LMFDB - Embedded newform 1152.2.k.c.865.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,2,Mod(289,1152)]

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([0, 3, 0]))

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

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

chi := DirichletCharacter("1152.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1152.k (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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.18939904.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 $ x^8 - 4x^7 + 14x^6 - 28x^5 + 43x^4 - 44x^3 + 30x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			865.2
Root			$0.500000 - 1.44392i$ of defining polynomial
Character	$\chi$	$=$	1152.865
Dual form			1152.2.k.c.289.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.334904 - 0.334904i) q^{5} +4.55765i q^{7} +O(q^{10})$	$q+(-0.334904 - 0.334904i) q^{5} +4.55765i q^{7} +(-2.47363 - 2.47363i) q^{11} +(0.0594122 - 0.0594122i) q^{13} -3.61706 q^{17} +(-2.55765 + 2.55765i) q^{19} +2.82843i q^{23} -4.77568i q^{25} +(-5.16333 + 5.16333i) q^{29} -0.557647 q^{31} +(1.52637 - 1.52637i) q^{35} +(-4.38607 - 4.38607i) q^{37} -9.27391i q^{41} +(1.61040 + 1.61040i) q^{43} -2.82843 q^{47} -13.7721 q^{49} +(-0.493523 - 0.493523i) q^{53} +1.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-2.72922 + 2.72922i) q^{61} -0.0397948 q^{65} +(-3.77568 + 3.77568i) q^{67} +9.11529i q^{71} +0.541560i q^{73} +(11.2739 - 11.2739i) q^{77} -10.9937 q^{79} +(-10.6417 + 10.6417i) q^{83} +(1.21137 + 1.21137i) q^{85} +14.6533i q^{89} +(0.270780 + 0.270780i) q^{91} +1.71313 q^{95} +4.31724 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} + 8 q^{19} - 16 q^{29} + 24 q^{31} + 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} + 16 q^{53} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{67} + 16 q^{77} - 24 q^{79} - 40 q^{83} + 16 q^{85} + 8 q^{91} + 48 q^{95}+O(q^{100}) $ 8 * q - 8 * q^11 + 8 * q^19 - 16 * q^29 + 24 * q^31 + 24 * q^35 + 16 * q^37 + 8 * q^43 - 8 * q^49 + 16 * q^53 + 32 * q^59 - 16 * q^61 + 16 * q^65 + 16 * q^67 + 16 * q^77 - 24 * q^79 - 40 * q^83 + 16 * q^85 + 8 * q^91 + 48 * q^95

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
$5$	−0.334904	−	0.334904i	−0.149774	−	0.149774i	0.628243	−	0.778017i	$-0.283774\pi$
$5$	−0.334904	−	0.334904i	−0.149774	−	0.149774i	−0.778017	+	0.628243i	$0.783774\pi$
$6$		0			0
$7$			4.55765i			1.72263i	0.508072	+	0.861314i	$0.330358\pi$
$7$			4.55765i			1.72263i	−0.508072	+	0.861314i	$0.669642\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.47363	−	2.47363i	−0.745826	−	0.745826i	0.227866	−	0.973692i	$-0.426825\pi$
$11$	−2.47363	−	2.47363i	−0.745826	−	0.745826i	−0.973692	+	0.227866i	$0.926825\pi$
$12$		0			0
$13$	0.0594122	−	0.0594122i	0.0164780	−	0.0164780i	−0.698820	−	0.715298i	$-0.746291\pi$
$13$	0.0594122	−	0.0594122i	0.0164780	−	0.0164780i	0.715298	+	0.698820i	$0.246291\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.61706			−0.877266			−0.438633	−	0.898666i	$-0.644537\pi$
$17$	−3.61706			−0.877266			−0.438633	+	0.898666i	$0.644537\pi$
$18$		0			0
$19$	−2.55765	+	2.55765i	−0.586765	+	0.586765i	−0.936754	−	0.349989i	$-0.886185\pi$
$19$	−2.55765	+	2.55765i	−0.586765	+	0.586765i	0.349989	+	0.936754i	$0.386185\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			2.82843i			0.589768i	0.955533	+	0.294884i	$0.0952810\pi$
$23$			2.82843i			0.589768i	−0.955533	+	0.294884i	$0.904719\pi$
$24$		0			0
$25$		−	4.77568i		−	0.955136i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.16333	+	5.16333i	−0.958807	+	0.958807i	−0.999184	−	0.0403780i	$-0.987144\pi$
$29$	−5.16333	+	5.16333i	−0.958807	+	0.958807i	0.0403780	+	0.999184i	$0.487144\pi$
$30$		0			0
$31$	−0.557647			−0.100156			−0.0500782	−	0.998745i	$-0.515947\pi$
$31$	−0.557647			−0.100156			−0.0500782	+	0.998745i	$0.515947\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.52637	−	1.52637i	0.258004	−	0.258004i
$36$		0			0
$37$	−4.38607	−	4.38607i	−0.721066	−	0.721066i	0.247756	−	0.968822i	$-0.420307\pi$
$37$	−4.38607	−	4.38607i	−0.721066	−	0.721066i	−0.968822	+	0.247756i	$0.920307\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	9.27391i		−	1.44834i	−0.689620	−	0.724171i	$-0.742223\pi$
$41$		−	9.27391i		−	1.44834i	0.689620	−	0.724171i	$-0.257777\pi$
$42$		0			0
$43$	1.61040	+	1.61040i	0.245583	+	0.245583i	0.819155	−	0.573572i	$-0.194443\pi$
$43$	1.61040	+	1.61040i	0.245583	+	0.245583i	−0.573572	+	0.819155i	$0.694443\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.82843			−0.412568			−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843			−0.412568			−0.206284	+	0.978492i	$0.566137\pi$
$48$		0			0
$49$	−13.7721			−1.96745
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.493523	−	0.493523i	−0.0677906	−	0.0677906i	0.672399	−	0.740189i	$-0.265264\pi$
$53$	−0.493523	−	0.493523i	−0.0677906	−	0.0677906i	−0.740189	+	0.672399i	$0.765264\pi$
$54$		0			0
$55$			1.65685i			0.223410i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	+	4.00000i	0.520756	+	0.520756i	0.917800	−	0.397044i	$-0.129964\pi$
$59$	4.00000	+	4.00000i	0.520756	+	0.520756i	−0.397044	+	0.917800i	$0.629964\pi$
$60$		0			0
$61$	−2.72922	+	2.72922i	−0.349441	+	0.349441i	−0.859901	−	0.510460i	$-0.829475\pi$
$61$	−2.72922	+	2.72922i	−0.349441	+	0.349441i	0.510460	+	0.859901i	$0.329475\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.0397948			−0.00493593
$66$		0			0
$67$	−3.77568	+	3.77568i	−0.461273	+	0.461273i	−0.899072	−	0.437800i	$-0.855758\pi$
$67$	−3.77568	+	3.77568i	−0.461273	+	0.461273i	0.437800	+	0.899072i	$0.355758\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			9.11529i			1.08179i	0.841091	+	0.540893i	$0.181914\pi$
$71$			9.11529i			1.08179i	−0.841091	+	0.540893i	$0.818086\pi$
$72$		0			0
$73$			0.541560i			0.0633848i	0.999498	+	0.0316924i	$0.0100897\pi$
$73$			0.541560i			0.0633848i	−0.999498	+	0.0316924i	$0.989910\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	11.2739	−	11.2739i	1.28478	−	1.28478i
$78$		0			0
$79$	−10.9937			−1.23689			−0.618445	−	0.785828i	$-0.712237\pi$
$79$	−10.9937			−1.23689			−0.618445	+	0.785828i	$0.712237\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.6417	+	10.6417i	−1.16807	+	1.16807i	−0.185415	+	0.982660i	$0.559363\pi$
$83$	−10.6417	+	10.6417i	−1.16807	+	1.16807i	−0.982660	+	0.185415i	$0.940637\pi$
$84$		0			0
$85$	1.21137	+	1.21137i	0.131391	+	0.131391i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			14.6533i			1.55325i	0.629964	+	0.776625i	$0.283070\pi$
$89$			14.6533i			1.55325i	−0.629964	+	0.776625i	$0.716930\pi$
$90$		0			0
$91$	0.270780	+	0.270780i	0.0283854	+	0.0283854i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.71313			0.175764
$96$		0			0
$97$	4.31724			0.438349			0.219175	−	0.975686i	$-0.429664\pi$
$97$	4.31724			0.438349			0.219175	+	0.975686i	$0.429664\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−0.453728	−	0.453728i	−0.0451477	−	0.0451477i	0.684173	−	0.729320i	$-0.260164\pi$
$101$	−0.453728	−	0.453728i	−0.0451477	−	0.0451477i	−0.729320	+	0.684173i	$0.760164\pi$
$102$		0			0
$103$		−	1.33686i		−	0.131724i	−0.997829	−	0.0658622i	$-0.979020\pi$
$103$		−	1.33686i		−	0.131724i	0.997829	−	0.0658622i	$-0.0209798\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.06255	+	6.06255i	0.586088	+	0.586088i	0.936570	−	0.350481i	$-0.113982\pi$
$107$	6.06255	+	6.06255i	0.586088	+	0.586088i	−0.350481	+	0.936570i	$0.613982\pi$
$108$		0			0
$109$	−5.71627	+	5.71627i	−0.547519	+	0.547519i	−0.925722	−	0.378203i	$-0.876542\pi$
$109$	−5.71627	+	5.71627i	−0.547519	+	0.547519i	0.378203	+	0.925722i	$0.376542\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.55136			0.898516			0.449258	−	0.893402i	$-0.351688\pi$
$113$	9.55136			0.898516			0.449258	+	0.893402i	$0.351688\pi$
$114$		0			0
$115$	0.947252	−	0.947252i	0.0883317	−	0.0883317i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	16.4853i		−	1.51120i
$120$		0			0
$121$			1.23765i			0.112514i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.27391	+	3.27391i	−0.292828	+	0.292828i
$126$		0			0
$127$	5.09921			0.452481			0.226241	−	0.974071i	$-0.427356\pi$
$127$	5.09921			0.452481			0.226241	+	0.974071i	$0.427356\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.11882	−	2.11882i	0.185123	−	0.185123i	−0.608461	−	0.793584i	$-0.708213\pi$
$131$	2.11882	−	2.11882i	0.185123	−	0.185123i	0.793584	+	0.608461i	$0.208213\pi$
$132$		0			0
$133$	−11.6569	−	11.6569i	−1.01078	−	1.01078i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3.37941i		−	0.288723i	−0.989525	−	0.144361i	$-0.953887\pi$
$137$		−	3.37941i		−	0.288723i	0.989525	−	0.144361i	$-0.0461127\pi$
$138$		0			0
$139$	−5.88118	−	5.88118i	−0.498835	−	0.498835i	0.412240	−	0.911075i	$-0.364746\pi$
$139$	−5.88118	−	5.88118i	−0.498835	−	0.498835i	−0.911075	+	0.412240i	$0.864746\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.293927			−0.0245794
$144$		0			0
$145$	3.45844			0.287208
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.99176	−	9.99176i	−0.818557	−	0.818557i	0.167342	−	0.985899i	$-0.446482\pi$
$149$	−9.99176	−	9.99176i	−0.818557	−	0.818557i	−0.985899	+	0.167342i	$0.946482\pi$
$150$		0			0
$151$		−	9.97685i		−	0.811905i	−0.913894	−	0.405952i	$-0.866940\pi$
$151$		−	9.97685i		−	0.811905i	0.913894	−	0.405952i	$-0.133060\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.186758	+	0.186758i	0.0150008	+	0.0150008i
$156$		0			0
$157$	16.1618	−	16.1618i	1.28985	−	1.28985i	0.354971	−	0.934877i	$-0.384491\pi$
$157$	16.1618	−	16.1618i	1.28985	−	1.28985i	0.934877	−	0.354971i	$-0.115509\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.8910			−1.01595
$162$		0			0
$163$	7.50490	−	7.50490i	0.587829	−	0.587829i	−0.349214	−	0.937043i	$-0.613551\pi$
$163$	7.50490	−	7.50490i	0.587829	−	0.587829i	0.937043	+	0.349214i	$0.113551\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	5.83822i		−	0.451775i	−0.974153	−	0.225888i	$-0.927472\pi$
$167$		−	5.83822i		−	0.451775i	0.974153	−	0.225888i	$-0.0725282\pi$
$168$		0			0
$169$			12.9929i			0.999457i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.62530	+	3.62530i	−0.275627	+	0.275627i	−0.831360	−	0.555734i	$-0.812437\pi$
$173$	−3.62530	+	3.62530i	−0.275627	+	0.275627i	0.555734	+	0.831360i	$0.312437\pi$
$174$		0			0
$175$	21.7659			1.64534
$176$		0			0
$177$		0			0
$178$		0			0
$179$	9.28334	−	9.28334i	0.693869	−	0.693869i	−0.269212	−	0.963081i	$-0.586763\pi$
$179$	9.28334	−	9.28334i	0.693869	−	0.693869i	0.963081	+	0.269212i	$0.0867632\pi$
$180$		0			0
$181$	10.8316	+	10.8316i	0.805104	+	0.805104i	0.983888	−	0.178785i	$-0.0572165\pi$
$181$	10.8316	+	10.8316i	0.805104	+	0.805104i	−0.178785	+	0.983888i	$0.557217\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.93783i			0.215993i
$186$		0			0
$187$	8.94725	+	8.94725i	0.654288	+	0.654288i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.63001			0.624446			0.312223	−	0.950009i	$-0.398926\pi$
$191$	8.63001			0.624446			0.312223	+	0.950009i	$0.398926\pi$
$192$		0			0
$193$	11.4514			0.824288			0.412144	−	0.911119i	$-0.364780\pi$
$193$	11.4514			0.824288			0.412144	+	0.911119i	$0.364780\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.48999	+	7.48999i	0.533640	+	0.533640i	0.921654	−	0.388014i	$-0.126839\pi$
$197$	7.48999	+	7.48999i	0.533640	+	0.533640i	−0.388014	+	0.921654i	$0.626839\pi$
$198$		0			0
$199$			3.68000i			0.260868i	0.991457	+	0.130434i	$0.0416371\pi$
$199$			3.68000i			0.260868i	−0.991457	+	0.130434i	$0.958363\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−23.5326	−	23.5326i	−1.65167	−	1.65167i
$204$		0			0
$205$	−3.10587	+	3.10587i	−0.216923	+	0.216923i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	12.6533			0.875249
$210$		0			0
$211$	−10.1188	+	10.1188i	−0.696609	+	0.696609i	−0.963677	−	0.267069i	$-0.913945\pi$
$211$	−10.1188	+	10.1188i	−0.696609	+	0.696609i	0.267069	+	0.963677i	$0.413945\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	1.07866i		−	0.0735637i
$216$		0			0
$217$		−	2.54156i		−	0.172532i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.214897	+	0.214897i	−0.0144556	+	0.0144556i
$222$		0			0
$223$	−4.86156			−0.325554			−0.162777	−	0.986663i	$-0.552045\pi$
$223$	−4.86156			−0.325554			−0.162777	+	0.986663i	$0.552045\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.6417	−	10.6417i	0.706312	−	0.706312i	−0.259445	−	0.965758i	$-0.583540\pi$
$227$	10.6417	−	10.6417i	0.706312	−	0.706312i	0.965758	+	0.259445i	$0.0835398\pi$
$228$		0			0
$229$	20.1712	+	20.1712i	1.33295	+	1.33295i	0.902720	+	0.430229i	$0.141567\pi$
$229$	20.1712	+	20.1712i	1.33295	+	1.33295i	0.430229	+	0.902720i	$0.358433\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.5702i			0.889014i	0.895775	+	0.444507i	$0.146621\pi$
$233$			13.5702i			0.889014i	−0.895775	+	0.444507i	$0.853379\pi$
$234$		0			0
$235$	0.947252	+	0.947252i	0.0617919	+	0.0617919i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−29.3629			−1.89933			−0.949665	−	0.313267i	$-0.898576\pi$
$239$	−29.3629			−1.89933			−0.949665	+	0.313267i	$0.898576\pi$
$240$		0			0
$241$	24.0063			1.54638			0.773190	−	0.634175i	$-0.218660\pi$
$241$	24.0063			1.54638			0.773190	+	0.634175i	$0.218660\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	4.61235	+	4.61235i	0.294672	+	0.294672i
$246$		0			0
$247$			0.303911i			0.0193374i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.7570	+	15.7570i	0.994571	+	0.994571i	0.999985	−	0.00541463i	$-0.00172354\pi$
$251$	15.7570	+	15.7570i	0.994571	+	0.994571i	−0.00541463	+	0.999985i	$0.501724\pi$
$252$		0			0
$253$	6.99647	−	6.99647i	0.439864	−	0.439864i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.66038			−0.540220			−0.270110	−	0.962829i	$-0.587060\pi$
$257$	−8.66038			−0.540220			−0.270110	+	0.962829i	$0.587060\pi$
$258$		0			0
$259$	19.9902	−	19.9902i	1.24213	−	1.24213i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			13.3208i			0.821394i	0.911772	+	0.410697i	$0.134715\pi$
$263$			13.3208i			0.821394i	−0.911772	+	0.410697i	$0.865285\pi$
$264$		0			0
$265$			0.330566i			0.0203065i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−11.6714	+	11.6714i	−0.711616	+	0.711616i	−0.966873	−	0.255257i	$-0.917840\pi$
$269$	−11.6714	+	11.6714i	−0.711616	+	0.711616i	0.255257	+	0.966873i	$0.417840\pi$
$270$		0			0
$271$	−21.9769			−1.33500			−0.667499	−	0.744610i	$-0.732635\pi$
$271$	−21.9769			−1.33500			−0.667499	+	0.744610i	$0.732635\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.8132	+	11.8132i	−0.712365	+	0.712365i
$276$		0			0
$277$	10.9504	+	10.9504i	0.657945	+	0.657945i	0.954893	−	0.296949i	$-0.0959690\pi$
$277$	10.9504	+	10.9504i	0.657945	+	0.657945i	−0.296949	+	0.954893i	$0.595969\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	22.8910i		−	1.36556i	−0.730624	−	0.682780i	$-0.760771\pi$
$281$		−	22.8910i		−	1.36556i	0.730624	−	0.682780i	$-0.239229\pi$
$282$		0			0
$283$	−4.48528	−	4.48528i	−0.266622	−	0.266622i	0.561115	−	0.827738i	$-0.310372\pi$
$283$	−4.48528	−	4.48528i	−0.266622	−	0.266622i	−0.827738	+	0.561115i	$0.810372\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	42.2672			2.49496
$288$		0			0
$289$	−3.91688			−0.230405
$290$		0			0
$291$		0			0
$292$		0			0
$293$	21.6221	+	21.6221i	1.26318	+	1.26318i	0.949543	+	0.313636i	$0.101547\pi$
$293$	21.6221	+	21.6221i	1.26318	+	1.26318i	0.313636	+	0.949543i	$0.398453\pi$
$294$		0			0
$295$		−	2.67923i		−	0.155991i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.168043	+	0.168043i	0.00971818	+	0.00971818i
$300$		0			0
$301$	−7.33962	+	7.33962i	−0.423048	+	0.423048i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.82805			0.104674
$306$		0			0
$307$	12.1118	−	12.1118i	0.691255	−	0.691255i	−0.271253	−	0.962508i	$-0.587438\pi$
$307$	12.1118	−	12.1118i	0.691255	−	0.691255i	0.962508	+	0.271253i	$0.0874380\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	26.8651i		−	1.52338i	−0.647943	−	0.761689i	$-0.724370\pi$
$311$		−	26.8651i		−	1.52338i	0.647943	−	0.761689i	$-0.275630\pi$
$312$		0			0
$313$		−	19.6890i		−	1.11289i	−0.830885	−	0.556445i	$-0.812165\pi$
$313$		−	19.6890i		−	1.11289i	0.830885	−	0.556445i	$-0.187835\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	21.3447	−	21.3447i	1.19884	−	1.19884i	0.224323	−	0.974515i	$-0.427983\pi$
$317$	21.3447	−	21.3447i	1.19884	−	1.19884i	0.974515	−	0.224323i	$-0.0720171\pi$
$318$		0			0
$319$	25.5443			1.43021
$320$		0			0
$321$		0			0
$322$		0			0
$323$	9.25116	−	9.25116i	0.514748	−	0.514748i
$324$		0			0
$325$	−0.283734	−	0.283734i	−0.0157387	−	0.0157387i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	12.8910i		−	0.710702i
$330$		0			0
$331$	−14.6926	−	14.6926i	−0.807576	−	0.807576i	0.176690	−	0.984266i	$-0.443461\pi$
$331$	−14.6926	−	14.6926i	−0.807576	−	0.807576i	−0.984266	+	0.176690i	$0.943461\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2.52898			0.138173
$336$		0			0
$337$	−23.0098			−1.25342			−0.626712	−	0.779251i	$-0.715600\pi$
$337$	−23.0098			−1.25342			−0.626712	+	0.779251i	$0.715600\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.37941	+	1.37941i	0.0746993	+	0.0746993i
$342$		0			0
$343$		−	30.8651i		−	1.66656i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.9026	−	10.9026i	−0.585284	−	0.585284i	0.351067	−	0.936350i	$-0.385819\pi$
$347$	−10.9026	−	10.9026i	−0.585284	−	0.585284i	−0.936350	+	0.351067i	$0.885819\pi$
$348$		0			0
$349$	−20.0563	+	20.0563i	−1.07359	+	1.07359i	−0.0765186	+	0.997068i	$0.524380\pi$
$349$	−20.0563	+	20.0563i	−1.07359	+	1.07359i	−0.997068	+	0.0765186i	$0.975620\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.2117			0.649965			0.324983	−	0.945720i	$-0.394642\pi$
$353$	12.2117			0.649965			0.324983	+	0.945720i	$0.394642\pi$
$354$		0			0
$355$	3.05275	−	3.05275i	0.162023	−	0.162023i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			33.4780i			1.76690i	0.468522	+	0.883452i	$0.344786\pi$
$359$			33.4780i			1.76690i	−0.468522	+	0.883452i	$0.655214\pi$
$360$		0			0
$361$			5.91688i			0.311415i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.181370	−	0.181370i	0.00949337	−	0.00949337i
$366$		0			0
$367$	−0.702379			−0.0366639			−0.0183319	−	0.999832i	$-0.505836\pi$
$367$	−0.702379			−0.0366639			−0.0183319	+	0.999832i	$0.505836\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.24930	−	2.24930i	0.116778	−	0.116778i
$372$		0			0
$373$	−18.9598	−	18.9598i	−0.981702	−	0.981702i	0.0181339	−	0.999836i	$-0.494227\pi$
$373$	−18.9598	−	18.9598i	−0.981702	−	0.981702i	−0.999836	+	0.0181339i	$0.994227\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			0.613530i			0.0315984i
$378$		0			0
$379$	1.77844	+	1.77844i	0.0913523	+	0.0913523i	0.751306	−	0.659954i	$-0.229424\pi$
$379$	1.77844	+	1.77844i	0.0913523	+	0.0913523i	−0.659954	+	0.751306i	$0.729424\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−25.4880			−1.30238			−0.651188	−	0.758916i	$-0.725729\pi$
$383$	−25.4880			−1.30238			−0.651188	+	0.758916i	$0.725729\pi$
$384$		0			0
$385$	−7.55136			−0.384853
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.7049	−	11.7049i	−0.593462	−	0.593462i	0.345103	−	0.938565i	$-0.387844\pi$
$389$	−11.7049	−	11.7049i	−0.593462	−	0.593462i	−0.938565	+	0.345103i	$0.887844\pi$
$390$		0			0
$391$		−	10.2306i		−	0.517383i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3.68184	+	3.68184i	0.185253	+	0.185253i
$396$		0			0
$397$	9.04646	−	9.04646i	0.454029	−	0.454029i	−0.442661	−	0.896689i	$-0.645965\pi$
$397$	9.04646	−	9.04646i	0.454029	−	0.454029i	0.896689	+	0.442661i	$0.145965\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0853			0.903137			0.451568	−	0.892237i	$-0.350865\pi$
$401$	18.0853			0.903137			0.451568	+	0.892237i	$0.350865\pi$
$402$		0			0
$403$	−0.0331311	+	0.0331311i	−0.00165038	+	0.00165038i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			21.6990i			1.07558i
$408$		0			0
$409$			25.2271i			1.24740i	0.781665	+	0.623699i	$0.214371\pi$
$409$			25.2271i			1.24740i	−0.781665	+	0.623699i	$0.785629\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−18.2306	+	18.2306i	−0.897069	+	0.897069i
$414$		0			0
$415$	7.12787			0.349894
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.25283	−	7.25283i	0.354324	−	0.354324i	−0.507392	−	0.861716i	$-0.669390\pi$
$419$	7.25283	−	7.25283i	0.354324	−	0.354324i	0.861716	+	0.507392i	$0.169390\pi$
$420$		0			0
$421$	−2.39550	−	2.39550i	−0.116749	−	0.116749i	0.646318	−	0.763068i	$-0.276308\pi$
$421$	−2.39550	−	2.39550i	−0.116749	−	0.116749i	−0.763068	+	0.646318i	$0.776308\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			17.2739i			0.837908i
$426$		0			0
$427$	−12.4388	−	12.4388i	−0.601957	−	0.601957i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.42454			0.213123			0.106561	−	0.994306i	$-0.466016\pi$
$431$	4.42454			0.213123			0.106561	+	0.994306i	$0.466016\pi$
$432$		0			0
$433$	7.31371			0.351474			0.175737	−	0.984437i	$-0.443769\pi$
$433$	7.31371			0.351474			0.175737	+	0.984437i	$0.443769\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−7.23412	−	7.23412i	−0.346055	−	0.346055i
$438$		0			0
$439$			29.6533i			1.41527i	0.706576	+	0.707637i	$0.250239\pi$
$439$			29.6533i			1.41527i	−0.706576	+	0.707637i	$0.749761\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.3056	+	10.3056i	0.489633	+	0.489633i	0.908190	−	0.418557i	$-0.137464\pi$
$443$	10.3056	+	10.3056i	0.489633	+	0.489633i	−0.418557	+	0.908190i	$0.637464\pi$
$444$		0			0
$445$	4.90746	−	4.90746i	0.232636	−	0.232636i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.48844			0.306208			0.153104	−	0.988210i	$-0.451073\pi$
$449$	6.48844			0.306208			0.153104	+	0.988210i	$0.451073\pi$
$450$		0			0
$451$	−22.9402	+	22.9402i	−1.08021	+	1.08021i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	0.181370i		−	0.00850278i
$456$		0			0
$457$			9.00353i			0.421167i	0.977576	+	0.210584i	$0.0675364\pi$
$457$			9.00353i			0.421167i	−0.977576	+	0.210584i	$0.932464\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.6218	−	14.6218i	0.681004	−	0.681004i	−0.279223	−	0.960226i	$-0.590077\pi$
$461$	14.6218	−	14.6218i	0.681004	−	0.681004i	0.960226	+	0.279223i	$0.0900767\pi$
$462$		0			0
$463$	−18.6435			−0.866437			−0.433219	−	0.901289i	$-0.642622\pi$
$463$	−18.6435			−0.866437			−0.433219	+	0.901289i	$0.642622\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−23.5138	+	23.5138i	−1.08809	+	1.08809i	−0.0923633	+	0.995725i	$0.529442\pi$
$467$	−23.5138	+	23.5138i	−1.08809	+	1.08809i	−0.995725	+	0.0923633i	$0.970558\pi$
$468$		0			0
$469$	−17.2082	−	17.2082i	−0.794601	−	0.794601i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	7.96703i		−	0.366325i
$474$		0			0
$475$	12.2145	+	12.2145i	0.560440	+	0.560440i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.08864			0.0497412			0.0248706	−	0.999691i	$-0.492083\pi$
$479$	1.08864			0.0497412			0.0248706	+	0.999691i	$0.492083\pi$
$480$		0			0
$481$	−0.521173			−0.0237634
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.44586	−	1.44586i	−0.0656531	−	0.0656531i
$486$		0			0
$487$			35.3298i			1.60095i	0.599369	+	0.800473i	$0.295418\pi$
$487$			35.3298i			1.60095i	−0.599369	+	0.800473i	$0.704582\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.8910	−	12.8910i	−0.581761	−	0.581761i	0.353626	−	0.935387i	$-0.384949\pi$
$491$	−12.8910	−	12.8910i	−0.581761	−	0.581761i	−0.935387	+	0.353626i	$0.884949\pi$
$492$		0			0
$493$	18.6761	−	18.6761i	0.841128	−	0.841128i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−41.5443			−1.86352
$498$		0			0
$499$	−14.3798	+	14.3798i	−0.643728	+	0.643728i	−0.951470	−	0.307742i	$-0.900427\pi$
$499$	−14.3798	+	14.3798i	−0.643728	+	0.643728i	0.307742	+	0.951470i	$0.400427\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	30.2969i		−	1.35087i	−0.737420	−	0.675435i	$-0.763956\pi$
$503$		−	30.2969i		−	1.35087i	0.737420	−	0.675435i	$-0.236044\pi$
$504$		0			0
$505$			0.303911i			0.0135239i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.5825	−	10.5825i	0.469063	−	0.469063i	−0.432548	−	0.901611i	$-0.642385\pi$
$509$	10.5825	−	10.5825i	0.469063	−	0.469063i	0.901611	+	0.432548i	$0.142385\pi$
$510$		0			0
$511$	−2.46824			−0.109188
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.447718	+	0.447718i	−0.0197288	+	0.0197288i
$516$		0			0
$517$	6.99647	+	6.99647i	0.307704	+	0.307704i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			24.9049i			1.09110i	0.838078	+	0.545551i	$0.183680\pi$
$521$			24.9049i			1.09110i	−0.838078	+	0.545551i	$0.816320\pi$
$522$		0			0
$523$	12.9008	+	12.9008i	0.564112	+	0.564112i	0.930473	−	0.366361i	$-0.119396\pi$
$523$	12.9008	+	12.9008i	0.564112	+	0.564112i	−0.366361	+	0.930473i	$0.619396\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.01704			0.0878638
$528$		0			0
$529$	15.0000			0.652174
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.550984	−	0.550984i	−0.0238657	−	0.0238657i
$534$		0			0
$535$		−	4.06074i		−	0.175561i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	34.0671	+	34.0671i	1.46738	+	1.46738i
$540$		0			0
$541$	−18.2767	+	18.2767i	−0.785776	+	0.785776i	−0.980799	−	0.195023i	$-0.937522\pi$
$541$	−18.2767	+	18.2767i	−0.785776	+	0.785776i	0.195023	+	0.980799i	$0.437522\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.82880			0.164008
$546$		0			0
$547$	−13.7355	+	13.7355i	−0.587287	+	0.587287i	−0.936896	−	0.349609i	$-0.886315\pi$
$547$	−13.7355	+	13.7355i	−0.587287	+	0.587287i	0.349609	+	0.936896i	$0.386315\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	26.4120i		−	1.12519i
$552$		0			0
$553$		−	50.1055i		−	2.13070i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−27.5525	+	27.5525i	−1.16744	+	1.16744i	−0.184631	+	0.982808i	$0.559109\pi$
$557$	−27.5525	+	27.5525i	−1.16744	+	1.16744i	−0.982808	+	0.184631i	$0.940891\pi$
$558$		0			0
$559$	0.191354			0.00809342
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−19.8928	+	19.8928i	−0.838383	+	0.838383i	−0.988646	−	0.150263i	$-0.951988\pi$
$563$	−19.8928	+	19.8928i	−0.838383	+	0.838383i	0.150263	+	0.988646i	$0.451988\pi$
$564$		0			0
$565$	−3.19879	−	3.19879i	−0.134574	−	0.134574i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			13.4849i			0.565317i	0.959221	+	0.282658i	$0.0912163\pi$
$569$			13.4849i			0.565317i	−0.959221	+	0.282658i	$0.908784\pi$
$570$		0			0
$571$	14.8284	+	14.8284i	0.620550	+	0.620550i	0.945672	−	0.325122i	$-0.105405\pi$
$571$	14.8284	+	14.8284i	0.620550	+	0.620550i	−0.325122	+	0.945672i	$0.605405\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	13.5077			0.563308
$576$		0			0
$577$	−11.6176			−0.483648			−0.241824	−	0.970320i	$-0.577746\pi$
$577$	−11.6176			−0.483648			−0.241824	+	0.970320i	$0.577746\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−48.5010	−	48.5010i	−2.01216	−	2.01216i
$582$		0			0
$583$			2.44158i			0.101120i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.0268	−	17.0268i	−0.702773	−	0.702773i	0.262232	−	0.965005i	$-0.415541\pi$
$587$	−17.0268	−	17.0268i	−0.702773	−	0.702773i	−0.965005	+	0.262232i	$0.915541\pi$
$588$		0			0
$589$	1.42627	−	1.42627i	0.0587682	−	0.0587682i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−41.5372			−1.70573			−0.852865	−	0.522132i	$-0.825137\pi$
$593$	−41.5372			−1.70573			−0.852865	+	0.522132i	$0.825137\pi$
$594$		0			0
$595$	−5.52099	+	5.52099i	−0.226338	+	0.226338i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	6.43160i		−	0.262788i	−0.991330	−	0.131394i	$-0.958055\pi$
$599$		−	6.43160i		−	0.262788i	0.991330	−	0.131394i	$-0.0419453\pi$
$600$		0			0
$601$		−	3.45844i		−	0.141073i	−0.997509	−	0.0705364i	$-0.977529\pi$
$601$		−	3.45844i		−	0.141073i	0.997509	−	0.0705364i	$-0.0224711\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.414494	−	0.414494i	0.0168516	−	0.0168516i
$606$		0			0
$607$	−30.1019			−1.22180			−0.610900	−	0.791708i	$-0.709192\pi$
$607$	−30.1019			−1.22180			−0.610900	+	0.791708i	$0.709192\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.168043	+	0.168043i	−0.00679829	+	0.00679829i
$612$		0			0
$613$	−2.50490	−	2.50490i	−0.101172	−	0.101172i	0.654709	−	0.755881i	$-0.272791\pi$
$613$	−2.50490	−	2.50490i	−0.101172	−	0.101172i	−0.755881	+	0.654709i	$0.772791\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	22.9098i		−	0.922315i	−0.887318	−	0.461157i	$-0.847434\pi$
$617$		−	22.9098i		−	0.922315i	0.887318	−	0.461157i	$-0.152566\pi$
$618$		0			0
$619$	28.6104	+	28.6104i	1.14995	+	1.14995i	0.986562	+	0.163386i	$0.0522415\pi$
$619$	28.6104	+	28.6104i	1.14995	+	1.14995i	0.163386	+	0.986562i	$0.447758\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−66.7847			−2.67567
$624$		0			0
$625$	−21.6855			−0.867420
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.8647	+	15.8647i	0.632567	+	0.632567i
$630$		0			0
$631$			11.1851i			0.445270i	0.974902	+	0.222635i	$0.0714659\pi$
$631$			11.1851i			0.445270i	−0.974902	+	0.222635i	$0.928534\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.70774	−	1.70774i	−0.0677698	−	0.0677698i
$636$		0			0
$637$	−0.818234	+	0.818234i	−0.0324196	+	0.0324196i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.69312			0.264362			0.132181	−	0.991226i	$-0.457802\pi$
$641$	6.69312			0.264362			0.132181	+	0.991226i	$0.457802\pi$
$642$		0			0
$643$	17.9410	−	17.9410i	0.707522	−	0.707522i	−0.258491	−	0.966014i	$-0.583225\pi$
$643$	17.9410	−	17.9410i	0.707522	−	0.707522i	0.966014	+	0.258491i	$0.0832253\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.72999i		−	0.264583i	−0.991211	−	0.132292i	$-0.957766\pi$
$647$		−	6.72999i		−	0.264583i	0.991211	−	0.132292i	$-0.0422335\pi$
$648$		0			0
$649$		−	19.7890i		−	0.776786i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26.1731	−	26.1731i	1.02423	−	1.02423i	0.0245347	−	0.999699i	$-0.492190\pi$
$653$	26.1731	−	26.1731i	1.02423	−	1.02423i	0.999699	−	0.0245347i	$-0.00781042\pi$
$654$		0			0
$655$	−1.41921			−0.0554529
$656$		0			0
$657$		0			0
$658$		0			0
$659$	13.9741	−	13.9741i	0.544353	−	0.544353i	−0.380449	−	0.924802i	$-0.624230\pi$
$659$	13.9741	−	13.9741i	0.544353	−	0.544353i	0.924802	+	0.380449i	$0.124230\pi$
$660$		0			0
$661$	−11.9241	−	11.9241i	−0.463794	−	0.463794i	0.436103	−	0.899897i	$-0.356358\pi$
$661$	−11.9241	−	11.9241i	−0.463794	−	0.463794i	−0.899897	+	0.436103i	$0.856358\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			7.80785i			0.302776i
$666$		0			0
$667$	−14.6041	−	14.6041i	−0.565473	−	0.565473i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.5021			0.521244
$672$		0			0
$673$	−37.3066			−1.43807			−0.719033	−	0.694976i	$-0.755415\pi$
$673$	−37.3066			−1.43807			−0.719033	+	0.694976i	$0.755415\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.447461	+	0.447461i	0.0171973	+	0.0171973i	0.715653	−	0.698456i	$-0.246129\pi$
$677$	0.447461	+	0.447461i	0.0171973	+	0.0171973i	−0.698456	+	0.715653i	$0.746129\pi$
$678$		0			0
$679$			19.6764i			0.755113i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.27521	−	4.27521i	−0.163586	−	0.163586i	0.620567	−	0.784153i	$-0.286902\pi$
$683$	−4.27521	−	4.27521i	−0.163586	−	0.163586i	−0.784153	+	0.620567i	$0.786902\pi$
$684$		0			0
$685$	−1.13178	+	1.13178i	−0.0432430	+	0.0432430i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.0586426			−0.00223410
$690$		0			0
$691$	−20.0786	+	20.0786i	−0.763827	+	0.763827i	−0.977012	−	0.213185i	$-0.931616\pi$
$691$	−20.0786	+	20.0786i	−0.763827	+	0.763827i	0.213185	+	0.977012i	$0.431616\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.93926i			0.149425i
$696$		0			0
$697$			33.5443i			1.27058i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.4467	+	10.4467i	−0.394565	+	0.394565i	−0.876311	−	0.481746i	$-0.840003\pi$
$701$	−10.4467	+	10.4467i	−0.394565	+	0.394565i	0.481746	+	0.876311i	$0.340003\pi$
$702$		0			0
$703$	22.4361			0.846192
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.06793	−	2.06793i	0.0777727	−	0.0777727i
$708$		0			0
$709$	−16.0916	−	16.0916i	−0.604332	−	0.604332i	0.337127	−	0.941459i	$-0.390545\pi$
$709$	−16.0916	−	16.0916i	−0.604332	−	0.604332i	−0.941459	+	0.337127i	$0.890545\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.57726i		−	0.0590690i
$714$		0			0
$715$	0.0984373	+	0.0984373i	0.00368135	+	0.00368135i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	30.9957			1.15594			0.577972	−	0.816057i	$-0.303844\pi$
$719$	30.9957			1.15594			0.577972	+	0.816057i	$0.303844\pi$
$720$		0			0
$721$	6.09292			0.226912
$722$		0			0
$723$		0			0
$724$		0			0
$725$	24.6584	+	24.6584i	0.915790	+	0.915790i
$726$		0			0
$727$		−	41.1117i		−	1.52475i	−0.647135	−	0.762375i	$-0.724033\pi$
$727$		−	41.1117i		−	1.52475i	0.647135	−	0.762375i	$-0.275967\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.82490	−	5.82490i	−0.215442	−	0.215442i
$732$		0			0
$733$	−0.146061	+	0.146061i	−0.00539490	+	0.00539490i	−0.709799	−	0.704404i	$-0.751214\pi$
$733$	−0.146061	+	0.146061i	−0.00539490	+	0.00539490i	0.704404	+	0.709799i	$0.251214\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.6792			0.688058
$738$		0			0
$739$	1.50766	−	1.50766i	0.0554601	−	0.0554601i	−0.678833	−	0.734293i	$-0.737514\pi$
$739$	1.50766	−	1.50766i	0.0554601	−	0.0554601i	0.734293	+	0.678833i	$0.237514\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			40.5175i			1.48644i	0.669046	+	0.743221i	$0.266703\pi$
$743$			40.5175i			1.48644i	−0.669046	+	0.743221i	$0.733297\pi$
$744$		0			0
$745$			6.69256i			0.245196i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−27.6309	+	27.6309i	−1.00961	+	1.00961i
$750$		0			0
$751$	−12.5843			−0.459208			−0.229604	−	0.973284i	$-0.573743\pi$
$751$	−12.5843			−0.459208			−0.229604	+	0.973284i	$0.573743\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−3.34129	+	3.34129i	−0.121602	+	0.121602i
$756$		0			0
$757$	7.49900	+	7.49900i	0.272556	+	0.272556i	0.830128	−	0.557572i	$-0.188267\pi$
$757$	7.49900	+	7.49900i	0.272556	+	0.272556i	−0.557572	+	0.830128i	$0.688267\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	42.8182i		−	1.55216i	−0.630635	−	0.776079i	$-0.717206\pi$
$761$		−	42.8182i		−	1.55216i	0.630635	−	0.776079i	$-0.282794\pi$
$762$		0			0
$763$	−26.0527	−	26.0527i	−0.943172	−	0.943172i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.475298			0.0171620
$768$		0			0
$769$	12.7455			0.459614			0.229807	−	0.973236i	$-0.426190\pi$
$769$	12.7455			0.459614			0.229807	+	0.973236i	$0.426190\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22.8765	−	22.8765i	−0.822809	−	0.822809i	0.163701	−	0.986510i	$-0.447657\pi$
$773$	−22.8765	−	22.8765i	−0.822809	−	0.822809i	−0.986510	+	0.163701i	$0.947657\pi$
$774$		0			0
$775$			2.66314i			0.0956630i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	23.7194	+	23.7194i	0.849836	+	0.849836i
$780$		0			0
$781$	22.5478	−	22.5478i	0.806825	−	0.806825i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.8253			−0.386370
$786$		0			0
$787$	−5.20470	+	5.20470i	−0.185528	+	0.185528i	−0.793759	−	0.608232i	$-0.791879\pi$
$787$	−5.20470	+	5.20470i	−0.185528	+	0.185528i	0.608232	+	0.793759i	$0.291879\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			43.5317i			1.54781i
$792$		0			0
$793$			0.324298i			0.0115162i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.0149	−	17.0149i	0.602698	−	0.602698i	−0.338330	−	0.941028i	$-0.609862\pi$
$797$	17.0149	−	17.0149i	0.602698	−	0.602698i	0.941028	+	0.338330i	$0.109862\pi$
$798$		0			0
$799$	10.2306			0.361932
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.33962	−	1.33962i	0.0472740	−	0.0472740i
$804$		0			0
$805$	4.31724	+	4.31724i	0.152163	+	0.152163i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	7.83586i		−	0.275494i	−0.990467	−	0.137747i	$-0.956014\pi$
$809$		−	7.83586i		−	0.275494i	0.990467	−	0.137747i	$-0.0439861\pi$
$810$		0			0
$811$	−32.3396	−	32.3396i	−1.13560	−	1.13560i	−0.989230	−	0.146366i	$-0.953242\pi$
$811$	−32.3396	−	32.3396i	−1.13560	−	1.13560i	−0.146366	−	0.989230i	$-0.546758\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.02684			−0.176083
$816$		0			0
$817$	−8.23765			−0.288199
$818$		0			0
$819$		0			0
$820$		0			0
$821$	19.3541	+	19.3541i	0.675464	+	0.675464i	0.958970	−	0.283507i	$-0.0914978\pi$
$821$	19.3541	+	19.3541i	0.675464	+	0.675464i	−0.283507	+	0.958970i	$0.591498\pi$
$822$		0			0
$823$			28.8560i			1.00586i	0.864328	+	0.502929i	$0.167744\pi$
$823$			28.8560i			1.00586i	−0.864328	+	0.502929i	$0.832256\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10.1984	−	10.1984i	−0.354634	−	0.354634i	0.507197	−	0.861830i	$-0.330682\pi$
$827$	−10.1984	−	10.1984i	−0.354634	−	0.354634i	−0.861830	+	0.507197i	$0.830682\pi$
$828$		0			0
$829$	−15.3794	+	15.3794i	−0.534148	+	0.534148i	−0.921804	−	0.387656i	$-0.873285\pi$
$829$	−15.3794	+	15.3794i	−0.534148	+	0.534148i	0.387656	+	0.921804i	$0.373285\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	49.8147			1.72598
$834$		0			0
$835$	−1.95524	+	1.95524i	−0.0676640	+	0.0676640i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			44.4557i			1.53478i	0.641181	+	0.767390i	$0.278445\pi$
$839$			44.4557i			1.53478i	−0.641181	+	0.767390i	$0.721555\pi$
$840$		0			0
$841$		−	24.3200i		−	0.838620i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.35139	−	4.35139i	0.149692	−	0.149692i
$846$		0			0
$847$	−5.64077			−0.193819
$848$		0			0
$849$		0			0
$850$		0			0
$851$	12.4057	−	12.4057i	0.425262	−	0.425262i
$852$		0			0
$853$	11.7131	+	11.7131i	0.401049	+	0.401049i	0.878603	−	0.477553i	$-0.158476\pi$
$853$	11.7131	+	11.7131i	0.401049	+	0.401049i	−0.477553	+	0.878603i	$0.658476\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			19.0888i			0.652062i	0.945359	+	0.326031i	$0.105711\pi$
$857$			19.0888i			0.652062i	−0.945359	+	0.326031i	$0.894289\pi$
$858$		0			0
$859$	38.1323	+	38.1323i	1.30106	+	1.30106i	0.927679	+	0.373379i	$0.121801\pi$
$859$	38.1323	+	38.1323i	1.30106	+	1.30106i	0.373379	+	0.927679i	$0.378199\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.64533			−0.124089			−0.0620443	−	0.998073i	$-0.519762\pi$
$863$	−3.64533			−0.124089			−0.0620443	+	0.998073i	$0.519762\pi$
$864$		0			0
$865$	2.42826			0.0825632
$866$		0			0
$867$		0			0
$868$		0			0
$869$	27.1943	+	27.1943i	0.922504	+	0.922504i
$870$		0			0
$871$			0.448643i			0.0152017i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−14.9213	−	14.9213i	−0.504433	−	0.504433i
$876$		0			0
$877$	−40.0563	+	40.0563i	−1.35260	+	1.35260i	−0.469866	+	0.882738i	$0.655698\pi$
$877$	−40.0563	+	40.0563i	−1.35260	+	1.35260i	−0.882738	+	0.469866i	$0.844302\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20.0118			0.674214			0.337107	−	0.941466i	$-0.390552\pi$
$881$	20.0118			0.674214			0.337107	+	0.941466i	$0.390552\pi$
$882$		0			0
$883$	10.6273	−	10.6273i	0.357636	−	0.357636i	−0.505305	−	0.862941i	$-0.668620\pi$
$883$	10.6273	−	10.6273i	0.357636	−	0.357636i	0.862941	+	0.505305i	$0.168620\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	26.1180i		−	0.876958i	−0.898742	−	0.438479i	$-0.855517\pi$
$887$		−	26.1180i		−	0.876958i	0.898742	−	0.438479i	$-0.144483\pi$
$888$		0			0
$889$			23.2404i			0.779458i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.23412	−	7.23412i	0.242081	−	0.242081i
$894$		0			0
$895$	−6.21805			−0.207847
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.87932	−	2.87932i	0.0960306	−	0.0960306i
$900$		0			0
$901$	1.78510	+	1.78510i	0.0594704	+	0.0594704i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	7.25507i		−	0.241167i
$906$		0			0
$907$	−36.2378	−	36.2378i	−1.20326	−	1.20326i	−0.973170	−	0.230087i	$-0.926099\pi$
$907$	−36.2378	−	36.2378i	−1.20326	−	1.20326i	−0.230087	−	0.973170i	$-0.573901\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	21.0535			0.697533			0.348767	−	0.937210i	$-0.386601\pi$
$911$	21.0535			0.697533			0.348767	+	0.937210i	$0.386601\pi$
$912$		0			0
$913$	52.6470			1.74236
$914$		0			0
$915$		0			0
$916$		0			0
$917$	9.65685	+	9.65685i	0.318897	+	0.318897i
$918$		0			0
$919$		−	17.8839i		−	0.589937i	−0.955507	−	0.294968i	$-0.904691\pi$
$919$		−	17.8839i		−	0.589937i	0.955507	−	0.294968i	$-0.0953091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.541560	+	0.541560i	0.0178257	+	0.0178257i
$924$		0			0
$925$	−20.9465	+	20.9465i	−0.688716	+	0.688716i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.2774			−0.337192			−0.168596	−	0.985685i	$-0.553923\pi$
$929$	−10.2774			−0.337192			−0.168596	+	0.985685i	$0.553923\pi$
$930$		0			0
$931$	35.2243	−	35.2243i	1.15443	−	1.15443i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5.99294i		−	0.195990i
$936$		0			0
$937$			13.5780i			0.443574i	0.975095	+	0.221787i	$0.0711890\pi$
$937$			13.5780i			0.443574i	−0.975095	+	0.221787i	$0.928811\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.95902	+	3.95902i	−0.129060	+	0.129060i	−0.768686	−	0.639626i	$-0.779089\pi$
$941$	−3.95902	+	3.95902i	−0.129060	+	0.129060i	0.639626	+	0.768686i	$0.279089\pi$
$942$		0			0
$943$	26.2306			0.854186
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−33.1708	+	33.1708i	−1.07791	+	1.07791i	−0.0812084	+	0.996697i	$0.525878\pi$
$947$	−33.1708	+	33.1708i	−1.07791	+	1.07791i	−0.996697	+	0.0812084i	$0.974122\pi$
$948$		0			0
$949$	0.0321752	+	0.0321752i	0.00104445	+	0.00104445i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	5.59115i		−	0.181115i	−0.995891	−	0.0905576i	$-0.971135\pi$
$953$		−	5.59115i		−	0.181115i	0.995891	−	0.0905576i	$-0.0288649\pi$
$954$		0			0
$955$	−2.89023	−	2.89023i	−0.0935255	−	0.0935255i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	15.4022			0.497362
$960$		0			0
$961$	−30.6890			−0.989969
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−3.83511	−	3.83511i	−0.123457	−	0.123457i
$966$		0			0
$967$		−	30.7561i		−	0.989048i	−0.869164	−	0.494524i	$-0.835342\pi$
$967$		−	30.7561i		−	0.989048i	0.869164	−	0.494524i	$-0.164658\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	8.03756	+	8.03756i	0.257938	+	0.257938i	0.824215	−	0.566277i	$-0.191617\pi$
$971$	8.03756	+	8.03756i	0.257938	+	0.257938i	−0.566277	+	0.824215i	$0.691617\pi$
$972$		0			0
$973$	26.8043	−	26.8043i	0.859307	−	0.859307i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22.8323			−0.730471			−0.365235	−	0.930915i	$-0.619012\pi$
$977$	−22.8323			−0.730471			−0.365235	+	0.930915i	$0.619012\pi$
$978$		0			0
$979$	36.2468	−	36.2468i	1.15845	−	1.15845i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	46.3557i		−	1.47852i	−0.673422	−	0.739258i	$-0.735176\pi$
$983$		−	46.3557i		−	1.47852i	0.673422	−	0.739258i	$-0.264824\pi$
$984$		0			0
$985$		−	5.01686i		−	0.159850i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.55489	+	4.55489i	−0.144837	+	0.144837i
$990$		0			0
$991$	3.43683			0.109175			0.0545873	−	0.998509i	$-0.482616\pi$
$991$	3.43683			0.109175			0.0545873	+	0.998509i	$0.482616\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.23245	−	1.23245i	0.0390712	−	0.0390712i
$996$		0			0
$997$	21.9430	+	21.9430i	0.694940	+	0.694940i	0.963315	−	0.268374i	$-0.0864864\pi$
$997$	21.9430	+	21.9430i	0.694940	+	0.694940i	−0.268374	+	0.963315i	$0.586486\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.k.c.865.2		8
3.2	odd	2		384.2.j.b.97.4		8
4.3	odd	2		1152.2.k.f.865.2		8
8.3	odd	2		576.2.k.b.433.3		8
8.5	even	2		144.2.k.b.37.1		8
12.11	even	2		384.2.j.a.97.2		8
16.3	odd	4		1152.2.k.f.289.2		8
16.5	even	4		144.2.k.b.109.1		8
16.11	odd	4		576.2.k.b.145.3		8
16.13	even	4	inner	1152.2.k.c.289.2		8
24.5	odd	2		48.2.j.a.37.4	yes	8
24.11	even	2		192.2.j.a.49.3		8
32.3	odd	8		9216.2.a.x.1.3		4
32.13	even	8		9216.2.a.bo.1.2		4
32.19	odd	8		9216.2.a.bn.1.2		4
32.29	even	8		9216.2.a.y.1.3		4
48.5	odd	4		48.2.j.a.13.4	✓	8
48.11	even	4		192.2.j.a.145.3		8
48.29	odd	4		384.2.j.b.289.4		8
48.35	even	4		384.2.j.a.289.2		8
96.5	odd	8		3072.2.d.f.1537.7		8
96.11	even	8		3072.2.d.i.1537.6		8
96.29	odd	8		3072.2.a.t.1.2		4
96.35	even	8		3072.2.a.n.1.2		4
96.53	odd	8		3072.2.d.f.1537.2		8
96.59	even	8		3072.2.d.i.1537.3		8
96.77	odd	8		3072.2.a.i.1.3		4
96.83	even	8		3072.2.a.o.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.4	✓	8	48.5	odd	4
48.2.j.a.37.4	yes	8	24.5	odd	2
144.2.k.b.37.1		8	8.5	even	2
144.2.k.b.109.1		8	16.5	even	4
192.2.j.a.49.3		8	24.11	even	2
192.2.j.a.145.3		8	48.11	even	4
384.2.j.a.97.2		8	12.11	even	2
384.2.j.a.289.2		8	48.35	even	4
384.2.j.b.97.4		8	3.2	odd	2
384.2.j.b.289.4		8	48.29	odd	4
576.2.k.b.145.3		8	16.11	odd	4
576.2.k.b.433.3		8	8.3	odd	2
1152.2.k.c.289.2		8	16.13	even	4	inner
1152.2.k.c.865.2		8	1.1	even	1	trivial
1152.2.k.f.289.2		8	16.3	odd	4
1152.2.k.f.865.2		8	4.3	odd	2
3072.2.a.i.1.3		4	96.77	odd	8
3072.2.a.n.1.2		4	96.35	even	8
3072.2.a.o.1.3		4	96.83	even	8
3072.2.a.t.1.2		4	96.29	odd	8
3072.2.d.f.1537.2		8	96.53	odd	8
3072.2.d.f.1537.7		8	96.5	odd	8
3072.2.d.i.1537.3		8	96.59	even	8
3072.2.d.i.1537.6		8	96.11	even	8
9216.2.a.x.1.3		4	32.3	odd	8
9216.2.a.y.1.3		4	32.29	even	8
9216.2.a.bn.1.2		4	32.19	odd	8
9216.2.a.bo.1.2		4	32.13	even	8

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.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.334904 - 0.334904i) q^{5} +4.55765i q^{7} +O(q^{10})\)	\(q+(-0.334904 - 0.334904i) q^{5} +4.55765i q^{7} +(-2.47363 - 2.47363i) q^{11} +(0.0594122 - 0.0594122i) q^{13} -3.61706 q^{17} +(-2.55765 + 2.55765i) q^{19} +2.82843i q^{23} -4.77568i q^{25} +(-5.16333 + 5.16333i) q^{29} -0.557647 q^{31} +(1.52637 - 1.52637i) q^{35} +(-4.38607 - 4.38607i) q^{37} -9.27391i q^{41} +(1.61040 + 1.61040i) q^{43} -2.82843 q^{47} -13.7721 q^{49} +(-0.493523 - 0.493523i) q^{53} +1.65685i q^{55} +(4.00000 + 4.00000i) q^{59} +(-2.72922 + 2.72922i) q^{61} -0.0397948 q^{65} +(-3.77568 + 3.77568i) q^{67} +9.11529i q^{71} +0.541560i q^{73} +(11.2739 - 11.2739i) q^{77} -10.9937 q^{79} +(-10.6417 + 10.6417i) q^{83} +(1.21137 + 1.21137i) q^{85} +14.6533i q^{89} +(0.270780 + 0.270780i) q^{91} +1.71313 q^{95} +4.31724 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} + 8 q^{19} - 16 q^{29} + 24 q^{31} + 24 q^{35} + 16 q^{37} + 8 q^{43} - 8 q^{49} + 16 q^{53} + 32 q^{59} - 16 q^{61} + 16 q^{65} + 16 q^{67} + 16 q^{77} - 24 q^{79} - 40 q^{83} + 16 q^{85} + 8 q^{91} + 48 q^{95}+O(q^{100}) \) 8 * q - 8 * q^11 + 8 * q^19 - 16 * q^29 + 24 * q^31 + 24 * q^35 + 16 * q^37 + 8 * q^43 - 8 * q^49 + 16 * q^53 + 32 * q^59 - 16 * q^61 + 16 * q^65 + 16 * q^67 + 16 * q^77 - 24 * q^79 - 40 * q^83 + 16 * q^85 + 8 * q^91 + 48 * q^95

Introduction

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