LMFDB - Embedded newform 4400.2.b.b.4049.1

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.b.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +8.00000 q^{29} +7.00000 q^{31} -3.00000i q^{33} -1.00000i q^{37} +4.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -12.0000i q^{57} -1.00000 q^{59} +4.00000 q^{61} -12.0000i q^{63} +5.00000i q^{67} +3.00000 q^{69} -3.00000 q^{71} -16.0000i q^{73} +2.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} -24.0000i q^{87} -15.0000 q^{89} -21.0000i q^{93} -7.00000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{9}+O(q^{10}) $ 2 * q - 12 * q^9	$ 2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) $ 2 * q - 12 * q^9 + 2 * q^11 + 8 * q^19 + 12 * q^21 + 16 * q^29 + 14 * q^31 + 8 * q^41 + 6 * q^49 - 36 * q^51 - 2 * q^59 + 8 * q^61 + 6 * q^69 - 6 * q^71 + 4 * q^79 + 18 * q^81 - 30 * q^89 - 12 * 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
$3$	− 3.00000i	− 1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	− 3.00000i	− 1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	6.00000	1.30931
$22$	0	0
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	9.00000i	1.73205i
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.00000i	− 0.164399i	−0.996616	−	0.0821995i	$-0.973806\pi$
$37$	− 1.00000i	− 0.164399i	0.996616	−	0.0821995i	$-0.0261945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	−18.0000	−2.52050
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 12.0000i	− 1.58944i
$58$	0	0
$59$	−1.00000	−0.130189	−0.0650945	−	0.997879i	$-0.520735\pi$
$59$	−1.00000	−0.130189	−0.0650945	+	0.997879i	$0.520735\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	− 12.0000i	− 1.51186i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.00000i	0.610847i	0.952217	+	0.305424i	$0.0987981\pi$
$67$	5.00000i	0.610847i	−0.952217	+	0.305424i	$0.901202\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	− 16.0000i	− 1.87266i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	− 16.0000i	− 1.87266i	0.351123	−	0.936329i	$-0.385800\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 24.0000i	− 2.57307i
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 21.0000i	− 2.17760i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 7.00000i	− 0.710742i	−0.934725	−	0.355371i	$-0.884354\pi$
$97$	− 7.00000i	− 0.710742i	0.934725	−	0.355371i	$-0.115646\pi$
$98$	0	0
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 16.0000i	− 1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	− 16.0000i	− 1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2.00000i	− 0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$	− 2.00000i	− 0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	7.00000i	0.658505i	0.944242	+	0.329252i	$0.106797\pi$
$113$	7.00000i	0.658505i	−0.944242	+	0.329252i	$0.893203\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.0000	1.10004
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	0	0
$129$	18.0000	1.58481
$130$	0	0
$131$	2.00000	0.174741	0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000	0.174741	0.0873704	+	0.996176i	$0.472154\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15.0000i	− 1.28154i	−0.767734	−	0.640768i	$-0.778616\pi$
$137$	− 15.0000i	− 1.28154i	0.767734	−	0.640768i	$-0.221384\pi$
$138$	0	0
$139$	−22.0000	−1.86602	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000	−1.86602	−0.933008	+	0.359856i	$0.882826\pi$
$140$	0	0
$141$	24.0000	2.02116
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.00000i	− 0.742307i
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	0	0
$153$	36.0000i	2.91043i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 11.0000i	− 0.877896i	−0.898513	−	0.438948i	$-0.855351\pi$
$157$	− 11.0000i	− 0.877896i	0.898513	−	0.438948i	$-0.144649\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−2.00000	−0.157622
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 16.0000i	− 1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$	− 16.0000i	− 1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	−24.0000	−1.83533
$172$	0	0
$173$	− 18.0000i	− 1.36851i	−0.729241	−	0.684257i	$-0.760127\pi$
$173$	− 18.0000i	− 1.36851i	0.729241	−	0.684257i	$-0.239873\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.00000i	0.225494i
$178$	0	0
$179$	−5.00000	−0.373718	−0.186859	−	0.982387i	$-0.559831\pi$
$179$	−5.00000	−0.373718	−0.186859	+	0.982387i	$0.559831\pi$
$180$	0	0
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	− 12.0000i	− 0.887066i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	−18.0000	−1.30931
$190$	0	0
$191$	9.00000	0.651217	0.325609	−	0.945505i	$-0.394431\pi$
$191$	9.00000	0.651217	0.325609	+	0.945505i	$0.394431\pi$
$192$	0	0
$193$	− 4.00000i	− 0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$	− 4.00000i	− 0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	15.0000	1.05802
$202$	0	0
$203$	16.0000i	1.12298i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.00000i	− 0.417029i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	9.00000i	0.616670i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	14.0000i	0.950382i
$218$	0	0
$219$	−48.0000	−3.24354
$220$	0	0
$221$	0	0
$222$	0	0
$223$	29.0000i	1.94198i	0.239113	+	0.970992i	$0.423143\pi$
$223$	29.0000i	1.94198i	−0.239113	+	0.970992i	$0.576857\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 18.0000i	− 1.19470i	−0.801980	−	0.597351i	$-0.796220\pi$
$227$	− 18.0000i	− 1.19470i	0.801980	−	0.597351i	$-0.203780\pi$
$228$	0	0
$229$	21.0000	1.38772	0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000	1.38772	0.693860	+	0.720110i	$0.255909\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	16.0000i	1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$	16.0000i	1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 6.00000i	− 0.389742i
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 10.0000i	− 0.623783i	−0.950118	−	0.311891i	$-0.899037\pi$
$257$	− 10.0000i	− 0.623783i	0.950118	−	0.311891i	$-0.100963\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	−48.0000	−2.97113
$262$	0	0
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	45.0000i	2.75396i
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	0	0
$279$	−42.0000	−2.51447
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000i	0.472225i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	−21.0000	−1.23104
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.00000i	0.522233i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	30.0000i	1.72345i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	0	0
$309$	−48.0000	−2.73062
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	9.00000i	0.508710i	0.967111	+	0.254355i	$0.0818632\pi$
$313$	9.00000i	0.508710i	−0.967111	+	0.254355i	$0.918137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 42.0000i	− 2.32261i
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	35.0000	1.92377	0.961887	−	0.273447i	$-0.0881639\pi$
$331$	35.0000	1.92377	0.961887	+	0.273447i	$0.0881639\pi$
$332$	0	0
$333$	6.00000i	0.328798i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	0	0
$339$	21.0000	1.14056
$340$	0	0
$341$	7.00000	0.379071
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.00000i	− 0.159674i	−0.996808	−	0.0798369i	$-0.974560\pi$
$353$	− 3.00000i	− 0.159674i	0.996808	−	0.0798369i	$-0.0254400\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 36.0000i	− 1.90532i
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	− 3.00000i	− 0.157459i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 33.0000i	− 1.72259i	−0.508109	−	0.861293i	$-0.669655\pi$
$367$	− 33.0000i	− 1.72259i	0.508109	−	0.861293i	$-0.330345\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	25.0000	1.28416	0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000	1.28416	0.642082	+	0.766636i	$0.278071\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	0	0
$383$	1.00000i	0.0510976i	0.999674	+	0.0255488i	$0.00813332\pi$
$383$	1.00000i	0.0510976i	−0.999674	+	0.0255488i	$0.991867\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 36.0000i	− 1.82998i
$388$	0	0
$389$	−13.0000	−0.659126	−0.329563	−	0.944134i	$-0.606901\pi$
$389$	−13.0000	−0.659126	−0.329563	+	0.944134i	$0.606901\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	− 6.00000i	− 0.302660i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 26.0000i	− 1.30490i	−0.757831	−	0.652451i	$-0.773741\pi$
$397$	− 26.0000i	− 1.30490i	0.757831	−	0.652451i	$-0.226259\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.00000i	− 0.0495682i
$408$	0	0
$409$	−34.0000	−1.68119	−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000	−1.68119	−0.840596	+	0.541663i	$0.817795\pi$
$410$	0	0
$411$	−45.0000	−2.21969
$412$	0	0
$413$	− 2.00000i	− 0.0984136i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	66.0000i	3.23203i
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	− 48.0000i	− 2.33384i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.0000	1.25238	0.626188	−	0.779672i	$-0.284614\pi$
$431$	26.0000	1.25238	0.626188	+	0.779672i	$0.284614\pi$
$432$	0	0
$433$	− 13.0000i	− 0.624740i	−0.949960	−	0.312370i	$-0.898877\pi$
$433$	− 13.0000i	− 0.624740i	0.949960	−	0.312370i	$-0.101123\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000i	0.191346i
$438$	0	0
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	54.0000i	2.55411i
$448$	0	0
$449$	21.0000	0.991051	0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000	0.991051	0.495526	+	0.868593i	$0.334975\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	− 54.0000i	− 2.53714i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.0000i	0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$	12.0000i	0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$	0	0
$459$	54.0000	2.52050
$460$	0	0
$461$	−28.0000	−1.30409	−0.652045	−	0.758180i	$-0.726089\pi$
$461$	−28.0000	−1.30409	−0.652045	+	0.758180i	$0.726089\pi$
$462$	0	0
$463$	27.0000i	1.25480i	0.778699	+	0.627398i	$0.215880\pi$
$463$	27.0000i	1.25480i	−0.778699	+	0.627398i	$0.784120\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.0000i	1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$	33.0000i	1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	−33.0000	−1.52056
$472$	0	0
$473$	6.00000i	0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.00000i	0.273009i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.00000i	− 0.407829i	−0.978989	−	0.203914i	$-0.934634\pi$
$487$	− 9.00000i	− 0.407829i	0.978989	−	0.203914i	$-0.0653664\pi$
$488$	0	0
$489$	−12.0000	−0.542659
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	− 48.0000i	− 2.16181i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−48.0000	−2.14448
$502$	0	0
$503$	− 30.0000i	− 1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$	− 30.0000i	− 1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 39.0000i	− 1.73205i
$508$	0	0
$509$	13.0000	0.576215	0.288107	−	0.957598i	$-0.406974\pi$
$509$	13.0000	0.576215	0.288107	+	0.957598i	$0.406974\pi$
$510$	0	0
$511$	32.0000	1.41560
$512$	0	0
$513$	36.0000i	1.58944i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	0	0
$519$	−54.0000	−2.37034
$520$	0	0
$521$	37.0000	1.62100	0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000	1.62100	0.810500	+	0.585739i	$0.199196\pi$
$522$	0	0
$523$	44.0000i	1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$	44.0000i	1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 42.0000i	− 1.82955i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	6.00000	0.260378
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.0000i	0.647298i
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	0	0
$543$	15.0000i	0.643712i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	−24.0000	−1.02430
$550$	0	0
$551$	32.0000	1.36325
$552$	0	0
$553$	4.00000i	0.170097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.0000i	1.77960i	0.456354	+	0.889799i	$0.349155\pi$
$557$	42.0000i	1.77960i	−0.456354	+	0.889799i	$0.650845\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	18.0000i	0.755929i
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	− 27.0000i	− 1.12794i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 23.0000i	− 0.957503i	−0.877951	−	0.478751i	$-0.841090\pi$
$577$	− 23.0000i	− 0.957503i	0.877951	−	0.478751i	$-0.158910\pi$
$578$	0	0
$579$	−12.0000	−0.498703
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	− 2.00000i	− 0.0828315i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	28.0000	1.15372
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	4.00000i	0.164260i	0.996622	+	0.0821302i	$0.0261723\pi$
$593$	4.00000i	0.164260i	−0.996622	+	0.0821302i	$0.973828\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 24.0000i	− 0.982255i
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	0	0
$603$	− 30.0000i	− 1.22169i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.0000i	0.730597i	0.930890	+	0.365299i	$0.119033\pi$
$607$	18.0000i	0.730597i	−0.930890	+	0.365299i	$0.880967\pi$
$608$	0	0
$609$	48.0000	1.94506
$610$	0	0
$611$	0	0
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	−3.00000	−0.120580	−0.0602901	−	0.998181i	$-0.519203\pi$
$619$	−3.00000	−0.120580	−0.0602901	+	0.998181i	$0.519203\pi$
$620$	0	0
$621$	−9.00000	−0.361158
$622$	0	0
$623$	− 30.0000i	− 1.20192i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 12.0000i	− 0.479234i
$628$	0	0
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	0	0
$633$	60.0000i	2.38479i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	18.0000	0.712069
$640$	0	0
$641$	−9.00000	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$641$	−9.00000	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$642$	0	0
$643$	7.00000i	0.276053i	0.990429	+	0.138027i	$0.0440759\pi$
$643$	7.00000i	0.276053i	−0.990429	+	0.138027i	$0.955924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 15.0000i	− 0.589711i	−0.955542	−	0.294855i	$-0.904729\pi$
$647$	− 15.0000i	− 0.589711i	0.955542	−	0.294855i	$-0.0952715\pi$
$648$	0	0
$649$	−1.00000	−0.0392534
$650$	0	0
$651$	42.0000	1.64611
$652$	0	0
$653$	− 11.0000i	− 0.430463i	−0.976563	−	0.215232i	$-0.930949\pi$
$653$	− 11.0000i	− 0.430463i	0.976563	−	0.215232i	$-0.0690506\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	96.0000i	3.74532i
$658$	0	0
$659$	22.0000	0.856998	0.428499	−	0.903542i	$-0.359042\pi$
$659$	22.0000	0.856998	0.428499	+	0.903542i	$0.359042\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8.00000i	0.309761i
$668$	0	0
$669$	87.0000	3.36361
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	0	0
$681$	−54.0000	−2.06928
$682$	0	0
$683$	8.00000i	0.306111i	0.988218	+	0.153056i	$0.0489114\pi$
$683$	8.00000i	0.306111i	−0.988218	+	0.153056i	$0.951089\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 63.0000i	− 2.40360i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	29.0000	1.10321	0.551606	−	0.834105i	$-0.314015\pi$
$691$	29.0000	1.10321	0.551606	+	0.834105i	$0.314015\pi$
$692$	0	0
$693$	− 12.0000i	− 0.455842i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 24.0000i	− 0.909065i
$698$	0	0
$699$	48.0000	1.81553
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	−19.0000	−0.713560	−0.356780	−	0.934188i	$-0.616125\pi$
$709$	−19.0000	−0.713560	−0.356780	+	0.934188i	$0.616125\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	0	0
$713$	7.00000i	0.262152i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.00000i	0.224074i
$718$	0	0
$719$	−23.0000	−0.857755	−0.428878	−	0.903363i	$-0.641091\pi$
$719$	−23.0000	−0.857755	−0.428878	+	0.903363i	$0.641091\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	35.0000i	1.29808i	0.760755	+	0.649039i	$0.224829\pi$
$727$	35.0000i	1.29808i	−0.760755	+	0.649039i	$0.775171\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	36.0000	1.33151
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.00000i	0.184177i
$738$	0	0
$739$	−26.0000	−0.956425	−0.478213	−	0.878244i	$-0.658715\pi$
$739$	−26.0000	−0.956425	−0.478213	+	0.878244i	$0.658715\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 36.0000i	− 1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$	− 36.0000i	− 1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−23.0000	−0.839282	−0.419641	−	0.907690i	$-0.637844\pi$
$751$	−23.0000	−0.839282	−0.419641	+	0.907690i	$0.637844\pi$
$752$	0	0
$753$	− 39.0000i	− 1.42124i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	10.0000i	0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$	10.0000i	0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	0	0
$759$	3.00000	0.108893
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	28.0000i	1.01367i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.00000i	− 0.215249i
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	−3.00000	−0.107348
$782$	0	0
$783$	72.0000i	2.57307i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	0	0
$789$	42.0000	1.49524
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.00000i	0.0354218i	0.999843	+	0.0177109i	$0.00563785\pi$
$797$	1.00000i	0.0354218i	−0.999843	+	0.0177109i	$0.994362\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	90.0000	3.17999
$802$	0	0
$803$	− 16.0000i	− 0.564628i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	78.0000i	2.74573i
$808$	0	0
$809$	−12.0000	−0.421898	−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000	−0.421898	−0.210949	+	0.977497i	$0.567655\pi$
$810$	0	0
$811$	26.0000	0.912983	0.456492	−	0.889728i	$-0.349106\pi$
$811$	26.0000	0.912983	0.456492	+	0.889728i	$0.349106\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.0000i	0.839654i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−34.0000	−1.18661	−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000	−1.18661	−0.593304	+	0.804978i	$0.702177\pi$
$822$	0	0
$823$	− 31.0000i	− 1.08059i	−0.841475	−	0.540296i	$-0.818312\pi$
$823$	− 31.0000i	− 1.08059i	0.841475	−	0.540296i	$-0.181688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000i	0.556375i	0.960527	+	0.278187i	$0.0897336\pi$
$827$	16.0000i	0.556375i	−0.960527	+	0.278187i	$0.910266\pi$
$828$	0	0
$829$	35.0000	1.21560	0.607800	−	0.794090i	$-0.292052\pi$
$829$	35.0000	1.21560	0.607800	+	0.794090i	$0.292052\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	63.0000i	2.17760i
$838$	0	0
$839$	21.0000	0.725001	0.362500	−	0.931984i	$-0.381923\pi$
$839$	21.0000	0.725001	0.362500	+	0.931984i	$0.381923\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	− 18.0000i	− 0.619953i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	1.00000	0.0342796
$852$	0	0
$853$	26.0000i	0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$	26.0000i	0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 24.0000i	− 0.819824i	−0.912125	−	0.409912i	$-0.865559\pi$
$857$	− 24.0000i	− 0.819824i	0.912125	−	0.409912i	$-0.134441\pi$
$858$	0	0
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	24.0000	0.817918
$862$	0	0
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	57.0000i	1.93582i
$868$	0	0
$869$	2.00000	0.0678454
$870$	0	0
$871$	0	0
$872$	0	0
$873$	42.0000i	1.42148i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	56.0000i	1.89099i	0.325643	+	0.945493i	$0.394419\pi$
$877$	56.0000i	1.89099i	−0.325643	+	0.945493i	$0.605581\pi$
$878$	0	0
$879$	−36.0000	−1.21425
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	0	0
$883$	− 28.0000i	− 0.942275i	−0.882060	−	0.471138i	$-0.843844\pi$
$883$	− 28.0000i	− 0.942275i	0.882060	−	0.471138i	$-0.156156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.0000i	1.14161i	0.821086	+	0.570804i	$0.193368\pi$
$887$	34.0000i	1.14161i	−0.821086	+	0.570804i	$0.806632\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	0	0
$891$	9.00000	0.301511
$892$	0	0
$893$	32.0000i	1.07084i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56.0000	1.86770
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	36.0000i	1.19800i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	0	0
$909$	60.0000	1.99007
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	− 2.00000i	− 0.0661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000i	0.132092i
$918$	0	0
$919$	−50.0000	−1.64935	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	−50.0000	−1.64935	−0.824674	+	0.565608i	$0.808641\pi$
$920$	0	0
$921$	12.0000	0.395413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	96.0000i	3.15305i
$928$	0	0
$929$	22.0000	0.721797	0.360898	−	0.932605i	$-0.382470\pi$
$929$	22.0000	0.721797	0.360898	+	0.932605i	$0.382470\pi$
$930$	0	0
$931$	12.0000	0.393284
$932$	0	0
$933$	− 36.0000i	− 1.17859i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.00000i	0.130674i	0.997863	+	0.0653372i	$0.0208123\pi$
$937$	4.00000i	0.130674i	−0.997863	+	0.0653372i	$0.979188\pi$
$938$	0	0
$939$	27.0000	0.881112
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	4.00000i	0.130258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 47.0000i	− 1.52729i	−0.645634	−	0.763647i	$-0.723407\pi$
$947$	− 47.0000i	− 1.52729i	0.645634	−	0.763647i	$-0.276593\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−45.0000	−1.45922
$952$	0	0
$953$	34.0000i	1.10137i	0.834714	+	0.550684i	$0.185633\pi$
$953$	34.0000i	1.10137i	−0.834714	+	0.550684i	$0.814367\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	0	0
$959$	30.0000	0.968751
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	12.0000i	0.386695i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000i	0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$	8.00000i	0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$	0	0
$969$	−72.0000	−2.31297
$970$	0	0
$971$	−53.0000	−1.70085	−0.850425	−	0.526096i	$-0.823655\pi$
$971$	−53.0000	−1.70085	−0.850425	+	0.526096i	$0.823655\pi$
$972$	0	0
$973$	− 44.0000i	− 1.41058i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 11.0000i	− 0.351921i	−0.984397	−	0.175961i	$-0.943697\pi$
$977$	− 11.0000i	− 0.351921i	0.984397	−	0.175961i	$-0.0563031\pi$
$978$	0	0
$979$	−15.0000	−0.479402
$980$	0	0
$981$	−84.0000	−2.68191
$982$	0	0
$983$	25.0000i	0.797376i	0.917087	+	0.398688i	$0.130534\pi$
$983$	25.0000i	0.797376i	−0.917087	+	0.398688i	$0.869466\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	48.0000i	1.52786i
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	− 105.000i	− 3.33207i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	0	0
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.b.4049.1		2
4.3	odd	2		2200.2.b.a.1849.2		2
5.2	odd	4		4400.2.a.a.1.1		1
5.3	odd	4		176.2.a.c.1.1		1
5.4	even	2	inner	4400.2.b.b.4049.2		2
15.8	even	4		1584.2.a.q.1.1		1
20.3	even	4		88.2.a.a.1.1	✓	1
20.7	even	4		2200.2.a.k.1.1		1
20.19	odd	2		2200.2.b.a.1849.1		2
35.13	even	4		8624.2.a.c.1.1		1
40.3	even	4		704.2.a.l.1.1		1
40.13	odd	4		704.2.a.b.1.1		1
55.43	even	4		1936.2.a.l.1.1		1
60.23	odd	4		792.2.a.g.1.1		1
80.3	even	4		2816.2.c.i.1409.1		2
80.13	odd	4		2816.2.c.d.1409.2		2
80.43	even	4		2816.2.c.i.1409.2		2
80.53	odd	4		2816.2.c.d.1409.1		2
120.53	even	4		6336.2.a.k.1.1		1
120.83	odd	4		6336.2.a.h.1.1		1
140.83	odd	4		4312.2.a.l.1.1		1
220.3	even	20		968.2.i.j.9.1		4
220.43	odd	4		968.2.a.a.1.1		1
220.63	odd	20		968.2.i.i.9.1		4
220.83	odd	20		968.2.i.i.729.1		4
220.103	even	20		968.2.i.j.753.1		4
220.123	odd	20		968.2.i.i.81.1		4
220.163	even	20		968.2.i.j.81.1		4
220.183	odd	20		968.2.i.i.753.1		4
220.203	even	20		968.2.i.j.729.1		4
440.43	odd	4		7744.2.a.bk.1.1		1
440.373	even	4		7744.2.a.b.1.1		1
660.263	even	4		8712.2.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.a.1.1	✓	1	20.3	even	4
176.2.a.c.1.1		1	5.3	odd	4
704.2.a.b.1.1		1	40.13	odd	4
704.2.a.l.1.1		1	40.3	even	4
792.2.a.g.1.1		1	60.23	odd	4
968.2.a.a.1.1		1	220.43	odd	4
968.2.i.i.9.1		4	220.63	odd	20
968.2.i.i.81.1		4	220.123	odd	20
968.2.i.i.729.1		4	220.83	odd	20
968.2.i.i.753.1		4	220.183	odd	20
968.2.i.j.9.1		4	220.3	even	20
968.2.i.j.81.1		4	220.163	even	20
968.2.i.j.729.1		4	220.203	even	20
968.2.i.j.753.1		4	220.103	even	20
1584.2.a.q.1.1		1	15.8	even	4
1936.2.a.l.1.1		1	55.43	even	4
2200.2.a.k.1.1		1	20.7	even	4
2200.2.b.a.1849.1		2	20.19	odd	2
2200.2.b.a.1849.2		2	4.3	odd	2
2816.2.c.d.1409.1		2	80.53	odd	4
2816.2.c.d.1409.2		2	80.13	odd	4
2816.2.c.i.1409.1		2	80.3	even	4
2816.2.c.i.1409.2		2	80.43	even	4
4312.2.a.l.1.1		1	140.83	odd	4
4400.2.a.a.1.1		1	5.2	odd	4
4400.2.b.b.4049.1		2	1.1	even	1	trivial
4400.2.b.b.4049.2		2	5.4	even	2	inner
6336.2.a.h.1.1		1	120.83	odd	4
6336.2.a.k.1.1		1	120.53	even	4
7744.2.a.b.1.1		1	440.373	even	4
7744.2.a.bk.1.1		1	440.43	odd	4
8624.2.a.c.1.1		1	35.13	even	4
8712.2.a.x.1.1		1	660.263	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +2.00000i q^{7} -6.00000 q^{9} +1.00000 q^{11} -6.00000i q^{17} +4.00000 q^{19} +6.00000 q^{21} +1.00000i q^{23} +9.00000i q^{27} +8.00000 q^{29} +7.00000 q^{31} -3.00000i q^{33} -1.00000i q^{37} +4.00000 q^{41} +6.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -18.0000 q^{51} -2.00000i q^{53} -12.0000i q^{57} -1.00000 q^{59} +4.00000 q^{61} -12.0000i q^{63} +5.00000i q^{67} +3.00000 q^{69} -3.00000 q^{71} -16.0000i q^{73} +2.00000i q^{77} +2.00000 q^{79} +9.00000 q^{81} -2.00000i q^{83} -24.0000i q^{87} -15.0000 q^{89} -21.0000i q^{93} -7.00000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{9}+O(q^{10}) \) 2 * q - 12 * q^9	\( 2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99}+O(q^{100}) \) 2 * q - 12 * q^9 + 2 * q^11 + 8 * q^19 + 12 * q^21 + 16 * q^29 + 14 * q^31 + 8 * q^41 + 6 * q^49 - 36 * q^51 - 2 * q^59 + 8 * q^61 + 6 * q^69 - 6 * q^71 + 4 * q^79 + 18 * q^81 - 30 * q^89 - 12 * q^99

Introduction

L-functions

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