LMFDB - Embedded newform 6525.2.a.cf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] 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("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,0,0,16,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 148x^{8} - 502x^{6} + 792x^{4} - 496x^{2} + 45 $ x^12 - 20x^10 + 148x^8 - 502x^6 + 792x^4 - 496*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1305)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35513$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35513 q^{2} -0.163621 q^{4} -1.26969 q^{7} +2.93199 q^{8} -0.474782 q^{11} -0.407685 q^{13} +1.72060 q^{14} -3.64599 q^{16} -2.97132 q^{17} +5.80491 q^{19} +0.643392 q^{22} +3.06516 q^{23} +0.552466 q^{26} +0.207747 q^{28} -1.00000 q^{29} -4.49210 q^{31} -0.923188 q^{32} +4.02653 q^{34} +7.20325 q^{37} -7.86641 q^{38} +12.0668 q^{41} -1.03847 q^{43} +0.0776841 q^{44} -4.15369 q^{46} -2.97132 q^{47} -5.38789 q^{49} +0.0667056 q^{52} -7.87199 q^{53} -3.72271 q^{56} +1.35513 q^{58} -4.45891 q^{59} -1.71609 q^{61} +6.08739 q^{62} +8.54301 q^{64} -15.7754 q^{67} +0.486169 q^{68} +9.00806 q^{71} -0.677990 q^{73} -9.76134 q^{74} -0.949802 q^{76} +0.602826 q^{77} +1.63957 q^{79} -16.3521 q^{82} +15.5423 q^{83} +1.40726 q^{86} -1.39206 q^{88} +13.4175 q^{89} +0.517633 q^{91} -0.501523 q^{92} +4.02653 q^{94} +7.10733 q^{97} +7.30129 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) $ 12 * q + 16 * q^4 + 12 * q^11 + 16 * q^14 + 16 * q^16 - 20 * q^19 + 56 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 32 * q^41 + 68 * q^44 + 20 * q^46 - 4 * q^49 + 76 * q^56 + 44 * q^59 - 52 * q^61 + 36 * q^64 + 20 * q^71 + 20 * q^74 - 28 * q^76 - 4 * q^79 + 36 * q^86 + 68 * q^89 + 48 * q^91 + 4 * 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 $

$2$	−1.35513	−0.958222	−0.479111	−	0.877754i	$-0.659041\pi$
$2$	−1.35513	−0.958222	−0.479111	+	0.877754i	$0.659041\pi$
$3$	0	0
$3$	0	0
$4$	−0.163621	−0.0818103
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.26969	−0.479897	−0.239949	−	0.970786i	$-0.577131\pi$
$7$	−1.26969	−0.479897	−0.239949	+	0.970786i	$0.577131\pi$
$8$	2.93199	1.03661
$9$	0	0
$10$	0	0
$11$	−0.474782	−0.143152	−0.0715761	−	0.997435i	$-0.522803\pi$
$11$	−0.474782	−0.143152	−0.0715761	+	0.997435i	$0.522803\pi$
$12$	0	0
$13$	−0.407685	−0.113071	−0.0565357	−	0.998401i	$-0.518005\pi$
$13$	−0.407685	−0.113071	−0.0565357	+	0.998401i	$0.518005\pi$
$14$	1.72060	0.459848
$15$	0	0
$16$	−3.64599	−0.911497
$17$	−2.97132	−0.720651	−0.360326	−	0.932827i	$-0.617334\pi$
$17$	−2.97132	−0.720651	−0.360326	+	0.932827i	$0.617334\pi$
$18$	0	0
$19$	5.80491	1.33174	0.665868	−	0.746069i	$-0.268061\pi$
$19$	5.80491	1.33174	0.665868	+	0.746069i	$0.268061\pi$
$20$	0	0
$21$	0	0
$22$	0.643392	0.137172
$23$	3.06516	0.639130	0.319565	−	0.947564i	$-0.396463\pi$
$23$	3.06516	0.639130	0.319565	+	0.947564i	$0.396463\pi$
$24$	0	0
$25$	0	0
$26$	0.552466	0.108348
$27$	0	0
$28$	0.207747	0.0392605
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−4.49210	−0.806805	−0.403403	−	0.915023i	$-0.632173\pi$
$31$	−4.49210	−0.806805	−0.403403	+	0.915023i	$0.632173\pi$
$32$	−0.923188	−0.163198
$33$	0	0
$34$	4.02653	0.690544
$35$	0	0
$36$	0	0
$37$	7.20325	1.18421	0.592103	−	0.805862i	$-0.298298\pi$
$37$	7.20325	1.18421	0.592103	+	0.805862i	$0.298298\pi$
$38$	−7.86641	−1.27610
$39$	0	0
$40$	0	0
$41$	12.0668	1.88452	0.942259	−	0.334886i	$-0.108698\pi$
$41$	12.0668	1.88452	0.942259	+	0.334886i	$0.108698\pi$
$42$	0	0
$43$	−1.03847	−0.158365	−0.0791827	−	0.996860i	$-0.525231\pi$
$43$	−1.03847	−0.158365	−0.0791827	+	0.996860i	$0.525231\pi$
$44$	0.0776841	0.0117113
$45$	0	0
$46$	−4.15369	−0.612428
$47$	−2.97132	−0.433412	−0.216706	−	0.976237i	$-0.569531\pi$
$47$	−2.97132	−0.433412	−0.216706	+	0.976237i	$0.569531\pi$
$48$	0	0
$49$	−5.38789	−0.769698
$50$	0	0
$51$	0	0
$52$	0.0667056	0.00925040
$53$	−7.87199	−1.08130	−0.540651	−	0.841247i	$-0.681822\pi$
$53$	−7.87199	−1.08130	−0.540651	+	0.841247i	$0.681822\pi$
$54$	0	0
$55$	0	0
$56$	−3.72271	−0.497469
$57$	0	0
$58$	1.35513	0.177937
$59$	−4.45891	−0.580501	−0.290250	−	0.956951i	$-0.593739\pi$
$59$	−4.45891	−0.580501	−0.290250	+	0.956951i	$0.593739\pi$
$60$	0	0
$61$	−1.71609	−0.219722	−0.109861	−	0.993947i	$-0.535041\pi$
$61$	−1.71609	−0.219722	−0.109861	+	0.993947i	$0.535041\pi$
$62$	6.08739	0.773099
$63$	0	0
$64$	8.54301	1.06788
$65$	0	0
$66$	0	0
$67$	−15.7754	−1.92727	−0.963636	−	0.267218i	$-0.913896\pi$
$67$	−15.7754	−1.92727	−0.963636	+	0.267218i	$0.913896\pi$
$68$	0.486169	0.0589567
$69$	0	0
$70$	0	0
$71$	9.00806	1.06906	0.534530	−	0.845149i	$-0.320489\pi$
$71$	9.00806	1.06906	0.534530	+	0.845149i	$0.320489\pi$
$72$	0	0
$73$	−0.677990	−0.0793527	−0.0396763	−	0.999213i	$-0.512633\pi$
$73$	−0.677990	−0.0793527	−0.0396763	+	0.999213i	$0.512633\pi$
$74$	−9.76134	−1.13473
$75$	0	0
$76$	−0.949802	−0.108950
$77$	0.602826	0.0686984
$78$	0	0
$79$	1.63957	0.184466	0.0922329	−	0.995737i	$-0.470600\pi$
$79$	1.63957	0.184466	0.0922329	+	0.995737i	$0.470600\pi$
$80$	0	0
$81$	0	0
$82$	−16.3521	−1.80579
$83$	15.5423	1.70599	0.852995	−	0.521919i	$-0.174784\pi$
$83$	15.5423	1.70599	0.852995	+	0.521919i	$0.174784\pi$
$84$	0	0
$85$	0	0
$86$	1.40726	0.151749
$87$	0	0
$88$	−1.39206	−0.148394
$89$	13.4175	1.42225	0.711126	−	0.703065i	$-0.248186\pi$
$89$	13.4175	1.42225	0.711126	+	0.703065i	$0.248186\pi$
$90$	0	0
$91$	0.517633	0.0542627
$92$	−0.501523	−0.0522874
$93$	0	0
$94$	4.02653	0.415305
$95$	0	0
$96$	0	0
$97$	7.10733	0.721640	0.360820	−	0.932636i	$-0.382497\pi$
$97$	7.10733	0.721640	0.360820	+	0.932636i	$0.382497\pi$
$98$	7.30129	0.737542
$99$	0	0
$100$	0	0
$101$	4.68433	0.466108	0.233054	−	0.972464i	$-0.425128\pi$
$101$	4.68433	0.466108	0.233054	+	0.972464i	$0.425128\pi$
$102$	0	0
$103$	−15.5180	−1.52904	−0.764519	−	0.644601i	$-0.777024\pi$
$103$	−15.5180	−1.52904	−0.764519	+	0.644601i	$0.777024\pi$
$104$	−1.19533	−0.117212
$105$	0	0
$106$	10.6676	1.03613
$107$	−0.676108	−0.0653618	−0.0326809	−	0.999466i	$-0.510405\pi$
$107$	−0.676108	−0.0653618	−0.0326809	+	0.999466i	$0.510405\pi$
$108$	0	0
$109$	−7.44734	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$109$	−7.44734	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$110$	0	0
$111$	0	0
$112$	4.62927	0.437425
$113$	−14.1017	−1.32658	−0.663290	−	0.748363i	$-0.730840\pi$
$113$	−14.1017	−1.32658	−0.663290	+	0.748363i	$0.730840\pi$
$114$	0	0
$115$	0	0
$116$	0.163621	0.0151918
$117$	0	0
$118$	6.04241	0.556249
$119$	3.77265	0.345839
$120$	0	0
$121$	−10.7746	−0.979507
$122$	2.32552	0.210543
$123$	0	0
$124$	0.735000	0.0660050
$125$	0	0
$126$	0	0
$127$	11.5508	1.02497	0.512486	−	0.858696i	$-0.328725\pi$
$127$	11.5508	1.02497	0.512486	+	0.858696i	$0.328725\pi$
$128$	−9.73053	−0.860065
$129$	0	0
$130$	0	0
$131$	10.2432	0.894954	0.447477	−	0.894296i	$-0.352323\pi$
$131$	10.2432	0.894954	0.447477	+	0.894296i	$0.352323\pi$
$132$	0	0
$133$	−7.37043	−0.639097
$134$	21.3777	1.84676
$135$	0	0
$136$	−8.71188	−0.747038
$137$	−2.06577	−0.176491	−0.0882455	−	0.996099i	$-0.528126\pi$
$137$	−2.06577	−0.176491	−0.0882455	+	0.996099i	$0.528126\pi$
$138$	0	0
$139$	3.48237	0.295370	0.147685	−	0.989034i	$-0.452818\pi$
$139$	3.48237	0.295370	0.147685	+	0.989034i	$0.452818\pi$
$140$	0	0
$141$	0	0
$142$	−12.2071	−1.02440
$143$	0.193562	0.0161864
$144$	0	0
$145$	0	0
$146$	0.918765	0.0760375
$147$	0	0
$148$	−1.17860	−0.0968802
$149$	10.8558	0.889343	0.444672	−	0.895694i	$-0.353320\pi$
$149$	10.8558	0.889343	0.444672	+	0.895694i	$0.353320\pi$
$150$	0	0
$151$	11.3555	0.924097	0.462048	−	0.886855i	$-0.347115\pi$
$151$	11.3555	0.924097	0.462048	+	0.886855i	$0.347115\pi$
$152$	17.0199	1.38050
$153$	0	0
$154$	−0.816908	−0.0658283
$155$	0	0
$156$	0	0
$157$	−16.9813	−1.35526	−0.677628	−	0.735405i	$-0.736992\pi$
$157$	−16.9813	−1.35526	−0.677628	+	0.735405i	$0.736992\pi$
$158$	−2.22183	−0.176759
$159$	0	0
$160$	0	0
$161$	−3.89180	−0.306717
$162$	0	0
$163$	−7.78532	−0.609793	−0.304897	−	0.952385i	$-0.598622\pi$
$163$	−7.78532	−0.609793	−0.304897	+	0.952385i	$0.598622\pi$
$164$	−1.97438	−0.154173
$165$	0	0
$166$	−21.0619	−1.63472
$167$	2.06389	0.159709	0.0798544	−	0.996807i	$-0.474554\pi$
$167$	2.06389	0.159709	0.0798544	+	0.996807i	$0.474554\pi$
$168$	0	0
$169$	−12.8338	−0.987215
$170$	0	0
$171$	0	0
$172$	0.169915	0.0129559
$173$	0.522120	0.0396960	0.0198480	−	0.999803i	$-0.493682\pi$
$173$	0.522120	0.0396960	0.0198480	+	0.999803i	$0.493682\pi$
$174$	0	0
$175$	0	0
$176$	1.73105	0.130483
$177$	0	0
$178$	−18.1825	−1.36283
$179$	10.4805	0.783346	0.391673	−	0.920104i	$-0.371896\pi$
$179$	10.4805	0.783346	0.391673	+	0.920104i	$0.371896\pi$
$180$	0	0
$181$	12.2724	0.912197	0.456099	−	0.889929i	$-0.349246\pi$
$181$	12.2724	0.912197	0.456099	+	0.889929i	$0.349246\pi$
$182$	−0.701461	−0.0519957
$183$	0	0
$184$	8.98701	0.662531
$185$	0	0
$186$	0	0
$187$	1.41073	0.103163
$188$	0.486169	0.0354575
$189$	0	0
$190$	0	0
$191$	9.69478	0.701490	0.350745	−	0.936471i	$-0.385928\pi$
$191$	9.69478	0.701490	0.350745	+	0.936471i	$0.385928\pi$
$192$	0	0
$193$	−11.0857	−0.797963	−0.398982	−	0.916959i	$-0.630636\pi$
$193$	−11.0857	−0.797963	−0.398982	+	0.916959i	$0.630636\pi$
$194$	−9.63136	−0.691491
$195$	0	0
$196$	0.881569	0.0629692
$197$	2.00431	0.142801	0.0714007	−	0.997448i	$-0.477253\pi$
$197$	2.00431	0.142801	0.0714007	+	0.997448i	$0.477253\pi$
$198$	0	0
$199$	−18.1170	−1.28428	−0.642140	−	0.766588i	$-0.721953\pi$
$199$	−18.1170	−1.28428	−0.642140	+	0.766588i	$0.721953\pi$
$200$	0	0
$201$	0	0
$202$	−6.34788	−0.446635
$203$	1.26969	0.0891147
$204$	0	0
$205$	0	0
$206$	21.0290	1.46516
$207$	0	0
$208$	1.48641	0.103064
$209$	−2.75607	−0.190641
$210$	0	0
$211$	18.4829	1.27241	0.636206	−	0.771519i	$-0.280503\pi$
$211$	18.4829	1.27241	0.636206	+	0.771519i	$0.280503\pi$
$212$	1.28802	0.0884615
$213$	0	0
$214$	0.916215	0.0626312
$215$	0	0
$216$	0	0
$217$	5.70357	0.387184
$218$	10.0921	0.683525
$219$	0	0
$220$	0	0
$221$	1.21136	0.0814851
$222$	0	0
$223$	−21.6226	−1.44796	−0.723978	−	0.689823i	$-0.757688\pi$
$223$	−21.6226	−1.44796	−0.723978	+	0.689823i	$0.757688\pi$
$224$	1.17216	0.0783184
$225$	0	0
$226$	19.1097	1.27116
$227$	−19.5621	−1.29838	−0.649191	−	0.760625i	$-0.724892\pi$
$227$	−19.5621	−1.29838	−0.649191	+	0.760625i	$0.724892\pi$
$228$	0	0
$229$	10.7779	0.712220	0.356110	−	0.934444i	$-0.384103\pi$
$229$	10.7779	0.712220	0.356110	+	0.934444i	$0.384103\pi$
$230$	0	0
$231$	0	0
$232$	−2.93199	−0.192494
$233$	4.25222	0.278572	0.139286	−	0.990252i	$-0.455519\pi$
$233$	4.25222	0.278572	0.139286	+	0.990252i	$0.455519\pi$
$234$	0	0
$235$	0	0
$236$	0.729569	0.0474909
$237$	0	0
$238$	−5.11244	−0.331390
$239$	16.8385	1.08919	0.544596	−	0.838699i	$-0.316683\pi$
$239$	16.8385	1.08919	0.544596	+	0.838699i	$0.316683\pi$
$240$	0	0
$241$	21.8396	1.40681	0.703405	−	0.710789i	$-0.251662\pi$
$241$	21.8396	1.40681	0.703405	+	0.710789i	$0.251662\pi$
$242$	14.6010	0.938586
$243$	0	0
$244$	0.280787	0.0179755
$245$	0	0
$246$	0	0
$247$	−2.36657	−0.150581
$248$	−13.1708	−0.836346
$249$	0	0
$250$	0	0
$251$	2.24264	0.141554	0.0707772	−	0.997492i	$-0.477452\pi$
$251$	2.24264	0.141554	0.0707772	+	0.997492i	$0.477452\pi$
$252$	0	0
$253$	−1.45528	−0.0914929
$254$	−15.6529	−0.982150
$255$	0	0
$256$	−3.89989	−0.243743
$257$	−1.27685	−0.0796479	−0.0398239	−	0.999207i	$-0.512680\pi$
$257$	−1.27685	−0.0796479	−0.0398239	+	0.999207i	$0.512680\pi$
$258$	0	0
$259$	−9.14589	−0.568298
$260$	0	0
$261$	0	0
$262$	−13.8809	−0.857564
$263$	19.6767	1.21332	0.606658	−	0.794963i	$-0.292510\pi$
$263$	19.6767	1.21332	0.606658	+	0.794963i	$0.292510\pi$
$264$	0	0
$265$	0	0
$266$	9.98789	0.612397
$267$	0	0
$268$	2.58118	0.157671
$269$	−11.7353	−0.715516	−0.357758	−	0.933814i	$-0.616459\pi$
$269$	−11.7353	−0.715516	−0.357758	+	0.933814i	$0.616459\pi$
$270$	0	0
$271$	−9.92348	−0.602809	−0.301404	−	0.953496i	$-0.597455\pi$
$271$	−9.92348	−0.602809	−0.301404	+	0.953496i	$0.597455\pi$
$272$	10.8334	0.656871
$273$	0	0
$274$	2.79940	0.169118
$275$	0	0
$276$	0	0
$277$	1.50995	0.0907242	0.0453621	−	0.998971i	$-0.485556\pi$
$277$	1.50995	0.0907242	0.0453621	+	0.998971i	$0.485556\pi$
$278$	−4.71906	−0.283031
$279$	0	0
$280$	0	0
$281$	25.3285	1.51097	0.755487	−	0.655164i	$-0.227400\pi$
$281$	25.3285	1.51097	0.755487	+	0.655164i	$0.227400\pi$
$282$	0	0
$283$	2.09287	0.124408	0.0622041	−	0.998063i	$-0.480187\pi$
$283$	2.09287	0.124408	0.0622041	+	0.998063i	$0.480187\pi$
$284$	−1.47390	−0.0874601
$285$	0	0
$286$	−0.262301	−0.0155102
$287$	−15.3211	−0.904375
$288$	0	0
$289$	−8.17125	−0.480662
$290$	0	0
$291$	0	0
$292$	0.110933	0.00649186
$293$	10.8526	0.634013	0.317007	−	0.948423i	$-0.397322\pi$
$293$	10.8526	0.634013	0.317007	+	0.948423i	$0.397322\pi$
$294$	0	0
$295$	0	0
$296$	21.1198	1.22757
$297$	0	0
$298$	−14.7111	−0.852189
$299$	−1.24962	−0.0722674
$300$	0	0
$301$	1.31854	0.0759991
$302$	−15.3882	−0.885490
$303$	0	0
$304$	−21.1646	−1.21387
$305$	0	0
$306$	0	0
$307$	16.8956	0.964281	0.482141	−	0.876094i	$-0.339859\pi$
$307$	16.8956	0.964281	0.482141	+	0.876094i	$0.339859\pi$
$308$	−0.0986347	−0.00562023
$309$	0	0
$310$	0	0
$311$	−10.6643	−0.604717	−0.302359	−	0.953194i	$-0.597774\pi$
$311$	−10.6643	−0.604717	−0.302359	+	0.953194i	$0.597774\pi$
$312$	0	0
$313$	−13.8702	−0.783991	−0.391995	−	0.919967i	$-0.628215\pi$
$313$	−13.8702	−0.783991	−0.391995	+	0.919967i	$0.628215\pi$
$314$	23.0119	1.29864
$315$	0	0
$316$	−0.268267	−0.0150912
$317$	1.33953	0.0752355	0.0376178	−	0.999292i	$-0.488023\pi$
$317$	1.33953	0.0752355	0.0376178	+	0.999292i	$0.488023\pi$
$318$	0	0
$319$	0.474782	0.0265827
$320$	0	0
$321$	0	0
$322$	5.27390	0.293903
$323$	−17.2482	−0.959718
$324$	0	0
$325$	0	0
$326$	10.5501	0.584317
$327$	0	0
$328$	35.3797	1.95352
$329$	3.77265	0.207993
$330$	0	0
$331$	−3.00072	−0.164934	−0.0824672	−	0.996594i	$-0.526280\pi$
$331$	−3.00072	−0.164934	−0.0824672	+	0.996594i	$0.526280\pi$
$332$	−2.54304	−0.139567
$333$	0	0
$334$	−2.79685	−0.153037
$335$	0	0
$336$	0	0
$337$	24.2763	1.32241	0.661207	−	0.750203i	$-0.270044\pi$
$337$	24.2763	1.32241	0.661207	+	0.750203i	$0.270044\pi$
$338$	17.3915	0.945971
$339$	0	0
$340$	0	0
$341$	2.13277	0.115496
$342$	0	0
$343$	15.7288	0.849274
$344$	−3.04479	−0.164164
$345$	0	0
$346$	−0.707540	−0.0380376
$347$	14.0792	0.755809	0.377904	−	0.925845i	$-0.376645\pi$
$347$	14.0792	0.755809	0.377904	+	0.925845i	$0.376645\pi$
$348$	0	0
$349$	−17.9641	−0.961597	−0.480798	−	0.876831i	$-0.659653\pi$
$349$	−17.9641	−0.961597	−0.480798	+	0.876831i	$0.659653\pi$
$350$	0	0
$351$	0	0
$352$	0.438313	0.0233622
$353$	9.49061	0.505134	0.252567	−	0.967579i	$-0.418725\pi$
$353$	9.49061	0.505134	0.252567	+	0.967579i	$0.418725\pi$
$354$	0	0
$355$	0	0
$356$	−2.19538	−0.116355
$357$	0	0
$358$	−14.2024	−0.750620
$359$	−4.09130	−0.215931	−0.107965	−	0.994155i	$-0.534434\pi$
$359$	−4.09130	−0.215931	−0.107965	+	0.994155i	$0.534434\pi$
$360$	0	0
$361$	14.6969	0.773523
$362$	−16.6307	−0.874088
$363$	0	0
$364$	−0.0846954	−0.00443925
$365$	0	0
$366$	0	0
$367$	18.8383	0.983351	0.491676	−	0.870778i	$-0.336385\pi$
$367$	18.8383	0.983351	0.491676	+	0.870778i	$0.336385\pi$
$368$	−11.1755	−0.582565
$369$	0	0
$370$	0	0
$371$	9.99498	0.518914
$372$	0	0
$373$	7.23999	0.374872	0.187436	−	0.982277i	$-0.439982\pi$
$373$	7.23999	0.374872	0.187436	+	0.982277i	$0.439982\pi$
$374$	−1.91172	−0.0988529
$375$	0	0
$376$	−8.71188	−0.449281
$377$	0.407685	0.0209968
$378$	0	0
$379$	20.2420	1.03976	0.519881	−	0.854239i	$-0.325976\pi$
$379$	20.2420	1.03976	0.519881	+	0.854239i	$0.325976\pi$
$380$	0	0
$381$	0	0
$382$	−13.1377	−0.672183
$383$	−4.34201	−0.221866	−0.110933	−	0.993828i	$-0.535384\pi$
$383$	−4.34201	−0.221866	−0.110933	+	0.993828i	$0.535384\pi$
$384$	0	0
$385$	0	0
$386$	15.0225	0.764626
$387$	0	0
$388$	−1.16290	−0.0590375
$389$	−0.508687	−0.0257915	−0.0128957	−	0.999917i	$-0.504105\pi$
$389$	−0.508687	−0.0257915	−0.0128957	+	0.999917i	$0.504105\pi$
$390$	0	0
$391$	−9.10757	−0.460590
$392$	−15.7972	−0.797881
$393$	0	0
$394$	−2.71611	−0.136835
$395$	0	0
$396$	0	0
$397$	31.2135	1.56656	0.783280	−	0.621669i	$-0.213545\pi$
$397$	31.2135	1.56656	0.783280	+	0.621669i	$0.213545\pi$
$398$	24.5509	1.23063
$399$	0	0
$400$	0	0
$401$	−5.60957	−0.280129	−0.140064	−	0.990142i	$-0.544731\pi$
$401$	−5.60957	−0.280129	−0.140064	+	0.990142i	$0.544731\pi$
$402$	0	0
$403$	1.83136	0.0912267
$404$	−0.766452	−0.0381324
$405$	0	0
$406$	−1.72060	−0.0853917
$407$	−3.41997	−0.169522
$408$	0	0
$409$	−9.68080	−0.478685	−0.239342	−	0.970935i	$-0.576932\pi$
$409$	−9.68080	−0.478685	−0.239342	+	0.970935i	$0.576932\pi$
$410$	0	0
$411$	0	0
$412$	2.53907	0.125091
$413$	5.66143	0.278581
$414$	0	0
$415$	0	0
$416$	0.376370	0.0184530
$417$	0	0
$418$	3.73483	0.182677
$419$	−18.2788	−0.892979	−0.446489	−	0.894789i	$-0.647326\pi$
$419$	−18.2788	−0.892979	−0.446489	+	0.894789i	$0.647326\pi$
$420$	0	0
$421$	26.0517	1.26968	0.634842	−	0.772642i	$-0.281065\pi$
$421$	26.0517	1.26968	0.634842	+	0.772642i	$0.281065\pi$
$422$	−25.0467	−1.21925
$423$	0	0
$424$	−23.0806	−1.12089
$425$	0	0
$426$	0	0
$427$	2.17890	0.105444
$428$	0.110625	0.00534727
$429$	0	0
$430$	0	0
$431$	6.19278	0.298296	0.149148	−	0.988815i	$-0.452347\pi$
$431$	6.19278	0.298296	0.149148	+	0.988815i	$0.452347\pi$
$432$	0	0
$433$	35.1538	1.68938	0.844692	−	0.535253i	$-0.179784\pi$
$433$	35.1538	1.68938	0.844692	+	0.535253i	$0.179784\pi$
$434$	−7.72909	−0.371008
$435$	0	0
$436$	1.21854	0.0583574
$437$	17.7930	0.851153
$438$	0	0
$439$	−20.1018	−0.959408	−0.479704	−	0.877430i	$-0.659256\pi$
$439$	−20.1018	−0.959408	−0.479704	+	0.877430i	$0.659256\pi$
$440$	0	0
$441$	0	0
$442$	−1.64156	−0.0780808
$443$	−20.8513	−0.990676	−0.495338	−	0.868700i	$-0.664956\pi$
$443$	−20.8513	−0.990676	−0.495338	+	0.868700i	$0.664956\pi$
$444$	0	0
$445$	0	0
$446$	29.3014	1.38746
$447$	0	0
$448$	−10.8470	−0.512471
$449$	39.5199	1.86506	0.932529	−	0.361094i	$-0.117597\pi$
$449$	39.5199	1.86506	0.932529	+	0.361094i	$0.117597\pi$
$450$	0	0
$451$	−5.72910	−0.269773
$452$	2.30733	0.108528
$453$	0	0
$454$	26.5092	1.24414
$455$	0	0
$456$	0	0
$457$	25.6115	1.19806	0.599028	−	0.800728i	$-0.295554\pi$
$457$	25.6115	1.19806	0.599028	+	0.800728i	$0.295554\pi$
$458$	−14.6054	−0.682465
$459$	0	0
$460$	0	0
$461$	−0.911675	−0.0424609	−0.0212305	−	0.999775i	$-0.506758\pi$
$461$	−0.911675	−0.0424609	−0.0212305	+	0.999775i	$0.506758\pi$
$462$	0	0
$463$	25.9744	1.20713	0.603567	−	0.797312i	$-0.293746\pi$
$463$	25.9744	1.20713	0.603567	+	0.797312i	$0.293746\pi$
$464$	3.64599	0.169261
$465$	0	0
$466$	−5.76231	−0.266934
$467$	15.8506	0.733479	0.366739	−	0.930324i	$-0.380474\pi$
$467$	15.8506	0.733479	0.366739	+	0.930324i	$0.380474\pi$
$468$	0	0
$469$	20.0299	0.924893
$470$	0	0
$471$	0	0
$472$	−13.0735	−0.601756
$473$	0.493048	0.0226704
$474$	0	0
$475$	0	0
$476$	−0.617284	−0.0282932
$477$	0	0
$478$	−22.8184	−1.04369
$479$	16.3087	0.745162	0.372581	−	0.928000i	$-0.378473\pi$
$479$	16.3087	0.745162	0.372581	+	0.928000i	$0.378473\pi$
$480$	0	0
$481$	−2.93666	−0.133900
$482$	−29.5955	−1.34804
$483$	0	0
$484$	1.76294	0.0801338
$485$	0	0
$486$	0	0
$487$	9.93950	0.450402	0.225201	−	0.974312i	$-0.427696\pi$
$487$	9.93950	0.450402	0.225201	+	0.974312i	$0.427696\pi$
$488$	−5.03155	−0.227767
$489$	0	0
$490$	0	0
$491$	38.0638	1.71779	0.858897	−	0.512149i	$-0.171150\pi$
$491$	38.0638	1.71779	0.858897	+	0.512149i	$0.171150\pi$
$492$	0	0
$493$	2.97132	0.133822
$494$	3.20702	0.144290
$495$	0	0
$496$	16.3781	0.735401
$497$	−11.4374	−0.513039
$498$	0	0
$499$	44.3795	1.98670	0.993349	−	0.115143i	$-0.0367327\pi$
$499$	44.3795	1.98670	0.993349	+	0.115143i	$0.0367327\pi$
$500$	0	0
$501$	0	0
$502$	−3.03908	−0.135641
$503$	3.18073	0.141822	0.0709108	−	0.997483i	$-0.477409\pi$
$503$	3.18073	0.141822	0.0709108	+	0.997483i	$0.477409\pi$
$504$	0	0
$505$	0	0
$506$	1.97210	0.0876705
$507$	0	0
$508$	−1.88995	−0.0838532
$509$	30.5882	1.35580	0.677899	−	0.735155i	$-0.262891\pi$
$509$	30.5882	1.35580	0.677899	+	0.735155i	$0.262891\pi$
$510$	0	0
$511$	0.860836	0.0380811
$512$	24.7459	1.09363
$513$	0	0
$514$	1.73030	0.0763203
$515$	0	0
$516$	0	0
$517$	1.41073	0.0620439
$518$	12.3939	0.544555
$519$	0	0
$520$	0	0
$521$	28.7858	1.26113	0.630564	−	0.776138i	$-0.282824\pi$
$521$	28.7858	1.26113	0.630564	+	0.776138i	$0.282824\pi$
$522$	0	0
$523$	27.2018	1.18945	0.594726	−	0.803928i	$-0.297261\pi$
$523$	27.2018	1.18945	0.594726	+	0.803928i	$0.297261\pi$
$524$	−1.67600	−0.0732164
$525$	0	0
$526$	−26.6645	−1.16263
$527$	13.3475	0.581425
$528$	0	0
$529$	−13.6048	−0.591513
$530$	0	0
$531$	0	0
$532$	1.20595	0.0522847
$533$	−4.91945	−0.213085
$534$	0	0
$535$	0	0
$536$	−46.2533	−1.99784
$537$	0	0
$538$	15.9029	0.685623
$539$	2.55807	0.110184
$540$	0	0
$541$	5.84584	0.251332	0.125666	−	0.992073i	$-0.459893\pi$
$541$	5.84584	0.251332	0.125666	+	0.992073i	$0.459893\pi$
$542$	13.4476	0.577625
$543$	0	0
$544$	2.74309	0.117609
$545$	0	0
$546$	0	0
$547$	−18.9324	−0.809489	−0.404745	−	0.914430i	$-0.632640\pi$
$547$	−18.9324	−0.809489	−0.404745	+	0.914430i	$0.632640\pi$
$548$	0.338003	0.0144388
$549$	0	0
$550$	0	0
$551$	−5.80491	−0.247297
$552$	0	0
$553$	−2.08174	−0.0885247
$554$	−2.04618	−0.0869340
$555$	0	0
$556$	−0.569787	−0.0241643
$557$	29.6529	1.25643	0.628217	−	0.778038i	$-0.283785\pi$
$557$	29.6529	1.25643	0.628217	+	0.778038i	$0.283785\pi$
$558$	0	0
$559$	0.423369	0.0179066
$560$	0	0
$561$	0	0
$562$	−34.3235	−1.44785
$563$	−19.3966	−0.817471	−0.408736	−	0.912653i	$-0.634030\pi$
$563$	−19.3966	−0.817471	−0.408736	+	0.912653i	$0.634030\pi$
$564$	0	0
$565$	0	0
$566$	−2.83611	−0.119211
$567$	0	0
$568$	26.4115	1.10820
$569$	−6.00700	−0.251826	−0.125913	−	0.992041i	$-0.540186\pi$
$569$	−6.00700	−0.251826	−0.125913	+	0.992041i	$0.540186\pi$
$570$	0	0
$571$	−28.1469	−1.17791	−0.588957	−	0.808165i	$-0.700461\pi$
$571$	−28.1469	−1.17791	−0.588957	+	0.808165i	$0.700461\pi$
$572$	−0.0316706	−0.00132422
$573$	0	0
$574$	20.7621	0.866592
$575$	0	0
$576$	0	0
$577$	32.9966	1.37367	0.686834	−	0.726815i	$-0.259000\pi$
$577$	32.9966	1.37367	0.686834	+	0.726815i	$0.259000\pi$
$578$	11.0731	0.460581
$579$	0	0
$580$	0	0
$581$	−19.7339	−0.818700
$582$	0	0
$583$	3.73748	0.154791
$584$	−1.98786	−0.0822581
$585$	0	0
$586$	−14.7066	−0.607526
$587$	7.81241	0.322453	0.161226	−	0.986917i	$-0.448455\pi$
$587$	7.81241	0.322453	0.161226	+	0.986917i	$0.448455\pi$
$588$	0	0
$589$	−26.0762	−1.07445
$590$	0	0
$591$	0	0
$592$	−26.2629	−1.07940
$593$	−33.5131	−1.37622	−0.688108	−	0.725608i	$-0.741559\pi$
$593$	−33.5131	−1.37622	−0.688108	+	0.725608i	$0.741559\pi$
$594$	0	0
$595$	0	0
$596$	−1.77623	−0.0727574
$597$	0	0
$598$	1.69340	0.0692482
$599$	27.1213	1.10815	0.554074	−	0.832468i	$-0.313073\pi$
$599$	27.1213	1.10815	0.554074	+	0.832468i	$0.313073\pi$
$600$	0	0
$601$	−17.4641	−0.712376	−0.356188	−	0.934414i	$-0.615924\pi$
$601$	−17.4641	−0.712376	−0.356188	+	0.934414i	$0.615924\pi$
$602$	−1.78679	−0.0728240
$603$	0	0
$604$	−1.85799	−0.0756006
$605$	0	0
$606$	0	0
$607$	−32.9273	−1.33648	−0.668239	−	0.743947i	$-0.732951\pi$
$607$	−32.9273	−1.33648	−0.668239	+	0.743947i	$0.732951\pi$
$608$	−5.35902	−0.217337
$609$	0	0
$610$	0	0
$611$	1.21136	0.0490065
$612$	0	0
$613$	−2.97228	−0.120049	−0.0600245	−	0.998197i	$-0.519118\pi$
$613$	−2.97228	−0.120049	−0.0600245	+	0.998197i	$0.519118\pi$
$614$	−22.8957	−0.923996
$615$	0	0
$616$	1.76748	0.0712138
$617$	−0.0642578	−0.00258692	−0.00129346	−	0.999999i	$-0.500412\pi$
$617$	−0.0642578	−0.00258692	−0.00129346	+	0.999999i	$0.500412\pi$
$618$	0	0
$619$	7.45588	0.299677	0.149839	−	0.988710i	$-0.452125\pi$
$619$	7.45588	0.299677	0.149839	+	0.988710i	$0.452125\pi$
$620$	0	0
$621$	0	0
$622$	14.4515	0.579454
$623$	−17.0360	−0.682535
$624$	0	0
$625$	0	0
$626$	18.7960	0.751237
$627$	0	0
$628$	2.77849	0.110874
$629$	−21.4032	−0.853400
$630$	0	0
$631$	−38.0735	−1.51568	−0.757841	−	0.652439i	$-0.773746\pi$
$631$	−38.0735	−1.51568	−0.757841	+	0.652439i	$0.773746\pi$
$632$	4.80720	0.191220
$633$	0	0
$634$	−1.81524	−0.0720924
$635$	0	0
$636$	0	0
$637$	2.19656	0.0870309
$638$	−0.643392	−0.0254721
$639$	0	0
$640$	0	0
$641$	−4.49665	−0.177607	−0.0888035	−	0.996049i	$-0.528304\pi$
$641$	−4.49665	−0.177607	−0.0888035	+	0.996049i	$0.528304\pi$
$642$	0	0
$643$	20.9607	0.826610	0.413305	−	0.910593i	$-0.364374\pi$
$643$	20.9607	0.826610	0.413305	+	0.910593i	$0.364374\pi$
$644$	0.636778	0.0250926
$645$	0	0
$646$	23.3736	0.919623
$647$	37.8398	1.48764	0.743819	−	0.668381i	$-0.233013\pi$
$647$	37.8398	1.48764	0.743819	+	0.668381i	$0.233013\pi$
$648$	0	0
$649$	2.11701	0.0831000
$650$	0	0
$651$	0	0
$652$	1.27384	0.0498873
$653$	−48.0703	−1.88114	−0.940568	−	0.339606i	$-0.889706\pi$
$653$	−48.0703	−1.88114	−0.940568	+	0.339606i	$0.889706\pi$
$654$	0	0
$655$	0	0
$656$	−43.9954	−1.71773
$657$	0	0
$658$	−5.11244	−0.199304
$659$	26.0861	1.01617	0.508084	−	0.861307i	$-0.330354\pi$
$659$	26.0861	1.01617	0.508084	+	0.861307i	$0.330354\pi$
$660$	0	0
$661$	5.00120	0.194524	0.0972620	−	0.995259i	$-0.468992\pi$
$661$	5.00120	0.194524	0.0972620	+	0.995259i	$0.468992\pi$
$662$	4.06636	0.158044
$663$	0	0
$664$	45.5699	1.76845
$665$	0	0
$666$	0	0
$667$	−3.06516	−0.118683
$668$	−0.337695	−0.0130658
$669$	0	0
$670$	0	0
$671$	0.814768	0.0314538
$672$	0	0
$673$	−26.1113	−1.00652	−0.503258	−	0.864136i	$-0.667866\pi$
$673$	−26.1113	−1.00652	−0.503258	+	0.864136i	$0.667866\pi$
$674$	−32.8976	−1.26717
$675$	0	0
$676$	2.09987	0.0807643
$677$	−32.0992	−1.23367	−0.616836	−	0.787092i	$-0.711586\pi$
$677$	−32.0992	−1.23367	−0.616836	+	0.787092i	$0.711586\pi$
$678$	0	0
$679$	−9.02410	−0.346313
$680$	0	0
$681$	0	0
$682$	−2.89018	−0.110671
$683$	4.70468	0.180020	0.0900098	−	0.995941i	$-0.471310\pi$
$683$	4.70468	0.180020	0.0900098	+	0.995941i	$0.471310\pi$
$684$	0	0
$685$	0	0
$686$	−21.3145	−0.813793
$687$	0	0
$688$	3.78625	0.144349
$689$	3.20929	0.122264
$690$	0	0
$691$	0.881335	0.0335276	0.0167638	−	0.999859i	$-0.494664\pi$
$691$	0.881335	0.0335276	0.0167638	+	0.999859i	$0.494664\pi$
$692$	−0.0854295	−0.00324754
$693$	0	0
$694$	−19.0791	−0.724233
$695$	0	0
$696$	0	0
$697$	−35.8543	−1.35808
$698$	24.3437	0.921423
$699$	0	0
$700$	0	0
$701$	−45.1606	−1.70569	−0.852847	−	0.522161i	$-0.825126\pi$
$701$	−45.1606	−1.70569	−0.852847	+	0.522161i	$0.825126\pi$
$702$	0	0
$703$	41.8142	1.57705
$704$	−4.05607	−0.152869
$705$	0	0
$706$	−12.8610	−0.484031
$707$	−5.94764	−0.223684
$708$	0	0
$709$	18.8061	0.706277	0.353139	−	0.935571i	$-0.385114\pi$
$709$	18.8061	0.706277	0.353139	+	0.935571i	$0.385114\pi$
$710$	0	0
$711$	0	0
$712$	39.3399	1.47433
$713$	−13.7690	−0.515653
$714$	0	0
$715$	0	0
$716$	−1.71482	−0.0640857
$717$	0	0
$718$	5.54425	0.206910
$719$	−18.7384	−0.698824	−0.349412	−	0.936969i	$-0.613619\pi$
$719$	−18.7384	−0.698824	−0.349412	+	0.936969i	$0.613619\pi$
$720$	0	0
$721$	19.7031	0.733782
$722$	−19.9163	−0.741207
$723$	0	0
$724$	−2.00801	−0.0746271
$725$	0	0
$726$	0	0
$727$	−20.2155	−0.749752	−0.374876	−	0.927075i	$-0.622315\pi$
$727$	−20.2155	−0.749752	−0.374876	+	0.927075i	$0.622315\pi$
$728$	1.51769	0.0562495
$729$	0	0
$730$	0	0
$731$	3.08563	0.114126
$732$	0	0
$733$	45.1963	1.66936	0.834682	−	0.550732i	$-0.185651\pi$
$733$	45.1963	1.66936	0.834682	+	0.550732i	$0.185651\pi$
$734$	−25.5284	−0.942269
$735$	0	0
$736$	−2.82972	−0.104305
$737$	7.48988	0.275893
$738$	0	0
$739$	42.2473	1.55409	0.777046	−	0.629443i	$-0.216717\pi$
$739$	42.2473	1.55409	0.777046	+	0.629443i	$0.216717\pi$
$740$	0	0
$741$	0	0
$742$	−13.5445	−0.497235
$743$	38.5919	1.41580	0.707900	−	0.706313i	$-0.249643\pi$
$743$	38.5919	1.41580	0.707900	+	0.706313i	$0.249643\pi$
$744$	0	0
$745$	0	0
$746$	−9.81113	−0.359211
$747$	0	0
$748$	−0.230825	−0.00843978
$749$	0.858447	0.0313670
$750$	0	0
$751$	−1.50634	−0.0549670	−0.0274835	−	0.999622i	$-0.508749\pi$
$751$	−1.50634	−0.0549670	−0.0274835	+	0.999622i	$0.508749\pi$
$752$	10.8334	0.395053
$753$	0	0
$754$	−0.552466	−0.0201196
$755$	0	0
$756$	0	0
$757$	17.3890	0.632016	0.316008	−	0.948757i	$-0.397657\pi$
$757$	17.3890	0.632016	0.316008	+	0.948757i	$0.397657\pi$
$758$	−27.4306	−0.996324
$759$	0	0
$760$	0	0
$761$	9.21457	0.334028	0.167014	−	0.985955i	$-0.446587\pi$
$761$	9.21457	0.334028	0.167014	+	0.985955i	$0.446587\pi$
$762$	0	0
$763$	9.45581	0.342323
$764$	−1.58626	−0.0573890
$765$	0	0
$766$	5.88399	0.212597
$767$	1.81783	0.0656381
$768$	0	0
$769$	25.6939	0.926547	0.463273	−	0.886215i	$-0.346675\pi$
$769$	25.6939	0.926547	0.463273	+	0.886215i	$0.346675\pi$
$770$	0	0
$771$	0	0
$772$	1.81384	0.0652816
$773$	42.9070	1.54326	0.771628	−	0.636074i	$-0.219443\pi$
$773$	42.9070	1.54326	0.771628	+	0.636074i	$0.219443\pi$
$774$	0	0
$775$	0	0
$776$	20.8386	0.748062
$777$	0	0
$778$	0.689338	0.0247140
$779$	70.0466	2.50968
$780$	0	0
$781$	−4.27687	−0.153038
$782$	12.3420	0.441347
$783$	0	0
$784$	19.6442	0.701578
$785$	0	0
$786$	0	0
$787$	22.2120	0.791772	0.395886	−	0.918300i	$-0.370438\pi$
$787$	22.2120	0.791772	0.395886	+	0.918300i	$0.370438\pi$
$788$	−0.327947	−0.0116826
$789$	0	0
$790$	0	0
$791$	17.9048	0.636622
$792$	0	0
$793$	0.699623	0.0248443
$794$	−42.2984	−1.50111
$795$	0	0
$796$	2.96431	0.105067
$797$	−46.2386	−1.63785	−0.818927	−	0.573897i	$-0.805431\pi$
$797$	−46.2386	−1.63785	−0.818927	+	0.573897i	$0.805431\pi$
$798$	0	0
$799$	8.82875	0.312339
$800$	0	0
$801$	0	0
$802$	7.60171	0.268426
$803$	0.321897	0.0113595
$804$	0	0
$805$	0	0
$806$	−2.48174	−0.0874154
$807$	0	0
$808$	13.7344	0.483175
$809$	23.9257	0.841184	0.420592	−	0.907250i	$-0.361822\pi$
$809$	23.9257	0.841184	0.420592	+	0.907250i	$0.361822\pi$
$810$	0	0
$811$	−52.0210	−1.82670	−0.913352	−	0.407170i	$-0.866516\pi$
$811$	−52.0210	−1.82670	−0.913352	+	0.407170i	$0.866516\pi$
$812$	−0.207747	−0.00729050
$813$	0	0
$814$	4.63451	0.162440
$815$	0	0
$816$	0	0
$817$	−6.02823	−0.210901
$818$	13.1188	0.458686
$819$	0	0
$820$	0	0
$821$	15.0444	0.525052	0.262526	−	0.964925i	$-0.415444\pi$
$821$	15.0444	0.525052	0.262526	+	0.964925i	$0.415444\pi$
$822$	0	0
$823$	−44.8652	−1.56390	−0.781952	−	0.623339i	$-0.785776\pi$
$823$	−44.8652	−1.56390	−0.781952	+	0.623339i	$0.785776\pi$
$824$	−45.4987	−1.58502
$825$	0	0
$826$	−7.67198	−0.266942
$827$	−41.5569	−1.44508	−0.722538	−	0.691331i	$-0.757025\pi$
$827$	−41.5569	−1.44508	−0.722538	+	0.691331i	$0.757025\pi$
$828$	0	0
$829$	−25.3645	−0.880946	−0.440473	−	0.897766i	$-0.645189\pi$
$829$	−25.3645	−0.880946	−0.440473	+	0.897766i	$0.645189\pi$
$830$	0	0
$831$	0	0
$832$	−3.48286	−0.120746
$833$	16.0092	0.554684
$834$	0	0
$835$	0	0
$836$	0.450949	0.0155964
$837$	0	0
$838$	24.7702	0.855672
$839$	−3.34480	−0.115475	−0.0577376	−	0.998332i	$-0.518389\pi$
$839$	−3.34480	−0.115475	−0.0577376	+	0.998332i	$0.518389\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−35.3035	−1.21664
$843$	0	0
$844$	−3.02417	−0.104096
$845$	0	0
$846$	0	0
$847$	13.6804	0.470063
$848$	28.7012	0.985603
$849$	0	0
$850$	0	0
$851$	22.0791	0.756862
$852$	0	0
$853$	45.8873	1.57115	0.785575	−	0.618766i	$-0.212367\pi$
$853$	45.8873	1.57115	0.785575	+	0.618766i	$0.212367\pi$
$854$	−2.95269	−0.101039
$855$	0	0
$856$	−1.98234	−0.0677550
$857$	51.8755	1.77203	0.886017	−	0.463652i	$-0.153461\pi$
$857$	51.8755	1.77203	0.886017	+	0.463652i	$0.153461\pi$
$858$	0	0
$859$	31.7698	1.08397	0.541987	−	0.840387i	$-0.317672\pi$
$859$	31.7698	1.08397	0.541987	+	0.840387i	$0.317672\pi$
$860$	0	0
$861$	0	0
$862$	−8.39203	−0.285834
$863$	39.9472	1.35982	0.679909	−	0.733296i	$-0.262019\pi$
$863$	39.9472	1.35982	0.679909	+	0.733296i	$0.262019\pi$
$864$	0	0
$865$	0	0
$866$	−47.6380	−1.61880
$867$	0	0
$868$	−0.933222	−0.0316756
$869$	−0.778438	−0.0264067
$870$	0	0
$871$	6.43139	0.217920
$872$	−21.8355	−0.739444
$873$	0	0
$874$	−24.1118	−0.815594
$875$	0	0
$876$	0	0
$877$	−44.8875	−1.51574	−0.757872	−	0.652403i	$-0.773761\pi$
$877$	−44.8875	−1.51574	−0.757872	+	0.652403i	$0.773761\pi$
$878$	27.2406	0.919326
$879$	0	0
$880$	0	0
$881$	−3.90189	−0.131458	−0.0657290	−	0.997838i	$-0.520937\pi$
$881$	−3.90189	−0.131458	−0.0657290	+	0.997838i	$0.520937\pi$
$882$	0	0
$883$	52.7256	1.77436	0.887178	−	0.461427i	$-0.152662\pi$
$883$	52.7256	1.77436	0.887178	+	0.461427i	$0.152662\pi$
$884$	−0.198204	−0.00666632
$885$	0	0
$886$	28.2563	0.949288
$887$	20.7833	0.697834	0.348917	−	0.937154i	$-0.386550\pi$
$887$	20.7833	0.697834	0.348917	+	0.937154i	$0.386550\pi$
$888$	0	0
$889$	−14.6660	−0.491881
$890$	0	0
$891$	0	0
$892$	3.53790	0.118458
$893$	−17.2482	−0.577190
$894$	0	0
$895$	0	0
$896$	12.3547	0.412743
$897$	0	0
$898$	−53.5546	−1.78714
$899$	4.49210	0.149820
$900$	0	0
$901$	23.3902	0.779241
$902$	7.76368	0.258502
$903$	0	0
$904$	−41.3461	−1.37515
$905$	0	0
$906$	0	0
$907$	−26.2116	−0.870342	−0.435171	−	0.900348i	$-0.643312\pi$
$907$	−26.2116	−0.870342	−0.435171	+	0.900348i	$0.643312\pi$
$908$	3.20076	0.106221
$909$	0	0
$910$	0	0
$911$	31.4264	1.04120	0.520601	−	0.853800i	$-0.325708\pi$
$911$	31.4264	1.04120	0.520601	+	0.853800i	$0.325708\pi$
$912$	0	0
$913$	−7.37921	−0.244216
$914$	−34.7070	−1.14800
$915$	0	0
$916$	−1.76348	−0.0582669
$917$	−13.0057	−0.429486
$918$	0	0
$919$	−21.0083	−0.693000	−0.346500	−	0.938050i	$-0.612630\pi$
$919$	−21.0083	−0.693000	−0.346500	+	0.938050i	$0.612630\pi$
$920$	0	0
$921$	0	0
$922$	1.23544	0.0406870
$923$	−3.67245	−0.120880
$924$	0	0
$925$	0	0
$926$	−35.1988	−1.15670
$927$	0	0
$928$	0.923188	0.0303051
$929$	37.3920	1.22679	0.613395	−	0.789776i	$-0.289803\pi$
$929$	37.3920	1.22679	0.613395	+	0.789776i	$0.289803\pi$
$930$	0	0
$931$	−31.2762	−1.02504
$932$	−0.695750	−0.0227901
$933$	0	0
$934$	−21.4796	−0.702836
$935$	0	0
$936$	0	0
$937$	−13.8377	−0.452057	−0.226029	−	0.974121i	$-0.572574\pi$
$937$	−13.8377	−0.452057	−0.226029	+	0.974121i	$0.572574\pi$
$938$	−27.1431	−0.886253
$939$	0	0
$940$	0	0
$941$	14.5708	0.474995	0.237497	−	0.971388i	$-0.423673\pi$
$941$	14.5708	0.474995	0.237497	+	0.971388i	$0.423673\pi$
$942$	0	0
$943$	36.9867	1.20445
$944$	16.2571	0.529125
$945$	0	0
$946$	−0.668144	−0.0217232
$947$	43.4267	1.41118	0.705589	−	0.708622i	$-0.250683\pi$
$947$	43.4267	1.41118	0.705589	+	0.708622i	$0.250683\pi$
$948$	0	0
$949$	0.276406	0.00897252
$950$	0	0
$951$	0	0
$952$	11.0614	0.358501
$953$	51.0500	1.65367	0.826836	−	0.562442i	$-0.190138\pi$
$953$	51.0500	1.65367	0.826836	+	0.562442i	$0.190138\pi$
$954$	0	0
$955$	0	0
$956$	−2.75512	−0.0891071
$957$	0	0
$958$	−22.1004	−0.714030
$959$	2.62289	0.0846976
$960$	0	0
$961$	−10.8210	−0.349065
$962$	3.97955	0.128306
$963$	0	0
$964$	−3.57340	−0.115091
$965$	0	0
$966$	0	0
$967$	56.0658	1.80295	0.901477	−	0.432828i	$-0.142484\pi$
$967$	56.0658	1.80295	0.901477	+	0.432828i	$0.142484\pi$
$968$	−31.5910	−1.01537
$969$	0	0
$970$	0	0
$971$	−6.58898	−0.211450	−0.105725	−	0.994395i	$-0.533716\pi$
$971$	−6.58898	−0.211450	−0.105725	+	0.994395i	$0.533716\pi$
$972$	0	0
$973$	−4.42152	−0.141748
$974$	−13.4693	−0.431585
$975$	0	0
$976$	6.25683	0.200276
$977$	29.3327	0.938437	0.469218	−	0.883082i	$-0.344536\pi$
$977$	29.3327	0.938437	0.469218	+	0.883082i	$0.344536\pi$
$978$	0	0
$979$	−6.37039	−0.203599
$980$	0	0
$981$	0	0
$982$	−51.5814	−1.64603
$983$	13.6719	0.436064	0.218032	−	0.975942i	$-0.430036\pi$
$983$	13.6719	0.436064	0.218032	+	0.975942i	$0.430036\pi$
$984$	0	0
$985$	0	0
$986$	−4.02653	−0.128231
$987$	0	0
$988$	0.387220	0.0123191
$989$	−3.18308	−0.101216
$990$	0	0
$991$	19.7210	0.626459	0.313230	−	0.949677i	$-0.398589\pi$
$991$	19.7210	0.626459	0.313230	+	0.949677i	$0.398589\pi$
$992$	4.14705	0.131669
$993$	0	0
$994$	15.4992	0.491606
$995$	0	0
$996$	0	0
$997$	53.4618	1.69315	0.846576	−	0.532268i	$-0.178660\pi$
$997$	53.4618	1.69315	0.846576	+	0.532268i	$0.178660\pi$
$998$	−60.1400	−1.90370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.cf.1.4		12
3.2	odd	2		6525.2.a.ce.1.9		12
5.2	odd	4		1305.2.c.l.784.4	yes	12
5.3	odd	4		1305.2.c.l.784.9	yes	12
5.4	even	2	inner	6525.2.a.cf.1.9		12
15.2	even	4		1305.2.c.k.784.9	yes	12
15.8	even	4		1305.2.c.k.784.4	✓	12
15.14	odd	2		6525.2.a.ce.1.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.c.k.784.4	✓	12	15.8	even	4
1305.2.c.k.784.9	yes	12	15.2	even	4
1305.2.c.l.784.4	yes	12	5.2	odd	4
1305.2.c.l.784.9	yes	12	5.3	odd	4
6525.2.a.ce.1.4		12	15.14	odd	2
6525.2.a.ce.1.9		12	3.2	odd	2
6525.2.a.cf.1.4		12	1.1	even	1	trivial
6525.2.a.cf.1.9		12	5.4	even	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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-1.35513 q^{2} -0.163621 q^{4} -1.26969 q^{7} +2.93199 q^{8} -0.474782 q^{11} -0.407685 q^{13} +1.72060 q^{14} -3.64599 q^{16} -2.97132 q^{17} +5.80491 q^{19} +0.643392 q^{22} +3.06516 q^{23} +0.552466 q^{26} +0.207747 q^{28} -1.00000 q^{29} -4.49210 q^{31} -0.923188 q^{32} +4.02653 q^{34} +7.20325 q^{37} -7.86641 q^{38} +12.0668 q^{41} -1.03847 q^{43} +0.0776841 q^{44} -4.15369 q^{46} -2.97132 q^{47} -5.38789 q^{49} +0.0667056 q^{52} -7.87199 q^{53} -3.72271 q^{56} +1.35513 q^{58} -4.45891 q^{59} -1.71609 q^{61} +6.08739 q^{62} +8.54301 q^{64} -15.7754 q^{67} +0.486169 q^{68} +9.00806 q^{71} -0.677990 q^{73} -9.76134 q^{74} -0.949802 q^{76} +0.602826 q^{77} +1.63957 q^{79} -16.3521 q^{82} +15.5423 q^{83} +1.40726 q^{86} -1.39206 q^{88} +13.4175 q^{89} +0.517633 q^{91} -0.501523 q^{92} +4.02653 q^{94} +7.10733 q^{97} +7.30129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{4} + 12 q^{11} + 16 q^{14} + 16 q^{16} - 20 q^{19} + 56 q^{26} - 12 q^{29} - 16 q^{31} + 4 q^{34} + 32 q^{41} + 68 q^{44} + 20 q^{46} - 4 q^{49} + 76 q^{56} + 44 q^{59} - 52 q^{61} + 36 q^{64}+ \cdots + 4 q^{94}+O(q^{100}) \) 12 * q + 16 * q^4 + 12 * q^11 + 16 * q^14 + 16 * q^16 - 20 * q^19 + 56 * q^26 - 12 * q^29 - 16 * q^31 + 4 * q^34 + 32 * q^41 + 68 * q^44 + 20 * q^46 - 4 * q^49 + 76 * q^56 + 44 * q^59 - 52 * q^61 + 36 * q^64 + 20 * q^71 + 20 * q^74 - 28 * q^76 - 4 * q^79 + 36 * q^86 + 68 * q^89 + 48 * q^91 + 4 * q^94

Introduction

L-functions

Modular forms

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