LMFDB - Embedded newform 3120.2.l.i.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1249,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3120.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (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:	$24.9133254306$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3120.1249
Dual form			3120.2.l.i.1249.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +1.00000i q^{27} +10.0000 q^{29} -4.00000 q^{31} -6.00000i q^{33} +(4.00000 - 8.00000i) q^{35} +6.00000i q^{37} +1.00000 q^{39} +10.0000 q^{41} +(-2.00000 - 1.00000i) q^{45} +8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} -2.00000i q^{57} -6.00000 q^{59} -6.00000 q^{61} +4.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -12.0000i q^{67} +6.00000 q^{69} +2.00000i q^{73} +(4.00000 - 3.00000i) q^{75} -24.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +(-4.00000 + 8.00000i) q^{85} -10.0000i q^{87} -14.0000 q^{89} +4.00000 q^{91} +4.00000i q^{93} +(4.00000 + 2.00000i) q^{95} -14.0000i q^{97} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^9	$ 2 q + 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 20 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{39} + 20 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} + 24 q^{55} - 12 q^{59} - 12 q^{61} - 2 q^{65} + 12 q^{69} + 8 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 + 12 * q^11 + 2 * q^15 + 4 * q^19 - 8 * q^21 + 6 * q^25 + 20 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^39 + 20 * q^41 - 4 * q^45 - 18 * q^49 + 8 * q^51 + 24 * q^55 - 12 * q^59 - 12 * q^61 - 2 * q^65 + 12 * q^69 + 8 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^85 - 28 * q^89 + 8 * q^91 + 8 * q^95 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	4.00000i		−	1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$		−	4.00000i		−	1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	1.00000	−	2.00000i	0.258199	−	0.516398i
$16$		0			0
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$		−	6.00000i		−	1.04447i
$34$		0			0
$35$	4.00000	−	8.00000i	0.676123	−	1.35225i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	10.0000			1.56174			0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000			1.56174			0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		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
$49$	−9.00000			−1.28571
$50$		0			0
$51$	4.00000			0.560112
$52$		0			0
$53$		−	6.00000i		−	0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$		−	6.00000i		−	0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$		0			0
$55$	12.0000	+	6.00000i	1.61808	+	0.809040i
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$			4.00000i			0.503953i
$64$		0			0
$65$	−1.00000	+	2.00000i	−0.124035	+	0.248069i
$66$		0			0
$67$		−	12.0000i		−	1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$		−	12.0000i		−	1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$		0			0
$69$	6.00000			0.722315
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	4.00000	−	3.00000i	0.461880	−	0.346410i
$76$		0			0
$77$		−	24.0000i		−	2.73505i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	−4.00000	+	8.00000i	−0.433861	+	0.867722i
$86$		0			0
$87$		−	10.0000i		−	1.07211i
$88$		0			0
$89$	−14.0000			−1.48400			−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000			−1.48400			−0.741999	+	0.670402i	$0.766122\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$			4.00000i			0.414781i
$94$		0			0
$95$	4.00000	+	2.00000i	0.410391	+	0.205196i
$96$		0			0
$97$		−	14.0000i		−	1.42148i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$		−	14.0000i		−	1.42148i	0.703452	−	0.710742i	$-0.251641\pi$
$98$		0			0
$99$	−6.00000			−0.603023
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$			2.00000i			0.197066i	0.995134	+	0.0985329i	$0.0314150\pi$
$103$			2.00000i			0.197066i	−0.995134	+	0.0985329i	$0.968585\pi$
$104$		0			0
$105$	−8.00000	−	4.00000i	−0.780720	−	0.390360i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$	6.00000			0.569495
$112$		0			0
$113$		−	4.00000i		−	0.376288i	−0.982141	−	0.188144i	$-0.939753\pi$
$113$		−	4.00000i		−	0.376288i	0.982141	−	0.188144i	$-0.0602472\pi$
$114$		0			0
$115$	−6.00000	+	12.0000i	−0.559503	+	1.11901i
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$	16.0000			1.46672
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		−	10.0000i		−	0.901670i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	10.0000i		−	0.887357i	−0.896186	−	0.443678i	$-0.853673\pi$
$127$		−	10.0000i		−	0.887357i	0.896186	−	0.443678i	$-0.146327\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		−	8.00000i		−	0.693688i
$134$		0			0
$135$	−1.00000	+	2.00000i	−0.0860663	+	0.172133i
$136$		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
$141$	8.00000			0.673722
$142$		0			0
$143$			6.00000i			0.501745i
$144$		0			0
$145$	20.0000	+	10.0000i	1.66091	+	0.830455i
$146$		0			0
$147$			9.00000i			0.742307i
$148$		0			0
$149$	−16.0000			−1.31077			−0.655386	−	0.755295i	$-0.727494\pi$
$149$	−16.0000			−1.31077			−0.655386	+	0.755295i	$0.727494\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	−8.00000	−	4.00000i	−0.642575	−	0.321288i
$156$		0			0
$157$		−	6.00000i		−	0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$		−	6.00000i		−	0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	24.0000			1.89146
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$	6.00000	−	12.0000i	0.467099	−	0.934199i
$166$		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
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	−2.00000			−0.152944
$172$		0			0
$173$		−	22.0000i		−	1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$		−	22.0000i		−	1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$			6.00000i			0.450988i
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$		0			0
$185$	−6.00000	+	12.0000i	−0.441129	+	0.882258i
$186$		0			0
$187$			24.0000i			1.75505i
$188$		0			0
$189$	4.00000			0.290957
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		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
$195$	2.00000	+	1.00000i	0.143223	+	0.0716115i
$196$		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	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−12.0000			−0.846415
$202$		0			0
$203$		−	40.0000i		−	2.80745i
$204$		0			0
$205$	20.0000	+	10.0000i	1.39686	+	0.698430i
$206$		0			0
$207$		−	6.00000i		−	0.417029i
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	2.00000			0.135147
$220$		0			0
$221$	−4.00000			−0.269069
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$		0			0
$227$		−	4.00000i		−	0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		0			0
$229$	−8.00000			−0.528655			−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000			−0.528655			−0.264327	+	0.964433i	$0.585150\pi$
$230$		0			0
$231$	−24.0000			−1.57908
$232$		0			0
$233$		−	12.0000i		−	0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$		−	12.0000i		−	0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$		0			0
$235$	−8.00000	+	16.0000i	−0.521862	+	1.04372i
$236$		0			0
$237$			8.00000i			0.519656i
$238$		0			0
$239$	−4.00000			−0.258738			−0.129369	−	0.991596i	$-0.541295\pi$
$239$	−4.00000			−0.258738			−0.129369	+	0.991596i	$0.541295\pi$
$240$		0			0
$241$	18.0000			1.15948			0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000			1.15948			0.579741	+	0.814801i	$0.303154\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−18.0000	−	9.00000i	−1.14998	−	0.574989i
$246$		0			0
$247$			2.00000i			0.127257i
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$			36.0000i			2.26330i
$254$		0			0
$255$	8.00000	+	4.00000i	0.500979	+	0.250490i
$256$		0			0
$257$		−	28.0000i		−	1.74659i	−0.487190	−	0.873296i	$-0.661978\pi$
$257$		−	28.0000i		−	1.74659i	0.487190	−	0.873296i	$-0.338022\pi$
$258$		0			0
$259$	24.0000			1.49129
$260$		0			0
$261$	−10.0000			−0.618984
$262$		0			0
$263$			10.0000i			0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$			10.0000i			0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$		0			0
$265$	6.00000	−	12.0000i	0.368577	−	0.737154i
$266$		0			0
$267$			14.0000i			0.856786i
$268$		0			0
$269$	−10.0000			−0.609711			−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000			−0.609711			−0.304855	+	0.952399i	$0.598608\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		−	4.00000i		−	0.242091i
$274$		0			0
$275$	18.0000	+	24.0000i	1.08544	+	1.44725i
$276$		0			0
$277$			14.0000i			0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$			14.0000i			0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$		0			0
$279$	4.00000			0.239474
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$	2.00000	−	4.00000i	0.118470	−	0.236940i
$286$		0			0
$287$		−	40.0000i		−	2.36113i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	−14.0000			−0.820695
$292$		0			0
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	−12.0000	−	6.00000i	−0.698667	−	0.349334i
$296$		0			0
$297$			6.00000i			0.348155i
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	10.0000i		−	0.574485i
$304$		0			0
$305$	−12.0000	−	6.00000i	−0.687118	−	0.343559i
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	2.00000			0.113776
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$	−4.00000	+	8.00000i	−0.225374	+	0.450749i
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$	60.0000			3.35936
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.00000i			0.445132i
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$	32.0000			1.76422
$330$		0			0
$331$	−2.00000			−0.109930			−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000			−0.109930			−0.0549650	+	0.998488i	$0.517505\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$	12.0000	−	24.0000i	0.655630	−	1.31126i
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	−4.00000			−0.217250
$340$		0			0
$341$	−24.0000			−1.29967
$342$		0			0
$343$			8.00000i			0.431959i
$344$		0			0
$345$	12.0000	+	6.00000i	0.646058	+	0.323029i
$346$		0			0
$347$			16.0000i			0.858925i	0.903085	+	0.429463i	$0.141297\pi$
$347$			16.0000i			0.858925i	−0.903085	+	0.429463i	$0.858703\pi$
$348$		0			0
$349$	4.00000			0.214115			0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000			0.214115			0.107058	+	0.994253i	$0.465857\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	16.0000i		−	0.846810i
$358$		0			0
$359$	−20.0000			−1.05556			−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000			−1.05556			−0.527780	+	0.849381i	$0.676975\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		−	25.0000i		−	1.31216i
$364$		0			0
$365$	−2.00000	+	4.00000i	−0.104685	+	0.209370i
$366$		0			0
$367$		−	18.0000i		−	0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$		−	18.0000i		−	0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$		0			0
$369$	−10.0000			−0.520579
$370$		0			0
$371$	−24.0000			−1.24602
$372$		0			0
$373$		−	18.0000i		−	0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$		−	18.0000i		−	0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$		0			0
$375$	11.0000	−	2.00000i	0.568038	−	0.103280i
$376$		0			0
$377$			10.0000i			0.515026i
$378$		0			0
$379$	10.0000			0.513665			0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000			0.513665			0.256833	+	0.966456i	$0.417321\pi$
$380$		0			0
$381$	−10.0000			−0.512316
$382$		0			0
$383$		−	20.0000i		−	1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$		−	20.0000i		−	1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$		0			0
$385$	24.0000	−	48.0000i	1.22315	−	2.44631i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−30.0000			−1.52106			−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000			−1.52106			−0.760530	+	0.649303i	$0.775061\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$		−	8.00000i		−	0.403547i
$394$		0			0
$395$	−16.0000	−	8.00000i	−0.805047	−	0.402524i
$396$		0			0
$397$			10.0000i			0.501886i	0.968002	+	0.250943i	$0.0807406\pi$
$397$			10.0000i			0.501886i	−0.968002	+	0.250943i	$0.919259\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$		0			0
$407$			36.0000i			1.78445i
$408$		0			0
$409$	34.0000			1.68119			0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000			1.68119			0.840596	+	0.541663i	$0.182205\pi$
$410$		0			0
$411$	14.0000			0.690569
$412$		0			0
$413$			24.0000i			1.18096i
$414$		0			0
$415$	−4.00000	+	8.00000i	−0.196352	+	0.392705i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$	8.00000			0.390826			0.195413	−	0.980721i	$-0.437395\pi$
$419$	8.00000			0.390826			0.195413	+	0.980721i	$0.437395\pi$
$420$		0			0
$421$	4.00000			0.194948			0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000			0.194948			0.0974740	+	0.995238i	$0.468924\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	−16.0000	+	12.0000i	−0.776114	+	0.582086i
$426$		0			0
$427$			24.0000i			1.16144i
$428$		0			0
$429$	6.00000			0.289683
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$			16.0000i			0.768911i	0.923144	+	0.384455i	$0.125611\pi$
$433$			16.0000i			0.768911i	−0.923144	+	0.384455i	$0.874389\pi$
$434$		0			0
$435$	10.0000	−	20.0000i	0.479463	−	0.958927i
$436$		0			0
$437$			12.0000i			0.574038i
$438$		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
$441$	9.00000			0.428571
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−28.0000	−	14.0000i	−1.32733	−	0.663664i
$446$		0			0
$447$			16.0000i			0.756774i
$448$		0			0
$449$	−38.0000			−1.79333			−0.896665	−	0.442709i	$-0.854018\pi$
$449$	−38.0000			−1.79333			−0.896665	+	0.442709i	$0.854018\pi$
$450$		0			0
$451$	60.0000			2.82529
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$	8.00000	+	4.00000i	0.375046	+	0.187523i
$456$		0			0
$457$		−	14.0000i		−	0.654892i	−0.944870	−	0.327446i	$-0.893812\pi$
$457$		−	14.0000i		−	0.654892i	0.944870	−	0.327446i	$-0.106188\pi$
$458$		0			0
$459$	−4.00000			−0.186704
$460$		0			0
$461$	−4.00000			−0.186299			−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000			−0.186299			−0.0931493	+	0.995652i	$0.529693\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$	−4.00000	+	8.00000i	−0.185496	+	0.370991i
$466$		0			0
$467$			8.00000i			0.370196i	0.982720	+	0.185098i	$0.0592602\pi$
$467$			8.00000i			0.370196i	−0.982720	+	0.185098i	$0.940740\pi$
$468$		0			0
$469$	−48.0000			−2.21643
$470$		0			0
$471$	−6.00000			−0.276465
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.00000	+	8.00000i	0.275299	+	0.367065i
$476$		0			0
$477$			6.00000i			0.274721i
$478$		0			0
$479$	−20.0000			−0.913823			−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000			−0.913823			−0.456912	+	0.889512i	$0.651044\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$		−	24.0000i		−	1.09204i
$484$		0			0
$485$	14.0000	−	28.0000i	0.635707	−	1.27141i
$486$		0			0
$487$			32.0000i			1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$			32.0000i			1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$		0			0
$489$	4.00000			0.180886
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$			40.0000i			1.80151i
$494$		0			0
$495$	−12.0000	−	6.00000i	−0.539360	−	0.269680i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−18.0000			−0.805791			−0.402895	−	0.915246i	$-0.631996\pi$
$499$	−18.0000			−0.805791			−0.402895	+	0.915246i	$0.631996\pi$
$500$		0			0
$501$	−16.0000			−0.714827
$502$		0			0
$503$			14.0000i			0.624229i	0.950044	+	0.312115i	$0.101037\pi$
$503$			14.0000i			0.624229i	−0.950044	+	0.312115i	$0.898963\pi$
$504$		0			0
$505$	20.0000	+	10.0000i	0.889988	+	0.444994i
$506$		0			0
$507$			1.00000i			0.0444116i
$508$		0			0
$509$	−16.0000			−0.709188			−0.354594	−	0.935020i	$-0.615381\pi$
$509$	−16.0000			−0.709188			−0.354594	+	0.935020i	$0.615381\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$			2.00000i			0.0883022i
$514$		0			0
$515$	−2.00000	+	4.00000i	−0.0881305	+	0.176261i
$516$		0			0
$517$			48.0000i			2.11104i
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$		−	16.0000i		−	0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$		0			0
$525$	−12.0000	−	16.0000i	−0.523723	−	0.698297i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	6.00000			0.260378
$532$		0			0
$533$			10.0000i			0.433148i
$534$		0			0
$535$		0			0
$536$		0			0
$537$			24.0000i			1.03568i
$538$		0			0
$539$	−54.0000			−2.32594
$540$		0			0
$541$	−32.0000			−1.37579			−0.687894	−	0.725811i	$-0.741464\pi$
$541$	−32.0000			−1.37579			−0.687894	+	0.725811i	$0.741464\pi$
$542$		0			0
$543$		−	6.00000i		−	0.257485i
$544$		0			0
$545$	−8.00000	−	4.00000i	−0.342682	−	0.171341i
$546$		0			0
$547$			36.0000i			1.53925i	0.638497	+	0.769624i	$0.279557\pi$
$547$			36.0000i			1.53925i	−0.638497	+	0.769624i	$0.720443\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$	20.0000			0.852029
$552$		0			0
$553$			32.0000i			1.36078i
$554$		0			0
$555$	12.0000	+	6.00000i	0.509372	+	0.254686i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	24.0000			1.01328
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	4.00000	−	8.00000i	0.168281	−	0.336563i
$566$		0			0
$567$		−	4.00000i		−	0.167984i
$568$		0			0
$569$	42.0000			1.76073			0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000			1.76073			0.880366	+	0.474295i	$0.157297\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$		−	2.00000i		−	0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$		−	2.00000i		−	0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$		0			0
$579$	−2.00000			−0.0831172
$580$		0			0
$581$	16.0000			0.663792
$582$		0			0
$583$		−	36.0000i		−	1.49097i
$584$		0			0
$585$	1.00000	−	2.00000i	0.0413449	−	0.0826898i
$586$		0			0
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	−8.00000			−0.329634
$590$		0			0
$591$	6.00000			0.246807
$592$		0			0
$593$			14.0000i			0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$			14.0000i			0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$		0			0
$595$	32.0000	+	16.0000i	1.31187	+	0.655936i
$596$		0			0
$597$			8.00000i			0.327418i
$598$		0			0
$599$	−32.0000			−1.30748			−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000			−1.30748			−0.653742	+	0.756717i	$0.726802\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$			12.0000i			0.488678i
$604$		0			0
$605$	50.0000	+	25.0000i	2.03279	+	1.01639i
$606$		0			0
$607$			42.0000i			1.70473i	0.522949	+	0.852364i	$0.324832\pi$
$607$			42.0000i			1.70473i	−0.522949	+	0.852364i	$0.675168\pi$
$608$		0			0
$609$	−40.0000			−1.62088
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$		−	38.0000i		−	1.53481i	−0.641165	−	0.767403i	$-0.721549\pi$
$613$		−	38.0000i		−	1.53481i	0.641165	−	0.767403i	$-0.278451\pi$
$614$		0			0
$615$	10.0000	−	20.0000i	0.403239	−	0.806478i
$616$		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
$619$	−22.0000			−0.884255			−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000			−0.884255			−0.442127	+	0.896952i	$0.645776\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$		0			0
$623$			56.0000i			2.24359i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	12.0000i		−	0.479234i
$628$		0			0
$629$	−24.0000			−0.956943
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$			4.00000i			0.158986i
$634$		0			0
$635$	10.0000	−	20.0000i	0.396838	−	0.793676i
$636$		0			0
$637$		−	9.00000i		−	0.356593i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			20.0000i			0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$			20.0000i			0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	6.00000i		−	0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$		−	6.00000i		−	0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$		0			0
$651$	16.0000			0.627089
$652$		0			0
$653$		−	10.0000i		−	0.391330i	−0.980671	−	0.195665i	$-0.937313\pi$
$653$		−	10.0000i		−	0.391330i	0.980671	−	0.195665i	$-0.0626866\pi$
$654$		0			0
$655$	16.0000	+	8.00000i	0.625172	+	0.312586i
$656$		0			0
$657$		−	2.00000i		−	0.0780274i
$658$		0			0
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$			4.00000i			0.155347i
$664$		0			0
$665$	8.00000	−	16.0000i	0.310227	−	0.620453i
$666$		0			0
$667$			60.0000i			2.32321i
$668$		0			0
$669$	−8.00000			−0.309298
$670$		0			0
$671$	−36.0000			−1.38976
$672$		0			0
$673$			44.0000i			1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$			44.0000i			1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$		0			0
$675$	−4.00000	+	3.00000i	−0.153960	+	0.115470i
$676$		0			0
$677$		−	6.00000i		−	0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$		−	6.00000i		−	0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$		0			0
$679$	−56.0000			−2.14908
$680$		0			0
$681$	−4.00000			−0.153280
$682$		0			0
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−14.0000	+	28.0000i	−0.534913	+	1.06983i
$686$		0			0
$687$			8.00000i			0.305219i
$688$		0			0
$689$	6.00000			0.228582
$690$		0			0
$691$	−18.0000			−0.684752			−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000			−0.684752			−0.342376	+	0.939563i	$0.611232\pi$
$692$		0			0
$693$			24.0000i			0.911685i
$694$		0			0
$695$	32.0000	+	16.0000i	1.21383	+	0.606915i
$696$		0			0
$697$			40.0000i			1.51511i
$698$		0			0
$699$	−12.0000			−0.453882
$700$		0			0
$701$	−6.00000			−0.226617			−0.113308	−	0.993560i	$-0.536145\pi$
$701$	−6.00000			−0.226617			−0.113308	+	0.993560i	$0.536145\pi$
$702$		0			0
$703$			12.0000i			0.452589i
$704$		0			0
$705$	16.0000	+	8.00000i	0.602595	+	0.301297i
$706$		0			0
$707$		−	40.0000i		−	1.50435i
$708$		0			0
$709$	8.00000			0.300446			0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000			0.300446			0.150223	+	0.988652i	$0.452001\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		−	24.0000i		−	0.898807i
$714$		0			0
$715$	−6.00000	+	12.0000i	−0.224387	+	0.448775i
$716$		0			0
$717$			4.00000i			0.149383i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		−	18.0000i		−	0.669427i
$724$		0			0
$725$	30.0000	+	40.0000i	1.11417	+	1.48556i
$726$		0			0
$727$			38.0000i			1.40934i	0.709534	+	0.704671i	$0.248905\pi$
$727$			38.0000i			1.40934i	−0.709534	+	0.704671i	$0.751095\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	18.0000i		−	0.664845i	−0.943131	−	0.332423i	$-0.892134\pi$
$733$		−	18.0000i		−	0.664845i	0.943131	−	0.332423i	$-0.107866\pi$
$734$		0			0
$735$	−9.00000	+	18.0000i	−0.331970	+	0.663940i
$736$		0			0
$737$		−	72.0000i		−	2.65215i
$738$		0			0
$739$	−2.00000			−0.0735712			−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000			−0.0735712			−0.0367856	+	0.999323i	$0.511712\pi$
$740$		0			0
$741$	2.00000			0.0734718
$742$		0			0
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$	−32.0000	−	16.0000i	−1.17239	−	0.586195i
$746$		0			0
$747$		−	4.00000i		−	0.146352i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	48.0000			1.75154			0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000			1.75154			0.875772	+	0.482724i	$0.160353\pi$
$752$		0			0
$753$		−	20.0000i		−	0.728841i
$754$		0			0
$755$	−32.0000	−	16.0000i	−1.16460	−	0.582300i
$756$		0			0
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$		0			0
$759$	36.0000			1.30672
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$			16.0000i			0.579239i
$764$		0			0
$765$	4.00000	−	8.00000i	0.144620	−	0.289241i
$766$		0			0
$767$		−	6.00000i		−	0.216647i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$	−28.0000			−1.00840
$772$		0			0
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$	−12.0000	−	16.0000i	−0.431053	−	0.574737i
$776$		0			0
$777$		−	24.0000i		−	0.860995i
$778$		0			0
$779$	20.0000			0.716574
$780$		0			0
$781$		0			0
$782$		0			0
$783$			10.0000i			0.357371i
$784$		0			0
$785$	6.00000	−	12.0000i	0.214149	−	0.428298i
$786$		0			0
$787$		−	12.0000i		−	0.427754i	−0.976861	−	0.213877i	$-0.931391\pi$
$787$		−	12.0000i		−	0.427754i	0.976861	−	0.213877i	$-0.0686091\pi$
$788$		0			0
$789$	10.0000			0.356009
$790$		0			0
$791$	−16.0000			−0.568895
$792$		0			0
$793$		−	6.00000i		−	0.213066i
$794$		0			0
$795$	−12.0000	−	6.00000i	−0.425596	−	0.212798i
$796$		0			0
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		0			0
$801$	14.0000			0.494666
$802$		0			0
$803$			12.0000i			0.423471i
$804$		0			0
$805$	48.0000	+	24.0000i	1.69178	+	0.845889i
$806$		0			0
$807$			10.0000i			0.352017i
$808$		0			0
$809$	2.00000			0.0703163			0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000			0.0703163			0.0351581	+	0.999382i	$0.488807\pi$
$810$		0			0
$811$	50.0000			1.75574			0.877869	−	0.478901i	$-0.158965\pi$
$811$	50.0000			1.75574			0.877869	+	0.478901i	$0.158965\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	−4.00000	+	8.00000i	−0.140114	+	0.280228i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−4.00000			−0.139771
$820$		0			0
$821$	−28.0000			−0.977207			−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000			−0.977207			−0.488603	+	0.872506i	$0.662493\pi$
$822$		0			0
$823$			14.0000i			0.488009i	0.969774	+	0.244005i	$0.0784612\pi$
$823$			14.0000i			0.488009i	−0.969774	+	0.244005i	$0.921539\pi$
$824$		0			0
$825$	24.0000	−	18.0000i	0.835573	−	0.626680i
$826$		0			0
$827$		−	44.0000i		−	1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$		−	44.0000i		−	1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$		0			0
$829$	−26.0000			−0.903017			−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000			−0.903017			−0.451509	+	0.892267i	$0.649114\pi$
$830$		0			0
$831$	14.0000			0.485655
$832$		0			0
$833$		−	36.0000i		−	1.24733i
$834$		0			0
$835$	16.0000	−	32.0000i	0.553703	−	1.10741i
$836$		0			0
$837$		−	4.00000i		−	0.138260i
$838$		0			0
$839$	40.0000			1.38095			0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000			1.38095			0.690477	+	0.723355i	$0.257401\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$			18.0000i			0.619953i
$844$		0			0
$845$	−2.00000	−	1.00000i	−0.0688021	−	0.0344010i
$846$		0			0
$847$		−	100.000i		−	3.43604i
$848$		0			0
$849$	12.0000			0.411839
$850$		0			0
$851$	−36.0000			−1.23406
$852$		0			0
$853$		−	26.0000i		−	0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$		−	26.0000i		−	0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$		0			0
$855$	−4.00000	−	2.00000i	−0.136797	−	0.0683986i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	28.0000			0.955348			0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000			0.955348			0.477674	+	0.878537i	$0.341480\pi$
$860$		0			0
$861$	−40.0000			−1.36320
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	22.0000	−	44.0000i	0.748022	−	1.49604i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	−48.0000			−1.62829
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$			14.0000i			0.473828i
$874$		0			0
$875$	44.0000	−	8.00000i	1.48747	−	0.270449i
$876$		0			0
$877$		−	18.0000i		−	0.607817i	−0.952701	−	0.303908i	$-0.901708\pi$
$877$		−	18.0000i		−	0.607817i	0.952701	−	0.303908i	$-0.0982917\pi$
$878$		0			0
$879$	18.0000			0.607125
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$		−	40.0000i		−	1.34611i	−0.739594	−	0.673054i	$-0.764982\pi$
$883$		−	40.0000i		−	1.34611i	0.739594	−	0.673054i	$-0.235018\pi$
$884$		0			0
$885$	−6.00000	+	12.0000i	−0.201688	+	0.403376i
$886$		0			0
$887$		−	6.00000i		−	0.201460i	−0.994914	−	0.100730i	$-0.967882\pi$
$887$		−	6.00000i		−	0.201460i	0.994914	−	0.100730i	$-0.0321179\pi$
$888$		0			0
$889$	−40.0000			−1.34156
$890$		0			0
$891$	6.00000			0.201008
$892$		0			0
$893$			16.0000i			0.535420i
$894$		0			0
$895$	−48.0000	−	24.0000i	−1.60446	−	0.802232i
$896$		0			0
$897$			6.00000i			0.200334i
$898$		0			0
$899$	−40.0000			−1.33407
$900$		0			0
$901$	24.0000			0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	12.0000	+	6.00000i	0.398893	+	0.199447i
$906$		0			0
$907$			40.0000i			1.32818i	0.747653	+	0.664089i	$0.231180\pi$
$907$			40.0000i			1.32818i	−0.747653	+	0.664089i	$0.768820\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$			24.0000i			0.794284i
$914$		0			0
$915$	−6.00000	+	12.0000i	−0.198354	+	0.396708i
$916$		0			0
$917$		−	32.0000i		−	1.05673i
$918$		0			0
$919$	−56.0000			−1.84727			−0.923635	−	0.383274i	$-0.874797\pi$
$919$	−56.0000			−1.84727			−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$	−4.00000			−0.131804
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−24.0000	+	18.0000i	−0.789115	+	0.591836i
$926$		0			0
$927$		−	2.00000i		−	0.0656886i
$928$		0			0
$929$	−46.0000			−1.50921			−0.754606	−	0.656179i	$-0.772172\pi$
$929$	−46.0000			−1.50921			−0.754606	+	0.656179i	$0.772172\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$		−	8.00000i		−	0.261908i
$934$		0			0
$935$	−24.0000	+	48.0000i	−0.784884	+	1.56977i
$936$		0			0
$937$		−	48.0000i		−	1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$		−	48.0000i		−	1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$		0			0
$939$	−8.00000			−0.261070
$940$		0			0
$941$	−8.00000			−0.260793			−0.130396	−	0.991462i	$-0.541625\pi$
$941$	−8.00000			−0.260793			−0.130396	+	0.991462i	$0.541625\pi$
$942$		0			0
$943$			60.0000i			1.95387i
$944$		0			0
$945$	8.00000	+	4.00000i	0.260240	+	0.130120i
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	−2.00000			−0.0649227
$950$		0			0
$951$	−2.00000			−0.0648544
$952$		0			0
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	60.0000i		−	1.93952i
$958$		0			0
$959$	56.0000			1.80833
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.00000	−	4.00000i	0.0643823	−	0.128765i
$966$		0			0
$967$		−	4.00000i		−	0.128631i	−0.997930	−	0.0643157i	$-0.979514\pi$
$967$		−	4.00000i		−	0.128631i	0.997930	−	0.0643157i	$-0.0204865\pi$
$968$		0			0
$969$	8.00000			0.256997
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$		−	64.0000i		−	2.05175i
$974$		0			0
$975$	3.00000	+	4.00000i	0.0960769	+	0.128103i
$976$		0			0
$977$		−	58.0000i		−	1.85558i	−0.373097	−	0.927792i	$-0.621704\pi$
$977$		−	58.0000i		−	1.85558i	0.373097	−	0.927792i	$-0.378296\pi$
$978$		0			0
$979$	−84.0000			−2.68465
$980$		0			0
$981$	4.00000			0.127710
$982$		0			0
$983$			28.0000i			0.893061i	0.894768	+	0.446531i	$0.147341\pi$
$983$			28.0000i			0.893061i	−0.894768	+	0.446531i	$0.852659\pi$
$984$		0			0
$985$	−6.00000	+	12.0000i	−0.191176	+	0.382352i
$986$		0			0
$987$		−	32.0000i		−	1.01857i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$			2.00000i			0.0634681i
$994$		0			0
$995$	−16.0000	−	8.00000i	−0.507234	−	0.253617i
$996$		0			0
$997$			10.0000i			0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$			10.0000i			0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$		0			0
$999$	−6.00000			−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.l.i.1249.1		2
4.3	odd	2		390.2.e.d.79.1	✓	2
5.4	even	2	inner	3120.2.l.i.1249.2		2
12.11	even	2		1170.2.e.c.469.2		2
20.3	even	4		1950.2.a.d.1.1		1
20.7	even	4		1950.2.a.v.1.1		1
20.19	odd	2		390.2.e.d.79.2	yes	2
60.23	odd	4		5850.2.a.cc.1.1		1
60.47	odd	4		5850.2.a.e.1.1		1
60.59	even	2		1170.2.e.c.469.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.d.79.1	✓	2	4.3	odd	2
390.2.e.d.79.2	yes	2	20.19	odd	2
1170.2.e.c.469.1		2	60.59	even	2
1170.2.e.c.469.2		2	12.11	even	2
1950.2.a.d.1.1		1	20.3	even	4
1950.2.a.v.1.1		1	20.7	even	4
3120.2.l.i.1249.1		2	1.1	even	1	trivial
3120.2.l.i.1249.2		2	5.4	even	2	inner
5850.2.a.e.1.1		1	60.47	odd	4
5850.2.a.cc.1.1		1	60.23	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -4.00000i q^{7} -1.00000 q^{9} +6.00000 q^{11} +1.00000i q^{13} +(1.00000 - 2.00000i) q^{15} +4.00000i q^{17} +2.00000 q^{19} -4.00000 q^{21} +6.00000i q^{23} +(3.00000 + 4.00000i) q^{25} +1.00000i q^{27} +10.0000 q^{29} -4.00000 q^{31} -6.00000i q^{33} +(4.00000 - 8.00000i) q^{35} +6.00000i q^{37} +1.00000 q^{39} +10.0000 q^{41} +(-2.00000 - 1.00000i) q^{45} +8.00000i q^{47} -9.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} +(12.0000 + 6.00000i) q^{55} -2.00000i q^{57} -6.00000 q^{59} -6.00000 q^{61} +4.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} -12.0000i q^{67} +6.00000 q^{69} +2.00000i q^{73} +(4.00000 - 3.00000i) q^{75} -24.0000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +4.00000i q^{83} +(-4.00000 + 8.00000i) q^{85} -10.0000i q^{87} -14.0000 q^{89} +4.00000 q^{91} +4.00000i q^{93} +(4.00000 + 2.00000i) q^{95} -14.0000i q^{97} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^9	\( 2 q + 4 q^{5} - 2 q^{9} + 12 q^{11} + 2 q^{15} + 4 q^{19} - 8 q^{21} + 6 q^{25} + 20 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{39} + 20 q^{41} - 4 q^{45} - 18 q^{49} + 8 q^{51} + 24 q^{55} - 12 q^{59} - 12 q^{61} - 2 q^{65} + 12 q^{69} + 8 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{85} - 28 q^{89} + 8 q^{91} + 8 q^{95} - 12 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 + 12 * q^11 + 2 * q^15 + 4 * q^19 - 8 * q^21 + 6 * q^25 + 20 * q^29 - 8 * q^31 + 8 * q^35 + 2 * q^39 + 20 * q^41 - 4 * q^45 - 18 * q^49 + 8 * q^51 + 24 * q^55 - 12 * q^59 - 12 * q^61 - 2 * q^65 + 12 * q^69 + 8 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^85 - 28 * q^89 + 8 * q^91 + 8 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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