LMFDB - Embedded newform 4400.2.a.bh.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,0,0,-5,0,1,0,2,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278 q^{3} -4.30278 q^{7} -1.30278 q^{9} +1.00000 q^{11} +5.00000 q^{13} -3.90833 q^{17} +1.00000 q^{19} -5.60555 q^{21} +3.69722 q^{23} -5.60555 q^{27} -9.90833 q^{29} +4.21110 q^{31} +1.30278 q^{33} +9.60555 q^{37} +6.51388 q^{39} +1.60555 q^{41} +7.21110 q^{43} +3.00000 q^{47} +11.5139 q^{49} -5.09167 q^{51} +2.30278 q^{53} +1.30278 q^{57} -0.211103 q^{59} +2.90833 q^{61} +5.60555 q^{63} +4.00000 q^{67} +4.81665 q^{69} -4.60555 q^{71} +2.90833 q^{73} -4.30278 q^{77} +0.0916731 q^{79} -3.39445 q^{81} -14.5139 q^{83} -12.9083 q^{87} +5.30278 q^{89} -21.5139 q^{91} +5.48612 q^{93} +11.6972 q^{97} -1.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 5 q^{7} + q^{9} + 2 q^{11} + 10 q^{13} + 3 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} - 4 q^{27} - 9 q^{29} - 6 q^{31} - q^{33} + 12 q^{37} - 5 q^{39} - 4 q^{41} + 6 q^{47} + 5 q^{49} - 21 q^{51}+ \cdots + q^{99}+O(q^{100}) $ 2 * q - q^3 - 5 * q^7 + q^9 + 2 * q^11 + 10 * q^13 + 3 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 - 4 * q^27 - 9 * q^29 - 6 * q^31 - q^33 + 12 * q^37 - 5 * q^39 - 4 * q^41 + 6 * q^47 + 5 * q^49 - 21 * q^51 + q^53 - q^57 + 14 * q^59 - 5 * q^61 + 4 * q^63 + 8 * q^67 - 12 * q^69 - 2 * q^71 - 5 * q^73 - 5 * q^77 + 11 * q^79 - 14 * q^81 - 11 * q^83 - 15 * q^87 + 7 * q^89 - 25 * q^91 + 29 * q^93 + 27 * q^97 + q^99

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
$3$	1.30278	0.752158	0.376079	−	0.926588i	$-0.377272\pi$
$3$	1.30278	0.752158	0.376079	+	0.926588i	$0.377272\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.30278	−1.62630	−0.813148	−	0.582057i	$-0.802248\pi$
$7$	−4.30278	−1.62630	−0.813148	+	0.582057i	$0.802248\pi$
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.90833	−0.947909	−0.473954	−	0.880549i	$-0.657174\pi$
$17$	−3.90833	−0.947909	−0.473954	+	0.880549i	$0.657174\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−5.60555	−1.22323
$22$	0	0
$23$	3.69722	0.770925	0.385462	−	0.922724i	$-0.374042\pi$
$23$	3.69722	0.770925	0.385462	+	0.922724i	$0.374042\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.60555	−1.07879
$28$	0	0
$29$	−9.90833	−1.83993	−0.919965	−	0.392000i	$-0.871783\pi$
$29$	−9.90833	−1.83993	−0.919965	+	0.392000i	$0.871783\pi$
$30$	0	0
$31$	4.21110	0.756336	0.378168	−	0.925737i	$-0.376554\pi$
$31$	4.21110	0.756336	0.378168	+	0.925737i	$0.376554\pi$
$32$	0	0
$33$	1.30278	0.226784
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.60555	1.57914	0.789571	−	0.613659i	$-0.210303\pi$
$37$	9.60555	1.57914	0.789571	+	0.613659i	$0.210303\pi$
$38$	0	0
$39$	6.51388	1.04306
$40$	0	0
$41$	1.60555	0.250745	0.125372	−	0.992110i	$-0.459987\pi$
$41$	1.60555	0.250745	0.125372	+	0.992110i	$0.459987\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	−5.09167	−0.712977
$52$	0	0
$53$	2.30278	0.316311	0.158155	−	0.987414i	$-0.449445\pi$
$53$	2.30278	0.316311	0.158155	+	0.987414i	$0.449445\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.30278	0.172557
$58$	0	0
$59$	−0.211103	−0.0274832	−0.0137416	−	0.999906i	$-0.504374\pi$
$59$	−0.211103	−0.0274832	−0.0137416	+	0.999906i	$0.504374\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	0	0
$63$	5.60555	0.706233
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	4.81665	0.579857
$70$	0	0
$71$	−4.60555	−0.546578	−0.273289	−	0.961932i	$-0.588112\pi$
$71$	−4.60555	−0.546578	−0.273289	+	0.961932i	$0.588112\pi$
$72$	0	0
$73$	2.90833	0.340394	0.170197	−	0.985410i	$-0.445560\pi$
$73$	2.90833	0.340394	0.170197	+	0.985410i	$0.445560\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.30278	−0.490347
$78$	0	0
$79$	0.0916731	0.0103140	0.00515701	−	0.999987i	$-0.498358\pi$
$79$	0.0916731	0.0103140	0.00515701	+	0.999987i	$0.498358\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−14.5139	−1.59311	−0.796553	−	0.604569i	$-0.793345\pi$
$83$	−14.5139	−1.59311	−0.796553	+	0.604569i	$0.793345\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.9083	−1.38392
$88$	0	0
$89$	5.30278	0.562093	0.281047	−	0.959694i	$-0.409318\pi$
$89$	5.30278	0.562093	0.281047	+	0.959694i	$0.409318\pi$
$90$	0	0
$91$	−21.5139	−2.25527
$92$	0	0
$93$	5.48612	0.568884
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.6972	1.18767	0.593837	−	0.804586i	$-0.297613\pi$
$97$	11.6972	1.18767	0.593837	+	0.804586i	$0.297613\pi$
$98$	0	0
$99$	−1.30278	−0.130934
$100$	0	0
$101$	17.5139	1.74270	0.871348	−	0.490666i	$-0.163246\pi$
$101$	17.5139	1.74270	0.871348	+	0.490666i	$0.163246\pi$
$102$	0	0
$103$	7.90833	0.779231	0.389615	−	0.920978i	$-0.372608\pi$
$103$	7.90833	0.779231	0.389615	+	0.920978i	$0.372608\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	−6.51388	−0.623916	−0.311958	−	0.950096i	$-0.600985\pi$
$109$	−6.51388	−0.623916	−0.311958	+	0.950096i	$0.600985\pi$
$110$	0	0
$111$	12.5139	1.18776
$112$	0	0
$113$	10.8167	1.01755	0.508773	−	0.860901i	$-0.330099\pi$
$113$	10.8167	1.01755	0.508773	+	0.860901i	$0.330099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.51388	−0.602208
$118$	0	0
$119$	16.8167	1.54158
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.09167	0.188600
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.1194	1.51910	0.759552	−	0.650447i	$-0.225418\pi$
$127$	17.1194	1.51910	0.759552	+	0.650447i	$0.225418\pi$
$128$	0	0
$129$	9.39445	0.827135
$130$	0	0
$131$	−0.908327	−0.0793609	−0.0396804	−	0.999212i	$-0.512634\pi$
$131$	−0.908327	−0.0793609	−0.0396804	+	0.999212i	$0.512634\pi$
$132$	0	0
$133$	−4.30278	−0.373098
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.09167	−0.178704	−0.0893518	−	0.996000i	$-0.528480\pi$
$137$	−2.09167	−0.178704	−0.0893518	+	0.996000i	$0.528480\pi$
$138$	0	0
$139$	−8.21110	−0.696457	−0.348228	−	0.937410i	$-0.613217\pi$
$139$	−8.21110	−0.696457	−0.348228	+	0.937410i	$0.613217\pi$
$140$	0	0
$141$	3.90833	0.329141
$142$	0	0
$143$	5.00000	0.418121
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.0000	1.23718
$148$	0	0
$149$	2.78890	0.228475	0.114238	−	0.993453i	$-0.463557\pi$
$149$	2.78890	0.228475	0.114238	+	0.993453i	$0.463557\pi$
$150$	0	0
$151$	20.8167	1.69404	0.847018	−	0.531565i	$-0.178396\pi$
$151$	20.8167	1.69404	0.847018	+	0.531565i	$0.178396\pi$
$152$	0	0
$153$	5.09167	0.411637
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.78890	0.382196	0.191098	−	0.981571i	$-0.438795\pi$
$157$	4.78890	0.382196	0.191098	+	0.981571i	$0.438795\pi$
$158$	0	0
$159$	3.00000	0.237915
$160$	0	0
$161$	−15.9083	−1.25375
$162$	0	0
$163$	−5.69722	−0.446241	−0.223121	−	0.974791i	$-0.571624\pi$
$163$	−5.69722	−0.446241	−0.223121	+	0.974791i	$0.571624\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.4222	1.19341	0.596703	−	0.802462i	$-0.296477\pi$
$167$	15.4222	1.19341	0.596703	+	0.802462i	$0.296477\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	−1.30278	−0.0996257
$172$	0	0
$173$	16.8167	1.27855	0.639273	−	0.768980i	$-0.279235\pi$
$173$	16.8167	1.27855	0.639273	+	0.768980i	$0.279235\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−0.275019	−0.0206717
$178$	0	0
$179$	5.51388	0.412127	0.206063	−	0.978539i	$-0.433935\pi$
$179$	5.51388	0.412127	0.206063	+	0.978539i	$0.433935\pi$
$180$	0	0
$181$	−9.09167	−0.675779	−0.337889	−	0.941186i	$-0.609713\pi$
$181$	−9.09167	−0.675779	−0.337889	+	0.941186i	$0.609713\pi$
$182$	0	0
$183$	3.78890	0.280083
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.90833	−0.285805
$188$	0	0
$189$	24.1194	1.75443
$190$	0	0
$191$	−6.69722	−0.484594	−0.242297	−	0.970202i	$-0.577901\pi$
$191$	−6.69722	−0.484594	−0.242297	+	0.970202i	$0.577901\pi$
$192$	0	0
$193$	−1.21110	−0.0871771	−0.0435885	−	0.999050i	$-0.513879\pi$
$193$	−1.21110	−0.0871771	−0.0435885	+	0.999050i	$0.513879\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.69722	0.690899	0.345449	−	0.938437i	$-0.387727\pi$
$197$	9.69722	0.690899	0.345449	+	0.938437i	$0.387727\pi$
$198$	0	0
$199$	24.5139	1.73774	0.868871	−	0.495038i	$-0.164846\pi$
$199$	24.5139	1.73774	0.868871	+	0.495038i	$0.164846\pi$
$200$	0	0
$201$	5.21110	0.367563
$202$	0	0
$203$	42.6333	2.99227
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−4.81665	−0.334781
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−25.2389	−1.73751	−0.868757	−	0.495238i	$-0.835081\pi$
$211$	−25.2389	−1.73751	−0.868757	+	0.495238i	$0.835081\pi$
$212$	0	0
$213$	−6.00000	−0.411113
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−18.1194	−1.23003
$218$	0	0
$219$	3.78890	0.256030
$220$	0	0
$221$	−19.5416	−1.31451
$222$	0	0
$223$	−20.6333	−1.38171	−0.690854	−	0.722994i	$-0.742765\pi$
$223$	−20.6333	−1.38171	−0.690854	+	0.722994i	$0.742765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.30278	−0.351958	−0.175979	−	0.984394i	$-0.556309\pi$
$227$	−5.30278	−0.351958	−0.175979	+	0.984394i	$0.556309\pi$
$228$	0	0
$229$	13.7250	0.906972	0.453486	−	0.891263i	$-0.350180\pi$
$229$	13.7250	0.906972	0.453486	+	0.891263i	$0.350180\pi$
$230$	0	0
$231$	−5.60555	−0.368818
$232$	0	0
$233$	5.09167	0.333567	0.166783	−	0.985994i	$-0.446662\pi$
$233$	5.09167	0.333567	0.166783	+	0.985994i	$0.446662\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.119429	0.00775778
$238$	0	0
$239$	4.11943	0.266464	0.133232	−	0.991085i	$-0.457465\pi$
$239$	4.11943	0.266464	0.133232	+	0.991085i	$0.457465\pi$
$240$	0	0
$241$	−24.9361	−1.60627	−0.803137	−	0.595794i	$-0.796837\pi$
$241$	−24.9361	−1.60627	−0.803137	+	0.595794i	$0.796837\pi$
$242$	0	0
$243$	12.3944	0.795104
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.00000	0.318142
$248$	0	0
$249$	−18.9083	−1.19827
$250$	0	0
$251$	3.90833	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$251$	3.90833	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$252$	0	0
$253$	3.69722	0.232443
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−41.3305	−2.56815
$260$	0	0
$261$	12.9083	0.799005
$262$	0	0
$263$	1.18335	0.0729683	0.0364841	−	0.999334i	$-0.488384\pi$
$263$	1.18335	0.0729683	0.0364841	+	0.999334i	$0.488384\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.90833	0.422783
$268$	0	0
$269$	−23.7250	−1.44654	−0.723269	−	0.690567i	$-0.757361\pi$
$269$	−23.7250	−1.44654	−0.723269	+	0.690567i	$0.757361\pi$
$270$	0	0
$271$	−14.2111	−0.863263	−0.431632	−	0.902050i	$-0.642062\pi$
$271$	−14.2111	−0.863263	−0.431632	+	0.902050i	$0.642062\pi$
$272$	0	0
$273$	−28.0278	−1.69632
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.6056	1.29815	0.649076	−	0.760724i	$-0.275156\pi$
$277$	21.6056	1.29815	0.649076	+	0.760724i	$0.275156\pi$
$278$	0	0
$279$	−5.48612	−0.328446
$280$	0	0
$281$	−22.8167	−1.36113	−0.680564	−	0.732689i	$-0.738265\pi$
$281$	−22.8167	−1.36113	−0.680564	+	0.732689i	$0.738265\pi$
$282$	0	0
$283$	−2.69722	−0.160333	−0.0801667	−	0.996781i	$-0.525545\pi$
$283$	−2.69722	−0.160333	−0.0801667	+	0.996781i	$0.525545\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.90833	−0.407786
$288$	0	0
$289$	−1.72498	−0.101469
$290$	0	0
$291$	15.2389	0.893318
$292$	0	0
$293$	15.2111	0.888642	0.444321	−	0.895868i	$-0.353445\pi$
$293$	15.2111	0.888642	0.444321	+	0.895868i	$0.353445\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.60555	−0.325267
$298$	0	0
$299$	18.4861	1.06908
$300$	0	0
$301$	−31.0278	−1.78841
$302$	0	0
$303$	22.8167	1.31078
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.09167	0.347670	0.173835	−	0.984775i	$-0.444384\pi$
$307$	6.09167	0.347670	0.173835	+	0.984775i	$0.444384\pi$
$308$	0	0
$309$	10.3028	0.586104
$310$	0	0
$311$	16.8167	0.953585	0.476792	−	0.879016i	$-0.341799\pi$
$311$	16.8167	0.953585	0.476792	+	0.879016i	$0.341799\pi$
$312$	0	0
$313$	21.8167	1.23315	0.616575	−	0.787296i	$-0.288520\pi$
$313$	21.8167	1.23315	0.616575	+	0.787296i	$0.288520\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.90833	0.556507	0.278254	−	0.960508i	$-0.410244\pi$
$317$	9.90833	0.556507	0.278254	+	0.960508i	$0.410244\pi$
$318$	0	0
$319$	−9.90833	−0.554760
$320$	0	0
$321$	3.90833	0.218142
$322$	0	0
$323$	−3.90833	−0.217465
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.48612	−0.469284
$328$	0	0
$329$	−12.9083	−0.711659
$330$	0	0
$331$	14.3944	0.791190	0.395595	−	0.918425i	$-0.370538\pi$
$331$	14.3944	0.791190	0.395595	+	0.918425i	$0.370538\pi$
$332$	0	0
$333$	−12.5139	−0.685756
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.8444	1.46231	0.731154	−	0.682212i	$-0.238982\pi$
$337$	26.8444	1.46231	0.731154	+	0.682212i	$0.238982\pi$
$338$	0	0
$339$	14.0917	0.765355
$340$	0	0
$341$	4.21110	0.228044
$342$	0	0
$343$	−19.4222	−1.04870
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5.51388	−0.296000	−0.148000	−	0.988987i	$-0.547284\pi$
$347$	−5.51388	−0.296000	−0.148000	+	0.988987i	$0.547284\pi$
$348$	0	0
$349$	−26.8167	−1.43546	−0.717731	−	0.696320i	$-0.754819\pi$
$349$	−26.8167	−1.43546	−0.717731	+	0.696320i	$0.754819\pi$
$350$	0	0
$351$	−28.0278	−1.49601
$352$	0	0
$353$	−24.6333	−1.31110	−0.655549	−	0.755152i	$-0.727563\pi$
$353$	−24.6333	−1.31110	−0.655549	+	0.755152i	$0.727563\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	21.9083	1.15951
$358$	0	0
$359$	−15.2111	−0.802811	−0.401406	−	0.915900i	$-0.631478\pi$
$359$	−15.2111	−0.802811	−0.401406	+	0.915900i	$0.631478\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	1.30278	0.0683780
$364$	0	0
$365$	0	0
$366$	0	0
$367$	24.3028	1.26859	0.634297	−	0.773089i	$-0.281290\pi$
$367$	24.3028	1.26859	0.634297	+	0.773089i	$0.281290\pi$
$368$	0	0
$369$	−2.09167	−0.108888
$370$	0	0
$371$	−9.90833	−0.514415
$372$	0	0
$373$	−1.42221	−0.0736390	−0.0368195	−	0.999322i	$-0.511723\pi$
$373$	−1.42221	−0.0736390	−0.0368195	+	0.999322i	$0.511723\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−49.5416	−2.55152
$378$	0	0
$379$	−24.8167	−1.27475	−0.637373	−	0.770555i	$-0.719979\pi$
$379$	−24.8167	−1.27475	−0.637373	+	0.770555i	$0.719979\pi$
$380$	0	0
$381$	22.3028	1.14261
$382$	0	0
$383$	21.6333	1.10541	0.552705	−	0.833377i	$-0.313596\pi$
$383$	21.6333	1.10541	0.552705	+	0.833377i	$0.313596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.39445	−0.477547
$388$	0	0
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−14.4500	−0.730766
$392$	0	0
$393$	−1.18335	−0.0596919
$394$	0	0
$395$	0	0
$396$	0	0
$397$	25.3028	1.26991	0.634955	−	0.772549i	$-0.281019\pi$
$397$	25.3028	1.26991	0.634955	+	0.772549i	$0.281019\pi$
$398$	0	0
$399$	−5.60555	−0.280629
$400$	0	0
$401$	−27.2111	−1.35886	−0.679429	−	0.733741i	$-0.737772\pi$
$401$	−27.2111	−1.35886	−0.679429	+	0.733741i	$0.737772\pi$
$402$	0	0
$403$	21.0555	1.04885
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.60555	0.476129
$408$	0	0
$409$	8.21110	0.406013	0.203006	−	0.979177i	$-0.434929\pi$
$409$	8.21110	0.406013	0.203006	+	0.979177i	$0.434929\pi$
$410$	0	0
$411$	−2.72498	−0.134413
$412$	0	0
$413$	0.908327	0.0446958
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−10.6972	−0.523845
$418$	0	0
$419$	−13.6056	−0.664675	−0.332337	−	0.943161i	$-0.607837\pi$
$419$	−13.6056	−0.664675	−0.332337	+	0.943161i	$0.607837\pi$
$420$	0	0
$421$	4.30278	0.209704	0.104852	−	0.994488i	$-0.466563\pi$
$421$	4.30278	0.209704	0.104852	+	0.994488i	$0.466563\pi$
$422$	0	0
$423$	−3.90833	−0.190029
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.5139	−0.605589
$428$	0	0
$429$	6.51388	0.314493
$430$	0	0
$431$	−33.0000	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$431$	−33.0000	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.69722	0.176862
$438$	0	0
$439$	−20.6972	−0.987825	−0.493912	−	0.869512i	$-0.664434\pi$
$439$	−20.6972	−0.987825	−0.493912	+	0.869512i	$0.664434\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	1.39445	0.0662523	0.0331261	−	0.999451i	$-0.489454\pi$
$443$	1.39445	0.0662523	0.0331261	+	0.999451i	$0.489454\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.63331	0.171850
$448$	0	0
$449$	41.5139	1.95916	0.979581	−	0.201052i	$-0.0644361\pi$
$449$	41.5139	1.95916	0.979581	+	0.201052i	$0.0644361\pi$
$450$	0	0
$451$	1.60555	0.0756025
$452$	0	0
$453$	27.1194	1.27418
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−24.3028	−1.13684	−0.568418	−	0.822740i	$-0.692444\pi$
$457$	−24.3028	−1.13684	−0.568418	+	0.822740i	$0.692444\pi$
$458$	0	0
$459$	21.9083	1.02259
$460$	0	0
$461$	17.7889	0.828512	0.414256	−	0.910161i	$-0.364042\pi$
$461$	17.7889	0.828512	0.414256	+	0.910161i	$0.364042\pi$
$462$	0	0
$463$	−26.2111	−1.21813	−0.609067	−	0.793119i	$-0.708456\pi$
$463$	−26.2111	−1.21813	−0.609067	+	0.793119i	$0.708456\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.6333	−1.13989	−0.569947	−	0.821682i	$-0.693036\pi$
$467$	−24.6333	−1.13989	−0.569947	+	0.821682i	$0.693036\pi$
$468$	0	0
$469$	−17.2111	−0.794735
$470$	0	0
$471$	6.23886	0.287471
$472$	0	0
$473$	7.21110	0.331567
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.00000	−0.137361
$478$	0	0
$479$	13.1833	0.602362	0.301181	−	0.953567i	$-0.402619\pi$
$479$	13.1833	0.602362	0.301181	+	0.953567i	$0.402619\pi$
$480$	0	0
$481$	48.0278	2.18988
$482$	0	0
$483$	−20.7250	−0.943019
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10.2111	0.462709	0.231355	−	0.972869i	$-0.425684\pi$
$487$	10.2111	0.462709	0.231355	+	0.972869i	$0.425684\pi$
$488$	0	0
$489$	−7.42221	−0.335644
$490$	0	0
$491$	24.2111	1.09263	0.546316	−	0.837579i	$-0.316030\pi$
$491$	24.2111	1.09263	0.546316	+	0.837579i	$0.316030\pi$
$492$	0	0
$493$	38.7250	1.74409
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.8167	0.888898
$498$	0	0
$499$	21.5139	0.963093	0.481547	−	0.876420i	$-0.340075\pi$
$499$	21.5139	0.963093	0.481547	+	0.876420i	$0.340075\pi$
$500$	0	0
$501$	20.0917	0.897630
$502$	0	0
$503$	−16.6056	−0.740405	−0.370202	−	0.928951i	$-0.620712\pi$
$503$	−16.6056	−0.740405	−0.370202	+	0.928951i	$0.620712\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	15.6333	0.694300
$508$	0	0
$509$	−26.3028	−1.16585	−0.582925	−	0.812526i	$-0.698092\pi$
$509$	−26.3028	−1.16585	−0.582925	+	0.812526i	$0.698092\pi$
$510$	0	0
$511$	−12.5139	−0.553581
$512$	0	0
$513$	−5.60555	−0.247491
$514$	0	0
$515$	0	0
$516$	0	0
$517$	3.00000	0.131940
$518$	0	0
$519$	21.9083	0.961669
$520$	0	0
$521$	23.4500	1.02736	0.513681	−	0.857981i	$-0.328282\pi$
$521$	23.4500	1.02736	0.513681	+	0.857981i	$0.328282\pi$
$522$	0	0
$523$	3.57779	0.156446	0.0782230	−	0.996936i	$-0.475075\pi$
$523$	3.57779	0.156446	0.0782230	+	0.996936i	$0.475075\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.4584	−0.716938
$528$	0	0
$529$	−9.33053	−0.405675
$530$	0	0
$531$	0.275019	0.0119348
$532$	0	0
$533$	8.02776	0.347721
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.18335	0.309984
$538$	0	0
$539$	11.5139	0.495938
$540$	0	0
$541$	−6.72498	−0.289130	−0.144565	−	0.989495i	$-0.546178\pi$
$541$	−6.72498	−0.289130	−0.144565	+	0.989495i	$0.546178\pi$
$542$	0	0
$543$	−11.8444	−0.508292
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.1194	−0.774731	−0.387365	−	0.921926i	$-0.626615\pi$
$547$	−18.1194	−0.774731	−0.387365	+	0.921926i	$0.626615\pi$
$548$	0	0
$549$	−3.78890	−0.161706
$550$	0	0
$551$	−9.90833	−0.422109
$552$	0	0
$553$	−0.394449	−0.0167737
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.42221	−0.399232	−0.199616	−	0.979874i	$-0.563969\pi$
$557$	−9.42221	−0.399232	−0.199616	+	0.979874i	$0.563969\pi$
$558$	0	0
$559$	36.0555	1.52499
$560$	0	0
$561$	−5.09167	−0.214971
$562$	0	0
$563$	−18.9083	−0.796891	−0.398445	−	0.917192i	$-0.630450\pi$
$563$	−18.9083	−0.796891	−0.398445	+	0.917192i	$0.630450\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	14.6056	0.613375
$568$	0	0
$569$	15.1472	0.635003	0.317502	−	0.948258i	$-0.397156\pi$
$569$	15.1472	0.635003	0.317502	+	0.948258i	$0.397156\pi$
$570$	0	0
$571$	17.3305	0.725260	0.362630	−	0.931933i	$-0.381879\pi$
$571$	17.3305	0.725260	0.362630	+	0.931933i	$0.381879\pi$
$572$	0	0
$573$	−8.72498	−0.364491
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.3583	−1.30546	−0.652731	−	0.757589i	$-0.726377\pi$
$577$	−31.3583	−1.30546	−0.652731	+	0.757589i	$0.726377\pi$
$578$	0	0
$579$	−1.57779	−0.0655709
$580$	0	0
$581$	62.4500	2.59086
$582$	0	0
$583$	2.30278	0.0953712
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.5416	1.54951	0.774755	−	0.632262i	$-0.217873\pi$
$587$	37.5416	1.54951	0.774755	+	0.632262i	$0.217873\pi$
$588$	0	0
$589$	4.21110	0.173515
$590$	0	0
$591$	12.6333	0.519665
$592$	0	0
$593$	−13.6056	−0.558713	−0.279357	−	0.960187i	$-0.590121\pi$
$593$	−13.6056	−0.558713	−0.279357	+	0.960187i	$0.590121\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.9361	1.30706
$598$	0	0
$599$	14.0917	0.575770	0.287885	−	0.957665i	$-0.407048\pi$
$599$	14.0917	0.575770	0.287885	+	0.957665i	$0.407048\pi$
$600$	0	0
$601$	8.90833	0.363378	0.181689	−	0.983356i	$-0.441844\pi$
$601$	8.90833	0.363378	0.181689	+	0.983356i	$0.441844\pi$
$602$	0	0
$603$	−5.21110	−0.212213
$604$	0	0
$605$	0	0
$606$	0	0
$607$	7.21110	0.292690	0.146345	−	0.989234i	$-0.453249\pi$
$607$	7.21110	0.292690	0.146345	+	0.989234i	$0.453249\pi$
$608$	0	0
$609$	55.5416	2.25066
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	−41.1194	−1.66080	−0.830399	−	0.557169i	$-0.811888\pi$
$613$	−41.1194	−1.66080	−0.830399	+	0.557169i	$0.811888\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	10.6056	0.426963	0.213482	−	0.976947i	$-0.431520\pi$
$617$	10.6056	0.426963	0.213482	+	0.976947i	$0.431520\pi$
$618$	0	0
$619$	−17.4222	−0.700258	−0.350129	−	0.936702i	$-0.613862\pi$
$619$	−17.4222	−0.700258	−0.350129	+	0.936702i	$0.613862\pi$
$620$	0	0
$621$	−20.7250	−0.831665
$622$	0	0
$623$	−22.8167	−0.914130
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.30278	0.0520278
$628$	0	0
$629$	−37.5416	−1.49688
$630$	0	0
$631$	39.9361	1.58983	0.794915	−	0.606721i	$-0.207515\pi$
$631$	39.9361	1.58983	0.794915	+	0.606721i	$0.207515\pi$
$632$	0	0
$633$	−32.8806	−1.30689
$634$	0	0
$635$	0	0
$636$	0	0
$637$	57.5694	2.28098
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	42.2111	1.66724	0.833619	−	0.552340i	$-0.186265\pi$
$641$	42.2111	1.66724	0.833619	+	0.552340i	$0.186265\pi$
$642$	0	0
$643$	22.0000	0.867595	0.433798	−	0.901010i	$-0.357173\pi$
$643$	22.0000	0.867595	0.433798	+	0.901010i	$0.357173\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.2389	−0.677729	−0.338865	−	0.940835i	$-0.610043\pi$
$647$	−17.2389	−0.677729	−0.338865	+	0.940835i	$0.610043\pi$
$648$	0	0
$649$	−0.211103	−0.00828650
$650$	0	0
$651$	−23.6056	−0.925174
$652$	0	0
$653$	−19.1194	−0.748201	−0.374101	−	0.927388i	$-0.622049\pi$
$653$	−19.1194	−0.748201	−0.374101	+	0.927388i	$0.622049\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.78890	−0.147819
$658$	0	0
$659$	20.0917	0.782660	0.391330	−	0.920250i	$-0.372015\pi$
$659$	20.0917	0.782660	0.391330	+	0.920250i	$0.372015\pi$
$660$	0	0
$661$	12.8167	0.498510	0.249255	−	0.968438i	$-0.419814\pi$
$661$	12.8167	0.498510	0.249255	+	0.968438i	$0.419814\pi$
$662$	0	0
$663$	−25.4584	−0.988721
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.6333	−1.41845
$668$	0	0
$669$	−26.8806	−1.03926
$670$	0	0
$671$	2.90833	0.112275
$672$	0	0
$673$	−6.02776	−0.232353	−0.116176	−	0.993229i	$-0.537064\pi$
$673$	−6.02776	−0.232353	−0.116176	+	0.993229i	$0.537064\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.2389	−1.00844	−0.504221	−	0.863575i	$-0.668220\pi$
$677$	−26.2389	−1.00844	−0.504221	+	0.863575i	$0.668220\pi$
$678$	0	0
$679$	−50.3305	−1.93151
$680$	0	0
$681$	−6.90833	−0.264728
$682$	0	0
$683$	9.84441	0.376686	0.188343	−	0.982103i	$-0.439688\pi$
$683$	9.84441	0.376686	0.188343	+	0.982103i	$0.439688\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	17.8806	0.682186
$688$	0	0
$689$	11.5139	0.438644
$690$	0	0
$691$	26.5416	1.00969	0.504846	−	0.863210i	$-0.331549\pi$
$691$	26.5416	1.00969	0.504846	+	0.863210i	$0.331549\pi$
$692$	0	0
$693$	5.60555	0.212937
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−6.27502	−0.237683
$698$	0	0
$699$	6.63331	0.250895
$700$	0	0
$701$	−26.7889	−1.01180	−0.505901	−	0.862591i	$-0.668840\pi$
$701$	−26.7889	−1.01180	−0.505901	+	0.862591i	$0.668840\pi$
$702$	0	0
$703$	9.60555	0.362280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−75.3583	−2.83414
$708$	0	0
$709$	11.6333	0.436898	0.218449	−	0.975848i	$-0.429900\pi$
$709$	11.6333	0.436898	0.218449	+	0.975848i	$0.429900\pi$
$710$	0	0
$711$	−0.119429	−0.00447895
$712$	0	0
$713$	15.5694	0.583078
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.36669	0.200423
$718$	0	0
$719$	−28.8167	−1.07468	−0.537340	−	0.843366i	$-0.680571\pi$
$719$	−28.8167	−1.07468	−0.537340	+	0.843366i	$0.680571\pi$
$720$	0	0
$721$	−34.0278	−1.26726
$722$	0	0
$723$	−32.4861	−1.20817
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.330532	−0.0122588	−0.00612938	−	0.999981i	$-0.501951\pi$
$727$	−0.330532	−0.0122588	−0.00612938	+	0.999981i	$0.501951\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	−28.1833	−1.04240
$732$	0	0
$733$	12.3944	0.457799	0.228900	−	0.973450i	$-0.426487\pi$
$733$	12.3944	0.457799	0.228900	+	0.973450i	$0.426487\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−9.88057	−0.363463	−0.181731	−	0.983348i	$-0.558170\pi$
$739$	−9.88057	−0.363463	−0.181731	+	0.983348i	$0.558170\pi$
$740$	0	0
$741$	6.51388	0.239293
$742$	0	0
$743$	−44.3028	−1.62531	−0.812656	−	0.582744i	$-0.801979\pi$
$743$	−44.3028	−1.62531	−0.812656	+	0.582744i	$0.801979\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	18.9083	0.691820
$748$	0	0
$749$	−12.9083	−0.471660
$750$	0	0
$751$	5.66947	0.206882	0.103441	−	0.994636i	$-0.467015\pi$
$751$	5.66947	0.206882	0.103441	+	0.994636i	$0.467015\pi$
$752$	0	0
$753$	5.09167	0.185551
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−23.0555	−0.837967	−0.418983	−	0.907994i	$-0.637613\pi$
$757$	−23.0555	−0.837967	−0.418983	+	0.907994i	$0.637613\pi$
$758$	0	0
$759$	4.81665	0.174833
$760$	0	0
$761$	−42.4222	−1.53780	−0.768902	−	0.639367i	$-0.779197\pi$
$761$	−42.4222	−1.53780	−0.768902	+	0.639367i	$0.779197\pi$
$762$	0	0
$763$	28.0278	1.01467
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.05551	−0.0381124
$768$	0	0
$769$	−26.8167	−0.967033	−0.483517	−	0.875335i	$-0.660641\pi$
$769$	−26.8167	−0.967033	−0.483517	+	0.875335i	$0.660641\pi$
$770$	0	0
$771$	−23.4500	−0.844530
$772$	0	0
$773$	−22.1194	−0.795581	−0.397790	−	0.917476i	$-0.630223\pi$
$773$	−22.1194	−0.795581	−0.397790	+	0.917476i	$0.630223\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−53.8444	−1.93166
$778$	0	0
$779$	1.60555	0.0575248
$780$	0	0
$781$	−4.60555	−0.164800
$782$	0	0
$783$	55.5416	1.98490
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.21110	0.150110	0.0750548	−	0.997179i	$-0.476087\pi$
$787$	4.21110	0.150110	0.0750548	+	0.997179i	$0.476087\pi$
$788$	0	0
$789$	1.54163	0.0548836
$790$	0	0
$791$	−46.5416	−1.65483
$792$	0	0
$793$	14.5416	0.516389
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.5139	−0.514108	−0.257054	−	0.966397i	$-0.582752\pi$
$797$	−14.5139	−0.514108	−0.257054	+	0.966397i	$0.582752\pi$
$798$	0	0
$799$	−11.7250	−0.414800
$800$	0	0
$801$	−6.90833	−0.244094
$802$	0	0
$803$	2.90833	0.102633
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.9083	−1.08802
$808$	0	0
$809$	−3.63331	−0.127740	−0.0638701	−	0.997958i	$-0.520344\pi$
$809$	−3.63331	−0.127740	−0.0638701	+	0.997958i	$0.520344\pi$
$810$	0	0
$811$	54.8722	1.92682	0.963411	−	0.268028i	$-0.0863719\pi$
$811$	54.8722	1.92682	0.963411	+	0.268028i	$0.0863719\pi$
$812$	0	0
$813$	−18.5139	−0.649310
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110	0.252285
$818$	0	0
$819$	28.0278	0.979369
$820$	0	0
$821$	12.0000	0.418803	0.209401	−	0.977830i	$-0.432848\pi$
$821$	12.0000	0.418803	0.209401	+	0.977830i	$0.432848\pi$
$822$	0	0
$823$	10.4222	0.363295	0.181648	−	0.983364i	$-0.441857\pi$
$823$	10.4222	0.363295	0.181648	+	0.983364i	$0.441857\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.81665	0.271812	0.135906	−	0.990722i	$-0.456606\pi$
$827$	7.81665	0.271812	0.135906	+	0.990722i	$0.456606\pi$
$828$	0	0
$829$	−38.7527	−1.34594	−0.672969	−	0.739671i	$-0.734981\pi$
$829$	−38.7527	−1.34594	−0.672969	+	0.739671i	$0.734981\pi$
$830$	0	0
$831$	28.1472	0.976415
$832$	0	0
$833$	−45.0000	−1.55916
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−23.6056	−0.815927
$838$	0	0
$839$	−16.1194	−0.556505	−0.278252	−	0.960508i	$-0.589755\pi$
$839$	−16.1194	−0.556505	−0.278252	+	0.960508i	$0.589755\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	0	0
$843$	−29.7250	−1.02378
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.30278	−0.147845
$848$	0	0
$849$	−3.51388	−0.120596
$850$	0	0
$851$	35.5139	1.21740
$852$	0	0
$853$	19.7250	0.675370	0.337685	−	0.941259i	$-0.390356\pi$
$853$	19.7250	0.675370	0.337685	+	0.941259i	$0.390356\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	−48.6056	−1.65840	−0.829200	−	0.558952i	$-0.811204\pi$
$859$	−48.6056	−1.65840	−0.829200	+	0.558952i	$0.811204\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	19.6056	0.667381	0.333690	−	0.942683i	$-0.391706\pi$
$863$	19.6056	0.667381	0.333690	+	0.942683i	$0.391706\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.24726	−0.0763210
$868$	0	0
$869$	0.0916731	0.00310980
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	−15.2389	−0.515757
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.0000	0.675352	0.337676	−	0.941262i	$-0.390359\pi$
$877$	20.0000	0.675352	0.337676	+	0.941262i	$0.390359\pi$
$878$	0	0
$879$	19.8167	0.668399
$880$	0	0
$881$	−34.5416	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$881$	−34.5416	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$882$	0	0
$883$	12.4500	0.418975	0.209487	−	0.977811i	$-0.432821\pi$
$883$	12.4500	0.418975	0.209487	+	0.977811i	$0.432821\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.2389	−1.58613	−0.793063	−	0.609140i	$-0.791515\pi$
$887$	−47.2389	−1.58613	−0.793063	+	0.609140i	$0.791515\pi$
$888$	0	0
$889$	−73.6611	−2.47051
$890$	0	0
$891$	−3.39445	−0.113718
$892$	0	0
$893$	3.00000	0.100391
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.0833	0.804117
$898$	0	0
$899$	−41.7250	−1.39161
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	−40.4222	−1.34517
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	0	0
$909$	−22.8167	−0.756781
$910$	0	0
$911$	39.2111	1.29912	0.649561	−	0.760310i	$-0.274953\pi$
$911$	39.2111	1.29912	0.649561	+	0.760310i	$0.274953\pi$
$912$	0	0
$913$	−14.5139	−0.480339
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.90833	0.129064
$918$	0	0
$919$	−41.2111	−1.35943	−0.679714	−	0.733477i	$-0.737896\pi$
$919$	−41.2111	−1.35943	−0.679714	+	0.733477i	$0.737896\pi$
$920$	0	0
$921$	7.93608	0.261503
$922$	0	0
$923$	−23.0278	−0.757968
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−10.3028	−0.338388
$928$	0	0
$929$	46.3944	1.52215	0.761076	−	0.648662i	$-0.224671\pi$
$929$	46.3944	1.52215	0.761076	+	0.648662i	$0.224671\pi$
$930$	0	0
$931$	11.5139	0.377352
$932$	0	0
$933$	21.9083	0.717246
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.21110	0.170239	0.0851196	−	0.996371i	$-0.472873\pi$
$937$	5.21110	0.170239	0.0851196	+	0.996371i	$0.472873\pi$
$938$	0	0
$939$	28.4222	0.927524
$940$	0	0
$941$	52.3944	1.70801	0.854005	−	0.520265i	$-0.174167\pi$
$941$	52.3944	1.70801	0.854005	+	0.520265i	$0.174167\pi$
$942$	0	0
$943$	5.93608	0.193305
$944$	0	0
$945$	0	0
$946$	0	0
$947$	36.6333	1.19042	0.595211	−	0.803569i	$-0.297068\pi$
$947$	36.6333	1.19042	0.595211	+	0.803569i	$0.297068\pi$
$948$	0	0
$949$	14.5416	0.472041
$950$	0	0
$951$	12.9083	0.418581
$952$	0	0
$953$	−49.2666	−1.59590	−0.797951	−	0.602722i	$-0.794083\pi$
$953$	−49.2666	−1.59590	−0.797951	+	0.602722i	$0.794083\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−12.9083	−0.417267
$958$	0	0
$959$	9.00000	0.290625
$960$	0	0
$961$	−13.2666	−0.427955
$962$	0	0
$963$	−3.90833	−0.125944
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.09167	−0.131579	−0.0657897	−	0.997834i	$-0.520957\pi$
$967$	−4.09167	−0.131579	−0.0657897	+	0.997834i	$0.520957\pi$
$968$	0	0
$969$	−5.09167	−0.163568
$970$	0	0
$971$	30.3583	0.974244	0.487122	−	0.873334i	$-0.338047\pi$
$971$	30.3583	0.974244	0.487122	+	0.873334i	$0.338047\pi$
$972$	0	0
$973$	35.3305	1.13264
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.9722	−0.510997	−0.255499	−	0.966809i	$-0.582240\pi$
$977$	−15.9722	−0.510997	−0.255499	+	0.966809i	$0.582240\pi$
$978$	0	0
$979$	5.30278	0.169477
$980$	0	0
$981$	8.48612	0.270941
$982$	0	0
$983$	48.8444	1.55789	0.778947	−	0.627089i	$-0.215754\pi$
$983$	48.8444	1.55789	0.778947	+	0.627089i	$0.215754\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−16.8167	−0.535280
$988$	0	0
$989$	26.6611	0.847773
$990$	0	0
$991$	6.09167	0.193508	0.0967542	−	0.995308i	$-0.469154\pi$
$991$	6.09167	0.193508	0.0967542	+	0.995308i	$0.469154\pi$
$992$	0	0
$993$	18.7527	0.595100
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.2750	0.452094	0.226047	−	0.974116i	$-0.427420\pi$
$997$	14.2750	0.452094	0.226047	+	0.974116i	$0.427420\pi$
$998$	0	0
$999$	−53.8444	−1.70356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bh.1.2		2
4.3	odd	2		275.2.a.f.1.2	yes	2
5.2	odd	4		4400.2.b.y.4049.2		4
5.3	odd	4		4400.2.b.y.4049.3		4
5.4	even	2		4400.2.a.bs.1.1		2
12.11	even	2		2475.2.a.o.1.1		2
20.3	even	4		275.2.b.c.199.1		4
20.7	even	4		275.2.b.c.199.4		4
20.19	odd	2		275.2.a.e.1.1	✓	2
44.43	even	2		3025.2.a.h.1.1		2
60.23	odd	4		2475.2.c.k.199.4		4
60.47	odd	4		2475.2.c.k.199.1		4
60.59	even	2		2475.2.a.t.1.2		2
220.219	even	2		3025.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	20.19	odd	2
275.2.a.f.1.2	yes	2	4.3	odd	2
275.2.b.c.199.1		4	20.3	even	4
275.2.b.c.199.4		4	20.7	even	4
2475.2.a.o.1.1		2	12.11	even	2
2475.2.a.t.1.2		2	60.59	even	2
2475.2.c.k.199.1		4	60.47	odd	4
2475.2.c.k.199.4		4	60.23	odd	4
3025.2.a.h.1.1		2	44.43	even	2
3025.2.a.n.1.2		2	220.219	even	2
4400.2.a.bh.1.2		2	1.1	even	1	trivial
4400.2.a.bs.1.1		2	5.4	even	2
4400.2.b.y.4049.2		4	5.2	odd	4
4400.2.b.y.4049.3		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{3} -4.30278 q^{7} -1.30278 q^{9} +1.00000 q^{11} +5.00000 q^{13} -3.90833 q^{17} +1.00000 q^{19} -5.60555 q^{21} +3.69722 q^{23} -5.60555 q^{27} -9.90833 q^{29} +4.21110 q^{31} +1.30278 q^{33} +9.60555 q^{37} +6.51388 q^{39} +1.60555 q^{41} +7.21110 q^{43} +3.00000 q^{47} +11.5139 q^{49} -5.09167 q^{51} +2.30278 q^{53} +1.30278 q^{57} -0.211103 q^{59} +2.90833 q^{61} +5.60555 q^{63} +4.00000 q^{67} +4.81665 q^{69} -4.60555 q^{71} +2.90833 q^{73} -4.30278 q^{77} +0.0916731 q^{79} -3.39445 q^{81} -14.5139 q^{83} -12.9083 q^{87} +5.30278 q^{89} -21.5139 q^{91} +5.48612 q^{93} +11.6972 q^{97} -1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 5 q^{7} + q^{9} + 2 q^{11} + 10 q^{13} + 3 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} - 4 q^{27} - 9 q^{29} - 6 q^{31} - q^{33} + 12 q^{37} - 5 q^{39} - 4 q^{41} + 6 q^{47} + 5 q^{49} - 21 q^{51}+ \cdots + q^{99}+O(q^{100}) \) 2 * q - q^3 - 5 * q^7 + q^9 + 2 * q^11 + 10 * q^13 + 3 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 - 4 * q^27 - 9 * q^29 - 6 * q^31 - q^33 + 12 * q^37 - 5 * q^39 - 4 * q^41 + 6 * q^47 + 5 * q^49 - 21 * q^51 + q^53 - q^57 + 14 * q^59 - 5 * q^61 + 4 * q^63 + 8 * q^67 - 12 * q^69 - 2 * q^71 - 5 * q^73 - 5 * q^77 + 11 * q^79 - 14 * q^81 - 11 * q^83 - 15 * q^87 + 7 * q^89 - 25 * q^91 + 29 * q^93 + 27 * q^97 + q^99

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