LMFDB - Embedded newform 4732.2.g.a.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(337,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4732.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.g (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:	$37.7852102365$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 364)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4732.337
Dual form			4732.2.g.a.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{15} +2.00000 q^{17} -1.00000i q^{19} -2.00000i q^{21} +7.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -5.00000 q^{29} -9.00000i q^{31} -8.00000i q^{33} -1.00000 q^{35} +2.00000i q^{37} +2.00000i q^{41} -1.00000 q^{43} +1.00000i q^{45} -9.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} -4.00000 q^{55} +2.00000i q^{57} +14.0000 q^{61} +1.00000i q^{63} +10.0000i q^{67} -14.0000 q^{69} -14.0000i q^{71} -3.00000i q^{73} -8.00000 q^{75} -4.00000 q^{77} +5.00000 q^{79} -11.0000 q^{81} +5.00000i q^{83} +2.00000i q^{85} +10.0000 q^{87} +9.00000i q^{89} +18.0000i q^{93} +1.00000 q^{95} -1.00000i q^{97} +4.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^9	$ 2 q - 4 q^{3} + 2 q^{9} + 4 q^{17} + 14 q^{23} + 8 q^{25} + 8 q^{27} - 10 q^{29} - 2 q^{35} - 2 q^{43} - 2 q^{49} - 8 q^{51} + 6 q^{53} - 8 q^{55} + 28 q^{61} - 28 q^{69} - 16 q^{75} - 8 q^{77} + 10 q^{79} - 22 q^{81} + 20 q^{87} + 2 q^{95}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^9 + 4 * q^17 + 14 * q^23 + 8 * q^25 + 8 * q^27 - 10 * q^29 - 2 * q^35 - 2 * q^43 - 2 * q^49 - 8 * q^51 + 6 * q^53 - 8 * q^55 + 28 * q^61 - 28 * q^69 - 16 * q^75 - 8 * q^77 + 10 * q^79 - 22 * q^81 + 20 * q^87 + 2 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times$.

$n$	$2367$	$2705$	$4565$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	1.00000i	0.447214i	0.974679	+	0.223607i	$0.0717831\pi$
$5$	1.00000i	0.447214i	−0.974679	+	0.223607i	$0.928217\pi$
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 2.00000i	− 0.516398i
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i	−0.993399	−	0.114708i	$-0.963407\pi$
$19$	− 1.00000i	− 0.229416i	0.993399	−	0.114708i	$-0.0365932\pi$
$20$	0	0
$21$	− 2.00000i	− 0.436436i
$22$	0	0
$23$	7.00000	1.45960	0.729800	−	0.683660i	$-0.239613\pi$
$23$	7.00000	1.45960	0.729800	+	0.683660i	$0.239613\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	− 9.00000i	− 1.61645i	−0.588875	−	0.808224i	$-0.700429\pi$
$31$	− 9.00000i	− 1.61645i	0.588875	−	0.808224i	$-0.299571\pi$
$32$	0	0
$33$	− 8.00000i	− 1.39262i
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.00000i	0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$	2.00000i	0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	3.00000	0.412082	0.206041	−	0.978543i	$-0.433942\pi$
$53$	3.00000	0.412082	0.206041	+	0.978543i	$0.433942\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	14.0000	1.79252	0.896258	−	0.443533i	$-0.146275\pi$
$61$	14.0000	1.79252	0.896258	+	0.443533i	$0.146275\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.0000i	1.22169i	0.791748	+	0.610847i	$0.209171\pi$
$67$	10.0000i	1.22169i	−0.791748	+	0.610847i	$0.790829\pi$
$68$	0	0
$69$	−14.0000	−1.68540
$70$	0	0
$71$	− 14.0000i	− 1.66149i	−0.556650	−	0.830747i	$-0.687914\pi$
$71$	− 14.0000i	− 1.66149i	0.556650	−	0.830747i	$-0.312086\pi$
$72$	0	0
$73$	− 3.00000i	− 0.351123i	−0.984468	−	0.175562i	$-0.943826\pi$
$73$	− 3.00000i	− 0.351123i	0.984468	−	0.175562i	$-0.0561742\pi$
$74$	0	0
$75$	−8.00000	−0.923760
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	5.00000i	0.548821i	0.961613	+	0.274411i	$0.0884828\pi$
$83$	5.00000i	0.548821i	−0.961613	+	0.274411i	$0.911517\pi$
$84$	0	0
$85$	2.00000i	0.216930i
$86$	0	0
$87$	10.0000	1.07211
$88$	0	0
$89$	9.00000i	0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$	9.00000i	0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	18.0000i	1.86651i
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	− 1.00000i	− 0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$	− 1.00000i	− 0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$	0	0
$99$	4.00000i	0.402015i
$100$	0	0
$101$	12.0000	1.19404	0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000	1.19404	0.597022	+	0.802225i	$0.296350\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	− 4.00000i	− 0.383131i	−0.981480	−	0.191565i	$-0.938644\pi$
$109$	− 4.00000i	− 0.383131i	0.981480	−	0.191565i	$-0.0613564\pi$
$110$	0	0
$111$	− 4.00000i	− 0.379663i
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	7.00000i	0.652753i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.00000i	0.183340i
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 4.00000i	− 0.360668i
$124$	0	0
$125$	9.00000i	0.804984i
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	4.00000i	0.344265i
$136$	0	0
$137$	14.0000i	1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$	14.0000i	1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	18.0000i	1.51587i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 5.00000i	− 0.415227i
$146$	0	0
$147$	2.00000	0.164957
$148$	0	0
$149$	12.0000i	0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$	12.0000i	0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$	0	0
$151$	− 14.0000i	− 1.13930i	−0.821886	−	0.569652i	$-0.807078\pi$
$151$	− 14.0000i	− 1.13930i	0.821886	−	0.569652i	$-0.192922\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	9.00000	0.722897
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	7.00000i	0.551677i
$162$	0	0
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	0	0
$165$	8.00000	0.622799
$166$	0	0
$167$	− 3.00000i	− 0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$	− 3.00000i	− 0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 1.00000i	− 0.0764719i
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	4.00000i	0.302372i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	−28.0000	−2.06982
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	4.00000i	0.290957i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	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
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	− 20.0000i	− 1.41069i
$202$	0	0
$203$	− 5.00000i	− 0.350931i
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	7.00000	0.486534
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−23.0000	−1.58339	−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000	−1.58339	−0.791693	+	0.610920i	$0.790800\pi$
$212$	0	0
$213$	28.0000i	1.91853i
$214$	0	0
$215$	− 1.00000i	− 0.0681994i
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	21.0000i	1.40626i	0.711059	+	0.703132i	$0.248216\pi$
$223$	21.0000i	1.40626i	−0.711059	+	0.703132i	$0.751784\pi$
$224$	0	0
$225$	4.00000	0.266667
$226$	0	0
$227$	− 8.00000i	− 0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$	− 8.00000i	− 0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$	0	0
$229$	2.00000i	0.132164i	0.997814	+	0.0660819i	$0.0210498\pi$
$229$	2.00000i	0.132164i	−0.997814	+	0.0660819i	$0.978950\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	25.0000	1.63780	0.818902	−	0.573933i	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	−10.0000	−0.649570
$238$	0	0
$239$	12.0000i	0.776215i	0.921614	+	0.388108i	$0.126871\pi$
$239$	12.0000i	0.776215i	−0.921614	+	0.388108i	$0.873129\pi$
$240$	0	0
$241$	25.0000i	1.61039i	0.593009	+	0.805196i	$0.297940\pi$
$241$	25.0000i	1.61039i	−0.593009	+	0.805196i	$0.702060\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 10.0000i	− 0.633724i
$250$	0	0
$251$	16.0000	1.00991	0.504956	−	0.863145i	$-0.331509\pi$
$251$	16.0000	1.00991	0.504956	+	0.863145i	$0.331509\pi$
$252$	0	0
$253$	28.0000i	1.76034i
$254$	0	0
$255$	− 4.00000i	− 0.250490i
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	19.0000	1.17159	0.585795	−	0.810459i	$-0.300782\pi$
$263$	19.0000	1.17159	0.585795	+	0.810459i	$0.300782\pi$
$264$	0	0
$265$	3.00000i	0.184289i
$266$	0	0
$267$	− 18.0000i	− 1.10158i
$268$	0	0
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	16.0000i	0.964836i
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	0	0
$279$	− 9.00000i	− 0.538816i
$280$	0	0
$281$	8.00000i	0.477240i	0.971113	+	0.238620i	$0.0766950\pi$
$281$	8.00000i	0.477240i	−0.971113	+	0.238620i	$0.923305\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	2.00000i	0.117242i
$292$	0	0
$293$	− 5.00000i	− 0.292103i	−0.989277	−	0.146052i	$-0.953343\pi$
$293$	− 5.00000i	− 0.292103i	0.989277	−	0.146052i	$-0.0466565\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.0000i	0.928414i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 1.00000i	− 0.0576390i
$302$	0	0
$303$	−24.0000	−1.37876
$304$	0	0
$305$	14.0000i	0.801638i
$306$	0	0
$307$	27.0000i	1.54097i	0.637457	+	0.770486i	$0.279986\pi$
$307$	27.0000i	1.54097i	−0.637457	+	0.770486i	$0.720014\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	20.0000i	1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$317$	20.0000i	1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$318$	0	0
$319$	− 20.0000i	− 1.11979i
$320$	0	0
$321$	24.0000	1.33955
$322$	0	0
$323$	− 2.00000i	− 0.111283i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.00000i	0.442401i
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	30.0000i	1.64895i	0.565899	+	0.824475i	$0.308529\pi$
$331$	30.0000i	1.64895i	−0.565899	+	0.824475i	$0.691471\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−21.0000	−1.14394	−0.571971	−	0.820274i	$-0.693821\pi$
$337$	−21.0000	−1.14394	−0.571971	+	0.820274i	$0.693821\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	36.0000	1.94951
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	0	0
$345$	− 14.0000i	− 0.753735i
$346$	0	0
$347$	16.0000	0.858925	0.429463	−	0.903085i	$-0.358703\pi$
$347$	16.0000	0.858925	0.429463	+	0.903085i	$0.358703\pi$
$348$	0	0
$349$	− 35.0000i	− 1.87351i	−0.349990	−	0.936754i	$-0.613815\pi$
$349$	− 35.0000i	− 1.87351i	0.349990	−	0.936754i	$-0.386185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.00000i	− 0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$	− 2.00000i	− 0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	0	0
$357$	− 4.00000i	− 0.211702i
$358$	0	0
$359$	30.0000i	1.58334i	0.610949	+	0.791670i	$0.290788\pi$
$359$	30.0000i	1.58334i	−0.610949	+	0.791670i	$0.709212\pi$
$360$	0	0
$361$	18.0000	0.947368
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	3.00000	0.157027
$366$	0	0
$367$	24.0000	1.25279	0.626395	−	0.779506i	$-0.284530\pi$
$367$	24.0000	1.25279	0.626395	+	0.779506i	$0.284530\pi$
$368$	0	0
$369$	2.00000i	0.104116i
$370$	0	0
$371$	3.00000i	0.155752i
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	− 18.0000i	− 0.929516i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	28.0000i	1.43826i	0.694874	+	0.719132i	$0.255460\pi$
$379$	28.0000i	1.43826i	−0.694874	+	0.719132i	$0.744540\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	0	0
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	− 4.00000i	− 0.203859i
$386$	0	0
$387$	−1.00000	−0.0508329
$388$	0	0
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	14.0000	0.708010
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	5.00000i	0.251577i
$396$	0	0
$397$	25.0000i	1.25471i	0.778732	+	0.627357i	$0.215863\pi$
$397$	25.0000i	1.25471i	−0.778732	+	0.627357i	$0.784137\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	28.0000i	1.39825i	0.714998	+	0.699127i	$0.246428\pi$
$401$	28.0000i	1.39825i	−0.714998	+	0.699127i	$0.753572\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 11.0000i	− 0.546594i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	− 25.0000i	− 1.23617i	−0.786111	−	0.618085i	$-0.787909\pi$
$409$	− 25.0000i	− 1.23617i	0.786111	−	0.618085i	$-0.212091\pi$
$410$	0	0
$411$	− 28.0000i	− 1.38114i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.00000	−0.245440
$416$	0	0
$417$	−32.0000	−1.56705
$418$	0	0
$419$	−16.0000	−0.781651	−0.390826	−	0.920465i	$-0.627810\pi$
$419$	−16.0000	−0.781651	−0.390826	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	10.0000i	0.487370i	0.969854	+	0.243685i	$0.0783563\pi$
$421$	10.0000i	0.487370i	−0.969854	+	0.243685i	$0.921644\pi$
$422$	0	0
$423$	− 9.00000i	− 0.437595i
$424$	0	0
$425$	8.00000	0.388057
$426$	0	0
$427$	14.0000i	0.677507i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.0000i	0.674356i	0.941441	+	0.337178i	$0.109472\pi$
$431$	14.0000i	0.674356i	−0.941441	+	0.337178i	$0.890528\pi$
$432$	0	0
$433$	−24.0000	−1.15337	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000	−1.15337	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	10.0000i	0.479463i
$436$	0	0
$437$	− 7.00000i	− 0.334855i
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−39.0000	−1.85295	−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000	−1.85295	−0.926473	+	0.376361i	$0.877175\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	0	0
$447$	− 24.0000i	− 1.13516i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	−8.00000	−0.376705
$452$	0	0
$453$	28.0000i	1.31555i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	10.0000i	0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$	10.0000i	0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$	0	0
$463$	− 24.0000i	− 1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$	− 24.0000i	− 1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$	0	0
$465$	−18.0000	−0.834730
$466$	0	0
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−10.0000	−0.461757
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	− 4.00000i	− 0.183920i
$474$	0	0
$475$	− 4.00000i	− 0.183533i
$476$	0	0
$477$	3.00000	0.137361
$478$	0	0
$479$	− 27.0000i	− 1.23366i	−0.787096	−	0.616831i	$-0.788416\pi$
$479$	− 27.0000i	− 1.23366i	0.787096	−	0.616831i	$-0.211584\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 14.0000i	− 0.637022i
$484$	0	0
$485$	1.00000	0.0454077
$486$	0	0
$487$	− 34.0000i	− 1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$	− 34.0000i	− 1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$	0	0
$489$	48.0000i	2.17064i
$490$	0	0
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−10.0000	−0.450377
$494$	0	0
$495$	−4.00000	−0.179787
$496$	0	0
$497$	14.0000	0.627986
$498$	0	0
$499$	− 2.00000i	− 0.0895323i	−0.998997	−	0.0447661i	$-0.985746\pi$
$499$	− 2.00000i	− 0.0895323i	0.998997	−	0.0447661i	$-0.0142543\pi$
$500$	0	0
$501$	6.00000i	0.268060i
$502$	0	0
$503$	−34.0000	−1.51599	−0.757993	−	0.652263i	$-0.773820\pi$
$503$	−34.0000	−1.51599	−0.757993	+	0.652263i	$0.773820\pi$
$504$	0	0
$505$	12.0000i	0.533993i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.00000i	0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$	9.00000i	0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	− 4.00000i	− 0.176261i
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	−18.0000	−0.787085	−0.393543	−	0.919306i	$-0.628751\pi$
$523$	−18.0000	−0.787085	−0.393543	+	0.919306i	$0.628751\pi$
$524$	0	0
$525$	− 8.00000i	− 0.349149i
$526$	0	0
$527$	− 18.0000i	− 0.784092i
$528$	0	0
$529$	26.0000	1.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 12.0000i	− 0.518805i
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	− 4.00000i	− 0.172292i
$540$	0	0
$541$	38.0000i	1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$	38.0000i	1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$	0	0
$543$	−40.0000	−1.71656
$544$	0	0
$545$	4.00000	0.171341
$546$	0	0
$547$	23.0000	0.983409	0.491704	−	0.870762i	$-0.336374\pi$
$547$	23.0000	0.983409	0.491704	+	0.870762i	$0.336374\pi$
$548$	0	0
$549$	14.0000	0.597505
$550$	0	0
$551$	5.00000i	0.213007i
$552$	0	0
$553$	5.00000i	0.212622i
$554$	0	0
$555$	4.00000	0.169791
$556$	0	0
$557$	− 4.00000i	− 0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$	− 4.00000i	− 0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 16.0000i	− 0.675521i
$562$	0	0
$563$	40.0000	1.68580	0.842900	−	0.538071i	$-0.180847\pi$
$563$	40.0000	1.68580	0.842900	+	0.538071i	$0.180847\pi$
$564$	0	0
$565$	1.00000i	0.0420703i
$566$	0	0
$567$	− 11.0000i	− 0.461957i
$568$	0	0
$569$	37.0000	1.55112	0.775560	−	0.631273i	$-0.217467\pi$
$569$	37.0000	1.55112	0.775560	+	0.631273i	$0.217467\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	0	0
$573$	48.0000	2.00523
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	− 18.0000i	− 0.749350i	−0.927156	−	0.374675i	$-0.877754\pi$
$577$	− 18.0000i	− 0.749350i	0.927156	−	0.374675i	$-0.122246\pi$
$578$	0	0
$579$	4.00000i	0.166234i
$580$	0	0
$581$	−5.00000	−0.207435
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	−9.00000	−0.370839
$590$	0	0
$591$	− 12.0000i	− 0.493614i
$592$	0	0
$593$	− 41.0000i	− 1.68367i	−0.539736	−	0.841834i	$-0.681476\pi$
$593$	− 41.0000i	− 1.68367i	0.539736	−	0.841834i	$-0.318524\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	0	0
$597$	−28.0000	−1.14596
$598$	0	0
$599$	21.0000	0.858037	0.429018	−	0.903296i	$-0.358860\pi$
$599$	21.0000	0.858037	0.429018	+	0.903296i	$0.358860\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	0	0
$605$	− 5.00000i	− 0.203279i
$606$	0	0
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	0	0
$609$	10.0000i	0.405220i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 16.0000i	− 0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$	− 16.0000i	− 0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	34.0000i	1.36879i	0.729112	+	0.684394i	$0.239933\pi$
$617$	34.0000i	1.36879i	−0.729112	+	0.684394i	$0.760067\pi$
$618$	0	0
$619$	− 20.0000i	− 0.803868i	−0.915669	−	0.401934i	$-0.868338\pi$
$619$	− 20.0000i	− 0.803868i	0.915669	−	0.401934i	$-0.131662\pi$
$620$	0	0
$621$	28.0000	1.12360
$622$	0	0
$623$	−9.00000	−0.360577
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	−8.00000	−0.319489
$628$	0	0
$629$	4.00000i	0.159490i
$630$	0	0
$631$	2.00000i	0.0796187i	0.999207	+	0.0398094i	$0.0126751\pi$
$631$	2.00000i	0.0796187i	−0.999207	+	0.0398094i	$0.987325\pi$
$632$	0	0
$633$	46.0000	1.82834
$634$	0	0
$635$	8.00000i	0.317470i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 14.0000i	− 0.553831i
$640$	0	0
$641$	−41.0000	−1.61940	−0.809701	−	0.586842i	$-0.800371\pi$
$641$	−41.0000	−1.61940	−0.809701	+	0.586842i	$0.800371\pi$
$642$	0	0
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	2.00000i	0.0787499i
$646$	0	0
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−18.0000	−0.705476
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	− 12.0000i	− 0.468879i
$656$	0	0
$657$	− 3.00000i	− 0.117041i
$658$	0	0
$659$	−45.0000	−1.75295	−0.876476	−	0.481446i	$-0.840112\pi$
$659$	−45.0000	−1.75295	−0.876476	+	0.481446i	$0.840112\pi$
$660$	0	0
$661$	− 31.0000i	− 1.20576i	−0.797832	−	0.602880i	$-0.794020\pi$
$661$	− 31.0000i	− 1.20576i	0.797832	−	0.602880i	$-0.205980\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.00000i	0.0387783i
$666$	0	0
$667$	−35.0000	−1.35521
$668$	0	0
$669$	− 42.0000i	− 1.62381i
$670$	0	0
$671$	56.0000i	2.16186i
$672$	0	0
$673$	−21.0000	−0.809491	−0.404745	−	0.914429i	$-0.632640\pi$
$673$	−21.0000	−0.809491	−0.404745	+	0.914429i	$0.632640\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	16.0000	0.614930	0.307465	−	0.951559i	$-0.400519\pi$
$677$	16.0000	0.614930	0.307465	+	0.951559i	$0.400519\pi$
$678$	0	0
$679$	1.00000	0.0383765
$680$	0	0
$681$	16.0000i	0.613121i
$682$	0	0
$683$	− 4.00000i	− 0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$	− 4.00000i	− 0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	− 4.00000i	− 0.152610i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 35.0000i	− 1.33146i	−0.746191	−	0.665731i	$-0.768120\pi$
$691$	− 35.0000i	− 1.33146i	0.746191	−	0.665731i	$-0.231880\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	16.0000i	0.606915i
$696$	0	0
$697$	4.00000i	0.151511i
$698$	0	0
$699$	−50.0000	−1.89117
$700$	0	0
$701$	−45.0000	−1.69963	−0.849813	−	0.527084i	$-0.823285\pi$
$701$	−45.0000	−1.69963	−0.849813	+	0.527084i	$0.823285\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	−18.0000	−0.677919
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	12.0000i	0.450669i	0.974281	+	0.225335i	$0.0723476\pi$
$709$	12.0000i	0.450669i	−0.974281	+	0.225335i	$0.927652\pi$
$710$	0	0
$711$	5.00000	0.187515
$712$	0	0
$713$	− 63.0000i	− 2.35937i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	0	0
$719$	40.0000	1.49175	0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000	1.49175	0.745874	+	0.666087i	$0.232032\pi$
$720$	0	0
$721$	− 4.00000i	− 0.148968i
$722$	0	0
$723$	− 50.0000i	− 1.85952i
$724$	0	0
$725$	−20.0000	−0.742781
$726$	0	0
$727$	42.0000	1.55769	0.778847	−	0.627214i	$-0.215805\pi$
$727$	42.0000	1.55769	0.778847	+	0.627214i	$0.215805\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−2.00000	−0.0739727
$732$	0	0
$733$	− 21.0000i	− 0.775653i	−0.921732	−	0.387826i	$-0.873226\pi$
$733$	− 21.0000i	− 0.775653i	0.921732	−	0.387826i	$-0.126774\pi$
$734$	0	0
$735$	2.00000i	0.0737711i
$736$	0	0
$737$	−40.0000	−1.47342
$738$	0	0
$739$	8.00000i	0.294285i	0.989115	+	0.147142i	$0.0470076\pi$
$739$	8.00000i	0.294285i	−0.989115	+	0.147142i	$0.952992\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 8.00000i	− 0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$	− 8.00000i	− 0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	5.00000i	0.182940i
$748$	0	0
$749$	− 12.0000i	− 0.438470i
$750$	0	0
$751$	−25.0000	−0.912263	−0.456131	−	0.889912i	$-0.650765\pi$
$751$	−25.0000	−0.912263	−0.456131	+	0.889912i	$0.650765\pi$
$752$	0	0
$753$	−32.0000	−1.16614
$754$	0	0
$755$	14.0000	0.509512
$756$	0	0
$757$	−47.0000	−1.70824	−0.854122	−	0.520073i	$-0.825905\pi$
$757$	−47.0000	−1.70824	−0.854122	+	0.520073i	$0.825905\pi$
$758$	0	0
$759$	− 56.0000i	− 2.03267i
$760$	0	0
$761$	35.0000i	1.26875i	0.773026	+	0.634375i	$0.218742\pi$
$761$	35.0000i	1.26875i	−0.773026	+	0.634375i	$0.781258\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	0	0
$765$	2.00000i	0.0723102i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	37.0000i	1.33425i	0.744944	+	0.667127i	$0.232476\pi$
$769$	37.0000i	1.33425i	−0.744944	+	0.667127i	$0.767524\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	0	0
$773$	38.0000i	1.36677i	0.730061	+	0.683383i	$0.239492\pi$
$773$	38.0000i	1.36677i	−0.730061	+	0.683383i	$0.760508\pi$
$774$	0	0
$775$	− 36.0000i	− 1.29316i
$776$	0	0
$777$	4.00000	0.143499
$778$	0	0
$779$	2.00000	0.0716574
$780$	0	0
$781$	56.0000	2.00384
$782$	0	0
$783$	−20.0000	−0.714742
$784$	0	0
$785$	10.0000i	0.356915i
$786$	0	0
$787$	37.0000i	1.31891i	0.751745	+	0.659454i	$0.229212\pi$
$787$	37.0000i	1.31891i	−0.751745	+	0.659454i	$0.770788\pi$
$788$	0	0
$789$	−38.0000	−1.35284
$790$	0	0
$791$	1.00000i	0.0355559i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 6.00000i	− 0.212798i
$796$	0	0
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	− 18.0000i	− 0.636794i
$800$	0	0
$801$	9.00000i	0.317999i
$802$	0	0
$803$	12.0000	0.423471
$804$	0	0
$805$	−7.00000	−0.246718
$806$	0	0
$807$	−48.0000	−1.68968
$808$	0	0
$809$	45.0000	1.58212	0.791058	−	0.611741i	$-0.209531\pi$
$809$	45.0000	1.58212	0.791058	+	0.611741i	$0.209531\pi$
$810$	0	0
$811$	− 28.0000i	− 0.983213i	−0.870817	−	0.491606i	$-0.836410\pi$
$811$	− 28.0000i	− 0.983213i	0.870817	−	0.491606i	$-0.163590\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24.0000	0.840683
$816$	0	0
$817$	1.00000i	0.0349856i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34.0000i	1.18661i	0.804978	+	0.593304i	$0.202177\pi$
$821$	34.0000i	1.18661i	−0.804978	+	0.593304i	$0.797823\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	− 32.0000i	− 1.11410i
$826$	0	0
$827$	− 40.0000i	− 1.39094i	−0.718557	−	0.695468i	$-0.755197\pi$
$827$	− 40.0000i	− 1.39094i	0.718557	−	0.695468i	$-0.244803\pi$
$828$	0	0
$829$	−24.0000	−0.833554	−0.416777	−	0.909009i	$-0.636840\pi$
$829$	−24.0000	−0.833554	−0.416777	+	0.909009i	$0.636840\pi$
$830$	0	0
$831$	−46.0000	−1.59572
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	− 36.0000i	− 1.24434i
$838$	0	0
$839$	12.0000i	0.414286i	0.978311	+	0.207143i	$0.0664165\pi$
$839$	12.0000i	0.414286i	−0.978311	+	0.207143i	$0.933583\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	− 16.0000i	− 0.551069i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	0	0
$849$	−8.00000	−0.274559
$850$	0	0
$851$	14.0000i	0.479914i
$852$	0	0
$853$	− 37.0000i	− 1.26686i	−0.773802	−	0.633428i	$-0.781647\pi$
$853$	− 37.0000i	− 1.26686i	0.773802	−	0.633428i	$-0.218353\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−30.0000	−1.02478	−0.512390	−	0.858753i	$-0.671240\pi$
$857$	−30.0000	−1.02478	−0.512390	+	0.858753i	$0.671240\pi$
$858$	0	0
$859$	22.0000	0.750630	0.375315	−	0.926897i	$-0.377534\pi$
$859$	22.0000	0.750630	0.375315	+	0.926897i	$0.377534\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	0	0
$863$	− 48.0000i	− 1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$	− 48.0000i	− 1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$	0	0
$865$	− 2.00000i	− 0.0680020i
$866$	0	0
$867$	26.0000	0.883006
$868$	0	0
$869$	20.0000i	0.678454i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1.00000i	− 0.0338449i
$874$	0	0
$875$	−9.00000	−0.304256
$876$	0	0
$877$	14.0000i	0.472746i	0.971662	+	0.236373i	$0.0759588\pi$
$877$	14.0000i	0.472746i	−0.971662	+	0.236373i	$0.924041\pi$
$878$	0	0
$879$	10.0000i	0.337292i
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	8.00000i	0.268311i
$890$	0	0
$891$	− 44.0000i	− 1.47406i
$892$	0	0
$893$	−9.00000	−0.301174
$894$	0	0
$895$	− 9.00000i	− 0.300837i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	45.0000i	1.50083i
$900$	0	0
$901$	6.00000	0.199889
$902$	0	0
$903$	2.00000i	0.0665558i
$904$	0	0
$905$	20.0000i	0.664822i
$906$	0	0
$907$	3.00000	0.0996134	0.0498067	−	0.998759i	$-0.484139\pi$
$907$	3.00000	0.0996134	0.0498067	+	0.998759i	$0.484139\pi$
$908$	0	0
$909$	12.0000	0.398015
$910$	0	0
$911$	−21.0000	−0.695761	−0.347881	−	0.937539i	$-0.613099\pi$
$911$	−21.0000	−0.695761	−0.347881	+	0.937539i	$0.613099\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	0	0
$915$	− 28.0000i	− 0.925651i
$916$	0	0
$917$	− 12.0000i	− 0.396275i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	− 54.0000i	− 1.77936i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	8.00000i	0.263038i
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	− 3.00000i	− 0.0984268i	−0.998788	−	0.0492134i	$-0.984329\pi$
$929$	− 3.00000i	− 0.0984268i	0.998788	−	0.0492134i	$-0.0156714\pi$
$930$	0	0
$931$	1.00000i	0.0327737i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	24.0000	0.783210
$940$	0	0
$941$	35.0000i	1.14097i	0.821309	+	0.570484i	$0.193244\pi$
$941$	35.0000i	1.14097i	−0.821309	+	0.570484i	$0.806756\pi$
$942$	0	0
$943$	14.0000i	0.455903i
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	− 10.0000i	− 0.324956i	−0.986712	−	0.162478i	$-0.948051\pi$
$947$	− 10.0000i	− 0.324956i	0.986712	−	0.162478i	$-0.0519487\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 40.0000i	− 1.29709i
$952$	0	0
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	0	0
$955$	− 24.0000i	− 0.776622i
$956$	0	0
$957$	40.0000i	1.29302i
$958$	0	0
$959$	−14.0000	−0.452084
$960$	0	0
$961$	−50.0000	−1.61290
$962$	0	0
$963$	−12.0000	−0.386695
$964$	0	0
$965$	2.00000	0.0643823
$966$	0	0
$967$	− 12.0000i	− 0.385894i	−0.981209	−	0.192947i	$-0.938195\pi$
$967$	− 12.0000i	− 0.385894i	0.981209	−	0.192947i	$-0.0618045\pi$
$968$	0	0
$969$	4.00000i	0.128499i
$970$	0	0
$971$	30.0000	0.962746	0.481373	−	0.876516i	$-0.340138\pi$
$971$	30.0000	0.962746	0.481373	+	0.876516i	$0.340138\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 60.0000i	− 1.91957i	−0.280736	−	0.959785i	$-0.590579\pi$
$977$	− 60.0000i	− 1.91957i	0.280736	−	0.959785i	$-0.409421\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	− 4.00000i	− 0.127710i
$982$	0	0
$983$	− 11.0000i	− 0.350846i	−0.984493	−	0.175423i	$-0.943871\pi$
$983$	− 11.0000i	− 0.350846i	0.984493	−	0.175423i	$-0.0561292\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−18.0000	−0.572946
$988$	0	0
$989$	−7.00000	−0.222587
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	− 60.0000i	− 1.90404i
$994$	0	0
$995$	14.0000i	0.443830i
$996$	0	0
$997$	−32.0000	−1.01345	−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000	−1.01345	−0.506725	+	0.862108i	$0.669144\pi$
$998$	0	0
$999$	8.00000i	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.g.a.337.2		2
13.5	odd	4		364.2.a.a.1.1	✓	1
13.8	odd	4		4732.2.a.a.1.1		1
13.12	even	2	inner	4732.2.g.a.337.1		2
39.5	even	4		3276.2.a.b.1.1		1
52.31	even	4		1456.2.a.m.1.1		1
65.44	odd	4		9100.2.a.l.1.1		1
91.5	even	12		2548.2.j.c.1145.1		2
91.18	odd	12		2548.2.j.j.1353.1		2
91.31	even	12		2548.2.j.c.1353.1		2
91.44	odd	12		2548.2.j.j.1145.1		2
91.83	even	4		2548.2.a.i.1.1		1
104.5	odd	4		5824.2.a.bb.1.1		1
104.83	even	4		5824.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.a.a.1.1	✓	1	13.5	odd	4
1456.2.a.m.1.1		1	52.31	even	4
2548.2.a.i.1.1		1	91.83	even	4
2548.2.j.c.1145.1		2	91.5	even	12
2548.2.j.c.1353.1		2	91.31	even	12
2548.2.j.j.1145.1		2	91.44	odd	12
2548.2.j.j.1353.1		2	91.18	odd	12
3276.2.a.b.1.1		1	39.5	even	4
4732.2.a.a.1.1		1	13.8	odd	4
4732.2.g.a.337.1		2	13.12	even	2	inner
4732.2.g.a.337.2		2	1.1	even	1	trivial
5824.2.a.d.1.1		1	104.83	even	4
5824.2.a.bb.1.1		1	104.5	odd	4
9100.2.a.l.1.1		1	65.44	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{3} +1.00000i q^{5} +1.00000i q^{7} +1.00000 q^{9} +4.00000i q^{11} -2.00000i q^{15} +2.00000 q^{17} -1.00000i q^{19} -2.00000i q^{21} +7.00000 q^{23} +4.00000 q^{25} +4.00000 q^{27} -5.00000 q^{29} -9.00000i q^{31} -8.00000i q^{33} -1.00000 q^{35} +2.00000i q^{37} +2.00000i q^{41} -1.00000 q^{43} +1.00000i q^{45} -9.00000i q^{47} -1.00000 q^{49} -4.00000 q^{51} +3.00000 q^{53} -4.00000 q^{55} +2.00000i q^{57} +14.0000 q^{61} +1.00000i q^{63} +10.0000i q^{67} -14.0000 q^{69} -14.0000i q^{71} -3.00000i q^{73} -8.00000 q^{75} -4.00000 q^{77} +5.00000 q^{79} -11.0000 q^{81} +5.00000i q^{83} +2.00000i q^{85} +10.0000 q^{87} +9.00000i q^{89} +18.0000i q^{93} +1.00000 q^{95} -1.00000i q^{97} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^9	\( 2 q - 4 q^{3} + 2 q^{9} + 4 q^{17} + 14 q^{23} + 8 q^{25} + 8 q^{27} - 10 q^{29} - 2 q^{35} - 2 q^{43} - 2 q^{49} - 8 q^{51} + 6 q^{53} - 8 q^{55} + 28 q^{61} - 28 q^{69} - 16 q^{75} - 8 q^{77} + 10 q^{79} - 22 q^{81} + 20 q^{87} + 2 q^{95}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^9 + 4 * q^17 + 14 * q^23 + 8 * q^25 + 8 * q^27 - 10 * q^29 - 2 * q^35 - 2 * q^43 - 2 * q^49 - 8 * q^51 + 6 * q^53 - 8 * q^55 + 28 * q^61 - 28 * q^69 - 16 * q^75 - 8 * q^77 + 10 * q^79 - 22 * q^81 + 20 * q^87 + 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more