LMFDB - Embedded newform 539.2.q.d.312.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [539,2,Mod(214,539)] 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("539.214"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(539, base_ring=CyclotomicField(30)) chi = DirichletCharacter(H, H._module([20, 12])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.q (of order $15$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$4.30393666895$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{15})$
Coefficient field:	16.0.9234096523681640625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + 2 x^{14} - 5 x^{13} + 5 x^{12} + x^{11} + 6 x^{10} + 5 x^{9} - 21 x^{8} + 10 x^{7} + \cdots + 256 $ x^16 - x^15 + 2x^14 - 5x^13 + 5x^12 + x^11 + 6x^10 + 5x^9 - 21x^8 + 10x^7 + 24x^6 + 8x^5 + 80x^4 - 160x^3 + 128x^2 - 128*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{15}]$

Embedding invariants

Embedding label			312.1
Root			$-0.648523 - 1.25675i$ of defining polynomial
Character	$\chi$	$=$	539.312
Dual form			539.2.q.d.520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.295322 + 2.80980i) q^{2} +(-5.85146 - 1.24377i) q^{4} +(3.47668 - 10.7001i) q^{8} +(0.313585 - 2.98357i) q^{9} +(2.42220 - 2.26560i) q^{11} +(18.1084 + 8.06240i) q^{16} +(8.29061 + 1.76222i) q^{18} +(5.65055 + 7.47498i) q^{22} +(-3.75233 - 6.49922i) q^{23} +(-3.34565 - 3.71572i) q^{25} +(1.42225 + 4.37724i) q^{29} +(-16.7508 + 29.0132i) q^{32} +(-5.54579 + 17.0682i) q^{36} +(5.49419 - 6.10191i) q^{37} +6.59794 q^{43} +(-16.9913 + 10.2444i) q^{44} +(19.3696 - 8.62392i) q^{46} +(11.4285 - 8.30328i) q^{50} +(1.70800 - 0.760449i) q^{53} +(-12.7192 + 2.70355i) q^{58} +(-44.5014 - 32.3322i) q^{64} +(-3.04737 + 5.27819i) q^{67} +(7.95588 - 5.78028i) q^{71} +(-30.8343 - 13.7283i) q^{72} +(15.5226 + 17.2396i) q^{74} +(1.65186 - 15.7164i) q^{79} +(-8.80333 - 1.87121i) q^{81} +(-1.94851 + 18.5389i) q^{86} +(-15.8210 - 33.7946i) q^{88} +(13.8731 + 42.6969i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) $ 16 * q + 2 * q^2 - 2 * q^4 + 32 * q^8 - 6 * q^9 + 4 * q^11 + 28 * q^16 + 9 * q^18 - 8 * q^22 + 16 * q^23 - 10 * q^25 - 8 * q^29 - 100 * q^32 - 12 * q^36 + 18 * q^37 + 48 * q^43 - 9 * q^44 + 31 * q^46 + 20 * q^50 + 30 * q^53 - 71 * q^58 - 148 * q^64 + 8 * q^67 + 96 * q^71 - 57 * q^72 - 12 * q^74 - 24 * q^79 + 18 * q^81 - 29 * q^86 + 38 * q^88 + 142 * q^92 - 96 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/539\mathbb{Z}\right)^\times$.

$n$	$199$	$442$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{5}\right)$

Coefficient data

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

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

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

$2$	−0.295322	+	2.80980i	−0.208824	+	1.98683i	−0.0574802	+	0.998347i	$0.518307\pi$
$2$	−0.295322	+	2.80980i	−0.208824	+	1.98683i	−0.151344	+	0.988481i	$0.548360\pi$
$3$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$3$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$4$	−5.85146	−	1.24377i	−2.92573	−	0.621883i
$5$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$5$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	3.47668	−	10.7001i	1.22919	−	3.78306i
$9$	0.313585	−	2.98357i	0.104528	−	0.994522i
$10$		0			0
$11$	2.42220	−	2.26560i	0.730321	−	0.683104i
$11$	2.42220	−	2.26560i	0.730321	−	0.683104i
$12$		0			0
$13$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$13$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$14$		0			0
$15$		0			0
$16$	18.1084	+	8.06240i	4.52711	+	2.01560i
$17$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$17$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$18$	8.29061	+	1.76222i	1.95412	+	0.415360i
$19$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$19$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$20$		0			0
$21$		0			0
$22$	5.65055	+	7.47498i	1.20470	+	1.59367i
$23$	−3.75233	−	6.49922i	−0.782414	−	1.35518i	−0.930532	−	0.366211i	$-0.880655\pi$
$23$	−3.75233	−	6.49922i	−0.782414	−	1.35518i	0.148117	−	0.988970i	$-0.452679\pi$
$24$		0			0
$25$	−3.34565	−	3.71572i	−0.669131	−	0.743145i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.42225	+	4.37724i	0.264105	+	0.812833i	0.991898	+	0.127036i	$0.0405463\pi$
$29$	1.42225	+	4.37724i	0.264105	+	0.812833i	−0.727793	+	0.685797i	$0.759454\pi$
$30$		0			0
$31$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$31$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$32$	−16.7508	+	29.0132i	−2.96115	+	5.12886i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−5.54579	+	17.0682i	−0.924298	+	2.84470i
$37$	5.49419	−	6.10191i	0.903239	−	1.00315i	−0.0967305	−	0.995311i	$-0.530838\pi$
$37$	5.49419	−	6.10191i	0.903239	−	1.00315i	0.999969	−	0.00783774i	$-0.00249486\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$41$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$42$		0			0
$43$	6.59794			1.00618			0.503088	−	0.864235i	$-0.332197\pi$
$43$	6.59794			1.00618			0.503088	+	0.864235i	$0.332197\pi$
$44$	−16.9913	+	10.2444i	−2.56153	+	1.54440i
$45$		0			0
$46$	19.3696	−	8.62392i	2.85590	−	1.27153i
$47$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$47$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$48$		0			0
$49$		0			0
$50$	11.4285	−	8.30328i	1.61623	−	1.17426i
$51$		0			0
$52$		0			0
$53$	1.70800	−	0.760449i	0.234611	−	0.104456i	−0.286064	−	0.958211i	$-0.592347\pi$
$53$	1.70800	−	0.760449i	0.234611	−	0.104456i	0.520675	+	0.853755i	$0.325680\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−12.7192	+	2.70355i	−1.67011	+	0.354993i
$59$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$59$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$60$		0			0
$61$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$61$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$62$		0			0
$63$		0			0
$64$	−44.5014	−	32.3322i	−5.56268	−	4.04152i
$65$		0			0
$66$		0			0
$67$	−3.04737	+	5.27819i	−0.372295	+	0.644834i	−0.989918	−	0.141640i	$-0.954762\pi$
$67$	−3.04737	+	5.27819i	−0.372295	+	0.644834i	0.617623	+	0.786474i	$0.288096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.95588	−	5.78028i	0.944189	−	0.685993i	−0.00523645	−	0.999986i	$-0.501667\pi$
$71$	7.95588	−	5.78028i	0.944189	−	0.685993i	0.949425	+	0.313993i	$0.101667\pi$
$72$	−30.8343	−	13.7283i	−3.63385	−	1.61790i
$73$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$73$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$74$	15.5226	+	17.2396i	1.80447	+	2.00406i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.65186	−	15.7164i	0.185848	−	1.76823i	−0.362504	−	0.931982i	$-0.618078\pi$
$79$	1.65186	−	15.7164i	0.185848	−	1.76823i	0.548352	−	0.836247i	$-0.315255\pi$
$80$		0			0
$81$	−8.80333	−	1.87121i	−0.978148	−	0.207912i
$82$		0			0
$83$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$83$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$84$		0			0
$85$		0			0
$86$	−1.94851	+	18.5389i	−0.210114	+	1.99910i
$87$		0			0
$88$	−15.8210	−	33.7946i	−1.68652	−	3.60252i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$	13.8731	+	42.6969i	1.44637	+	4.45146i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$97$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$98$		0			0
$99$	−6.00000	−	7.93725i	−0.603023	−	0.797724i
$100$	14.9555	+	25.9036i	1.49555	+	2.59036i
$101$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$101$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$102$		0			0
$103$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$103$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$104$		0			0
$105$		0			0
$106$	1.63230	+	5.02371i	0.158543	+	0.487945i
$107$	1.12274	−	0.238646i	0.108539	−	0.0230707i	−0.153322	−	0.988176i	$-0.548997\pi$
$107$	1.12274	−	0.238646i	0.108539	−	0.0230707i	0.261861	+	0.965106i	$0.415664\pi$
$108$		0			0
$109$	2.25137	−	3.89948i	0.215642	−	0.373502i	−0.737829	−	0.674987i	$-0.764149\pi$
$109$	2.25137	−	3.89948i	0.215642	−	0.373502i	0.953471	+	0.301485i	$0.0974824\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.34450	+	13.3710i	−0.408696	+	1.25784i	0.509073	+	0.860724i	$0.329988\pi$
$113$	−4.34450	+	13.3710i	−0.408696	+	1.25784i	−0.917769	+	0.397114i	$0.870012\pi$
$114$		0			0
$115$		0			0
$116$	−2.87798	−	27.3822i	−0.267214	−	2.54237i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.734112	−	10.9755i	0.0667375	−	0.997771i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.21418	−	5.96795i	0.728890	−	0.529570i	−0.160322	−	0.987065i	$-0.551253\pi$
$127$	8.21418	−	5.96795i	0.728890	−	0.529570i	0.889212	+	0.457495i	$0.151253\pi$
$128$	59.1553	−	65.6986i	5.22864	−	5.80699i
$129$		0			0
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$		0			0
$134$	−13.9307	−	10.1213i	−1.20343	−	0.874343i
$135$		0			0
$136$		0			0
$137$	2.42718	+	23.0930i	0.207368	+	1.97297i	0.229857	+	0.973224i	$0.426174\pi$
$137$	2.42718	+	23.0930i	0.207368	+	1.97297i	−0.0224894	+	0.999747i	$0.507159\pi$
$138$		0			0
$139$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$139$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$140$		0			0
$141$		0			0
$142$	13.8919	+	24.0615i	1.16578	+	2.01919i
$143$		0			0
$144$	29.7332	−	51.4995i	2.47777	−	4.29162i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−39.7384	+	28.8716i	−3.26647	+	2.37323i
$149$	20.0980	+	8.94821i	1.64649	+	0.733066i	0.999565	−	0.0294862i	$-0.00938711\pi$
$149$	20.0980	+	8.94821i	1.64649	+	0.733066i	0.646927	+	0.762552i	$0.276054\pi$
$150$		0			0
$151$	−8.32996	−	9.25135i	−0.677882	−	0.752865i	0.301811	−	0.953368i	$-0.402409\pi$
$151$	−8.32996	−	9.25135i	−0.677882	−	0.752865i	−0.979693	+	0.200503i	$0.935742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$157$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$158$	43.6720	+	9.28277i	3.47436	+	0.738498i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	7.85752	−	24.1830i	0.617345	−	1.89999i
$163$	−0.932103	+	8.86836i	−0.0730079	+	0.694624i	0.895402	+	0.445259i	$0.146888\pi$
$163$	−0.932103	+	8.86836i	−0.0730079	+	0.694624i	−0.968410	+	0.249365i	$0.919778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$167$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$168$		0			0
$169$	−4.01722	−	12.3637i	−0.309017	−	0.951057i
$170$		0			0
$171$		0			0
$172$	−38.6076	−	8.20629i	−2.94380	−	0.625724i
$173$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$173$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$174$		0			0
$175$		0			0
$176$	62.1285	−	21.4977i	4.68311	−	1.62045i
$177$		0			0
$178$		0			0
$179$	−16.0100	−	17.7809i	−1.19664	−	1.32900i	−0.931038	−	0.364922i	$-0.881096\pi$
$179$	−16.0100	−	17.7809i	−1.19664	−	1.32900i	−0.265603	−	0.964082i	$-0.585571\pi$
$180$		0			0
$181$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$181$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$182$		0			0
$183$		0			0
$184$	−82.5881	+	17.5546i	−6.08847	+	1.29415i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−14.7366	+	16.3666i	−1.06630	+	1.18425i	−0.0840922	+	0.996458i	$0.526799\pi$
$191$	−14.7366	+	16.3666i	−1.06630	+	1.18425i	−0.982209	+	0.187790i	$0.939868\pi$
$192$		0			0
$193$	2.82262	+	26.8554i	0.203177	+	1.93310i	0.335805	+	0.941932i	$0.390992\pi$
$193$	2.82262	+	26.8554i	0.203177	+	1.93310i	−0.132628	+	0.991166i	$0.542342\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.0995			−1.28953			−0.644767	−	0.764379i	$-0.723046\pi$
$197$	−18.0995			−1.28953			−0.644767	+	0.764379i	$0.723046\pi$
$198$	24.0740	−	14.5147i	1.71087	−	1.03152i
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	−51.3905	+	22.8805i	−3.63385	+	1.61790i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−20.5675	+	9.15725i	−1.42954	+	0.636473i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−17.3570	−	12.6106i	−1.19490	−	0.868148i	−0.201129	−	0.979565i	$-0.564461\pi$
$211$	−17.3570	−	12.6106i	−1.19490	−	0.868148i	−0.993774	+	0.111417i	$0.964461\pi$
$212$	−10.9401	+	2.32539i	−0.751369	+	0.159708i
$213$		0			0
$214$	0.338977	+	3.22515i	0.0231720	+	0.220467i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	10.2919	+	7.47748i	0.697054	+	0.506439i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$223$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$224$		0			0
$225$	−12.1353	+	8.81678i	−0.809017	+	0.587785i
$226$	−36.2868	−	16.1559i	−2.41376	−	1.07468i
$227$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$227$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$228$		0			0
$229$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$229$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$230$		0			0
$231$		0			0
$232$	51.7817			3.39963
$233$	−2.29963	+	21.8795i	−0.150654	+	1.43337i	0.614191	+	0.789157i	$0.289482\pi$
$233$	−2.29963	+	21.8795i	−0.150654	+	1.43337i	−0.764844	+	0.644215i	$0.777184\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.805627	−	2.47947i	0.0521117	−	0.160383i	−0.921614	−	0.388108i	$-0.873129\pi$
$239$	0.805627	−	2.47947i	0.0521117	−	0.160383i	0.973726	+	0.227725i	$0.0731287\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	30.6221	+	5.30400i	1.96846	+	0.340954i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$251$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$252$		0			0
$253$	−23.8135	−	7.24115i	−1.49714	−	0.455247i
$254$	14.3429	+	24.8427i	0.899954	+	1.55877i
$255$		0			0
$256$	93.5167	+	103.861i	5.84479	+	6.49130i
$257$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$257$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	13.5058	−	2.87074i	0.835986	−	0.177694i
$262$		0			0
$263$	14.4994	−	25.1137i	0.894072	−	1.54858i	0.0591223	−	0.998251i	$-0.481170\pi$
$263$	14.4994	−	25.1137i	0.894072	−	1.54858i	0.834949	−	0.550327i	$-0.185497\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	24.3964	−	27.0949i	1.49025	−	1.65509i
$269$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$269$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$270$		0			0
$271$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$271$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$272$		0			0
$273$		0			0
$274$	−65.6036			−3.96326
$275$	−16.5222	−	1.42032i	−0.996325	−	0.0856484i
$276$		0			0
$277$	24.7616	−	11.0246i	1.48778	−	0.662402i	0.507796	−	0.861478i	$-0.330460\pi$
$277$	24.7616	−	11.0246i	1.48778	−	0.662402i	0.979984	+	0.199075i	$0.0637938\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.95218	+	5.05106i	−0.414733	+	0.301321i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−6.95218	+	5.05106i	−0.414733	+	0.301321i	0.360782	+	0.932650i	$0.382510\pi$
$282$		0			0
$283$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$283$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$284$	−53.7428	+	23.9278i	−3.18905	+	1.41986i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	81.3099	+	59.0751i	4.79123	+	3.48104i
$289$	16.6285	−	3.53450i	0.978148	−	0.207912i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$293$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$294$		0			0
$295$		0			0
$296$	−46.1897	−	80.0029i	−2.68472	−	4.65007i
$297$		0			0
$298$	−31.0780	+	53.8287i	−1.80030	+	3.11822i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	28.4545	−	20.6734i	1.63737	−	1.18962i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$311$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$312$		0			0
$313$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$313$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$314$		0			0
$315$		0			0
$316$	−29.2133	+	89.9092i	−1.64337	+	5.05779i
$317$	−3.52544	+	33.5424i	−0.198009	+	1.88393i	0.220024	+	0.975494i	$0.429386\pi$
$317$	−3.52544	+	33.5424i	−0.198009	+	1.88393i	−0.418033	+	0.908432i	$0.637280\pi$
$318$		0			0
$319$	13.3620	+	7.38030i	0.748131	+	0.413217i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	49.1850	+	21.8986i	2.73250	+	1.21659i
$325$		0			0
$326$	−24.6431	−	5.23804i	−1.36485	−	0.290108i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−3.04605	−	5.27591i	−0.167426	−	0.289990i	0.770088	−	0.637937i	$-0.220212\pi$
$331$	−3.04605	−	5.27591i	−0.167426	−	0.289990i	−0.937514	+	0.347947i	$0.886879\pi$
$332$		0			0
$333$	−16.4826	−	18.3057i	−0.903239	−	1.00315i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.08528	+	27.9616i	0.494907	+	1.52317i	0.817102	+	0.576493i	$0.195579\pi$
$337$	9.08528	+	27.9616i	0.494907	+	1.52317i	−0.322195	+	0.946673i	$0.604421\pi$
$338$	35.9260	−	7.63630i	1.95412	−	0.415360i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	22.9389	−	70.5987i	1.23678	−	3.80643i
$345$		0			0
$346$		0			0
$347$	2.61404	+	24.8709i	0.140329	+	1.33514i	0.807337	+	0.590091i	$0.200908\pi$
$347$	2.61404	+	24.8709i	0.140329	+	1.33514i	−0.667008	+	0.745051i	$0.732425\pi$
$348$		0			0
$349$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$349$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$350$		0			0
$351$		0			0
$352$	25.1585	+	108.226i	1.34096	+	5.76848i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	54.6888	−	39.7337i	2.89039	−	2.09999i
$359$	−21.9177	+	24.3421i	−1.15677	+	1.28473i	−0.204709	+	0.978823i	$0.565625\pi$
$359$	−21.9177	+	24.3421i	−1.15677	+	1.28473i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	−17.3574	+	7.72800i	−0.913545	+	0.406737i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$367$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$368$	−15.5495	−	147.944i	−0.810573	−	7.71209i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	31.4997	−	22.8859i	1.61803	−	1.17557i	0.808511	−	0.588481i	$-0.200274\pi$
$379$	31.4997	−	22.8859i	1.61803	−	1.17557i	0.809522	−	0.587090i	$-0.199726\pi$
$380$		0			0
$381$		0			0
$382$	−41.6349	−	46.2402i	−2.13023	−	2.36586i
$383$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$383$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$384$		0			0
$385$		0			0
$386$	−76.2920			−3.88316
$387$	2.06902	−	19.6854i	0.105174	−	1.00066i
$388$		0			0
$389$	21.3311	+	4.53407i	1.08153	+	0.229887i	0.714015	−	0.700130i	$-0.246875\pi$
$389$	21.3311	+	4.53407i	1.08153	+	0.229887i	0.367516	+	0.930017i	$0.380208\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	5.34517	−	50.8559i	0.269286	−	2.56208i
$395$		0			0
$396$	25.2367	+	53.9071i	1.26819	+	2.70893i
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	−30.6269	−	94.2600i	−1.53135	−	4.71300i
$401$	8.79150	+	3.91423i	0.439027	+	0.195467i	0.614333	−	0.789047i	$-0.289425\pi$
$401$	8.79150	+	3.91423i	0.439027	+	0.195467i	−0.175307	+	0.984514i	$0.556092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.516471	−	27.2277i	−0.0256005	−	1.34963i
$408$		0			0
$409$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$409$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−19.6560	−	60.4950i	−0.966040	−	2.97316i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−11.8136	+	36.3585i	−0.575759	+	1.77200i	0.0578225	+	0.998327i	$0.481584\pi$
$421$	−11.8136	+	36.3585i	−0.575759	+	1.77200i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	40.5591	−	45.0454i	1.97438	−	2.19278i
$423$		0			0
$424$	−2.19874	−	20.9196i	−0.106780	−	1.01595i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−6.86648			−0.331904
$429$		0			0
$430$		0			0
$431$	32.0208	−	14.2566i	1.54239	−	0.686715i	0.553157	−	0.833077i	$-0.313423\pi$
$431$	32.0208	−	14.2566i	1.54239	−	0.686715i	0.989231	+	0.146362i	$0.0467564\pi$
$432$		0			0
$433$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$433$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$434$		0			0
$435$		0			0
$436$	−18.0238	+	20.0175i	−0.863184	+	0.958663i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−37.1229	+	7.89071i	−1.76376	+	0.374899i	−0.971825	−	0.235705i	$-0.924260\pi$
$443$	−37.1229	+	7.89071i	−1.76376	+	0.374899i	−0.791936	+	0.610604i	$0.790927\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	32.0711	+	23.3010i	1.51353	+	1.09964i	0.964580	+	0.263790i	$0.0849724\pi$
$449$	32.0711	+	23.3010i	1.51353	+	1.09964i	0.548950	+	0.835855i	$0.315028\pi$
$450$	−21.1896	−	36.7014i	−0.998886	−	1.73012i
$451$		0			0
$452$	42.0521	−	72.8363i	1.97796	−	3.42593i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	35.0857	+	15.6211i	1.64124	+	0.730726i	0.999349	−	0.0360712i	$-0.0114843\pi$
$457$	35.0857	+	15.6211i	1.64124	+	0.730726i	0.641889	+	0.766798i	$0.278151\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	41.6915			1.93757			0.968784	−	0.247907i	$-0.0797429\pi$
$463$	41.6915			1.93757			0.968784	+	0.247907i	$0.0797429\pi$
$464$	−9.53629	+	90.7317i	−0.442711	+	4.21212i
$465$		0			0
$466$	−60.7978	−	12.9230i	−2.81640	−	0.598645i
$467$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$467$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	15.9815	−	14.9483i	0.734831	−	0.687323i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.73325	−	5.33439i	−0.0793599	−	0.244245i
$478$	6.72888	+	2.99589i	0.307772	+	0.137029i
$479$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$479$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−17.9466	+	63.3095i	−0.815752	+	2.87770i
$485$		0			0
$486$		0			0
$487$	−1.57611	−	1.75045i	−0.0714205	−	0.0793205i	0.706376	−	0.707837i	$-0.250329\pi$
$487$	−1.57611	−	1.75045i	−0.0714205	−	0.0793205i	−0.777796	+	0.628517i	$0.783662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13.5967	+	41.8465i	0.613613	+	1.88851i	0.420363	+	0.907356i	$0.361903\pi$
$491$	13.5967	+	41.8465i	0.613613	+	1.88851i	0.193249	+	0.981150i	$0.438097\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−9.08229	+	10.0869i	−0.406579	+	0.451552i	−0.911308	−	0.411726i	$-0.864926\pi$
$499$	−9.08229	+	10.0869i	−0.406579	+	0.451552i	0.504728	+	0.863278i	$0.331593\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$503$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$504$		0			0
$505$		0			0
$506$	27.3788	−	64.7727i	1.21714	−	2.87950i
$507$		0			0
$508$	−55.4877	+	24.7047i	−2.46187	+	1.09609i
$509$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$509$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$510$		0			0
$511$		0			0
$512$	−176.401	+	128.163i	−7.79590	+	5.66405i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$521$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$522$	4.07766	+	38.7963i	0.178474	+	1.69807i
$523$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$523$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$524$		0			0
$525$		0			0
$526$	66.2825	+	48.1570i	2.89005	+	2.09975i
$527$		0			0
$528$		0			0
$529$	−16.6599	+	28.8558i	−0.724344	+	1.25460i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	45.8826	+	50.9578i	1.98183	+	2.20104i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.25448	−	40.4787i	0.182915	−	1.74032i	−0.390101	−	0.920772i	$-0.627560\pi$
$541$	4.25448	−	40.4787i	0.182915	−	1.74032i	0.573016	−	0.819544i	$-0.305773\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	13.5967	−	41.8465i	0.581355	−	1.78923i	−0.0320849	−	0.999485i	$-0.510215\pi$
$547$	13.5967	−	41.8465i	0.581355	−	1.78923i	0.613440	−	0.789741i	$-0.289785\pi$
$548$	14.5198	−	138.147i	0.620256	−	5.90134i
$549$		0			0
$550$	8.87017	−	46.0046i	0.378225	−	1.96164i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	23.6642	+	72.8309i	1.00540	+	3.09429i
$555$		0			0
$556$		0			0
$557$	30.3169	+	6.44407i	1.28457	+	0.273044i	0.799094	−	0.601206i	$-0.205313\pi$
$557$	30.3169	+	6.44407i	1.28457	+	0.273044i	0.485476	+	0.874250i	$0.338646\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−12.1393	−	21.0259i	−0.512067	−	0.886925i
$563$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$563$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−34.1897	−	105.225i	−1.43457	−	4.41514i
$569$	−21.5192	+	4.57406i	−0.902134	+	0.191754i	−0.635552	−	0.772058i	$-0.719228\pi$
$569$	−21.5192	+	4.57406i	−0.902134	+	0.191754i	−0.266582	+	0.963812i	$0.585894\pi$
$570$		0			0
$571$	22.0283	−	38.1541i	0.921856	−	1.59670i	0.125314	−	0.992117i	$-0.460006\pi$
$571$	22.0283	−	38.1541i	0.921856	−	1.59670i	0.796542	−	0.604583i	$-0.206660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−11.5953	+	35.6868i	−0.483559	+	1.48824i
$576$	−110.420	+	122.634i	−4.60084	+	5.10975i
$577$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$577$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$578$	5.02047	+	47.7666i	0.208824	+	1.98683i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.41424	−	5.71160i	0.0999875	−	0.236550i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$587$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	148.687	−	66.1998i	6.11101	−	2.72080i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$	−106.473	−	77.3573i	−4.36131	−	3.16868i
$597$		0			0
$598$		0			0
$599$	2.64866	+	25.2003i	0.108221	+	1.02966i	0.905008	+	0.425395i	$0.139865\pi$
$599$	2.64866	+	25.2003i	0.108221	+	1.02966i	−0.796787	+	0.604261i	$0.793469\pi$
$600$		0			0
$601$		0			0		0.951057	−	0.309017i	$-0.100000\pi$
$601$		0			0		−0.951057	+	0.309017i	$0.900000\pi$
$602$		0			0
$603$	14.7922	+	10.7472i	0.602386	+	0.437659i
$604$	37.2359	+	64.4944i	1.51511	+	2.62424i
$605$		0			0
$606$		0			0
$607$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$607$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	33.0579	+	36.7145i	1.33520	+	1.48289i	0.717868	+	0.696179i	$0.245118\pi$
$613$	33.0579	+	36.7145i	1.33520	+	1.48289i	0.617327	+	0.786706i	$0.288215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	45.9166			1.84853			0.924266	−	0.381749i	$-0.124678\pi$
$617$	45.9166			1.84853			0.924266	+	0.381749i	$0.124678\pi$
$618$		0			0
$619$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$619$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.61321	+	24.8630i	−0.104528	+	0.994522i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−12.4683	−	38.3736i	−0.496357	−	1.52763i	−0.814832	−	0.579698i	$-0.803171\pi$
$631$	−12.4683	−	38.3736i	−0.496357	−	1.52763i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	−162.424	−	72.3158i	−6.46088	−	2.87657i
$633$		0			0
$634$	−93.2061	−	19.8116i	−3.70169	−	0.786818i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−24.6833	+	35.3651i	−0.977219	+	1.40012i
$639$	−14.7510	−	25.5495i	−0.583541	−	1.01072i
$640$		0			0
$641$	−33.2262	−	36.9015i	−1.31236	−	1.45752i	−0.801686	−	0.597746i	$-0.796063\pi$
$641$	−33.2262	−	36.9015i	−1.31236	−	1.45752i	−0.510672	−	0.859776i	$-0.670603\pi$
$642$		0			0
$643$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$643$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.406737	−	0.913545i	$-0.633333\pi$
$647$		0			0		0.406737	+	0.913545i	$0.366667\pi$
$648$	−50.6285	+	87.6911i	−1.98887	+	3.44483i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	16.4843	−	50.7336i	0.645576	−	1.98688i
$653$	3.60381	−	4.00244i	0.141028	−	0.156628i	−0.668493	−	0.743719i	$-0.733060\pi$
$653$	3.60381	−	4.00244i	0.141028	−	0.156628i	0.809521	+	0.587091i	$0.199727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.0000			1.71400			0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000			1.71400			0.856998	+	0.515319i	$0.172327\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	15.7238	−	7.00069i	0.611123	−	0.272089i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	56.3031	−	40.9066i	2.18170	−	1.58510i
$667$	23.1119	−	25.6684i	0.894896	−	0.993883i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.494817	−	0.359506i	−0.0190738	−	0.0138579i	0.578208	−	0.815890i	$-0.303752\pi$
$673$	−0.494817	−	0.359506i	−0.0190738	−	0.0138579i	−0.597281	+	0.802032i	$0.703752\pi$
$674$	−81.2496	+	17.2701i	−3.12962	+	0.665221i
$675$		0			0
$676$	8.12901	+	77.3424i	0.312654	+	2.97471i
$677$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$677$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	10.5507	−	18.2743i	0.403711	−	0.699249i	−0.590459	−	0.807067i	$-0.701053\pi$
$683$	10.5507	−	18.2743i	0.403711	−	0.699249i	0.994171	+	0.107819i	$0.0343867\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	119.478	+	53.1952i	4.55507	+	2.02805i
$689$		0			0
$690$		0			0
$691$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$691$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$692$		0			0
$693$		0			0
$694$	−70.6543			−2.68200
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.3604	+	47.2743i	−0.580153	+	1.78553i	0.0377695	+	0.999286i	$0.487975\pi$
$701$	−15.3604	+	47.2743i	−0.580153	+	1.78553i	−0.617922	+	0.786239i	$0.712025\pi$
$702$		0			0
$703$		0			0
$704$	−181.043	+	22.5074i	−6.82332	+	0.848280i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−23.9793	−	10.6763i	−0.900560	−	0.400955i	−0.0963826	−	0.995344i	$-0.530727\pi$
$709$	−23.9793	−	10.6763i	−0.900560	−	0.400955i	−0.804178	+	0.594389i	$0.797394\pi$
$710$		0			0
$711$	−46.3728	−	9.85685i	−1.73912	−	0.369661i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	71.5664	+	123.957i	2.67456	+	4.63248i
$717$		0			0
$718$	−61.9236	−	68.7731i	−2.31097	−	2.56659i
$719$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$719$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$720$		0			0
$721$		0			0
$722$	−16.5881	−	51.0529i	−0.617345	−	1.89999i
$723$		0			0
$724$		0			0
$725$	11.5063	−	19.9294i	0.427331	−	0.740160i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−8.34346	+	25.6785i	−0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$733$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$734$		0			0
$735$		0			0
$736$	251.417			9.26737
$737$	4.57694	+	19.6890i	0.168594	+	0.725252i
$738$		0			0
$739$	46.9559	−	20.9061i	1.72730	−	0.769045i	0.731072	−	0.682300i	$-0.239020\pi$
$739$	46.9559	−	20.9061i	1.72730	−	0.769045i	0.996231	−	0.0867442i	$-0.0276463\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.56653	+	6.22395i	−0.314275	+	0.228334i	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−8.56653	+	6.22395i	−0.314275	+	0.228334i	0.419453	+	0.907777i	$0.362222\pi$
$744$		0			0
$745$		0			0
$746$	56.7824	−	25.2812i	2.07895	−	0.925609i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−10.1040	+	2.14768i	−0.368702	+	0.0783700i	−0.388537	−	0.921433i	$-0.627019\pi$
$751$	−10.1040	+	2.14768i	−0.368702	+	0.0783700i	0.0198348	+	0.999803i	$0.493686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−30.3109	−	22.0222i	−1.10167	−	0.800410i	−0.120338	−	0.992733i	$-0.538398\pi$
$757$	−30.3109	−	22.0222i	−1.10167	−	0.800410i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	55.0022	+	95.2667i	1.99777	+	3.46024i
$759$		0			0
$760$		0			0
$761$		0			0		−0.994522	−	0.104528i	$-0.966667\pi$
$761$		0			0		0.994522	+	0.104528i	$0.0333333\pi$
$762$		0			0
$763$		0			0
$764$	106.587	−	77.4398i	3.85617	−	2.80167i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	16.8854	−	160.654i	0.607720	−	5.78207i
$773$		0			0		0.743145	−	0.669131i	$-0.233333\pi$
$773$		0			0		−0.743145	+	0.669131i	$0.766667\pi$
$774$	54.7009	+	11.6270i	1.96618	+	0.417925i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−19.0394	+	58.5972i	−0.682595	+	2.10081i
$779$		0			0
$780$		0			0
$781$	6.17492	−	32.0258i	0.220956	−	1.14597i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$787$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$788$	105.908	+	22.5115i	3.77283	+	0.801940i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−105.790	+	36.6054i	−3.75907	+	1.30072i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$797$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$798$		0			0
$799$		0			0
$800$	163.847	−	34.8268i	5.79288	−	1.23131i
$801$		0			0
$802$	−13.5945	+	23.5464i	−0.480039	+	0.831452i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.43577	+	51.7179i	0.191112	+	1.81831i	0.498688	+	0.866782i	$0.333815\pi$
$809$	5.43577	+	51.7179i	0.191112	+	1.81831i	−0.307576	+	0.951523i	$0.599518\pi$
$810$		0			0
$811$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$811$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$812$		0			0
$813$		0			0
$814$	76.6569	+	6.58975i	2.68682	+	0.230971i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	14.7209	−	16.3492i	0.513762	−	0.570591i	−0.429319	−	0.903153i	$-0.641246\pi$
$821$	14.7209	−	16.3492i	0.513762	−	0.570591i	0.943081	+	0.332562i	$0.107913\pi$
$822$		0			0
$823$	−49.2227	+	21.9154i	−1.71580	+	0.763921i	−0.718109	+	0.695930i	$0.754992\pi$
$823$	−49.2227	+	21.9154i	−1.71580	+	0.763921i	−0.997686	+	0.0679910i	$0.978341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.5967	−	25.8626i	−1.23782	−	0.899329i	−0.240369	−	0.970682i	$-0.577268\pi$
$827$	−35.5967	−	25.8626i	−1.23782	−	0.899329i	−0.997451	+	0.0713526i	$0.977268\pi$
$828$	131.740	−	28.0021i	4.57827	−	0.973140i
$829$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$829$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.951057	−	0.309017i	$-0.900000\pi$
$839$		0			0		0.951057	+	0.309017i	$0.100000\pi$
$840$		0			0
$841$	6.32408	−	4.59471i	0.218072	−	0.158438i
$842$	−98.6712	−	43.9312i	−3.40043	−	1.51397i
$843$		0			0
$844$	85.8790	+	95.3783i	2.95608	+	3.28305i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	37.0602			1.27265
$849$		0			0
$850$		0			0
$851$	−60.2737	−	12.8116i	−2.06615	−	0.439175i
$852$		0			0
$853$		0			0		−0.587785	−	0.809017i	$-0.700000\pi$
$853$		0			0		0.587785	+	0.809017i	$0.300000\pi$
$854$		0			0
$855$		0			0
$856$	1.34987	−	12.8431i	0.0461375	−	0.438969i
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$		0			0
$861$		0			0
$862$	30.6017	+	94.1823i	1.04230	+	3.20786i
$863$	7.30836	+	3.25389i	0.248780	+	0.110764i	0.527338	−	0.849655i	$-0.323190\pi$
$863$	7.30836	+	3.25389i	0.248780	+	0.110764i	−0.278559	+	0.960419i	$0.589857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−31.6059	−	41.8106i	−1.07216	−	1.41833i
$870$		0			0
$871$		0			0
$872$	−33.8976	−	37.6471i	−1.14792	−	1.27489i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	21.3131	−	4.53024i	0.719692	−	0.152975i	0.166515	−	0.986039i	$-0.446749\pi$
$877$	21.3131	−	4.53024i	0.719692	−	0.152975i	0.553177	+	0.833064i	$0.313415\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	−0.993863	−	0.110619i	$-0.964717\pi$
$883$	−3.70820	+	11.4127i	−0.124791	+	0.384067i	0.869072	+	0.494686i	$0.164717\pi$
$884$		0			0
$885$		0			0
$886$	−11.2081	−	106.638i	−0.376544	−	3.58258i
$887$		0			0		−0.743145	−	0.669131i	$-0.766667\pi$
$887$		0			0		0.743145	+	0.669131i	$0.233333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−25.5628	+	15.4124i	−0.856387	+	0.516334i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−74.9426	+	83.2321i	−2.50087	+	2.77749i
$899$		0			0
$900$	81.9750	−	36.4976i	2.73250	−	1.21659i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	127.967	+	92.9734i	4.25611	+	3.09225i
$905$		0			0
$906$		0			0
$907$	4.74881	+	45.1819i	0.157682	+	1.50024i	0.731829	+	0.681488i	$0.238667\pi$
$907$	4.74881	+	45.1819i	0.157682	+	1.50024i	−0.574148	+	0.818752i	$0.694667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.9443	−	9.40456i	−0.428863	−	0.311587i	0.352331	−	0.935875i	$-0.385389\pi$
$911$	−12.9443	−	9.40456i	−0.428863	−	0.311587i	−0.781194	+	0.624288i	$0.785389\pi$
$912$		0			0
$913$		0			0
$914$	−54.2538	+	93.9704i	−1.79456	+	3.10826i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	55.3651	+	24.6501i	1.82633	+	0.813133i	0.924922	+	0.380158i	$0.124130\pi$
$919$	55.3651	+	24.6501i	1.82633	+	0.813133i	0.901406	+	0.432976i	$0.142536\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−41.0547			−1.34987
$926$	−12.3124	+	117.145i	−0.404611	+	3.84961i
$927$		0			0
$928$	−150.821	−	32.0581i	−4.95096	−	1.05236i
$929$		0			0		0.406737	−	0.913545i	$-0.366667\pi$
$929$		0			0		−0.406737	+	0.913545i	$0.633333\pi$
$930$		0			0
$931$		0			0
$932$	40.6691	−	125.167i	1.33216	−	4.09997i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.587785	−	0.809017i	$-0.300000\pi$
$937$		0			0		−0.587785	+	0.809017i	$0.700000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.994522	−	0.104528i	$-0.0333333\pi$
$941$		0			0		−0.994522	+	0.104528i	$0.966667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	37.2820	+	49.3194i	1.21214	+	1.60351i
$947$	10.0000	+	17.3205i	0.324956	+	0.562841i	0.981504	−	0.191444i	$-0.0613171\pi$
$947$	10.0000	+	17.3205i	0.324956	+	0.562841i	−0.656547	+	0.754285i	$0.727984\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.682027	−	2.09906i	−0.0220930	−	0.0679954i	0.939402	−	0.342817i	$-0.111381\pi$
$953$	−0.682027	−	2.09906i	−0.0220930	−	0.0679954i	−0.961495	+	0.274822i	$0.911381\pi$
$954$	15.5004	−	3.29472i	0.501845	−	0.106670i
$955$		0			0
$956$	−7.79797	+	13.5065i	−0.252204	+	0.436831i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.7430	+	23.0375i	−0.669131	+	0.743145i
$962$		0			0
$963$	−0.359940	−	3.42460i	−0.0115989	−	0.110356i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−57.6533			−1.85401			−0.927003	−	0.375053i	$-0.877624\pi$
$967$	−57.6533			−1.85401			−0.927003	+	0.375053i	$0.877624\pi$
$968$	−114.887	−	46.0133i	−3.69260	−	1.47892i
$969$		0			0
$970$		0			0
$971$		0			0		−0.207912	−	0.978148i	$-0.566667\pi$
$971$		0			0		0.207912	+	0.978148i	$0.433333\pi$
$972$		0			0
$973$		0			0
$974$	5.38387	−	3.91161i	0.172510	−	0.125336i
$975$		0			0
$976$		0			0
$977$	−23.7936	+	10.5936i	−0.761225	+	0.338919i	−0.750367	−	0.661022i	$-0.770123\pi$
$977$	−23.7936	+	10.5936i	−0.761225	+	0.338919i	−0.0108586	+	0.999941i	$0.503456\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−10.9284	−	7.93992i	−0.348916	−	0.253502i
$982$	−121.596	+	25.8459i	−3.88027	+	0.824777i
$983$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$983$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−24.7576	−	42.8814i	−0.787246	−	1.36355i
$990$		0			0
$991$	12.0000	−	20.7846i	0.381193	−	0.660245i	−0.610040	−	0.792370i	$-0.708847\pi$
$991$	12.0000	−	20.7846i	0.381193	−	0.660245i	0.991233	+	0.132125i	$0.0421802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.207912	−	0.978148i	$-0.433333\pi$
$997$		0			0		−0.207912	+	0.978148i	$0.566667\pi$
$998$	−25.6600	−	28.4983i	−0.812253	−	0.902098i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.q.d.312.1		16
7.2	even	3	inner	539.2.q.d.422.2		16
7.3	odd	6		539.2.f.c.246.1	✓	8
7.4	even	3		539.2.f.c.246.1	✓	8
7.5	odd	6	inner	539.2.q.d.422.2		16
7.6	odd	2	CM	539.2.q.d.312.1		16
11.3	even	5	inner	539.2.q.d.410.2		16
77.3	odd	30		539.2.f.c.344.1	yes	8
77.17	even	30		5929.2.a.bg.1.1		4
77.25	even	15		539.2.f.c.344.1	yes	8
77.38	odd	30		5929.2.a.bc.1.4		4
77.39	odd	30		5929.2.a.bg.1.1		4
77.47	odd	30	inner	539.2.q.d.520.1		16
77.58	even	15	inner	539.2.q.d.520.1		16
77.60	even	15		5929.2.a.bc.1.4		4
77.69	odd	10	inner	539.2.q.d.410.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.f.c.246.1	✓	8	7.3	odd	6
539.2.f.c.246.1	✓	8	7.4	even	3
539.2.f.c.344.1	yes	8	77.3	odd	30
539.2.f.c.344.1	yes	8	77.25	even	15
539.2.q.d.312.1		16	1.1	even	1	trivial
539.2.q.d.312.1		16	7.6	odd	2	CM
539.2.q.d.410.2		16	11.3	even	5	inner
539.2.q.d.410.2		16	77.69	odd	10	inner
539.2.q.d.422.2		16	7.2	even	3	inner
539.2.q.d.422.2		16	7.5	odd	6	inner
539.2.q.d.520.1		16	77.47	odd	30	inner
539.2.q.d.520.1		16	77.58	even	15	inner
5929.2.a.bc.1.4		4	77.38	odd	30
5929.2.a.bc.1.4		4	77.60	even	15
5929.2.a.bg.1.1		4	77.17	even	30
5929.2.a.bg.1.1		4	77.39	odd	30

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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.q (of order \(15\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.295322 + 2.80980i) q^{2} +(-5.85146 - 1.24377i) q^{4} +(3.47668 - 10.7001i) q^{8} +(0.313585 - 2.98357i) q^{9} +(2.42220 - 2.26560i) q^{11} +(18.1084 + 8.06240i) q^{16} +(8.29061 + 1.76222i) q^{18} +(5.65055 + 7.47498i) q^{22} +(-3.75233 - 6.49922i) q^{23} +(-3.34565 - 3.71572i) q^{25} +(1.42225 + 4.37724i) q^{29} +(-16.7508 + 29.0132i) q^{32} +(-5.54579 + 17.0682i) q^{36} +(5.49419 - 6.10191i) q^{37} +6.59794 q^{43} +(-16.9913 + 10.2444i) q^{44} +(19.3696 - 8.62392i) q^{46} +(11.4285 - 8.30328i) q^{50} +(1.70800 - 0.760449i) q^{53} +(-12.7192 + 2.70355i) q^{58} +(-44.5014 - 32.3322i) q^{64} +(-3.04737 + 5.27819i) q^{67} +(7.95588 - 5.78028i) q^{71} +(-30.8343 - 13.7283i) q^{72} +(15.5226 + 17.2396i) q^{74} +(1.65186 - 15.7164i) q^{79} +(-8.80333 - 1.87121i) q^{81} +(-1.94851 + 18.5389i) q^{86} +(-15.8210 - 33.7946i) q^{88} +(13.8731 + 42.6969i) q^{92} +(-6.00000 - 7.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} - 2 q^{4} + 32 q^{8} - 6 q^{9} + 4 q^{11} + 28 q^{16} + 9 q^{18} - 8 q^{22} + 16 q^{23} - 10 q^{25} - 8 q^{29} - 100 q^{32} - 12 q^{36} + 18 q^{37} + 48 q^{43} - 9 q^{44} + 31 q^{46} + 20 q^{50}+ \cdots - 96 q^{99}+O(q^{100}) \) 16 * q + 2 * q^2 - 2 * q^4 + 32 * q^8 - 6 * q^9 + 4 * q^11 + 28 * q^16 + 9 * q^18 - 8 * q^22 + 16 * q^23 - 10 * q^25 - 8 * q^29 - 100 * q^32 - 12 * q^36 + 18 * q^37 + 48 * q^43 - 9 * q^44 + 31 * q^46 + 20 * q^50 + 30 * q^53 - 71 * q^58 - 148 * q^64 + 8 * q^67 + 96 * q^71 - 57 * q^72 - 12 * q^74 - 24 * q^79 + 18 * q^81 - 29 * q^86 + 38 * q^88 + 142 * q^92 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more