LMFDB - Embedded newform 6354.2.a.bl.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6354,2,Mod(1,6354)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6354.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6354 = 2 \cdot 3^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6354.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$50.7369454443$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 26x^{8} + 44x^{7} + 239x^{6} - 340x^{5} - 946x^{4} + 1056x^{3} + 1584x^{2} - 1024x - 1024 $ x^10 - 2x^9 - 26x^8 + 44x^7 + 239x^6 - 340x^5 - 946x^4 + 1056x^3 + 1584x^2 - 1024*x - 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2118)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.20839$ of defining polynomial
Character	$\chi$	$=$	6354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.25462 q^{5} +3.30920 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.25462 q^{5} +3.30920 q^{7} +1.00000 q^{8} -1.25462 q^{10} +3.08207 q^{11} +1.64133 q^{13} +3.30920 q^{14} +1.00000 q^{16} +7.25957 q^{17} +5.93727 q^{19} -1.25462 q^{20} +3.08207 q^{22} +9.00065 q^{23} -3.42593 q^{25} +1.64133 q^{26} +3.30920 q^{28} +4.86856 q^{29} -10.1895 q^{31} +1.00000 q^{32} +7.25957 q^{34} -4.15180 q^{35} -2.70148 q^{37} +5.93727 q^{38} -1.25462 q^{40} -6.19215 q^{41} -2.93581 q^{43} +3.08207 q^{44} +9.00065 q^{46} -3.36158 q^{47} +3.95083 q^{49} -3.42593 q^{50} +1.64133 q^{52} +2.04867 q^{53} -3.86683 q^{55} +3.30920 q^{56} +4.86856 q^{58} -5.98049 q^{59} -10.7015 q^{61} -10.1895 q^{62} +1.00000 q^{64} -2.05925 q^{65} -6.98394 q^{67} +7.25957 q^{68} -4.15180 q^{70} +10.0260 q^{71} -10.7814 q^{73} -2.70148 q^{74} +5.93727 q^{76} +10.1992 q^{77} +16.5082 q^{79} -1.25462 q^{80} -6.19215 q^{82} -9.52589 q^{83} -9.10801 q^{85} -2.93581 q^{86} +3.08207 q^{88} -11.9006 q^{89} +5.43151 q^{91} +9.00065 q^{92} -3.36158 q^{94} -7.44903 q^{95} +16.1700 q^{97} +3.95083 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8}+O(q^{10}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8	$ 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8} + 10 q^{10} + 2 q^{11} - 4 q^{13} + 10 q^{16} + 10 q^{17} + 14 q^{19} + 10 q^{20} + 2 q^{22} + 8 q^{23} + 20 q^{25} - 4 q^{26} + 14 q^{29} + 12 q^{31} + 10 q^{32} + 10 q^{34} + 6 q^{35} - 4 q^{37} + 14 q^{38} + 10 q^{40} + 2 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 8 q^{47} + 32 q^{49} + 20 q^{50} - 4 q^{52} + 36 q^{53} + 10 q^{55} + 14 q^{58} + 10 q^{59} - 6 q^{61} + 12 q^{62} + 10 q^{64} + 26 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} + 12 q^{71} - 8 q^{73} - 4 q^{74} + 14 q^{76} + 56 q^{77} + 28 q^{79} + 10 q^{80} + 2 q^{82} + 22 q^{83} - 4 q^{85} + 10 q^{86} + 2 q^{88} + 28 q^{89} + 8 q^{92} + 8 q^{94} + 16 q^{95} - 8 q^{97} + 32 q^{98}+O(q^{100}) $ 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8 + 10 * q^10 + 2 * q^11 - 4 * q^13 + 10 * q^16 + 10 * q^17 + 14 * q^19 + 10 * q^20 + 2 * q^22 + 8 * q^23 + 20 * q^25 - 4 * q^26 + 14 * q^29 + 12 * q^31 + 10 * q^32 + 10 * q^34 + 6 * q^35 - 4 * q^37 + 14 * q^38 + 10 * q^40 + 2 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 8 * q^47 + 32 * q^49 + 20 * q^50 - 4 * q^52 + 36 * q^53 + 10 * q^55 + 14 * q^58 + 10 * q^59 - 6 * q^61 + 12 * q^62 + 10 * q^64 + 26 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 + 12 * q^71 - 8 * q^73 - 4 * q^74 + 14 * q^76 + 56 * q^77 + 28 * q^79 + 10 * q^80 + 2 * q^82 + 22 * q^83 - 4 * q^85 + 10 * q^86 + 2 * q^88 + 28 * q^89 + 8 * q^92 + 8 * q^94 + 16 * q^95 - 8 * q^97 + 32 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.25462	−0.561084	−0.280542	−	0.959842i	$-0.590514\pi$
$5$	−1.25462	−0.561084	−0.280542	+	0.959842i	$0.590514\pi$
$6$	0	0
$7$	3.30920	1.25076	0.625381	−	0.780320i	$-0.284944\pi$
$7$	3.30920	1.25076	0.625381	+	0.780320i	$0.284944\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.25462	−0.396746
$11$	3.08207	0.929278	0.464639	−	0.885500i	$-0.346184\pi$
$11$	3.08207	0.929278	0.464639	+	0.885500i	$0.346184\pi$
$12$	0	0
$13$	1.64133	0.455224	0.227612	−	0.973752i	$-0.426908\pi$
$13$	1.64133	0.455224	0.227612	+	0.973752i	$0.426908\pi$
$14$	3.30920	0.884422
$15$	0	0
$16$	1.00000	0.250000
$17$	7.25957	1.76070	0.880352	−	0.474320i	$-0.157306\pi$
$17$	7.25957	1.76070	0.880352	+	0.474320i	$0.157306\pi$
$18$	0	0
$19$	5.93727	1.36210	0.681052	−	0.732235i	$-0.261523\pi$
$19$	5.93727	1.36210	0.681052	+	0.732235i	$0.261523\pi$
$20$	−1.25462	−0.280542
$21$	0	0
$22$	3.08207	0.657099
$23$	9.00065	1.87677	0.938383	−	0.345597i	$-0.112324\pi$
$23$	9.00065	1.87677	0.938383	+	0.345597i	$0.112324\pi$
$24$	0	0
$25$	−3.42593	−0.685185
$26$	1.64133	0.321892
$27$	0	0
$28$	3.30920	0.625381
$29$	4.86856	0.904069	0.452034	−	0.892001i	$-0.350698\pi$
$29$	4.86856	0.904069	0.452034	+	0.892001i	$0.350698\pi$
$30$	0	0
$31$	−10.1895	−1.83008	−0.915040	−	0.403363i	$-0.867841\pi$
$31$	−10.1895	−1.83008	−0.915040	+	0.403363i	$0.867841\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.25957	1.24501
$35$	−4.15180	−0.701782
$36$	0	0
$37$	−2.70148	−0.444121	−0.222060	−	0.975033i	$-0.571278\pi$
$37$	−2.70148	−0.444121	−0.222060	+	0.975033i	$0.571278\pi$
$38$	5.93727	0.963153
$39$	0	0
$40$	−1.25462	−0.198373
$41$	−6.19215	−0.967051	−0.483525	−	0.875330i	$-0.660644\pi$
$41$	−6.19215	−0.967051	−0.483525	+	0.875330i	$0.660644\pi$
$42$	0	0
$43$	−2.93581	−0.447706	−0.223853	−	0.974623i	$-0.571864\pi$
$43$	−2.93581	−0.447706	−0.223853	+	0.974623i	$0.571864\pi$
$44$	3.08207	0.464639
$45$	0	0
$46$	9.00065	1.32707
$47$	−3.36158	−0.490337	−0.245169	−	0.969480i	$-0.578843\pi$
$47$	−3.36158	−0.490337	−0.245169	+	0.969480i	$0.578843\pi$
$48$	0	0
$49$	3.95083	0.564405
$50$	−3.42593	−0.484499
$51$	0	0
$52$	1.64133	0.227612
$53$	2.04867	0.281407	0.140703	−	0.990052i	$-0.455064\pi$
$53$	2.04867	0.281407	0.140703	+	0.990052i	$0.455064\pi$
$54$	0	0
$55$	−3.86683	−0.521403
$56$	3.30920	0.442211
$57$	0	0
$58$	4.86856	0.639273
$59$	−5.98049	−0.778593	−0.389297	−	0.921112i	$-0.627282\pi$
$59$	−5.98049	−0.778593	−0.389297	+	0.921112i	$0.627282\pi$
$60$	0	0
$61$	−10.7015	−1.37018	−0.685091	−	0.728458i	$-0.740237\pi$
$61$	−10.7015	−1.37018	−0.685091	+	0.728458i	$0.740237\pi$
$62$	−10.1895	−1.29406
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.05925	−0.255419
$66$	0	0
$67$	−6.98394	−0.853224	−0.426612	−	0.904435i	$-0.640293\pi$
$67$	−6.98394	−0.853224	−0.426612	+	0.904435i	$0.640293\pi$
$68$	7.25957	0.880352
$69$	0	0
$70$	−4.15180	−0.496235
$71$	10.0260	1.18987	0.594934	−	0.803775i	$-0.297178\pi$
$71$	10.0260	1.18987	0.594934	+	0.803775i	$0.297178\pi$
$72$	0	0
$73$	−10.7814	−1.26187	−0.630935	−	0.775835i	$-0.717329\pi$
$73$	−10.7814	−1.26187	−0.630935	+	0.775835i	$0.717329\pi$
$74$	−2.70148	−0.314041
$75$	0	0
$76$	5.93727	0.681052
$77$	10.1992	1.16231
$78$	0	0
$79$	16.5082	1.85732	0.928658	−	0.370936i	$-0.120963\pi$
$79$	16.5082	1.85732	0.928658	+	0.370936i	$0.120963\pi$
$80$	−1.25462	−0.140271
$81$	0	0
$82$	−6.19215	−0.683808
$83$	−9.52589	−1.04560	−0.522801	−	0.852455i	$-0.675113\pi$
$83$	−9.52589	−1.04560	−0.522801	+	0.852455i	$0.675113\pi$
$84$	0	0
$85$	−9.10801	−0.987903
$86$	−2.93581	−0.316576
$87$	0	0
$88$	3.08207	0.328550
$89$	−11.9006	−1.26147	−0.630733	−	0.776000i	$-0.717246\pi$
$89$	−11.9006	−1.26147	−0.630733	+	0.776000i	$0.717246\pi$
$90$	0	0
$91$	5.43151	0.569377
$92$	9.00065	0.938383
$93$	0	0
$94$	−3.36158	−0.346721
$95$	−7.44903	−0.764254
$96$	0	0
$97$	16.1700	1.64182	0.820909	−	0.571059i	$-0.193467\pi$
$97$	16.1700	1.64182	0.820909	+	0.571059i	$0.193467\pi$
$98$	3.95083	0.399095
$99$	0	0
$100$	−3.42593	−0.342593
$101$	−5.82543	−0.579652	−0.289826	−	0.957079i	$-0.593597\pi$
$101$	−5.82543	−0.579652	−0.289826	+	0.957079i	$0.593597\pi$
$102$	0	0
$103$	8.56357	0.843794	0.421897	−	0.906644i	$-0.361364\pi$
$103$	8.56357	0.843794	0.421897	+	0.906644i	$0.361364\pi$
$104$	1.64133	0.160946
$105$	0	0
$106$	2.04867	0.198985
$107$	8.32880	0.805176	0.402588	−	0.915381i	$-0.368111\pi$
$107$	8.32880	0.805176	0.402588	+	0.915381i	$0.368111\pi$
$108$	0	0
$109$	−6.82558	−0.653772	−0.326886	−	0.945064i	$-0.605999\pi$
$109$	−6.82558	−0.653772	−0.326886	+	0.945064i	$0.605999\pi$
$110$	−3.86683	−0.368688
$111$	0	0
$112$	3.30920	0.312690
$113$	4.84290	0.455581	0.227791	−	0.973710i	$-0.426850\pi$
$113$	4.84290	0.455581	0.227791	+	0.973710i	$0.426850\pi$
$114$	0	0
$115$	−11.2924	−1.05302
$116$	4.86856	0.452034
$117$	0	0
$118$	−5.98049	−0.550549
$119$	24.0234	2.20222
$120$	0	0
$121$	−1.50086	−0.136442
$122$	−10.7015	−0.968864
$123$	0	0
$124$	−10.1895	−0.915040
$125$	10.5713	0.945530
$126$	0	0
$127$	−7.34600	−0.651852	−0.325926	−	0.945395i	$-0.605676\pi$
$127$	−7.34600	−0.651852	−0.325926	+	0.945395i	$0.605676\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.05925	−0.180608
$131$	0.626421	0.0547307	0.0273653	−	0.999625i	$-0.491288\pi$
$131$	0.626421	0.0547307	0.0273653	+	0.999625i	$0.491288\pi$
$132$	0	0
$133$	19.6476	1.70367
$134$	−6.98394	−0.603320
$135$	0	0
$136$	7.25957	0.622503
$137$	1.65755	0.141614	0.0708071	−	0.997490i	$-0.477443\pi$
$137$	1.65755	0.141614	0.0708071	+	0.997490i	$0.477443\pi$
$138$	0	0
$139$	3.40771	0.289038	0.144519	−	0.989502i	$-0.453836\pi$
$139$	3.40771	0.289038	0.144519	+	0.989502i	$0.453836\pi$
$140$	−4.15180	−0.350891
$141$	0	0
$142$	10.0260	0.841364
$143$	5.05871	0.423030
$144$	0	0
$145$	−6.10820	−0.507258
$146$	−10.7814	−0.892277
$147$	0	0
$148$	−2.70148	−0.222060
$149$	13.1176	1.07464	0.537319	−	0.843379i	$-0.319437\pi$
$149$	13.1176	1.07464	0.537319	+	0.843379i	$0.319437\pi$
$150$	0	0
$151$	−21.5801	−1.75617	−0.878084	−	0.478507i	$-0.841178\pi$
$151$	−21.5801	−1.75617	−0.878084	+	0.478507i	$0.841178\pi$
$152$	5.93727	0.481576
$153$	0	0
$154$	10.1992	0.821874
$155$	12.7839	1.02683
$156$	0	0
$157$	−2.59977	−0.207484	−0.103742	−	0.994604i	$-0.533082\pi$
$157$	−2.59977	−0.207484	−0.103742	+	0.994604i	$0.533082\pi$
$158$	16.5082	1.31332
$159$	0	0
$160$	−1.25462	−0.0991865
$161$	29.7850	2.34739
$162$	0	0
$163$	22.8231	1.78764	0.893822	−	0.448422i	$-0.148014\pi$
$163$	22.8231	1.78764	0.893822	+	0.448422i	$0.148014\pi$
$164$	−6.19215	−0.483525
$165$	0	0
$166$	−9.52589	−0.739352
$167$	−0.489625	−0.0378883	−0.0189442	−	0.999821i	$-0.506030\pi$
$167$	−0.489625	−0.0378883	−0.0189442	+	0.999821i	$0.506030\pi$
$168$	0	0
$169$	−10.3060	−0.792771
$170$	−9.10801	−0.698553
$171$	0	0
$172$	−2.93581	−0.223853
$173$	17.8205	1.35487	0.677435	−	0.735583i	$-0.263092\pi$
$173$	17.8205	1.35487	0.677435	+	0.735583i	$0.263092\pi$
$174$	0	0
$175$	−11.3371	−0.857003
$176$	3.08207	0.232320
$177$	0	0
$178$	−11.9006	−0.891991
$179$	−3.36324	−0.251381	−0.125690	−	0.992070i	$-0.540115\pi$
$179$	−3.36324	−0.251381	−0.125690	+	0.992070i	$0.540115\pi$
$180$	0	0
$181$	1.73271	0.128791	0.0643956	−	0.997924i	$-0.479488\pi$
$181$	1.73271	0.128791	0.0643956	+	0.997924i	$0.479488\pi$
$182$	5.43151	0.402610
$183$	0	0
$184$	9.00065	0.663537
$185$	3.38933	0.249189
$186$	0	0
$187$	22.3745	1.63618
$188$	−3.36158	−0.245169
$189$	0	0
$190$	−7.44903	−0.540409
$191$	6.97386	0.504611	0.252305	−	0.967648i	$-0.418811\pi$
$191$	6.97386	0.504611	0.252305	+	0.967648i	$0.418811\pi$
$192$	0	0
$193$	−16.8029	−1.20950	−0.604749	−	0.796416i	$-0.706726\pi$
$193$	−16.8029	−1.20950	−0.604749	+	0.796416i	$0.706726\pi$
$194$	16.1700	1.16094
$195$	0	0
$196$	3.95083	0.282202
$197$	−5.02522	−0.358032	−0.179016	−	0.983846i	$-0.557291\pi$
$197$	−5.02522	−0.358032	−0.179016	+	0.983846i	$0.557291\pi$
$198$	0	0
$199$	1.79520	0.127258	0.0636292	−	0.997974i	$-0.479732\pi$
$199$	1.79520	0.127258	0.0636292	+	0.997974i	$0.479732\pi$
$200$	−3.42593	−0.242249
$201$	0	0
$202$	−5.82543	−0.409876
$203$	16.1111	1.13077
$204$	0	0
$205$	7.76880	0.542597
$206$	8.56357	0.596652
$207$	0	0
$208$	1.64133	0.113806
$209$	18.2991	1.26577
$210$	0	0
$211$	23.9929	1.65174	0.825870	−	0.563861i	$-0.190685\pi$
$211$	23.9929	1.65174	0.825870	+	0.563861i	$0.190685\pi$
$212$	2.04867	0.140703
$213$	0	0
$214$	8.32880	0.569345
$215$	3.68332	0.251201
$216$	0	0
$217$	−33.7190	−2.28899
$218$	−6.82558	−0.462287
$219$	0	0
$220$	−3.86683	−0.260702
$221$	11.9154	0.801516
$222$	0	0
$223$	8.86863	0.593887	0.296944	−	0.954895i	$-0.404033\pi$
$223$	8.86863	0.593887	0.296944	+	0.954895i	$0.404033\pi$
$224$	3.30920	0.221106
$225$	0	0
$226$	4.84290	0.322145
$227$	23.3991	1.55305	0.776527	−	0.630084i	$-0.216980\pi$
$227$	23.3991	1.55305	0.776527	+	0.630084i	$0.216980\pi$
$228$	0	0
$229$	−0.561334	−0.0370940	−0.0185470	−	0.999828i	$-0.505904\pi$
$229$	−0.561334	−0.0370940	−0.0185470	+	0.999828i	$0.505904\pi$
$230$	−11.2924	−0.744600
$231$	0	0
$232$	4.86856	0.319637
$233$	4.15018	0.271887	0.135944	−	0.990717i	$-0.456593\pi$
$233$	4.15018	0.271887	0.135944	+	0.990717i	$0.456593\pi$
$234$	0	0
$235$	4.21751	0.275120
$236$	−5.98049	−0.389297
$237$	0	0
$238$	24.0234	1.55721
$239$	−9.38306	−0.606940	−0.303470	−	0.952841i	$-0.598145\pi$
$239$	−9.38306	−0.606940	−0.303470	+	0.952841i	$0.598145\pi$
$240$	0	0
$241$	−20.0655	−1.29253	−0.646267	−	0.763111i	$-0.723671\pi$
$241$	−20.0655	−1.29253	−0.646267	+	0.763111i	$0.723671\pi$
$242$	−1.50086	−0.0964787
$243$	0	0
$244$	−10.7015	−0.685091
$245$	−4.95680	−0.316678
$246$	0	0
$247$	9.74505	0.620063
$248$	−10.1895	−0.647031
$249$	0	0
$250$	10.5713	0.668591
$251$	−19.7121	−1.24422	−0.622109	−	0.782931i	$-0.713724\pi$
$251$	−19.7121	−1.24422	−0.622109	+	0.782931i	$0.713724\pi$
$252$	0	0
$253$	27.7406	1.74404
$254$	−7.34600	−0.460929
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.7803	−1.48337	−0.741687	−	0.670746i	$-0.765974\pi$
$257$	−23.7803	−1.48337	−0.741687	+	0.670746i	$0.765974\pi$
$258$	0	0
$259$	−8.93975	−0.555489
$260$	−2.05925	−0.127709
$261$	0	0
$262$	0.626421	0.0387004
$263$	−5.44068	−0.335487	−0.167743	−	0.985831i	$-0.553648\pi$
$263$	−5.44068	−0.335487	−0.167743	+	0.985831i	$0.553648\pi$
$264$	0	0
$265$	−2.57031	−0.157893
$266$	19.6476	1.20467
$267$	0	0
$268$	−6.98394	−0.426612
$269$	13.5740	0.827619	0.413809	−	0.910364i	$-0.364198\pi$
$269$	13.5740	0.827619	0.413809	+	0.910364i	$0.364198\pi$
$270$	0	0
$271$	0.528562	0.0321078	0.0160539	−	0.999871i	$-0.494890\pi$
$271$	0.528562	0.0321078	0.0160539	+	0.999871i	$0.494890\pi$
$272$	7.25957	0.440176
$273$	0	0
$274$	1.65755	0.100136
$275$	−10.5589	−0.636728
$276$	0	0
$277$	−16.0391	−0.963694	−0.481847	−	0.876255i	$-0.660034\pi$
$277$	−16.0391	−0.963694	−0.481847	+	0.876255i	$0.660034\pi$
$278$	3.40771	0.204381
$279$	0	0
$280$	−4.15180	−0.248117
$281$	−15.9800	−0.953288	−0.476644	−	0.879096i	$-0.658147\pi$
$281$	−15.9800	−0.953288	−0.476644	+	0.879096i	$0.658147\pi$
$282$	0	0
$283$	−5.80106	−0.344837	−0.172418	−	0.985024i	$-0.555158\pi$
$283$	−5.80106	−0.344837	−0.172418	+	0.985024i	$0.555158\pi$
$284$	10.0260	0.594934
$285$	0	0
$286$	5.05871	0.299128
$287$	−20.4911	−1.20955
$288$	0	0
$289$	35.7014	2.10008
$290$	−6.10820	−0.358686
$291$	0	0
$292$	−10.7814	−0.630935
$293$	−3.42291	−0.199968	−0.0999842	−	0.994989i	$-0.531879\pi$
$293$	−3.42291	−0.199968	−0.0999842	+	0.994989i	$0.531879\pi$
$294$	0	0
$295$	7.50325	0.436856
$296$	−2.70148	−0.157020
$297$	0	0
$298$	13.1176	0.759884
$299$	14.7731	0.854349
$300$	0	0
$301$	−9.71518	−0.559974
$302$	−21.5801	−1.24180
$303$	0	0
$304$	5.93727	0.340526
$305$	13.4263	0.768786
$306$	0	0
$307$	1.58208	0.0902941	0.0451471	−	0.998980i	$-0.485624\pi$
$307$	1.58208	0.0902941	0.0451471	+	0.998980i	$0.485624\pi$
$308$	10.1992	0.581153
$309$	0	0
$310$	12.7839	0.726077
$311$	−25.9540	−1.47172	−0.735859	−	0.677134i	$-0.763222\pi$
$311$	−25.9540	−1.47172	−0.735859	+	0.677134i	$0.763222\pi$
$312$	0	0
$313$	−16.9656	−0.958953	−0.479476	−	0.877555i	$-0.659173\pi$
$313$	−16.9656	−0.958953	−0.479476	+	0.877555i	$0.659173\pi$
$314$	−2.59977	−0.146713
$315$	0	0
$316$	16.5082	0.928658
$317$	−13.3264	−0.748483	−0.374241	−	0.927331i	$-0.622097\pi$
$317$	−13.3264	−0.748483	−0.374241	+	0.927331i	$0.622097\pi$
$318$	0	0
$319$	15.0052	0.840131
$320$	−1.25462	−0.0701355
$321$	0	0
$322$	29.7850	1.65985
$323$	43.1021	2.39826
$324$	0	0
$325$	−5.62309	−0.311913
$326$	22.8231	1.26405
$327$	0	0
$328$	−6.19215	−0.341904
$329$	−11.1242	−0.613295
$330$	0	0
$331$	−29.9030	−1.64362	−0.821809	−	0.569763i	$-0.807035\pi$
$331$	−29.9030	−1.64362	−0.821809	+	0.569763i	$0.807035\pi$
$332$	−9.52589	−0.522801
$333$	0	0
$334$	−0.489625	−0.0267911
$335$	8.76220	0.478730
$336$	0	0
$337$	−25.4746	−1.38769	−0.693846	−	0.720124i	$-0.744085\pi$
$337$	−25.4746	−1.38769	−0.693846	+	0.720124i	$0.744085\pi$
$338$	−10.3060	−0.560574
$339$	0	0
$340$	−9.10801	−0.493951
$341$	−31.4046	−1.70065
$342$	0	0
$343$	−10.0903	−0.544826
$344$	−2.93581	−0.158288
$345$	0	0
$346$	17.8205	0.958038
$347$	10.9204	0.586237	0.293119	−	0.956076i	$-0.405307\pi$
$347$	10.9204	0.586237	0.293119	+	0.956076i	$0.405307\pi$
$348$	0	0
$349$	−27.1416	−1.45286	−0.726429	−	0.687241i	$-0.758822\pi$
$349$	−27.1416	−1.45286	−0.726429	+	0.687241i	$0.758822\pi$
$350$	−11.3371	−0.605993
$351$	0	0
$352$	3.08207	0.164275
$353$	1.00000	0.0532246
$353$	1.00000	0.0532246
$354$	0	0
$355$	−12.5788	−0.667616
$356$	−11.9006	−0.630733
$357$	0	0
$358$	−3.36324	−0.177753
$359$	−12.0943	−0.638311	−0.319156	−	0.947702i	$-0.603399\pi$
$359$	−12.0943	−0.638311	−0.319156	+	0.947702i	$0.603399\pi$
$360$	0	0
$361$	16.2512	0.855327
$362$	1.73271	0.0910691
$363$	0	0
$364$	5.43151	0.284689
$365$	13.5266	0.708015
$366$	0	0
$367$	29.7130	1.55101	0.775504	−	0.631343i	$-0.217496\pi$
$367$	29.7130	1.55101	0.775504	+	0.631343i	$0.217496\pi$
$368$	9.00065	0.469191
$369$	0	0
$370$	3.38933	0.176203
$371$	6.77947	0.351973
$372$	0	0
$373$	−3.13723	−0.162440	−0.0812198	−	0.996696i	$-0.525882\pi$
$373$	−3.13723	−0.162440	−0.0812198	+	0.996696i	$0.525882\pi$
$374$	22.3745	1.15696
$375$	0	0
$376$	−3.36158	−0.173360
$377$	7.99093	0.411554
$378$	0	0
$379$	32.6733	1.67831	0.839157	−	0.543889i	$-0.183049\pi$
$379$	32.6733	1.67831	0.839157	+	0.543889i	$0.183049\pi$
$380$	−7.44903	−0.382127
$381$	0	0
$382$	6.97386	0.356814
$383$	−1.18754	−0.0606807	−0.0303403	−	0.999540i	$-0.509659\pi$
$383$	−1.18754	−0.0606807	−0.0303403	+	0.999540i	$0.509659\pi$
$384$	0	0
$385$	−12.7961	−0.652151
$386$	−16.8029	−0.855244
$387$	0	0
$388$	16.1700	0.820909
$389$	−12.1507	−0.616064	−0.308032	−	0.951376i	$-0.599670\pi$
$389$	−12.1507	−0.616064	−0.308032	+	0.951376i	$0.599670\pi$
$390$	0	0
$391$	65.3409	3.30443
$392$	3.95083	0.199547
$393$	0	0
$394$	−5.02522	−0.253167
$395$	−20.7115	−1.04211
$396$	0	0
$397$	−0.282877	−0.0141972	−0.00709860	−	0.999975i	$-0.502260\pi$
$397$	−0.282877	−0.0141972	−0.00709860	+	0.999975i	$0.502260\pi$
$398$	1.79520	0.0899853
$399$	0	0
$400$	−3.42593	−0.171296
$401$	−3.58908	−0.179230	−0.0896150	−	0.995976i	$-0.528564\pi$
$401$	−3.58908	−0.179230	−0.0896150	+	0.995976i	$0.528564\pi$
$402$	0	0
$403$	−16.7243	−0.833097
$404$	−5.82543	−0.289826
$405$	0	0
$406$	16.1111	0.799578
$407$	−8.32615	−0.412712
$408$	0	0
$409$	13.8793	0.686289	0.343145	−	0.939283i	$-0.388508\pi$
$409$	13.8793	0.686289	0.343145	+	0.939283i	$0.388508\pi$
$410$	7.76880	0.383674
$411$	0	0
$412$	8.56357	0.421897
$413$	−19.7907	−0.973835
$414$	0	0
$415$	11.9514	0.586670
$416$	1.64133	0.0804731
$417$	0	0
$418$	18.2991	0.895037
$419$	−23.0356	−1.12536	−0.562680	−	0.826675i	$-0.690230\pi$
$419$	−23.0356	−1.12536	−0.562680	+	0.826675i	$0.690230\pi$
$420$	0	0
$421$	−15.2794	−0.744672	−0.372336	−	0.928098i	$-0.621443\pi$
$421$	−15.2794	−0.744672	−0.372336	+	0.928098i	$0.621443\pi$
$422$	23.9929	1.16796
$423$	0	0
$424$	2.04867	0.0994923
$425$	−24.8707	−1.20641
$426$	0	0
$427$	−35.4133	−1.71377
$428$	8.32880	0.402588
$429$	0	0
$430$	3.68332	0.177626
$431$	11.6464	0.560987	0.280494	−	0.959856i	$-0.409502\pi$
$431$	11.6464	0.560987	0.280494	+	0.959856i	$0.409502\pi$
$432$	0	0
$433$	8.56395	0.411557	0.205779	−	0.978599i	$-0.434027\pi$
$433$	8.56395	0.411557	0.205779	+	0.978599i	$0.434027\pi$
$434$	−33.7190	−1.61856
$435$	0	0
$436$	−6.82558	−0.326886
$437$	53.4393	2.55635
$438$	0	0
$439$	26.8352	1.28077	0.640387	−	0.768052i	$-0.278774\pi$
$439$	26.8352	1.28077	0.640387	+	0.768052i	$0.278774\pi$
$440$	−3.86683	−0.184344
$441$	0	0
$442$	11.9154	0.566757
$443$	28.6577	1.36157	0.680784	−	0.732484i	$-0.261639\pi$
$443$	28.6577	1.36157	0.680784	+	0.732484i	$0.261639\pi$
$444$	0	0
$445$	14.9308	0.707788
$446$	8.86863	0.419942
$447$	0	0
$448$	3.30920	0.156345
$449$	37.6776	1.77812	0.889058	−	0.457794i	$-0.151360\pi$
$449$	37.6776	1.77812	0.889058	+	0.457794i	$0.151360\pi$
$450$	0	0
$451$	−19.0846	−0.898659
$452$	4.84290	0.227791
$453$	0	0
$454$	23.3991	1.09817
$455$	−6.81449	−0.319468
$456$	0	0
$457$	9.82451	0.459571	0.229786	−	0.973241i	$-0.426197\pi$
$457$	9.82451	0.459571	0.229786	+	0.973241i	$0.426197\pi$
$458$	−0.561334	−0.0262294
$459$	0	0
$460$	−11.2924	−0.526511
$461$	−21.6686	−1.00921	−0.504604	−	0.863351i	$-0.668361\pi$
$461$	−21.6686	−1.00921	−0.504604	+	0.863351i	$0.668361\pi$
$462$	0	0
$463$	−2.97672	−0.138340	−0.0691700	−	0.997605i	$-0.522035\pi$
$463$	−2.97672	−0.138340	−0.0691700	+	0.997605i	$0.522035\pi$
$464$	4.86856	0.226017
$465$	0	0
$466$	4.15018	0.192253
$467$	30.2661	1.40055	0.700274	−	0.713874i	$-0.253061\pi$
$467$	30.2661	1.40055	0.700274	+	0.713874i	$0.253061\pi$
$468$	0	0
$469$	−23.1113	−1.06718
$470$	4.21751	0.194539
$471$	0	0
$472$	−5.98049	−0.275274
$473$	−9.04835	−0.416044
$474$	0	0
$475$	−20.3407	−0.933293
$476$	24.0234	1.10111
$477$	0	0
$478$	−9.38306	−0.429171
$479$	−34.2744	−1.56604	−0.783019	−	0.621998i	$-0.786321\pi$
$479$	−34.2744	−1.56604	−0.783019	+	0.621998i	$0.786321\pi$
$480$	0	0
$481$	−4.43403	−0.202174
$482$	−20.0655	−0.913960
$483$	0	0
$484$	−1.50086	−0.0682208
$485$	−20.2873	−0.921198
$486$	0	0
$487$	−2.19372	−0.0994067	−0.0497034	−	0.998764i	$-0.515828\pi$
$487$	−2.19372	−0.0994067	−0.0497034	+	0.998764i	$0.515828\pi$
$488$	−10.7015	−0.484432
$489$	0	0
$490$	−4.95680	−0.223925
$491$	−18.9591	−0.855613	−0.427806	−	0.903870i	$-0.640713\pi$
$491$	−18.9591	−0.855613	−0.427806	+	0.903870i	$0.640713\pi$
$492$	0	0
$493$	35.3436	1.59180
$494$	9.74505	0.438451
$495$	0	0
$496$	−10.1895	−0.457520
$497$	33.1781	1.48824
$498$	0	0
$499$	−21.6217	−0.967918	−0.483959	−	0.875091i	$-0.660802\pi$
$499$	−21.6217	−0.967918	−0.483959	+	0.875091i	$0.660802\pi$
$500$	10.5713	0.472765
$501$	0	0
$502$	−19.7121	−0.879794
$503$	14.1688	0.631754	0.315877	−	0.948800i	$-0.397701\pi$
$503$	14.1688	0.631754	0.315877	+	0.948800i	$0.397701\pi$
$504$	0	0
$505$	7.30871	0.325233
$506$	27.7406	1.23322
$507$	0	0
$508$	−7.34600	−0.325926
$509$	−16.1316	−0.715020	−0.357510	−	0.933909i	$-0.616374\pi$
$509$	−16.1316	−0.715020	−0.357510	+	0.933909i	$0.616374\pi$
$510$	0	0
$511$	−35.6780	−1.57830
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.7803	−1.04890
$515$	−10.7440	−0.473439
$516$	0	0
$517$	−10.3606	−0.455660
$518$	−8.93975	−0.392790
$519$	0	0
$520$	−2.05925	−0.0903042
$521$	−4.94488	−0.216639	−0.108320	−	0.994116i	$-0.534547\pi$
$521$	−4.94488	−0.216639	−0.108320	+	0.994116i	$0.534547\pi$
$522$	0	0
$523$	−1.53501	−0.0671212	−0.0335606	−	0.999437i	$-0.510685\pi$
$523$	−1.53501	−0.0671212	−0.0335606	+	0.999437i	$0.510685\pi$
$524$	0.626421	0.0273653
$525$	0	0
$526$	−5.44068	−0.237225
$527$	−73.9711	−3.22223
$528$	0	0
$529$	58.0118	2.52225
$530$	−2.57031	−0.111647
$531$	0	0
$532$	19.6476	0.851834
$533$	−10.1634	−0.440225
$534$	0	0
$535$	−10.4495	−0.451771
$536$	−6.98394	−0.301660
$537$	0	0
$538$	13.5740	0.585215
$539$	12.1767	0.524489
$540$	0	0
$541$	12.0040	0.516091	0.258046	−	0.966133i	$-0.416921\pi$
$541$	12.0040	0.516091	0.258046	+	0.966133i	$0.416921\pi$
$542$	0.528562	0.0227037
$543$	0	0
$544$	7.25957	0.311252
$545$	8.56352	0.366821
$546$	0	0
$547$	11.6655	0.498781	0.249391	−	0.968403i	$-0.419770\pi$
$547$	11.6655	0.498781	0.249391	+	0.968403i	$0.419770\pi$
$548$	1.65755	0.0708071
$549$	0	0
$550$	−10.5589	−0.450234
$551$	28.9060	1.23144
$552$	0	0
$553$	54.6290	2.32306
$554$	−16.0391	−0.681434
$555$	0	0
$556$	3.40771	0.144519
$557$	5.77850	0.244843	0.122421	−	0.992478i	$-0.460934\pi$
$557$	5.77850	0.244843	0.122421	+	0.992478i	$0.460934\pi$
$558$	0	0
$559$	−4.81864	−0.203807
$560$	−4.15180	−0.175446
$561$	0	0
$562$	−15.9800	−0.674076
$563$	−11.9166	−0.502224	−0.251112	−	0.967958i	$-0.580796\pi$
$563$	−11.9166	−0.502224	−0.251112	+	0.967958i	$0.580796\pi$
$564$	0	0
$565$	−6.07600	−0.255619
$566$	−5.80106	−0.243837
$567$	0	0
$568$	10.0260	0.420682
$569$	12.6253	0.529278	0.264639	−	0.964348i	$-0.414747\pi$
$569$	12.6253	0.529278	0.264639	+	0.964348i	$0.414747\pi$
$570$	0	0
$571$	29.0667	1.21640	0.608201	−	0.793783i	$-0.291891\pi$
$571$	29.0667	1.21640	0.608201	+	0.793783i	$0.291891\pi$
$572$	5.05871	0.211515
$573$	0	0
$574$	−20.4911	−0.855281
$575$	−30.8356	−1.28593
$576$	0	0
$577$	−9.86507	−0.410688	−0.205344	−	0.978690i	$-0.565831\pi$
$577$	−9.86507	−0.410688	−0.205344	+	0.978690i	$0.565831\pi$
$578$	35.7014	1.48498
$579$	0	0
$580$	−6.10820	−0.253629
$581$	−31.5231	−1.30780
$582$	0	0
$583$	6.31414	0.261505
$584$	−10.7814	−0.446139
$585$	0	0
$586$	−3.42291	−0.141399
$587$	31.7403	1.31006	0.655030	−	0.755603i	$-0.272656\pi$
$587$	31.7403	1.31006	0.655030	+	0.755603i	$0.272656\pi$
$588$	0	0
$589$	−60.4976	−2.49276
$590$	7.50325	0.308904
$591$	0	0
$592$	−2.70148	−0.111030
$593$	25.7459	1.05726	0.528629	−	0.848853i	$-0.322706\pi$
$593$	25.7459	1.05726	0.528629	+	0.848853i	$0.322706\pi$
$594$	0	0
$595$	−30.1403	−1.23563
$596$	13.1176	0.537319
$597$	0	0
$598$	14.7731	0.604116
$599$	−22.0475	−0.900836	−0.450418	−	0.892818i	$-0.648725\pi$
$599$	−22.0475	−0.900836	−0.450418	+	0.892818i	$0.648725\pi$
$600$	0	0
$601$	40.8830	1.66765	0.833826	−	0.552028i	$-0.186146\pi$
$601$	40.8830	1.66765	0.833826	+	0.552028i	$0.186146\pi$
$602$	−9.71518	−0.395961
$603$	0	0
$604$	−21.5801	−0.878084
$605$	1.88301	0.0765551
$606$	0	0
$607$	−13.0857	−0.531132	−0.265566	−	0.964093i	$-0.585559\pi$
$607$	−13.0857	−0.531132	−0.265566	+	0.964093i	$0.585559\pi$
$608$	5.93727	0.240788
$609$	0	0
$610$	13.4263	0.543614
$611$	−5.51748	−0.223213
$612$	0	0
$613$	−1.52380	−0.0615458	−0.0307729	−	0.999526i	$-0.509797\pi$
$613$	−1.52380	−0.0615458	−0.0307729	+	0.999526i	$0.509797\pi$
$614$	1.58208	0.0638476
$615$	0	0
$616$	10.1992	0.410937
$617$	18.7173	0.753530	0.376765	−	0.926309i	$-0.377036\pi$
$617$	18.7173	0.753530	0.376765	+	0.926309i	$0.377036\pi$
$618$	0	0
$619$	−25.1833	−1.01220	−0.506101	−	0.862474i	$-0.668914\pi$
$619$	−25.1833	−1.01220	−0.506101	+	0.862474i	$0.668914\pi$
$620$	12.7839	0.513414
$621$	0	0
$622$	−25.9540	−1.04066
$623$	−39.3817	−1.57779
$624$	0	0
$625$	3.86659	0.154664
$626$	−16.9656	−0.678082
$627$	0	0
$628$	−2.59977	−0.103742
$629$	−19.6116	−0.781965
$630$	0	0
$631$	26.6558	1.06115	0.530575	−	0.847638i	$-0.321976\pi$
$631$	26.6558	1.06115	0.530575	+	0.847638i	$0.321976\pi$
$632$	16.5082	0.656661
$633$	0	0
$634$	−13.3264	−0.529257
$635$	9.21645	0.365743
$636$	0	0
$637$	6.48464	0.256931
$638$	15.0052	0.594063
$639$	0	0
$640$	−1.25462	−0.0495933
$641$	17.9349	0.708386	0.354193	−	0.935172i	$-0.384756\pi$
$641$	17.9349	0.708386	0.354193	+	0.935172i	$0.384756\pi$
$642$	0	0
$643$	26.8739	1.05980	0.529902	−	0.848059i	$-0.322229\pi$
$643$	26.8739	1.05980	0.529902	+	0.848059i	$0.322229\pi$
$644$	29.7850	1.17369
$645$	0	0
$646$	43.1021	1.69583
$647$	13.2575	0.521205	0.260603	−	0.965446i	$-0.416079\pi$
$647$	13.2575	0.521205	0.260603	+	0.965446i	$0.416079\pi$
$648$	0	0
$649$	−18.4323	−0.723530
$650$	−5.62309	−0.220556
$651$	0	0
$652$	22.8231	0.893822
$653$	11.3946	0.445905	0.222952	−	0.974829i	$-0.428431\pi$
$653$	11.3946	0.445905	0.222952	+	0.974829i	$0.428431\pi$
$654$	0	0
$655$	−0.785921	−0.0307085
$656$	−6.19215	−0.241763
$657$	0	0
$658$	−11.1242	−0.433665
$659$	−14.0180	−0.546062	−0.273031	−	0.962005i	$-0.588026\pi$
$659$	−14.0180	−0.546062	−0.273031	+	0.962005i	$0.588026\pi$
$660$	0	0
$661$	22.6613	0.881424	0.440712	−	0.897649i	$-0.354726\pi$
$661$	22.6613	0.881424	0.440712	+	0.897649i	$0.354726\pi$
$662$	−29.9030	−1.16221
$663$	0	0
$664$	−9.52589	−0.369676
$665$	−24.6504	−0.955900
$666$	0	0
$667$	43.8202	1.69673
$668$	−0.489625	−0.0189442
$669$	0	0
$670$	8.76220	0.338513
$671$	−32.9826	−1.27328
$672$	0	0
$673$	15.0011	0.578250	0.289125	−	0.957291i	$-0.406636\pi$
$673$	15.0011	0.578250	0.289125	+	0.957291i	$0.406636\pi$
$674$	−25.4746	−0.981246
$675$	0	0
$676$	−10.3060	−0.396385
$677$	17.1050	0.657398	0.328699	−	0.944435i	$-0.393390\pi$
$677$	17.1050	0.657398	0.328699	+	0.944435i	$0.393390\pi$
$678$	0	0
$679$	53.5100	2.05352
$680$	−9.10801	−0.349276
$681$	0	0
$682$	−31.4046	−1.20254
$683$	36.7452	1.40602	0.703008	−	0.711182i	$-0.251840\pi$
$683$	36.7452	1.40602	0.703008	+	0.711182i	$0.251840\pi$
$684$	0	0
$685$	−2.07960	−0.0794574
$686$	−10.0903	−0.385250
$687$	0	0
$688$	−2.93581	−0.111927
$689$	3.36255	0.128103
$690$	0	0
$691$	−18.5369	−0.705176	−0.352588	−	0.935779i	$-0.614698\pi$
$691$	−18.5369	−0.705176	−0.352588	+	0.935779i	$0.614698\pi$
$692$	17.8205	0.677435
$693$	0	0
$694$	10.9204	0.414532
$695$	−4.27539	−0.162175
$696$	0	0
$697$	−44.9523	−1.70269
$698$	−27.1416	−1.02733
$699$	0	0
$700$	−11.3371	−0.428502
$701$	44.8770	1.69498	0.847490	−	0.530812i	$-0.178113\pi$
$701$	44.8770	1.69498	0.847490	+	0.530812i	$0.178113\pi$
$702$	0	0
$703$	−16.0394	−0.604938
$704$	3.08207	0.116160
$705$	0	0
$706$	1.00000	0.0376355
$707$	−19.2775	−0.725006
$708$	0	0
$709$	−42.3335	−1.58987	−0.794934	−	0.606696i	$-0.792494\pi$
$709$	−42.3335	−1.58987	−0.794934	+	0.606696i	$0.792494\pi$
$710$	−12.5788	−0.472075
$711$	0	0
$712$	−11.9006	−0.445996
$713$	−91.7118	−3.43463
$714$	0	0
$715$	−6.34676	−0.237355
$716$	−3.36324	−0.125690
$717$	0	0
$718$	−12.0943	−0.451354
$719$	14.0853	0.525294	0.262647	−	0.964892i	$-0.415405\pi$
$719$	14.0853	0.525294	0.262647	+	0.964892i	$0.415405\pi$
$720$	0	0
$721$	28.3386	1.05539
$722$	16.2512	0.604807
$723$	0	0
$724$	1.73271	0.0643956
$725$	−16.6793	−0.619454
$726$	0	0
$727$	18.9778	0.703846	0.351923	−	0.936029i	$-0.385528\pi$
$727$	18.9778	0.703846	0.351923	+	0.936029i	$0.385528\pi$
$728$	5.43151	0.201305
$729$	0	0
$730$	13.5266	0.500642
$731$	−21.3127	−0.788278
$732$	0	0
$733$	0.661829	0.0244452	0.0122226	−	0.999925i	$-0.496109\pi$
$733$	0.661829	0.0244452	0.0122226	+	0.999925i	$0.496109\pi$
$734$	29.7130	1.09673
$735$	0	0
$736$	9.00065	0.331768
$737$	−21.5250	−0.792882
$738$	0	0
$739$	−37.8328	−1.39170	−0.695852	−	0.718186i	$-0.744973\pi$
$739$	−37.8328	−1.39170	−0.695852	+	0.718186i	$0.744973\pi$
$740$	3.38933	0.124594
$741$	0	0
$742$	6.77947	0.248882
$743$	−39.8062	−1.46035	−0.730173	−	0.683262i	$-0.760561\pi$
$743$	−39.8062	−1.46035	−0.730173	+	0.683262i	$0.760561\pi$
$744$	0	0
$745$	−16.4577	−0.602962
$746$	−3.13723	−0.114862
$747$	0	0
$748$	22.3745	0.818092
$749$	27.5617	1.00708
$750$	0	0
$751$	−43.4895	−1.58695	−0.793477	−	0.608600i	$-0.791731\pi$
$751$	−43.4895	−1.58695	−0.793477	+	0.608600i	$0.791731\pi$
$752$	−3.36158	−0.122584
$753$	0	0
$754$	7.99093	0.291013
$755$	27.0749	0.985357
$756$	0	0
$757$	−34.5926	−1.25729	−0.628644	−	0.777693i	$-0.716390\pi$
$757$	−34.5926	−1.25729	−0.628644	+	0.777693i	$0.716390\pi$
$758$	32.6733	1.18675
$759$	0	0
$760$	−7.44903	−0.270205
$761$	−28.4792	−1.03237	−0.516185	−	0.856477i	$-0.672648\pi$
$761$	−28.4792	−1.03237	−0.516185	+	0.856477i	$0.672648\pi$
$762$	0	0
$763$	−22.5872	−0.817713
$764$	6.97386	0.252305
$765$	0	0
$766$	−1.18754	−0.0429077
$767$	−9.81598	−0.354435
$768$	0	0
$769$	−40.9944	−1.47830	−0.739148	−	0.673543i	$-0.764772\pi$
$769$	−40.9944	−1.47830	−0.739148	+	0.673543i	$0.764772\pi$
$770$	−12.7961	−0.461140
$771$	0	0
$772$	−16.8029	−0.604749
$773$	0.678071	0.0243885	0.0121943	−	0.999926i	$-0.496118\pi$
$773$	0.678071	0.0243885	0.0121943	+	0.999926i	$0.496118\pi$
$774$	0	0
$775$	34.9083	1.25394
$776$	16.1700	0.580470
$777$	0	0
$778$	−12.1507	−0.435623
$779$	−36.7645	−1.31722
$780$	0	0
$781$	30.9008	1.10572
$782$	65.3409	2.33659
$783$	0	0
$784$	3.95083	0.141101
$785$	3.26172	0.116416
$786$	0	0
$787$	−9.51669	−0.339233	−0.169617	−	0.985510i	$-0.554253\pi$
$787$	−9.51669	−0.339233	−0.169617	+	0.985510i	$0.554253\pi$
$788$	−5.02522	−0.179016
$789$	0	0
$790$	−20.7115	−0.736883
$791$	16.0261	0.569824
$792$	0	0
$793$	−17.5647	−0.623740
$794$	−0.282877	−0.0100389
$795$	0	0
$796$	1.79520	0.0636292
$797$	17.5301	0.620950	0.310475	−	0.950582i	$-0.399512\pi$
$797$	17.5301	0.620950	0.310475	+	0.950582i	$0.399512\pi$
$798$	0	0
$799$	−24.4036	−0.863339
$800$	−3.42593	−0.121125
$801$	0	0
$802$	−3.58908	−0.126735
$803$	−33.2291	−1.17263
$804$	0	0
$805$	−37.3689	−1.31708
$806$	−16.7243	−0.589089
$807$	0	0
$808$	−5.82543	−0.204938
$809$	−48.3285	−1.69914	−0.849570	−	0.527476i	$-0.823139\pi$
$809$	−48.3285	−1.69914	−0.849570	+	0.527476i	$0.823139\pi$
$810$	0	0
$811$	−30.3626	−1.06617	−0.533087	−	0.846061i	$-0.678968\pi$
$811$	−30.3626	−1.06617	−0.533087	+	0.846061i	$0.678968\pi$
$812$	16.1111	0.565387
$813$	0	0
$814$	−8.32615	−0.291831
$815$	−28.6344	−1.00302
$816$	0	0
$817$	−17.4307	−0.609822
$818$	13.8793	0.485280
$819$	0	0
$820$	7.76880	0.271298
$821$	−18.9230	−0.660417	−0.330209	−	0.943908i	$-0.607119\pi$
$821$	−18.9230	−0.660417	−0.330209	+	0.943908i	$0.607119\pi$
$822$	0	0
$823$	0.779018	0.0271549	0.0135774	−	0.999908i	$-0.495678\pi$
$823$	0.779018	0.0271549	0.0135774	+	0.999908i	$0.495678\pi$
$824$	8.56357	0.298326
$825$	0	0
$826$	−19.7907	−0.688605
$827$	40.1285	1.39541	0.697703	−	0.716387i	$-0.254206\pi$
$827$	40.1285	1.39541	0.697703	+	0.716387i	$0.254206\pi$
$828$	0	0
$829$	−6.79176	−0.235888	−0.117944	−	0.993020i	$-0.537630\pi$
$829$	−6.79176	−0.235888	−0.117944	+	0.993020i	$0.537630\pi$
$830$	11.9514	0.414839
$831$	0	0
$832$	1.64133	0.0569030
$833$	28.6814	0.993750
$834$	0	0
$835$	0.614294	0.0212585
$836$	18.2991	0.632887
$837$	0	0
$838$	−23.0356	−0.795750
$839$	−33.4559	−1.15503	−0.577513	−	0.816382i	$-0.695977\pi$
$839$	−33.4559	−1.15503	−0.577513	+	0.816382i	$0.695977\pi$
$840$	0	0
$841$	−5.29714	−0.182660
$842$	−15.2794	−0.526562
$843$	0	0
$844$	23.9929	0.825870
$845$	12.9302	0.444811
$846$	0	0
$847$	−4.96664	−0.170656
$848$	2.04867	0.0703516
$849$	0	0
$850$	−24.8707	−0.853060
$851$	−24.3151	−0.833510
$852$	0	0
$853$	23.0564	0.789438	0.394719	−	0.918802i	$-0.370842\pi$
$853$	23.0564	0.789438	0.394719	+	0.918802i	$0.370842\pi$
$854$	−35.4133	−1.21182
$855$	0	0
$856$	8.32880	0.284673
$857$	52.1479	1.78134	0.890668	−	0.454653i	$-0.150237\pi$
$857$	52.1479	1.78134	0.890668	+	0.454653i	$0.150237\pi$
$858$	0	0
$859$	13.1084	0.447253	0.223626	−	0.974675i	$-0.428210\pi$
$859$	13.1084	0.447253	0.223626	+	0.974675i	$0.428210\pi$
$860$	3.68332	0.125600
$861$	0	0
$862$	11.6464	0.396678
$863$	51.1852	1.74236	0.871182	−	0.490961i	$-0.163354\pi$
$863$	51.1852	1.74236	0.871182	+	0.490961i	$0.163354\pi$
$864$	0	0
$865$	−22.3580	−0.760195
$866$	8.56395	0.291015
$867$	0	0
$868$	−33.7190	−1.14450
$869$	50.8794	1.72596
$870$	0	0
$871$	−11.4630	−0.388408
$872$	−6.82558	−0.231143
$873$	0	0
$874$	53.4393	1.80761
$875$	34.9827	1.18263
$876$	0	0
$877$	−57.0318	−1.92583	−0.962914	−	0.269807i	$-0.913040\pi$
$877$	−57.0318	−1.92583	−0.962914	+	0.269807i	$0.913040\pi$
$878$	26.8352	0.905644
$879$	0	0
$880$	−3.86683	−0.130351
$881$	40.8661	1.37681	0.688407	−	0.725324i	$-0.258310\pi$
$881$	40.8661	1.37681	0.688407	+	0.725324i	$0.258310\pi$
$882$	0	0
$883$	9.75002	0.328114	0.164057	−	0.986451i	$-0.447542\pi$
$883$	9.75002	0.328114	0.164057	+	0.986451i	$0.447542\pi$
$884$	11.9154	0.400758
$885$	0	0
$886$	28.6577	0.962774
$887$	−37.4371	−1.25702	−0.628508	−	0.777803i	$-0.716334\pi$
$887$	−37.4371	−1.25702	−0.628508	+	0.777803i	$0.716334\pi$
$888$	0	0
$889$	−24.3094	−0.815311
$890$	14.9308	0.500482
$891$	0	0
$892$	8.86863	0.296944
$893$	−19.9586	−0.667890
$894$	0	0
$895$	4.21960	0.141046
$896$	3.30920	0.110553
$897$	0	0
$898$	37.6776	1.25732
$899$	−49.6080	−1.65452
$900$	0	0
$901$	14.8725	0.495474
$902$	−19.0846	−0.635448
$903$	0	0
$904$	4.84290	0.161072
$905$	−2.17389	−0.0722626
$906$	0	0
$907$	40.1729	1.33392	0.666959	−	0.745094i	$-0.267596\pi$
$907$	40.1729	1.33392	0.666959	+	0.745094i	$0.267596\pi$
$908$	23.3991	0.776527
$909$	0	0
$910$	−6.81449	−0.225898
$911$	−36.9774	−1.22512	−0.612558	−	0.790426i	$-0.709859\pi$
$911$	−36.9774	−1.22512	−0.612558	+	0.790426i	$0.709859\pi$
$912$	0	0
$913$	−29.3594	−0.971656
$914$	9.82451	0.324966
$915$	0	0
$916$	−0.561334	−0.0185470
$917$	2.07296	0.0684550
$918$	0	0
$919$	−3.10948	−0.102572	−0.0512862	−	0.998684i	$-0.516332\pi$
$919$	−3.10948	−0.102572	−0.0512862	+	0.998684i	$0.516332\pi$
$920$	−11.2924	−0.372300
$921$	0	0
$922$	−21.6686	−0.713617
$923$	16.4560	0.541657
$924$	0	0
$925$	9.25507	0.304305
$926$	−2.97672	−0.0978212
$927$	0	0
$928$	4.86856	0.159818
$929$	9.31030	0.305461	0.152731	−	0.988268i	$-0.451193\pi$
$929$	9.31030	0.305461	0.152731	+	0.988268i	$0.451193\pi$
$930$	0	0
$931$	23.4572	0.768778
$932$	4.15018	0.135944
$933$	0	0
$934$	30.2661	0.990338
$935$	−28.0715	−0.918037
$936$	0	0
$937$	48.6528	1.58942	0.794709	−	0.606991i	$-0.207623\pi$
$937$	48.6528	1.58942	0.794709	+	0.606991i	$0.207623\pi$
$938$	−23.1113	−0.754610
$939$	0	0
$940$	4.21751	0.137560
$941$	9.92803	0.323644	0.161822	−	0.986820i	$-0.448263\pi$
$941$	9.92803	0.323644	0.161822	+	0.986820i	$0.448263\pi$
$942$	0	0
$943$	−55.7334	−1.81493
$944$	−5.98049	−0.194648
$945$	0	0
$946$	−9.04835	−0.294187
$947$	13.9709	0.453995	0.226997	−	0.973895i	$-0.427109\pi$
$947$	13.9709	0.453995	0.226997	+	0.973895i	$0.427109\pi$
$948$	0	0
$949$	−17.6959	−0.574434
$950$	−20.3407	−0.659938
$951$	0	0
$952$	24.0234	0.778603
$953$	−42.8000	−1.38643	−0.693215	−	0.720731i	$-0.743806\pi$
$953$	−42.8000	−1.38643	−0.693215	+	0.720731i	$0.743806\pi$
$954$	0	0
$955$	−8.74955	−0.283129
$956$	−9.38306	−0.303470
$957$	0	0
$958$	−34.2744	−1.10736
$959$	5.48518	0.177126
$960$	0	0
$961$	72.8250	2.34919
$962$	−4.43403	−0.142959
$963$	0	0
$964$	−20.0655	−0.646267
$965$	21.0812	0.678629
$966$	0	0
$967$	−39.1856	−1.26012	−0.630062	−	0.776545i	$-0.716971\pi$
$967$	−39.1856	−1.26012	−0.630062	+	0.776545i	$0.716971\pi$
$968$	−1.50086	−0.0482394
$969$	0	0
$970$	−20.2873	−0.651385
$971$	9.55256	0.306556	0.153278	−	0.988183i	$-0.451017\pi$
$971$	9.55256	0.306556	0.153278	+	0.988183i	$0.451017\pi$
$972$	0	0
$973$	11.2768	0.361518
$974$	−2.19372	−0.0702912
$975$	0	0
$976$	−10.7015	−0.342545
$977$	−49.6408	−1.58815	−0.794075	−	0.607820i	$-0.792044\pi$
$977$	−49.6408	−1.58815	−0.794075	+	0.607820i	$0.792044\pi$
$978$	0	0
$979$	−36.6786	−1.17225
$980$	−4.95680	−0.158339
$981$	0	0
$982$	−18.9591	−0.605010
$983$	−50.4427	−1.60887	−0.804437	−	0.594038i	$-0.797533\pi$
$983$	−50.4427	−1.60887	−0.804437	+	0.594038i	$0.797533\pi$
$984$	0	0
$985$	6.30475	0.200886
$986$	35.3436	1.12557
$987$	0	0
$988$	9.74505	0.310031
$989$	−26.4242	−0.840240
$990$	0	0
$991$	5.39760	0.171460	0.0857302	−	0.996318i	$-0.472678\pi$
$991$	5.39760	0.171460	0.0857302	+	0.996318i	$0.472678\pi$
$992$	−10.1895	−0.323516
$993$	0	0
$994$	33.1781	1.05235
$995$	−2.25230	−0.0714026
$996$	0	0
$997$	27.6505	0.875700	0.437850	−	0.899048i	$-0.355740\pi$
$997$	27.6505	0.875700	0.437850	+	0.899048i	$0.355740\pi$
$998$	−21.6217	−0.684421
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6354.2.a.bl.1.3		10
3.2	odd	2		2118.2.a.t.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2118.2.a.t.1.8	✓	10	3.2	odd	2
6354.2.a.bl.1.3		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6354 = 2 \cdot 3^{2} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6354.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.25462 q^{5} +3.30920 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.25462 q^{5} +3.30920 q^{7} +1.00000 q^{8} -1.25462 q^{10} +3.08207 q^{11} +1.64133 q^{13} +3.30920 q^{14} +1.00000 q^{16} +7.25957 q^{17} +5.93727 q^{19} -1.25462 q^{20} +3.08207 q^{22} +9.00065 q^{23} -3.42593 q^{25} +1.64133 q^{26} +3.30920 q^{28} +4.86856 q^{29} -10.1895 q^{31} +1.00000 q^{32} +7.25957 q^{34} -4.15180 q^{35} -2.70148 q^{37} +5.93727 q^{38} -1.25462 q^{40} -6.19215 q^{41} -2.93581 q^{43} +3.08207 q^{44} +9.00065 q^{46} -3.36158 q^{47} +3.95083 q^{49} -3.42593 q^{50} +1.64133 q^{52} +2.04867 q^{53} -3.86683 q^{55} +3.30920 q^{56} +4.86856 q^{58} -5.98049 q^{59} -10.7015 q^{61} -10.1895 q^{62} +1.00000 q^{64} -2.05925 q^{65} -6.98394 q^{67} +7.25957 q^{68} -4.15180 q^{70} +10.0260 q^{71} -10.7814 q^{73} -2.70148 q^{74} +5.93727 q^{76} +10.1992 q^{77} +16.5082 q^{79} -1.25462 q^{80} -6.19215 q^{82} -9.52589 q^{83} -9.10801 q^{85} -2.93581 q^{86} +3.08207 q^{88} -11.9006 q^{89} +5.43151 q^{91} +9.00065 q^{92} -3.36158 q^{94} -7.44903 q^{95} +16.1700 q^{97} +3.95083 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8}+O(q^{10}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8	\( 10 q + 10 q^{2} + 10 q^{4} + 10 q^{5} + 10 q^{8} + 10 q^{10} + 2 q^{11} - 4 q^{13} + 10 q^{16} + 10 q^{17} + 14 q^{19} + 10 q^{20} + 2 q^{22} + 8 q^{23} + 20 q^{25} - 4 q^{26} + 14 q^{29} + 12 q^{31} + 10 q^{32} + 10 q^{34} + 6 q^{35} - 4 q^{37} + 14 q^{38} + 10 q^{40} + 2 q^{41} + 10 q^{43} + 2 q^{44} + 8 q^{46} + 8 q^{47} + 32 q^{49} + 20 q^{50} - 4 q^{52} + 36 q^{53} + 10 q^{55} + 14 q^{58} + 10 q^{59} - 6 q^{61} + 12 q^{62} + 10 q^{64} + 26 q^{65} + 2 q^{67} + 10 q^{68} + 6 q^{70} + 12 q^{71} - 8 q^{73} - 4 q^{74} + 14 q^{76} + 56 q^{77} + 28 q^{79} + 10 q^{80} + 2 q^{82} + 22 q^{83} - 4 q^{85} + 10 q^{86} + 2 q^{88} + 28 q^{89} + 8 q^{92} + 8 q^{94} + 16 q^{95} - 8 q^{97} + 32 q^{98}+O(q^{100}) \) 10 * q + 10 * q^2 + 10 * q^4 + 10 * q^5 + 10 * q^8 + 10 * q^10 + 2 * q^11 - 4 * q^13 + 10 * q^16 + 10 * q^17 + 14 * q^19 + 10 * q^20 + 2 * q^22 + 8 * q^23 + 20 * q^25 - 4 * q^26 + 14 * q^29 + 12 * q^31 + 10 * q^32 + 10 * q^34 + 6 * q^35 - 4 * q^37 + 14 * q^38 + 10 * q^40 + 2 * q^41 + 10 * q^43 + 2 * q^44 + 8 * q^46 + 8 * q^47 + 32 * q^49 + 20 * q^50 - 4 * q^52 + 36 * q^53 + 10 * q^55 + 14 * q^58 + 10 * q^59 - 6 * q^61 + 12 * q^62 + 10 * q^64 + 26 * q^65 + 2 * q^67 + 10 * q^68 + 6 * q^70 + 12 * q^71 - 8 * q^73 - 4 * q^74 + 14 * q^76 + 56 * q^77 + 28 * q^79 + 10 * q^80 + 2 * q^82 + 22 * q^83 - 4 * q^85 + 10 * q^86 + 2 * q^88 + 28 * q^89 + 8 * q^92 + 8 * q^94 + 16 * q^95 - 8 * q^97 + 32 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more