LMFDB - Embedded newform 576.4.k.b.433.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [576,4,Mod(145,576)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(576, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("576.145"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,-40] 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:	no
Analytic conductor:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.2
Character	$\chi$	$=$	576.433
Dual form			576.4.k.b.145.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-11.7911 - 11.7911i) q^{5} +12.5754i q^{7} +(-17.0042 - 17.0042i) q^{11} +(-49.2384 + 49.2384i) q^{13} +51.8247 q^{17} +(11.6655 - 11.6655i) q^{19} -74.5524i q^{23} +153.058i q^{25} +(-211.183 + 211.183i) q^{29} +326.094 q^{31} +(148.277 - 148.277i) q^{35} +(110.757 + 110.757i) q^{37} -348.225i q^{41} +(-205.838 - 205.838i) q^{43} +254.983 q^{47} +184.861 q^{49} +(225.602 + 225.602i) q^{53} +400.995i q^{55} +(285.442 + 285.442i) q^{59} +(286.952 - 286.952i) q^{61} +1161.15 q^{65} +(627.335 - 627.335i) q^{67} +274.784i q^{71} +298.190i q^{73} +(213.834 - 213.834i) q^{77} +175.664 q^{79} +(125.254 - 125.254i) q^{83} +(-611.068 - 611.068i) q^{85} +900.271i q^{89} +(-619.190 - 619.190i) q^{91} -275.096 q^{95} +5.27858 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79}+ \cdots - 7728 q^{95}+O(q^{100}) $ 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−11.7911	−	11.7911i	−1.05462	−	1.05462i	−0.998419	−	0.0562056i	$-0.982100\pi$
$5$	−11.7911	−	11.7911i	−1.05462	−	1.05462i	−0.0562056	−	0.998419i	$-0.517900\pi$
$6$		0			0
$7$			12.5754i			0.679005i	0.940605	+	0.339503i	$0.110259\pi$
$7$			12.5754i			0.679005i	−0.940605	+	0.339503i	$0.889741\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.0042	−	17.0042i	−0.466087	−	0.466087i	0.434557	−	0.900644i	$-0.356905\pi$
$11$	−17.0042	−	17.0042i	−0.466087	−	0.466087i	−0.900644	+	0.434557i	$0.856905\pi$
$12$		0			0
$13$	−49.2384	+	49.2384i	−1.05048	+	1.05048i	−0.0518270	+	0.998656i	$0.516504\pi$
$13$	−49.2384	+	49.2384i	−1.05048	+	1.05048i	−0.998656	+	0.0518270i	$0.983496\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	51.8247			0.739372			0.369686	−	0.929157i	$-0.379465\pi$
$17$	51.8247			0.739372			0.369686	+	0.929157i	$0.379465\pi$
$18$		0			0
$19$	11.6655	−	11.6655i	0.140855	−	0.140855i	−0.633163	−	0.774018i	$-0.718244\pi$
$19$	11.6655	−	11.6655i	0.140855	−	0.140855i	0.774018	+	0.633163i	$0.218244\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	74.5524i		−	0.675881i	−0.941168	−	0.337940i	$-0.890270\pi$
$23$		−	74.5524i		−	0.675881i	0.941168	−	0.337940i	$-0.109730\pi$
$24$		0			0
$25$			153.058i			1.22447i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−211.183	+	211.183i	−1.35227	+	1.35227i	−0.469143	+	0.883122i	$0.655437\pi$
$29$	−211.183	+	211.183i	−1.35227	+	1.35227i	−0.883122	+	0.469143i	$0.844563\pi$
$30$		0			0
$31$	326.094			1.88930			0.944648	−	0.328084i	$-0.106403\pi$
$31$	326.094			1.88930			0.944648	+	0.328084i	$0.106403\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	148.277	−	148.277i	0.716096	−	0.716096i
$36$		0			0
$37$	110.757	+	110.757i	0.492117	+	0.492117i	0.908973	−	0.416855i	$-0.136868\pi$
$37$	110.757	+	110.757i	0.492117	+	0.492117i	−0.416855	+	0.908973i	$0.636868\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	348.225i		−	1.32643i	−0.748430	−	0.663214i	$-0.769192\pi$
$41$		−	348.225i		−	1.32643i	0.748430	−	0.663214i	$-0.230808\pi$
$42$		0			0
$43$	−205.838	−	205.838i	−0.729998	−	0.729998i	0.240621	−	0.970619i	$-0.422649\pi$
$43$	−205.838	−	205.838i	−0.729998	−	0.729998i	−0.970619	+	0.240621i	$0.922649\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	254.983			0.791342			0.395671	−	0.918392i	$-0.370512\pi$
$47$	254.983			0.791342			0.395671	+	0.918392i	$0.370512\pi$
$48$		0			0
$49$	184.861			0.538952
$50$		0			0
$51$		0			0
$52$		0			0
$53$	225.602	+	225.602i	0.584694	+	0.584694i	0.936189	−	0.351496i	$-0.114327\pi$
$53$	225.602	+	225.602i	0.584694	+	0.584694i	−0.351496	+	0.936189i	$0.614327\pi$
$54$		0			0
$55$			400.995i			0.983094i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	285.442	+	285.442i	0.629854	+	0.629854i	0.948031	−	0.318177i	$-0.103071\pi$
$59$	285.442	+	285.442i	0.629854	+	0.629854i	−0.318177	+	0.948031i	$0.603071\pi$
$60$		0			0
$61$	286.952	−	286.952i	0.602301	−	0.602301i	−0.338621	−	0.940923i	$-0.609961\pi$
$61$	286.952	−	286.952i	0.602301	−	0.602301i	0.940923	+	0.338621i	$0.109961\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1161.15			2.21573
$66$		0			0
$67$	627.335	−	627.335i	1.14390	−	1.14390i	0.156168	−	0.987731i	$-0.450086\pi$
$67$	627.335	−	627.335i	1.14390	−	1.14390i	0.987731	−	0.156168i	$-0.0499141\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			274.784i			0.459308i	0.973272	+	0.229654i	$0.0737595\pi$
$71$			274.784i			0.459308i	−0.973272	+	0.229654i	$0.926241\pi$
$72$		0			0
$73$			298.190i			0.478089i	0.971009	+	0.239044i	$0.0768341\pi$
$73$			298.190i			0.478089i	−0.971009	+	0.239044i	$0.923166\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	213.834	−	213.834i	0.316475	−	0.316475i
$78$		0			0
$79$	175.664			0.250174			0.125087	−	0.992146i	$-0.460079\pi$
$79$	175.664			0.250174			0.125087	+	0.992146i	$0.460079\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	125.254	−	125.254i	0.165644	−	0.165644i	−0.619418	−	0.785062i	$-0.712631\pi$
$83$	125.254	−	125.254i	0.165644	−	0.165644i	0.785062	+	0.619418i	$0.212631\pi$
$84$		0			0
$85$	−611.068	−	611.068i	−0.779760	−	0.779760i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			900.271i			1.07223i	0.844145	+	0.536116i	$0.180109\pi$
$89$			900.271i			1.07223i	−0.844145	+	0.536116i	$0.819891\pi$
$90$		0			0
$91$	−619.190	−	619.190i	−0.713283	−	0.713283i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−275.096			−0.297098
$96$		0			0
$97$	5.27858			0.00552534			0.00276267	−	0.999996i	$-0.499121\pi$
$97$	5.27858			0.00552534			0.00276267	+	0.999996i	$0.499121\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	459.109	+	459.109i	0.452307	+	0.452307i	0.896120	−	0.443812i	$-0.146374\pi$
$101$	459.109	+	459.109i	0.452307	+	0.452307i	−0.443812	+	0.896120i	$0.646374\pi$
$102$		0			0
$103$			791.173i			0.756860i	0.925630	+	0.378430i	$0.123536\pi$
$103$			791.173i			0.756860i	−0.925630	+	0.378430i	$0.876464\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1525.81	+	1525.81i	1.37855	+	1.37855i	0.847067	+	0.531487i	$0.178366\pi$
$107$	1525.81	+	1525.81i	1.37855	+	1.37855i	0.531487	+	0.847067i	$0.321634\pi$
$108$		0			0
$109$	−413.307	+	413.307i	−0.363189	+	0.363189i	−0.864986	−	0.501796i	$-0.832673\pi$
$109$	−413.307	+	413.307i	−0.363189	+	0.363189i	0.501796	+	0.864986i	$0.332673\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−210.248			−0.175031			−0.0875154	−	0.996163i	$-0.527893\pi$
$113$	−210.248			−0.175031			−0.0875154	+	0.996163i	$0.527893\pi$
$114$		0			0
$115$	−879.053	+	879.053i	−0.712801	+	0.712801i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			651.714i			0.502038i
$120$		0			0
$121$		−	752.715i		−	0.565526i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	330.838	−	330.838i	0.236729	−	0.236729i
$126$		0			0
$127$	−177.367			−0.123927			−0.0619637	−	0.998078i	$-0.519736\pi$
$127$	−177.367			−0.123927			−0.0619637	+	0.998078i	$0.519736\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	938.828	−	938.828i	0.626151	−	0.626151i	−0.320946	−	0.947097i	$-0.604001\pi$
$131$	938.828	−	938.828i	0.626151	−	0.626151i	0.947097	+	0.320946i	$0.104001\pi$
$132$		0			0
$133$	146.697	+	146.697i	0.0956411	+	0.0956411i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			307.429i			0.191718i	0.995395	+	0.0958592i	$0.0305598\pi$
$137$			307.429i			0.191718i	−0.995395	+	0.0958592i	$0.969440\pi$
$138$		0			0
$139$	1253.43	+	1253.43i	0.764852	+	0.764852i	0.977195	−	0.212343i	$-0.0681095\pi$
$139$	1253.43	+	1253.43i	0.764852	+	0.764852i	−0.212343	+	0.977195i	$0.568109\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1674.52			0.979233
$144$		0			0
$145$	4980.14			2.85226
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1686.80	−	1686.80i	−0.927436	−	0.927436i	0.0701033	−	0.997540i	$-0.477667\pi$
$149$	−1686.80	−	1686.80i	−0.927436	−	0.927436i	−0.997540	+	0.0701033i	$0.977667\pi$
$150$		0			0
$151$			1682.23i			0.906611i	0.891355	+	0.453306i	$0.149755\pi$
$151$			1682.23i			0.906611i	−0.891355	+	0.453306i	$0.850245\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3844.99	−	3844.99i	−1.99250	−	1.99250i
$156$		0			0
$157$	−1066.81	+	1066.81i	−0.542300	+	0.542300i	−0.924202	−	0.381903i	$-0.875269\pi$
$157$	−1066.81	+	1066.81i	−0.542300	+	0.542300i	0.381903	+	0.924202i	$0.375269\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	937.523			0.458927
$162$		0			0
$163$	2379.54	−	2379.54i	1.14343	−	1.14343i	0.155618	−	0.987817i	$-0.450263\pi$
$163$	2379.54	−	2379.54i	1.14343	−	1.14343i	0.987817	−	0.155618i	$-0.0497367\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			839.991i			0.389224i	0.980880	+	0.194612i	$0.0623448\pi$
$167$			839.991i			0.389224i	−0.980880	+	0.194612i	$0.937655\pi$
$168$		0			0
$169$		−	2651.84i		−	1.20703i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	450.278	−	450.278i	0.197885	−	0.197885i	−0.601208	−	0.799093i	$-0.705314\pi$
$173$	450.278	−	450.278i	0.197885	−	0.197885i	0.799093	+	0.601208i	$0.205314\pi$
$174$		0			0
$175$	−1924.76			−0.831419
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−647.759	+	647.759i	−0.270479	+	0.270479i	−0.829293	−	0.558814i	$-0.811257\pi$
$179$	−647.759	+	647.759i	−0.270479	+	0.270479i	0.558814	+	0.829293i	$0.311257\pi$
$180$		0			0
$181$	−755.750	−	755.750i	−0.310356	−	0.310356i	0.534691	−	0.845047i	$-0.320428\pi$
$181$	−755.750	−	755.750i	−0.310356	−	0.310356i	−0.845047	+	0.534691i	$0.820428\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	2611.89i		−	1.03800i
$186$		0			0
$187$	−881.237	−	881.237i	−0.344612	−	0.344612i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	760.485			0.288098			0.144049	−	0.989571i	$-0.453988\pi$
$191$	760.485			0.288098			0.144049	+	0.989571i	$0.453988\pi$
$192$		0			0
$193$	2184.99			0.814915			0.407458	−	0.913224i	$-0.366415\pi$
$193$	2184.99			0.814915			0.407458	+	0.913224i	$0.366415\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−83.4121	−	83.4121i	−0.0301668	−	0.0301668i	0.691862	−	0.722029i	$-0.256790\pi$
$197$	−83.4121	−	83.4121i	−0.0301668	−	0.0301668i	−0.722029	+	0.691862i	$0.756790\pi$
$198$		0			0
$199$			1734.18i			0.617754i	0.951102	+	0.308877i	$0.0999531\pi$
$199$			1734.18i			0.617754i	−0.951102	+	0.308877i	$0.900047\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2655.70	−	2655.70i	−0.918195	−	0.918195i
$204$		0			0
$205$	−4105.94	+	4105.94i	−1.39888	+	1.39888i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−396.723			−0.131301
$210$		0			0
$211$	365.900	−	365.900i	0.119382	−	0.119382i	−0.644892	−	0.764274i	$-0.723098\pi$
$211$	365.900	−	365.900i	0.119382	−	0.119382i	0.764274	+	0.644892i	$0.223098\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4854.09i			1.53975i
$216$		0			0
$217$			4100.75i			1.28284i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2551.77	+	2551.77i	−0.776698	+	0.776698i
$222$		0			0
$223$	4037.17			1.21233			0.606164	−	0.795340i	$-0.292708\pi$
$223$	4037.17			1.21233			0.606164	+	0.795340i	$0.292708\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1502.81	+	1502.81i	−0.439407	+	0.439407i	−0.891812	−	0.452406i	$-0.850566\pi$
$227$	−1502.81	+	1502.81i	−0.439407	+	0.439407i	0.452406	+	0.891812i	$0.350566\pi$
$228$		0			0
$229$	−3443.65	−	3443.65i	−0.993723	−	0.993723i	0.00625772	−	0.999980i	$-0.498008\pi$
$229$	−3443.65	−	3443.65i	−0.993723	−	0.993723i	−0.999980	+	0.00625772i	$0.998008\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1139.00i			0.320250i	0.987097	+	0.160125i	$0.0511898\pi$
$233$			1139.00i			0.320250i	−0.987097	+	0.160125i	$0.948810\pi$
$234$		0			0
$235$	−3006.52	−	3006.52i	−0.834569	−	0.834569i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4603.80			−1.24600			−0.623002	−	0.782220i	$-0.714087\pi$
$239$	−4603.80			−1.24600			−0.623002	+	0.782220i	$0.714087\pi$
$240$		0			0
$241$	−4681.58			−1.25132			−0.625658	−	0.780097i	$-0.715170\pi$
$241$	−4681.58			−1.25132			−0.625658	+	0.780097i	$0.715170\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2179.70	−	2179.70i	−0.568392	−	0.568392i
$246$		0			0
$247$			1148.78i			0.295931i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−476.440	−	476.440i	−0.119811	−	0.119811i	0.644659	−	0.764470i	$-0.276999\pi$
$251$	−476.440	−	476.440i	−0.119811	−	0.119811i	−0.764470	+	0.644659i	$0.776999\pi$
$252$		0			0
$253$	−1267.70	+	1267.70i	−0.315019	+	0.315019i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−925.326			−0.224592			−0.112296	−	0.993675i	$-0.535821\pi$
$257$	−925.326			−0.224592			−0.112296	+	0.993675i	$0.535821\pi$
$258$		0			0
$259$	−1392.81	+	1392.81i	−0.334150	+	0.334150i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	3477.65i		−	0.815365i	−0.913124	−	0.407683i	$-0.866337\pi$
$263$		−	3477.65i		−	0.815365i	0.913124	−	0.407683i	$-0.133663\pi$
$264$		0			0
$265$		−	5320.17i		−	1.23327i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3773.37	+	3773.37i	−0.855266	+	0.855266i	−0.990776	−	0.135510i	$-0.956733\pi$
$269$	−3773.37	+	3773.37i	−0.855266	+	0.855266i	0.135510	+	0.990776i	$0.456733\pi$
$270$		0			0
$271$	−8007.09			−1.79482			−0.897410	−	0.441198i	$-0.854554\pi$
$271$	−8007.09			−1.79482			−0.897410	+	0.441198i	$0.854554\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2602.63	−	2602.63i	0.570708	−	0.570708i
$276$		0			0
$277$	4550.73	+	4550.73i	0.987101	+	0.987101i	0.999918	−	0.0128173i	$-0.00407998\pi$
$277$	4550.73	+	4550.73i	0.987101	+	0.987101i	−0.0128173	+	0.999918i	$0.504080\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3333.05i			0.707592i	0.935323	+	0.353796i	$0.115109\pi$
$281$			3333.05i			0.707592i	−0.935323	+	0.353796i	$0.884891\pi$
$282$		0			0
$283$	−6242.23	−	6242.23i	−1.31117	−	1.31117i	−0.920552	−	0.390621i	$-0.872260\pi$
$283$	−6242.23	−	6242.23i	−1.31117	−	1.31117i	−0.390621	−	0.920552i	$-0.627740\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4379.05			0.900651
$288$		0			0
$289$	−2227.20			−0.453329
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2458.24	+	2458.24i	0.490143	+	0.490143i	0.908351	−	0.418208i	$-0.137342\pi$
$293$	2458.24	+	2458.24i	0.490143	+	0.490143i	−0.418208	+	0.908351i	$0.637342\pi$
$294$		0			0
$295$		−	6731.33i		−	1.32852i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3670.84	+	3670.84i	0.710001	+	0.710001i
$300$		0			0
$301$	2588.48	−	2588.48i	0.495673	−	0.495673i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6766.93			−1.27040
$306$		0			0
$307$	3008.49	−	3008.49i	0.559295	−	0.559295i	−0.369812	−	0.929107i	$-0.620578\pi$
$307$	3008.49	−	3008.49i	0.559295	−	0.559295i	0.929107	+	0.369812i	$0.120578\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	4507.09i		−	0.821779i	−0.911685	−	0.410890i	$-0.865218\pi$
$311$		−	4507.09i		−	0.821779i	0.911685	−	0.410890i	$-0.134782\pi$
$312$		0			0
$313$			5237.61i			0.945837i	0.881106	+	0.472919i	$0.156800\pi$
$313$			5237.61i			0.945837i	−0.881106	+	0.472919i	$0.843200\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	989.530	−	989.530i	0.175323	−	0.175323i	−0.613990	−	0.789314i	$-0.710437\pi$
$317$	989.530	−	989.530i	0.175323	−	0.175323i	0.789314	+	0.613990i	$0.210437\pi$
$318$		0			0
$319$	7181.99			1.26055
$320$		0			0
$321$		0			0
$322$		0			0
$323$	604.559	−	604.559i	0.104144	−	0.104144i
$324$		0			0
$325$	−7536.35	−	7536.35i	−1.28628	−	1.28628i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			3206.50i			0.537325i
$330$		0			0
$331$	5652.55	+	5652.55i	0.938647	+	0.938647i	0.998224	−	0.0595765i	$-0.0189750\pi$
$331$	5652.55	+	5652.55i	0.938647	+	0.938647i	−0.0595765	+	0.998224i	$0.518975\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14793.9			−2.41277
$336$		0			0
$337$	291.315			0.0470888			0.0235444	−	0.999723i	$-0.492505\pi$
$337$	291.315			0.0470888			0.0235444	+	0.999723i	$0.492505\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5544.96	−	5544.96i	−0.880577	−	0.880577i
$342$		0			0
$343$			6638.03i			1.04496i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6530.87	+	6530.87i	1.01036	+	1.01036i	0.999946	+	0.0104158i	$0.00331552\pi$
$347$	6530.87	+	6530.87i	1.01036	+	1.01036i	0.0104158	+	0.999946i	$0.496684\pi$
$348$		0			0
$349$	7267.79	−	7267.79i	1.11472	−	1.11472i	0.122213	−	0.992504i	$-0.461001\pi$
$349$	7267.79	−	7267.79i	1.11472	−	1.11472i	0.992504	−	0.122213i	$-0.0389990\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	8253.63			1.24447			0.622233	−	0.782832i	$-0.286226\pi$
$353$	8253.63			1.24447			0.622233	+	0.782832i	$0.286226\pi$
$354$		0			0
$355$	3240.00	−	3240.00i	0.484398	−	0.484398i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	4827.70i		−	0.709738i	−0.934916	−	0.354869i	$-0.884525\pi$
$359$		−	4827.70i		−	0.709738i	0.934916	−	0.354869i	$-0.115475\pi$
$360$		0			0
$361$			6586.83i			0.960320i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3515.97	−	3515.97i	0.504204	−	0.504204i
$366$		0			0
$367$	2556.37			0.363600			0.181800	−	0.983336i	$-0.441808\pi$
$367$	2556.37			0.363600			0.181800	+	0.983336i	$0.441808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2837.02	+	2837.02i	−0.397010	+	0.397010i
$372$		0			0
$373$	1141.57	+	1141.57i	0.158468	+	0.158468i	0.781887	−	0.623420i	$-0.214257\pi$
$373$	1141.57	+	1141.57i	0.158468	+	0.158468i	−0.623420	+	0.781887i	$0.714257\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	20796.6i		−	2.84106i
$378$		0			0
$379$	2602.11	+	2602.11i	0.352668	+	0.352668i	0.861101	−	0.508433i	$-0.169775\pi$
$379$	2602.11	+	2602.11i	0.352668	+	0.352668i	−0.508433	+	0.861101i	$0.669775\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−370.858			−0.0494777			−0.0247388	−	0.999694i	$-0.507875\pi$
$383$	−370.858			−0.0494777			−0.0247388	+	0.999694i	$0.507875\pi$
$384$		0			0
$385$	−5042.65			−0.667526
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6457.55	+	6457.55i	0.841673	+	0.841673i	0.989077	−	0.147403i	$-0.0470914\pi$
$389$	6457.55	+	6457.55i	0.841673	+	0.841673i	−0.147403	+	0.989077i	$0.547091\pi$
$390$		0			0
$391$		−	3863.66i		−	0.499728i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2071.27	−	2071.27i	−0.263840	−	0.263840i
$396$		0			0
$397$	−232.353	+	232.353i	−0.0293739	+	0.0293739i	−0.721641	−	0.692267i	$-0.756612\pi$
$397$	−232.353	+	232.353i	−0.0293739	+	0.0293739i	0.692267	+	0.721641i	$0.256612\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	119.615			0.0148960			0.00744801	−	0.999972i	$-0.497629\pi$
$401$	119.615			0.0148960			0.00744801	+	0.999972i	$0.497629\pi$
$402$		0			0
$403$	−16056.4	+	16056.4i	−1.98467	+	1.98467i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	3766.67i		−	0.458739i
$408$		0			0
$409$			626.952i			0.0757965i	0.999282	+	0.0378983i	$0.0120663\pi$
$409$			626.952i			0.0757965i	−0.999282	+	0.0378983i	$0.987934\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3589.54	+	3589.54i	−0.427674	+	0.427674i
$414$		0			0
$415$	−2953.76			−0.349384
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4981.58	+	4981.58i	−0.580826	+	0.580826i	−0.935130	−	0.354304i	$-0.884718\pi$
$419$	−4981.58	+	4981.58i	−0.580826	+	0.580826i	0.354304	+	0.935130i	$0.384718\pi$
$420$		0			0
$421$	10770.4	+	10770.4i	1.24683	+	1.24683i	0.957110	+	0.289723i	$0.0935632\pi$
$421$	10770.4	+	10770.4i	1.24683	+	1.24683i	0.289723	+	0.957110i	$0.406437\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			7932.20i			0.905337i
$426$		0			0
$427$	3608.52	+	3608.52i	0.408966	+	0.408966i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	7546.94			0.843442			0.421721	−	0.906726i	$-0.361426\pi$
$431$	7546.94			0.843442			0.421721	+	0.906726i	$0.361426\pi$
$432$		0			0
$433$	5614.17			0.623094			0.311547	−	0.950231i	$-0.399153\pi$
$433$	5614.17			0.623094			0.311547	+	0.950231i	$0.399153\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−869.689	−	869.689i	−0.0952010	−	0.0952010i
$438$		0			0
$439$			7198.92i			0.782656i	0.920251	+	0.391328i	$0.127984\pi$
$439$			7198.92i			0.782656i	−0.920251	+	0.391328i	$0.872016\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5582.95	−	5582.95i	−0.598767	−	0.598767i	0.341217	−	0.939984i	$-0.389161\pi$
$443$	−5582.95	−	5582.95i	−0.598767	−	0.598767i	−0.939984	+	0.341217i	$0.889161\pi$
$444$		0			0
$445$	10615.2	−	10615.2i	1.13080	−	1.13080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1224.53			−0.128706			−0.0643530	−	0.997927i	$-0.520498\pi$
$449$	−1224.53			−0.128706			−0.0643530	+	0.997927i	$0.520498\pi$
$450$		0			0
$451$	−5921.28	+	5921.28i	−0.618231	+	0.618231i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			14601.8i			1.50449i
$456$		0			0
$457$		−	11182.3i		−	1.14461i	−0.820040	−	0.572307i	$-0.806049\pi$
$457$		−	11182.3i		−	1.14461i	0.820040	−	0.572307i	$-0.193951\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	11334.9	−	11334.9i	1.14516	−	1.14516i	0.157670	−	0.987492i	$-0.449602\pi$
$461$	11334.9	−	11334.9i	1.14516	−	1.14516i	0.987492	−	0.157670i	$-0.0503982\pi$
$462$		0			0
$463$	−3013.37			−0.302469			−0.151234	−	0.988498i	$-0.548325\pi$
$463$	−3013.37			−0.302469			−0.151234	+	0.988498i	$0.548325\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−843.917	+	843.917i	−0.0836228	+	0.0836228i	−0.747681	−	0.664058i	$-0.768833\pi$
$467$	−843.917	+	843.917i	−0.0836228	+	0.0836228i	0.664058	+	0.747681i	$0.268833\pi$
$468$		0			0
$469$	7888.96	+	7888.96i	0.776713	+	0.776713i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			7000.20i			0.680485i
$474$		0			0
$475$	1785.50	+	1785.50i	0.172472	+	0.172472i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10569.4			1.00821			0.504103	−	0.863644i	$-0.331823\pi$
$479$	10569.4			1.00821			0.504103	+	0.863644i	$0.331823\pi$
$480$		0			0
$481$	−10907.0			−1.03392
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−62.2400	−	62.2400i	−0.00582716	−	0.00582716i
$486$		0			0
$487$		−	1579.10i		−	0.146932i	−0.997298	−	0.0734661i	$-0.976594\pi$
$487$		−	1579.10i		−	0.146932i	0.997298	−	0.0734661i	$-0.0234061\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	8650.50	+	8650.50i	0.795095	+	0.795095i	0.982318	−	0.187222i	$-0.0599485\pi$
$491$	8650.50	+	8650.50i	0.795095	+	0.795095i	−0.187222	+	0.982318i	$0.559948\pi$
$492$		0			0
$493$	−10944.5	+	10944.5i	−0.999828	+	0.999828i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3455.51			−0.311873
$498$		0			0
$499$	11942.3	−	11942.3i	1.07137	−	1.07137i	0.0741168	−	0.997250i	$-0.476386\pi$
$499$	11942.3	−	11942.3i	1.07137	−	1.07137i	0.997250	−	0.0741168i	$-0.0236137\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	3290.09i		−	0.291646i	−0.989311	−	0.145823i	$-0.953417\pi$
$503$		−	3290.09i		−	0.291646i	0.989311	−	0.145823i	$-0.0465830\pi$
$504$		0			0
$505$		−	10826.8i		−	0.954029i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1392.67	+	1392.67i	−0.121275	+	0.121275i	−0.765140	−	0.643864i	$-0.777330\pi$
$509$	−1392.67	+	1392.67i	−0.121275	+	0.121275i	0.643864	+	0.765140i	$0.277330\pi$
$510$		0			0
$511$	−3749.84			−0.324625
$512$		0			0
$513$		0			0
$514$		0			0
$515$	9328.77	−	9328.77i	0.798203	−	0.798203i
$516$		0			0
$517$	−4335.78	−	4335.78i	−0.368834	−	0.368834i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	1175.72i		−	0.0988661i	−0.998777	−	0.0494331i	$-0.984259\pi$
$521$		−	1175.72i		−	0.0988661i	0.998777	−	0.0494331i	$-0.0157414\pi$
$522$		0			0
$523$	9451.05	+	9451.05i	0.790183	+	0.790183i	0.981524	−	0.191341i	$-0.0612836\pi$
$523$	9451.05	+	9451.05i	0.790183	+	0.790183i	−0.191341	+	0.981524i	$0.561284\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16899.7			1.39689
$528$		0			0
$529$	6608.93			0.543185
$530$		0			0
$531$		0			0
$532$		0			0
$533$	17146.0	+	17146.0i	1.39339	+	1.39339i
$534$		0			0
$535$		−	35981.7i		−	2.90771i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−3143.40	−	3143.40i	−0.251198	−	0.251198i
$540$		0			0
$541$	2893.97	−	2893.97i	0.229984	−	0.229984i	−0.582702	−	0.812686i	$-0.698004\pi$
$541$	2893.97	−	2893.97i	0.229984	−	0.229984i	0.812686	+	0.582702i	$0.198004\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9746.66			0.766057
$546$		0			0
$547$	956.857	−	956.857i	0.0747939	−	0.0747939i	−0.668720	−	0.743514i	$-0.733158\pi$
$547$	956.857	−	956.857i	0.0747939	−	0.0747939i	0.743514	+	0.668720i	$0.233158\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			4927.09i			0.380946i
$552$		0			0
$553$			2209.04i			0.169870i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8482.07	−	8482.07i	0.645236	−	0.645236i	−0.306602	−	0.951838i	$-0.599192\pi$
$557$	8482.07	−	8482.07i	0.645236	−	0.645236i	0.951838	+	0.306602i	$0.0991919\pi$
$558$		0			0
$559$	20270.2			1.53370
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9461.78	−	9461.78i	0.708289	−	0.708289i	−0.257886	−	0.966175i	$-0.583026\pi$
$563$	9461.78	−	9461.78i	0.708289	−	0.708289i	0.966175	+	0.257886i	$0.0830260\pi$
$564$		0			0
$565$	2479.05	+	2479.05i	0.184592	+	0.184592i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			7197.61i			0.530298i	0.964207	+	0.265149i	$0.0854211\pi$
$569$			7197.61i			0.530298i	−0.964207	+	0.265149i	$0.914579\pi$
$570$		0			0
$571$	−990.936	−	990.936i	−0.0726259	−	0.0726259i	0.669861	−	0.742487i	$-0.266354\pi$
$571$	−990.936	−	990.936i	−0.0726259	−	0.0726259i	−0.742487	+	0.669861i	$0.766354\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	11410.9			0.827594
$576$		0			0
$577$	−14836.1			−1.07043			−0.535214	−	0.844717i	$-0.679769\pi$
$577$	−14836.1			−1.07043			−0.535214	+	0.844717i	$0.679769\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1575.11	+	1575.11i	0.112473	+	0.112473i
$582$		0			0
$583$		−	7672.35i		−	0.545036i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4038.29	+	4038.29i	0.283949	+	0.283949i	0.834682	−	0.550733i	$-0.185652\pi$
$587$	4038.29	+	4038.29i	0.283949	+	0.283949i	−0.550733	+	0.834682i	$0.685652\pi$
$588$		0			0
$589$	3804.04	−	3804.04i	0.266116	−	0.266116i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11081.1			−0.767361			−0.383681	−	0.923466i	$-0.625344\pi$
$593$	−11081.1			−0.767361			−0.383681	+	0.923466i	$0.625344\pi$
$594$		0			0
$595$	7684.40	−	7684.40i	0.529461	−	0.529461i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	7036.50i		−	0.479973i	−0.970776	−	0.239986i	$-0.922857\pi$
$599$		−	7036.50i		−	0.479973i	0.970776	−	0.239986i	$-0.0771430\pi$
$600$		0			0
$601$			24290.7i			1.64865i	0.566117	+	0.824325i	$0.308445\pi$
$601$			24290.7i			1.64865i	−0.566117	+	0.824325i	$0.691555\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−8875.31	+	8875.31i	−0.596418	+	0.596418i
$606$		0			0
$607$	−10931.5			−0.730967			−0.365484	−	0.930818i	$-0.619096\pi$
$607$	−10931.5			−0.730967			−0.365484	+	0.930818i	$0.619096\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12555.0	+	12555.0i	−0.831292	+	0.831292i
$612$		0			0
$613$	−2217.95	−	2217.95i	−0.146137	−	0.146137i	0.630253	−	0.776390i	$-0.282951\pi$
$613$	−2217.95	−	2217.95i	−0.146137	−	0.146137i	−0.776390	+	0.630253i	$0.782951\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26271.5i			1.71418i	0.515164	+	0.857092i	$0.327731\pi$
$617$			26271.5i			1.71418i	−0.515164	+	0.857092i	$0.672269\pi$
$618$		0			0
$619$	11171.8	+	11171.8i	0.725417	+	0.725417i	0.969703	−	0.244287i	$-0.0785538\pi$
$619$	11171.8	+	11171.8i	0.725417	+	0.725417i	−0.244287	+	0.969703i	$0.578554\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11321.2			−0.728050
$624$		0			0
$625$	11330.4			0.725148
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5739.95	+	5739.95i	0.363858	+	0.363858i
$630$		0			0
$631$		−	17415.1i		−	1.09871i	−0.835590	−	0.549354i	$-0.814874\pi$
$631$		−	17415.1i		−	1.09871i	0.835590	−	0.549354i	$-0.185126\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2091.34	+	2091.34i	0.130697	+	0.130697i
$636$		0			0
$637$	−9102.24	+	9102.24i	−0.566160	+	0.566160i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2727.28			0.168051			0.0840257	−	0.996464i	$-0.473222\pi$
$641$	2727.28			0.168051			0.0840257	+	0.996464i	$0.473222\pi$
$642$		0			0
$643$	−11681.7	+	11681.7i	−0.716458	+	0.716458i	−0.967878	−	0.251420i	$-0.919103\pi$
$643$	−11681.7	+	11681.7i	−0.716458	+	0.716458i	0.251420	+	0.967878i	$0.419103\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	4399.67i		−	0.267340i	−0.991026	−	0.133670i	$-0.957324\pi$
$647$		−	4399.67i		−	0.267340i	0.991026	−	0.133670i	$-0.0426762\pi$
$648$		0			0
$649$		−	9707.43i		−	0.587134i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	22924.8	−	22924.8i	1.37384	−	1.37384i	0.519167	−	0.854673i	$-0.326242\pi$
$653$	22924.8	−	22924.8i	1.37384	−	1.37384i	0.854673	−	0.519167i	$-0.173758\pi$
$654$		0			0
$655$	−22139.6			−1.32071
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−11508.4	+	11508.4i	−0.680276	+	0.680276i	−0.960062	−	0.279786i	$-0.909736\pi$
$659$	−11508.4	+	11508.4i	−0.680276	+	0.680276i	0.279786	+	0.960062i	$0.409736\pi$
$660$		0			0
$661$	10207.0	+	10207.0i	0.600615	+	0.600615i	0.940476	−	0.339861i	$-0.110380\pi$
$661$	10207.0	+	10207.0i	0.600615	+	0.600615i	−0.339861	+	0.940476i	$0.610380\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3459.43i		−	0.201731i
$666$		0			0
$667$	15744.2	+	15744.2i	0.913970	+	0.913970i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9758.76			−0.561450
$672$		0			0
$673$	−23908.4			−1.36939			−0.684695	−	0.728830i	$-0.740065\pi$
$673$	−23908.4			−1.36939			−0.684695	+	0.728830i	$0.740065\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−10411.1	−	10411.1i	−0.591035	−	0.591035i	0.346876	−	0.937911i	$-0.387242\pi$
$677$	−10411.1	−	10411.1i	−0.591035	−	0.591035i	−0.937911	+	0.346876i	$0.887242\pi$
$678$		0			0
$679$			66.3800i			0.00375174i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11126.8	+	11126.8i	0.623358	+	0.623358i	0.946389	−	0.323030i	$-0.104702\pi$
$683$	11126.8	+	11126.8i	0.623358	+	0.623358i	−0.323030	+	0.946389i	$0.604702\pi$
$684$		0			0
$685$	3624.91	−	3624.91i	0.202191	−	0.202191i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22216.5			−1.22842
$690$		0			0
$691$	−2722.85	+	2722.85i	−0.149901	+	0.149901i	−0.778074	−	0.628173i	$-0.783803\pi$
$691$	−2722.85	+	2722.85i	−0.149901	+	0.149901i	0.628173	+	0.778074i	$0.283803\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	29558.5i		−	1.61326i
$696$		0			0
$697$		−	18046.6i		−	0.980724i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2736.38	+	2736.38i	−0.147435	+	0.147435i	−0.776971	−	0.629536i	$-0.783245\pi$
$701$	−2736.38	+	2736.38i	−0.147435	+	0.147435i	0.629536	+	0.776971i	$0.283245\pi$
$702$		0			0
$703$	2584.06			0.138634
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−5773.45	+	5773.45i	−0.307119	+	0.307119i
$708$		0			0
$709$	−783.090	−	783.090i	−0.0414803	−	0.0414803i	0.686062	−	0.727543i	$-0.259338\pi$
$709$	−783.090	−	783.090i	−0.0414803	−	0.0414803i	−0.727543	+	0.686062i	$0.759338\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	24311.1i		−	1.27694i
$714$		0			0
$715$	−19744.4	−	19744.4i	−1.03272	−	1.03272i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11021.5			−0.571671			−0.285835	−	0.958279i	$-0.592271\pi$
$719$	−11021.5			−0.571671			−0.285835	+	0.958279i	$0.592271\pi$
$720$		0			0
$721$	−9949.27			−0.513912
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−32323.3	−	32323.3i	−1.65580	−	1.65580i
$726$		0			0
$727$			21740.4i			1.10909i	0.832154	+	0.554544i	$0.187107\pi$
$727$			21740.4i			1.10909i	−0.832154	+	0.554544i	$0.812893\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−10667.5	−	10667.5i	−0.539741	−	0.539741i
$732$		0			0
$733$	13194.8	−	13194.8i	0.664886	−	0.664886i	−0.291642	−	0.956528i	$-0.594202\pi$
$733$	13194.8	−	13194.8i	0.664886	−	0.664886i	0.956528	+	0.291642i	$0.0942015\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21334.7			−1.06631
$738$		0			0
$739$	−21786.6	+	21786.6i	−1.08448	+	1.08448i	−0.0883973	+	0.996085i	$0.528175\pi$
$739$	−21786.6	+	21786.6i	−1.08448	+	1.08448i	−0.996085	+	0.0883973i	$0.971825\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			7418.24i			0.366284i	0.983086	+	0.183142i	$0.0586268\pi$
$743$			7418.24i			0.366284i	−0.983086	+	0.183142i	$0.941373\pi$
$744$		0			0
$745$			39778.3i			1.95619i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−19187.5	+	19187.5i	−0.936045	+	0.936045i
$750$		0			0
$751$	9320.68			0.452885			0.226442	−	0.974025i	$-0.427291\pi$
$751$	9320.68			0.452885			0.226442	+	0.974025i	$0.427291\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	19835.3	−	19835.3i	0.956135	−	0.956135i
$756$		0			0
$757$	2105.29	+	2105.29i	0.101081	+	0.101081i	0.755839	−	0.654758i	$-0.227229\pi$
$757$	2105.29	+	2105.29i	0.101081	+	0.101081i	−0.654758	+	0.755839i	$0.727229\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	13011.0i		−	0.619776i	−0.950773	−	0.309888i	$-0.899708\pi$
$761$		−	13011.0i		−	0.619776i	0.950773	−	0.309888i	$-0.100292\pi$
$762$		0			0
$763$	−5197.48	−	5197.48i	−0.246607	−	0.246607i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−28109.4			−1.32330
$768$		0			0
$769$	26438.9			1.23981			0.619904	−	0.784677i	$-0.287171\pi$
$769$	26438.9			1.23981			0.619904	+	0.784677i	$0.287171\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21614.3	+	21614.3i	1.00571	+	1.00571i	0.999984	+	0.00572318i	$0.00182175\pi$
$773$	21614.3	+	21614.3i	1.00571	+	1.00571i	0.00572318	+	0.999984i	$0.498178\pi$
$774$		0			0
$775$			49911.4i			2.31338i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4062.20	−	4062.20i	−0.186834	−	0.186834i
$780$		0			0
$781$	4672.48	−	4672.48i	0.214077	−	0.214077i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	25157.8			1.14385
$786$		0			0
$787$	20949.4	−	20949.4i	0.948874	−	0.948874i	−0.0498808	−	0.998755i	$-0.515884\pi$
$787$	20949.4	−	20949.4i	0.948874	−	0.948874i	0.998755	+	0.0498808i	$0.0158842\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	2643.94i		−	0.118847i
$792$		0			0
$793$			28258.1i			1.26541i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1576.38	+	1576.38i	−0.0700603	+	0.0700603i	−0.741269	−	0.671208i	$-0.765776\pi$
$797$	−1576.38	+	1576.38i	−0.0700603	+	0.0700603i	0.671208	+	0.741269i	$0.265776\pi$
$798$		0			0
$799$	13214.4			0.585097
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5070.48	−	5070.48i	0.222831	−	0.222831i
$804$		0			0
$805$	−11054.4	−	11054.4i	−0.483995	−	0.483995i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	8102.89i		−	0.352142i	−0.984378	−	0.176071i	$-0.943661\pi$
$809$		−	8102.89i		−	0.352142i	0.984378	−	0.176071i	$-0.0563388\pi$
$810$		0			0
$811$	−20649.6	−	20649.6i	−0.894090	−	0.894090i	0.100816	−	0.994905i	$-0.467855\pi$
$811$	−20649.6	−	20649.6i	−0.894090	−	0.894090i	−0.994905	+	0.100816i	$0.967855\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−56114.6			−2.41179
$816$		0			0
$817$	−4802.38			−0.205647
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13405.1	−	13405.1i	−0.569844	−	0.569844i	0.362240	−	0.932085i	$-0.382012\pi$
$821$	−13405.1	−	13405.1i	−0.569844	−	0.569844i	−0.932085	+	0.362240i	$0.882012\pi$
$822$		0			0
$823$		−	17203.4i		−	0.728643i	−0.931273	−	0.364321i	$-0.881301\pi$
$823$		−	17203.4i		−	0.728643i	0.931273	−	0.364321i	$-0.118699\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−11127.2	−	11127.2i	−0.467872	−	0.467872i	0.433353	−	0.901224i	$-0.357330\pi$
$827$	−11127.2	−	11127.2i	−0.467872	−	0.467872i	−0.901224	+	0.433353i	$0.857330\pi$
$828$		0			0
$829$	−5752.09	+	5752.09i	−0.240987	+	0.240987i	−0.817259	−	0.576271i	$-0.804507\pi$
$829$	−5752.09	+	5752.09i	−0.240987	+	0.240987i	0.576271	+	0.817259i	$0.304507\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9580.34			0.398486
$834$		0			0
$835$	9904.39	−	9904.39i	0.410486	−	0.410486i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			45875.9i			1.88774i	0.330318	+	0.943870i	$0.392844\pi$
$839$			45875.9i			1.88774i	−0.330318	+	0.943870i	$0.607156\pi$
$840$		0			0
$841$		−	64807.5i		−	2.65724i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−31268.1	+	31268.1i	−1.27296	+	1.27296i
$846$		0			0
$847$	9465.66			0.383995
$848$		0			0
$849$		0			0
$850$		0			0
$851$	8257.21	−	8257.21i	0.332613	−	0.332613i
$852$		0			0
$853$	−27266.2	−	27266.2i	−1.09446	−	1.09446i	−0.995046	−	0.0994157i	$-0.968303\pi$
$853$	−27266.2	−	27266.2i	−1.09446	−	1.09446i	−0.0994157	−	0.995046i	$-0.531697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	28960.0i		−	1.15432i	−0.816629	−	0.577162i	$-0.804160\pi$
$857$		−	28960.0i		−	1.15432i	0.816629	−	0.577162i	$-0.195840\pi$
$858$		0			0
$859$	−17255.5	−	17255.5i	−0.685388	−	0.685388i	0.275821	−	0.961209i	$-0.411050\pi$
$859$	−17255.5	−	17255.5i	−0.685388	−	0.685388i	−0.961209	+	0.275821i	$0.911050\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	39885.9			1.57327			0.786635	−	0.617418i	$-0.211821\pi$
$863$	39885.9			1.57327			0.786635	+	0.617418i	$0.211821\pi$
$864$		0			0
$865$	−10618.5			−0.417388
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2987.03	−	2987.03i	−0.116603	−	0.116603i
$870$		0			0
$871$			61778.0i			2.40329i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4160.41	+	4160.41i	0.160740	+	0.160740i
$876$		0			0
$877$	8855.51	−	8855.51i	0.340968	−	0.340968i	−0.515763	−	0.856731i	$-0.672491\pi$
$877$	8855.51	−	8855.51i	0.340968	−	0.340968i	0.856731	+	0.515763i	$0.172491\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	43343.8			1.65753			0.828767	−	0.559593i	$-0.189043\pi$
$881$	43343.8			1.65753			0.828767	+	0.559593i	$0.189043\pi$
$882$		0			0
$883$	−11229.3	+	11229.3i	−0.427968	+	0.427968i	−0.887936	−	0.459968i	$-0.847861\pi$
$883$	−11229.3	+	11229.3i	−0.427968	+	0.427968i	0.459968	+	0.887936i	$0.347861\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	36912.9i		−	1.39731i	−0.715459	−	0.698655i	$-0.753782\pi$
$887$		−	36912.9i		−	1.39731i	0.715459	−	0.698655i	$-0.246218\pi$
$888$		0			0
$889$		−	2230.45i		−	0.0841473i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2974.49	−	2974.49i	0.111464	−	0.111464i
$894$		0			0
$895$	15275.5			0.570508
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−68865.5	+	68865.5i	−2.55483	+	2.55483i
$900$		0			0
$901$	11691.7	+	11691.7i	0.432306	+	0.432306i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			17822.2i			0.654618i
$906$		0			0
$907$	−14588.0	−	14588.0i	−0.534053	−	0.534053i	0.387723	−	0.921776i	$-0.373262\pi$
$907$	−14588.0	−	14588.0i	−0.534053	−	0.534053i	−0.921776	+	0.387723i	$0.873262\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	50418.8			1.83365			0.916823	−	0.399295i	$-0.130745\pi$
$911$	50418.8			1.83365			0.916823	+	0.399295i	$0.130745\pi$
$912$		0			0
$913$	−4259.69			−0.154409
$914$		0			0
$915$		0			0
$916$		0			0
$917$	11806.1	+	11806.1i	0.425160	+	0.425160i
$918$		0			0
$919$		−	53290.4i		−	1.91283i	−0.292018	−	0.956413i	$-0.594327\pi$
$919$		−	53290.4i		−	1.91283i	0.292018	−	0.956413i	$-0.405673\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−13529.9	−	13529.9i	−0.482495	−	0.482495i
$924$		0			0
$925$	−16952.3	+	16952.3i	−0.602581	+	0.602581i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44859.2			−1.58427			−0.792133	−	0.610349i	$-0.791029\pi$
$929$	−44859.2			−1.58427			−0.792133	+	0.610349i	$0.791029\pi$
$930$		0			0
$931$	2156.48	−	2156.48i	0.0759140	−	0.0759140i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			20781.4i			0.726872i
$936$		0			0
$937$		−	22463.9i		−	0.783204i	−0.920135	−	0.391602i	$-0.871921\pi$
$937$		−	22463.9i		−	0.783204i	0.920135	−	0.391602i	$-0.128079\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−20653.7	+	20653.7i	−0.715507	+	0.715507i	−0.967682	−	0.252175i	$-0.918854\pi$
$941$	−20653.7	+	20653.7i	−0.715507	+	0.715507i	0.252175	+	0.967682i	$0.418854\pi$
$942$		0			0
$943$	−25961.0			−0.896507
$944$		0			0
$945$		0			0
$946$		0			0
$947$	33123.2	−	33123.2i	1.13660	−	1.13660i	0.147544	−	0.989055i	$-0.452863\pi$
$947$	33123.2	−	33123.2i	1.13660	−	1.13660i	0.989055	−	0.147544i	$-0.0471368\pi$
$948$		0			0
$949$	−14682.4	−	14682.4i	−0.502224	−	0.502224i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	10871.9i		−	0.369544i	−0.982781	−	0.184772i	$-0.940845\pi$
$953$		−	10871.9i		−	0.369544i	0.982781	−	0.184772i	$-0.0591546\pi$
$954$		0			0
$955$	−8966.93	−	8966.93i	−0.303836	−	0.303836i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3866.02			−0.130178
$960$		0			0
$961$	76546.3			2.56944
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−25763.3	−	25763.3i	−0.859430	−	0.859430i
$966$		0			0
$967$		−	6501.58i		−	0.216212i	−0.994139	−	0.108106i	$-0.965521\pi$
$967$		−	6501.58i		−	0.216212i	0.994139	−	0.108106i	$-0.0344786\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16761.7	−	16761.7i	−0.553975	−	0.553975i	0.373611	−	0.927586i	$-0.378120\pi$
$971$	−16761.7	−	16761.7i	−0.553975	−	0.553975i	−0.927586	+	0.373611i	$0.878120\pi$
$972$		0			0
$973$	−15762.3	+	15762.3i	−0.519338	+	0.519338i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7407.34			0.242561			0.121280	−	0.992618i	$-0.461300\pi$
$977$	7407.34			0.242561			0.121280	+	0.992618i	$0.461300\pi$
$978$		0			0
$979$	15308.4	−	15308.4i	0.499753	−	0.499753i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			35694.8i			1.15818i	0.815265	+	0.579088i	$0.196591\pi$
$983$			35694.8i			1.15818i	−0.815265	+	0.579088i	$0.803409\pi$
$984$		0			0
$985$			1967.04i			0.0636294i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−15345.7	+	15345.7i	−0.493392	+	0.493392i
$990$		0			0
$991$	−46662.2			−1.49573			−0.747867	−	0.663849i	$-0.768922\pi$
$991$	−46662.2			−1.49573			−0.747867	+	0.663849i	$0.768922\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	20447.9	−	20447.9i	0.651498	−	0.651498i
$996$		0			0
$997$	−22390.7	−	22390.7i	−0.711255	−	0.711255i	0.255543	−	0.966798i	$-0.417746\pi$
$997$	−22390.7	−	22390.7i	−0.711255	−	0.711255i	−0.966798	+	0.255543i	$0.917746\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.b.433.2		24
3.2	odd	2		192.4.j.a.49.12		24
4.3	odd	2		144.4.k.b.37.1		24
12.11	even	2		48.4.j.a.37.12	yes	24
16.3	odd	4		144.4.k.b.109.1		24
16.13	even	4	inner	576.4.k.b.145.2		24
24.5	odd	2		384.4.j.a.97.6		24
24.11	even	2		384.4.j.b.97.7		24
48.5	odd	4		384.4.j.a.289.6		24
48.11	even	4		384.4.j.b.289.7		24
48.29	odd	4		192.4.j.a.145.12		24
48.35	even	4		48.4.j.a.13.12	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.12	✓	24	48.35	even	4
48.4.j.a.37.12	yes	24	12.11	even	2
144.4.k.b.37.1		24	4.3	odd	2
144.4.k.b.109.1		24	16.3	odd	4
192.4.j.a.49.12		24	3.2	odd	2
192.4.j.a.145.12		24	48.29	odd	4
384.4.j.a.97.6		24	24.5	odd	2
384.4.j.a.289.6		24	48.5	odd	4
384.4.j.b.97.7		24	24.11	even	2
384.4.j.b.289.7		24	48.11	even	4
576.4.k.b.145.2		24	16.13	even	4	inner
576.4.k.b.433.2		24	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-11.7911 - 11.7911i) q^{5} +12.5754i q^{7} +(-17.0042 - 17.0042i) q^{11} +(-49.2384 + 49.2384i) q^{13} +51.8247 q^{17} +(11.6655 - 11.6655i) q^{19} -74.5524i q^{23} +153.058i q^{25} +(-211.183 + 211.183i) q^{29} +326.094 q^{31} +(148.277 - 148.277i) q^{35} +(110.757 + 110.757i) q^{37} -348.225i q^{41} +(-205.838 - 205.838i) q^{43} +254.983 q^{47} +184.861 q^{49} +(225.602 + 225.602i) q^{53} +400.995i q^{55} +(285.442 + 285.442i) q^{59} +(286.952 - 286.952i) q^{61} +1161.15 q^{65} +(627.335 - 627.335i) q^{67} +274.784i q^{71} +298.190i q^{73} +(213.834 - 213.834i) q^{77} +175.664 q^{79} +(125.254 - 125.254i) q^{83} +(-611.068 - 611.068i) q^{85} +900.271i q^{89} +(-619.190 - 619.190i) q^{91} -275.096 q^{95} +5.27858 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79}+ \cdots - 7728 q^{95}+O(q^{100}) \) 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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