LMFDB - Embedded newform 1152.2.k.d.289.1

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:	8.0.629407744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 2x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.1
Root			$0.767178 + 1.18804i$ of defining polynomial
Character	$\chi$	$=$	1152.289
Dual form			1152.2.k.d.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.37608 + 2.37608i) q^{5} -3.64575i q^{7} +O(q^{10})$	$q+(-2.37608 + 2.37608i) q^{5} -3.64575i q^{7} +(0.841723 - 0.841723i) q^{11} +(2.64575 + 2.64575i) q^{13} -3.06871 q^{17} +(1.64575 + 1.64575i) q^{19} +7.82087i q^{23} -6.29150i q^{25} +(-0.692633 - 0.692633i) q^{29} +0.354249 q^{31} +(8.66259 + 8.66259i) q^{35} +(-4.64575 + 4.64575i) q^{37} +6.43560i q^{41} +(-5.64575 + 5.64575i) q^{43} +11.1878 q^{47} -6.29150 q^{49} +(-5.44479 + 5.44479i) q^{53} +4.00000i q^{55} +(-7.82087 + 7.82087i) q^{59} +(-4.64575 - 4.64575i) q^{61} -12.5730 q^{65} +(4.00000 + 4.00000i) q^{67} -3.36689i q^{71} +7.29150i q^{73} +(-3.06871 - 3.06871i) q^{77} -4.35425 q^{79} +(-0.841723 - 0.841723i) q^{83} +(7.29150 - 7.29150i) q^{85} +9.50432i q^{89} +(9.64575 - 9.64575i) q^{91} -7.82087 q^{95} +10.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{19} + 24 q^{31} - 16 q^{37} - 24 q^{43} - 8 q^{49} - 16 q^{61} + 32 q^{67} - 56 q^{79} + 16 q^{85} + 56 q^{91}+O(q^{100}) $ 8 * q - 8 * q^19 + 24 * q^31 - 16 * q^37 - 24 * q^43 - 8 * q^49 - 16 * q^61 + 32 * q^67 - 56 * q^79 + 16 * q^85 + 56 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$5$	−2.37608	+	2.37608i	−1.06261	+	1.06261i	−0.0647108	+	0.997904i	$0.520612\pi$
$5$	−2.37608	+	2.37608i	−1.06261	+	1.06261i	−0.997904	+	0.0647108i	$0.979388\pi$
$6$		0			0
$7$		−	3.64575i		−	1.37796i	−0.724778	−	0.688982i	$-0.758058\pi$
$7$		−	3.64575i		−	1.37796i	0.724778	−	0.688982i	$-0.241942\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.841723	−	0.841723i	0.253789	−	0.253789i	−0.568733	−	0.822522i	$-0.692566\pi$
$11$	0.841723	−	0.841723i	0.253789	−	0.253789i	0.822522	+	0.568733i	$0.192566\pi$
$12$		0			0
$13$	2.64575	+	2.64575i	0.733799	+	0.733799i	0.971370	−	0.237571i	$-0.0763512\pi$
$13$	2.64575	+	2.64575i	0.733799	+	0.733799i	−0.237571	+	0.971370i	$0.576351\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.06871			−0.744272			−0.372136	−	0.928178i	$-0.621375\pi$
$17$	−3.06871			−0.744272			−0.372136	+	0.928178i	$0.621375\pi$
$18$		0			0
$19$	1.64575	+	1.64575i	0.377561	+	0.377561i	0.870222	−	0.492660i	$-0.163976\pi$
$19$	1.64575	+	1.64575i	0.377561	+	0.377561i	−0.492660	+	0.870222i	$0.663976\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.82087i			1.63076i	0.578923	+	0.815382i	$0.303473\pi$
$23$			7.82087i			1.63076i	−0.578923	+	0.815382i	$0.696527\pi$
$24$		0			0
$25$		−	6.29150i		−	1.25830i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.692633	−	0.692633i	−0.128619	−	0.128619i	0.639867	−	0.768486i	$-0.278989\pi$
$29$	−0.692633	−	0.692633i	−0.128619	−	0.128619i	−0.768486	+	0.639867i	$0.778989\pi$
$30$		0			0
$31$	0.354249			0.0636249			0.0318125	−	0.999494i	$-0.489872\pi$
$31$	0.354249			0.0636249			0.0318125	+	0.999494i	$0.489872\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	8.66259	+	8.66259i	1.46425	+	1.46425i
$36$		0			0
$37$	−4.64575	+	4.64575i	−0.763757	+	0.763757i	−0.976999	−	0.213242i	$-0.931598\pi$
$37$	−4.64575	+	4.64575i	−0.763757	+	0.763757i	0.213242	+	0.976999i	$0.431598\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			6.43560i			1.00507i	0.864556	+	0.502536i	$0.167600\pi$
$41$			6.43560i			1.00507i	−0.864556	+	0.502536i	$0.832400\pi$
$42$		0			0
$43$	−5.64575	+	5.64575i	−0.860969	+	0.860969i	−0.991451	−	0.130482i	$-0.958348\pi$
$43$	−5.64575	+	5.64575i	−0.860969	+	0.860969i	0.130482	+	0.991451i	$0.458348\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	11.1878			1.63190			0.815951	−	0.578121i	$-0.196214\pi$
$47$	11.1878			1.63190			0.815951	+	0.578121i	$0.196214\pi$
$48$		0			0
$49$	−6.29150			−0.898786
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.44479	+	5.44479i	−0.747900	+	0.747900i	−0.974084	−	0.226185i	$-0.927375\pi$
$53$	−5.44479	+	5.44479i	−0.747900	+	0.747900i	0.226185	+	0.974084i	$0.427375\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−7.82087	+	7.82087i	−1.01819	+	1.01819i	−0.0183591	+	0.999831i	$0.505844\pi$
$59$	−7.82087	+	7.82087i	−1.01819	+	1.01819i	−0.999831	+	0.0183591i	$0.994156\pi$
$60$		0			0
$61$	−4.64575	−	4.64575i	−0.594828	−	0.594828i	0.344104	−	0.938932i	$-0.388183\pi$
$61$	−4.64575	−	4.64575i	−0.594828	−	0.594828i	−0.938932	+	0.344104i	$0.888183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.5730			−1.55949
$66$		0			0
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	3.36689i		−	0.399577i	−0.979839	−	0.199788i	$-0.935975\pi$
$71$		−	3.36689i		−	0.399577i	0.979839	−	0.199788i	$-0.0640254\pi$
$72$		0			0
$73$			7.29150i			0.853406i	0.904392	+	0.426703i	$0.140325\pi$
$73$			7.29150i			0.853406i	−0.904392	+	0.426703i	$0.859675\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.06871	−	3.06871i	−0.349712	−	0.349712i
$78$		0			0
$79$	−4.35425			−0.489891			−0.244946	−	0.969537i	$-0.578770\pi$
$79$	−4.35425			−0.489891			−0.244946	+	0.969537i	$0.578770\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.841723	−	0.841723i	−0.0923911	−	0.0923911i	0.659401	−	0.751792i	$-0.270810\pi$
$83$	−0.841723	−	0.841723i	−0.0923911	−	0.0923911i	−0.751792	+	0.659401i	$0.770810\pi$
$84$		0			0
$85$	7.29150	−	7.29150i	0.790875	−	0.790875i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			9.50432i			1.00746i	0.863862	+	0.503728i	$0.168039\pi$
$89$			9.50432i			1.00746i	−0.863862	+	0.503728i	$0.831961\pi$
$90$		0			0
$91$	9.64575	−	9.64575i	1.01115	−	1.01115i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−7.82087			−0.802404
$96$		0			0
$97$	10.5830			1.07454			0.537271	−	0.843410i	$-0.319455\pi$
$97$	10.5830			1.07454			0.537271	+	0.843410i	$0.319455\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.37608	+	2.37608i	−0.236429	+	0.236429i	−0.815370	−	0.578941i	$-0.803466\pi$
$101$	−2.37608	+	2.37608i	−0.236429	+	0.236429i	0.578941	+	0.815370i	$0.303466\pi$
$102$		0			0
$103$		−	1.06275i		−	0.104715i	−0.998628	−	0.0523577i	$-0.983326\pi$
$103$		−	1.06275i		−	0.104715i	0.998628	−	0.0523577i	$-0.0166736\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.50432	−	9.50432i	0.918817	−	0.918817i	−0.0781266	−	0.996943i	$-0.524894\pi$
$107$	9.50432	−	9.50432i	0.918817	−	0.918817i	0.996943	+	0.0781266i	$0.0248938\pi$
$108$		0			0
$109$	1.35425	+	1.35425i	0.129713	+	0.129713i	0.768983	−	0.639269i	$-0.220763\pi$
$109$	1.35425	+	1.35425i	0.129713	+	0.129713i	−0.639269	+	0.768983i	$0.720763\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−18.5830	−	18.5830i	−1.73287	−	1.73287i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			11.1878i			1.02558i
$120$		0			0
$121$			9.58301i			0.871182i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.06871	+	3.06871i	0.274474	+	0.274474i
$126$		0			0
$127$	14.9373			1.32547			0.662733	−	0.748855i	$-0.269396\pi$
$127$	14.9373			1.32547			0.662733	+	0.748855i	$0.269396\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.68345	+	1.68345i	0.147083	+	0.147083i	0.776814	−	0.629730i	$-0.216835\pi$
$131$	1.68345	+	1.68345i	0.147083	+	0.147083i	−0.629730	+	0.776814i	$0.716835\pi$
$132$		0			0
$133$	6.00000	−	6.00000i	0.520266	−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	15.9399i		−	1.36184i	−0.732358	−	0.680920i	$-0.761580\pi$
$137$		−	15.9399i		−	1.36184i	0.732358	−	0.680920i	$-0.238420\pi$
$138$		0			0
$139$	6.58301	−	6.58301i	0.558363	−	0.558363i	−0.370478	−	0.928841i	$-0.620806\pi$
$139$	6.58301	−	6.58301i	0.558363	−	0.558363i	0.928841	+	0.370478i	$0.120806\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.45398			0.372460
$144$		0			0
$145$	3.29150			0.273344
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.76135	−	3.76135i	0.308141	−	0.308141i	−0.536047	−	0.844188i	$-0.680083\pi$
$149$	3.76135	−	3.76135i	0.308141	−	0.308141i	0.844188	+	0.536047i	$0.180083\pi$
$150$		0			0
$151$			15.6458i			1.27323i	0.771180	+	0.636617i	$0.219667\pi$
$151$			15.6458i			1.27323i	−0.771180	+	0.636617i	$0.780333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.841723	+	0.841723i	−0.0676088	+	0.0676088i
$156$		0			0
$157$	1.35425	+	1.35425i	0.108081	+	0.108081i	0.759079	−	0.650998i	$-0.225650\pi$
$157$	1.35425	+	1.35425i	0.108081	+	0.108081i	−0.650998	+	0.759079i	$0.725650\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	28.5129			2.24714
$162$		0			0
$163$	6.35425	+	6.35425i	0.497703	+	0.497703i	0.910722	−	0.413019i	$-0.135526\pi$
$163$	6.35425	+	6.35425i	0.497703	+	0.497703i	−0.413019	+	0.910722i	$0.635526\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	4.45398i		−	0.344659i	−0.985039	−	0.172330i	$-0.944871\pi$
$167$		−	4.45398i		−	0.344659i	0.985039	−	0.172330i	$-0.0551294\pi$
$168$		0			0
$169$			1.00000i			0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.44479	+	5.44479i	0.413960	+	0.413960i	0.883115	−	0.469156i	$-0.155442\pi$
$173$	5.44479	+	5.44479i	0.413960	+	0.413960i	−0.469156	+	0.883115i	$0.655442\pi$
$174$		0			0
$175$	−22.9373			−1.73389
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.50432	−	9.50432i	−0.710386	−	0.710386i	0.256230	−	0.966616i	$-0.417520\pi$
$179$	−9.50432	−	9.50432i	−0.710386	−	0.710386i	−0.966616	+	0.256230i	$0.917520\pi$
$180$		0			0
$181$	−4.64575	+	4.64575i	−0.345316	+	0.345316i	−0.858361	−	0.513045i	$-0.828517\pi$
$181$	−4.64575	+	4.64575i	−0.345316	+	0.345316i	0.513045	+	0.858361i	$0.328517\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	22.0773i		−	1.62316i
$186$		0			0
$187$	−2.58301	+	2.58301i	−0.188888	+	0.188888i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.1878			−0.809518			−0.404759	−	0.914423i	$-0.632645\pi$
$191$	−11.1878			−0.809518			−0.404759	+	0.914423i	$0.632645\pi$
$192$		0			0
$193$	−19.8745			−1.43060			−0.715299	−	0.698818i	$-0.753710\pi$
$193$	−19.8745			−1.43060			−0.715299	+	0.698818i	$0.753710\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	8.81168	−	8.81168i	0.627806	−	0.627806i	−0.319709	−	0.947516i	$-0.603585\pi$
$197$	8.81168	−	8.81168i	0.627806	−	0.627806i	0.947516	+	0.319709i	$0.103585\pi$
$198$		0			0
$199$		−	8.35425i		−	0.592217i	−0.955154	−	0.296108i	$-0.904311\pi$
$199$		−	8.35425i		−	0.592217i	0.955154	−	0.296108i	$-0.0956890\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.52517	+	2.52517i	−0.177232	+	0.177232i
$204$		0			0
$205$	−15.2915	−	15.2915i	−1.06800	−	1.06800i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.77053			0.191642
$210$		0			0
$211$	6.58301	+	6.58301i	0.453193	+	0.453193i	0.896413	−	0.443220i	$-0.146164\pi$
$211$	6.58301	+	6.58301i	0.453193	+	0.453193i	−0.443220	+	0.896413i	$0.646164\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	26.8295i		−	1.82976i
$216$		0			0
$217$		−	1.29150i		−	0.0876729i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.11905	−	8.11905i	−0.546146	−	0.546146i
$222$		0			0
$223$	19.6458			1.31558			0.657788	−	0.753203i	$-0.271492\pi$
$223$	19.6458			1.31558			0.657788	+	0.753203i	$0.271492\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.1669	−	18.1669i	−1.20578	−	1.20578i	−0.972381	−	0.233399i	$-0.925015\pi$
$227$	−18.1669	−	18.1669i	−1.20578	−	1.20578i	−0.233399	−	0.972381i	$-0.574985\pi$
$228$		0			0
$229$	−2.06275	+	2.06275i	−0.136310	+	0.136310i	−0.771970	−	0.635659i	$-0.780728\pi$
$229$	−2.06275	+	2.06275i	−0.136310	+	0.136310i	0.635659	+	0.771970i	$0.280728\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.7792i			1.42680i	0.700757	+	0.713400i	$0.252846\pi$
$233$			21.7792i			1.42680i	−0.700757	+	0.713400i	$0.747154\pi$
$234$		0			0
$235$	−26.5830	+	26.5830i	−1.73408	+	1.73408i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−23.4626			−1.51767			−0.758835	−	0.651283i	$-0.774231\pi$
$239$	−23.4626			−1.51767			−0.758835	+	0.651283i	$0.774231\pi$
$240$		0			0
$241$	9.29150			0.598518			0.299259	−	0.954172i	$-0.403260\pi$
$241$	9.29150			0.598518			0.299259	+	0.954172i	$0.403260\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.9491	−	14.9491i	0.955063	−	0.955063i
$246$		0			0
$247$			8.70850i			0.554108i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.1166	−	13.1166i	0.827911	−	0.827911i	−0.159317	−	0.987228i	$-0.550929\pi$
$251$	13.1166	−	13.1166i	0.827911	−	0.827911i	0.987228	+	0.159317i	$0.0509291\pi$
$252$		0			0
$253$	6.58301	+	6.58301i	0.413870	+	0.413870i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−16.2381			−1.01290			−0.506452	−	0.862268i	$-0.669043\pi$
$257$	−16.2381			−1.01290			−0.506452	+	0.862268i	$0.669043\pi$
$258$		0			0
$259$	16.9373	+	16.9373i	1.05243	+	1.05243i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	15.6417i		−	0.964511i	−0.876031	−	0.482256i	$-0.839818\pi$
$263$		−	15.6417i		−	0.964511i	0.876031	−	0.482256i	$-0.160182\pi$
$264$		0			0
$265$		−	25.8745i		−	1.58946i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	8.51350	+	8.51350i	0.519077	+	0.519077i	0.917292	−	0.398215i	$-0.130370\pi$
$269$	8.51350	+	8.51350i	0.519077	+	0.519077i	−0.398215	+	0.917292i	$0.630370\pi$
$270$		0			0
$271$	−11.6458			−0.707429			−0.353715	−	0.935353i	$-0.615082\pi$
$271$	−11.6458			−0.707429			−0.353715	+	0.935353i	$0.615082\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.29570	−	5.29570i	−0.319343	−	0.319343i
$276$		0			0
$277$	−20.5203	+	20.5203i	−1.23294	+	1.23294i	−0.270115	+	0.962828i	$0.587062\pi$
$277$	−20.5203	+	20.5203i	−1.23294	+	1.23294i	−0.962828	+	0.270115i	$0.912938\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			9.50432i			0.566980i	0.958975	+	0.283490i	$0.0914923\pi$
$281$			9.50432i			0.566980i	−0.958975	+	0.283490i	$0.908508\pi$
$282$		0			0
$283$	18.5830	−	18.5830i	1.10465	−	1.10465i	0.110803	−	0.993842i	$-0.464658\pi$
$283$	18.5830	−	18.5830i	1.10465	−	1.10465i	0.993842	−	0.110803i	$-0.0353421\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	23.4626			1.38495
$288$		0			0
$289$	−7.58301			−0.446059
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.9491	−	14.9491i	0.873336	−	0.873336i	−0.119498	−	0.992834i	$-0.538129\pi$
$293$	14.9491	−	14.9491i	0.873336	−	0.873336i	0.992834	+	0.119498i	$0.0381286\pi$
$294$		0			0
$295$		−	37.1660i		−	2.16389i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.6921	+	20.6921i	−1.19665	+	1.19665i
$300$		0			0
$301$	20.5830	+	20.5830i	1.18638	+	1.18638i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	22.0773			1.26415
$306$		0			0
$307$	−20.0000	−	20.0000i	−1.14146	−	1.14146i	−0.988183	−	0.153277i	$-0.951017\pi$
$307$	−20.0000	−	20.0000i	−1.14146	−	1.14146i	−0.153277	−	0.988183i	$-0.548983\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	15.6417i		−	0.886962i	−0.896284	−	0.443481i	$-0.853743\pi$
$311$		−	15.6417i		−	0.886962i	0.896284	−	0.443481i	$-0.146257\pi$
$312$		0			0
$313$		−	1.29150i		−	0.0730000i	−0.999334	−	0.0365000i	$-0.988379\pi$
$313$		−	1.29150i		−	0.0730000i	0.999334	−	0.0365000i	$-0.0116209\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.37608	+	2.37608i	0.133454	+	0.133454i	0.770678	−	0.637224i	$-0.219918\pi$
$317$	2.37608	+	2.37608i	0.133454	+	0.133454i	−0.637224	+	0.770678i	$0.719918\pi$
$318$		0			0
$319$	−1.16601			−0.0652841
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.05034	−	5.05034i	−0.281008	−	0.281008i
$324$		0			0
$325$	16.6458	−	16.6458i	0.923340	−	0.923340i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	40.7878i		−	2.24870i
$330$		0			0
$331$	−8.00000	+	8.00000i	−0.439720	+	0.439720i	−0.891918	−	0.452198i	$-0.850640\pi$
$331$	−8.00000	+	8.00000i	−0.439720	+	0.439720i	0.452198	+	0.891918i	$0.350640\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−19.0086			−1.03855
$336$		0			0
$337$	15.2915			0.832981			0.416491	−	0.909140i	$-0.363260\pi$
$337$	15.2915			0.832981			0.416491	+	0.909140i	$0.363260\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.298179	−	0.298179i	0.0161473	−	0.0161473i
$342$		0			0
$343$		−	2.58301i		−	0.139469i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.841723	−	0.841723i	0.0451861	−	0.0451861i	−0.684153	−	0.729339i	$-0.739828\pi$
$347$	0.841723	−	0.841723i	0.0451861	−	0.0451861i	0.729339	+	0.684153i	$0.239828\pi$
$348$		0			0
$349$	23.2288	+	23.2288i	1.24341	+	1.24341i	0.958579	+	0.284828i	$0.0919366\pi$
$349$	23.2288	+	23.2288i	1.24341	+	1.24341i	0.284828	+	0.958579i	$0.408063\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	28.5129			1.51759			0.758796	−	0.651329i	$-0.225788\pi$
$353$	28.5129			1.51759			0.758796	+	0.651329i	$0.225788\pi$
$354$		0			0
$355$	8.00000	+	8.00000i	0.424596	+	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			19.0086i			1.00324i	0.865089	+	0.501619i	$0.167262\pi$
$359$			19.0086i			1.00324i	−0.865089	+	0.501619i	$0.832738\pi$
$360$		0			0
$361$		−	13.5830i		−	0.714895i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17.3252	−	17.3252i	−0.906842	−	0.906842i
$366$		0			0
$367$	12.8118			0.668769			0.334384	−	0.942437i	$-0.391472\pi$
$367$	12.8118			0.668769			0.334384	+	0.942437i	$0.391472\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.8504	+	19.8504i	1.03058	+	1.03058i
$372$		0			0
$373$	3.93725	−	3.93725i	0.203863	−	0.203863i	−0.597790	−	0.801653i	$-0.703954\pi$
$373$	3.93725	−	3.93725i	0.203863	−	0.203863i	0.801653	+	0.597790i	$0.203954\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	3.66507i		−	0.188761i
$378$		0			0
$379$	13.6458	−	13.6458i	0.700935	−	0.700935i	−0.263676	−	0.964611i	$-0.584935\pi$
$379$	13.6458	−	13.6458i	0.700935	−	0.700935i	0.964611	+	0.263676i	$0.0849350\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	11.1878			0.571668			0.285834	−	0.958279i	$-0.407729\pi$
$383$	11.1878			0.571668			0.285834	+	0.958279i	$0.407729\pi$
$384$		0			0
$385$	14.5830			0.743219
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.42642	+	7.42642i	−0.376534	+	0.376534i	−0.869850	−	0.493316i	$-0.835785\pi$
$389$	−7.42642	+	7.42642i	−0.376534	+	0.376534i	0.493316	+	0.869850i	$0.335785\pi$
$390$		0			0
$391$		−	24.0000i		−	1.21373i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.3460	−	10.3460i	0.520566	−	0.520566i
$396$		0			0
$397$	−23.9373	−	23.9373i	−1.20138	−	1.20138i	−0.973748	−	0.227628i	$-0.926903\pi$
$397$	−23.9373	−	23.9373i	−1.20138	−	1.20138i	−0.227628	−	0.973748i	$-0.573097\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.20614			−0.459733			−0.229866	−	0.973222i	$-0.573829\pi$
$401$	−9.20614			−0.459733			−0.229866	+	0.973222i	$0.573829\pi$
$402$		0			0
$403$	0.937254	+	0.937254i	0.0466879	+	0.0466879i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			7.82087i			0.387666i
$408$		0			0
$409$			17.1660i			0.848805i	0.905474	+	0.424402i	$0.139516\pi$
$409$			17.1660i			0.848805i	−0.905474	+	0.424402i	$0.860484\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.5129	+	28.5129i	1.40303	+	1.40303i
$414$		0			0
$415$	4.00000			0.196352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−13.1166	−	13.1166i	−0.640786	−	0.640786i	0.309962	−	0.950749i	$-0.399684\pi$
$419$	−13.1166	−	13.1166i	−0.640786	−	0.640786i	−0.950749	+	0.309962i	$0.899684\pi$
$420$		0			0
$421$	−16.6458	+	16.6458i	−0.811264	+	0.811264i	−0.984823	−	0.173559i	$-0.944473\pi$
$421$	−16.6458	+	16.6458i	−0.811264	+	0.811264i	0.173559	+	0.984823i	$0.444473\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			19.3068i			0.936518i
$426$		0			0
$427$	−16.9373	+	16.9373i	−0.819651	+	0.819651i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.08709			0.0523632			0.0261816	−	0.999657i	$-0.491665\pi$
$431$	1.08709			0.0523632			0.0261816	+	0.999657i	$0.491665\pi$
$432$		0			0
$433$	−4.00000			−0.192228			−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.8712	+	12.8712i	−0.615713	+	0.615713i
$438$		0			0
$439$			37.5203i			1.79074i	0.445319	+	0.895372i	$0.353090\pi$
$439$			37.5203i			1.79074i	−0.445319	+	0.895372i	$0.646910\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	23.2172	−	23.2172i	1.10308	−	1.10308i	0.109048	−	0.994036i	$-0.465220\pi$
$443$	23.2172	−	23.2172i	1.10308	−	1.10308i	0.994036	−	0.109048i	$-0.0347803\pi$
$444$		0			0
$445$	−22.5830	−	22.5830i	−1.07054	−	1.07054i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−37.7191			−1.78007			−0.890037	−	0.455889i	$-0.849322\pi$
$449$	−37.7191			−1.78007			−0.890037	+	0.455889i	$0.849322\pi$
$450$		0			0
$451$	5.41699	+	5.41699i	0.255076	+	0.255076i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			45.8381i			2.14892i
$456$		0			0
$457$			26.5830i			1.24350i	0.783216	+	0.621750i	$0.213578\pi$
$457$			26.5830i			1.24350i	−0.783216	+	0.621750i	$0.786422\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8.81168	−	8.81168i	−0.410401	−	0.410401i	0.471477	−	0.881878i	$-0.343721\pi$
$461$	−8.81168	−	8.81168i	−0.410401	−	0.410401i	−0.881878	+	0.471477i	$0.843721\pi$
$462$		0			0
$463$	−30.9373			−1.43778			−0.718888	−	0.695126i	$-0.755349\pi$
$463$	−30.9373			−1.43778			−0.718888	+	0.695126i	$0.755349\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	11.4331	+	11.4331i	0.529062	+	0.529062i	0.920293	−	0.391231i	$-0.127951\pi$
$467$	11.4331	+	11.4331i	0.529062	+	0.529062i	−0.391231	+	0.920293i	$0.627951\pi$
$468$		0			0
$469$	14.5830	−	14.5830i	0.673381	−	0.673381i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			9.50432i			0.437009i
$474$		0			0
$475$	10.3542	−	10.3542i	0.475086	−	0.475086i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.2748			−0.560852			−0.280426	−	0.959876i	$-0.590476\pi$
$479$	−12.2748			−0.560852			−0.280426	+	0.959876i	$0.590476\pi$
$480$		0			0
$481$	−24.5830			−1.12089
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−25.1461	+	25.1461i	−1.14182	+	1.14182i
$486$		0			0
$487$			18.2288i			0.826024i	0.910726	+	0.413012i	$0.135523\pi$
$487$			18.2288i			0.826024i	−0.910726	+	0.413012i	$0.864477\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.9583	+	13.9583i	−0.629929	+	0.629929i	−0.948050	−	0.318121i	$-0.896948\pi$
$491$	−13.9583	+	13.9583i	−0.629929	+	0.629929i	0.318121	+	0.948050i	$0.396948\pi$
$492$		0			0
$493$	2.12549	+	2.12549i	0.0957274	+	0.0957274i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.2748			−0.550602
$498$		0			0
$499$	−10.5830	−	10.5830i	−0.473760	−	0.473760i	0.429369	−	0.903129i	$-0.358736\pi$
$499$	−10.5830	−	10.5830i	−0.473760	−	0.473760i	−0.903129	+	0.429369i	$0.858736\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	25.7424i		−	1.14780i	−0.818926	−	0.573899i	$-0.805430\pi$
$503$		−	25.7424i		−	1.14780i	0.818926	−	0.573899i	$-0.194570\pi$
$504$		0			0
$505$		−	11.2915i		−	0.502465i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−27.2240	−	27.2240i	−1.20668	−	1.20668i	−0.972096	−	0.234585i	$-0.924627\pi$
$509$	−27.2240	−	27.2240i	−1.20668	−	1.20668i	−0.234585	−	0.972096i	$-0.575373\pi$
$510$		0			0
$511$	26.5830			1.17596
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.52517	+	2.52517i	0.111272	+	0.111272i
$516$		0			0
$517$	9.41699	−	9.41699i	0.414159	−	0.414159i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.7105i			0.819720i	0.912148	+	0.409860i	$0.134422\pi$
$521$			18.7105i			0.819720i	−0.912148	+	0.409860i	$0.865578\pi$
$522$		0			0
$523$	−3.06275	+	3.06275i	−0.133925	+	0.133925i	−0.770891	−	0.636967i	$-0.780189\pi$
$523$	−3.06275	+	3.06275i	−0.133925	+	0.133925i	0.636967	+	0.770891i	$0.280189\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.08709			−0.0473543
$528$		0			0
$529$	−38.1660			−1.65939
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−17.0270	+	17.0270i	−0.737522	+	0.737522i
$534$		0			0
$535$			45.1660i			1.95270i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5.29570	+	5.29570i	−0.228102	+	0.228102i
$540$		0			0
$541$	−8.52026	−	8.52026i	−0.366315	−	0.366315i	0.499817	−	0.866131i	$-0.333401\pi$
$541$	−8.52026	−	8.52026i	−0.366315	−	0.366315i	−0.866131	+	0.499817i	$0.833401\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−6.43560			−0.275671
$546$		0			0
$547$	−0.937254	−	0.937254i	−0.0400741	−	0.0400741i	0.686786	−	0.726860i	$-0.259021\pi$
$547$	−0.937254	−	0.937254i	−0.0400741	−	0.0400741i	−0.726860	+	0.686786i	$0.759021\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2.27980i		−	0.0971229i
$552$		0			0
$553$			15.8745i			0.675053i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.7516	+	24.7516i	1.04876	+	1.04876i	0.998749	+	0.0500103i	$0.0159254\pi$
$557$	24.7516	+	24.7516i	1.04876	+	1.04876i	0.0500103	+	0.998749i	$0.484075\pi$
$558$		0			0
$559$	−29.8745			−1.26356
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.4835	+	16.4835i	0.694695	+	0.694695i	0.963261	−	0.268566i	$-0.0865498\pi$
$563$	16.4835	+	16.4835i	0.694695	+	0.694695i	−0.268566	+	0.963261i	$0.586550\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11.9767i		−	0.502088i	−0.967976	−	0.251044i	$-0.919226\pi$
$569$		−	11.9767i		−	0.502088i	0.967976	−	0.251044i	$-0.0807739\pi$
$570$		0			0
$571$	4.00000	−	4.00000i	0.167395	−	0.167395i	−0.618438	−	0.785833i	$-0.712234\pi$
$571$	4.00000	−	4.00000i	0.167395	−	0.167395i	0.785833	+	0.618438i	$0.212234\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	49.2050			2.05199
$576$		0			0
$577$	35.0405			1.45876			0.729378	−	0.684111i	$-0.239810\pi$
$577$	35.0405			1.45876			0.729378	+	0.684111i	$0.239810\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.06871	+	3.06871i	−0.127312	+	0.127312i
$582$		0			0
$583$			9.16601i			0.379617i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−19.0086	+	19.0086i	−0.784570	+	0.784570i	−0.980598	−	0.196028i	$-0.937196\pi$
$587$	−19.0086	+	19.0086i	−0.784570	+	0.784570i	0.196028	+	0.980598i	$0.437196\pi$
$588$		0			0
$589$	0.583005	+	0.583005i	0.0240223	+	0.0240223i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−12.2748			−0.504068			−0.252034	−	0.967718i	$-0.581099\pi$
$593$	−12.2748			−0.504068			−0.252034	+	0.967718i	$0.581099\pi$
$594$		0			0
$595$	−26.5830	−	26.5830i	−1.08980	−	1.08980i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			30.1964i			1.23379i	0.787045	+	0.616896i	$0.211610\pi$
$599$			30.1964i			1.23379i	−0.787045	+	0.616896i	$0.788390\pi$
$600$		0			0
$601$			7.29150i			0.297427i	0.988880	+	0.148713i	$0.0475132\pi$
$601$			7.29150i			0.297427i	−0.988880	+	0.148713i	$0.952487\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−22.7700	−	22.7700i	−0.925731	−	0.925731i
$606$		0			0
$607$	44.1033			1.79010			0.895048	−	0.445970i	$-0.147141\pi$
$607$	44.1033			1.79010			0.895048	+	0.445970i	$0.147141\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.6000	+	29.6000i	1.19749	+	1.19749i
$612$		0			0
$613$	−21.3542	+	21.3542i	−0.862490	+	0.862490i	−0.991627	−	0.129137i	$-0.958779\pi$
$613$	−21.3542	+	21.3542i	−0.862490	+	0.862490i	0.129137	+	0.991627i	$0.458779\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			25.7424i			1.03635i	0.855274	+	0.518175i	$0.173389\pi$
$617$			25.7424i			1.03635i	−0.855274	+	0.518175i	$0.826611\pi$
$618$		0			0
$619$	21.1660	−	21.1660i	0.850734	−	0.850734i	−0.139490	−	0.990224i	$-0.544546\pi$
$619$	21.1660	−	21.1660i	0.850734	−	0.850734i	0.990224	+	0.139490i	$0.0445462\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	34.6504			1.38824
$624$		0			0
$625$	16.8745			0.674980
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14.2565	−	14.2565i	0.568443	−	0.568443i
$630$		0			0
$631$		−	18.2288i		−	0.725675i	−0.931852	−	0.362838i	$-0.881808\pi$
$631$		−	18.2288i		−	0.725675i	0.931852	−	0.362838i	$-0.118192\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−35.4921	+	35.4921i	−1.40846	+	1.40846i
$636$		0			0
$637$	−16.6458	−	16.6458i	−0.659529	−	0.659529i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−9.20614			−0.363621			−0.181810	−	0.983334i	$-0.558196\pi$
$641$	−9.20614			−0.363621			−0.181810	+	0.983334i	$0.558196\pi$
$642$		0			0
$643$	1.64575	+	1.64575i	0.0649021	+	0.0649021i	0.738813	−	0.673911i	$-0.235387\pi$
$643$	1.64575	+	1.64575i	0.0649021	+	0.0649021i	−0.673911	+	0.738813i	$0.735387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.82087i			0.307470i	0.988112	+	0.153735i	$0.0491302\pi$
$647$			7.82087i			0.307470i	−0.988112	+	0.153735i	$0.950870\pi$
$648$		0			0
$649$			13.1660i			0.516811i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.42642	+	7.42642i	0.290618	+	0.290618i	0.837324	−	0.546706i	$-0.184119\pi$
$653$	7.42642	+	7.42642i	0.290618	+	0.290618i	−0.546706	+	0.837324i	$0.684119\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		0			0
$657$		0			0
$658$		0			0
$659$	24.0590	+	24.0590i	0.937204	+	0.937204i	0.998142	−	0.0609372i	$-0.0194089\pi$
$659$	24.0590	+	24.0590i	0.937204	+	0.937204i	−0.0609372	+	0.998142i	$0.519409\pi$
$660$		0			0
$661$	29.2288	−	29.2288i	1.13687	−	1.13687i	0.147858	−	0.989009i	$-0.452762\pi$
$661$	29.2288	−	29.2288i	1.13687	−	1.13687i	0.989009	−	0.147858i	$-0.0472380\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			28.5129i			1.10568i
$666$		0			0
$667$	5.41699	−	5.41699i	0.209747	−	0.209747i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7.82087			−0.301921
$672$		0			0
$673$	20.0000			0.770943			0.385472	−	0.922720i	$-0.374039\pi$
$673$	20.0000			0.770943			0.385472	+	0.922720i	$0.374039\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.74297	−	5.74297i	0.220720	−	0.220720i	−0.588081	−	0.808802i	$-0.700117\pi$
$677$	5.74297	−	5.74297i	0.220720	−	0.220720i	0.808802	+	0.588081i	$0.200117\pi$
$678$		0			0
$679$		−	38.5830i		−	1.48068i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	35.4921	−	35.4921i	1.35807	−	1.35807i	0.481769	−	0.876298i	$-0.339994\pi$
$683$	35.4921	−	35.4921i	1.35807	−	1.35807i	0.876298	−	0.481769i	$-0.160006\pi$
$684$		0			0
$685$	37.8745	+	37.8745i	1.44711	+	1.44711i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−28.8111			−1.09762
$690$		0			0
$691$	1.64575	+	1.64575i	0.0626073	+	0.0626073i	0.737717	−	0.675110i	$-0.235904\pi$
$691$	1.64575	+	1.64575i	0.0626073	+	0.0626073i	−0.675110	+	0.737717i	$0.735904\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			31.2835i			1.18665i
$696$		0			0
$697$		−	19.7490i		−	0.748047i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	16.6326	+	16.6326i	0.628203	+	0.628203i	0.947616	−	0.319413i	$-0.103486\pi$
$701$	16.6326	+	16.6326i	0.628203	+	0.628203i	−0.319413	+	0.947616i	$0.603486\pi$
$702$		0			0
$703$	−15.2915			−0.576730
$704$		0			0
$705$		0			0
$706$		0			0
$707$	8.66259	+	8.66259i	0.325790	+	0.325790i
$708$		0			0
$709$	35.2288	−	35.2288i	1.32304	−	1.32304i	0.411744	−	0.911299i	$-0.364920\pi$
$709$	35.2288	−	35.2288i	1.32304	−	1.32304i	0.911299	−	0.411744i	$-0.135080\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			2.77053i			0.103757i
$714$		0			0
$715$	−10.5830	+	10.5830i	−0.395782	+	0.395782i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	10.1007			0.376692			0.188346	−	0.982103i	$-0.439687\pi$
$719$	10.1007			0.376692			0.188346	+	0.982103i	$0.439687\pi$
$720$		0			0
$721$	−3.87451			−0.144294
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.35770	+	4.35770i	−0.161841	+	0.161841i
$726$		0			0
$727$		−	35.3948i		−	1.31272i	−0.754448	−	0.656360i	$-0.772095\pi$
$727$		−	35.3948i		−	1.31272i	0.754448	−	0.656360i	$-0.227905\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	17.3252	−	17.3252i	0.640795	−	0.640795i
$732$		0			0
$733$	7.35425	+	7.35425i	0.271635	+	0.271635i	0.829758	−	0.558123i	$-0.188478\pi$
$733$	7.35425	+	7.35425i	0.271635	+	0.271635i	−0.558123	+	0.829758i	$0.688478\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.73378			0.248042
$738$		0			0
$739$	33.1660	+	33.1660i	1.22003	+	1.22003i	0.967620	+	0.252411i	$0.0812236\pi$
$739$	33.1660	+	33.1660i	1.22003	+	1.22003i	0.252411	+	0.967620i	$0.418776\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	14.5547i		−	0.533958i	−0.963702	−	0.266979i	$-0.913974\pi$
$743$		−	14.5547i		−	0.533958i	0.963702	−	0.266979i	$-0.0860255\pi$
$744$		0			0
$745$			17.8745i			0.654871i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−34.6504	−	34.6504i	−1.26610	−	1.26610i
$750$		0			0
$751$	2.93725			0.107182			0.0535910	−	0.998563i	$-0.482933\pi$
$751$	2.93725			0.107182			0.0535910	+	0.998563i	$0.482933\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−37.1755	−	37.1755i	−1.35296	−	1.35296i
$756$		0			0
$757$	−19.2288	+	19.2288i	−0.698881	+	0.698881i	−0.964169	−	0.265288i	$-0.914533\pi$
$757$	−19.2288	+	19.2288i	−0.698881	+	0.698881i	0.265288	+	0.964169i	$0.414533\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			6.43560i			0.233290i	0.993174	+	0.116645i	$0.0372140\pi$
$761$			6.43560i			0.233290i	−0.993174	+	0.116645i	$0.962786\pi$
$762$		0			0
$763$	4.93725	−	4.93725i	0.178741	−	0.178741i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−41.3842			−1.49430
$768$		0			0
$769$	6.70850			0.241915			0.120957	−	0.992658i	$-0.461404\pi$
$769$	6.70850			0.241915			0.120957	+	0.992658i	$0.461404\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−14.6509	+	14.6509i	−0.526957	+	0.526957i	−0.919664	−	0.392707i	$-0.871539\pi$
$773$	−14.6509	+	14.6509i	−0.526957	+	0.526957i	0.392707	+	0.919664i	$0.371539\pi$
$774$		0			0
$775$		−	2.22876i		−	0.0800593i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.5914	+	10.5914i	−0.379476	+	0.379476i
$780$		0			0
$781$	−2.83399	−	2.83399i	−0.101408	−	0.101408i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.43560			−0.229697
$786$		0			0
$787$	−20.2288	−	20.2288i	−0.721077	−	0.721077i	0.247747	−	0.968825i	$-0.420310\pi$
$787$	−20.2288	−	20.2288i	−0.721077	−	0.721077i	−0.968825	+	0.247747i	$0.920310\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	24.5830i		−	0.872968i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	14.6509	+	14.6509i	0.518962	+	0.518962i	0.917257	−	0.398295i	$-0.130398\pi$
$797$	14.6509	+	14.6509i	0.518962	+	0.518962i	−0.398295	+	0.917257i	$0.630398\pi$
$798$		0			0
$799$	−34.3320			−1.21458
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.13742	+	6.13742i	0.216585	+	0.216585i
$804$		0			0
$805$	−67.7490	+	67.7490i	−2.38784	+	2.38784i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	44.4529i		−	1.56288i	−0.623981	−	0.781440i	$-0.714486\pi$
$809$		−	44.4529i		−	1.56288i	0.623981	−	0.781440i	$-0.285514\pi$
$810$		0			0
$811$	−10.3542	+	10.3542i	−0.363587	+	0.363587i	−0.865132	−	0.501545i	$-0.832765\pi$
$811$	−10.3542	+	10.3542i	−0.363587	+	0.363587i	0.501545	+	0.865132i	$0.332765\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−30.1964			−1.05773
$816$		0			0
$817$	−18.5830			−0.650137
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30.8890	+	30.8890i	−1.07803	+	1.07803i	−0.0813488	+	0.996686i	$0.525923\pi$
$821$	−30.8890	+	30.8890i	−1.07803	+	1.07803i	−0.996686	+	0.0813488i	$0.974077\pi$
$822$		0			0
$823$			5.77124i			0.201173i	0.994928	+	0.100586i	$0.0320719\pi$
$823$			5.77124i			0.201173i	−0.994928	+	0.100586i	$0.967928\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−7.82087	+	7.82087i	−0.271958	+	0.271958i	−0.829888	−	0.557930i	$-0.811596\pi$
$827$	−7.82087	+	7.82087i	−0.271958	+	0.271958i	0.557930	+	0.829888i	$0.311596\pi$
$828$		0			0
$829$	−9.35425	−	9.35425i	−0.324886	−	0.324886i	0.525752	−	0.850638i	$-0.323784\pi$
$829$	−9.35425	−	9.35425i	−0.324886	−	0.324886i	−0.850638	+	0.525752i	$0.823784\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	19.3068			0.668941
$834$		0			0
$835$	10.5830	+	10.5830i	0.366240	+	0.366240i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	40.1914i		−	1.38756i	−0.720186	−	0.693781i	$-0.755943\pi$
$839$		−	40.1914i		−	1.38756i	0.720186	−	0.693781i	$-0.244057\pi$
$840$		0			0
$841$		−	28.0405i		−	0.966914i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.37608	−	2.37608i	−0.0817396	−	0.0817396i
$846$		0			0
$847$	34.9373			1.20046
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−36.3338	−	36.3338i	−1.24551	−	1.24551i
$852$		0			0
$853$	3.93725	−	3.93725i	0.134809	−	0.134809i	−0.636482	−	0.771291i	$-0.719611\pi$
$853$	3.93725	−	3.93725i	0.134809	−	0.134809i	0.771291	+	0.636482i	$0.219611\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	15.9399i		−	0.544497i	−0.962227	−	0.272249i	$-0.912233\pi$
$857$		−	15.9399i		−	0.544497i	0.962227	−	0.272249i	$-0.0877673\pi$
$858$		0			0
$859$	−32.2288	+	32.2288i	−1.09963	+	1.09963i	−0.105178	+	0.994453i	$0.533541\pi$
$859$	−32.2288	+	32.2288i	−1.09963	+	1.09963i	−0.994453	+	0.105178i	$0.966459\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	13.3619			0.454846			0.227423	−	0.973796i	$-0.426970\pi$
$863$	13.3619			0.454846			0.227423	+	0.973796i	$0.426970\pi$
$864$		0			0
$865$	−25.8745			−0.879760
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.66507	+	3.66507i	−0.124329	+	0.124329i
$870$		0			0
$871$			21.1660i			0.717183i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	11.1878	−	11.1878i	0.378215	−	0.378215i
$876$		0			0
$877$	6.06275	+	6.06275i	0.204724	+	0.204724i	0.802021	−	0.597296i	$-0.203758\pi$
$877$	6.06275	+	6.06275i	0.204724	+	0.204724i	−0.597296	+	0.802021i	$0.703758\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	40.7878			1.37418			0.687088	−	0.726574i	$-0.258889\pi$
$881$	40.7878			1.37418			0.687088	+	0.726574i	$0.258889\pi$
$882$		0			0
$883$	33.3948	+	33.3948i	1.12382	+	1.12382i	0.991161	+	0.132662i	$0.0423525\pi$
$883$	33.3948	+	33.3948i	1.12382	+	1.12382i	0.132662	+	0.991161i	$0.457648\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			19.0086i			0.638247i	0.947713	+	0.319124i	$0.103389\pi$
$887$			19.0086i			0.638247i	−0.947713	+	0.319124i	$0.896611\pi$
$888$		0			0
$889$		−	54.4575i		−	1.82645i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.4123	+	18.4123i	0.616143	+	0.616143i
$894$		0			0
$895$	45.1660			1.50973
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.245364	−	0.245364i	−0.00818336	−	0.00818336i
$900$		0			0
$901$	16.7085	−	16.7085i	0.556641	−	0.556641i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	22.0773i		−	0.733876i
$906$		0			0
$907$	−42.1033	+	42.1033i	−1.39802	+	1.39802i	−0.592294	+	0.805722i	$0.701778\pi$
$907$	−42.1033	+	42.1033i	−1.39802	+	1.39802i	−0.805722	+	0.592294i	$0.798222\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	45.8381			1.51869			0.759343	−	0.650691i	$-0.225521\pi$
$911$	45.8381			1.51869			0.759343	+	0.650691i	$0.225521\pi$
$912$		0			0
$913$	−1.41699			−0.0468957
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.13742	−	6.13742i	0.202676	−	0.202676i
$918$		0			0
$919$			16.1033i			0.531198i	0.964084	+	0.265599i	$0.0855697\pi$
$919$			16.1033i			0.531198i	−0.964084	+	0.265599i	$0.914430\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.90796	−	8.90796i	0.293209	−	0.293209i
$924$		0			0
$925$	29.2288	+	29.2288i	0.961036	+	0.961036i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.3068			−0.633436			−0.316718	−	0.948520i	$-0.602581\pi$
$929$	−19.3068			−0.633436			−0.316718	+	0.948520i	$0.602581\pi$
$930$		0			0
$931$	−10.3542	−	10.3542i	−0.339347	−	0.339347i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	12.2748i		−	0.401430i
$936$		0			0
$937$		−	11.1660i		−	0.364778i	−0.983226	−	0.182389i	$-0.941617\pi$
$937$		−	11.1660i		−	0.364778i	0.983226	−	0.182389i	$-0.0583830\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	25.8387	+	25.8387i	0.842317	+	0.842317i	0.989160	−	0.146843i	$-0.0469111\pi$
$941$	25.8387	+	25.8387i	0.842317	+	0.842317i	−0.146843	+	0.989160i	$0.546911\pi$
$942$		0			0
$943$	−50.3320			−1.63904
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.50432	−	9.50432i	−0.308849	−	0.308849i	0.535614	−	0.844463i	$-0.320080\pi$
$947$	−9.50432	−	9.50432i	−0.308849	−	0.308849i	−0.844463	+	0.535614i	$0.820080\pi$
$948$		0			0
$949$	−19.2915	+	19.2915i	−0.626229	+	0.626229i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24.8479i			0.804902i	0.915442	+	0.402451i	$0.131842\pi$
$953$			24.8479i			0.804902i	−0.915442	+	0.402451i	$0.868158\pi$
$954$		0			0
$955$	26.5830	−	26.5830i	0.860206	−	0.860206i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−58.1130			−1.87657
$960$		0			0
$961$	−30.8745			−0.995952
$962$		0			0
$963$		0			0
$964$		0			0
$965$	47.2234	−	47.2234i	1.52018	−	1.52018i
$966$		0			0
$967$		−	23.3948i		−	0.752325i	−0.926554	−	0.376162i	$-0.877243\pi$
$967$		−	23.3948i		−	0.752325i	0.926554	−	0.376162i	$-0.122757\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.4418	−	30.4418i	0.976923	−	0.976923i	−0.0228172	−	0.999740i	$-0.507264\pi$
$971$	30.4418	−	30.4418i	0.976923	−	0.976923i	0.999740	+	0.0228172i	$0.00726356\pi$
$972$		0			0
$973$	−24.0000	−	24.0000i	−0.769405	−	0.769405i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	43.8565			1.40309			0.701547	−	0.712623i	$-0.252493\pi$
$977$	43.8565			1.40309			0.701547	+	0.712623i	$0.252493\pi$
$978$		0			0
$979$	8.00000	+	8.00000i	0.255681	+	0.255681i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			6.73378i			0.214774i	0.994217	+	0.107387i	$0.0342484\pi$
$983$			6.73378i			0.214774i	−0.994217	+	0.107387i	$0.965752\pi$
$984$		0			0
$985$			41.8745i			1.33423i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−44.1547	−	44.1547i	−1.40404	−	1.40404i
$990$		0			0
$991$	−18.9373			−0.601562			−0.300781	−	0.953693i	$-0.597247\pi$
$991$	−18.9373			−0.601562			−0.300781	+	0.953693i	$0.597247\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	19.8504	+	19.8504i	0.629299	+	0.629299i
$996$		0			0
$997$	0.520259	−	0.520259i	0.0164768	−	0.0164768i	−0.698820	−	0.715297i	$-0.746291\pi$
$997$	0.520259	−	0.520259i	0.0164768	−	0.0164768i	0.715297	+	0.698820i	$0.246291\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.2.k.d.289.1		8
3.2	odd	2	inner	1152.2.k.d.289.4		8
4.3	odd	2		1152.2.k.e.289.1		8
8.3	odd	2		144.2.k.c.109.3	yes	8
8.5	even	2		576.2.k.c.145.4		8
12.11	even	2		1152.2.k.e.289.4		8
16.3	odd	4		144.2.k.c.37.3	yes	8
16.5	even	4	inner	1152.2.k.d.865.1		8
16.11	odd	4		1152.2.k.e.865.1		8
16.13	even	4		576.2.k.c.433.4		8
24.5	odd	2		576.2.k.c.145.1		8
24.11	even	2		144.2.k.c.109.2	yes	8
32.5	even	8		9216.2.a.bt.1.2		8
32.11	odd	8		9216.2.a.bq.1.7		8
32.21	even	8		9216.2.a.bt.1.7		8
32.27	odd	8		9216.2.a.bq.1.2		8
48.5	odd	4	inner	1152.2.k.d.865.4		8
48.11	even	4		1152.2.k.e.865.4		8
48.29	odd	4		576.2.k.c.433.1		8
48.35	even	4		144.2.k.c.37.2	✓	8
96.5	odd	8		9216.2.a.bt.1.8		8
96.11	even	8		9216.2.a.bq.1.1		8
96.53	odd	8		9216.2.a.bt.1.1		8
96.59	even	8		9216.2.a.bq.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.k.c.37.2	✓	8	48.35	even	4
144.2.k.c.37.3	yes	8	16.3	odd	4
144.2.k.c.109.2	yes	8	24.11	even	2
144.2.k.c.109.3	yes	8	8.3	odd	2
576.2.k.c.145.1		8	24.5	odd	2
576.2.k.c.145.4		8	8.5	even	2
576.2.k.c.433.1		8	48.29	odd	4
576.2.k.c.433.4		8	16.13	even	4
1152.2.k.d.289.1		8	1.1	even	1	trivial
1152.2.k.d.289.4		8	3.2	odd	2	inner
1152.2.k.d.865.1		8	16.5	even	4	inner
1152.2.k.d.865.4		8	48.5	odd	4	inner
1152.2.k.e.289.1		8	4.3	odd	2
1152.2.k.e.289.4		8	12.11	even	2
1152.2.k.e.865.1		8	16.11	odd	4
1152.2.k.e.865.4		8	48.11	even	4
9216.2.a.bq.1.1		8	96.11	even	8
9216.2.a.bq.1.2		8	32.27	odd	8
9216.2.a.bq.1.7		8	32.11	odd	8
9216.2.a.bq.1.8		8	96.59	even	8
9216.2.a.bt.1.1		8	96.53	odd	8
9216.2.a.bt.1.2		8	32.5	even	8
9216.2.a.bt.1.7		8	32.21	even	8
9216.2.a.bt.1.8		8	96.5	odd	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)

\(f(q)\)	\(=\)	\(q+(-2.37608 + 2.37608i) q^{5} -3.64575i q^{7} +O(q^{10})\)	\(q+(-2.37608 + 2.37608i) q^{5} -3.64575i q^{7} +(0.841723 - 0.841723i) q^{11} +(2.64575 + 2.64575i) q^{13} -3.06871 q^{17} +(1.64575 + 1.64575i) q^{19} +7.82087i q^{23} -6.29150i q^{25} +(-0.692633 - 0.692633i) q^{29} +0.354249 q^{31} +(8.66259 + 8.66259i) q^{35} +(-4.64575 + 4.64575i) q^{37} +6.43560i q^{41} +(-5.64575 + 5.64575i) q^{43} +11.1878 q^{47} -6.29150 q^{49} +(-5.44479 + 5.44479i) q^{53} +4.00000i q^{55} +(-7.82087 + 7.82087i) q^{59} +(-4.64575 - 4.64575i) q^{61} -12.5730 q^{65} +(4.00000 + 4.00000i) q^{67} -3.36689i q^{71} +7.29150i q^{73} +(-3.06871 - 3.06871i) q^{77} -4.35425 q^{79} +(-0.841723 - 0.841723i) q^{83} +(7.29150 - 7.29150i) q^{85} +9.50432i q^{89} +(9.64575 - 9.64575i) q^{91} -7.82087 q^{95} +10.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{19} + 24 q^{31} - 16 q^{37} - 24 q^{43} - 8 q^{49} - 16 q^{61} + 32 q^{67} - 56 q^{79} + 16 q^{85} + 56 q^{91}+O(q^{100}) \) 8 * q - 8 * q^19 + 24 * q^31 - 16 * q^37 - 24 * q^43 - 8 * q^49 - 16 * q^61 + 32 * q^67 - 56 * q^79 + 16 * q^85 + 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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