LMFDB - Embedded newform 576.2.l.a.143.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,2,Mod(143,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 2]))

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

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

chi := DirichletCharacter("576.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

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:	no
Analytic conductor:	$4.59938315643$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 6x^{12} - 12x^{10} + 33x^{8} - 48x^{6} + 96x^{4} - 256x^{2} + 256 $ x^16 - 4x^14 + 6x^12 - 12x^10 + 33x^8 - 48x^6 + 96x^4 - 256*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			143.1
Root			$-0.517174 - 1.31626i$ of defining polynomial
Character	$\chi$	$=$	576.143
Dual form			576.2.l.a.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.63251 + 2.63251i) q^{5} +0.207188 q^{7} +O(q^{10})$	$q+(-2.63251 + 2.63251i) q^{5} +0.207188 q^{7} +(-3.66686 - 3.66686i) q^{11} +(0.255601 - 0.255601i) q^{13} -0.654483i q^{17} +(-4.46733 - 4.46733i) q^{19} +3.48934i q^{23} -8.86025i q^{25} +(-4.33973 - 4.33973i) q^{29} -6.16426i q^{31} +(-0.545426 + 0.545426i) q^{35} +(4.39291 + 4.39291i) q^{37} -0.0684664 q^{41} +(-5.65306 + 5.65306i) q^{43} -9.14619 q^{47} -6.95707 q^{49} +(-1.51131 + 1.51131i) q^{53} +19.3061 q^{55} +(2.53542 + 2.53542i) q^{59} +(-5.46733 + 5.46733i) q^{61} +1.34575i q^{65} +(4.77135 + 4.77135i) q^{67} +5.94986i q^{71} -6.93467i q^{73} +(-0.759730 - 0.759730i) q^{77} +4.72748i q^{79} +(4.32777 - 4.32777i) q^{83} +(1.72294 + 1.72294i) q^{85} +11.9443 q^{89} +(0.0529576 - 0.0529576i) q^{91} +23.5206 q^{95} -0.925579 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} + 64 q^{55} - 32 q^{61} + 16 q^{67} - 32 q^{85} - 48 q^{91}+O(q^{100}) $ 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 + 64 * q^55 - 32 * q^61 + 16 * q^67 - 32 * q^85 - 48 * q^91

Display 100 coefficients 10 coefficients

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
$5$	−2.63251	+	2.63251i	−1.17730	+	1.17730i	−0.196865	+	0.980431i	$0.563076\pi$
$5$	−2.63251	+	2.63251i	−1.17730	+	1.17730i	−0.980431	+	0.196865i	$0.936924\pi$
$6$		0			0
$7$	0.207188			0.0783098			0.0391549	−	0.999233i	$-0.487533\pi$
$7$	0.207188			0.0783098			0.0391549	+	0.999233i	$0.487533\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.66686	−	3.66686i	−1.10560	−	1.10560i	−0.993722	−	0.111878i	$-0.964313\pi$
$11$	−3.66686	−	3.66686i	−1.10560	−	1.10560i	−0.111878	−	0.993722i	$-0.535687\pi$
$12$		0			0
$13$	0.255601	−	0.255601i	0.0708910	−	0.0708910i	−0.670772	−	0.741663i	$-0.734037\pi$
$13$	0.255601	−	0.255601i	0.0708910	−	0.0708910i	0.741663	+	0.670772i	$0.234037\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	0.654483i		−	0.158735i	−0.996845	−	0.0793677i	$-0.974710\pi$
$17$		−	0.654483i		−	0.158735i	0.996845	−	0.0793677i	$-0.0252901\pi$
$18$		0			0
$19$	−4.46733	−	4.46733i	−1.02488	−	1.02488i	−0.999683	−	0.0251941i	$-0.991980\pi$
$19$	−4.46733	−	4.46733i	−1.02488	−	1.02488i	−0.0251941	−	0.999683i	$-0.508020\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.48934i			0.727577i	0.931482	+	0.363789i	$0.118517\pi$
$23$			3.48934i			0.727577i	−0.931482	+	0.363789i	$0.881483\pi$
$24$		0			0
$25$		−	8.86025i		−	1.77205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.33973	−	4.33973i	−0.805869	−	0.805869i	0.178137	−	0.984006i	$-0.442993\pi$
$29$	−4.33973	−	4.33973i	−0.805869	−	0.805869i	−0.984006	+	0.178137i	$0.942993\pi$
$30$		0			0
$31$		−	6.16426i		−	1.10713i	−0.832805	−	0.553567i	$-0.813266\pi$
$31$		−	6.16426i		−	1.10713i	0.832805	−	0.553567i	$-0.186734\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.545426	+	0.545426i	−0.0921938	+	0.0921938i
$36$		0			0
$37$	4.39291	+	4.39291i	0.722190	+	0.722190i	0.969051	−	0.246861i	$-0.0793990\pi$
$37$	4.39291	+	4.39291i	0.722190	+	0.722190i	−0.246861	+	0.969051i	$0.579399\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.0684664			−0.0106927			−0.00534633	−	0.999986i	$-0.501702\pi$
$41$	−0.0684664			−0.0106927			−0.00534633	+	0.999986i	$0.501702\pi$
$42$		0			0
$43$	−5.65306	+	5.65306i	−0.862083	+	0.862083i	−0.991580	−	0.129497i	$-0.958664\pi$
$43$	−5.65306	+	5.65306i	−0.862083	+	0.862083i	0.129497	+	0.991580i	$0.458664\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.14619			−1.33411			−0.667055	−	0.745009i	$-0.732445\pi$
$47$	−9.14619			−1.33411			−0.667055	+	0.745009i	$0.732445\pi$
$48$		0			0
$49$	−6.95707			−0.993868
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.51131	+	1.51131i	−0.207594	+	0.207594i	−0.803244	−	0.595650i	$-0.796895\pi$
$53$	−1.51131	+	1.51131i	−0.207594	+	0.207594i	0.595650	+	0.803244i	$0.296895\pi$
$54$		0			0
$55$	19.3061			2.60324
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.53542	+	2.53542i	0.330083	+	0.330083i	0.852618	−	0.522535i	$-0.175013\pi$
$59$	2.53542	+	2.53542i	0.330083	+	0.330083i	−0.522535	+	0.852618i	$0.675013\pi$
$60$		0			0
$61$	−5.46733	+	5.46733i	−0.700020	+	0.700020i	−0.964415	−	0.264394i	$-0.914828\pi$
$61$	−5.46733	+	5.46733i	−0.700020	+	0.700020i	0.264394	+	0.964415i	$0.414828\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1.34575i			0.166919i
$66$		0			0
$67$	4.77135	+	4.77135i	0.582913	+	0.582913i	0.935703	−	0.352790i	$-0.114767\pi$
$67$	4.77135	+	4.77135i	0.582913	+	0.582913i	−0.352790	+	0.935703i	$0.614767\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.94986i			0.706119i	0.935601	+	0.353059i	$0.114859\pi$
$71$			5.94986i			0.706119i	−0.935601	+	0.353059i	$0.885141\pi$
$72$		0			0
$73$		−	6.93467i		−	0.811641i	−0.913953	−	0.405821i	$-0.866986\pi$
$73$		−	6.93467i		−	0.811641i	0.913953	−	0.405821i	$-0.133014\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.759730	−	0.759730i	−0.0865793	−	0.0865793i
$78$		0			0
$79$			4.72748i			0.531883i	0.963989	+	0.265942i	$0.0856828\pi$
$79$			4.72748i			0.531883i	−0.963989	+	0.265942i	$0.914317\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.32777	−	4.32777i	0.475035	−	0.475035i	−0.428505	−	0.903539i	$-0.640960\pi$
$83$	4.32777	−	4.32777i	0.475035	−	0.475035i	0.903539	+	0.428505i	$0.140960\pi$
$84$		0			0
$85$	1.72294	+	1.72294i	0.186879	+	0.186879i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.9443			1.26609			0.633045	−	0.774115i	$-0.281805\pi$
$89$	11.9443			1.26609			0.633045	+	0.774115i	$0.281805\pi$
$90$		0			0
$91$	0.0529576	−	0.0529576i	0.00555146	−	0.00555146i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	23.5206			2.41317
$96$		0			0
$97$	−0.925579			−0.0939783			−0.0469892	−	0.998895i	$-0.514963\pi$
$97$	−0.925579			−0.0939783			−0.0469892	+	0.998895i	$0.514963\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.27104	−	2.27104i	0.225977	−	0.225977i	−0.585033	−	0.811010i	$-0.698918\pi$
$101$	2.27104	−	2.27104i	0.225977	−	0.225977i	0.811010	+	0.585033i	$0.198918\pi$
$102$		0			0
$103$	−14.1643			−1.39565			−0.697823	−	0.716270i	$-0.745848\pi$
$103$	−14.1643			−1.39565			−0.697823	+	0.716270i	$0.745848\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.85318	−	8.85318i	−0.855870	−	0.855870i	0.134979	−	0.990848i	$-0.456903\pi$
$107$	−8.85318	−	8.85318i	−0.855870	−	0.855870i	−0.990848	+	0.134979i	$0.956903\pi$
$108$		0			0
$109$	−2.66998	+	2.66998i	−0.255737	+	0.255737i	−0.823318	−	0.567580i	$-0.807880\pi$
$109$	−2.66998	+	2.66998i	−0.255737	+	0.255737i	0.567580	+	0.823318i	$0.307880\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	8.39289i		−	0.789536i	−0.918781	−	0.394768i	$-0.870825\pi$
$113$		−	8.39289i		−	0.789536i	0.918781	−	0.394768i	$-0.129175\pi$
$114$		0			0
$115$	−9.18572	−	9.18572i	−0.856573	−	0.856573i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	0.135601i		−	0.0124305i
$120$		0			0
$121$			15.8917i			1.44470i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.1621	+	10.1621i	0.908930	+	0.908930i
$126$		0			0
$127$			9.64397i			0.855764i	0.903835	+	0.427882i	$0.140740\pi$
$127$			9.64397i			0.855764i	−0.903835	+	0.427882i	$0.859260\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	5.07084	−	5.07084i	0.443041	−	0.443041i	−0.449992	−	0.893033i	$-0.648573\pi$
$131$	5.07084	−	5.07084i	0.443041	−	0.443041i	0.893033	+	0.449992i	$0.148573\pi$
$132$		0			0
$133$	−0.925579	−	0.925579i	−0.0802579	−	0.0802579i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	11.9075			1.01732			0.508662	−	0.860966i	$-0.330140\pi$
$137$	11.9075			1.01732			0.508662	+	0.860966i	$0.330140\pi$
$138$		0			0
$139$	−0.771348	+	0.771348i	−0.0654249	+	0.0654249i	−0.739062	−	0.673637i	$-0.764731\pi$
$139$	−0.771348	+	0.771348i	−0.0654249	+	0.0654249i	0.673637	+	0.739062i	$0.264731\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.87451			−0.156754
$144$		0			0
$145$	22.8488			1.89749
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.02434	+	3.02434i	−0.247764	+	0.247764i	−0.820052	−	0.572289i	$-0.806056\pi$
$149$	−3.02434	+	3.02434i	−0.247764	+	0.247764i	0.572289	+	0.820052i	$0.306056\pi$
$150$		0			0
$151$	−6.57864			−0.535362			−0.267681	−	0.963508i	$-0.586257\pi$
$151$	−6.57864			−0.535362			−0.267681	+	0.963508i	$0.586257\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.2275	+	16.2275i	1.30342	+	1.30342i
$156$		0			0
$157$	7.46733	−	7.46733i	0.595958	−	0.595958i	−0.343276	−	0.939234i	$-0.611537\pi$
$157$	7.46733	−	7.46733i	0.595958	−	0.595958i	0.939234	+	0.343276i	$0.111537\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			0.722950i			0.0569764i
$162$		0			0
$163$	14.0674	+	14.0674i	1.10185	+	1.10185i	0.994188	+	0.107659i	$0.0343354\pi$
$163$	14.0674	+	14.0674i	1.10185	+	1.10185i	0.107659	+	0.994188i	$0.465665\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.55006i		−	0.506859i	−0.967354	−	0.253430i	$-0.918441\pi$
$167$		−	6.55006i		−	0.506859i	0.967354	−	0.253430i	$-0.0815586\pi$
$168$		0			0
$169$			12.8693i			0.989949i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.955645	+	0.955645i	0.0726563	+	0.0726563i	0.742501	−	0.669845i	$-0.233639\pi$
$173$	0.955645	+	0.955645i	0.0726563	+	0.0726563i	−0.669845	+	0.742501i	$0.733639\pi$
$174$		0			0
$175$		−	1.83574i		−	0.138769i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.77829	+	8.77829i	−0.656120	+	0.656120i	−0.954460	−	0.298340i	$-0.903567\pi$
$179$	−8.77829	+	8.77829i	−0.656120	+	0.656120i	0.298340	+	0.954460i	$0.403567\pi$
$180$		0			0
$181$	6.25560	+	6.25560i	0.464975	+	0.464975i	0.900282	−	0.435307i	$-0.143360\pi$
$181$	6.25560	+	6.25560i	0.464975	+	0.464975i	−0.435307	+	0.900282i	$0.643360\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−23.1288			−1.70046
$186$		0			0
$187$	−2.39990	+	2.39990i	−0.175498	+	0.175498i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.13608			−0.516349			−0.258174	−	0.966098i	$-0.583121\pi$
$191$	−7.13608			−0.516349			−0.258174	+	0.966098i	$0.583121\pi$
$192$		0			0
$193$	−10.9347			−0.787095			−0.393547	−	0.919304i	$-0.628752\pi$
$193$	−10.9347			−0.787095			−0.393547	+	0.919304i	$0.628752\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.3277	−	10.3277i	0.735819	−	0.735819i	−0.235947	−	0.971766i	$-0.575819\pi$
$197$	10.3277	−	10.3277i	0.735819	−	0.735819i	0.971766	+	0.235947i	$0.0758192\pi$
$198$		0			0
$199$	−7.18478			−0.509316			−0.254658	−	0.967031i	$-0.581963\pi$
$199$	−7.18478			−0.509316			−0.254658	+	0.967031i	$0.581963\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−0.899142	−	0.899142i	−0.0631074	−	0.0631074i
$204$		0			0
$205$	0.180239	−	0.180239i	0.0125884	−	0.0125884i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			32.7622i			2.26621i
$210$		0			0
$211$	0.163320	+	0.163320i	0.0112434	+	0.0112434i	0.712706	−	0.701463i	$-0.247469\pi$
$211$	0.163320	+	0.163320i	0.0112434	+	0.0112434i	−0.701463	+	0.712706i	$0.747469\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	29.7635i		−	2.02985i
$216$		0			0
$217$		−	1.27716i		−	0.0866994i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.167287	−	0.167287i	−0.0112529	−	0.0112529i
$222$		0			0
$223$		−	0.621565i		−	0.0416230i	−0.999783	−	0.0208115i	$-0.993375\pi$
$223$		−	0.621565i		−	0.0416230i	0.999783	−	0.0208115i	$-0.00662499\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.9257	−	10.9257i	0.725164	−	0.725164i	−0.244489	−	0.969652i	$-0.578620\pi$
$227$	10.9257	−	10.9257i	0.725164	−	0.725164i	0.969652	+	0.244489i	$0.0786201\pi$
$228$		0			0
$229$	−17.9761	−	17.9761i	−1.18789	−	1.18789i	−0.977649	−	0.210245i	$-0.932574\pi$
$229$	−17.9761	−	17.9761i	−1.18789	−	1.18789i	−0.210245	−	0.977649i	$-0.567426\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.5409			−1.41119			−0.705595	−	0.708616i	$-0.749320\pi$
$233$	−21.5409			−1.41119			−0.705595	+	0.708616i	$0.749320\pi$
$234$		0			0
$235$	24.0775	−	24.0775i	1.57064	−	1.57064i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.5420			−1.07001			−0.535005	−	0.844849i	$-0.679690\pi$
$239$	−16.5420			−1.07001			−0.535005	+	0.844849i	$0.679690\pi$
$240$		0			0
$241$	−24.9008			−1.60400			−0.802002	−	0.597322i	$-0.796232\pi$
$241$	−24.9008			−1.60400			−0.802002	+	0.597322i	$0.796232\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	18.3146	−	18.3146i	1.17008	−	1.17008i
$246$		0			0
$247$	−2.28371			−0.145309
$248$		0			0
$249$		0			0
$250$		0			0
$251$	8.73770	+	8.73770i	0.551519	+	0.551519i	0.926879	−	0.375360i	$-0.122481\pi$
$251$	8.73770	+	8.73770i	0.551519	+	0.551519i	−0.375360	+	0.926879i	$0.622481\pi$
$252$		0			0
$253$	12.7949	−	12.7949i	0.804409	−	0.804409i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.7732i		−	1.17104i	−0.810659	−	0.585519i	$-0.800891\pi$
$257$		−	18.7732i		−	1.17104i	0.810659	−	0.585519i	$-0.199109\pi$
$258$		0			0
$259$	0.910160	+	0.910160i	0.0565546	+	0.0565546i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			19.1523i			1.18098i	0.807045	+	0.590490i	$0.201065\pi$
$263$			19.1523i			1.18098i	−0.807045	+	0.590490i	$0.798935\pi$
$264$		0			0
$265$		−	7.95707i		−	0.488799i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.69478	−	4.69478i	−0.286246	−	0.286246i	0.549348	−	0.835594i	$-0.314876\pi$
$269$	−4.69478	−	4.69478i	−0.286246	−	0.286246i	−0.835594	+	0.549348i	$0.814876\pi$
$270$		0			0
$271$		−	25.2478i		−	1.53369i	−0.641831	−	0.766846i	$-0.721825\pi$
$271$		−	25.2478i		−	1.53369i	0.641831	−	0.766846i	$-0.278175\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−32.4893	+	32.4893i	−1.95918	+	1.95918i
$276$		0			0
$277$	−21.1473	−	21.1473i	−1.27062	−	1.27062i	−0.945763	−	0.324858i	$-0.894684\pi$
$277$	−21.1473	−	21.1473i	−1.27062	−	1.27062i	−0.324858	−	0.945763i	$-0.605316\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.34812			−0.378697			−0.189348	−	0.981910i	$-0.560638\pi$
$281$	−6.34812			−0.378697			−0.189348	+	0.981910i	$0.560638\pi$
$282$		0			0
$283$	−2.56322	+	2.56322i	−0.152368	+	0.152368i	−0.779175	−	0.626807i	$-0.784361\pi$
$283$	−2.56322	+	2.56322i	−0.152368	+	0.152368i	0.626807	+	0.779175i	$0.284361\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.0141854			−0.000837339
$288$		0			0
$289$	16.5717			0.974803
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.7102	+	11.7102i	−0.684119	+	0.684119i	−0.960926	−	0.276806i	$-0.910724\pi$
$293$	−11.7102	+	11.7102i	−0.684119	+	0.684119i	0.276806	+	0.960926i	$0.410724\pi$
$294$		0			0
$295$	−13.3490			−0.777211
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.891879	+	0.891879i	0.0515787	+	0.0515787i
$300$		0			0
$301$	−1.17125	+	1.17125i	−0.0675096	+	0.0675096i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	28.7857i		−	1.64826i
$306$		0			0
$307$	11.5572	+	11.5572i	0.659603	+	0.659603i	0.955286	−	0.295683i	$-0.0955473\pi$
$307$	11.5572	+	11.5572i	0.659603	+	0.659603i	−0.295683	+	0.955286i	$0.595547\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	13.9316i		−	0.789991i	−0.918683	−	0.394995i	$-0.870746\pi$
$311$		−	13.9316i		−	0.789991i	0.918683	−	0.394995i	$-0.129254\pi$
$312$		0			0
$313$			17.8264i			1.00761i	0.863818	+	0.503804i	$0.168067\pi$
$313$			17.8264i			1.00761i	−0.863818	+	0.503804i	$0.831933\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.6127	−	11.6127i	−0.652237	−	0.652237i	0.301294	−	0.953531i	$-0.402581\pi$
$317$	−11.6127	−	11.6127i	−0.652237	−	0.652237i	−0.953531	+	0.301294i	$0.902581\pi$
$318$		0			0
$319$			31.8264i			1.78194i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.92379	+	2.92379i	−0.162684	+	0.162684i
$324$		0			0
$325$	−2.26469	−	2.26469i	−0.125622	−	0.125622i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.89498			−0.104474
$330$		0			0
$331$	23.5572	−	23.5572i	1.29482	−	1.29482i	0.363050	−	0.931770i	$-0.381736\pi$
$331$	23.5572	−	23.5572i	1.29482	−	1.29482i	0.931770	−	0.363050i	$-0.118264\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−25.1213			−1.37252
$336$		0			0
$337$	22.8488			1.24465			0.622327	−	0.782757i	$-0.286187\pi$
$337$	22.8488			1.24465			0.622327	+	0.782757i	$0.286187\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−22.6035	+	22.6035i	−1.22405	+	1.22405i
$342$		0			0
$343$	−2.89174			−0.156139
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−18.0542	−	18.0542i	−0.969198	−	0.969198i	0.0303420	−	0.999540i	$-0.490340\pi$
$347$	−18.0542	−	18.0542i	−0.969198	−	0.969198i	−0.999540	+	0.0303420i	$0.990340\pi$
$348$		0			0
$349$	−3.32758	+	3.32758i	−0.178121	+	0.178121i	−0.790536	−	0.612415i	$-0.790198\pi$
$349$	−3.32758	+	3.32758i	−0.178121	+	0.178121i	0.612415	+	0.790536i	$0.290198\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.9229i			1.00717i	0.863947	+	0.503583i	$0.167985\pi$
$353$			18.9229i			1.00717i	−0.863947	+	0.503583i	$0.832015\pi$
$354$		0			0
$355$	−15.6631	−	15.6631i	−0.831310	−	0.831310i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			32.0808i			1.69316i	0.532262	+	0.846580i	$0.321342\pi$
$359$			32.0808i			1.69316i	−0.532262	+	0.846580i	$0.678658\pi$
$360$		0			0
$361$			20.9141i			1.10074i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.2556	+	18.2556i	0.955542	+	0.955542i
$366$		0			0
$367$		−	19.0989i		−	0.996956i	−0.866902	−	0.498478i	$-0.833892\pi$
$367$		−	19.0989i		−	0.996956i	0.866902	−	0.498478i	$-0.166108\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.313125	+	0.313125i	−0.0162566	+	0.0162566i
$372$		0			0
$373$	3.51026	+	3.51026i	0.181754	+	0.181754i	0.792120	−	0.610365i	$-0.208977\pi$
$373$	3.51026	+	3.51026i	0.181754	+	0.181754i	−0.610365	+	0.792120i	$0.708977\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.21848			−0.114258
$378$		0			0
$379$	−9.07536	+	9.07536i	−0.466170	+	0.466170i	−0.900671	−	0.434501i	$-0.856925\pi$
$379$	−9.07536	+	9.07536i	−0.466170	+	0.466170i	0.434501	+	0.900671i	$0.356925\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	28.3176			1.44696			0.723481	−	0.690344i	$-0.242541\pi$
$383$	28.3176			1.44696			0.723481	+	0.690344i	$0.242541\pi$
$384$		0			0
$385$	4.00000			0.203859
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.490254	+	0.490254i	−0.0248569	+	0.0248569i	−0.719426	−	0.694569i	$-0.755595\pi$
$389$	−0.490254	+	0.490254i	−0.0248569	+	0.0248569i	0.694569	+	0.719426i	$0.255595\pi$
$390$		0			0
$391$	2.28371			0.115492
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12.4452	−	12.4452i	−0.626183	−	0.626183i
$396$		0			0
$397$	24.4559	−	24.4559i	1.22741	−	1.22741i	0.262465	−	0.964942i	$-0.415465\pi$
$397$	24.4559	−	24.4559i	1.22741	−	1.22741i	0.964942	−	0.262465i	$-0.0845354\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	0.517550i		−	0.0258452i	−0.999916	−	0.0129226i	$-0.995886\pi$
$401$		−	0.517550i		−	0.0258452i	0.999916	−	0.0129226i	$-0.00411351\pi$
$402$		0			0
$403$	−1.57559	−	1.57559i	−0.0784859	−	0.0784859i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	32.2164i		−	1.59691i
$408$		0			0
$409$		−	36.5602i		−	1.80779i	−0.427758	−	0.903893i	$-0.640696\pi$
$409$		−	36.5602i		−	1.80779i	0.427758	−	0.903893i	$-0.359304\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	0.525309	+	0.525309i	0.0258488	+	0.0258488i
$414$		0			0
$415$			22.7858i			1.11851i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.9433	−	20.9433i	1.02315	−	1.02315i	0.0234208	−	0.999726i	$-0.492544\pi$
$419$	20.9433	−	20.9433i	1.02315	−	1.02315i	0.999726	−	0.0234208i	$-0.00745575\pi$
$420$		0			0
$421$	14.2218	+	14.2218i	0.693126	+	0.693126i	0.962919	−	0.269792i	$-0.0869550\pi$
$421$	14.2218	+	14.2218i	0.693126	+	0.693126i	−0.269792	+	0.962919i	$0.586955\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.79888			−0.281287
$426$		0			0
$427$	−1.13277	+	1.13277i	−0.0548184	+	0.0548184i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.9032			1.34405			0.672024	−	0.740529i	$-0.265425\pi$
$431$	27.9032			1.34405			0.672024	+	0.740529i	$0.265425\pi$
$432$		0			0
$433$	−28.4634			−1.36786			−0.683932	−	0.729546i	$-0.739731\pi$
$433$	−28.4634			−1.36786			−0.683932	+	0.729546i	$0.739731\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	15.5880	−	15.5880i	0.745677	−	0.745677i
$438$		0			0
$439$	−18.2702			−0.871988			−0.435994	−	0.899950i	$-0.643603\pi$
$439$	−18.2702			−0.871988			−0.435994	+	0.899950i	$0.643603\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.9646	+	13.9646i	0.663479	+	0.663479i	0.956198	−	0.292720i	$-0.0945602\pi$
$443$	13.9646	+	13.9646i	0.663479	+	0.663479i	−0.292720	+	0.956198i	$0.594560\pi$
$444$		0			0
$445$	−31.4434	+	31.4434i	−1.49056	+	1.49056i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			23.8072i			1.12353i	0.827296	+	0.561766i	$0.189878\pi$
$449$			23.8072i			1.12353i	−0.827296	+	0.561766i	$0.810122\pi$
$450$		0			0
$451$	0.251057	+	0.251057i	0.0118218	+	0.0118218i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			0.278823i			0.0130714i
$456$		0			0
$457$		−	13.9662i		−	0.653310i	−0.945144	−	0.326655i	$-0.894079\pi$
$457$		−	13.9662i		−	0.653310i	0.945144	−	0.326655i	$-0.105921\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.4422	−	19.4422i	−0.905514	−	0.905514i	0.0903920	−	0.995906i	$-0.471188\pi$
$461$	−19.4422	−	19.4422i	−0.905514	−	0.905514i	−0.995906	+	0.0903920i	$0.971188\pi$
$462$		0			0
$463$		−	27.9277i		−	1.29791i	−0.760827	−	0.648955i	$-0.775206\pi$
$463$		−	27.9277i		−	1.29791i	0.760827	−	0.648955i	$-0.224794\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.32777	−	4.32777i	0.200265	−	0.200265i	−0.599848	−	0.800114i	$-0.704772\pi$
$467$	4.32777	−	4.32777i	0.200265	−	0.200265i	0.800114	+	0.599848i	$0.204772\pi$
$468$		0			0
$469$	0.988567	+	0.988567i	0.0456478	+	0.0456478i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	41.4580			1.90624
$474$		0			0
$475$	−39.5817	+	39.5817i	−1.81613	+	1.81613i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.0898			−1.37484			−0.687420	−	0.726260i	$-0.741257\pi$
$479$	−30.0898			−1.37484			−0.687420	+	0.726260i	$0.741257\pi$
$480$		0			0
$481$	2.24567			0.102394
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.43660	−	2.43660i	0.110640	−	0.110640i
$486$		0			0
$487$	27.3397			1.23888			0.619440	−	0.785044i	$-0.287360\pi$
$487$	27.3397			1.23888			0.619440	+	0.785044i	$0.287360\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3.78368	+	3.78368i	0.170755	+	0.170755i	0.787311	−	0.616556i	$-0.211473\pi$
$491$	3.78368	+	3.78368i	0.170755	+	0.170755i	−0.616556	+	0.787311i	$0.711473\pi$
$492$		0			0
$493$	−2.84028	+	2.84028i	−0.127920	+	0.127920i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.23274i			0.0552960i
$498$		0			0
$499$	−15.2002	−	15.2002i	−0.680455	−	0.680455i	0.279648	−	0.960103i	$-0.409782\pi$
$499$	−15.2002	−	15.2002i	−0.680455	−	0.680455i	−0.960103	+	0.279648i	$0.909782\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.6164i		−	0.562537i	−0.959629	−	0.281268i	$-0.909245\pi$
$503$		−	12.6164i		−	0.562537i	0.959629	−	0.281268i	$-0.0907551\pi$
$504$		0			0
$505$			11.9571i			0.532083i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	26.3204	+	26.3204i	1.16663	+	1.16663i	0.982994	+	0.183639i	$0.0587876\pi$
$509$	26.3204	+	26.3204i	1.16663	+	1.16663i	0.183639	+	0.982994i	$0.441212\pi$
$510$		0			0
$511$		−	1.43678i		−	0.0635595i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	37.2876	−	37.2876i	1.64309	−	1.64309i
$516$		0			0
$517$	33.5378	+	33.5378i	1.47499	+	1.47499i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.60170			0.420658			0.210329	−	0.977631i	$-0.432546\pi$
$521$	9.60170			0.420658			0.210329	+	0.977631i	$0.432546\pi$
$522$		0			0
$523$	−11.2532	+	11.2532i	−0.492066	+	0.492066i	−0.908957	−	0.416890i	$-0.863120\pi$
$523$	−11.2532	+	11.2532i	−0.492066	+	0.492066i	0.416890	+	0.908957i	$0.363120\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.03441			−0.175741
$528$		0			0
$529$	10.8245			0.470632
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.0175001	+	0.0175001i	−0.000758013	+	0.000758013i
$534$		0			0
$535$	46.6122			2.01522
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.5106	+	25.5106i	1.09882	+	1.09882i
$540$		0			0
$541$	4.66998	−	4.66998i	0.200778	−	0.200778i	−0.599555	−	0.800333i	$-0.704656\pi$
$541$	4.66998	−	4.66998i	0.200778	−	0.200778i	0.800333	+	0.599555i	$0.204656\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	14.0575i		−	0.602157i
$546$		0			0
$547$	18.6755	+	18.6755i	0.798505	+	0.798505i	0.982860	−	0.184355i	$-0.0590195\pi$
$547$	18.6755	+	18.6755i	0.798505	+	0.798505i	−0.184355	+	0.982860i	$0.559019\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			38.7741i			1.65183i
$552$		0			0
$553$			0.979478i			0.0416516i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	3.20568	+	3.20568i	0.135829	+	0.135829i	0.771752	−	0.635923i	$-0.219381\pi$
$557$	3.20568	+	3.20568i	0.135829	+	0.135829i	−0.635923	+	0.771752i	$0.719381\pi$
$558$		0			0
$559$			2.88986i			0.122228i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.4157	+	32.4157i	−1.36616	+	1.36616i	−0.500319	+	0.865841i	$0.666784\pi$
$563$	−32.4157	+	32.4157i	−1.36616	+	1.36616i	−0.865841	+	0.500319i	$0.833216\pi$
$564$		0			0
$565$	22.0944	+	22.0944i	0.929518	+	0.929518i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.1238			0.466335			0.233168	−	0.972437i	$-0.425091\pi$
$569$	11.1238			0.466335			0.233168	+	0.972437i	$0.425091\pi$
$570$		0			0
$571$	−16.1349	+	16.1349i	−0.675223	+	0.675223i	−0.958915	−	0.283692i	$-0.908441\pi$
$571$	−16.1349	+	16.1349i	−0.675223	+	0.675223i	0.283692	+	0.958915i	$0.408441\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	30.9164			1.28930
$576$		0			0
$577$	−10.9795			−0.457082			−0.228541	−	0.973534i	$-0.573395\pi$
$577$	−10.9795			−0.457082			−0.228541	+	0.973534i	$0.573395\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.896663	−	0.896663i	0.0371999	−	0.0371999i
$582$		0			0
$583$	11.0835			0.459032
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7.40862	−	7.40862i	−0.305786	−	0.305786i	0.537486	−	0.843273i	$-0.319374\pi$
$587$	−7.40862	−	7.40862i	−0.305786	−	0.305786i	−0.843273	+	0.537486i	$0.819374\pi$
$588$		0			0
$589$	−27.5378	+	27.5378i	−1.13468	+	1.13468i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	12.3933i		−	0.508934i	−0.967081	−	0.254467i	$-0.918100\pi$
$593$		−	12.3933i		−	0.508934i	0.967081	−	0.254467i	$-0.0819000\pi$
$594$		0			0
$595$	0.356972	+	0.356972i	0.0146344	+	0.0146344i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			41.2488i			1.68538i	0.538399	+	0.842690i	$0.319029\pi$
$599$			41.2488i			1.68538i	−0.538399	+	0.842690i	$0.680971\pi$
$600$		0			0
$601$		−	20.4615i		−	0.834642i	−0.908759	−	0.417321i	$-0.862969\pi$
$601$		−	20.4615i		−	0.834642i	0.908759	−	0.417321i	$-0.137031\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−41.8352	−	41.8352i	−1.70084	−	1.70084i
$606$		0			0
$607$			30.1643i			1.22433i	0.790730	+	0.612165i	$0.209701\pi$
$607$			30.1643i			1.22433i	−0.790730	+	0.612165i	$0.790299\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.33778	+	2.33778i	−0.0945764	+	0.0945764i
$612$		0			0
$613$	−1.80074	−	1.80074i	−0.0727312	−	0.0727312i	0.669805	−	0.742537i	$-0.266377\pi$
$613$	−1.80074	−	1.80074i	−0.0727312	−	0.0727312i	−0.742537	+	0.669805i	$0.766377\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.8098			1.24036			0.620178	−	0.784461i	$-0.287060\pi$
$617$	30.8098			1.24036			0.620178	+	0.784461i	$0.287060\pi$
$618$		0			0
$619$	10.0429	−	10.0429i	0.403659	−	0.403659i	−0.475861	−	0.879520i	$-0.657864\pi$
$619$	10.0429	−	10.0429i	0.403659	−	0.403659i	0.879520	+	0.475861i	$0.157864\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.47471			0.0991472
$624$		0			0
$625$	−9.20274			−0.368110
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.87509	−	2.87509i	0.114637	−	0.114637i
$630$		0			0
$631$	4.91925			0.195832			0.0979161	−	0.995195i	$-0.468782\pi$
$631$	4.91925			0.195832			0.0979161	+	0.995195i	$0.468782\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−25.3879	−	25.3879i	−1.00749	−	1.00749i
$636$		0			0
$637$	−1.77824	+	1.77824i	−0.0704563	+	0.0704563i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	15.0833i		−	0.595756i	−0.954604	−	0.297878i	$-0.903721\pi$
$641$		−	15.0833i		−	0.595756i	0.954604	−	0.297878i	$-0.0962789\pi$
$642$		0			0
$643$	2.26109	+	2.26109i	0.0891686	+	0.0891686i	0.750284	−	0.661116i	$-0.229917\pi$
$643$	2.26109	+	2.26109i	0.0891686	+	0.0891686i	−0.661116	+	0.750284i	$0.729917\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			12.1851i			0.479046i	0.970891	+	0.239523i	$0.0769911\pi$
$647$			12.1851i			0.479046i	−0.970891	+	0.239523i	$0.923009\pi$
$648$		0			0
$649$		−	18.5941i		−	0.729881i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.4822	+	11.4822i	0.449335	+	0.449335i	0.895133	−	0.445798i	$-0.147080\pi$
$653$	11.4822	+	11.4822i	0.449335	+	0.449335i	−0.445798	+	0.895133i	$0.647080\pi$
$654$		0			0
$655$			26.6981i			1.04318i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	16.6502	−	16.6502i	0.648599	−	0.648599i	−0.304056	−	0.952654i	$-0.598341\pi$
$659$	16.6502	−	16.6502i	0.648599	−	0.648599i	0.952654	+	0.304056i	$0.0983409\pi$
$660$		0			0
$661$	−14.6386	−	14.6386i	−0.569375	−	0.569375i	0.362578	−	0.931953i	$-0.381896\pi$
$661$	−14.6386	−	14.6386i	−0.569375	−	0.569375i	−0.931953	+	0.362578i	$0.881896\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.87320			0.188974
$666$		0			0
$667$	15.1428	−	15.1428i	0.586331	−	0.586331i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	40.0959			1.54789
$672$		0			0
$673$	−20.5492			−0.792115			−0.396058	−	0.918226i	$-0.629622\pi$
$673$	−20.5492			−0.792115			−0.396058	+	0.918226i	$0.629622\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−6.72550	+	6.72550i	−0.258482	+	0.258482i	−0.824437	−	0.565954i	$-0.808508\pi$
$677$	−6.72550	+	6.72550i	−0.258482	+	0.258482i	0.565954	+	0.824437i	$0.308508\pi$
$678$		0			0
$679$	−0.191769			−0.00735942
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−30.5399	−	30.5399i	−1.16858	−	1.16858i	−0.982543	−	0.186033i	$-0.940437\pi$
$683$	−30.5399	−	30.5399i	−1.16858	−	1.16858i	−0.186033	−	0.982543i	$-0.559563\pi$
$684$		0			0
$685$	−31.3466	+	31.3466i	−1.19769	+	1.19769i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			0.772584i			0.0294331i
$690$		0			0
$691$	−10.8672	−	10.8672i	−0.413409	−	0.413409i	0.469515	−	0.882924i	$-0.344429\pi$
$691$	−10.8672	−	10.8672i	−0.413409	−	0.413409i	−0.882924	+	0.469515i	$0.844429\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	4.06117i		−	0.154049i
$696$		0			0
$697$			0.0448101i			0.00169730i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20.5506	+	20.5506i	0.776184	+	0.776184i	0.979180	−	0.202995i	$-0.0650677\pi$
$701$	20.5506	+	20.5506i	0.776184	+	0.776184i	−0.202995	+	0.979180i	$0.565068\pi$
$702$		0			0
$703$		−	39.2492i		−	1.48031i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.470532	−	0.470532i	0.0176962	−	0.0176962i
$708$		0			0
$709$	−33.4311	−	33.4311i	−1.25553	−	1.25553i	−0.953205	−	0.302325i	$-0.902237\pi$
$709$	−33.4311	−	33.4311i	−1.25553	−	1.25553i	−0.302325	−	0.953205i	$-0.597763\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.5092			0.805525
$714$		0			0
$715$	4.93467	−	4.93467i	0.184546	−	0.184546i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.15381			−0.341380			−0.170690	−	0.985325i	$-0.554600\pi$
$719$	−9.15381			−0.341380			−0.170690	+	0.985325i	$0.554600\pi$
$720$		0			0
$721$	−2.93467			−0.109293
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−38.4511	+	38.4511i	−1.42804	+	1.42804i
$726$		0			0
$727$	37.6052			1.39470			0.697351	−	0.716730i	$-0.254362\pi$
$727$	37.6052			1.39470			0.697351	+	0.716730i	$0.254362\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.69983	+	3.69983i	0.136843	+	0.136843i
$732$		0			0
$733$	33.4129	−	33.4129i	1.23413	−	1.23413i	0.271771	−	0.962362i	$-0.412391\pi$
$733$	33.4129	−	33.4129i	1.23413	−	1.23413i	0.962362	−	0.271771i	$-0.0876092\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	34.9917i		−	1.28894i
$738$		0			0
$739$	−21.9860	−	21.9860i	−0.808769	−	0.808769i	0.175678	−	0.984448i	$-0.443788\pi$
$739$	−21.9860	−	21.9860i	−0.808769	−	0.808769i	−0.984448	+	0.175678i	$0.943788\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20.6315i		−	0.756896i	−0.925623	−	0.378448i	$-0.876458\pi$
$743$		−	20.6315i		−	0.756896i	0.925623	−	0.378448i	$-0.123542\pi$
$744$		0			0
$745$		−	15.9232i		−	0.583382i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.83428	−	1.83428i	−0.0670230	−	0.0670230i
$750$		0			0
$751$		−	21.1419i		−	0.771477i	−0.922608	−	0.385739i	$-0.873947\pi$
$751$		−	21.1419i		−	0.771477i	0.922608	−	0.385739i	$-0.126053\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	17.3183	−	17.3183i	0.630279	−	0.630279i
$756$		0			0
$757$	16.9156	+	16.9156i	0.614810	+	0.614810i	0.944195	−	0.329386i	$-0.106842\pi$
$757$	16.9156	+	16.9156i	0.614810	+	0.614810i	−0.329386	+	0.944195i	$0.606842\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−29.0688			−1.05374			−0.526871	−	0.849945i	$-0.676635\pi$
$761$	−29.0688			−1.05374			−0.526871	+	0.849945i	$0.676635\pi$
$762$		0			0
$763$	−0.553188	+	0.553188i	−0.0200267	+	0.0200267i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.29611			0.0467999
$768$		0			0
$769$	17.9123			0.645933			0.322966	−	0.946410i	$-0.395320\pi$
$769$	17.9123			0.645933			0.322966	+	0.946410i	$0.395320\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−19.1937	+	19.1937i	−0.690351	+	0.690351i	−0.962309	−	0.271958i	$-0.912329\pi$
$773$	−19.1937	+	19.1937i	−0.690351	+	0.690351i	0.271958	+	0.962309i	$0.412329\pi$
$774$		0			0
$775$	−54.6169			−1.96190
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.305862	+	0.305862i	0.0109587	+	0.0109587i
$780$		0			0
$781$	21.8173	−	21.8173i	0.780685	−	0.780685i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			39.3157i			1.40324i
$786$		0			0
$787$	−5.09354	−	5.09354i	−0.181565	−	0.181565i	0.610472	−	0.792038i	$-0.290980\pi$
$787$	−5.09354	−	5.09354i	−0.181565	−	0.181565i	−0.792038	+	0.610472i	$0.790980\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.73891i		−	0.0618284i
$792$		0			0
$793$			2.79491i			0.0992503i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−22.2946	−	22.2946i	−0.789714	−	0.789714i	0.191733	−	0.981447i	$-0.438589\pi$
$797$	−22.2946	−	22.2946i	−0.789714	−	0.789714i	−0.981447	+	0.191733i	$0.938589\pi$
$798$		0			0
$799$			5.98603i			0.211770i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−25.4285	+	25.4285i	−0.897351	+	0.897351i
$804$		0			0
$805$	−1.90317	−	1.90317i	−0.0670781	−	0.0670781i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16.9021			−0.594246			−0.297123	−	0.954839i	$-0.596027\pi$
$809$	−16.9021			−0.594246			−0.297123	+	0.954839i	$0.596027\pi$
$810$		0			0
$811$	−38.2508	+	38.2508i	−1.34317	+	1.34317i	−0.450281	+	0.892887i	$0.648676\pi$
$811$	−38.2508	+	38.2508i	−1.34317	+	1.34317i	−0.892887	+	0.450281i	$0.851324\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−74.0654			−2.59440
$816$		0			0
$817$	50.5082			1.76706
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.6774	−	15.6774i	0.547144	−	0.547144i	−0.378470	−	0.925614i	$-0.623550\pi$
$821$	15.6774	−	15.6774i	0.547144	−	0.547144i	0.925614	+	0.378470i	$0.123550\pi$
$822$		0			0
$823$	0.813334			0.0283510			0.0141755	−	0.999900i	$-0.495488\pi$
$823$	0.813334			0.0283510			0.0141755	+	0.999900i	$0.495488\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.7929	+	12.7929i	0.444854	+	0.444854i	0.893639	−	0.448786i	$-0.148143\pi$
$827$	12.7929	+	12.7929i	0.444854	+	0.444854i	−0.448786	+	0.893639i	$0.648143\pi$
$828$		0			0
$829$	−29.2623	+	29.2623i	−1.01632	+	1.01632i	−0.0164588	+	0.999865i	$0.505239\pi$
$829$	−29.2623	+	29.2623i	−1.01632	+	1.01632i	−0.999865	+	0.0164588i	$0.994761\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			4.55329i			0.157762i
$834$		0			0
$835$	17.2431	+	17.2431i	0.596723	+	0.596723i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1.42409i		−	0.0491652i	−0.999698	−	0.0245826i	$-0.992174\pi$
$839$		−	1.42409i		−	0.0491652i	0.999698	−	0.0245826i	$-0.00782567\pi$
$840$		0			0
$841$			8.66659i			0.298848i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−33.8787	−	33.8787i	−1.16546	−	1.16546i
$846$		0			0
$847$			3.29258i			0.113134i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15.3284	+	15.3284i	−0.525449	+	0.525449i
$852$		0			0
$853$	26.0796	+	26.0796i	0.892947	+	0.892947i	0.994800	−	0.101852i	$-0.0324769\pi$
$853$	26.0796	+	26.0796i	0.892947	+	0.892947i	−0.101852	+	0.994800i	$0.532477\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−46.9249			−1.60293			−0.801463	−	0.598045i	$-0.795944\pi$
$857$	−46.9249			−1.60293			−0.801463	+	0.598045i	$0.795944\pi$
$858$		0			0
$859$	29.1799	−	29.1799i	0.995605	−	0.995605i	−0.00438503	−	0.999990i	$-0.501396\pi$
$859$	29.1799	−	29.1799i	0.995605	−	0.995605i	0.999990	+	0.00438503i	$0.00139580\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−21.9276			−0.746424			−0.373212	−	0.927746i	$-0.621744\pi$
$863$	−21.9276			−0.746424			−0.373212	+	0.927746i	$0.621744\pi$
$864$		0			0
$865$	−5.03149			−0.171076
$866$		0			0
$867$		0			0
$868$		0			0
$869$	17.3350	−	17.3350i	0.588050	−	0.588050i
$870$		0			0
$871$	2.43913			0.0826466
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.10548	+	2.10548i	0.0711781	+	0.0711781i
$876$		0			0
$877$	29.1540	−	29.1540i	0.984460	−	0.984460i	−0.0154208	−	0.999881i	$-0.504909\pi$
$877$	29.1540	−	29.1540i	0.984460	−	0.984460i	0.999881	+	0.0154208i	$0.00490879\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			22.0859i			0.744094i	0.928214	+	0.372047i	$0.121344\pi$
$881$			22.0859i			0.744094i	−0.928214	+	0.372047i	$0.878656\pi$
$882$		0			0
$883$	−15.8038	−	15.8038i	−0.531840	−	0.531840i	0.389280	−	0.921120i	$-0.372724\pi$
$883$	−15.8038	−	15.8038i	−0.531840	−	0.531840i	−0.921120	+	0.389280i	$0.872724\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	28.0131i		−	0.940587i	−0.882510	−	0.470293i	$-0.844148\pi$
$887$		−	28.0131i		−	0.940587i	0.882510	−	0.470293i	$-0.155852\pi$
$888$		0			0
$889$			1.99812i			0.0670147i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	40.8591	+	40.8591i	1.36730	+	1.36730i
$894$		0			0
$895$		−	46.2179i		−	1.54489i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.7513	+	26.7513i	−0.892204	+	0.892204i
$900$		0			0
$901$	0.989125	+	0.989125i	0.0329525	+	0.0329525i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−32.9359			−1.09483
$906$		0			0
$907$	2.86723	−	2.86723i	0.0952049	−	0.0952049i	−0.657900	−	0.753105i	$-0.728555\pi$
$907$	2.86723	−	2.86723i	0.0952049	−	0.0952049i	0.753105	+	0.657900i	$0.228555\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−13.6528			−0.452338			−0.226169	−	0.974088i	$-0.572620\pi$
$911$	−13.6528			−0.452338			−0.226169	+	0.974088i	$0.572620\pi$
$912$		0			0
$913$	−31.7387			−1.05040
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.05062	−	1.05062i	0.0346945	−	0.0346945i
$918$		0			0
$919$	−16.4461			−0.542506			−0.271253	−	0.962508i	$-0.587438\pi$
$919$	−16.4461			−0.542506			−0.271253	+	0.962508i	$0.587438\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.52079	+	1.52079i	0.0500575	+	0.0500575i
$924$		0			0
$925$	38.9223	−	38.9223i	1.27976	−	1.27976i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			58.8247i			1.92997i	0.262297	+	0.964987i	$0.415520\pi$
$929$			58.8247i			1.92997i	−0.262297	+	0.964987i	$0.584480\pi$
$930$		0			0
$931$	31.0796	+	31.0796i	1.01859	+	1.01859i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	12.6355i		−	0.413226i
$936$		0			0
$937$		−	31.5717i		−	1.03140i	−0.856769	−	0.515700i	$-0.827532\pi$
$937$		−	31.5717i		−	1.03140i	0.856769	−	0.515700i	$-0.172468\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	5.30353	+	5.30353i	0.172890	+	0.172890i	0.788248	−	0.615358i	$-0.210988\pi$
$941$	5.30353	+	5.30353i	0.172890	+	0.172890i	−0.615358	+	0.788248i	$0.710988\pi$
$942$		0			0
$943$		−	0.238902i		−	0.00777973i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11.9011	−	11.9011i	0.386732	−	0.386732i	−0.486788	−	0.873520i	$-0.661832\pi$
$947$	11.9011	−	11.9011i	0.386732	−	0.386732i	0.873520	+	0.486788i	$0.161832\pi$
$948$		0			0
$949$	−1.77251	−	1.77251i	−0.0575381	−	0.0575381i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−18.1504			−0.587948			−0.293974	−	0.955813i	$-0.594978\pi$
$953$	−18.1504			−0.587948			−0.293974	+	0.955813i	$0.594978\pi$
$954$		0			0
$955$	18.7858	−	18.7858i	0.607895	−	0.607895i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.46709			0.0796665
$960$		0			0
$961$	−6.99812			−0.225746
$962$		0			0
$963$		0			0
$964$		0			0
$965$	28.7857	−	28.7857i	0.926643	−	0.926643i
$966$		0			0
$967$	−48.0626			−1.54559			−0.772794	−	0.634657i	$-0.781141\pi$
$967$	−48.0626			−1.54559			−0.772794	+	0.634657i	$0.781141\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−8.81259	−	8.81259i	−0.282810	−	0.282810i	0.551419	−	0.834228i	$-0.314087\pi$
$971$	−8.81259	−	8.81259i	−0.282810	−	0.282810i	−0.834228	+	0.551419i	$0.814087\pi$
$972$		0			0
$973$	−0.159814	+	0.159814i	−0.00512341	+	0.00512341i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	40.9404i		−	1.30980i	−0.755716	−	0.654900i	$-0.772711\pi$
$977$		−	40.9404i		−	1.30980i	0.755716	−	0.654900i	$-0.227289\pi$
$978$		0			0
$979$	−43.7980	−	43.7980i	−1.39979	−	1.39979i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			15.0884i			0.481246i	0.970619	+	0.240623i	$0.0773518\pi$
$983$			15.0884i			0.481246i	−0.970619	+	0.240623i	$0.922648\pi$
$984$		0			0
$985$			54.3757i			1.73255i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.7254	−	19.7254i	−0.627232	−	0.627232i
$990$		0			0
$991$			3.48435i			0.110684i	0.998467	+	0.0553420i	$0.0176249\pi$
$991$			3.48435i			0.110684i	−0.998467	+	0.0553420i	$0.982375\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	18.9140	−	18.9140i	0.599615	−	0.599615i
$996$		0			0
$997$	−10.5079	−	10.5079i	−0.332789	−	0.332789i	0.520856	−	0.853645i	$-0.325613\pi$
$997$	−10.5079	−	10.5079i	−0.332789	−	0.332789i	−0.853645	+	0.520856i	$0.825613\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.l.a.143.1		16
3.2	odd	2	inner	576.2.l.a.143.8		16
4.3	odd	2		144.2.l.a.107.5	yes	16
8.3	odd	2		1152.2.l.a.287.8		16
8.5	even	2		1152.2.l.b.287.8		16
12.11	even	2		144.2.l.a.107.4	yes	16
16.3	odd	4	inner	576.2.l.a.431.8		16
16.5	even	4		1152.2.l.a.863.1		16
16.11	odd	4		1152.2.l.b.863.1		16
16.13	even	4		144.2.l.a.35.4	✓	16
24.5	odd	2		1152.2.l.b.287.1		16
24.11	even	2		1152.2.l.a.287.1		16
48.5	odd	4		1152.2.l.a.863.8		16
48.11	even	4		1152.2.l.b.863.8		16
48.29	odd	4		144.2.l.a.35.5	yes	16
48.35	even	4	inner	576.2.l.a.431.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.l.a.35.4	✓	16	16.13	even	4
144.2.l.a.35.5	yes	16	48.29	odd	4
144.2.l.a.107.4	yes	16	12.11	even	2
144.2.l.a.107.5	yes	16	4.3	odd	2
576.2.l.a.143.1		16	1.1	even	1	trivial
576.2.l.a.143.8		16	3.2	odd	2	inner
576.2.l.a.431.1		16	48.35	even	4	inner
576.2.l.a.431.8		16	16.3	odd	4	inner
1152.2.l.a.287.1		16	24.11	even	2
1152.2.l.a.287.8		16	8.3	odd	2
1152.2.l.a.863.1		16	16.5	even	4
1152.2.l.a.863.8		16	48.5	odd	4
1152.2.l.b.287.1		16	24.5	odd	2
1152.2.l.b.287.8		16	8.5	even	2
1152.2.l.b.863.1		16	16.11	odd	4
1152.2.l.b.863.8		16	48.11	even	4

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

\(f(q)\)	\(=\)	\(q+(-2.63251 + 2.63251i) q^{5} +0.207188 q^{7} +O(q^{10})\)	\(q+(-2.63251 + 2.63251i) q^{5} +0.207188 q^{7} +(-3.66686 - 3.66686i) q^{11} +(0.255601 - 0.255601i) q^{13} -0.654483i q^{17} +(-4.46733 - 4.46733i) q^{19} +3.48934i q^{23} -8.86025i q^{25} +(-4.33973 - 4.33973i) q^{29} -6.16426i q^{31} +(-0.545426 + 0.545426i) q^{35} +(4.39291 + 4.39291i) q^{37} -0.0684664 q^{41} +(-5.65306 + 5.65306i) q^{43} -9.14619 q^{47} -6.95707 q^{49} +(-1.51131 + 1.51131i) q^{53} +19.3061 q^{55} +(2.53542 + 2.53542i) q^{59} +(-5.46733 + 5.46733i) q^{61} +1.34575i q^{65} +(4.77135 + 4.77135i) q^{67} +5.94986i q^{71} -6.93467i q^{73} +(-0.759730 - 0.759730i) q^{77} +4.72748i q^{79} +(4.32777 - 4.32777i) q^{83} +(1.72294 + 1.72294i) q^{85} +11.9443 q^{89} +(0.0529576 - 0.0529576i) q^{91} +23.5206 q^{95} -0.925579 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 16 q^{19} + 32 q^{43} + 16 q^{49} + 64 q^{55} - 32 q^{61} + 16 q^{67} - 32 q^{85} - 48 q^{91}+O(q^{100}) \) 16 * q - 16 * q^19 + 32 * q^43 + 16 * q^49 + 64 * q^55 - 32 * q^61 + 16 * q^67 - 32 * q^85 - 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more