LMFDB - Embedded newform 1152.2.l.a.287.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(287,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, 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("1152.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.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:	$9.19876631285$
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			287.1
Root			$-0.517174 - 1.31626i$ of defining polynomial
Character	$\chi$	$=$	1152.287
Dual form			1152.2.l.a.863.1

$q$-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 $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{4}\right)$

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	$-0.512467\pi$
$7$	−0.207188			−0.0783098			−0.0391549	+	0.999233i	$0.512467\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.66686	+	3.66686i	1.10560	+	1.10560i	0.993722	+	0.111878i	$0.0356867\pi$
$11$	3.66686	+	3.66686i	1.10560	+	1.10560i	0.111878	+	0.993722i	$0.464313\pi$
$12$		0			0
$13$	−0.255601	+	0.255601i	−0.0708910	+	0.0708910i	−0.741663	−	0.670772i	$-0.765963\pi$
$13$	−0.255601	+	0.255601i	−0.0708910	+	0.0708910i	0.670772	+	0.741663i	$0.265963\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			0.654483i			0.158735i	0.996845	+	0.0793677i	$0.0252901\pi$
$17$			0.654483i			0.158735i	−0.996845	+	0.0793677i	$0.974710\pi$
$18$		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	$0.186734\pi$
$31$			6.16426i			1.10713i	−0.832805	+	0.553567i	$0.813266\pi$
$32$		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	0.246861	−	0.969051i	$-0.420601\pi$
$37$	−4.39291	−	4.39291i	−0.722190	−	0.722190i	−0.969051	+	0.246861i	$0.920601\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.0684664			0.0106927			0.00534633	−	0.999986i	$-0.498298\pi$
$41$	0.0684664			0.0106927			0.00534633	+	0.999986i	$0.498298\pi$
$42$		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	0.522535	−	0.852618i	$-0.324987\pi$
$59$	−2.53542	−	2.53542i	−0.330083	−	0.330083i	−0.852618	+	0.522535i	$0.824987\pi$
$60$		0			0
$61$	5.46733	−	5.46733i	0.700020	−	0.700020i	−0.264394	−	0.964415i	$-0.585172\pi$
$61$	5.46733	−	5.46733i	0.700020	−	0.700020i	0.964415	+	0.264394i	$0.0851720\pi$
$62$		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	$-0.914317\pi$
$79$		−	4.72748i		−	0.531883i	0.963989	−	0.265942i	$-0.0856828\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.32777	+	4.32777i	−0.475035	+	0.475035i	−0.903539	−	0.428505i	$-0.859040\pi$
$83$	−4.32777	+	4.32777i	−0.475035	+	0.475035i	0.428505	+	0.903539i	$0.359040\pi$
$84$		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	$-0.718195\pi$
$89$	−11.9443			−1.26609			−0.633045	+	0.774115i	$0.718195\pi$
$90$		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	$-0.254152\pi$
$103$	14.1643			1.39565			0.697823	+	0.716270i	$0.254152\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.85318	+	8.85318i	0.855870	+	0.855870i	0.990848	−	0.134979i	$-0.0430967\pi$
$107$	8.85318	+	8.85318i	0.855870	+	0.855870i	−0.134979	+	0.990848i	$0.543097\pi$
$108$		0			0
$109$	2.66998	−	2.66998i	0.255737	−	0.255737i	−0.567580	−	0.823318i	$-0.692120\pi$
$109$	2.66998	−	2.66998i	0.255737	−	0.255737i	0.823318	+	0.567580i	$0.192120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			8.39289i			0.789536i	0.918781	+	0.394768i	$0.129175\pi$
$113$			8.39289i			0.789536i	−0.918781	+	0.394768i	$0.870825\pi$
$114$		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	$-0.859260\pi$
$127$		−	9.64397i		−	0.855764i	0.903835	−	0.427882i	$-0.140740\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.07084	+	5.07084i	−0.443041	+	0.443041i	−0.893033	−	0.449992i	$-0.851427\pi$
$131$	−5.07084	+	5.07084i	−0.443041	+	0.443041i	0.449992	+	0.893033i	$0.351427\pi$
$132$		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	$-0.669860\pi$
$137$	−11.9075			−1.01732			−0.508662	+	0.860966i	$0.669860\pi$
$138$		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	$-0.413743\pi$
$151$	6.57864			0.535362			0.267681	+	0.963508i	$0.413743\pi$
$152$		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	−0.939234	−	0.343276i	$-0.888463\pi$
$157$	−7.46733	+	7.46733i	−0.595958	+	0.595958i	0.343276	+	0.939234i	$0.388463\pi$
$158$		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	−0.298340	−	0.954460i	$-0.596433\pi$
$179$	8.77829	−	8.77829i	0.656120	−	0.656120i	0.954460	+	0.298340i	$0.0964328\pi$
$180$		0			0
$181$	−6.25560	−	6.25560i	−0.464975	−	0.464975i	0.435307	−	0.900282i	$-0.356640\pi$
$181$	−6.25560	−	6.25560i	−0.464975	−	0.464975i	−0.900282	+	0.435307i	$0.856640\pi$
$182$		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	$-0.418037\pi$
$199$	7.18478			0.509316			0.254658	+	0.967031i	$0.418037\pi$
$200$		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	$0.00662499\pi$
$223$			0.621565i			0.0416230i	−0.999783	+	0.0208115i	$0.993375\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.9257	+	10.9257i	−0.725164	+	0.725164i	−0.969652	−	0.244489i	$-0.921380\pi$
$227$	−10.9257	+	10.9257i	−0.725164	+	0.725164i	0.244489	+	0.969652i	$0.421380\pi$
$228$		0			0
$229$	17.9761	+	17.9761i	1.18789	+	1.18789i	0.977649	+	0.210245i	$0.0674262\pi$
$229$	17.9761	+	17.9761i	1.18789	+	1.18789i	0.210245	+	0.977649i	$0.432574\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.5409			1.41119			0.705595	−	0.708616i	$-0.250680\pi$
$233$	21.5409			1.41119			0.705595	+	0.708616i	$0.250680\pi$
$234$		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	0.375360	−	0.926879i	$-0.377519\pi$
$251$	−8.73770	−	8.73770i	−0.551519	−	0.551519i	−0.926879	+	0.375360i	$0.877519\pi$
$252$		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	$0.199109\pi$
$257$			18.7732i			1.17104i	−0.810659	+	0.585519i	$0.800891\pi$
$258$		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	$0.278175\pi$
$271$			25.2478i			1.53369i	−0.641831	+	0.766846i	$0.721825\pi$
$272$		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	$0.105316\pi$
$277$	21.1473	+	21.1473i	1.27062	+	1.27062i	0.324858	+	0.945763i	$0.394684\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.34812			0.378697			0.189348	−	0.981910i	$-0.439362\pi$
$281$	6.34812			0.378697			0.189348	+	0.981910i	$0.439362\pi$
$282$		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	0.999540	−	0.0303420i	$-0.00965963\pi$
$347$	18.0542	+	18.0542i	0.969198	+	0.969198i	−0.0303420	+	0.999540i	$0.509660\pi$
$348$		0			0
$349$	3.32758	−	3.32758i	0.178121	−	0.178121i	−0.612415	−	0.790536i	$-0.709802\pi$
$349$	3.32758	−	3.32758i	0.178121	−	0.178121i	0.790536	+	0.612415i	$0.209802\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	18.9229i		−	1.00717i	−0.863947	−	0.503583i	$-0.832015\pi$
$353$		−	18.9229i		−	1.00717i	0.863947	−	0.503583i	$-0.167985\pi$
$354$		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	$0.166108\pi$
$367$			19.0989i			0.996956i	−0.866902	+	0.498478i	$0.833892\pi$
$368$		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	0.610365	−	0.792120i	$-0.291023\pi$
$373$	−3.51026	−	3.51026i	−0.181754	−	0.181754i	−0.792120	+	0.610365i	$0.791023\pi$
$374$		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	$0.584535\pi$
$397$	−24.4559	+	24.4559i	−1.22741	+	1.22741i	−0.964942	+	0.262465i	$0.915465\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			0.517550i			0.0258452i	0.999916	+	0.0129226i	$0.00411351\pi$
$401$			0.517550i			0.0258452i	−0.999916	+	0.0129226i	$0.995886\pi$
$402$		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	$0.507456\pi$
$419$	−20.9433	+	20.9433i	−1.02315	+	1.02315i	−0.999726	+	0.0234208i	$0.992544\pi$
$420$		0			0
$421$	−14.2218	−	14.2218i	−0.693126	−	0.693126i	0.269792	−	0.962919i	$-0.413045\pi$
$421$	−14.2218	−	14.2218i	−0.693126	−	0.693126i	−0.962919	+	0.269792i	$0.913045\pi$
$422$		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	$-0.356397\pi$
$439$	18.2702			0.871988			0.435994	+	0.899950i	$0.356397\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−13.9646	−	13.9646i	−0.663479	−	0.663479i	0.292720	−	0.956198i	$-0.405440\pi$
$443$	−13.9646	−	13.9646i	−0.663479	−	0.663479i	−0.956198	+	0.292720i	$0.905440\pi$
$444$		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	$-0.810122\pi$
$449$		−	23.8072i		−	1.12353i	0.827296	−	0.561766i	$-0.189878\pi$
$450$		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	$0.224794\pi$
$463$			27.9277i			1.29791i	−0.760827	+	0.648955i	$0.775206\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.32777	+	4.32777i	−0.200265	+	0.200265i	−0.800114	−	0.599848i	$-0.795228\pi$
$467$	−4.32777	+	4.32777i	−0.200265	+	0.200265i	0.599848	+	0.800114i	$0.295228\pi$
$468$		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	$-0.712640\pi$
$487$	−27.3397			−1.23888			−0.619440	+	0.785044i	$0.712640\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.78368	−	3.78368i	−0.170755	−	0.170755i	0.616556	−	0.787311i	$-0.288527\pi$
$491$	−3.78368	−	3.78368i	−0.170755	−	0.170755i	−0.787311	+	0.616556i	$0.788527\pi$
$492$		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	$-0.567454\pi$
$521$	−9.60170			−0.420658			−0.210329	+	0.977631i	$0.567454\pi$
$522$		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	−0.800333	−	0.599555i	$-0.795344\pi$
$541$	−4.66998	+	4.66998i	−0.200778	+	0.200778i	0.599555	+	0.800333i	$0.295344\pi$
$542$		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	$-0.333216\pi$
$563$	32.4157	−	32.4157i	1.36616	−	1.36616i	0.865841	−	0.500319i	$-0.166784\pi$
$564$		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	$-0.574909\pi$
$569$	−11.1238			−0.466335			−0.233168	+	0.972437i	$0.574909\pi$
$570$		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	0.843273	−	0.537486i	$-0.180626\pi$
$587$	7.40862	+	7.40862i	0.305786	+	0.305786i	−0.537486	+	0.843273i	$0.680626\pi$
$588$		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	$0.0819000\pi$
$593$			12.3933i			0.508934i	−0.967081	+	0.254467i	$0.918100\pi$
$594$		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	$-0.790299\pi$
$607$		−	30.1643i		−	1.22433i	0.790730	−	0.612165i	$-0.209701\pi$
$608$		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	0.742537	−	0.669805i	$-0.233623\pi$
$613$	1.80074	+	1.80074i	0.0727312	+	0.0727312i	−0.669805	+	0.742537i	$0.733623\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−30.8098			−1.24036			−0.620178	−	0.784461i	$-0.712940\pi$
$617$	−30.8098			−1.24036			−0.620178	+	0.784461i	$0.712940\pi$
$618$		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	$-0.531218\pi$
$631$	−4.91925			−0.195832			−0.0979161	+	0.995195i	$0.531218\pi$
$632$		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	$0.0962789\pi$
$641$			15.0833i			0.595756i	−0.954604	+	0.297878i	$0.903721\pi$
$642$		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	−0.952654	−	0.304056i	$-0.901659\pi$
$659$	−16.6502	+	16.6502i	−0.648599	+	0.648599i	0.304056	+	0.952654i	$0.401659\pi$
$660$		0			0
$661$	14.6386	+	14.6386i	0.569375	+	0.569375i	0.931953	−	0.362578i	$-0.118104\pi$
$661$	14.6386	+	14.6386i	0.569375	+	0.569375i	−0.362578	+	0.931953i	$0.618104\pi$
$662$		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	$0.0595632\pi$
$683$	30.5399	+	30.5399i	1.16858	+	1.16858i	0.186033	+	0.982543i	$0.440437\pi$
$684$		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	$0.0977626\pi$
$709$	33.4311	+	33.4311i	1.25553	+	1.25553i	0.302325	+	0.953205i	$0.402237\pi$
$710$		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	$-0.745638\pi$
$727$	−37.6052			−1.39470			−0.697351	+	0.716730i	$0.745638\pi$
$728$		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	$0.587609\pi$
$733$	−33.4129	+	33.4129i	−1.23413	+	1.23413i	−0.962362	+	0.271771i	$0.912391\pi$
$734$		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	$0.126053\pi$
$751$			21.1419i			0.771477i	−0.922608	+	0.385739i	$0.873947\pi$
$752$		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	0.329386	−	0.944195i	$-0.393158\pi$
$757$	−16.9156	−	16.9156i	−0.614810	−	0.614810i	−0.944195	+	0.329386i	$0.893158\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	29.0688			1.05374			0.526871	−	0.849945i	$-0.323365\pi$
$761$	29.0688			1.05374			0.526871	+	0.849945i	$0.323365\pi$
$762$		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	$-0.403973\pi$
$809$	16.9021			0.594246			0.297123	+	0.954839i	$0.403973\pi$
$810$		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	$-0.504512\pi$
$823$	−0.813334			−0.0283510			−0.0141755	+	0.999900i	$0.504512\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.7929	−	12.7929i	−0.444854	−	0.444854i	0.448786	−	0.893639i	$-0.351857\pi$
$827$	−12.7929	−	12.7929i	−0.444854	−	0.444854i	−0.893639	+	0.448786i	$0.851857\pi$
$828$		0			0
$829$	29.2623	−	29.2623i	1.01632	−	1.01632i	0.0164588	−	0.999865i	$-0.494761\pi$
$829$	29.2623	−	29.2623i	1.01632	−	1.01632i	0.999865	−	0.0164588i	$-0.00523922\pi$
$830$		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	0.101852	−	0.994800i	$-0.467523\pi$
$853$	−26.0796	−	26.0796i	−0.892947	−	0.892947i	−0.994800	+	0.101852i	$0.967523\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	46.9249			1.60293			0.801463	−	0.598045i	$-0.204056\pi$
$857$	46.9249			1.60293			0.801463	+	0.598045i	$0.204056\pi$
$858$		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	−0.999881	−	0.0154208i	$-0.995091\pi$
$877$	−29.1540	+	29.1540i	−0.984460	+	0.984460i	0.0154208	+	0.999881i	$0.495091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	22.0859i		−	0.744094i	−0.928214	−	0.372047i	$-0.878656\pi$
$881$		−	22.0859i		−	0.744094i	0.928214	−	0.372047i	$-0.121344\pi$
$882$		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	$-0.412562\pi$
$919$	16.4461			0.542506			0.271253	+	0.962508i	$0.412562\pi$
$920$		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	$-0.584480\pi$
$929$		−	58.8247i		−	1.92997i	0.262297	−	0.964987i	$-0.415520\pi$
$930$		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	−0.873520	−	0.486788i	$-0.838168\pi$
$947$	−11.9011	+	11.9011i	−0.386732	+	0.386732i	0.486788	+	0.873520i	$0.338168\pi$
$948$		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	$-0.405022\pi$
$953$	18.1504			0.587948			0.293974	+	0.955813i	$0.405022\pi$
$954$		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	$-0.218859\pi$
$967$	48.0626			1.54559			0.772794	+	0.634657i	$0.218859\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.81259	+	8.81259i	0.282810	+	0.282810i	0.834228	−	0.551419i	$-0.185913\pi$
$971$	8.81259	+	8.81259i	0.282810	+	0.282810i	−0.551419	+	0.834228i	$0.685913\pi$
$972$		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	$0.227289\pi$
$977$			40.9404i			1.30980i	−0.755716	+	0.654900i	$0.772711\pi$
$978$		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	$-0.982375\pi$
$991$		−	3.48435i		−	0.110684i	0.998467	−	0.0553420i	$-0.0176249\pi$
$992$		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	0.853645	−	0.520856i	$-0.174387\pi$
$997$	10.5079	+	10.5079i	0.332789	+	0.332789i	−0.520856	+	0.853645i	$0.674387\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.l.a.35.4	✓	16	48.11	even	4
144.2.l.a.35.5	yes	16	16.11	odd	4
144.2.l.a.107.4	yes	16	8.5	even	2
144.2.l.a.107.5	yes	16	24.5	odd	2
576.2.l.a.143.1		16	24.11	even	2
576.2.l.a.143.8		16	8.3	odd	2
576.2.l.a.431.1		16	16.5	even	4
576.2.l.a.431.8		16	48.5	odd	4
1152.2.l.a.287.1		16	1.1	even	1	trivial
1152.2.l.a.287.8		16	3.2	odd	2	inner
1152.2.l.a.863.1		16	48.35	even	4	inner
1152.2.l.a.863.8		16	16.3	odd	4	inner
1152.2.l.b.287.1		16	4.3	odd	2
1152.2.l.b.287.8		16	12.11	even	2
1152.2.l.b.863.1		16	48.29	odd	4
1152.2.l.b.863.8		16	16.13	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1152.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

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