	$-0.547089\pi$
$109$	−3.07768	−0.294789	−0.147394	+	0.989078i	$0.547089\pi$
$110$	8.50651	0.811064
$111$	0	0
$112$	−1.09017	−0.103011
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	1.23607	0.115264
$116$	6.88191	0.638969
$117$	0	0
$118$	−16.7082	−1.53811
$119$	1.52786	0.140059
$120$	0	0
$121$	−9.09017	−0.826379
$122$	−18.0171	−1.63119
$123$	0	0
$124$	3.07768	0.276384
$125$	1.52786	0.136656
$126$	0	0
$127$	12.4822	1.10762	0.553810	−	0.832643i	$-0.313173\pi$
$127$	12.4822	1.10762	0.553810	+	0.832643i	$0.313173\pi$
$128$	5.70634	0.504374
$129$	0	0
$130$	−4.47214	−0.392232
$131$	−6.09017	−0.532101	−0.266050	−	0.963959i	$-0.585719\pi$
$131$	−6.09017	−0.532101	−0.266050	+	0.963959i	$0.585719\pi$
$132$	0	0
$133$	0	0
$134$	24.7984	2.14225
$135$	0	0
$136$	−4.70228	−0.403217
$137$	−8.70820	−0.743992	−0.371996	−	0.928234i	$-0.621327\pi$
$137$	−8.70820	−0.743992	−0.371996	+	0.928234i	$0.621327\pi$
$138$	0	0
$139$	−12.0902	−1.02547	−0.512737	−	0.858545i	$-0.671369\pi$
$139$	−12.0902	−1.02547	−0.512737	+	0.858545i	$0.671369\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	−19.2705	−1.61715
$143$	−1.00406	−0.0839635
$144$	0	0
$145$	13.7638	1.14302
$146$	17.1190	1.41678
$147$	0	0
$148$	−10.6861	−0.878395
$149$	−6.03444	−0.494361	−0.247180	−	0.968969i	$-0.579504\pi$
$149$	−6.03444	−0.494361	−0.247180	+	0.968969i	$0.579504\pi$
$150$	0	0
$151$	−17.2905	−1.40708	−0.703542	−	0.710654i	$-0.748399\pi$
$151$	−17.2905	−1.40708	−0.703542	+	0.710654i	$0.748399\pi$
$152$	0	0
$153$	0	0
$154$	0.620541	0.0500047
$155$	6.15537	0.494411
$156$	0	0
$157$	−4.56231	−0.364112	−0.182056	−	0.983288i	$-0.558275\pi$
$157$	−4.56231	−0.364112	−0.182056	+	0.983288i	$0.558275\pi$
$158$	−8.94427	−0.711568
$159$	0	0
$160$	−23.7234	−1.87550
$161$	0.0901699	0.00710639
$162$	0	0
$163$	15.4721	1.21187	0.605936	−	0.795514i	$-0.292799\pi$
$163$	15.4721	1.21187	0.605936	+	0.795514i	$0.292799\pi$
$164$	3.52671	0.275390
$165$	0	0
$166$	6.15537	0.477749
$167$	−1.62460	−0.125715	−0.0628576	−	0.998023i	$-0.520021\pi$
$167$	−1.62460	−0.125715	−0.0628576	+	0.998023i	$0.520021\pi$
$168$	0	0
$169$	−12.4721	−0.959395
$170$	39.8384	3.05546
$171$	0	0
$172$	15.0902	1.15061
$173$	16.6700	1.26740	0.633698	−	0.773581i	$-0.281536\pi$
$173$	16.6700	1.26740	0.633698	+	0.773581i	$0.281536\pi$
$174$	0	0
$175$	1.29180	0.0976506
$176$	−6.38197	−0.481059
$177$	0	0
$178$	−14.1459	−1.06028
$179$	8.67802	0.648626	0.324313	−	0.945950i	$-0.394867\pi$
$179$	8.67802	0.648626	0.324313	+	0.945950i	$0.394867\pi$
$180$	0	0
$181$	−5.81234	−0.432028	−0.216014	−	0.976390i	$-0.569306\pi$
$181$	−5.81234	−0.432028	−0.216014	+	0.976390i	$0.569306\pi$
$182$	−0.326238	−0.0241824
$183$	0	0
$184$	−0.277515	−0.0204586
$185$	−21.3723	−1.57132
$186$	0	0
$187$	8.94427	0.654070
$188$	18.5623	1.35380
$189$	0	0
$190$	0	0
$191$	17.0000	1.23008	0.615038	−	0.788497i	$-0.289140\pi$
$191$	17.0000	1.23008	0.615038	+	0.788497i	$0.289140\pi$
$192$	0	0
$193$	8.50651	0.612312	0.306156	−	0.951981i	$-0.400957\pi$
$193$	8.50651	0.612312	0.306156	+	0.951981i	$0.400957\pi$
$194$	−8.41641	−0.604263
$195$	0	0
$196$	−11.2361	−0.802576
$197$	−11.4721	−0.817356	−0.408678	−	0.912679i	$-0.634010\pi$
$197$	−11.4721	−0.817356	−0.408678	+	0.912679i	$0.634010\pi$
$198$	0	0
$199$	−1.05573	−0.0748386	−0.0374193	−	0.999300i	$-0.511914\pi$
$199$	−1.05573	−0.0748386	−0.0374193	+	0.999300i	$0.511914\pi$
$200$	−3.97574	−0.281127
$201$	0	0
$202$	−7.50245	−0.527871
$203$	1.00406	0.0704710
$204$	0	0
$205$	7.05342	0.492632
$206$	21.1803	1.47570
$207$	0	0
$208$	3.35520	0.232641
$209$	0	0
$210$	0	0
$211$	0.620541	0.0427198	0.0213599	−	0.999772i	$-0.493200\pi$
$211$	0.620541	0.0427198	0.0213599	+	0.999772i	$0.493200\pi$
$212$	8.05748	0.553390
$213$	0	0
$214$	−11.3820	−0.778055
$215$	30.1803	2.05828
$216$	0	0
$217$	0.449028	0.0304820
$218$	−5.85410	−0.396490
$219$	0	0
$220$	7.23607	0.487856
$221$	−4.70228	−0.316310
$222$	0	0
$223$	−16.2865	−1.09062	−0.545311	−	0.838233i	$-0.683589\pi$
$223$	−16.2865	−1.09062	−0.545311	+	0.838233i	$0.683589\pi$
$224$	−1.73060	−0.115631
$225$	0	0
$226$	0	0
$227$	−15.7314	−1.04413	−0.522066	−	0.852905i	$-0.674839\pi$
$227$	−15.7314	−1.04413	−0.522066	+	0.852905i	$0.674839\pi$
$228$	0	0
$229$	−7.09017	−0.468532	−0.234266	−	0.972173i	$-0.575269\pi$
$229$	−7.09017	−0.468532	−0.234266	+	0.972173i	$0.575269\pi$
$230$	2.35114	0.155030
$231$	0	0
$232$	−3.09017	−0.202880
$233$	−4.52786	−0.296630	−0.148315	−	0.988940i	$-0.547385\pi$
$233$	−4.52786	−0.296630	−0.148315	+	0.988940i	$0.547385\pi$
$234$	0	0
$235$	37.1246	2.42174
$236$	−14.2128	−0.925178
$237$	0	0
$238$	2.90617	0.188379
$239$	9.14590	0.591599	0.295799	−	0.955250i	$-0.404414\pi$
$239$	9.14590	0.591599	0.295799	+	0.955250i	$0.404414\pi$
$240$	0	0
$241$	−27.1441	−1.74851	−0.874253	−	0.485470i	$-0.838648\pi$
$241$	−27.1441	−1.74851	−0.874253	+	0.485470i	$0.838648\pi$
$242$	−17.2905	−1.11148
$243$	0	0
$244$	−15.3262	−0.981162
$245$	−22.4721	−1.43569
$246$	0	0
$247$	0	0
$248$	−1.38197	−0.0877549
$249$	0	0
$250$	2.90617	0.183802
$251$	16.4164	1.03619	0.518097	−	0.855322i	$-0.326641\pi$
$251$	16.4164	1.03619	0.518097	+	0.855322i	$0.326641\pi$
$252$	0	0
$253$	0.527864	0.0331865
$254$	23.7426	1.48975
$255$	0	0
$256$	20.2705	1.26691
$257$	10.8576	0.677282	0.338641	−	0.940916i	$-0.390033\pi$
$257$	10.8576	0.677282	0.338641	+	0.940916i	$0.390033\pi$
$258$	0	0
$259$	−1.55909	−0.0968769
$260$	−3.80423	−0.235928
$261$	0	0
$262$	−11.5842	−0.715674
$263$	23.8885	1.47303	0.736515	−	0.676421i	$-0.236470\pi$
$263$	23.8885	1.47303	0.736515	+	0.676421i	$0.236470\pi$
$264$	0	0
$265$	16.1150	0.989934
$266$	0	0
$267$	0	0
$268$	21.0948	1.28857
$269$	−25.7970	−1.57287	−0.786437	−	0.617671i	$-0.788076\pi$
$269$	−25.7970	−1.57287	−0.786437	+	0.617671i	$0.788076\pi$
$270$	0	0
$271$	−19.2705	−1.17060	−0.585300	−	0.810817i	$-0.699023\pi$
$271$	−19.2705	−1.17060	−0.585300	+	0.810817i	$0.699023\pi$
$272$	−29.8885	−1.81226
$273$	0	0
$274$	−16.5640	−1.00067
$275$	7.56231	0.456024
$276$	0	0
$277$	−14.7639	−0.887079	−0.443539	−	0.896255i	$-0.646277\pi$
$277$	−14.7639	−0.887079	−0.443539	+	0.896255i	$0.646277\pi$
$278$	−22.9969	−1.37926
$279$	0	0
$280$	−0.555029	−0.0331693
$281$	15.6659	0.934551	0.467276	−	0.884112i	$-0.345236\pi$
$281$	15.6659	0.934551	0.467276	+	0.884112i	$0.345236\pi$
$282$	0	0
$283$	−12.0902	−0.718686	−0.359343	−	0.933206i	$-0.616999\pi$
$283$	−12.0902	−0.718686	−0.359343	+	0.933206i	$0.616999\pi$
$284$	−16.3925	−0.972714
$285$	0	0
$286$	−1.90983	−0.112931
$287$	0.514540	0.0303724
$288$	0	0
$289$	24.8885	1.46403
$290$	26.1803	1.53736
$291$	0	0
$292$	14.5623	0.852194
$293$	15.2824	0.892808	0.446404	−	0.894831i	$-0.352704\pi$
$293$	15.2824	0.892808	0.446404	+	0.894831i	$0.352704\pi$
$294$	0	0
$295$	−28.4257	−1.65501
$296$	4.79837	0.278900
$297$	0	0
$298$	−11.4782	−0.664914
$299$	−0.277515	−0.0160491
$300$	0	0
$301$	2.20163	0.126900
$302$	−32.8885	−1.89252
$303$	0	0
$304$	0	0
$305$	−30.6525	−1.75516
$306$	0	0
$307$	−27.6336	−1.57714	−0.788568	−	0.614948i	$-0.789177\pi$
$307$	−27.6336	−1.57714	−0.788568	+	0.614948i	$0.789177\pi$
$308$	0.527864	0.0300778
$309$	0	0
$310$	11.7082	0.664981
$311$	14.4721	0.820640	0.410320	−	0.911942i	$-0.365417\pi$
$311$	14.4721	0.820640	0.410320	+	0.911942i	$0.365417\pi$
$312$	0	0
$313$	5.38197	0.304207	0.152103	−	0.988365i	$-0.451395\pi$
$313$	5.38197	0.304207	0.152103	+	0.988365i	$0.451395\pi$
$314$	−8.67802	−0.489729
$315$	0	0
$316$	−7.60845	−0.428009
$317$	−31.3319	−1.75977	−0.879886	−	0.475184i	$-0.842381\pi$
$317$	−31.3319	−1.75977	−0.879886	+	0.475184i	$0.842381\pi$
$318$	0	0
$319$	5.87785	0.329097
$320$	−15.2361	−0.851722
$321$	0	0
$322$	0.171513	0.00955807
$323$	0	0
$324$	0	0
$325$	−3.97574	−0.220534
$326$	29.4298	1.62996
$327$	0	0
$328$	−1.58359	−0.0874392
$329$	2.70820	0.149308
$330$	0	0
$331$	20.2622	1.11371	0.556856	−	0.830609i	$-0.312008\pi$
$331$	20.2622	1.11371	0.556856	+	0.830609i	$0.312008\pi$
$332$	5.23607	0.287367
$333$	0	0
$334$	−3.09017	−0.169087
$335$	42.1895	2.30506
$336$	0	0
$337$	−0.555029	−0.0302344	−0.0151172	−	0.999886i	$-0.504812\pi$
$337$	−0.555029	−0.0302344	−0.0151172	+	0.999886i	$0.504812\pi$
$338$	−23.7234	−1.29038
$339$	0	0
$340$	33.8885	1.83786
$341$	2.62866	0.142350
$342$	0	0
$343$	−3.29180	−0.177740
$344$	−6.77591	−0.365333
$345$	0	0
$346$	31.7082	1.70464
$347$	−28.8328	−1.54783	−0.773913	−	0.633292i	$-0.781703\pi$
$347$	−28.8328	−1.54783	−0.773913	+	0.633292i	$0.781703\pi$
$348$	0	0
$349$	13.6180	0.728957	0.364478	−	0.931212i	$-0.381247\pi$
$349$	13.6180	0.728957	0.364478	+	0.931212i	$0.381247\pi$
$350$	2.45714	0.131340
$351$	0	0
$352$	−10.1311	−0.539990
$353$	−25.3820	−1.35095	−0.675473	−	0.737385i	$-0.736060\pi$
$353$	−25.3820	−1.35095	−0.675473	+	0.737385i	$0.736060\pi$
$354$	0	0
$355$	−32.7849	−1.74004
$356$	−12.0332	−0.637759
$357$	0	0
$358$	16.5066	0.872400
$359$	35.3820	1.86739	0.933694	−	0.358071i	$-0.116565\pi$
$359$	35.3820	1.86739	0.933694	+	0.358071i	$0.116565\pi$
$360$	0	0
$361$	0	0
$362$	−11.0557	−0.581076
$363$	0	0
$364$	−0.277515	−0.0145457
$365$	29.1246	1.52445
$366$	0	0
$367$	12.5279	0.653949	0.326975	−	0.945033i	$-0.393971\pi$
$367$	12.5279	0.653949	0.326975	+	0.945033i	$0.393971\pi$
$368$	−1.76393	−0.0919513
$369$	0	0
$370$	−40.6525	−2.11342
$371$	1.17557	0.0610326
$372$	0	0
$373$	−36.7607	−1.90340	−0.951698	−	0.307035i	$-0.900663\pi$
$373$	−36.7607	−1.90340	−0.951698	+	0.307035i	$0.900663\pi$
$374$	17.0130	0.879722
$375$	0	0
$376$	−8.33499	−0.429845
$377$	−3.09017	−0.159152
$378$	0	0
$379$	−21.3723	−1.09782	−0.548910	−	0.835882i	$-0.684957\pi$
$379$	−21.3723	−1.09782	−0.548910	+	0.835882i	$0.684957\pi$
$380$	0	0
$381$	0	0
$382$	32.3359	1.65445
$383$	11.3067	0.577744	0.288872	−	0.957368i	$-0.406720\pi$
$383$	11.3067	0.577744	0.288872	+	0.957368i	$0.406720\pi$
$384$	0	0
$385$	1.05573	0.0538049
$386$	16.1803	0.823558
$387$	0	0
$388$	−7.15942	−0.363465
$389$	−1.90983	−0.0968322	−0.0484161	−	0.998827i	$-0.515417\pi$
$389$	−1.90983	−0.0968322	−0.0484161	+	0.998827i	$0.515417\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	5.04531	0.254827
$393$	0	0
$394$	−21.8213	−1.09934
$395$	−15.2169	−0.765646
$396$	0	0
$397$	1.29180	0.0648334	0.0324167	−	0.999474i	$-0.489680\pi$
$397$	1.29180	0.0648334	0.0324167	+	0.999474i	$0.489680\pi$
$398$	−2.00811	−0.100658
$399$	0	0
$400$	−25.2705	−1.26353
$401$	−4.70228	−0.234821	−0.117410	−	0.993083i	$-0.537459\pi$
$401$	−4.70228	−0.234821	−0.117410	+	0.993083i	$0.537459\pi$
$402$	0	0
$403$	−1.38197	−0.0688406
$404$	−6.38197	−0.317515
$405$	0	0
$406$	1.90983	0.0947833
$407$	−9.12705	−0.452411
$408$	0	0
$409$	−24.0664	−1.19001	−0.595004	−	0.803722i	$-0.702850\pi$
$409$	−24.0664	−1.19001	−0.595004	+	0.803722i	$0.702850\pi$
$410$	13.4164	0.662589
$411$	0	0
$412$	18.0171	0.887637
$413$	−2.07363	−0.102036
$414$	0	0
$415$	10.4721	0.514057
$416$	5.32624	0.261140
$417$	0	0
$418$	0	0
$419$	6.58359	0.321630	0.160815	−	0.986985i	$-0.448588\pi$
$419$	6.58359	0.321630	0.160815	+	0.986985i	$0.448588\pi$
$420$	0	0
$421$	32.4014	1.57915	0.789575	−	0.613655i	$-0.210301\pi$
$421$	32.4014	1.57915	0.789575	+	0.613655i	$0.210301\pi$
$422$	1.18034	0.0574580
$423$	0	0
$424$	−3.61803	−0.175707
$425$	35.4164	1.71795
$426$	0	0
$427$	−2.23607	−0.108211
$428$	−9.68208	−0.468001
$429$	0	0
$430$	57.4064	2.76838
$431$	−12.4822	−0.601249	−0.300624	−	0.953743i	$-0.597195\pi$
$431$	−12.4822	−0.601249	−0.300624	+	0.953743i	$0.597195\pi$
$432$	0	0
$433$	18.9151	0.909003	0.454502	−	0.890746i	$-0.349817\pi$
$433$	18.9151	0.909003	0.454502	+	0.890746i	$0.349817\pi$
$434$	0.854102	0.0409982
$435$	0	0
$436$	−4.97980	−0.238489
$437$	0	0
$438$	0	0
$439$	−5.98385	−0.285594	−0.142797	−	0.989752i	$-0.545610\pi$
$439$	−5.98385	−0.285594	−0.142797	+	0.989752i	$0.545610\pi$
$440$	−3.24920	−0.154899
$441$	0	0
$442$	−8.94427	−0.425436
$443$	−15.1803	−0.721240	−0.360620	−	0.932713i	$-0.617435\pi$
$443$	−15.1803	−0.721240	−0.360620	+	0.932713i	$0.617435\pi$
$444$	0	0
$445$	−24.0664	−1.14086
$446$	−30.9787	−1.46688
$447$	0	0
$448$	−1.11146	−0.0525114
$449$	−24.2380	−1.14386	−0.571930	−	0.820303i	$-0.693805\pi$
$449$	−24.2380	−1.14386	−0.571930	+	0.820303i	$0.693805\pi$
$450$	0	0
$451$	3.01217	0.141838
$452$	0	0
$453$	0	0
$454$	−29.9230	−1.40436
$455$	−0.555029	−0.0260202
$456$	0	0
$457$	6.29180	0.294318	0.147159	−	0.989113i	$-0.452987\pi$
$457$	6.29180	0.294318	0.147159	+	0.989113i	$0.452987\pi$
$458$	−13.4863	−0.630174
$459$	0	0
$460$	2.00000	0.0932505
$461$	11.8197	0.550496	0.275248	−	0.961373i	$-0.411240\pi$
$461$	11.8197	0.550496	0.275248	+	0.961373i	$0.411240\pi$
$462$	0	0
$463$	10.1459	0.471520	0.235760	−	0.971811i	$-0.424242\pi$
$463$	10.1459	0.471520	0.235760	+	0.971811i	$0.424242\pi$
$464$	−19.6417	−0.911842
$465$	0	0
$466$	−8.61251	−0.398967
$467$	−4.23607	−0.196022	−0.0980109	−	0.995185i	$-0.531248\pi$
$467$	−4.23607	−0.196022	−0.0980109	+	0.995185i	$0.531248\pi$
$468$	0	0
$469$	3.07768	0.142114
$470$	70.6152	3.25724
$471$	0	0
$472$	6.38197	0.293754
$473$	12.8885	0.592616
$474$	0	0
$475$	0	0
$476$	2.47214	0.113310
$477$	0	0
$478$	17.3965	0.795699
$479$	15.9098	0.726939	0.363469	−	0.931606i	$-0.381592\pi$
$479$	15.9098	0.726939	0.363469	+	0.931606i	$0.381592\pi$
$480$	0	0
$481$	4.79837	0.218787
$482$	−51.6312	−2.35174
$483$	0	0
$484$	−14.7082	−0.668555
$485$	−14.3188	−0.650185
$486$	0	0
$487$	16.6700	0.755389	0.377695	−	0.925930i	$-0.376717\pi$
$487$	16.6700	0.755389	0.377695	+	0.925930i	$0.376717\pi$
$488$	6.88191	0.311529
$489$	0	0
$490$	−42.7445	−1.93100
$491$	−17.2705	−0.779407	−0.389704	−	0.920940i	$-0.627423\pi$
$491$	−17.2705	−0.779407	−0.389704	+	0.920940i	$0.627423\pi$
$492$	0	0
$493$	27.5276	1.23978
$494$	0	0
$495$	0	0
$496$	−8.78402	−0.394414
$497$	−2.39163	−0.107279
$498$	0	0
$499$	30.4721	1.36412	0.682060	−	0.731296i	$-0.261084\pi$
$499$	30.4721	1.36412	0.682060	+	0.731296i	$0.261084\pi$
$500$	2.47214	0.110557
$501$	0	0
$502$	31.2259	1.39368
$503$	−20.5967	−0.918364	−0.459182	−	0.888342i	$-0.651857\pi$
$503$	−20.5967	−0.918364	−0.459182	+	0.888342i	$0.651857\pi$
$504$	0	0
$505$	−12.7639	−0.567988
$506$	1.00406	0.0446358
$507$	0	0
$508$	20.1967	0.896084
$509$	18.1886	0.806195	0.403097	−	0.915157i	$-0.367934\pi$
$509$	18.1886	0.806195	0.403097	+	0.915157i	$0.367934\pi$
$510$	0	0
$511$	2.12461	0.0939873
$512$	27.1441	1.19961
$513$	0	0
$514$	20.6525	0.910942
$515$	36.0341	1.58785
$516$	0	0
$517$	15.8541	0.697263
$518$	−2.96556	−0.130299
$519$	0	0
$520$	1.70820	0.0749097
$521$	−21.8213	−0.956008	−0.478004	−	0.878358i	$-0.658640\pi$
$521$	−21.8213	−0.956008	−0.478004	+	0.878358i	$0.658640\pi$
$522$	0	0
$523$	34.4095	1.50462	0.752312	−	0.658807i	$-0.228938\pi$
$523$	34.4095	1.50462	0.752312	+	0.658807i	$0.228938\pi$
$524$	−9.85410	−0.430478
$525$	0	0
$526$	45.4387	1.98122
$527$	12.3107	0.536264
$528$	0	0
$529$	−22.8541	−0.993657
$530$	30.6525	1.33146
$531$	0	0
$532$	0	0
$533$	−1.58359	−0.0685930
$534$	0	0
$535$	−19.3642	−0.837186
$536$	−9.47214	−0.409134
$537$	0	0
$538$	−49.0689	−2.11551
$539$	−9.59675	−0.413361
$540$	0	0
$541$	6.70820	0.288408	0.144204	−	0.989548i	$-0.453938\pi$
$541$	6.70820	0.288408	0.144204	+	0.989548i	$0.453938\pi$
$542$	−36.6547	−1.57445
$543$	0	0
$544$	−47.4468	−2.03427
$545$	−9.95959	−0.426622
$546$	0	0
$547$	−20.7112	−0.885549	−0.442774	−	0.896633i	$-0.646006\pi$
$547$	−20.7112	−0.885549	−0.442774	+	0.896633i	$0.646006\pi$
$548$	−14.0902	−0.601902
$549$	0	0
$550$	14.3844	0.613351
$551$	0	0
$552$	0	0
$553$	−1.11006	−0.0472045
$554$	−28.0827	−1.19312
$555$	0	0
$556$	−19.5623	−0.829627
$557$	39.3607	1.66777	0.833883	−	0.551942i	$-0.186113\pi$
$557$	39.3607	1.66777	0.833883	+	0.551942i	$0.186113\pi$
$558$	0	0
$559$	−6.77591	−0.286590
$560$	−3.52786	−0.149079
$561$	0	0
$562$	29.7984	1.25697
$563$	−21.7153	−0.915191	−0.457595	−	0.889161i	$-0.651289\pi$
$563$	−21.7153	−0.915191	−0.457595	+	0.889161i	$0.651289\pi$
$564$	0	0
$565$	0	0
$566$	−22.9969	−0.966631
$567$	0	0
$568$	7.36068	0.308847
$569$	−25.4540	−1.06709	−0.533544	−	0.845772i	$-0.679140\pi$
$569$	−25.4540	−1.06709	−0.533544	+	0.845772i	$0.679140\pi$
$570$	0	0
$571$	−46.1033	−1.92936	−0.964682	−	0.263417i	$-0.915150\pi$
$571$	−46.1033	−1.92936	−0.964682	+	0.263417i	$0.915150\pi$
$572$	−1.62460	−0.0679279
$573$	0	0
$574$	0.978714	0.0408507
$575$	2.09017	0.0871661
$576$	0	0
$577$	−15.2918	−0.636606	−0.318303	−	0.947989i	$-0.603113\pi$
$577$	−15.2918	−0.636606	−0.318303	+	0.947989i	$0.603113\pi$
$578$	47.3408	1.96912
$579$	0	0
$580$	22.2703	0.924725
$581$	0.763932	0.0316932
$582$	0	0
$583$	6.88191	0.285020
$584$	−6.53888	−0.270581
$585$	0	0
$586$	29.0689	1.20082
$587$	−16.8197	−0.694222	−0.347111	−	0.937824i	$-0.612837\pi$
$587$	−16.8197	−0.694222	−0.347111	+	0.937824i	$0.612837\pi$
$588$	0	0
$589$	0	0
$590$	−54.0689	−2.22598
$591$	0	0
$592$	30.4993	1.25351
$593$	−17.2918	−0.710089	−0.355044	−	0.934849i	$-0.615534\pi$
$593$	−17.2918	−0.710089	−0.355044	+	0.934849i	$0.615534\pi$
$594$	0	0
$595$	4.94427	0.202695
$596$	−9.76393	−0.399946
$597$	0	0
$598$	−0.527864	−0.0215860
$599$	−4.87380	−0.199138	−0.0995689	−	0.995031i	$-0.531746\pi$
$599$	−4.87380	−0.199138	−0.0995689	+	0.995031i	$0.531746\pi$
$600$	0	0
$601$	−9.74759	−0.397613	−0.198806	−	0.980039i	$-0.563706\pi$
$601$	−9.74759	−0.397613	−0.198806	+	0.980039i	$0.563706\pi$
$602$	4.18774	0.170680
$603$	0	0
$604$	−27.9767	−1.13835
$605$	−29.4164	−1.19595
$606$	0	0
$607$	−27.9767	−1.13554	−0.567769	−	0.823188i	$-0.692193\pi$
$607$	−27.9767	−1.13554	−0.567769	+	0.823188i	$0.692193\pi$
$608$	0	0
$609$	0	0
$610$	−58.3045	−2.36068
$611$	−8.33499	−0.337198
$612$	0	0
$613$	−33.4721	−1.35193	−0.675963	−	0.736935i	$-0.736272\pi$
$613$	−33.4721	−1.35193	−0.675963	+	0.736935i	$0.736272\pi$
$614$	−52.5623	−2.12124
$615$	0	0
$616$	−0.237026	−0.00955004
$617$	16.2705	0.655026	0.327513	−	0.944847i	$-0.393789\pi$
$617$	16.2705	0.655026	0.327513	+	0.944847i	$0.393789\pi$
$618$	0	0
$619$	−34.7771	−1.39781	−0.698905	−	0.715215i	$-0.746329\pi$
$619$	−34.7771	−1.39781	−0.698905	+	0.715215i	$0.746329\pi$
$620$	9.95959	0.399987
$621$	0	0
$622$	27.5276	1.10376
$623$	−1.75562	−0.0703376
$624$	0	0
$625$	−22.4164	−0.896656
$626$	10.2371	0.409157
$627$	0	0
$628$	−7.38197	−0.294573
$629$	−42.7445	−1.70434
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	3.41641	0.135897
$633$	0	0
$634$	−59.5967	−2.36689
$635$	40.3934	1.60296
$636$	0	0
$637$	5.04531	0.199902
$638$	11.1803	0.442634
$639$	0	0
$640$	18.4661	0.729937
$641$	46.7858	1.84793	0.923964	−	0.382480i	$-0.124930\pi$
$641$	46.7858	1.84793	0.923964	+	0.382480i	$0.124930\pi$
$642$	0	0
$643$	−5.70820	−0.225110	−0.112555	−	0.993646i	$-0.535903\pi$
$643$	−5.70820	−0.225110	−0.112555	+	0.993646i	$0.535903\pi$
$644$	0.145898	0.00574919
$645$	0	0
$646$	0	0
$647$	−8.12461	−0.319411	−0.159706	−	0.987165i	$-0.551055\pi$
$647$	−8.12461	−0.319411	−0.159706	+	0.987165i	$0.551055\pi$
$648$	0	0
$649$	−12.1392	−0.476506
$650$	−7.56231	−0.296618
$651$	0	0
$652$	25.0344	0.980424
$653$	22.7426	0.889989	0.444994	−	0.895533i	$-0.353206\pi$
$653$	22.7426	0.889989	0.444994	+	0.895533i	$0.353206\pi$
$654$	0	0
$655$	−19.7082	−0.770063
$656$	−10.0656	−0.392995
$657$	0	0
$658$	5.15131	0.200819
$659$	37.8303	1.47366	0.736829	−	0.676079i	$-0.236322\pi$
$659$	37.8303	1.47366	0.736829	+	0.676079i	$0.236322\pi$
$660$	0	0
$661$	33.6830	1.31012	0.655059	−	0.755578i	$-0.272644\pi$
$661$	33.6830	1.31012	0.655059	+	0.755578i	$0.272644\pi$
$662$	38.5410	1.49794
$663$	0	0
$664$	−2.35114	−0.0912420
$665$	0	0
$666$	0	0
$667$	1.62460	0.0629047
$668$	−2.62866	−0.101706
$669$	0	0
$670$	80.2492	3.10030
$671$	−13.0902	−0.505340
$672$	0	0
$673$	9.57608	0.369131	0.184565	−	0.982820i	$-0.440912\pi$
$673$	9.57608	0.369131	0.184565	+	0.982820i	$0.440912\pi$
$674$	−1.05573	−0.0406651
$675$	0	0
$676$	−20.1803	−0.776167
$677$	−15.1109	−0.580759	−0.290380	−	0.956912i	$-0.593782\pi$
$677$	−15.1109	−0.580759	−0.290380	+	0.956912i	$0.593782\pi$
$678$	0	0
$679$	−1.04455	−0.0400860
$680$	−15.2169	−0.583542
$681$	0	0
$682$	5.00000	0.191460
$683$	−40.2219	−1.53905	−0.769524	−	0.638618i	$-0.779506\pi$
$683$	−40.2219	−1.53905	−0.769524	+	0.638618i	$0.779506\pi$
$684$	0	0
$685$	−28.1803	−1.07672
$686$	−6.26137	−0.239060
$687$	0	0
$688$	−43.0689	−1.64199
$689$	−3.61803	−0.137836
$690$	0	0
$691$	−16.3050	−0.620270	−0.310135	−	0.950693i	$-0.600374\pi$
$691$	−16.3050	−0.620270	−0.310135	+	0.950693i	$0.600374\pi$
$692$	26.9726	1.02534
$693$	0	0
$694$	−54.8433	−2.08182
$695$	−39.1246	−1.48408
$696$	0	0
$697$	14.1068	0.534335
$698$	25.9030	0.980445
$699$	0	0
$700$	2.09017	0.0790010
$701$	−8.74265	−0.330205	−0.165103	−	0.986276i	$-0.552796\pi$
$701$	−8.74265	−0.330205	−0.165103	+	0.986276i	$0.552796\pi$
$702$	0	0
$703$	0	0
$704$	−6.50658	−0.245226
$705$	0	0
$706$	−48.2794	−1.81702
$707$	−0.931116	−0.0350182
$708$	0	0
$709$	−13.4164	−0.503864	−0.251932	−	0.967745i	$-0.581066\pi$
$709$	−13.4164	−0.503864	−0.251932	+	0.967745i	$0.581066\pi$
$710$	−62.3607	−2.34035
$711$	0	0
$712$	5.40325	0.202495
$713$	0.726543	0.0272092
$714$	0	0
$715$	−3.24920	−0.121513
$716$	14.0413	0.524749
$717$	0	0
$718$	67.3005	2.51163
$719$	−14.6180	−0.545161	−0.272580	−	0.962133i	$-0.587877\pi$
$719$	−14.6180	−0.545161	−0.272580	+	0.962133i	$0.587877\pi$
$720$	0	0
$721$	2.62866	0.0978962
$722$	0	0
$723$	0	0
$724$	−9.40456	−0.349518
$725$	23.2744	0.864389
$726$	0	0
$727$	−36.9443	−1.37019	−0.685094	−	0.728455i	$-0.740239\pi$
$727$	−36.9443	−1.37019	−0.685094	+	0.728455i	$0.740239\pi$
$728$	0.124612	0.00461842
$729$	0	0
$730$	55.3983	2.05038
$731$	60.3607	2.23252
$732$	0	0
$733$	12.8197	0.473505	0.236752	−	0.971570i	$-0.423917\pi$
$733$	12.8197	0.473505	0.236752	+	0.971570i	$0.423917\pi$
$734$	23.8294	0.879560
$735$	0	0
$736$	−2.80017	−0.103216
$737$	18.0171	0.663667
$738$	0	0
$739$	19.4721	0.716294	0.358147	−	0.933665i	$-0.383409\pi$
$739$	19.4721	0.716294	0.358147	+	0.933665i	$0.383409\pi$
$740$	−34.5811	−1.27123
$741$	0	0
$742$	2.23607	0.0820886
$743$	−36.7607	−1.34862	−0.674309	−	0.738449i	$-0.735559\pi$
$743$	−36.7607	−1.34862	−0.674309	+	0.738449i	$0.735559\pi$
$744$	0	0
$745$	−19.5279	−0.715446
$746$	−69.9230	−2.56006
$747$	0	0
$748$	14.4721	0.529154
$749$	−1.41260	−0.0516151
$750$	0	0
$751$	12.0737	0.440576	0.220288	−	0.975435i	$-0.429300\pi$
$751$	12.0737	0.440576	0.220288	+	0.975435i	$0.429300\pi$
$752$	−52.9787	−1.93193
$753$	0	0
$754$	−5.87785	−0.214059
$755$	−55.9533	−2.03635
$756$	0	0
$757$	−37.2705	−1.35462	−0.677310	−	0.735698i	$-0.736854\pi$
$757$	−37.2705	−1.35462	−0.677310	+	0.735698i	$0.736854\pi$
$758$	−40.6525	−1.47656
$759$	0	0
$760$	0	0
$761$	23.9443	0.867979	0.433990	−	0.900918i	$-0.357105\pi$
$761$	23.9443	0.867979	0.433990	+	0.900918i	$0.357105\pi$
$762$	0	0
$763$	−0.726543	−0.0263026
$764$	27.5066	0.995153
$765$	0	0
$766$	21.5066	0.777064
$767$	6.38197	0.230439
$768$	0	0
$769$	33.1591	1.19575	0.597873	−	0.801591i	$-0.296013\pi$
$769$	33.1591	1.19575	0.597873	+	0.801591i	$0.296013\pi$
$770$	2.00811	0.0723674
$771$	0	0
$772$	13.7638	0.495371
$773$	12.3762	0.445143	0.222571	−	0.974916i	$-0.428555\pi$
$773$	12.3762	0.445143	0.222571	+	0.974916i	$0.428555\pi$
$774$	0	0
$775$	10.4086	0.373889
$776$	3.21478	0.115404
$777$	0	0
$778$	−3.63271	−0.130239
$779$	0	0
$780$	0	0
$781$	−14.0008	−0.500990
$782$	4.70228	0.168153
$783$	0	0
$784$	32.0689	1.14532
$785$	−14.7639	−0.526947
$786$	0	0
$787$	−19.3642	−0.690258	−0.345129	−	0.938555i	$-0.612165\pi$
$787$	−19.3642	−0.690258	−0.345129	+	0.938555i	$0.612165\pi$
$788$	−18.5623	−0.661255
$789$	0	0
$790$	−28.9443	−1.02979
$791$	0	0
$792$	0	0
$793$	6.88191	0.244384
$794$	2.45714	0.0872007
$795$	0	0
$796$	−1.70820	−0.0605457
$797$	−13.2088	−0.467879	−0.233940	−	0.972251i	$-0.575162\pi$
$797$	−13.2088	−0.467879	−0.233940	+	0.972251i	$0.575162\pi$
$798$	0	0
$799$	74.2492	2.62675
$800$	−40.1159	−1.41831
$801$	0	0
$802$	−8.94427	−0.315833
$803$	12.4377	0.438917
$804$	0	0
$805$	0.291796	0.0102845
$806$	−2.62866	−0.0925904
$807$	0	0
$808$	2.86568	0.100814
$809$	34.2705	1.20489	0.602443	−	0.798162i	$-0.294194\pi$
$809$	34.2705	1.20489	0.602443	+	0.798162i	$0.294194\pi$
$810$	0	0
$811$	22.2048	0.779717	0.389858	−	0.920875i	$-0.372524\pi$
$811$	22.2048	0.779717	0.389858	+	0.920875i	$0.372524\pi$
$812$	1.62460	0.0570122
$813$	0	0
$814$	−17.3607	−0.608492
$815$	50.0689	1.75384
$816$	0	0
$817$	0	0
$818$	−45.7771	−1.60056
$819$	0	0
$820$	11.4127	0.398548
$821$	−10.0344	−0.350204	−0.175102	−	0.984550i	$-0.556026\pi$
$821$	−10.0344	−0.350204	−0.175102	+	0.984550i	$0.556026\pi$
$822$	0	0
$823$	50.7214	1.76804	0.884018	−	0.467453i	$-0.154828\pi$
$823$	50.7214	1.76804	0.884018	+	0.467453i	$0.154828\pi$
$824$	−8.09017	−0.281834
$825$	0	0
$826$	−3.94427	−0.137239
$827$	29.7073	1.03302	0.516511	−	0.856280i	$-0.327230\pi$
$827$	29.7073	1.03302	0.516511	+	0.856280i	$0.327230\pi$
$828$	0	0
$829$	15.1109	0.524823	0.262412	−	0.964956i	$-0.415482\pi$
$829$	15.1109	0.524823	0.262412	+	0.964956i	$0.415482\pi$
$830$	19.9192	0.691405
$831$	0	0
$832$	3.42071	0.118592
$833$	−44.9443	−1.55723
$834$	0	0
$835$	−5.25731	−0.181937
$836$	0	0
$837$	0	0
$838$	12.5227	0.432591
$839$	16.9475	0.585093	0.292546	−	0.956251i	$-0.405497\pi$
$839$	16.9475	0.585093	0.292546	+	0.956251i	$0.405497\pi$
$840$	0	0
$841$	−10.9098	−0.376201
$842$	61.6312	2.12395
$843$	0	0
$844$	1.00406	0.0345611
$845$	−40.3607	−1.38845
$846$	0	0
$847$	−2.14590	−0.0737339
$848$	−22.9969	−0.789716
$849$	0	0
$850$	67.3660	2.31064
$851$	−2.52265	−0.0864755
$852$	0	0
$853$	40.3050	1.38002	0.690008	−	0.723802i	$-0.257607\pi$
$853$	40.3050	1.38002	0.690008	+	0.723802i	$0.257607\pi$
$854$	−4.25325	−0.145543
$855$	0	0
$856$	4.34752	0.148595
$857$	55.3328	1.89013	0.945066	−	0.326879i	$-0.105997\pi$
$857$	55.3328	1.89013	0.945066	+	0.326879i	$0.105997\pi$
$858$	0	0
$859$	38.9443	1.32876	0.664381	−	0.747394i	$-0.268695\pi$
$859$	38.9443	1.32876	0.664381	+	0.747394i	$0.268695\pi$
$860$	48.8328	1.66519
$861$	0	0
$862$	−23.7426	−0.808678
$863$	32.5729	1.10880	0.554398	−	0.832252i	$-0.312949\pi$
$863$	32.5729	1.10880	0.554398	+	0.832252i	$0.312949\pi$
$864$	0	0
$865$	53.9452	1.83419
$866$	35.9787	1.22261
$867$	0	0
$868$	0.726543	0.0246605
$869$	−6.49839	−0.220443
$870$	0	0
$871$	−9.47214	−0.320951
$872$	2.23607	0.0757228
$873$	0	0
$874$	0	0
$875$	0.360680	0.0121932
$876$	0	0
$877$	−9.74759	−0.329153	−0.164576	−	0.986364i	$-0.552626\pi$
$877$	−9.74759	−0.329153	−0.164576	+	0.986364i	$0.552626\pi$
$878$	−11.3820	−0.384123
$879$	0	0
$880$	−20.6525	−0.696195
$881$	−40.5755	−1.36702	−0.683511	−	0.729940i	$-0.739548\pi$
$881$	−40.5755	−1.36702	−0.683511	+	0.729940i	$0.739548\pi$
$882$	0	0
$883$	13.9656	0.469978	0.234989	−	0.971998i	$-0.424495\pi$
$883$	13.9656	0.469978	0.234989	+	0.971998i	$0.424495\pi$
$884$	−7.60845	−0.255900
$885$	0	0
$886$	−28.8747	−0.970065
$887$	16.4580	0.552605	0.276303	−	0.961071i	$-0.410891\pi$
$887$	16.4580	0.552605	0.276303	+	0.961071i	$0.410891\pi$
$888$	0	0
$889$	2.94666	0.0988278
$890$	−45.7771	−1.53445
$891$	0	0
$892$	−26.3521	−0.882332
$893$	0	0
$894$	0	0
$895$	28.0827	0.938700
$896$	1.34708	0.0450029
$897$	0	0
$898$	−46.1033	−1.53849
$899$	8.09017	0.269822
$900$	0	0
$901$	32.2299	1.07373
$902$	5.72949	0.190771
$903$	0	0
$904$	0	0
$905$	−18.8091	−0.625237
$906$	0	0
$907$	28.9807	0.962289	0.481145	−	0.876641i	$-0.340221\pi$
$907$	28.9807	0.962289	0.481145	+	0.876641i	$0.340221\pi$
$908$	−25.4540	−0.844721
$909$	0	0
$910$	−1.05573	−0.0349970
$911$	−25.4540	−0.843329	−0.421665	−	0.906752i	$-0.638554\pi$
$911$	−25.4540	−0.843329	−0.421665	+	0.906752i	$0.638554\pi$
$912$	0	0
$913$	4.47214	0.148006
$914$	11.9677	0.395857
$915$	0	0
$916$	−11.4721	−0.379050
$917$	−1.43769	−0.0474768
$918$	0	0
$919$	−47.3607	−1.56228	−0.781142	−	0.624353i	$-0.785363\pi$
$919$	−47.3607	−1.56228	−0.781142	+	0.624353i	$0.785363\pi$
$920$	−0.898056	−0.0296080
$921$	0	0
$922$	22.4823	0.740416
$923$	7.36068	0.242280
$924$	0	0
$925$	−36.1401	−1.18828
$926$	19.2986	0.634193
$927$	0	0
$928$	−31.1803	−1.02354
$929$	−16.5066	−0.541563	−0.270782	−	0.962641i	$-0.587282\pi$
$929$	−16.5066	−0.541563	−0.270782	+	0.962641i	$0.587282\pi$
$930$	0	0
$931$	0	0
$932$	−7.32624	−0.239979
$933$	0	0
$934$	−8.05748	−0.263649
$935$	28.9443	0.946579
$936$	0	0
$937$	−17.2016	−0.561953	−0.280976	−	0.959715i	$-0.590658\pi$
$937$	−17.2016	−0.561953	−0.280976	+	0.959715i	$0.590658\pi$
$938$	5.85410	0.191143
$939$	0	0
$940$	60.0689	1.95923
$941$	−27.8052	−0.906422	−0.453211	−	0.891403i	$-0.649722\pi$
$941$	−27.8052	−0.906422	−0.453211	+	0.891403i	$0.649722\pi$
$942$	0	0
$943$	0.832544	0.0271114
$944$	40.5649	1.32028
$945$	0	0
$946$	24.5155	0.797067
$947$	28.1803	0.915738	0.457869	−	0.889020i	$-0.348613\pi$
$947$	28.1803	0.915738	0.457869	+	0.889020i	$0.348613\pi$
$948$	0	0
$949$	−6.53888	−0.212261
$950$	0	0
$951$	0	0
$952$	−1.11006	−0.0359772
$953$	20.4337	0.661913	0.330957	−	0.943646i	$-0.392629\pi$
$953$	20.4337	0.661913	0.330957	+	0.943646i	$0.392629\pi$
$954$	0	0
$955$	55.0132	1.78018
$956$	14.7984	0.478614
$957$	0	0
$958$	30.2623	0.977730
$959$	−2.05573	−0.0663829
$960$	0	0
$961$	−27.3820	−0.883289
$962$	9.12705	0.294268
$963$	0	0
$964$	−43.9201	−1.41457
$965$	27.5276	0.886146
$966$	0	0
$967$	20.2918	0.652540	0.326270	−	0.945277i	$-0.394208\pi$
$967$	20.2918	0.652540	0.326270	+	0.945277i	$0.394208\pi$
$968$	6.60440	0.212273
$969$	0	0
$970$	−27.2361	−0.874497
$971$	44.6467	1.43278	0.716390	−	0.697700i	$-0.245793\pi$
$971$	44.6467	1.43278	0.716390	+	0.697700i	$0.245793\pi$
$972$	0	0
$973$	−2.85410	−0.0914983
$974$	31.7082	1.01600
$975$	0	0
$976$	43.7426	1.40017
$977$	−9.57608	−0.306366	−0.153183	−	0.988198i	$-0.548952\pi$
$977$	−9.57608	−0.306366	−0.153183	+	0.988198i	$0.548952\pi$
$978$	0	0
$979$	−10.2776	−0.328473
$980$	−36.3607	−1.16150
$981$	0	0
$982$	−32.8505	−1.04830
$983$	39.2178	1.25085	0.625427	−	0.780282i	$-0.284925\pi$
$983$	39.2178	1.25085	0.625427	+	0.780282i	$0.284925\pi$
$984$	0	0
$985$	−37.1246	−1.18289
$986$	52.3607	1.66750
$987$	0	0
$988$	0	0
$989$	3.56231	0.113275
$990$	0	0
$991$	55.3328	1.75770	0.878852	−	0.477094i	$-0.158310\pi$
$991$	55.3328	1.75770	0.878852	+	0.477094i	$0.158310\pi$
$992$	−13.9443	−0.442731
$993$	0	0
$994$	−4.54915	−0.144290
$995$	−3.41641	−0.108307
$996$	0	0
$997$	−33.1246	−1.04907	−0.524533	−	0.851390i	$-0.675760\pi$
$997$	−33.1246	−1.04907	−0.524533	+	0.851390i	$0.675760\pi$
$998$	57.9614	1.83474
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bc.1.4		4
3.2	odd	2		361.2.a.i.1.1	✓	4
12.11	even	2		5776.2.a.bu.1.1		4
15.14	odd	2		9025.2.a.bj.1.4		4
19.18	odd	2	inner	3249.2.a.bc.1.1		4
57.2	even	18		361.2.e.m.99.1		24
57.5	odd	18		361.2.e.m.234.1		24
57.8	even	6		361.2.c.j.292.1		8
57.11	odd	6		361.2.c.j.292.4		8
57.14	even	18		361.2.e.m.234.4		24
57.17	odd	18		361.2.e.m.99.4		24
57.23	odd	18		361.2.e.m.54.1		24
57.26	odd	6		361.2.c.j.68.4		8
57.29	even	18		361.2.e.m.62.1		24
57.32	even	18		361.2.e.m.245.4		24
57.35	odd	18		361.2.e.m.28.1		24
57.41	even	18		361.2.e.m.28.4		24
57.44	odd	18		361.2.e.m.245.1		24
57.47	odd	18		361.2.e.m.62.4		24
57.50	even	6		361.2.c.j.68.1		8
57.53	even	18		361.2.e.m.54.4		24
57.56	even	2		361.2.a.i.1.4	yes	4
228.227	odd	2		5776.2.a.bu.1.4		4
285.284	even	2		9025.2.a.bj.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
361.2.a.i.1.1	✓	4	3.2	odd	2
361.2.a.i.1.4	yes	4	57.56	even	2
361.2.c.j.68.1		8	57.50	even	6
361.2.c.j.68.4		8	57.26	odd	6
361.2.c.j.292.1		8	57.8	even	6
361.2.c.j.292.4		8	57.11	odd	6
361.2.e.m.28.1		24	57.35	odd	18
361.2.e.m.28.4		24	57.41	even	18
361.2.e.m.54.1		24	57.23	odd	18
361.2.e.m.54.4		24	57.53	even	18
361.2.e.m.62.1		24	57.29	even	18
361.2.e.m.62.4		24	57.47	odd	18
361.2.e.m.99.1		24	57.2	even	18
361.2.e.m.99.4		24	57.17	odd	18
361.2.e.m.234.1		24	57.5	odd	18
361.2.e.m.234.4		24	57.14	even	18
361.2.e.m.245.1		24	57.44	odd	18
361.2.e.m.245.4		24	57.32	even	18
3249.2.a.bc.1.1		4	19.18	odd	2	inner
3249.2.a.bc.1.4		4	1.1	even	1	trivial
5776.2.a.bu.1.1		4	12.11	even	2
5776.2.a.bu.1.4		4	228.227	odd	2
9025.2.a.bj.1.1		4	285.284	even	2
9025.2.a.bj.1.4		4	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90211 q^{2} +1.61803 q^{4} +3.23607 q^{5} +0.236068 q^{7} -0.726543 q^{8} +6.15537 q^{10} +1.38197 q^{11} -0.726543 q^{13} +0.449028 q^{14} -4.61803 q^{16} +6.47214 q^{17} +5.23607 q^{20} +2.62866 q^{22} +0.381966 q^{23} +5.47214 q^{25} -1.38197 q^{26} +0.381966 q^{28} +4.25325 q^{29} +1.90211 q^{31} -7.33094 q^{32} +12.3107 q^{34} +0.763932 q^{35} -6.60440 q^{37} -2.35114 q^{40} +2.17963 q^{41} +9.32624 q^{43} +2.23607 q^{44} +0.726543 q^{46} +11.4721 q^{47} -6.94427 q^{49} +10.4086 q^{50} -1.17557 q^{52} +4.97980 q^{53} +4.47214 q^{55} -0.171513 q^{56} +8.09017 q^{58} -8.78402 q^{59} -9.47214 q^{61} +3.61803 q^{62} -4.70820 q^{64} -2.35114 q^{65} +13.0373 q^{67} +10.4721 q^{68} +1.45309 q^{70} -10.1311 q^{71} +9.00000 q^{73} -12.5623 q^{74} +0.326238 q^{77} -4.70228 q^{79} -14.9443 q^{80} +4.14590 q^{82} +3.23607 q^{83} +20.9443 q^{85} +17.7396 q^{86} -1.00406 q^{88} -7.43694 q^{89} -0.171513 q^{91} +0.618034 q^{92} +21.8213 q^{94} -4.42477 q^{97} -13.2088 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 8 q^{7} + 10 q^{11} - 14 q^{16} + 8 q^{17} + 12 q^{20} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 6 q^{28} + 12 q^{35} + 6 q^{43} + 28 q^{47} + 8 q^{49} + 10 q^{58} - 20 q^{61} + 10 q^{62}+ \cdots - 2 q^{92}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 8 * q^7 + 10 * q^11 - 14 * q^16 + 8 * q^17 + 12 * q^20 + 6 * q^23 + 4 * q^25 - 10 * q^26 + 6 * q^28 + 12 * q^35 + 6 * q^43 + 28 * q^47 + 8 * q^49 + 10 * q^58 - 20 * q^61 + 10 * q^62 + 8 * q^64 + 24 * q^68 + 36 * q^73 - 10 * q^74 - 30 * q^77 - 24 * q^80 + 30 * q^82 + 4 * q^83 + 48 * q^85 - 2 * q^92

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