LMFDB - Embedded newform 3249.2.a.bj.1.1

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)

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);

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")

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,12,0,0,-6,0,0,12,0,0,18,0,0,-12,0,0,0,0,0,-18,0,0,12,0, 0,6,0,0,24,0,0,30,0,0,24,0,0,42,0,0,18,0,0,6,0,0,0,0,0,60,0,0,0,0,0,-24, 0,0,-18,0,0,-30,0,0,0,0,0,42,0,0,0,0,0,0,0,0,54,0,0,-66,0,0,-48,0,0,0, 0,0,18] 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:	$6$
Coefficient field:	6.6.21415104.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 45x^{2} - 51 $ x^6 - 12x^4 + 45x^2 - 51
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 171)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.35204$ 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)

$f(q)$	$=$	$q-2.35204 q^{2} +3.53209 q^{4} -3.60353 q^{5} +0.184793 q^{7} -3.60353 q^{8} +8.47565 q^{10} +3.16889 q^{11} +6.06418 q^{13} -0.434639 q^{14} +1.41147 q^{16} +3.31984 q^{17} -12.7280 q^{20} -7.45336 q^{22} -4.42039 q^{23} +7.98545 q^{25} -14.2632 q^{26} +0.652704 q^{28} +7.58928 q^{29} +7.41147 q^{31} +3.88722 q^{32} -7.80840 q^{34} -0.665906 q^{35} +1.77332 q^{37} +12.9855 q^{40} -2.21930 q^{41} +6.75877 q^{43} +11.1928 q^{44} +10.3969 q^{46} -4.80260 q^{47} -6.96585 q^{49} -18.7821 q^{50} +21.4192 q^{52} +0.150949 q^{53} -11.4192 q^{55} -0.665906 q^{56} -17.8503 q^{58} +7.15464 q^{59} -5.92127 q^{61} -17.4321 q^{62} -11.9659 q^{64} -21.8525 q^{65} -1.38919 q^{67} +11.7260 q^{68} +1.56624 q^{70} -9.02594 q^{71} -8.70233 q^{73} -4.17091 q^{74} +0.585588 q^{77} +2.87164 q^{79} -5.08629 q^{80} +5.21987 q^{82} +7.30559 q^{83} -11.9632 q^{85} -15.8969 q^{86} -11.4192 q^{88} +7.33981 q^{89} +1.12061 q^{91} -15.6132 q^{92} +11.2959 q^{94} +0.837496 q^{97} +16.3840 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 6 q^{7} + 12 q^{10} + 18 q^{13} - 12 q^{16} - 18 q^{22} + 12 q^{25} + 6 q^{28} + 24 q^{31} + 30 q^{34} + 24 q^{37} + 42 q^{40} + 18 q^{43} + 6 q^{46} + 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots + 6 q^{94}+O(q^{100}) $ 6 * q + 12 * q^4 - 6 * q^7 + 12 * q^10 + 18 * q^13 - 12 * q^16 - 18 * q^22 + 12 * q^25 + 6 * q^28 + 24 * q^31 + 30 * q^34 + 24 * q^37 + 42 * q^40 + 18 * q^43 + 6 * q^46 + 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 + 42 * q^70 + 54 * q^79 - 66 * q^82 - 48 * q^85 + 18 * q^91 + 6 * q^94

Coefficient data

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

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

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

$2$	−2.35204	−1.66314	−0.831571	−	0.555418i	$-0.812558\pi$
$2$	−2.35204	−1.66314	−0.831571	+	0.555418i	$0.812558\pi$
$3$	0	0
$3$	0	0
$4$	3.53209	1.76604
$5$	−3.60353	−1.61155	−0.805775	−	0.592222i	$-0.798251\pi$
$5$	−3.60353	−1.61155	−0.805775	+	0.592222i	$0.798251\pi$
$6$	0	0
$7$	0.184793	0.0698450	0.0349225	−	0.999390i	$-0.488882\pi$
$7$	0.184793	0.0698450	0.0349225	+	0.999390i	$0.488882\pi$
$8$	−3.60353	−1.27404
$9$	0	0
$10$	8.47565	2.68024
$11$	3.16889	0.955457	0.477729	−	0.878507i	$-0.341460\pi$
$11$	3.16889	0.955457	0.477729	+	0.878507i	$0.341460\pi$
$12$	0	0
$13$	6.06418	1.68190	0.840950	−	0.541113i	$-0.181997\pi$
$13$	6.06418	1.68190	0.840950	+	0.541113i	$0.181997\pi$
$14$	−0.434639	−0.116162
$15$	0	0
$16$	1.41147	0.352869
$17$	3.31984	0.805180	0.402590	−	0.915380i	$-0.368110\pi$
$17$	3.31984	0.805180	0.402590	+	0.915380i	$0.368110\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−12.7280	−2.84607
$21$	0	0
$22$	−7.45336	−1.58906
$23$	−4.42039	−0.921715	−0.460857	−	0.887474i	$-0.652458\pi$
$23$	−4.42039	−0.921715	−0.460857	+	0.887474i	$0.652458\pi$
$24$	0	0
$25$	7.98545	1.59709
$26$	−14.2632	−2.79724
$27$	0	0
$28$	0.652704	0.123349
$29$	7.58928	1.40929	0.704647	−	0.709558i	$-0.251105\pi$
$29$	7.58928	1.40929	0.704647	+	0.709558i	$0.251105\pi$
$30$	0	0
$31$	7.41147	1.33114	0.665570	−	0.746335i	$-0.268188\pi$
$31$	7.41147	1.33114	0.665570	+	0.746335i	$0.268188\pi$
$32$	3.88722	0.687171
$33$	0	0
$34$	−7.80840	−1.33913
$35$	−0.665906	−0.112559
$36$	0	0
$37$	1.77332	0.291532	0.145766	−	0.989319i	$-0.453435\pi$
$37$	1.77332	0.291532	0.145766	+	0.989319i	$0.453435\pi$
$38$	0	0
$39$	0	0
$40$	12.9855	2.05318
$41$	−2.21930	−0.346596	−0.173298	−	0.984869i	$-0.555442\pi$
$41$	−2.21930	−0.346596	−0.173298	+	0.984869i	$0.555442\pi$
$42$	0	0
$43$	6.75877	1.03070	0.515351	−	0.856979i	$-0.327661\pi$
$43$	6.75877	1.03070	0.515351	+	0.856979i	$0.327661\pi$
$44$	11.1928	1.68738
$45$	0	0
$46$	10.3969	1.53294
$47$	−4.80260	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$47$	−4.80260	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$48$	0	0
$49$	−6.96585	−0.995122
$50$	−18.7821	−2.65619
$51$	0	0
$52$	21.4192	2.97031
$53$	0.150949	0.0207344	0.0103672	−	0.999946i	$-0.496700\pi$
$53$	0.150949	0.0207344	0.0103672	+	0.999946i	$0.496700\pi$
$54$	0	0
$55$	−11.4192	−1.53977
$56$	−0.665906	−0.0889854
$57$	0	0
$58$	−17.8503	−2.34386
$59$	7.15464	0.931455	0.465728	−	0.884928i	$-0.345793\pi$
$59$	7.15464	0.931455	0.465728	+	0.884928i	$0.345793\pi$
$60$	0	0
$61$	−5.92127	−0.758141	−0.379071	−	0.925368i	$-0.623756\pi$
$61$	−5.92127	−0.758141	−0.379071	+	0.925368i	$0.623756\pi$
$62$	−17.4321	−2.21388
$63$	0	0
$64$	−11.9659	−1.49573
$65$	−21.8525	−2.71046
$66$	0	0
$67$	−1.38919	−0.169716	−0.0848580	−	0.996393i	$-0.527044\pi$
$67$	−1.38919	−0.169716	−0.0848580	+	0.996393i	$0.527044\pi$
$68$	11.7260	1.42198
$69$	0	0
$70$	1.56624	0.187201
$71$	−9.02594	−1.07118	−0.535591	−	0.844477i	$-0.679911\pi$
$71$	−9.02594	−1.07118	−0.535591	+	0.844477i	$0.679911\pi$
$72$	0	0
$73$	−8.70233	−1.01853	−0.509266	−	0.860609i	$-0.670083\pi$
$73$	−8.70233	−1.01853	−0.509266	+	0.860609i	$0.670083\pi$
$74$	−4.17091	−0.484859
$75$	0	0
$76$	0	0
$77$	0.585588	0.0667339
$78$	0	0
$79$	2.87164	0.323085	0.161543	−	0.986866i	$-0.448353\pi$
$79$	2.87164	0.323085	0.161543	+	0.986866i	$0.448353\pi$
$80$	−5.08629	−0.568665
$81$	0	0
$82$	5.21987	0.576439
$83$	7.30559	0.801893	0.400946	−	0.916102i	$-0.368681\pi$
$83$	7.30559	0.801893	0.400946	+	0.916102i	$0.368681\pi$
$84$	0	0
$85$	−11.9632	−1.29759
$86$	−15.8969	−1.71421
$87$	0	0
$88$	−11.4192	−1.21729
$89$	7.33981	0.778018	0.389009	−	0.921234i	$-0.372817\pi$
$89$	7.33981	0.778018	0.389009	+	0.921234i	$0.372817\pi$
$90$	0	0
$91$	1.12061	0.117472
$92$	−15.6132	−1.62779
$93$	0	0
$94$	11.2959	1.16508
$95$	0	0
$96$	0	0
$97$	0.837496	0.0850349	0.0425174	−	0.999096i	$-0.486462\pi$
$97$	0.837496	0.0850349	0.0425174	+	0.999096i	$0.486462\pi$
$98$	16.3840	1.65503
$99$	0	0
$100$	28.2053	2.82053
$101$	4.32186	0.430041	0.215021	−	0.976609i	$-0.431018\pi$
$101$	4.32186	0.430041	0.215021	+	0.976609i	$0.431018\pi$
$102$	0	0
$103$	6.85710	0.675650	0.337825	−	0.941209i	$-0.390309\pi$
$103$	6.85710	0.675650	0.337825	+	0.941209i	$0.390309\pi$
$104$	−21.8525	−2.14281
$105$	0	0
$106$	−0.355037	−0.0344843
$107$	−11.9111	−1.15149	−0.575747	−	0.817628i	$-0.695289\pi$
$107$	−11.9111	−1.15149	−0.575747	+	0.817628i	$0.695289\pi$
$108$	0	0
$109$	−6.61587	−0.633685	−0.316843	−	0.948478i	$-0.602623\pi$
$109$	−6.61587	−0.633685	−0.316843	+	0.948478i	$0.602623\pi$
$110$	26.8584	2.56085
$111$	0	0
$112$	0.260830	0.0246461
$113$	5.57336	0.524297	0.262149	−	0.965027i	$-0.415569\pi$
$113$	5.57336	0.524297	0.262149	+	0.965027i	$0.415569\pi$
$114$	0	0
$115$	15.9290	1.48539
$116$	26.8060	2.48888
$117$	0	0
$118$	−16.8280	−1.54914
$119$	0.613482	0.0562378
$120$	0	0
$121$	−0.958111	−0.0871010
$122$	13.9271	1.26090
$123$	0	0
$124$	26.1780	2.35085
$125$	−10.7582	−0.962241
$126$	0	0
$127$	−4.81790	−0.427519	−0.213760	−	0.976886i	$-0.568571\pi$
$127$	−4.81790	−0.427519	−0.213760	+	0.976886i	$0.568571\pi$
$128$	20.3697	1.80044
$129$	0	0
$130$	51.3979	4.50789
$131$	15.6132	1.36413	0.682066	−	0.731291i	$-0.261082\pi$
$131$	15.6132	1.36413	0.682066	+	0.731291i	$0.261082\pi$
$132$	0	0
$133$	0	0
$134$	3.26742	0.282262
$135$	0	0
$136$	−11.9632	−1.02583
$137$	−5.38819	−0.460344	−0.230172	−	0.973150i	$-0.573929\pi$
$137$	−5.38819	−0.460344	−0.230172	+	0.973150i	$0.573929\pi$
$138$	0	0
$139$	−22.9786	−1.94902	−0.974512	−	0.224337i	$-0.927978\pi$
$139$	−22.9786	−1.94902	−0.974512	+	0.224337i	$0.927978\pi$
$140$	−2.35204	−0.198784
$141$	0	0
$142$	21.2294	1.78153
$143$	19.2167	1.60698
$144$	0	0
$145$	−27.3482	−2.27115
$146$	20.4682	1.69396
$147$	0	0
$148$	6.26352	0.514858
$149$	15.0276	1.23111	0.615555	−	0.788094i	$-0.288932\pi$
$149$	15.0276	1.23111	0.615555	+	0.788094i	$0.288932\pi$
$150$	0	0
$151$	14.7219	1.19805	0.599027	−	0.800729i	$-0.295554\pi$
$151$	14.7219	1.19805	0.599027	+	0.800729i	$0.295554\pi$
$152$	0	0
$153$	0	0
$154$	−1.37733	−0.110988
$155$	−26.7075	−2.14520
$156$	0	0
$157$	−8.79561	−0.701966	−0.350983	−	0.936382i	$-0.614152\pi$
$157$	−8.79561	−0.701966	−0.350983	+	0.936382i	$0.614152\pi$
$158$	−6.75422	−0.537337
$159$	0	0
$160$	−14.0077	−1.10741
$161$	−0.816855	−0.0643772
$162$	0	0
$163$	0.908481	0.0711577	0.0355789	−	0.999367i	$-0.488673\pi$
$163$	0.908481	0.0711577	0.0355789	+	0.999367i	$0.488673\pi$
$164$	−7.83876	−0.612104
$165$	0	0
$166$	−17.1830	−1.33366
$167$	−14.4141	−1.11540	−0.557700	−	0.830043i	$-0.688316\pi$
$167$	−14.4141	−1.11540	−0.557700	+	0.830043i	$0.688316\pi$
$168$	0	0
$169$	23.7743	1.82879
$170$	28.1378	2.15807
$171$	0	0
$172$	23.8726	1.82027
$173$	−4.28765	−0.325984	−0.162992	−	0.986627i	$-0.552114\pi$
$173$	−4.28765	−0.325984	−0.162992	+	0.986627i	$0.552114\pi$
$174$	0	0
$175$	1.47565	0.111549
$176$	4.47281	0.337151
$177$	0	0
$178$	−17.2635	−1.29396
$179$	−0.0461008	−0.00344574	−0.00172287	−	0.999999i	$-0.500548\pi$
$179$	−0.0461008	−0.00344574	−0.00172287	+	0.999999i	$0.500548\pi$
$180$	0	0
$181$	3.11381	0.231447	0.115724	−	0.993281i	$-0.463081\pi$
$181$	3.11381	0.231447	0.115724	+	0.993281i	$0.463081\pi$
$182$	−2.63573	−0.195373
$183$	0	0
$184$	15.9290	1.17430
$185$	−6.39021	−0.469818
$186$	0	0
$187$	10.5202	0.769315
$188$	−16.9632	−1.23717
$189$	0	0
$190$	0	0
$191$	23.9550	1.73333	0.866663	−	0.498895i	$-0.166261\pi$
$191$	23.9550	1.73333	0.866663	+	0.498895i	$0.166261\pi$
$192$	0	0
$193$	−5.69965	−0.410269	−0.205135	−	0.978734i	$-0.565763\pi$
$193$	−5.69965	−0.410269	−0.205135	+	0.978734i	$0.565763\pi$
$194$	−1.96982	−0.141425
$195$	0	0
$196$	−24.6040	−1.75743
$197$	2.58331	0.184053	0.0920265	−	0.995757i	$-0.470666\pi$
$197$	2.58331	0.184053	0.0920265	+	0.995757i	$0.470666\pi$
$198$	0	0
$199$	17.2618	1.22365	0.611827	−	0.790992i	$-0.290435\pi$
$199$	17.2618	1.22365	0.611827	+	0.790992i	$0.290435\pi$
$200$	−28.7758	−2.03476
$201$	0	0
$202$	−10.1652	−0.715220
$203$	1.40244	0.0984322
$204$	0	0
$205$	7.99731	0.558556
$206$	−16.1282	−1.12370
$207$	0	0
$208$	8.55943	0.593490
$209$	0	0
$210$	0	0
$211$	4.61081	0.317422	0.158711	−	0.987325i	$-0.449266\pi$
$211$	4.61081	0.317422	0.158711	+	0.987325i	$0.449266\pi$
$212$	0.533164	0.0366179
$213$	0	0
$214$	28.0155	1.91510
$215$	−24.3555	−1.66103
$216$	0	0
$217$	1.36959	0.0929735
$218$	15.5608	1.05391
$219$	0	0
$220$	−40.3337	−2.71930
$221$	20.1321	1.35423
$222$	0	0
$223$	17.9213	1.20010	0.600049	−	0.799964i	$-0.295148\pi$
$223$	17.9213	1.20010	0.600049	+	0.799964i	$0.295148\pi$
$224$	0.718330	0.0479954
$225$	0	0
$226$	−13.1088	−0.871981
$227$	−26.5223	−1.76035	−0.880174	−	0.474650i	$-0.842574\pi$
$227$	−26.5223	−1.76035	−0.880174	+	0.474650i	$0.842574\pi$
$228$	0	0
$229$	−3.36959	−0.222668	−0.111334	−	0.993783i	$-0.535512\pi$
$229$	−3.36959	−0.222668	−0.111334	+	0.993783i	$0.535512\pi$
$230$	−37.4657	−2.47041
$231$	0	0
$232$	−27.3482	−1.79550
$233$	−18.3475	−1.20198	−0.600991	−	0.799256i	$-0.705227\pi$
$233$	−18.3475	−1.20198	−0.600991	+	0.799256i	$0.705227\pi$
$234$	0	0
$235$	17.3063	1.12894
$236$	25.2708	1.64499
$237$	0	0
$238$	−1.44293	−0.0935315
$239$	9.59332	0.620540	0.310270	−	0.950648i	$-0.399580\pi$
$239$	9.59332	0.620540	0.310270	+	0.950648i	$0.399580\pi$
$240$	0	0
$241$	−6.12836	−0.394762	−0.197381	−	0.980327i	$-0.563244\pi$
$241$	−6.12836	−0.394762	−0.197381	+	0.980327i	$0.563244\pi$
$242$	2.25351	0.144861
$243$	0	0
$244$	−20.9145	−1.33891
$245$	25.1017	1.60369
$246$	0	0
$247$	0	0
$248$	−26.7075	−1.69593
$249$	0	0
$250$	25.3037	1.60034
$251$	22.1019	1.39506	0.697531	−	0.716555i	$-0.254282\pi$
$251$	22.1019	1.39506	0.697531	+	0.716555i	$0.254282\pi$
$252$	0	0
$253$	−14.0077	−0.880659
$254$	11.3319	0.711025
$255$	0	0
$256$	−23.9786	−1.49867
$257$	16.9171	1.05526	0.527631	−	0.849474i	$-0.323081\pi$
$257$	16.9171	1.05526	0.527631	+	0.849474i	$0.323081\pi$
$258$	0	0
$259$	0.327696	0.0203620
$260$	−77.1849	−4.78680
$261$	0	0
$262$	−36.7229	−2.26875
$263$	−8.07635	−0.498009	−0.249004	−	0.968502i	$-0.580103\pi$
$263$	−8.07635	−0.498009	−0.249004	+	0.968502i	$0.580103\pi$
$264$	0	0
$265$	−0.543948	−0.0334145
$266$	0	0
$267$	0	0
$268$	−4.90673	−0.299726
$269$	11.0419	0.673234	0.336617	−	0.941642i	$-0.390717\pi$
$269$	11.0419	0.673234	0.336617	+	0.941642i	$0.390717\pi$
$270$	0	0
$271$	−2.02229	−0.122845	−0.0614226	−	0.998112i	$-0.519564\pi$
$271$	−2.02229	−0.122845	−0.0614226	+	0.998112i	$0.519564\pi$
$272$	4.68587	0.284123
$273$	0	0
$274$	12.6732	0.765618
$275$	25.3051	1.52595
$276$	0	0
$277$	20.8057	1.25009	0.625047	−	0.780587i	$-0.285080\pi$
$277$	20.8057	1.25009	0.625047	+	0.780587i	$0.285080\pi$
$278$	54.0467	3.24150
$279$	0	0
$280$	2.39961	0.143404
$281$	−8.32582	−0.496677	−0.248338	−	0.968673i	$-0.579884\pi$
$281$	−8.32582	−0.496677	−0.248338	+	0.968673i	$0.579884\pi$
$282$	0	0
$283$	11.1625	0.663542	0.331771	−	0.943360i	$-0.392354\pi$
$283$	11.1625	0.663542	0.331771	+	0.943360i	$0.392354\pi$
$284$	−31.8804	−1.89176
$285$	0	0
$286$	−45.1985	−2.67264
$287$	−0.410110	−0.0242080
$288$	0	0
$289$	−5.97864	−0.351685
$290$	64.3241	3.77724
$291$	0	0
$292$	−30.7374	−1.79877
$293$	3.88722	0.227094	0.113547	−	0.993533i	$-0.463779\pi$
$293$	3.88722	0.227094	0.113547	+	0.993533i	$0.463779\pi$
$294$	0	0
$295$	−25.7820	−1.50109
$296$	−6.39021	−0.371424
$297$	0	0
$298$	−35.3455	−2.04751
$299$	−26.8060	−1.55023
$300$	0	0
$301$	1.24897	0.0719895
$302$	−34.6266	−1.99254
$303$	0	0
$304$	0	0
$305$	21.3375	1.22178
$306$	0	0
$307$	−17.1361	−0.978009	−0.489004	−	0.872281i	$-0.662640\pi$
$307$	−17.1361	−0.978009	−0.489004	+	0.872281i	$0.662640\pi$
$308$	2.06835	0.117855
$309$	0	0
$310$	62.8171	3.56777
$311$	6.67390	0.378442	0.189221	−	0.981934i	$-0.439404\pi$
$311$	6.67390	0.378442	0.189221	+	0.981934i	$0.439404\pi$
$312$	0	0
$313$	−21.4757	−1.21388	−0.606938	−	0.794749i	$-0.707602\pi$
$313$	−21.4757	−1.21388	−0.606938	+	0.794749i	$0.707602\pi$
$314$	20.6876	1.16747
$315$	0	0
$316$	10.1429	0.570583
$317$	33.5260	1.88301	0.941504	−	0.337000i	$-0.109412\pi$
$317$	33.5260	1.88301	0.941504	+	0.337000i	$0.109412\pi$
$318$	0	0
$319$	24.0496	1.34652
$320$	43.1193	2.41044
$321$	0	0
$322$	1.92127	0.107068
$323$	0	0
$324$	0	0
$325$	48.4252	2.68615
$326$	−2.13678	−0.118345
$327$	0	0
$328$	7.99731	0.441578
$329$	−0.887485	−0.0489286
$330$	0	0
$331$	24.0077	1.31958	0.659792	−	0.751448i	$-0.270644\pi$
$331$	24.0077	1.31958	0.659792	+	0.751448i	$0.270644\pi$
$332$	25.8040	1.41618
$333$	0	0
$334$	33.9026	1.85507
$335$	5.00598	0.273506
$336$	0	0
$337$	28.2814	1.54058	0.770292	−	0.637691i	$-0.220110\pi$
$337$	28.2814	1.54058	0.770292	+	0.637691i	$0.220110\pi$
$338$	−55.9180	−3.04154
$339$	0	0
$340$	−42.2550	−2.29160
$341$	23.4862	1.27185
$342$	0	0
$343$	−2.58079	−0.139349
$344$	−24.3555	−1.31316
$345$	0	0
$346$	10.0847	0.542157
$347$	22.6232	1.21448	0.607239	−	0.794519i	$-0.292277\pi$
$347$	22.6232	1.21448	0.607239	+	0.794519i	$0.292277\pi$
$348$	0	0
$349$	17.6304	0.943734	0.471867	−	0.881670i	$-0.343580\pi$
$349$	17.6304	0.943734	0.471867	+	0.881670i	$0.343580\pi$
$350$	−3.47079	−0.185522
$351$	0	0
$352$	12.3182	0.656562
$353$	−25.6252	−1.36389	−0.681945	−	0.731404i	$-0.738866\pi$
$353$	−25.6252	−1.36389	−0.681945	+	0.731404i	$0.738866\pi$
$354$	0	0
$355$	32.5253	1.72626
$356$	25.9249	1.37401
$357$	0	0
$358$	0.108431	0.00573075
$359$	−32.1942	−1.69915	−0.849573	−	0.527472i	$-0.823140\pi$
$359$	−32.1942	−1.69915	−0.849573	+	0.527472i	$0.823140\pi$
$360$	0	0
$361$	0	0
$362$	−7.32380	−0.384930
$363$	0	0
$364$	3.95811	0.207461
$365$	31.3591	1.64141
$366$	0	0
$367$	19.2959	1.00724	0.503619	−	0.863926i	$-0.332001\pi$
$367$	19.2959	1.00724	0.503619	+	0.863926i	$0.332001\pi$
$368$	−6.23926	−0.325244
$369$	0	0
$370$	15.0300	0.781374
$371$	0.0278942	0.00144819
$372$	0	0
$373$	15.2763	0.790977	0.395489	−	0.918471i	$-0.370575\pi$
$373$	15.2763	0.790977	0.395489	+	0.918471i	$0.370575\pi$
$374$	−24.7440	−1.27948
$375$	0	0
$376$	17.3063	0.892506
$377$	46.0228	2.37029
$378$	0	0
$379$	−20.9718	−1.07725	−0.538625	−	0.842545i	$-0.681056\pi$
$379$	−20.9718	−1.07725	−0.538625	+	0.842545i	$0.681056\pi$
$380$	0	0
$381$	0	0
$382$	−56.3432	−2.88277
$383$	4.95355	0.253115	0.126557	−	0.991959i	$-0.459607\pi$
$383$	4.95355	0.253115	0.126557	+	0.991959i	$0.459607\pi$
$384$	0	0
$385$	−2.11019	−0.107545
$386$	13.4058	0.682337
$387$	0	0
$388$	2.95811	0.150175
$389$	32.9286	1.66954	0.834772	−	0.550596i	$-0.185599\pi$
$389$	32.9286	1.66954	0.834772	+	0.550596i	$0.185599\pi$
$390$	0	0
$391$	−14.6750	−0.742146
$392$	25.1017	1.26783
$393$	0	0
$394$	−6.07604	−0.306106
$395$	−10.3481	−0.520668
$396$	0	0
$397$	−30.7766	−1.54463	−0.772317	−	0.635237i	$-0.780902\pi$
$397$	−30.7766	−1.54463	−0.772317	+	0.635237i	$0.780902\pi$
$398$	−40.6003	−2.03511
$399$	0	0
$400$	11.2713	0.563563
$401$	−15.2771	−0.762901	−0.381451	−	0.924389i	$-0.624575\pi$
$401$	−15.2771	−0.762901	−0.381451	+	0.924389i	$0.624575\pi$
$402$	0	0
$403$	44.9445	2.23884
$404$	15.2652	0.759472
$405$	0	0
$406$	−3.29860	−0.163707
$407$	5.61946	0.278546
$408$	0	0
$409$	12.6450	0.625253	0.312627	−	0.949876i	$-0.398791\pi$
$409$	12.6450	0.625253	0.312627	+	0.949876i	$0.398791\pi$
$410$	−18.8100	−0.928959
$411$	0	0
$412$	24.2199	1.19323
$413$	1.32212	0.0650575
$414$	0	0
$415$	−26.3259	−1.29229
$416$	23.5728	1.15575
$417$	0	0
$418$	0	0
$419$	23.2828	1.13744	0.568720	−	0.822531i	$-0.307439\pi$
$419$	23.2828	1.13744	0.568720	+	0.822531i	$0.307439\pi$
$420$	0	0
$421$	−22.8999	−1.11607	−0.558037	−	0.829816i	$-0.688445\pi$
$421$	−22.8999	−1.11607	−0.558037	+	0.829816i	$0.688445\pi$
$422$	−10.8448	−0.527917
$423$	0	0
$424$	−0.543948	−0.0264165
$425$	26.5104	1.28595
$426$	0	0
$427$	−1.09421	−0.0529524
$428$	−42.0712	−2.03359
$429$	0	0
$430$	57.2850	2.76253
$431$	−30.0552	−1.44771	−0.723855	−	0.689952i	$-0.757631\pi$
$431$	−30.0552	−1.44771	−0.723855	+	0.689952i	$0.757631\pi$
$432$	0	0
$433$	−5.52435	−0.265483	−0.132742	−	0.991151i	$-0.542378\pi$
$433$	−5.52435	−0.265483	−0.132742	+	0.991151i	$0.542378\pi$
$434$	−3.22132	−0.154628
$435$	0	0
$436$	−23.3678	−1.11912
$437$	0	0
$438$	0	0
$439$	17.6527	0.842518	0.421259	−	0.906940i	$-0.361588\pi$
$439$	17.6527	0.842518	0.421259	+	0.906940i	$0.361588\pi$
$440$	41.1495	1.96173
$441$	0	0
$442$	−47.3515	−2.25228
$443$	8.41246	0.399688	0.199844	−	0.979828i	$-0.435956\pi$
$443$	8.41246	0.399688	0.199844	+	0.979828i	$0.435956\pi$
$444$	0	0
$445$	−26.4492	−1.25381
$446$	−42.1515	−1.99593
$447$	0	0
$448$	−2.21120	−0.104469
$449$	14.9473	0.705407	0.352703	−	0.935735i	$-0.385263\pi$
$449$	14.9473	0.705407	0.352703	+	0.935735i	$0.385263\pi$
$450$	0	0
$451$	−7.03272	−0.331158
$452$	19.6856	0.925932
$453$	0	0
$454$	62.3816	2.92771
$455$	−4.03817	−0.189312
$456$	0	0
$457$	31.2267	1.46072	0.730361	−	0.683061i	$-0.239352\pi$
$457$	31.2267	1.46072	0.730361	+	0.683061i	$0.239352\pi$
$458$	7.92540	0.370329
$459$	0	0
$460$	56.2627	2.62326
$461$	−1.33181	−0.0620287	−0.0310143	−	0.999519i	$-0.509874\pi$
$461$	−1.33181	−0.0620287	−0.0310143	+	0.999519i	$0.509874\pi$
$462$	0	0
$463$	−5.21719	−0.242463	−0.121232	−	0.992624i	$-0.538684\pi$
$463$	−5.21719	−0.242463	−0.121232	+	0.992624i	$0.538684\pi$
$464$	10.7121	0.497296
$465$	0	0
$466$	43.1539	1.99907
$467$	−20.5668	−0.951716	−0.475858	−	0.879522i	$-0.657862\pi$
$467$	−20.5668	−0.951716	−0.475858	+	0.879522i	$0.657862\pi$
$468$	0	0
$469$	−0.256711	−0.0118538
$470$	−40.7052	−1.87759
$471$	0	0
$472$	−25.7820	−1.18671
$473$	21.4178	0.984793
$474$	0	0
$475$	0	0
$476$	2.16687	0.0993185
$477$	0	0
$478$	−22.5639	−1.03205
$479$	15.6013	0.712842	0.356421	−	0.934325i	$-0.383997\pi$
$479$	15.6013	0.712842	0.356421	+	0.934325i	$0.383997\pi$
$480$	0	0
$481$	10.7537	0.490327
$482$	14.4141	0.656546
$483$	0	0
$484$	−3.38413	−0.153824
$485$	−3.01795	−0.137038
$486$	0	0
$487$	−22.2327	−1.00746	−0.503729	−	0.863862i	$-0.668039\pi$
$487$	−22.2327	−1.00746	−0.503729	+	0.863862i	$0.668039\pi$
$488$	21.3375	0.965904
$489$	0	0
$490$	−59.0401	−2.66716
$491$	−32.3270	−1.45889	−0.729447	−	0.684037i	$-0.760223\pi$
$491$	−32.3270	−1.45889	−0.729447	+	0.684037i	$0.760223\pi$
$492$	0	0
$493$	25.1952	1.13474
$494$	0	0
$495$	0	0
$496$	10.4611	0.469717
$497$	−1.66793	−0.0748167
$498$	0	0
$499$	−4.19934	−0.187988	−0.0939941	−	0.995573i	$-0.529963\pi$
$499$	−4.19934	−0.187988	−0.0939941	+	0.995573i	$0.529963\pi$
$500$	−37.9988	−1.69936
$501$	0	0
$502$	−51.9846	−2.32019
$503$	15.6314	0.696970	0.348485	−	0.937314i	$-0.386696\pi$
$503$	15.6314	0.696970	0.348485	+	0.937314i	$0.386696\pi$
$504$	0	0
$505$	−15.5740	−0.693033
$506$	32.9468	1.46466
$507$	0	0
$508$	−17.0172	−0.755018
$509$	43.5722	1.93130	0.965652	−	0.259840i	$-0.0836698\pi$
$509$	43.5722	1.93130	0.965652	+	0.259840i	$0.0836698\pi$
$510$	0	0
$511$	−1.60813	−0.0711393
$512$	15.6593	0.692050
$513$	0	0
$514$	−39.7897	−1.75505
$515$	−24.7098	−1.08884
$516$	0	0
$517$	−15.2189	−0.669328
$518$	−0.770754	−0.0338650
$519$	0	0
$520$	78.7461	3.45324
$521$	7.97782	0.349515	0.174757	−	0.984612i	$-0.444086\pi$
$521$	7.97782	0.349515	0.174757	+	0.984612i	$0.444086\pi$
$522$	0	0
$523$	11.7784	0.515032	0.257516	−	0.966274i	$-0.417096\pi$
$523$	11.7784	0.515032	0.257516	+	0.966274i	$0.417096\pi$
$524$	55.1472	2.40912
$525$	0	0
$526$	18.9959	0.828260
$527$	24.6049	1.07181
$528$	0	0
$529$	−3.46017	−0.150442
$530$	1.27939	0.0555731
$531$	0	0
$532$	0	0
$533$	−13.4582	−0.582940
$534$	0	0
$535$	42.9222	1.85569
$536$	5.00598	0.216225
$537$	0	0
$538$	−25.9709	−1.11969
$539$	−22.0740	−0.950796
$540$	0	0
$541$	−3.08141	−0.132480	−0.0662402	−	0.997804i	$-0.521100\pi$
$541$	−3.08141	−0.132480	−0.0662402	+	0.997804i	$0.521100\pi$
$542$	4.75650	0.204309
$543$	0	0
$544$	12.9050	0.553296
$545$	23.8405	1.02121
$546$	0	0
$547$	−12.0787	−0.516449	−0.258225	−	0.966085i	$-0.583137\pi$
$547$	−12.0787	−0.516449	−0.258225	+	0.966085i	$0.583137\pi$
$548$	−19.0316	−0.812988
$549$	0	0
$550$	−59.5185	−2.53788
$551$	0	0
$552$	0	0
$553$	0.530658	0.0225659
$554$	−48.9359	−2.07909
$555$	0	0
$556$	−81.1626	−3.44206
$557$	−36.1960	−1.53367	−0.766836	−	0.641843i	$-0.778170\pi$
$557$	−36.1960	−1.53367	−0.766836	+	0.641843i	$0.778170\pi$
$558$	0	0
$559$	40.9864	1.73354
$560$	−0.939909	−0.0397184
$561$	0	0
$562$	19.5827	0.826044
$563$	−28.2951	−1.19250	−0.596248	−	0.802800i	$-0.703343\pi$
$563$	−28.2951	−1.19250	−0.596248	+	0.802800i	$0.703343\pi$
$564$	0	0
$565$	−20.0838	−0.844931
$566$	−26.2546	−1.10357
$567$	0	0
$568$	32.5253	1.36473
$569$	25.3757	1.06380	0.531902	−	0.846806i	$-0.321477\pi$
$569$	25.3757	1.06380	0.531902	+	0.846806i	$0.321477\pi$
$570$	0	0
$571$	41.8025	1.74938	0.874691	−	0.484682i	$-0.161065\pi$
$571$	41.8025	1.74938	0.874691	+	0.484682i	$0.161065\pi$
$572$	67.8752	2.83801
$573$	0	0
$574$	0.964594	0.0402614
$575$	−35.2988	−1.47206
$576$	0	0
$577$	35.2746	1.46850	0.734250	−	0.678880i	$-0.237534\pi$
$577$	35.2746	1.46850	0.734250	+	0.678880i	$0.237534\pi$
$578$	14.0620	0.584902
$579$	0	0
$580$	−96.5964	−4.01095
$581$	1.35002	0.0560082
$582$	0	0
$583$	0.478340	0.0198108
$584$	31.3591	1.29765
$585$	0	0
$586$	−9.14290	−0.377690
$587$	−23.4219	−0.966724	−0.483362	−	0.875421i	$-0.660584\pi$
$587$	−23.4219	−0.966724	−0.483362	+	0.875421i	$0.660584\pi$
$588$	0	0
$589$	0	0
$590$	60.6403	2.49652
$591$	0	0
$592$	2.50299	0.102872
$593$	5.38187	0.221007	0.110503	−	0.993876i	$-0.464754\pi$
$593$	5.38187	0.221007	0.110503	+	0.993876i	$0.464754\pi$
$594$	0	0
$595$	−2.21070	−0.0906300
$596$	53.0789	2.17420
$597$	0	0
$598$	63.0488	2.57826
$599$	39.0533	1.59567	0.797837	−	0.602873i	$-0.205978\pi$
$599$	39.0533	1.59567	0.797837	+	0.602873i	$0.205978\pi$
$600$	0	0
$601$	28.7965	1.17464	0.587318	−	0.809356i	$-0.300184\pi$
$601$	28.7965	1.17464	0.587318	+	0.809356i	$0.300184\pi$
$602$	−2.93763	−0.119729
$603$	0	0
$604$	51.9992	2.11582
$605$	3.45258	0.140368
$606$	0	0
$607$	32.4766	1.31818	0.659092	−	0.752062i	$-0.270941\pi$
$607$	32.4766	1.31818	0.659092	+	0.752062i	$0.270941\pi$
$608$	0	0
$609$	0	0
$610$	−50.1867	−2.03200
$611$	−29.1238	−1.17822
$612$	0	0
$613$	34.1266	1.37836	0.689180	−	0.724590i	$-0.257971\pi$
$613$	34.1266	1.37836	0.689180	+	0.724590i	$0.257971\pi$
$614$	40.3048	1.62657
$615$	0	0
$616$	−2.11019	−0.0850218
$617$	−17.8849	−0.720020	−0.360010	−	0.932948i	$-0.617227\pi$
$617$	−17.8849	−0.720020	−0.360010	+	0.932948i	$0.617227\pi$
$618$	0	0
$619$	−10.6628	−0.428574	−0.214287	−	0.976771i	$-0.568743\pi$
$619$	−10.6628	−0.428574	−0.214287	+	0.976771i	$0.568743\pi$
$620$	−94.3332	−3.78851
$621$	0	0
$622$	−15.6973	−0.629404
$623$	1.35634	0.0543407
$624$	0	0
$625$	−1.15982	−0.0463926
$626$	50.5116	2.01885
$627$	0	0
$628$	−31.0669	−1.23970
$629$	5.88714	0.234736
$630$	0	0
$631$	−30.8607	−1.22855	−0.614273	−	0.789094i	$-0.710551\pi$
$631$	−30.8607	−1.22855	−0.614273	+	0.789094i	$0.710551\pi$
$632$	−10.3481	−0.411624
$633$	0	0
$634$	−78.8545	−3.13171
$635$	17.3614	0.688968
$636$	0	0
$637$	−42.2422	−1.67370
$638$	−56.5657	−2.23946
$639$	0	0
$640$	−73.4029	−2.90150
$641$	42.7553	1.68873	0.844367	−	0.535765i	$-0.179977\pi$
$641$	42.7553	1.68873	0.844367	+	0.535765i	$0.179977\pi$
$642$	0	0
$643$	−25.2550	−0.995958	−0.497979	−	0.867189i	$-0.665924\pi$
$643$	−25.2550	−0.995958	−0.497979	+	0.867189i	$0.665924\pi$
$644$	−2.88520	−0.113693
$645$	0	0
$646$	0	0
$647$	−6.93939	−0.272815	−0.136408	−	0.990653i	$-0.543556\pi$
$647$	−6.93939	−0.272815	−0.136408	+	0.990653i	$0.543556\pi$
$648$	0	0
$649$	22.6723	0.889966
$650$	−113.898	−4.46745
$651$	0	0
$652$	3.20884	0.125668
$653$	−35.4840	−1.38859	−0.694297	−	0.719688i	$-0.744285\pi$
$653$	−35.4840	−1.38859	−0.694297	+	0.719688i	$0.744285\pi$
$654$	0	0
$655$	−56.2627	−2.19837
$656$	−3.13248	−0.122303
$657$	0	0
$658$	2.08740	0.0813753
$659$	30.6226	1.19289	0.596444	−	0.802655i	$-0.296580\pi$
$659$	30.6226	1.19289	0.596444	+	0.802655i	$0.296580\pi$
$660$	0	0
$661$	6.67736	0.259719	0.129860	−	0.991532i	$-0.458547\pi$
$661$	6.67736	0.259719	0.129860	+	0.991532i	$0.458547\pi$
$662$	−56.4671	−2.19466
$663$	0	0
$664$	−26.3259	−1.02164
$665$	0	0
$666$	0	0
$667$	−33.5476	−1.29897
$668$	−50.9120	−1.96984
$669$	0	0
$670$	−11.7743	−0.454879
$671$	−18.7639	−0.724372
$672$	0	0
$673$	3.68779	0.142154	0.0710768	−	0.997471i	$-0.477356\pi$
$673$	3.68779	0.142154	0.0710768	+	0.997471i	$0.477356\pi$
$674$	−66.5189	−2.56221
$675$	0	0
$676$	83.9728	3.22972
$677$	−9.93500	−0.381833	−0.190916	−	0.981606i	$-0.561146\pi$
$677$	−9.93500	−0.381833	−0.190916	+	0.981606i	$0.561146\pi$
$678$	0	0
$679$	0.154763	0.00593926
$680$	43.1097	1.65318
$681$	0	0
$682$	−55.2404	−2.11526
$683$	9.84280	0.376624	0.188312	−	0.982109i	$-0.439698\pi$
$683$	9.84280	0.376624	0.188312	+	0.982109i	$0.439698\pi$
$684$	0	0
$685$	19.4165	0.741867
$686$	6.07011	0.231758
$687$	0	0
$688$	9.53983	0.363703
$689$	0.915379	0.0348732
$690$	0	0
$691$	−12.2763	−0.467013	−0.233506	−	0.972355i	$-0.575020\pi$
$691$	−12.2763	−0.467013	−0.233506	+	0.972355i	$0.575020\pi$
$692$	−15.1443	−0.575702
$693$	0	0
$694$	−53.2107	−2.01985
$695$	82.8043	3.14095
$696$	0	0
$697$	−7.36772	−0.279072
$698$	−41.4674	−1.56957
$699$	0	0
$700$	5.21213	0.197000
$701$	9.34605	0.352995	0.176498	−	0.984301i	$-0.443523\pi$
$701$	9.34605	0.352995	0.176498	+	0.984301i	$0.443523\pi$
$702$	0	0
$703$	0	0
$704$	−37.9185	−1.42911
$705$	0	0
$706$	60.2714	2.26834
$707$	0.798648	0.0300363
$708$	0	0
$709$	−30.7006	−1.15298	−0.576492	−	0.817103i	$-0.695579\pi$
$709$	−30.7006	−1.15298	−0.576492	+	0.817103i	$0.695579\pi$
$710$	−76.5007	−2.87102
$711$	0	0
$712$	−26.4492	−0.991227
$713$	−32.7616	−1.22693
$714$	0	0
$715$	−69.2481	−2.58973
$716$	−0.162832	−0.00608532
$717$	0	0
$718$	75.7220	2.82592
$719$	5.18482	0.193361	0.0966806	−	0.995315i	$-0.469177\pi$
$719$	5.18482	0.193361	0.0966806	+	0.995315i	$0.469177\pi$
$720$	0	0
$721$	1.26714	0.0471908
$722$	0	0
$723$	0	0
$724$	10.9982	0.408747
$725$	60.6038	2.25077
$726$	0	0
$727$	−15.6682	−0.581101	−0.290550	−	0.956860i	$-0.593838\pi$
$727$	−15.6682	−0.581101	−0.290550	+	0.956860i	$0.593838\pi$
$728$	−4.03817	−0.149665
$729$	0	0
$730$	−73.7579	−2.72990
$731$	22.4381	0.829901
$732$	0	0
$733$	−26.2576	−0.969848	−0.484924	−	0.874556i	$-0.661153\pi$
$733$	−26.2576	−0.969848	−0.484924	+	0.874556i	$0.661153\pi$
$734$	−45.3847	−1.67518
$735$	0	0
$736$	−17.1830	−0.633375
$737$	−4.40218	−0.162156
$738$	0	0
$739$	−10.2257	−0.376160	−0.188080	−	0.982154i	$-0.560226\pi$
$739$	−10.2257	−0.376160	−0.188080	+	0.982154i	$0.560226\pi$
$740$	−22.5708	−0.829719
$741$	0	0
$742$	−0.0656082	−0.00240855
$743$	36.3469	1.33344	0.666720	−	0.745309i	$-0.267698\pi$
$743$	36.3469	1.33344	0.666720	+	0.745309i	$0.267698\pi$
$744$	0	0
$745$	−54.1525	−1.98399
$746$	−35.9305	−1.31551
$747$	0	0
$748$	37.1584	1.35865
$749$	−2.20109	−0.0804261
$750$	0	0
$751$	−34.5381	−1.26031	−0.630156	−	0.776468i	$-0.717009\pi$
$751$	−34.5381	−1.26031	−0.630156	+	0.776468i	$0.717009\pi$
$752$	−6.77875	−0.247196
$753$	0	0
$754$	−108.247	−3.94214
$755$	−53.0510	−1.93072
$756$	0	0
$757$	8.08647	0.293908	0.146954	−	0.989143i	$-0.453053\pi$
$757$	8.08647	0.293908	0.146954	+	0.989143i	$0.453053\pi$
$758$	49.3266	1.79162
$759$	0	0
$760$	0	0
$761$	−22.1362	−0.802435	−0.401218	−	0.915983i	$-0.631413\pi$
$761$	−22.1362	−0.802435	−0.401218	+	0.915983i	$0.631413\pi$
$762$	0	0
$763$	−1.22256	−0.0442597
$764$	84.6113	3.06113
$765$	0	0
$766$	−11.6509	−0.420966
$767$	43.3870	1.56661
$768$	0	0
$769$	−8.03952	−0.289913	−0.144956	−	0.989438i	$-0.546304\pi$
$769$	−8.03952	−0.289913	−0.144956	+	0.989438i	$0.546304\pi$
$770$	4.96324	0.178863
$771$	0	0
$772$	−20.1317	−0.724554
$773$	10.9776	0.394835	0.197418	−	0.980319i	$-0.436744\pi$
$773$	10.9776	0.394835	0.197418	+	0.980319i	$0.436744\pi$
$774$	0	0
$775$	59.1840	2.12595
$776$	−3.01795	−0.108338
$777$	0	0
$778$	−77.4492	−2.77669
$779$	0	0
$780$	0	0
$781$	−28.6023	−1.02347
$782$	34.5162	1.23430
$783$	0	0
$784$	−9.83212	−0.351147
$785$	31.6953	1.13125
$786$	0	0
$787$	−7.11617	−0.253664	−0.126832	−	0.991924i	$-0.540481\pi$
$787$	−7.11617	−0.253664	−0.126832	+	0.991924i	$0.540481\pi$
$788$	9.12447	0.325046
$789$	0	0
$790$	24.3391	0.865945
$791$	1.02991	0.0366196
$792$	0	0
$793$	−35.9077	−1.27512
$794$	72.3878	2.56895
$795$	0	0
$796$	60.9701	2.16103
$797$	−43.6001	−1.54439	−0.772197	−	0.635383i	$-0.780842\pi$
$797$	−43.6001	−1.54439	−0.772197	+	0.635383i	$0.780842\pi$
$798$	0	0
$799$	−15.9439	−0.564054
$800$	31.0412	1.09747
$801$	0	0
$802$	35.9323	1.26881
$803$	−27.5768	−0.973163
$804$	0	0
$805$	2.94356	0.103747
$806$	−105.711	−3.72352
$807$	0	0
$808$	−15.5740	−0.547891
$809$	12.9496	0.455283	0.227641	−	0.973745i	$-0.426899\pi$
$809$	12.9496	0.455283	0.227641	+	0.973745i	$0.426899\pi$
$810$	0	0
$811$	30.6509	1.07630	0.538150	−	0.842849i	$-0.319123\pi$
$811$	30.6509	1.07630	0.538150	+	0.842849i	$0.319123\pi$
$812$	4.95355	0.173836
$813$	0	0
$814$	−13.2172	−0.463262
$815$	−3.27374	−0.114674
$816$	0	0
$817$	0	0
$818$	−29.7414	−1.03989
$819$	0	0
$820$	28.2472	0.986436
$821$	−18.7115	−0.653035	−0.326517	−	0.945191i	$-0.605875\pi$
$821$	−18.7115	−0.653035	−0.326517	+	0.945191i	$0.605875\pi$
$822$	0	0
$823$	−18.7178	−0.652462	−0.326231	−	0.945290i	$-0.605779\pi$
$823$	−18.7178	−0.652462	−0.326231	+	0.945290i	$0.605779\pi$
$824$	−24.7098	−0.860806
$825$	0	0
$826$	−3.10969	−0.108200
$827$	−38.9548	−1.35459	−0.677295	−	0.735712i	$-0.736848\pi$
$827$	−38.9548	−1.35459	−0.677295	+	0.735712i	$0.736848\pi$
$828$	0	0
$829$	22.6860	0.787918	0.393959	−	0.919128i	$-0.371105\pi$
$829$	22.6860	0.787918	0.393959	+	0.919128i	$0.371105\pi$
$830$	61.9196	2.14926
$831$	0	0
$832$	−72.5631	−2.51567
$833$	−23.1255	−0.801252
$834$	0	0
$835$	51.9418	1.79752
$836$	0	0
$837$	0	0
$838$	−54.7621	−1.89172
$839$	24.3030	0.839034	0.419517	−	0.907748i	$-0.362199\pi$
$839$	24.3030	0.839034	0.419517	+	0.907748i	$0.362199\pi$
$840$	0	0
$841$	28.5972	0.986110
$842$	53.8615	1.85619
$843$	0	0
$844$	16.2858	0.560581
$845$	−85.6713	−2.94718
$846$	0	0
$847$	−0.177052	−0.00608357
$848$	0.213060	0.00731651
$849$	0	0
$850$	−62.3536	−2.13871
$851$	−7.83876	−0.268709
$852$	0	0
$853$	50.1361	1.71663	0.858314	−	0.513125i	$-0.171512\pi$
$853$	50.1361	1.71663	0.858314	+	0.513125i	$0.171512\pi$
$854$	2.57362	0.0880674
$855$	0	0
$856$	42.9222	1.46705
$857$	−20.0399	−0.684550	−0.342275	−	0.939600i	$-0.611198\pi$
$857$	−20.0399	−0.684550	−0.342275	+	0.939600i	$0.611198\pi$
$858$	0	0
$859$	14.9094	0.508703	0.254351	−	0.967112i	$-0.418138\pi$
$859$	14.9094	0.508703	0.254351	+	0.967112i	$0.418138\pi$
$860$	−86.0256	−2.93345
$861$	0	0
$862$	70.6911	2.40775
$863$	−21.1929	−0.721414	−0.360707	−	0.932679i	$-0.617465\pi$
$863$	−21.1929	−0.721414	−0.360707	+	0.932679i	$0.617465\pi$
$864$	0	0
$865$	15.4507	0.525339
$866$	12.9935	0.441537
$867$	0	0
$868$	4.83750	0.164195
$869$	9.09994	0.308694
$870$	0	0
$871$	−8.42427	−0.285445
$872$	23.8405	0.807341
$873$	0	0
$874$	0	0
$875$	−1.98803	−0.0672077
$876$	0	0
$877$	−50.5303	−1.70629	−0.853144	−	0.521675i	$-0.825307\pi$
$877$	−50.5303	−1.70629	−0.853144	+	0.521675i	$0.825307\pi$
$878$	−41.5199	−1.40123
$879$	0	0
$880$	−16.1179	−0.543335
$881$	−22.7378	−0.766054	−0.383027	−	0.923737i	$-0.625118\pi$
$881$	−22.7378	−0.766054	−0.383027	+	0.923737i	$0.625118\pi$
$882$	0	0
$883$	37.7975	1.27199	0.635993	−	0.771695i	$-0.280591\pi$
$883$	37.7975	1.27199	0.635993	+	0.771695i	$0.280591\pi$
$884$	71.1084	2.39164
$885$	0	0
$886$	−19.7864	−0.664738
$887$	−26.7536	−0.898298	−0.449149	−	0.893457i	$-0.648273\pi$
$887$	−26.7536	−0.898298	−0.449149	+	0.893457i	$0.648273\pi$
$888$	0	0
$889$	−0.890311	−0.0298601
$890$	62.2097	2.08527
$891$	0	0
$892$	63.2995	2.11943
$893$	0	0
$894$	0	0
$895$	0.166126	0.00555297
$896$	3.76417	0.125752
$897$	0	0
$898$	−35.1566	−1.17319
$899$	56.2478	1.87597
$900$	0	0
$901$	0.501126	0.0166949
$902$	16.5412	0.550763
$903$	0	0
$904$	−20.0838	−0.667977
$905$	−11.2207	−0.372989
$906$	0	0
$907$	16.8093	0.558145	0.279072	−	0.960270i	$-0.409973\pi$
$907$	16.8093	0.558145	0.279072	+	0.960270i	$0.409973\pi$
$908$	−93.6792	−3.10885
$909$	0	0
$910$	9.49794	0.314854
$911$	31.9168	1.05745	0.528726	−	0.848793i	$-0.322670\pi$
$911$	31.9168	1.05745	0.528726	+	0.848793i	$0.322670\pi$
$912$	0	0
$913$	23.1506	0.766174
$914$	−73.4464	−2.42939
$915$	0	0
$916$	−11.9017	−0.393242
$917$	2.88520	0.0952778
$918$	0	0
$919$	−14.9409	−0.492854	−0.246427	−	0.969161i	$-0.579257\pi$
$919$	−14.9409	−0.492854	−0.246427	+	0.969161i	$0.579257\pi$
$920$	−57.4007	−1.89245
$921$	0	0
$922$	3.13247	0.103163
$923$	−54.7349	−1.80162
$924$	0	0
$925$	14.1607	0.465603
$926$	12.2710	0.403251
$927$	0	0
$928$	29.5012	0.968426
$929$	−32.0775	−1.05243	−0.526214	−	0.850352i	$-0.676389\pi$
$929$	−32.0775	−1.05243	−0.526214	+	0.850352i	$0.676389\pi$
$930$	0	0
$931$	0	0
$932$	−64.8049	−2.12275
$933$	0	0
$934$	48.3738	1.58284
$935$	−37.9100	−1.23979
$936$	0	0
$937$	−21.2567	−0.694426	−0.347213	−	0.937786i	$-0.612872\pi$
$937$	−21.2567	−0.694426	−0.347213	+	0.937786i	$0.612872\pi$
$938$	0.603795	0.0197146
$939$	0	0
$940$	61.1275	1.99376
$941$	−3.85301	−0.125604	−0.0628022	−	0.998026i	$-0.520004\pi$
$941$	−3.85301	−0.125604	−0.0628022	+	0.998026i	$0.520004\pi$
$942$	0	0
$943$	9.81016	0.319463
$944$	10.0986	0.328681
$945$	0	0
$946$	−50.3756	−1.63785
$947$	−12.1264	−0.394055	−0.197028	−	0.980398i	$-0.563129\pi$
$947$	−12.1264	−0.394055	−0.197028	+	0.980398i	$0.563129\pi$
$948$	0	0
$949$	−52.7725	−1.71307
$950$	0	0
$951$	0	0
$952$	−2.21070	−0.0716493
$953$	−23.0515	−0.746713	−0.373356	−	0.927688i	$-0.621793\pi$
$953$	−23.0515	−0.746713	−0.373356	+	0.927688i	$0.621793\pi$
$954$	0	0
$955$	−86.3228	−2.79334
$956$	33.8845	1.09590
$957$	0	0
$958$	−36.6949	−1.18556
$959$	−0.995698	−0.0321527
$960$	0	0
$961$	23.9299	0.771934
$962$	−25.2932	−0.815484
$963$	0	0
$964$	−21.6459	−0.697167
$965$	20.5389	0.661169
$966$	0	0
$967$	−23.1652	−0.744942	−0.372471	−	0.928044i	$-0.621489\pi$
$967$	−23.1652	−0.744942	−0.372471	+	0.928044i	$0.621489\pi$
$968$	3.45258	0.110970
$969$	0	0
$970$	7.09833	0.227914
$971$	31.5625	1.01289	0.506445	−	0.862272i	$-0.330959\pi$
$971$	31.5625	1.01289	0.506445	+	0.862272i	$0.330959\pi$
$972$	0	0
$973$	−4.24628	−0.136130
$974$	52.2921	1.67555
$975$	0	0
$976$	−8.35773	−0.267524
$977$	−36.6306	−1.17192	−0.585958	−	0.810341i	$-0.699282\pi$
$977$	−36.6306	−1.17192	−0.585958	+	0.810341i	$0.699282\pi$
$978$	0	0
$979$	23.2591	0.743363
$980$	88.6614	2.83218
$981$	0	0
$982$	76.0343	2.42635
$983$	36.7291	1.17148	0.585739	−	0.810500i	$-0.300804\pi$
$983$	36.7291	1.17148	0.585739	+	0.810500i	$0.300804\pi$
$984$	0	0
$985$	−9.30903	−0.296610
$986$	−59.2602	−1.88723
$987$	0	0
$988$	0	0
$989$	−29.8764	−0.950014
$990$	0	0
$991$	14.8716	0.472413	0.236207	−	0.971703i	$-0.424096\pi$
$991$	14.8716	0.472413	0.236207	+	0.971703i	$0.424096\pi$
$992$	28.8101	0.914720
$993$	0	0
$994$	3.92303	0.124431
$995$	−62.2033	−1.97198
$996$	0	0
$997$	17.6905	0.560263	0.280131	−	0.959962i	$-0.409622\pi$
$997$	17.6905	0.560263	0.280131	+	0.959962i	$0.409622\pi$
$998$	9.87701	0.312651
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bj.1.1		6
3.2	odd	2	inner	3249.2.a.bj.1.6		6
19.3	odd	18		171.2.u.d.28.2	yes	12
19.13	odd	18		171.2.u.d.55.2	yes	12
19.18	odd	2		3249.2.a.bi.1.6		6
57.32	even	18		171.2.u.d.55.1	yes	12
57.41	even	18		171.2.u.d.28.1	✓	12
57.56	even	2		3249.2.a.bi.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.u.d.28.1	✓	12	57.41	even	18
171.2.u.d.28.2	yes	12	19.3	odd	18
171.2.u.d.55.1	yes	12	57.32	even	18
171.2.u.d.55.2	yes	12	19.13	odd	18
3249.2.a.bi.1.1		6	57.56	even	2
3249.2.a.bi.1.6		6	19.18	odd	2
3249.2.a.bj.1.1		6	1.1	even	1	trivial
3249.2.a.bj.1.6		6	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.35204 q^{2} +3.53209 q^{4} -3.60353 q^{5} +0.184793 q^{7} -3.60353 q^{8} +8.47565 q^{10} +3.16889 q^{11} +6.06418 q^{13} -0.434639 q^{14} +1.41147 q^{16} +3.31984 q^{17} -12.7280 q^{20} -7.45336 q^{22} -4.42039 q^{23} +7.98545 q^{25} -14.2632 q^{26} +0.652704 q^{28} +7.58928 q^{29} +7.41147 q^{31} +3.88722 q^{32} -7.80840 q^{34} -0.665906 q^{35} +1.77332 q^{37} +12.9855 q^{40} -2.21930 q^{41} +6.75877 q^{43} +11.1928 q^{44} +10.3969 q^{46} -4.80260 q^{47} -6.96585 q^{49} -18.7821 q^{50} +21.4192 q^{52} +0.150949 q^{53} -11.4192 q^{55} -0.665906 q^{56} -17.8503 q^{58} +7.15464 q^{59} -5.92127 q^{61} -17.4321 q^{62} -11.9659 q^{64} -21.8525 q^{65} -1.38919 q^{67} +11.7260 q^{68} +1.56624 q^{70} -9.02594 q^{71} -8.70233 q^{73} -4.17091 q^{74} +0.585588 q^{77} +2.87164 q^{79} -5.08629 q^{80} +5.21987 q^{82} +7.30559 q^{83} -11.9632 q^{85} -15.8969 q^{86} -11.4192 q^{88} +7.33981 q^{89} +1.12061 q^{91} -15.6132 q^{92} +11.2959 q^{94} +0.837496 q^{97} +16.3840 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 6 q^{7} + 12 q^{10} + 18 q^{13} - 12 q^{16} - 18 q^{22} + 12 q^{25} + 6 q^{28} + 24 q^{31} + 30 q^{34} + 24 q^{37} + 42 q^{40} + 18 q^{43} + 6 q^{46} + 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots + 6 q^{94}+O(q^{100}) \) 6 * q + 12 * q^4 - 6 * q^7 + 12 * q^10 + 18 * q^13 - 12 * q^16 - 18 * q^22 + 12 * q^25 + 6 * q^28 + 24 * q^31 + 30 * q^34 + 24 * q^37 + 42 * q^40 + 18 * q^43 + 6 * q^46 + 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 + 42 * q^70 + 54 * q^79 - 66 * q^82 - 48 * q^85 + 18 * q^91 + 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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