LMFDB - Embedded newform 4400.2.b.y.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, 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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.y.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +O(q^{10})$	$q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +1.00000 q^{11} +5.00000i q^{13} +3.90833i q^{17} -1.00000 q^{19} -5.60555 q^{21} +3.69722i q^{23} +5.60555i q^{27} +9.90833 q^{29} +4.21110 q^{31} +1.30278i q^{33} -9.60555i q^{37} -6.51388 q^{39} +1.60555 q^{41} +7.21110i q^{43} -3.00000i q^{47} -11.5139 q^{49} -5.09167 q^{51} +2.30278i q^{53} -1.30278i q^{57} +0.211103 q^{59} +2.90833 q^{61} +5.60555i q^{63} -4.00000i q^{67} -4.81665 q^{69} -4.60555 q^{71} +2.90833i q^{73} +4.30278i q^{77} -0.0916731 q^{79} -3.39445 q^{81} -14.5139i q^{83} +12.9083i q^{87} -5.30278 q^{89} -21.5139 q^{91} +5.48612i q^{93} -11.6972i q^{97} +1.30278 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^9	$ 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 - 8 * q^21 + 18 * q^29 - 12 * q^31 + 10 * q^39 - 8 * q^41 - 10 * q^49 - 42 * q^51 - 28 * q^59 - 10 * q^61 + 24 * q^69 - 4 * q^71 - 22 * q^79 - 28 * q^81 - 14 * q^89 - 50 * q^91 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	1.30278i	0.752158i	0.926588	+	0.376079i	$0.122728\pi$
$3$	1.30278i	0.752158i	−0.926588	+	0.376079i	$0.877272\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.30278i	1.62630i	0.582057	+	0.813148i	$0.302248\pi$
$7$	4.30278i	1.62630i	−0.582057	+	0.813148i	$0.697752\pi$
$8$	0	0
$9$	1.30278	0.434259
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.90833i	0.947909i	0.880549	+	0.473954i	$0.157174\pi$
$17$	3.90833i	0.947909i	−0.880549	+	0.473954i	$0.842826\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−5.60555	−1.22323
$22$	0	0
$23$	3.69722i	0.770925i	0.922724	+	0.385462i	$0.125958\pi$
$23$	3.69722i	0.770925i	−0.922724	+	0.385462i	$0.874042\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.60555i	1.07879i
$28$	0	0
$29$	9.90833	1.83993	0.919965	−	0.392000i	$-0.128217\pi$
$29$	9.90833	1.83993	0.919965	+	0.392000i	$0.128217\pi$
$30$	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
$33$	1.30278i	0.226784i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.60555i	− 1.57914i	−0.613659	−	0.789571i	$-0.710303\pi$
$37$	− 9.60555i	− 1.57914i	0.613659	−	0.789571i	$-0.289697\pi$
$38$	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
$43$	7.21110i	1.09968i	0.835269	+	0.549841i	$0.185312\pi$
$43$	7.21110i	1.09968i	−0.835269	+	0.549841i	$0.814688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	0	0
$49$	−11.5139	−1.64484
$50$	0	0
$51$	−5.09167	−0.712977
$52$	0	0
$53$	2.30278i	0.316311i	0.987414	+	0.158155i	$0.0505547\pi$
$53$	2.30278i	0.316311i	−0.987414	+	0.158155i	$0.949445\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.30278i	− 0.172557i
$58$	0	0
$59$	0.211103	0.0274832	0.0137416	−	0.999906i	$-0.495626\pi$
$59$	0.211103	0.0274832	0.0137416	+	0.999906i	$0.495626\pi$
$60$	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
$63$	5.60555i	0.706233i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	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
$73$	2.90833i	0.340394i	0.985410	+	0.170197i	$0.0544404\pi$
$73$	2.90833i	0.340394i	−0.985410	+	0.170197i	$0.945560\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.30278i	0.490347i
$78$	0	0
$79$	−0.0916731	−0.0103140	−0.00515701	−	0.999987i	$-0.501642\pi$
$79$	−0.0916731	−0.0103140	−0.00515701	+	0.999987i	$0.501642\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	− 14.5139i	− 1.59311i	−0.604569	−	0.796553i	$-0.706655\pi$
$83$	− 14.5139i	− 1.59311i	0.604569	−	0.796553i	$-0.293345\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.9083i	1.38392i
$88$	0	0
$89$	−5.30278	−0.562093	−0.281047	−	0.959694i	$-0.590682\pi$
$89$	−5.30278	−0.562093	−0.281047	+	0.959694i	$0.590682\pi$
$90$	0	0
$91$	−21.5139	−2.25527
$92$	0	0
$93$	5.48612i	0.568884i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.6972i	− 1.18767i	−0.804586	−	0.593837i	$-0.797613\pi$
$97$	− 11.6972i	− 1.18767i	0.804586	−	0.593837i	$-0.202387\pi$
$98$	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
$103$	7.90833i	0.779231i	0.920978	+	0.389615i	$0.127392\pi$
$103$	7.90833i	0.779231i	−0.920978	+	0.389615i	$0.872608\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	0	0
$109$	6.51388	0.623916	0.311958	−	0.950096i	$-0.399015\pi$
$109$	6.51388	0.623916	0.311958	+	0.950096i	$0.399015\pi$
$110$	0	0
$111$	12.5139	1.18776
$112$	0	0
$113$	10.8167i	1.01755i	0.860901	+	0.508773i	$0.169901\pi$
$113$	10.8167i	1.01755i	−0.860901	+	0.508773i	$0.830099\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.51388i	0.602208i
$118$	0	0
$119$	−16.8167	−1.54158
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.09167i	0.188600i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 17.1194i	− 1.51910i	−0.650447	−	0.759552i	$-0.725418\pi$
$127$	− 17.1194i	− 1.51910i	0.650447	−	0.759552i	$-0.274582\pi$
$128$	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
$133$	− 4.30278i	− 0.373098i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.09167i	0.178704i	0.996000	+	0.0893518i	$0.0284796\pi$
$137$	2.09167i	0.178704i	−0.996000	+	0.0893518i	$0.971520\pi$
$138$	0	0
$139$	8.21110	0.696457	0.348228	−	0.937410i	$-0.386783\pi$
$139$	8.21110	0.696457	0.348228	+	0.937410i	$0.386783\pi$
$140$	0	0
$141$	3.90833	0.329141
$142$	0	0
$143$	5.00000i	0.418121i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 15.0000i	− 1.23718i
$148$	0	0
$149$	−2.78890	−0.228475	−0.114238	−	0.993453i	$-0.536443\pi$
$149$	−2.78890	−0.228475	−0.114238	+	0.993453i	$0.536443\pi$
$150$	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
$153$	5.09167i	0.411637i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.78890i	− 0.382196i	−0.981571	−	0.191098i	$-0.938795\pi$
$157$	− 4.78890i	− 0.382196i	0.981571	−	0.191098i	$-0.0612048\pi$
$158$	0	0
$159$	−3.00000	−0.237915
$160$	0	0
$161$	−15.9083	−1.25375
$162$	0	0
$163$	− 5.69722i	− 0.446241i	−0.974791	−	0.223121i	$-0.928376\pi$
$163$	− 5.69722i	− 0.446241i	0.974791	−	0.223121i	$-0.0716243\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 15.4222i	− 1.19341i	−0.802462	−	0.596703i	$-0.796477\pi$
$167$	− 15.4222i	− 1.19341i	0.802462	−	0.596703i	$-0.203523\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.30278	−0.0996257
$172$	0	0
$173$	16.8167i	1.27855i	0.768980	+	0.639273i	$0.220765\pi$
$173$	16.8167i	1.27855i	−0.768980	+	0.639273i	$0.779235\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.275019i	0.0206717i
$178$	0	0
$179$	−5.51388	−0.412127	−0.206063	−	0.978539i	$-0.566065\pi$
$179$	−5.51388	−0.412127	−0.206063	+	0.978539i	$0.566065\pi$
$180$	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
$183$	3.78890i	0.280083i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.90833i	0.285805i
$188$	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
$193$	− 1.21110i	− 0.0871771i	−0.999050	−	0.0435885i	$-0.986121\pi$
$193$	− 1.21110i	− 0.0871771i	0.999050	−	0.0435885i	$-0.0138791\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 9.69722i	− 0.690899i	−0.938437	−	0.345449i	$-0.887727\pi$
$197$	− 9.69722i	− 0.690899i	0.938437	−	0.345449i	$-0.112273\pi$
$198$	0	0
$199$	−24.5139	−1.73774	−0.868871	−	0.495038i	$-0.835154\pi$
$199$	−24.5139	−1.73774	−0.868871	+	0.495038i	$0.835154\pi$
$200$	0	0
$201$	5.21110	0.367563
$202$	0	0
$203$	42.6333i	2.99227i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.81665i	0.334781i
$208$	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
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	18.1194i	1.23003i
$218$	0	0
$219$	−3.78890	−0.256030
$220$	0	0
$221$	−19.5416	−1.31451
$222$	0	0
$223$	− 20.6333i	− 1.38171i	−0.722994	−	0.690854i	$-0.757235\pi$
$223$	− 20.6333i	− 1.38171i	0.722994	−	0.690854i	$-0.242765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.30278i	0.351958i	0.984394	+	0.175979i	$0.0563090\pi$
$227$	5.30278i	0.351958i	−0.984394	+	0.175979i	$0.943691\pi$
$228$	0	0
$229$	−13.7250	−0.906972	−0.453486	−	0.891263i	$-0.649820\pi$
$229$	−13.7250	−0.906972	−0.453486	+	0.891263i	$0.649820\pi$
$230$	0	0
$231$	−5.60555	−0.368818
$232$	0	0
$233$	5.09167i	0.333567i	0.985994	+	0.166783i	$0.0533380\pi$
$233$	5.09167i	0.333567i	−0.985994	+	0.166783i	$0.946662\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 0.119429i	− 0.00775778i
$238$	0	0
$239$	−4.11943	−0.266464	−0.133232	−	0.991085i	$-0.542535\pi$
$239$	−4.11943	−0.266464	−0.133232	+	0.991085i	$0.542535\pi$
$240$	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
$243$	12.3944i	0.795104i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.00000i	− 0.318142i
$248$	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
$253$	3.69722i	0.232443i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	41.3305	2.56815
$260$	0	0
$261$	12.9083	0.799005
$262$	0	0
$263$	1.18335i	0.0729683i	0.999334	+	0.0364841i	$0.0116158\pi$
$263$	1.18335i	0.0729683i	−0.999334	+	0.0364841i	$0.988384\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6.90833i	− 0.422783i
$268$	0	0
$269$	23.7250	1.44654	0.723269	−	0.690567i	$-0.242639\pi$
$269$	23.7250	1.44654	0.723269	+	0.690567i	$0.242639\pi$
$270$	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
$273$	− 28.0278i	− 1.69632i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 21.6056i	− 1.29815i	−0.760724	−	0.649076i	$-0.775156\pi$
$277$	− 21.6056i	− 1.29815i	0.760724	−	0.649076i	$-0.224844\pi$
$278$	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
$283$	− 2.69722i	− 0.160333i	−0.996781	−	0.0801667i	$-0.974455\pi$
$283$	− 2.69722i	− 0.160333i	0.996781	−	0.0801667i	$-0.0255453\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.90833i	0.407786i
$288$	0	0
$289$	1.72498	0.101469
$290$	0	0
$291$	15.2389	0.893318
$292$	0	0
$293$	15.2111i	0.888642i	0.895868	+	0.444321i	$0.146555\pi$
$293$	15.2111i	0.888642i	−0.895868	+	0.444321i	$0.853445\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.60555i	0.325267i
$298$	0	0
$299$	−18.4861	−1.06908
$300$	0	0
$301$	−31.0278	−1.78841
$302$	0	0
$303$	22.8167i	1.31078i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6.09167i	− 0.347670i	−0.984775	−	0.173835i	$-0.944384\pi$
$307$	− 6.09167i	− 0.347670i	0.984775	−	0.173835i	$-0.0556160\pi$
$308$	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
$313$	21.8167i	1.23315i	0.787296	+	0.616575i	$0.211480\pi$
$313$	21.8167i	1.23315i	−0.787296	+	0.616575i	$0.788520\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9.90833i	− 0.556507i	−0.960508	−	0.278254i	$-0.910244\pi$
$317$	− 9.90833i	− 0.556507i	0.960508	−	0.278254i	$-0.0897555\pi$
$318$	0	0
$319$	9.90833	0.554760
$320$	0	0
$321$	3.90833	0.218142
$322$	0	0
$323$	− 3.90833i	− 0.217465i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.48612i	0.469284i
$328$	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
$333$	− 12.5139i	− 0.685756i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 26.8444i	− 1.46231i	−0.682212	−	0.731154i	$-0.738982\pi$
$337$	− 26.8444i	− 1.46231i	0.682212	−	0.731154i	$-0.261018\pi$
$338$	0	0
$339$	−14.0917	−0.765355
$340$	0	0
$341$	4.21110	0.228044
$342$	0	0
$343$	− 19.4222i	− 1.04870i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.51388i	0.296000i	0.988987	+	0.148000i	$0.0472836\pi$
$347$	5.51388i	0.296000i	−0.988987	+	0.148000i	$0.952716\pi$
$348$	0	0
$349$	26.8167	1.43546	0.717731	−	0.696320i	$-0.245181\pi$
$349$	26.8167	1.43546	0.717731	+	0.696320i	$0.245181\pi$
$350$	0	0
$351$	−28.0278	−1.49601
$352$	0	0
$353$	− 24.6333i	− 1.31110i	−0.755152	−	0.655549i	$-0.772437\pi$
$353$	− 24.6333i	− 1.31110i	0.755152	−	0.655549i	$-0.227563\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 21.9083i	− 1.15951i
$358$	0	0
$359$	15.2111	0.802811	0.401406	−	0.915900i	$-0.368522\pi$
$359$	15.2111	0.802811	0.401406	+	0.915900i	$0.368522\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	1.30278i	0.0683780i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 24.3028i	− 1.26859i	−0.773089	−	0.634297i	$-0.781290\pi$
$367$	− 24.3028i	− 1.26859i	0.773089	−	0.634297i	$-0.218710\pi$
$368$	0	0
$369$	2.09167	0.108888
$370$	0	0
$371$	−9.90833	−0.514415
$372$	0	0
$373$	− 1.42221i	− 0.0736390i	−0.999322	−	0.0368195i	$-0.988277\pi$
$373$	− 1.42221i	− 0.0736390i	0.999322	−	0.0368195i	$-0.0117227\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	49.5416i	2.55152i
$378$	0	0
$379$	24.8167	1.27475	0.637373	−	0.770555i	$-0.280021\pi$
$379$	24.8167	1.27475	0.637373	+	0.770555i	$0.280021\pi$
$380$	0	0
$381$	22.3028	1.14261
$382$	0	0
$383$	21.6333i	1.10541i	0.833377	+	0.552705i	$0.186404\pi$
$383$	21.6333i	1.10541i	−0.833377	+	0.552705i	$0.813596\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.39445i	0.477547i
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−14.4500	−0.730766
$392$	0	0
$393$	− 1.18335i	− 0.0596919i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 25.3028i	− 1.26991i	−0.772549	−	0.634955i	$-0.781019\pi$
$397$	− 25.3028i	− 1.26991i	0.772549	−	0.634955i	$-0.218981\pi$
$398$	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
$403$	21.0555i	1.04885i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9.60555i	− 0.476129i
$408$	0	0
$409$	−8.21110	−0.406013	−0.203006	−	0.979177i	$-0.565071\pi$
$409$	−8.21110	−0.406013	−0.203006	+	0.979177i	$0.565071\pi$
$410$	0	0
$411$	−2.72498	−0.134413
$412$	0	0
$413$	0.908327i	0.0446958i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.6972i	0.523845i
$418$	0	0
$419$	13.6056	0.664675	0.332337	−	0.943161i	$-0.392163\pi$
$419$	13.6056	0.664675	0.332337	+	0.943161i	$0.392163\pi$
$420$	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
$423$	− 3.90833i	− 0.190029i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	12.5139i	0.605589i
$428$	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
$433$	5.00000i	0.240285i	0.992757	+	0.120142i	$0.0383351\pi$
$433$	5.00000i	0.240285i	−0.992757	+	0.120142i	$0.961665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.69722i	− 0.176862i
$438$	0	0
$439$	20.6972	0.987825	0.493912	−	0.869512i	$-0.335566\pi$
$439$	20.6972	0.987825	0.493912	+	0.869512i	$0.335566\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	1.39445i	0.0662523i	0.999451	+	0.0331261i	$0.0105463\pi$
$443$	1.39445i	0.0662523i	−0.999451	+	0.0331261i	$0.989454\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.63331i	− 0.171850i
$448$	0	0
$449$	−41.5139	−1.95916	−0.979581	−	0.201052i	$-0.935564\pi$
$449$	−41.5139	−1.95916	−0.979581	+	0.201052i	$0.935564\pi$
$450$	0	0
$451$	1.60555	0.0756025
$452$	0	0
$453$	27.1194i	1.27418i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.3028i	1.13684i	0.822740	+	0.568418i	$0.192444\pi$
$457$	24.3028i	1.13684i	−0.822740	+	0.568418i	$0.807556\pi$
$458$	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
$463$	− 26.2111i	− 1.21813i	−0.793119	−	0.609067i	$-0.791544\pi$
$463$	− 26.2111i	− 1.21813i	0.793119	−	0.609067i	$-0.208456\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.6333i	1.13989i	0.821682	+	0.569947i	$0.193036\pi$
$467$	24.6333i	1.13989i	−0.821682	+	0.569947i	$0.806964\pi$
$468$	0	0
$469$	17.2111	0.794735
$470$	0	0
$471$	6.23886	0.287471
$472$	0	0
$473$	7.21110i	0.331567i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.00000i	0.137361i
$478$	0	0
$479$	−13.1833	−0.602362	−0.301181	−	0.953567i	$-0.597381\pi$
$479$	−13.1833	−0.602362	−0.301181	+	0.953567i	$0.597381\pi$
$480$	0	0
$481$	48.0278	2.18988
$482$	0	0
$483$	− 20.7250i	− 0.943019i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 10.2111i	− 0.462709i	−0.972869	−	0.231355i	$-0.925684\pi$
$487$	− 10.2111i	− 0.462709i	0.972869	−	0.231355i	$-0.0743158\pi$
$488$	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
$493$	38.7250i	1.74409i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 19.8167i	− 0.888898i
$498$	0	0
$499$	−21.5139	−0.963093	−0.481547	−	0.876420i	$-0.659925\pi$
$499$	−21.5139	−0.963093	−0.481547	+	0.876420i	$0.659925\pi$
$500$	0	0
$501$	20.0917	0.897630
$502$	0	0
$503$	− 16.6056i	− 0.740405i	−0.928951	−	0.370202i	$-0.879288\pi$
$503$	− 16.6056i	− 0.740405i	0.928951	−	0.370202i	$-0.120712\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 15.6333i	− 0.694300i
$508$	0	0
$509$	26.3028	1.16585	0.582925	−	0.812526i	$-0.301908\pi$
$509$	26.3028	1.16585	0.582925	+	0.812526i	$0.301908\pi$
$510$	0	0
$511$	−12.5139	−0.553581
$512$	0	0
$513$	− 5.60555i	− 0.247491i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.00000i	− 0.131940i
$518$	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
$523$	3.57779i	0.156446i	0.996936	+	0.0782230i	$0.0249246\pi$
$523$	3.57779i	0.156446i	−0.996936	+	0.0782230i	$0.975075\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.4584i	0.716938i
$528$	0	0
$529$	9.33053	0.405675
$530$	0	0
$531$	0.275019	0.0119348
$532$	0	0
$533$	8.02776i	0.347721i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 7.18335i	− 0.309984i
$538$	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
$543$	− 11.8444i	− 0.508292i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	18.1194i	0.774731i	0.921926	+	0.387365i	$0.126615\pi$
$547$	18.1194i	0.774731i	−0.921926	+	0.387365i	$0.873385\pi$
$548$	0	0
$549$	3.78890	0.161706
$550$	0	0
$551$	−9.90833	−0.422109
$552$	0	0
$553$	− 0.394449i	− 0.0167737i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.42221i	0.399232i	0.979874	+	0.199616i	$0.0639694\pi$
$557$	9.42221i	0.399232i	−0.979874	+	0.199616i	$0.936031\pi$
$558$	0	0
$559$	−36.0555	−1.52499
$560$	0	0
$561$	−5.09167	−0.214971
$562$	0	0
$563$	− 18.9083i	− 0.796891i	−0.917192	−	0.398445i	$-0.869550\pi$
$563$	− 18.9083i	− 0.796891i	0.917192	−	0.398445i	$-0.130450\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 14.6056i	− 0.613375i
$568$	0	0
$569$	−15.1472	−0.635003	−0.317502	−	0.948258i	$-0.602844\pi$
$569$	−15.1472	−0.635003	−0.317502	+	0.948258i	$0.602844\pi$
$570$	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
$573$	− 8.72498i	− 0.364491i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.3583i	1.30546i	0.757589	+	0.652731i	$0.226377\pi$
$577$	31.3583i	1.30546i	−0.757589	+	0.652731i	$0.773623\pi$
$578$	0	0
$579$	1.57779	0.0655709
$580$	0	0
$581$	62.4500	2.59086
$582$	0	0
$583$	2.30278i	0.0953712i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 37.5416i	− 1.54951i	−0.632262	−	0.774755i	$-0.717873\pi$
$587$	− 37.5416i	− 1.54951i	0.632262	−	0.774755i	$-0.282127\pi$
$588$	0	0
$589$	−4.21110	−0.173515
$590$	0	0
$591$	12.6333	0.519665
$592$	0	0
$593$	− 13.6056i	− 0.558713i	−0.960187	−	0.279357i	$-0.909879\pi$
$593$	− 13.6056i	− 0.558713i	0.960187	−	0.279357i	$-0.0901211\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 31.9361i	− 1.30706i
$598$	0	0
$599$	−14.0917	−0.575770	−0.287885	−	0.957665i	$-0.592952\pi$
$599$	−14.0917	−0.575770	−0.287885	+	0.957665i	$0.592952\pi$
$600$	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
$603$	− 5.21110i	− 0.212213i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 7.21110i	− 0.292690i	−0.989234	−	0.146345i	$-0.953249\pi$
$607$	− 7.21110i	− 0.292690i	0.989234	−	0.146345i	$-0.0467509\pi$
$608$	0	0
$609$	−55.5416	−2.25066
$610$	0	0
$611$	15.0000	0.606835
$612$	0	0
$613$	− 41.1194i	− 1.66080i	−0.557169	−	0.830399i	$-0.688112\pi$
$613$	− 41.1194i	− 1.66080i	0.557169	−	0.830399i	$-0.311888\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 10.6056i	− 0.426963i	−0.976947	−	0.213482i	$-0.931520\pi$
$617$	− 10.6056i	− 0.426963i	0.976947	−	0.213482i	$-0.0684804\pi$
$618$	0	0
$619$	17.4222	0.700258	0.350129	−	0.936702i	$-0.386138\pi$
$619$	17.4222	0.700258	0.350129	+	0.936702i	$0.386138\pi$
$620$	0	0
$621$	−20.7250	−0.831665
$622$	0	0
$623$	− 22.8167i	− 0.914130i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.30278i	− 0.0520278i
$628$	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
$633$	− 32.8806i	− 1.30689i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 57.5694i	− 2.28098i
$638$	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
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.2389i	0.677729i	0.940835	+	0.338865i	$0.110043\pi$
$647$	17.2389i	0.677729i	−0.940835	+	0.338865i	$0.889957\pi$
$648$	0	0
$649$	0.211103	0.00828650
$650$	0	0
$651$	−23.6056	−0.925174
$652$	0	0
$653$	− 19.1194i	− 0.748201i	−0.927388	−	0.374101i	$-0.877951\pi$
$653$	− 19.1194i	− 0.748201i	0.927388	−	0.374101i	$-0.122049\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.78890i	0.147819i
$658$	0	0
$659$	−20.0917	−0.782660	−0.391330	−	0.920250i	$-0.627985\pi$
$659$	−20.0917	−0.782660	−0.391330	+	0.920250i	$0.627985\pi$
$660$	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
$663$	− 25.4584i	− 0.988721i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.6333i	1.41845i
$668$	0	0
$669$	26.8806	1.03926
$670$	0	0
$671$	2.90833	0.112275
$672$	0	0
$673$	− 6.02776i	− 0.232353i	−0.993229	−	0.116176i	$-0.962936\pi$
$673$	− 6.02776i	− 0.232353i	0.993229	−	0.116176i	$-0.0370638\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.2389i	1.00844i	0.863575	+	0.504221i	$0.168220\pi$
$677$	26.2389i	1.00844i	−0.863575	+	0.504221i	$0.831780\pi$
$678$	0	0
$679$	50.3305	1.93151
$680$	0	0
$681$	−6.90833	−0.264728
$682$	0	0
$683$	9.84441i	0.376686i	0.982103	+	0.188343i	$0.0603116\pi$
$683$	9.84441i	0.376686i	−0.982103	+	0.188343i	$0.939688\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.8806i	− 0.682186i
$688$	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
$693$	5.60555i	0.212937i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.27502i	0.237683i
$698$	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
$703$	9.60555i	0.362280i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	75.3583i	2.83414i
$708$	0	0
$709$	−11.6333	−0.436898	−0.218449	−	0.975848i	$-0.570100\pi$
$709$	−11.6333	−0.436898	−0.218449	+	0.975848i	$0.570100\pi$
$710$	0	0
$711$	−0.119429	−0.00447895
$712$	0	0
$713$	15.5694i	0.583078i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 5.36669i	− 0.200423i
$718$	0	0
$719$	28.8167	1.07468	0.537340	−	0.843366i	$-0.319429\pi$
$719$	28.8167	1.07468	0.537340	+	0.843366i	$0.319429\pi$
$720$	0	0
$721$	−34.0278	−1.26726
$722$	0	0
$723$	− 32.4861i	− 1.20817i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0.330532i	0.0122588i	0.999981	+	0.00612938i	$0.00195105\pi$
$727$	0.330532i	0.0122588i	−0.999981	+	0.00612938i	$0.998049\pi$
$728$	0	0
$729$	−26.3305	−0.975205
$730$	0	0
$731$	−28.1833	−1.04240
$732$	0	0
$733$	12.3944i	0.457799i	0.973450	+	0.228900i	$0.0735128\pi$
$733$	12.3944i	0.457799i	−0.973450	+	0.228900i	$0.926487\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	9.88057	0.363463	0.181731	−	0.983348i	$-0.441830\pi$
$739$	9.88057	0.363463	0.181731	+	0.983348i	$0.441830\pi$
$740$	0	0
$741$	6.51388	0.239293
$742$	0	0
$743$	− 44.3028i	− 1.62531i	−0.582744	−	0.812656i	$-0.698021\pi$
$743$	− 44.3028i	− 1.62531i	0.582744	−	0.812656i	$-0.301979\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 18.9083i	− 0.691820i
$748$	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
$753$	5.09167i	0.185551i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23.0555i	0.837967i	0.907994	+	0.418983i	$0.137613\pi$
$757$	23.0555i	0.837967i	−0.907994	+	0.418983i	$0.862387\pi$
$758$	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
$763$	28.0278i	1.01467i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.05551i	0.0381124i
$768$	0	0
$769$	26.8167	0.967033	0.483517	−	0.875335i	$-0.339359\pi$
$769$	26.8167	0.967033	0.483517	+	0.875335i	$0.339359\pi$
$770$	0	0
$771$	−23.4500	−0.844530
$772$	0	0
$773$	− 22.1194i	− 0.795581i	−0.917476	−	0.397790i	$-0.869777\pi$
$773$	− 22.1194i	− 0.795581i	0.917476	−	0.397790i	$-0.130223\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	53.8444i	1.93166i
$778$	0	0
$779$	−1.60555	−0.0575248
$780$	0	0
$781$	−4.60555	−0.164800
$782$	0	0
$783$	55.5416i	1.98490i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.21110i	− 0.150110i	−0.997179	−	0.0750548i	$-0.976087\pi$
$787$	− 4.21110i	− 0.150110i	0.997179	−	0.0750548i	$-0.0239132\pi$
$788$	0	0
$789$	−1.54163	−0.0548836
$790$	0	0
$791$	−46.5416	−1.65483
$792$	0	0
$793$	14.5416i	0.516389i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.5139i	0.514108i	0.966397	+	0.257054i	$0.0827518\pi$
$797$	14.5139i	0.514108i	−0.966397	+	0.257054i	$0.917248\pi$
$798$	0	0
$799$	11.7250	0.414800
$800$	0	0
$801$	−6.90833	−0.244094
$802$	0	0
$803$	2.90833i	0.102633i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.9083i	1.08802i
$808$	0	0
$809$	3.63331	0.127740	0.0638701	−	0.997958i	$-0.479656\pi$
$809$	3.63331	0.127740	0.0638701	+	0.997958i	$0.479656\pi$
$810$	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
$813$	− 18.5139i	− 0.649310i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.21110i	− 0.252285i
$818$	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
$823$	10.4222i	0.363295i	0.983364	+	0.181648i	$0.0581430\pi$
$823$	10.4222i	0.363295i	−0.983364	+	0.181648i	$0.941857\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.81665i	− 0.271812i	−0.990722	−	0.135906i	$-0.956606\pi$
$827$	− 7.81665i	− 0.271812i	0.990722	−	0.135906i	$-0.0433945\pi$
$828$	0	0
$829$	38.7527	1.34594	0.672969	−	0.739671i	$-0.265019\pi$
$829$	38.7527	1.34594	0.672969	+	0.739671i	$0.265019\pi$
$830$	0	0
$831$	28.1472	0.976415
$832$	0	0
$833$	− 45.0000i	− 1.55916i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	23.6056i	0.815927i
$838$	0	0
$839$	16.1194	0.556505	0.278252	−	0.960508i	$-0.410245\pi$
$839$	16.1194	0.556505	0.278252	+	0.960508i	$0.410245\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	0	0
$843$	− 29.7250i	− 1.02378i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.30278i	0.147845i
$848$	0	0
$849$	3.51388	0.120596
$850$	0	0
$851$	35.5139	1.21740
$852$	0	0
$853$	19.7250i	0.675370i	0.941259	+	0.337685i	$0.109644\pi$
$853$	19.7250i	0.675370i	−0.941259	+	0.337685i	$0.890356\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	0	0
$859$	48.6056	1.65840	0.829200	−	0.558952i	$-0.188796\pi$
$859$	48.6056	1.65840	0.829200	+	0.558952i	$0.188796\pi$
$860$	0	0
$861$	−9.00000	−0.306719
$862$	0	0
$863$	19.6056i	0.667381i	0.942683	+	0.333690i	$0.108294\pi$
$863$	19.6056i	0.667381i	−0.942683	+	0.333690i	$0.891706\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.24726i	0.0763210i
$868$	0	0
$869$	−0.0916731	−0.00310980
$870$	0	0
$871$	20.0000	0.677674
$872$	0	0
$873$	− 15.2389i	− 0.515757i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 20.0000i	− 0.675352i	−0.941262	−	0.337676i	$-0.890359\pi$
$877$	− 20.0000i	− 0.675352i	0.941262	−	0.337676i	$-0.109641\pi$
$878$	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
$883$	12.4500i	0.418975i	0.977811	+	0.209487i	$0.0671795\pi$
$883$	12.4500i	0.418975i	−0.977811	+	0.209487i	$0.932821\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	47.2389i	1.58613i	0.609140	+	0.793063i	$0.291515\pi$
$887$	47.2389i	1.58613i	−0.609140	+	0.793063i	$0.708485\pi$
$888$	0	0
$889$	73.6611	2.47051
$890$	0	0
$891$	−3.39445	−0.113718
$892$	0	0
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24.0833i	− 0.804117i
$898$	0	0
$899$	41.7250	1.39161
$900$	0	0
$901$	−9.00000	−0.299833
$902$	0	0
$903$	− 40.4222i	− 1.34517i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	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
$913$	− 14.5139i	− 0.480339i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.90833i	− 0.129064i
$918$	0	0
$919$	41.2111	1.35943	0.679714	−	0.733477i	$-0.262104\pi$
$919$	41.2111	1.35943	0.679714	+	0.733477i	$0.262104\pi$
$920$	0	0
$921$	7.93608	0.261503
$922$	0	0
$923$	− 23.0278i	− 0.757968i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10.3028i	0.338388i
$928$	0	0
$929$	−46.3944	−1.52215	−0.761076	−	0.648662i	$-0.775329\pi$
$929$	−46.3944	−1.52215	−0.761076	+	0.648662i	$0.775329\pi$
$930$	0	0
$931$	11.5139	0.377352
$932$	0	0
$933$	21.9083i	0.717246i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 5.21110i	− 0.170239i	−0.996371	−	0.0851196i	$-0.972873\pi$
$937$	− 5.21110i	− 0.170239i	0.996371	−	0.0851196i	$-0.0271273\pi$
$938$	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
$943$	5.93608i	0.193305i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.6333i	− 1.19042i	−0.803569	−	0.595211i	$-0.797068\pi$
$947$	− 36.6333i	− 1.19042i	0.803569	−	0.595211i	$-0.202932\pi$
$948$	0	0
$949$	−14.5416	−0.472041
$950$	0	0
$951$	12.9083	0.418581
$952$	0	0
$953$	− 49.2666i	− 1.59590i	−0.602722	−	0.797951i	$-0.705917\pi$
$953$	− 49.2666i	− 1.59590i	0.602722	−	0.797951i	$-0.294083\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	12.9083i	0.417267i
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	−13.2666	−0.427955
$962$	0	0
$963$	− 3.90833i	− 0.125944i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.09167i	0.131579i	0.997834	+	0.0657897i	$0.0209566\pi$
$967$	4.09167i	0.131579i	−0.997834	+	0.0657897i	$0.979043\pi$
$968$	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
$973$	35.3305i	1.13264i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.9722i	0.510997i	0.966809	+	0.255499i	$0.0822396\pi$
$977$	15.9722i	0.510997i	−0.966809	+	0.255499i	$0.917760\pi$
$978$	0	0
$979$	−5.30278	−0.169477
$980$	0	0
$981$	8.48612	0.270941
$982$	0	0
$983$	48.8444i	1.55789i	0.627089	+	0.778947i	$0.284246\pi$
$983$	48.8444i	1.55789i	−0.627089	+	0.778947i	$0.715754\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.8167i	0.535280i
$988$	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
$993$	18.7527i	0.595100i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 14.2750i	− 0.452094i	−0.974116	−	0.226047i	$-0.927420\pi$
$997$	− 14.2750i	− 0.452094i	0.974116	−	0.226047i	$-0.0725804\pi$
$998$	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.b.y.4049.3		4
4.3	odd	2		275.2.b.c.199.1		4
5.2	odd	4		4400.2.a.bh.1.2		2
5.3	odd	4		4400.2.a.bs.1.1		2
5.4	even	2	inner	4400.2.b.y.4049.2		4
12.11	even	2		2475.2.c.k.199.4		4
20.3	even	4		275.2.a.e.1.1	✓	2
20.7	even	4		275.2.a.f.1.2	yes	2
20.19	odd	2		275.2.b.c.199.4		4
60.23	odd	4		2475.2.a.t.1.2		2
60.47	odd	4		2475.2.a.o.1.1		2
60.59	even	2		2475.2.c.k.199.1		4
220.43	odd	4		3025.2.a.n.1.2		2
220.87	odd	4		3025.2.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.e.1.1	✓	2	20.3	even	4
275.2.a.f.1.2	yes	2	20.7	even	4
275.2.b.c.199.1		4	4.3	odd	2
275.2.b.c.199.4		4	20.19	odd	2
2475.2.a.o.1.1		2	60.47	odd	4
2475.2.a.t.1.2		2	60.23	odd	4
2475.2.c.k.199.1		4	60.59	even	2
2475.2.c.k.199.4		4	12.11	even	2
3025.2.a.h.1.1		2	220.87	odd	4
3025.2.a.n.1.2		2	220.43	odd	4
4400.2.a.bh.1.2		2	5.2	odd	4
4400.2.a.bs.1.1		2	5.3	odd	4
4400.2.b.y.4049.2		4	5.4	even	2	inner
4400.2.b.y.4049.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +O(q^{10})\)	\(q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +1.00000 q^{11} +5.00000i q^{13} +3.90833i q^{17} -1.00000 q^{19} -5.60555 q^{21} +3.69722i q^{23} +5.60555i q^{27} +9.90833 q^{29} +4.21110 q^{31} +1.30278i q^{33} -9.60555i q^{37} -6.51388 q^{39} +1.60555 q^{41} +7.21110i q^{43} -3.00000i q^{47} -11.5139 q^{49} -5.09167 q^{51} +2.30278i q^{53} -1.30278i q^{57} +0.211103 q^{59} +2.90833 q^{61} +5.60555i q^{63} -4.00000i q^{67} -4.81665 q^{69} -4.60555 q^{71} +2.90833i q^{73} +4.30278i q^{77} -0.0916731 q^{79} -3.39445 q^{81} -14.5139i q^{83} +12.9083i q^{87} -5.30278 q^{89} -21.5139 q^{91} +5.48612i q^{93} -11.6972i q^{97} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^9	\( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91} - 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^9 + 4 * q^11 - 4 * q^19 - 8 * q^21 + 18 * q^29 - 12 * q^31 + 10 * q^39 - 8 * q^41 - 10 * q^49 - 42 * q^51 - 28 * q^59 - 10 * q^61 + 24 * q^69 - 4 * q^71 - 22 * q^79 - 28 * q^81 - 14 * q^89 - 50 * q^91 - 2 * q^99

Introduction

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