LMFDB - Embedded newform 1560.2.l.b.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1249,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, 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("1560.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1560.l (of order $2$, degree $1$, 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:	$12.4566627153$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1560.1249
Dual form			1560.2.l.b.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -1.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +7.00000i q^{17} +1.00000 q^{21} -7.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} +4.00000 q^{29} +8.00000 q^{31} +3.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} +5.00000i q^{37} +1.00000 q^{39} -3.00000 q^{41} -8.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -6.00000i q^{47} +6.00000 q^{49} -7.00000 q^{51} -11.0000i q^{53} +(3.00000 - 6.00000i) q^{55} +4.00000 q^{59} +1.00000 q^{61} +1.00000i q^{63} +(-2.00000 - 1.00000i) q^{65} -12.0000i q^{67} +7.00000 q^{69} -9.00000 q^{71} +6.00000i q^{73} +(4.00000 - 3.00000i) q^{75} -3.00000i q^{77} +13.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +(14.0000 + 7.00000i) q^{85} +4.00000i q^{87} -3.00000 q^{89} -1.00000 q^{91} +8.00000i q^{93} +19.0000i q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{9} + 6 q^{11} + 4 q^{15} + 2 q^{21} - 6 q^{25} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 2 q^{39} - 6 q^{41} - 2 q^{45} + 12 q^{49} - 14 q^{51} + 6 q^{55} + 8 q^{59} + 2 q^{61} - 4 q^{65} + 14 q^{69} - 18 q^{71} + 8 q^{75} + 26 q^{79} + 2 q^{81} + 28 q^{85} - 6 q^{89} - 2 q^{91} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 + 6 * q^11 + 4 * q^15 + 2 * q^21 - 6 * q^25 + 8 * q^29 + 16 * q^31 - 4 * q^35 + 2 * q^39 - 6 * q^41 - 2 * q^45 + 12 * q^49 - 14 * q^51 + 6 * q^55 + 8 * q^59 + 2 * q^61 - 4 * q^65 + 14 * q^69 - 18 * q^71 + 8 * q^75 + 26 * q^79 + 2 * q^81 + 28 * q^85 - 6 * q^89 - 2 * q^91 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1560\mathbb{Z}\right)^\times$.

$n$	$391$	$521$	$781$	$937$	$1081$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$			7.00000i			1.69775i	0.528594	+	0.848875i	$0.322719\pi$
$17$			7.00000i			1.69775i	−0.528594	+	0.848875i	$0.677281\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	1.00000			0.218218
$22$		0			0
$23$		−	7.00000i		−	1.45960i	−0.683660	−	0.729800i	$-0.739613\pi$
$23$		−	7.00000i		−	1.45960i	0.683660	−	0.729800i	$-0.260387\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$			3.00000i			0.522233i
$34$		0			0
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$			5.00000i			0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$			5.00000i			0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$		0			0
$39$	1.00000			0.160128
$40$		0			0
$41$	−3.00000			−0.468521			−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000			−0.468521			−0.234261	+	0.972174i	$0.575267\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$	−7.00000			−0.980196
$52$		0			0
$53$		−	11.0000i		−	1.51097i	−0.655168	−	0.755483i	$-0.727402\pi$
$53$		−	11.0000i		−	1.51097i	0.655168	−	0.755483i	$-0.272598\pi$
$54$		0			0
$55$	3.00000	−	6.00000i	0.404520	−	0.809040i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000			0.520756			0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000			0.520756			0.260378	+	0.965507i	$0.416153\pi$
$60$		0			0
$61$	1.00000			0.128037			0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000			0.128037			0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−2.00000	−	1.00000i	−0.248069	−	0.124035i
$66$		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
$69$	7.00000			0.842701
$70$		0			0
$71$	−9.00000			−1.06810			−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000			−1.06810			−0.534052	+	0.845452i	$0.679331\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	4.00000	−	3.00000i	0.461880	−	0.346410i
$76$		0			0
$77$		−	3.00000i		−	0.341882i
$78$		0			0
$79$	13.0000			1.46261			0.731307	−	0.682048i	$-0.238911\pi$
$79$	13.0000			1.46261			0.731307	+	0.682048i	$0.238911\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		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
$85$	14.0000	+	7.00000i	1.51851	+	0.759257i
$86$		0			0
$87$			4.00000i			0.428845i
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$			8.00000i			0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			19.0000i			1.92916i	0.263795	+	0.964579i	$0.415026\pi$
$97$			19.0000i			1.92916i	−0.263795	+	0.964579i	$0.584974\pi$
$98$		0			0
$99$	−3.00000			−0.301511
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$		0			0
$105$	1.00000	−	2.00000i	0.0975900	−	0.195180i
$106$		0			0
$107$			15.0000i			1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$			15.0000i			1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$	−5.00000			−0.474579
$112$		0			0
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$	−14.0000	−	7.00000i	−1.30551	−	0.652753i
$116$		0			0
$117$			1.00000i			0.0924500i
$118$		0			0
$119$	7.00000			0.641689
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		−	3.00000i		−	0.270501i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	6.00000i		−	0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$		−	6.00000i		−	0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.00000	−	1.00000i	−0.172133	−	0.0860663i
$136$		0			0
$137$		−	6.00000i		−	0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$		−	6.00000i		−	0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$		0			0
$139$	11.0000			0.933008			0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000			0.933008			0.466504	+	0.884519i	$0.345513\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$		−	3.00000i		−	0.250873i
$144$		0			0
$145$	4.00000	−	8.00000i	0.332182	−	0.664364i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	1.00000			0.0819232			0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000			0.0819232			0.0409616	+	0.999161i	$0.486958\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		−	7.00000i		−	0.565916i
$154$		0			0
$155$	8.00000	−	16.0000i	0.642575	−	1.28515i
$156$		0			0
$157$			16.0000i			1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$157$			16.0000i			1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$158$		0			0
$159$	11.0000			0.872357
$160$		0			0
$161$	−7.00000			−0.551677
$162$		0			0
$163$			21.0000i			1.64485i	0.568876	+	0.822423i	$0.307379\pi$
$163$			21.0000i			1.64485i	−0.568876	+	0.822423i	$0.692621\pi$
$164$		0			0
$165$	6.00000	+	3.00000i	0.467099	+	0.233550i
$166$		0			0
$167$		−	14.0000i		−	1.08335i	−0.840587	−	0.541676i	$-0.817790\pi$
$167$		−	14.0000i		−	1.08335i	0.840587	−	0.541676i	$-0.182210\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$	−4.00000	+	3.00000i	−0.302372	+	0.226779i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	1.00000			0.0743294			0.0371647	−	0.999309i	$-0.488167\pi$
$181$	1.00000			0.0743294			0.0371647	+	0.999309i	$0.488167\pi$
$182$		0			0
$183$			1.00000i			0.0739221i
$184$		0			0
$185$	10.0000	+	5.00000i	0.735215	+	0.367607i
$186$		0			0
$187$			21.0000i			1.53567i
$188$		0			0
$189$	−1.00000			−0.0727393
$190$		0			0
$191$	6.00000			0.434145			0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000			0.434145			0.217072	+	0.976156i	$0.430349\pi$
$192$		0			0
$193$			11.0000i			0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$			11.0000i			0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$		0			0
$195$	1.00000	−	2.00000i	0.0716115	−	0.143223i
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		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
$203$		−	4.00000i		−	0.280745i
$204$		0			0
$205$	−3.00000	+	6.00000i	−0.209529	+	0.419058i
$206$		0			0
$207$			7.00000i			0.486534i
$208$		0			0
$209$		0			0
$210$		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
$213$		−	9.00000i		−	0.616670i
$214$		0			0
$215$	−16.0000	−	8.00000i	−1.09119	−	0.545595i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		0			0
$219$	−6.00000			−0.405442
$220$		0			0
$221$	7.00000			0.470871
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$		0			0
$231$	3.00000			0.197386
$232$		0			0
$233$			11.0000i			0.720634i	0.932830	+	0.360317i	$0.117331\pi$
$233$			11.0000i			0.720634i	−0.932830	+	0.360317i	$0.882669\pi$
$234$		0			0
$235$	−12.0000	−	6.00000i	−0.782794	−	0.391397i
$236$		0			0
$237$			13.0000i			0.844441i
$238$		0			0
$239$	−27.0000			−1.74648			−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000			−1.74648			−0.873242	+	0.487286i	$0.837987\pi$
$240$		0			0
$241$	20.0000			1.28831			0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000			1.28831			0.644157	+	0.764894i	$0.277208\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	6.00000	−	12.0000i	0.383326	−	0.766652i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.00000			0.126745
$250$		0			0
$251$	−22.0000			−1.38863			−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000			−1.38863			−0.694314	+	0.719672i	$0.744292\pi$
$252$		0			0
$253$		−	21.0000i		−	1.32026i
$254$		0			0
$255$	−7.00000	+	14.0000i	−0.438357	+	0.876714i
$256$		0			0
$257$		−	26.0000i		−	1.62184i	−0.585160	−	0.810918i	$-0.698968\pi$
$257$		−	26.0000i		−	1.62184i	0.585160	−	0.810918i	$-0.301032\pi$
$258$		0			0
$259$	5.00000			0.310685
$260$		0			0
$261$	−4.00000			−0.247594
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−22.0000	−	11.0000i	−1.35145	−	0.675725i
$266$		0			0
$267$		−	3.00000i		−	0.183597i
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	22.0000			1.33640			0.668202	−	0.743980i	$-0.267064\pi$
$271$	22.0000			1.33640			0.668202	+	0.743980i	$0.267064\pi$
$272$		0			0
$273$		−	1.00000i		−	0.0605228i
$274$		0			0
$275$	−9.00000	−	12.0000i	−0.542720	−	0.723627i
$276$		0			0
$277$			26.0000i			1.56219i	0.624413	+	0.781094i	$0.285338\pi$
$277$			26.0000i			1.56219i	−0.624413	+	0.781094i	$0.714662\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		0			0
$281$	−22.0000			−1.31241			−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000			−1.31241			−0.656205	+	0.754583i	$0.727839\pi$
$282$		0			0
$283$		−	14.0000i		−	0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$		−	14.0000i		−	0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.00000i			0.177084i
$288$		0			0
$289$	−32.0000			−1.88235
$290$		0			0
$291$	−19.0000			−1.11380
$292$		0			0
$293$			24.0000i			1.40209i	0.713115	+	0.701047i	$0.247284\pi$
$293$			24.0000i			1.40209i	−0.713115	+	0.701047i	$0.752716\pi$
$294$		0			0
$295$	4.00000	−	8.00000i	0.232889	−	0.465778i
$296$		0			0
$297$		−	3.00000i		−	0.174078i
$298$		0			0
$299$	−7.00000			−0.404820
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$			14.0000i			0.804279i
$304$		0			0
$305$	1.00000	−	2.00000i	0.0572598	−	0.114520i
$306$		0			0
$307$			11.0000i			0.627803i	0.949456	+	0.313902i	$0.101636\pi$
$307$			11.0000i			0.627803i	−0.949456	+	0.313902i	$0.898364\pi$
$308$		0			0
$309$	−10.0000			−0.568880
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$			16.0000i			0.904373i	0.891923	+	0.452187i	$0.149356\pi$
$313$			16.0000i			0.904373i	−0.891923	+	0.452187i	$0.850644\pi$
$314$		0			0
$315$	2.00000	+	1.00000i	0.112687	+	0.0563436i
$316$		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
$319$	12.0000			0.671871
$320$		0			0
$321$	−15.0000			−0.837218
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.00000			−0.330791
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$		−	5.00000i		−	0.273998i
$334$		0			0
$335$	−24.0000	−	12.0000i	−1.31126	−	0.655630i
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$	18.0000			0.977626
$340$		0			0
$341$	24.0000			1.29967
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$	7.00000	−	14.0000i	0.376867	−	0.753735i
$346$		0			0
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	−1.00000			−0.0533761
$352$		0			0
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		0			0
$355$	−9.00000	+	18.0000i	−0.477670	+	0.955341i
$356$		0			0
$357$			7.00000i			0.370479i
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	12.0000	+	6.00000i	0.628109	+	0.314054i
$366$		0			0
$367$			18.0000i			0.939592i	0.882775	+	0.469796i	$0.155673\pi$
$367$			18.0000i			0.939592i	−0.882775	+	0.469796i	$0.844327\pi$
$368$		0			0
$369$	3.00000			0.156174
$370$		0			0
$371$	−11.0000			−0.571092
$372$		0			0
$373$		−	2.00000i		−	0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$		−	2.00000i		−	0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$		0			0
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		0			0
$383$			14.0000i			0.715367i	0.933843	+	0.357683i	$0.116433\pi$
$383$			14.0000i			0.715367i	−0.933843	+	0.357683i	$0.883567\pi$
$384$		0			0
$385$	−6.00000	−	3.00000i	−0.305788	−	0.152894i
$386$		0			0
$387$			8.00000i			0.406663i
$388$		0			0
$389$	−28.0000			−1.41966			−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000			−1.41966			−0.709828	+	0.704375i	$0.751227\pi$
$390$		0			0
$391$	49.0000			2.47804
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13.0000	−	26.0000i	0.654101	−	1.30820i
$396$		0			0
$397$			33.0000i			1.65622i	0.560564	+	0.828111i	$0.310584\pi$
$397$			33.0000i			1.65622i	−0.560564	+	0.828111i	$0.689416\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−34.0000			−1.69788			−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000			−1.69788			−0.848939	+	0.528490i	$0.822758\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$			15.0000i			0.743522i
$408$		0			0
$409$	−24.0000			−1.18672			−0.593362	−	0.804936i	$-0.702200\pi$
$409$	−24.0000			−1.18672			−0.593362	+	0.804936i	$0.702200\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$		0			0
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	−4.00000	−	2.00000i	−0.196352	−	0.0981761i
$416$		0			0
$417$			11.0000i			0.538672i
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	18.0000			0.877266			0.438633	−	0.898666i	$-0.355463\pi$
$421$	18.0000			0.877266			0.438633	+	0.898666i	$0.355463\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	28.0000	−	21.0000i	1.35820	−	1.01865i
$426$		0			0
$427$		−	1.00000i		−	0.0483934i
$428$		0			0
$429$	3.00000			0.144841
$430$		0			0
$431$	−36.0000			−1.73406			−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000			−1.73406			−0.867029	+	0.498257i	$0.833974\pi$
$432$		0			0
$433$		−	30.0000i		−	1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$		−	30.0000i		−	1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$		0			0
$435$	8.00000	+	4.00000i	0.383571	+	0.191785i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−13.0000			−0.620456			−0.310228	−	0.950662i	$-0.600405\pi$
$439$	−13.0000			−0.620456			−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$	−6.00000			−0.285714
$442$		0			0
$443$			9.00000i			0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$			9.00000i			0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$		0			0
$445$	−3.00000	+	6.00000i	−0.142214	+	0.284427i
$446$		0			0
$447$			1.00000i			0.0472984i
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	−9.00000			−0.423793
$452$		0			0
$453$		−	2.00000i		−	0.0939682i
$454$		0			0
$455$	−1.00000	+	2.00000i	−0.0468807	+	0.0937614i
$456$		0			0
$457$			3.00000i			0.140334i	0.997535	+	0.0701670i	$0.0223532\pi$
$457$			3.00000i			0.140334i	−0.997535	+	0.0701670i	$0.977647\pi$
$458$		0			0
$459$	7.00000			0.326732
$460$		0			0
$461$	−23.0000			−1.07122			−0.535608	−	0.844466i	$-0.679918\pi$
$461$	−23.0000			−1.07122			−0.535608	+	0.844466i	$0.679918\pi$
$462$		0			0
$463$			29.0000i			1.34774i	0.738848	+	0.673872i	$0.235370\pi$
$463$			29.0000i			1.34774i	−0.738848	+	0.673872i	$0.764630\pi$
$464$		0			0
$465$	16.0000	+	8.00000i	0.741982	+	0.370991i
$466$		0			0
$467$		−	5.00000i		−	0.231372i	−0.993286	−	0.115686i	$-0.963093\pi$
$467$		−	5.00000i		−	0.231372i	0.993286	−	0.115686i	$-0.0369067\pi$
$468$		0			0
$469$	−12.0000			−0.554109
$470$		0			0
$471$	−16.0000			−0.737241
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			11.0000i			0.503655i
$478$		0			0
$479$	−5.00000			−0.228456			−0.114228	−	0.993455i	$-0.536439\pi$
$479$	−5.00000			−0.228456			−0.114228	+	0.993455i	$0.536439\pi$
$480$		0			0
$481$	5.00000			0.227980
$482$		0			0
$483$		−	7.00000i		−	0.318511i
$484$		0			0
$485$	38.0000	+	19.0000i	1.72549	+	0.862746i
$486$		0			0
$487$		−	27.0000i		−	1.22349i	−0.791056	−	0.611743i	$-0.790469\pi$
$487$		−	27.0000i		−	1.22349i	0.791056	−	0.611743i	$-0.209531\pi$
$488$		0			0
$489$	−21.0000			−0.949653
$490$		0			0
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$			28.0000i			1.26106i
$494$		0			0
$495$	−3.00000	+	6.00000i	−0.134840	+	0.269680i
$496$		0			0
$497$			9.00000i			0.403705i
$498$		0			0
$499$	18.0000			0.805791			0.402895	−	0.915246i	$-0.368004\pi$
$499$	18.0000			0.805791			0.402895	+	0.915246i	$0.368004\pi$
$500$		0			0
$501$	14.0000			0.625474
$502$		0			0
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$		0			0
$505$	14.0000	−	28.0000i	0.622992	−	1.24598i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	−11.0000			−0.487566			−0.243783	−	0.969830i	$-0.578389\pi$
$509$	−11.0000			−0.487566			−0.243783	+	0.969830i	$0.578389\pi$
$510$		0			0
$511$	6.00000			0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.0000	+	10.0000i	0.881305	+	0.440653i
$516$		0			0
$517$		−	18.0000i		−	0.791639i
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		0			0
$525$	−3.00000	−	4.00000i	−0.130931	−	0.174574i
$526$		0			0
$527$			56.0000i			2.43940i
$528$		0			0
$529$	−26.0000			−1.13043
$530$		0			0
$531$	−4.00000			−0.173585
$532$		0			0
$533$			3.00000i			0.129944i
$534$		0			0
$535$	30.0000	+	15.0000i	1.29701	+	0.648507i
$536$		0			0
$537$		−	10.0000i		−	0.431532i
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$	20.0000			0.859867			0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000			0.859867			0.429934	+	0.902861i	$0.358537\pi$
$542$		0			0
$543$			1.00000i			0.0429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$	−1.00000			−0.0426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	13.0000i		−	0.552816i
$554$		0			0
$555$	−5.00000	+	10.0000i	−0.212238	+	0.424476i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	−21.0000			−0.886621
$562$		0			0
$563$		−	27.0000i		−	1.13791i	−0.822367	−	0.568957i	$-0.807347\pi$
$563$		−	27.0000i		−	1.13791i	0.822367	−	0.568957i	$-0.192653\pi$
$564$		0			0
$565$	−36.0000	−	18.0000i	−1.51453	−	0.757266i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	36.0000			1.50920			0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000			1.50920			0.754599	+	0.656186i	$0.227831\pi$
$570$		0			0
$571$	19.0000			0.795125			0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000			0.795125			0.397563	+	0.917575i	$0.369856\pi$
$572$		0			0
$573$			6.00000i			0.250654i
$574$		0			0
$575$	−28.0000	+	21.0000i	−1.16768	+	0.875761i
$576$		0			0
$577$		−	37.0000i		−	1.54033i	−0.637845	−	0.770165i	$-0.720174\pi$
$577$		−	37.0000i		−	1.54033i	0.637845	−	0.770165i	$-0.279826\pi$
$578$		0			0
$579$	−11.0000			−0.457144
$580$		0			0
$581$	−2.00000			−0.0829740
$582$		0			0
$583$		−	33.0000i		−	1.36672i
$584$		0			0
$585$	2.00000	+	1.00000i	0.0826898	+	0.0413449i
$586$		0			0
$587$			26.0000i			1.07313i	0.843857	+	0.536567i	$0.180279\pi$
$587$			26.0000i			1.07313i	−0.843857	+	0.536567i	$0.819721\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−2.00000			−0.0822690
$592$		0			0
$593$		−	36.0000i		−	1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	7.00000	−	14.0000i	0.286972	−	0.573944i
$596$		0			0
$597$		−	8.00000i		−	0.327418i
$598$		0			0
$599$	6.00000			0.245153			0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000			0.245153			0.122577	+	0.992459i	$0.460884\pi$
$600$		0			0
$601$	23.0000			0.938190			0.469095	−	0.883148i	$-0.344580\pi$
$601$	23.0000			0.938190			0.469095	+	0.883148i	$0.344580\pi$
$602$		0			0
$603$			12.0000i			0.488678i
$604$		0			0
$605$	−2.00000	+	4.00000i	−0.0813116	+	0.162623i
$606$		0			0
$607$			8.00000i			0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$			8.00000i			0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$		0			0
$609$	4.00000			0.162088
$610$		0			0
$611$	−6.00000			−0.242734
$612$		0			0
$613$			9.00000i			0.363507i	0.983344	+	0.181753i	$0.0581772\pi$
$613$			9.00000i			0.363507i	−0.983344	+	0.181753i	$0.941823\pi$
$614$		0			0
$615$	−6.00000	−	3.00000i	−0.241943	−	0.120972i
$616$		0			0
$617$			36.0000i			1.44931i	0.689114	+	0.724653i	$0.258000\pi$
$617$			36.0000i			1.44931i	−0.689114	+	0.724653i	$0.742000\pi$
$618$		0			0
$619$	2.00000			0.0803868			0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000			0.0803868			0.0401934	+	0.999192i	$0.487203\pi$
$620$		0			0
$621$	−7.00000			−0.280900
$622$		0			0
$623$			3.00000i			0.120192i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−35.0000			−1.39554
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		−	4.00000i		−	0.158986i
$634$		0			0
$635$	−12.0000	−	6.00000i	−0.476205	−	0.238103i
$636$		0			0
$637$		−	6.00000i		−	0.237729i
$638$		0			0
$639$	9.00000			0.356034
$640$		0			0
$641$	−12.0000			−0.473972			−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000			−0.473972			−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$		−	19.0000i		−	0.749287i	−0.927169	−	0.374643i	$-0.877765\pi$
$643$		−	19.0000i		−	0.749287i	0.927169	−	0.374643i	$-0.122235\pi$
$644$		0			0
$645$	8.00000	−	16.0000i	0.315000	−	0.629999i
$646$		0			0
$647$			9.00000i			0.353827i	0.984226	+	0.176913i	$0.0566112\pi$
$647$			9.00000i			0.353827i	−0.984226	+	0.176913i	$0.943389\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	8.00000			0.313545
$652$		0			0
$653$			26.0000i			1.01746i	0.860927	+	0.508729i	$0.169885\pi$
$653$			26.0000i			1.01746i	−0.860927	+	0.508729i	$0.830115\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	6.00000i		−	0.234082i
$658$		0			0
$659$	−16.0000			−0.623272			−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000			−0.623272			−0.311636	+	0.950202i	$0.600877\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$			7.00000i			0.271857i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	28.0000i		−	1.08416i
$668$		0			0
$669$	16.0000			0.618596
$670$		0			0
$671$	3.00000			0.115814
$672$		0			0
$673$		−	34.0000i		−	1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$		−	34.0000i		−	1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$		0			0
$675$	−4.00000	+	3.00000i	−0.153960	+	0.115470i
$676$		0			0
$677$		−	5.00000i		−	0.192166i	−0.995373	−	0.0960828i	$-0.969369\pi$
$677$		−	5.00000i		−	0.192166i	0.995373	−	0.0960828i	$-0.0306314\pi$
$678$		0			0
$679$	19.0000			0.729153
$680$		0			0
$681$		0			0
$682$		0			0
$683$			30.0000i			1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$			30.0000i			1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$		0			0
$685$	−12.0000	−	6.00000i	−0.458496	−	0.229248i
$686$		0			0
$687$		−	22.0000i		−	0.839352i
$688$		0			0
$689$	−11.0000			−0.419067
$690$		0			0
$691$	2.00000			0.0760836			0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000			0.0760836			0.0380418	+	0.999276i	$0.487888\pi$
$692$		0			0
$693$			3.00000i			0.113961i
$694$		0			0
$695$	11.0000	−	22.0000i	0.417254	−	0.834508i
$696$		0			0
$697$		−	21.0000i		−	0.795432i
$698$		0			0
$699$	−11.0000			−0.416058
$700$		0			0
$701$	36.0000			1.35970			0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000			1.35970			0.679851	+	0.733351i	$0.262045\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	6.00000	−	12.0000i	0.225973	−	0.451946i
$706$		0			0
$707$		−	14.0000i		−	0.526524i
$708$		0			0
$709$	4.00000			0.150223			0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000			0.150223			0.0751116	+	0.997175i	$0.476069\pi$
$710$		0			0
$711$	−13.0000			−0.487538
$712$		0			0
$713$		−	56.0000i		−	2.09722i
$714$		0			0
$715$	−6.00000	−	3.00000i	−0.224387	−	0.112194i
$716$		0			0
$717$		−	27.0000i		−	1.00833i
$718$		0			0
$719$	22.0000			0.820462			0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000			0.820462			0.410231	+	0.911982i	$0.365448\pi$
$720$		0			0
$721$	10.0000			0.372419
$722$		0			0
$723$			20.0000i			0.743808i
$724$		0			0
$725$	−12.0000	−	16.0000i	−0.445669	−	0.594225i
$726$		0			0
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	56.0000			2.07123
$732$		0			0
$733$		−	17.0000i		−	0.627909i	−0.949438	−	0.313955i	$-0.898346\pi$
$733$		−	17.0000i		−	0.627909i	0.949438	−	0.313955i	$-0.101654\pi$
$734$		0			0
$735$	12.0000	+	6.00000i	0.442627	+	0.221313i
$736$		0			0
$737$		−	36.0000i		−	1.32608i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$	1.00000	−	2.00000i	0.0366372	−	0.0732743i
$746$		0			0
$747$			2.00000i			0.0731762i
$748$		0			0
$749$	15.0000			0.548088
$750$		0			0
$751$	17.0000			0.620339			0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000			0.620339			0.310169	+	0.950681i	$0.399614\pi$
$752$		0			0
$753$		−	22.0000i		−	0.801725i
$754$		0			0
$755$	−2.00000	+	4.00000i	−0.0727875	+	0.145575i
$756$		0			0
$757$			38.0000i			1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$			38.0000i			1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$		0			0
$759$	21.0000			0.762252
$760$		0			0
$761$	14.0000			0.507500			0.253750	−	0.967270i	$-0.418336\pi$
$761$	14.0000			0.507500			0.253750	+	0.967270i	$0.418336\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−14.0000	−	7.00000i	−0.506171	−	0.253086i
$766$		0			0
$767$		−	4.00000i		−	0.144432i
$768$		0			0
$769$	−18.0000			−0.649097			−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000			−0.649097			−0.324548	+	0.945869i	$0.605212\pi$
$770$		0			0
$771$	26.0000			0.936367
$772$		0			0
$773$			34.0000i			1.22290i	0.791285	+	0.611448i	$0.209412\pi$
$773$			34.0000i			1.22290i	−0.791285	+	0.611448i	$0.790588\pi$
$774$		0			0
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$			5.00000i			0.179374i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−27.0000			−0.966136
$782$		0			0
$783$		−	4.00000i		−	0.142948i
$784$		0			0
$785$	32.0000	+	16.0000i	1.14213	+	0.571064i
$786$		0			0
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$	−18.0000			−0.640006
$792$		0			0
$793$		−	1.00000i		−	0.0355110i
$794$		0			0
$795$	11.0000	−	22.0000i	0.390130	−	0.780260i
$796$		0			0
$797$		−	39.0000i		−	1.38145i	−0.723117	−	0.690725i	$-0.757291\pi$
$797$		−	39.0000i		−	1.38145i	0.723117	−	0.690725i	$-0.242709\pi$
$798$		0			0
$799$	42.0000			1.48585
$800$		0			0
$801$	3.00000			0.106000
$802$		0			0
$803$			18.0000i			0.635206i
$804$		0			0
$805$	−7.00000	+	14.0000i	−0.246718	+	0.493435i
$806$		0			0
$807$			6.00000i			0.211210i
$808$		0			0
$809$	12.0000			0.421898			0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000			0.421898			0.210949	+	0.977497i	$0.432345\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$			22.0000i			0.771574i
$814$		0			0
$815$	42.0000	+	21.0000i	1.47120	+	0.735598i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	1.00000			0.0349428
$820$		0			0
$821$	−35.0000			−1.22151			−0.610754	−	0.791820i	$-0.709134\pi$
$821$	−35.0000			−1.22151			−0.610754	+	0.791820i	$0.709134\pi$
$822$		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
$825$	12.0000	−	9.00000i	0.417786	−	0.313340i
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	−50.0000			−1.73657			−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000			−1.73657			−0.868286	+	0.496064i	$0.834778\pi$
$830$		0			0
$831$	−26.0000			−0.901930
$832$		0			0
$833$			42.0000i			1.45521i
$834$		0			0
$835$	−28.0000	−	14.0000i	−0.968980	−	0.484490i
$836$		0			0
$837$		−	8.00000i		−	0.276520i
$838$		0			0
$839$	5.00000			0.172619			0.0863096	−	0.996268i	$-0.472493\pi$
$839$	5.00000			0.172619			0.0863096	+	0.996268i	$0.472493\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		−	22.0000i		−	0.757720i
$844$		0			0
$845$	−1.00000	+	2.00000i	−0.0344010	+	0.0688021i
$846$		0			0
$847$			2.00000i			0.0687208i
$848$		0			0
$849$	14.0000			0.480479
$850$		0			0
$851$	35.0000			1.19978
$852$		0			0
$853$			13.0000i			0.445112i	0.974920	+	0.222556i	$0.0714399\pi$
$853$			13.0000i			0.445112i	−0.974920	+	0.222556i	$0.928560\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	23.0000i		−	0.785665i	−0.919610	−	0.392833i	$-0.871495\pi$
$857$		−	23.0000i		−	0.785665i	0.919610	−	0.392833i	$-0.128505\pi$
$858$		0			0
$859$	23.0000			0.784750			0.392375	−	0.919805i	$-0.371654\pi$
$859$	23.0000			0.784750			0.392375	+	0.919805i	$0.371654\pi$
$860$		0			0
$861$	−3.00000			−0.102240
$862$		0			0
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$	12.0000	+	6.00000i	0.408012	+	0.204006i
$866$		0			0
$867$		−	32.0000i		−	1.08678i
$868$		0			0
$869$	39.0000			1.32298
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		−	19.0000i		−	0.643053i
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$	−24.0000			−0.809500
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	26.0000i		−	0.874970i	−0.899226	−	0.437485i	$-0.855869\pi$
$883$		−	26.0000i		−	0.874970i	0.899226	−	0.437485i	$-0.144131\pi$
$884$		0			0
$885$	8.00000	+	4.00000i	0.268917	+	0.134459i
$886$		0			0
$887$			53.0000i			1.77957i	0.456384	+	0.889783i	$0.349144\pi$
$887$			53.0000i			1.77957i	−0.456384	+	0.889783i	$0.650856\pi$
$888$		0			0
$889$	−6.00000			−0.201234
$890$		0			0
$891$	3.00000			0.100504
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−10.0000	+	20.0000i	−0.334263	+	0.668526i
$896$		0			0
$897$		−	7.00000i		−	0.233723i
$898$		0			0
$899$	32.0000			1.06726
$900$		0			0
$901$	77.0000			2.56524
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	1.00000	−	2.00000i	0.0332411	−	0.0664822i
$906$		0			0
$907$			42.0000i			1.39459i	0.716786	+	0.697294i	$0.245613\pi$
$907$			42.0000i			1.39459i	−0.716786	+	0.697294i	$0.754387\pi$
$908$		0			0
$909$	−14.0000			−0.464351
$910$		0			0
$911$	38.0000			1.25900			0.629498	−	0.777002i	$-0.283261\pi$
$911$	38.0000			1.25900			0.629498	+	0.777002i	$0.283261\pi$
$912$		0			0
$913$		−	6.00000i		−	0.198571i
$914$		0			0
$915$	2.00000	+	1.00000i	0.0661180	+	0.0330590i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−11.0000			−0.362857			−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000			−0.362857			−0.181428	+	0.983404i	$0.558072\pi$
$920$		0			0
$921$	−11.0000			−0.362462
$922$		0			0
$923$			9.00000i			0.296239i
$924$		0			0
$925$	20.0000	−	15.0000i	0.657596	−	0.493197i
$926$		0			0
$927$		−	10.0000i		−	0.328443i
$928$		0			0
$929$	−15.0000			−0.492134			−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000			−0.492134			−0.246067	+	0.969253i	$0.579138\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	20.0000i		−	0.654771i
$934$		0			0
$935$	42.0000	+	21.0000i	1.37355	+	0.686773i
$936$		0			0
$937$		−	26.0000i		−	0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$		−	26.0000i		−	0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$		0			0
$939$	−16.0000			−0.522140
$940$		0			0
$941$	−1.00000			−0.0325991			−0.0162995	−	0.999867i	$-0.505189\pi$
$941$	−1.00000			−0.0325991			−0.0162995	+	0.999867i	$0.505189\pi$
$942$		0			0
$943$			21.0000i			0.683854i
$944$		0			0
$945$	−1.00000	+	2.00000i	−0.0325300	+	0.0650600i
$946$		0			0
$947$		−	18.0000i		−	0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$		−	18.0000i		−	0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$		0			0
$949$	6.00000			0.194768
$950$		0			0
$951$	−20.0000			−0.648544
$952$		0			0
$953$			57.0000i			1.84641i	0.384307	+	0.923206i	$0.374441\pi$
$953$			57.0000i			1.84641i	−0.384307	+	0.923206i	$0.625559\pi$
$954$		0			0
$955$	6.00000	−	12.0000i	0.194155	−	0.388311i
$956$		0			0
$957$			12.0000i			0.387905i
$958$		0			0
$959$	−6.00000			−0.193750
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	15.0000i		−	0.483368i
$964$		0			0
$965$	22.0000	+	11.0000i	0.708205	+	0.354103i
$966$		0			0
$967$			16.0000i			0.514525i	0.966342	+	0.257263i	$0.0828206\pi$
$967$			16.0000i			0.514525i	−0.966342	+	0.257263i	$0.917179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6.00000			−0.192549			−0.0962746	−	0.995355i	$-0.530693\pi$
$971$	−6.00000			−0.192549			−0.0962746	+	0.995355i	$0.530693\pi$
$972$		0			0
$973$		−	11.0000i		−	0.352644i
$974$		0			0
$975$	−3.00000	−	4.00000i	−0.0960769	−	0.128103i
$976$		0			0
$977$		−	34.0000i		−	1.08776i	−0.839164	−	0.543878i	$-0.816955\pi$
$977$		−	34.0000i		−	1.08776i	0.839164	−	0.543878i	$-0.183045\pi$
$978$		0			0
$979$	−9.00000			−0.287641
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	40.0000i		−	1.27580i	−0.770118	−	0.637901i	$-0.779803\pi$
$983$		−	40.0000i		−	1.27580i	0.770118	−	0.637901i	$-0.220197\pi$
$984$		0			0
$985$	4.00000	+	2.00000i	0.127451	+	0.0637253i
$986$		0			0
$987$		−	6.00000i		−	0.190982i
$988$		0			0
$989$	−56.0000			−1.78070
$990$		0			0
$991$	17.0000			0.540023			0.270011	−	0.962857i	$-0.412973\pi$
$991$	17.0000			0.540023			0.270011	+	0.962857i	$0.412973\pi$
$992$		0			0
$993$		−	4.00000i		−	0.126936i
$994$		0			0
$995$	−8.00000	+	16.0000i	−0.253617	+	0.507234i
$996$		0			0
$997$		−	14.0000i		−	0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$		−	14.0000i		−	0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$		0			0
$999$	5.00000			0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.2.l.b.1249.2	yes	2
3.2	odd	2		4680.2.l.b.2809.2		2
4.3	odd	2		3120.2.l.e.1249.1		2
5.2	odd	4		7800.2.a.v.1.1		1
5.3	odd	4		7800.2.a.b.1.1		1
5.4	even	2	inner	1560.2.l.b.1249.1	✓	2
15.14	odd	2		4680.2.l.b.2809.1		2
20.19	odd	2		3120.2.l.e.1249.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.b.1249.1	✓	2	5.4	even	2	inner
1560.2.l.b.1249.2	yes	2	1.1	even	1	trivial
3120.2.l.e.1249.1		2	4.3	odd	2
3120.2.l.e.1249.2		2	20.19	odd	2
4680.2.l.b.2809.1		2	15.14	odd	2
4680.2.l.b.2809.2		2	3.2	odd	2
7800.2.a.b.1.1		1	5.3	odd	4
7800.2.a.v.1.1		1	5.2	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1560.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -1.00000i q^{13} +(2.00000 + 1.00000i) q^{15} +7.00000i q^{17} +1.00000 q^{21} -7.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} +4.00000 q^{29} +8.00000 q^{31} +3.00000i q^{33} +(-2.00000 - 1.00000i) q^{35} +5.00000i q^{37} +1.00000 q^{39} -3.00000 q^{41} -8.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} -6.00000i q^{47} +6.00000 q^{49} -7.00000 q^{51} -11.0000i q^{53} +(3.00000 - 6.00000i) q^{55} +4.00000 q^{59} +1.00000 q^{61} +1.00000i q^{63} +(-2.00000 - 1.00000i) q^{65} -12.0000i q^{67} +7.00000 q^{69} -9.00000 q^{71} +6.00000i q^{73} +(4.00000 - 3.00000i) q^{75} -3.00000i q^{77} +13.0000 q^{79} +1.00000 q^{81} -2.00000i q^{83} +(14.0000 + 7.00000i) q^{85} +4.00000i q^{87} -3.00000 q^{89} -1.00000 q^{91} +8.00000i q^{93} +19.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} + 6 q^{11} + 4 q^{15} + 2 q^{21} - 6 q^{25} + 8 q^{29} + 16 q^{31} - 4 q^{35} + 2 q^{39} - 6 q^{41} - 2 q^{45} + 12 q^{49} - 14 q^{51} + 6 q^{55} + 8 q^{59} + 2 q^{61} - 4 q^{65} + 14 q^{69} - 18 q^{71} + 8 q^{75} + 26 q^{79} + 2 q^{81} + 28 q^{85} - 6 q^{89} - 2 q^{91} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 + 6 * q^11 + 4 * q^15 + 2 * q^21 - 6 * q^25 + 8 * q^29 + 16 * q^31 - 4 * q^35 + 2 * q^39 - 6 * q^41 - 2 * q^45 + 12 * q^49 - 14 * q^51 + 6 * q^55 + 8 * q^59 + 2 * q^61 - 4 * q^65 + 14 * q^69 - 18 * q^71 + 8 * q^75 + 26 * q^79 + 2 * q^81 + 28 * q^85 - 6 * q^89 - 2 * q^91 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more