LMFDB - Embedded newform 960.2.d.d.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(289,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("960.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.d (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:	$7.66563859404$
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:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			289.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	960.289
Dual form			960.2.d.d.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} -4.00000i q^{17} -2.00000i q^{19} +2.00000i q^{21} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +4.00000i q^{29} -8.00000 q^{31} +2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +10.0000 q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +(1.00000 + 2.00000i) q^{45} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{51} +10.0000 q^{53} +(-4.00000 + 2.00000i) q^{55} -2.00000i q^{57} -6.00000i q^{59} -8.00000i q^{61} +2.00000i q^{63} +(2.00000 + 4.00000i) q^{65} +12.0000 q^{67} +6.00000i q^{69} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(8.00000 - 4.00000i) q^{85} +4.00000i q^{87} +10.0000 q^{89} +4.00000i q^{91} -8.00000 q^{93} +(4.00000 - 2.00000i) q^{95} -16.0000i q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{25} + 2 q^{27} - 16 q^{31} - 8 q^{35} + 20 q^{37} + 4 q^{39} - 12 q^{41} - 8 q^{43} + 2 q^{45} + 6 q^{49} + 20 q^{53} - 8 q^{55} + 4 q^{65} + 24 q^{67} - 6 q^{75} - 8 q^{77} - 32 q^{79} + 2 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{89} - 16 q^{93} + 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^25 + 2 * q^27 - 16 * q^31 - 8 * q^35 + 20 * q^37 + 4 * q^39 - 12 * q^41 - 8 * q^43 + 2 * q^45 + 6 * q^49 + 20 * q^53 - 8 * q^55 + 4 * q^65 + 24 * q^67 - 6 * q^75 - 8 * q^77 - 32 * q^79 + 2 * q^81 - 8 * q^83 + 16 * q^85 + 20 * q^89 - 16 * q^93 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.00000i			0.603023i	0.953463	+	0.301511i	$0.0974911\pi$
$11$			2.00000i			0.603023i	−0.953463	+	0.301511i	$0.902509\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$	1.00000	+	2.00000i	0.258199	+	0.516398i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$			2.00000i			0.436436i
$22$		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
$27$	1.00000			0.192450
$28$		0			0
$29$			4.00000i			0.742781i	0.928477	+	0.371391i	$0.121119\pi$
$29$			4.00000i			0.742781i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$	−8.00000			−1.43684			−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000			−1.43684			−0.718421	+	0.695608i	$0.755135\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	−4.00000	+	2.00000i	−0.676123	+	0.338062i
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$	2.00000			0.320256
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		−	4.00000i		−	0.560112i
$52$		0			0
$53$	10.0000			1.37361			0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000			1.37361			0.686803	+	0.726844i	$0.259014\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		0			0
$61$		−	8.00000i		−	1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$		−	8.00000i		−	1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$	2.00000	+	4.00000i	0.248069	+	0.496139i
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−3.00000	+	4.00000i	−0.346410	+	0.461880i
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	8.00000	−	4.00000i	0.867722	−	0.433861i
$86$		0			0
$87$			4.00000i			0.428845i
$88$		0			0
$89$	10.0000			1.06000			0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000			1.06000			0.529999	+	0.847998i	$0.322192\pi$
$90$		0			0
$91$			4.00000i			0.419314i
$92$		0			0
$93$	−8.00000			−0.829561
$94$		0			0
$95$	4.00000	−	2.00000i	0.410391	−	0.205196i
$96$		0			0
$97$		−	16.0000i		−	1.62455i	−0.583272	−	0.812277i	$-0.698228\pi$
$97$		−	16.0000i		−	1.62455i	0.583272	−	0.812277i	$-0.301772\pi$
$98$		0			0
$99$			2.00000i			0.201008i
$100$		0			0
$101$			12.0000i			1.19404i	0.802225	+	0.597022i	$0.203650\pi$
$101$			12.0000i			1.19404i	−0.802225	+	0.597022i	$0.796350\pi$
$102$		0			0
$103$		−	6.00000i		−	0.591198i	−0.955312	−	0.295599i	$-0.904481\pi$
$103$		−	6.00000i		−	0.591198i	0.955312	−	0.295599i	$-0.0955191\pi$
$104$		0			0
$105$	−4.00000	+	2.00000i	−0.390360	+	0.195180i
$106$		0			0
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$		−	16.0000i		−	1.53252i	−0.642529	−	0.766261i	$-0.722115\pi$
$109$		−	16.0000i		−	1.53252i	0.642529	−	0.766261i	$-0.277885\pi$
$110$		0			0
$111$	10.0000			0.949158
$112$		0			0
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	−12.0000	+	6.00000i	−1.11901	+	0.559503i
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	−6.00000			−0.541002
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$		−	14.0000i		−	1.24230i	−0.783692	−	0.621150i	$-0.786666\pi$
$127$		−	14.0000i		−	1.24230i	0.783692	−	0.621150i	$-0.213334\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$			10.0000i			0.873704i	0.899533	+	0.436852i	$0.143907\pi$
$131$			10.0000i			0.873704i	−0.899533	+	0.436852i	$0.856093\pi$
$132$		0			0
$133$	4.00000			0.346844
$134$		0			0
$135$	1.00000	+	2.00000i	0.0860663	+	0.172133i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$		−	2.00000i		−	0.169638i	−0.996396	−	0.0848189i	$-0.972969\pi$
$139$		−	2.00000i		−	0.169638i	0.996396	−	0.0848189i	$-0.0270312\pi$
$140$		0			0
$141$			6.00000i			0.505291i
$142$		0			0
$143$			4.00000i			0.334497i
$144$		0			0
$145$	−8.00000	+	4.00000i	−0.664364	+	0.332182i
$146$		0			0
$147$	3.00000			0.247436
$148$		0			0
$149$			4.00000i			0.327693i	0.986486	+	0.163846i	$0.0523901\pi$
$149$			4.00000i			0.327693i	−0.986486	+	0.163846i	$0.947610\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		−	4.00000i		−	0.323381i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$	10.0000			0.793052
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$		0			0
$165$	−4.00000	+	2.00000i	−0.311400	+	0.155700i
$166$		0			0
$167$		−	10.0000i		−	0.773823i	−0.922117	−	0.386912i	$-0.873542\pi$
$167$		−	10.0000i		−	0.773823i	0.922117	−	0.386912i	$-0.126458\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$		−	2.00000i		−	0.152944i
$172$		0			0
$173$	−22.0000			−1.67263			−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000			−1.67263			−0.836315	+	0.548250i	$0.815294\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$		0			0
$179$		−	6.00000i		−	0.448461i	−0.974536	−	0.224231i	$-0.928013\pi$
$179$		−	6.00000i		−	0.448461i	0.974536	−	0.224231i	$-0.0719869\pi$
$180$		0			0
$181$		−	8.00000i		−	0.594635i	−0.954779	−	0.297318i	$-0.903908\pi$
$181$		−	8.00000i		−	0.594635i	0.954779	−	0.297318i	$-0.0960920\pi$
$182$		0			0
$183$		−	8.00000i		−	0.591377i
$184$		0			0
$185$	10.0000	+	20.0000i	0.735215	+	1.47043i
$186$		0			0
$187$	8.00000			0.585018
$188$		0			0
$189$			2.00000i			0.145479i
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$			24.0000i			1.72756i	0.503871	+	0.863779i	$0.331909\pi$
$193$			24.0000i			1.72756i	−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$	2.00000	+	4.00000i	0.143223	+	0.286446i
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$	12.0000			0.846415
$202$		0			0
$203$	−8.00000			−0.561490
$204$		0			0
$205$	−6.00000	−	12.0000i	−0.419058	−	0.838116i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$		−	26.0000i		−	1.78991i	−0.446153	−	0.894957i	$-0.647206\pi$
$211$		−	26.0000i		−	1.78991i	0.446153	−	0.894957i	$-0.352794\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	8.00000i		−	0.538138i
$222$		0			0
$223$		−	22.0000i		−	1.47323i	−0.676313	−	0.736614i	$-0.736423\pi$
$223$		−	22.0000i		−	1.47323i	0.676313	−	0.736614i	$-0.263577\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$	12.0000			0.796468			0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000			0.796468			0.398234	+	0.917284i	$0.369623\pi$
$228$		0			0
$229$		−	8.00000i		−	0.528655i	−0.964433	−	0.264327i	$-0.914850\pi$
$229$		−	8.00000i		−	0.528655i	0.964433	−	0.264327i	$-0.0851500\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		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
$235$	−12.0000	+	6.00000i	−0.782794	+	0.391397i
$236$		0			0
$237$	−16.0000			−1.03931
$238$		0			0
$239$	8.00000			0.517477			0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000			0.517477			0.258738	+	0.965947i	$0.416693\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	3.00000	+	6.00000i	0.191663	+	0.383326i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$			26.0000i			1.64111i	0.571571	+	0.820553i	$0.306334\pi$
$251$			26.0000i			1.64111i	−0.571571	+	0.820553i	$0.693666\pi$
$252$		0			0
$253$	−12.0000			−0.754434
$254$		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
$259$			20.0000i			1.24274i
$260$		0			0
$261$			4.00000i			0.247594i
$262$		0			0
$263$		−	10.0000i		−	0.616626i	−0.951285	−	0.308313i	$-0.900236\pi$
$263$		−	10.0000i		−	0.616626i	0.951285	−	0.308313i	$-0.0997645\pi$
$264$		0			0
$265$	10.0000	+	20.0000i	0.614295	+	1.22859i
$266$		0			0
$267$	10.0000			0.611990
$268$		0			0
$269$		−	12.0000i		−	0.731653i	−0.930683	−	0.365826i	$-0.880786\pi$
$269$		−	12.0000i		−	0.731653i	0.930683	−	0.365826i	$-0.119214\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$			4.00000i			0.242091i
$274$		0			0
$275$	−8.00000	−	6.00000i	−0.482418	−	0.361814i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		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
$283$	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
$285$	4.00000	−	2.00000i	0.236940	−	0.118470i
$286$		0			0
$287$		−	12.0000i		−	0.708338i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		−	16.0000i		−	0.937937i
$292$		0			0
$293$	2.00000			0.116841			0.0584206	−	0.998292i	$-0.481394\pi$
$293$	2.00000			0.116841			0.0584206	+	0.998292i	$0.481394\pi$
$294$		0			0
$295$	12.0000	−	6.00000i	0.698667	−	0.349334i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$			12.0000i			0.693978i
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$		0			0
$303$			12.0000i			0.689382i
$304$		0			0
$305$	16.0000	−	8.00000i	0.916157	−	0.458079i
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		−	6.00000i		−	0.341328i
$310$		0			0
$311$	−16.0000			−0.907277			−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000			−0.907277			−0.453638	+	0.891186i	$0.649874\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$		0			0
$315$	−4.00000	+	2.00000i	−0.225374	+	0.112687i
$316$		0			0
$317$	2.00000			0.112331			0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000			0.112331			0.0561656	+	0.998421i	$0.482113\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$	20.0000			1.11629
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	−6.00000	+	8.00000i	−0.332820	+	0.443760i
$326$		0			0
$327$		−	16.0000i		−	0.884802i
$328$		0			0
$329$	−12.0000			−0.661581
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$		0			0
$333$	10.0000			0.547997
$334$		0			0
$335$	12.0000	+	24.0000i	0.655630	+	1.31126i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$			12.0000i			0.651751i
$340$		0			0
$341$		−	16.0000i		−	0.866449i
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$	−12.0000	+	6.00000i	−0.646058	+	0.323029i
$346$		0			0
$347$	−4.00000			−0.214731			−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000			−0.214731			−0.107366	+	0.994220i	$0.534242\pi$
$348$		0			0
$349$			16.0000i			0.856460i	0.903670	+	0.428230i	$0.140863\pi$
$349$			16.0000i			0.856460i	−0.903670	+	0.428230i	$0.859137\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$			20.0000i			1.06449i	0.846590	+	0.532246i	$0.178652\pi$
$353$			20.0000i			1.06449i	−0.846590	+	0.532246i	$0.821348\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	8.00000			0.423405
$358$		0			0
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	7.00000			0.367405
$364$		0			0
$365$		0			0
$366$		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
$369$	−6.00000			−0.312348
$370$		0			0
$371$			20.0000i			1.03835i
$372$		0			0
$373$	10.0000			0.517780			0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000			0.517780			0.258890	+	0.965907i	$0.416643\pi$
$374$		0			0
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$		0			0
$377$			8.00000i			0.412021i
$378$		0			0
$379$		−	2.00000i		−	0.102733i	−0.998680	−	0.0513665i	$-0.983642\pi$
$379$		−	2.00000i		−	0.102733i	0.998680	−	0.0513665i	$-0.0163577\pi$
$380$		0			0
$381$		−	14.0000i		−	0.717242i
$382$		0			0
$383$		−	10.0000i		−	0.510976i	−0.966812	−	0.255488i	$-0.917764\pi$
$383$		−	10.0000i		−	0.510976i	0.966812	−	0.255488i	$-0.0822362\pi$
$384$		0			0
$385$	−4.00000	−	8.00000i	−0.203859	−	0.407718i
$386$		0			0
$387$	−4.00000			−0.203331
$388$		0			0
$389$			20.0000i			1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$389$			20.0000i			1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$			10.0000i			0.504433i
$394$		0			0
$395$	−16.0000	−	32.0000i	−0.805047	−	1.61009i
$396$		0			0
$397$	2.00000			0.100377			0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000			0.100377			0.0501886	+	0.998740i	$0.484018\pi$
$398$		0			0
$399$	4.00000			0.200250
$400$		0			0
$401$	−14.0000			−0.699127			−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000			−0.699127			−0.349563	+	0.936913i	$0.613670\pi$
$402$		0			0
$403$	−16.0000			−0.797017
$404$		0			0
$405$	1.00000	+	2.00000i	0.0496904	+	0.0993808i
$406$		0			0
$407$			20.0000i			0.991363i
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$		−	12.0000i		−	0.591916i
$412$		0			0
$413$	12.0000			0.590481
$414$		0			0
$415$	−4.00000	−	8.00000i	−0.196352	−	0.392705i
$416$		0			0
$417$		−	2.00000i		−	0.0979404i
$418$		0			0
$419$		−	30.0000i		−	1.46560i	−0.680446	−	0.732798i	$-0.738214\pi$
$419$		−	30.0000i		−	1.46560i	0.680446	−	0.732798i	$-0.261786\pi$
$420$		0			0
$421$			16.0000i			0.779792i	0.920859	+	0.389896i	$0.127489\pi$
$421$			16.0000i			0.779792i	−0.920859	+	0.389896i	$0.872511\pi$
$422$		0			0
$423$			6.00000i			0.291730i
$424$		0			0
$425$	16.0000	+	12.0000i	0.776114	+	0.582086i
$426$		0			0
$427$	16.0000			0.774294
$428$		0			0
$429$			4.00000i			0.193122i
$430$		0			0
$431$	−32.0000			−1.54139			−0.770693	−	0.637207i	$-0.780090\pi$
$431$	−32.0000			−1.54139			−0.770693	+	0.637207i	$0.780090\pi$
$432$		0			0
$433$			8.00000i			0.384455i	0.981350	+	0.192228i	$0.0615712\pi$
$433$			8.00000i			0.384455i	−0.981350	+	0.192228i	$0.938429\pi$
$434$		0			0
$435$	−8.00000	+	4.00000i	−0.383571	+	0.191785i
$436$		0			0
$437$	12.0000			0.574038
$438$		0			0
$439$	24.0000			1.14546			0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000			1.14546			0.572729	+	0.819745i	$0.305885\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$		0			0
$445$	10.0000	+	20.0000i	0.474045	+	0.948091i
$446$		0			0
$447$			4.00000i			0.189194i
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$		−	12.0000i		−	0.565058i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.00000	+	4.00000i	−0.375046	+	0.187523i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		−	4.00000i		−	0.186704i
$460$		0			0
$461$			28.0000i			1.30409i	0.758180	+	0.652045i	$0.226089\pi$
$461$			28.0000i			1.30409i	−0.758180	+	0.652045i	$0.773911\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$		0			0
$465$	−8.00000	−	16.0000i	−0.370991	−	0.741982i
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$			24.0000i			1.10822i
$470$		0			0
$471$	−14.0000			−0.645086
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	8.00000	+	6.00000i	0.367065	+	0.275299i
$476$		0			0
$477$	10.0000			0.457869
$478$		0			0
$479$	24.0000			1.09659			0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000			1.09659			0.548294	+	0.836286i	$0.315277\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$		0			0
$483$	−12.0000			−0.546019
$484$		0			0
$485$	32.0000	−	16.0000i	1.45305	−	0.726523i
$486$		0			0
$487$			26.0000i			1.17817i	0.808070	+	0.589086i	$0.200512\pi$
$487$			26.0000i			1.17817i	−0.808070	+	0.589086i	$0.799488\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$			26.0000i			1.17336i	0.809818	+	0.586682i	$0.199566\pi$
$491$			26.0000i			1.17336i	−0.809818	+	0.586682i	$0.800434\pi$
$492$		0			0
$493$	16.0000			0.720604
$494$		0			0
$495$	−4.00000	+	2.00000i	−0.179787	+	0.0898933i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			30.0000i			1.34298i	0.741012	+	0.671492i	$0.234346\pi$
$499$			30.0000i			1.34298i	−0.741012	+	0.671492i	$0.765654\pi$
$500$		0			0
$501$		−	10.0000i		−	0.446767i
$502$		0			0
$503$		−	2.00000i		−	0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$		−	2.00000i		−	0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$		0			0
$505$	−24.0000	+	12.0000i	−1.06799	+	0.533993i
$506$		0			0
$507$	−9.00000			−0.399704
$508$		0			0
$509$		−	12.0000i		−	0.531891i	−0.963988	−	0.265945i	$-0.914316\pi$
$509$		−	12.0000i		−	0.531891i	0.963988	−	0.265945i	$-0.0856841\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		−	2.00000i		−	0.0883022i
$514$		0			0
$515$	12.0000	−	6.00000i	0.528783	−	0.264392i
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−22.0000			−0.965693
$520$		0			0
$521$	−22.0000			−0.963837			−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000			−0.963837			−0.481919	+	0.876216i	$0.660060\pi$
$522$		0			0
$523$	−12.0000			−0.524723			−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000			−0.524723			−0.262362	+	0.964970i	$0.584501\pi$
$524$		0			0
$525$	−8.00000	−	6.00000i	−0.349149	−	0.261861i
$526$		0			0
$527$			32.0000i			1.39394i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		−	6.00000i		−	0.260378i
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	20.0000	+	40.0000i	0.864675	+	1.72935i
$536$		0			0
$537$		−	6.00000i		−	0.258919i
$538$		0			0
$539$			6.00000i			0.258438i
$540$		0			0
$541$			40.0000i			1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$541$			40.0000i			1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$542$		0			0
$543$		−	8.00000i		−	0.343313i
$544$		0			0
$545$	32.0000	−	16.0000i	1.37073	−	0.685365i
$546$		0			0
$547$	−28.0000			−1.19719			−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000			−1.19719			−0.598597	+	0.801050i	$0.704275\pi$
$548$		0			0
$549$		−	8.00000i		−	0.341432i
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$	10.0000	+	20.0000i	0.424476	+	0.848953i
$556$		0			0
$557$	42.0000			1.77960			0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000			1.77960			0.889799	+	0.456354i	$0.150845\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	8.00000			0.337760
$562$		0			0
$563$	20.0000			0.842900			0.421450	−	0.906852i	$-0.361521\pi$
$563$	20.0000			0.842900			0.421450	+	0.906852i	$0.361521\pi$
$564$		0			0
$565$	−24.0000	+	12.0000i	−1.00969	+	0.504844i
$566$		0			0
$567$			2.00000i			0.0839921i
$568$		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
$571$		−	10.0000i		−	0.418487i	−0.977864	−	0.209243i	$-0.932900\pi$
$571$		−	10.0000i		−	0.418487i	0.977864	−	0.209243i	$-0.0671001\pi$
$572$		0			0
$573$	24.0000			1.00261
$574$		0			0
$575$	−24.0000	−	18.0000i	−1.00087	−	0.750652i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$			24.0000i			0.997406i
$580$		0			0
$581$		−	8.00000i		−	0.331896i
$582$		0			0
$583$			20.0000i			0.828315i
$584$		0			0
$585$	2.00000	+	4.00000i	0.0826898	+	0.165380i
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$			16.0000i			0.659269i
$590$		0			0
$591$	−6.00000			−0.246807
$592$		0			0
$593$			28.0000i			1.14982i	0.818216	+	0.574911i	$0.194963\pi$
$593$			28.0000i			1.14982i	−0.818216	+	0.574911i	$0.805037\pi$
$594$		0			0
$595$	8.00000	+	16.0000i	0.327968	+	0.655936i
$596$		0			0
$597$	8.00000			0.327418
$598$		0			0
$599$	−40.0000			−1.63436			−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000			−1.63436			−0.817178	+	0.576386i	$0.804463\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$		0			0
$605$	7.00000	+	14.0000i	0.284590	+	0.569181i
$606$		0			0
$607$		−	38.0000i		−	1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$		−	38.0000i		−	1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$		0			0
$609$	−8.00000			−0.324176
$610$		0			0
$611$			12.0000i			0.485468i
$612$		0			0
$613$	−6.00000			−0.242338			−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000			−0.242338			−0.121169	+	0.992632i	$0.538664\pi$
$614$		0			0
$615$	−6.00000	−	12.0000i	−0.241943	−	0.483887i
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$		−	18.0000i		−	0.723481i	−0.932279	−	0.361741i	$-0.882183\pi$
$619$		−	18.0000i		−	0.723481i	0.932279	−	0.361741i	$-0.117817\pi$
$620$		0			0
$621$			6.00000i			0.240772i
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	4.00000			0.159745
$628$		0			0
$629$		−	40.0000i		−	1.59490i
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		0			0
$633$		−	26.0000i		−	1.03341i
$634$		0			0
$635$	28.0000	−	14.0000i	1.11115	−	0.555573i
$636$		0			0
$637$	6.00000			0.237729
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−46.0000			−1.81689			−0.908445	−	0.418004i	$-0.862730\pi$
$641$	−46.0000			−1.81689			−0.908445	+	0.418004i	$0.862730\pi$
$642$		0			0
$643$	12.0000			0.473234			0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000			0.473234			0.236617	+	0.971603i	$0.423961\pi$
$644$		0			0
$645$	−4.00000	−	8.00000i	−0.157500	−	0.315000i
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		−	16.0000i		−	0.627089i
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$	−20.0000	+	10.0000i	−0.781465	+	0.390732i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	30.0000i		−	1.16863i	−0.811525	−	0.584317i	$-0.801362\pi$
$659$		−	30.0000i		−	1.16863i	0.811525	−	0.584317i	$-0.198638\pi$
$660$		0			0
$661$			24.0000i			0.933492i	0.884391	+	0.466746i	$0.154574\pi$
$661$			24.0000i			0.933492i	−0.884391	+	0.466746i	$0.845426\pi$
$662$		0			0
$663$		−	8.00000i		−	0.310694i
$664$		0			0
$665$	4.00000	+	8.00000i	0.155113	+	0.310227i
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$		−	22.0000i		−	0.850569i
$670$		0			0
$671$	16.0000			0.617673
$672$		0			0
$673$			40.0000i			1.54189i	0.636904	+	0.770943i	$0.280215\pi$
$673$			40.0000i			1.54189i	−0.636904	+	0.770943i	$0.719785\pi$
$674$		0			0
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$		0			0
$677$	−38.0000			−1.46046			−0.730229	−	0.683202i	$-0.760587\pi$
$677$	−38.0000			−1.46046			−0.730229	+	0.683202i	$0.760587\pi$
$678$		0			0
$679$	32.0000			1.22805
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		0			0
$685$	24.0000	−	12.0000i	0.916993	−	0.458496i
$686$		0			0
$687$		−	8.00000i		−	0.305219i
$688$		0			0
$689$	20.0000			0.761939
$690$		0			0
$691$		−	2.00000i		−	0.0760836i	−0.999276	−	0.0380418i	$-0.987888\pi$
$691$		−	2.00000i		−	0.0760836i	0.999276	−	0.0380418i	$-0.0121120\pi$
$692$		0			0
$693$	−4.00000			−0.151947
$694$		0			0
$695$	4.00000	−	2.00000i	0.151729	−	0.0758643i
$696$		0			0
$697$			24.0000i			0.909065i
$698$		0			0
$699$		−	12.0000i		−	0.453882i
$700$		0			0
$701$		−	44.0000i		−	1.66186i	−0.556379	−	0.830929i	$-0.687810\pi$
$701$		−	44.0000i		−	1.66186i	0.556379	−	0.830929i	$-0.312190\pi$
$702$		0			0
$703$		−	20.0000i		−	0.754314i
$704$		0			0
$705$	−12.0000	+	6.00000i	−0.451946	+	0.225973i
$706$		0			0
$707$	−24.0000			−0.902613
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$	−16.0000			−0.600047
$712$		0			0
$713$		−	48.0000i		−	1.79761i
$714$		0			0
$715$	−8.00000	+	4.00000i	−0.299183	+	0.149592i
$716$		0			0
$717$	8.00000			0.298765
$718$		0			0
$719$	−32.0000			−1.19340			−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000			−1.19340			−0.596699	+	0.802465i	$0.703521\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$	2.00000			0.0743808
$724$		0			0
$725$	−16.0000	−	12.0000i	−0.594225	−	0.445669i
$726$		0			0
$727$			26.0000i			0.964287i	0.876092	+	0.482143i	$0.160142\pi$
$727$			26.0000i			0.964287i	−0.876092	+	0.482143i	$0.839858\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			16.0000i			0.591781i
$732$		0			0
$733$	−14.0000			−0.517102			−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000			−0.517102			−0.258551	+	0.965998i	$0.583245\pi$
$734$		0			0
$735$	3.00000	+	6.00000i	0.110657	+	0.221313i
$736$		0			0
$737$			24.0000i			0.884051i
$738$		0			0
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$		0			0
$741$		−	4.00000i		−	0.146944i
$742$		0			0
$743$		−	34.0000i		−	1.24734i	−0.781688	−	0.623670i	$-0.785641\pi$
$743$		−	34.0000i		−	1.24734i	0.781688	−	0.623670i	$-0.214359\pi$
$744$		0			0
$745$	−8.00000	+	4.00000i	−0.293097	+	0.146549i
$746$		0			0
$747$	−4.00000			−0.146352
$748$		0			0
$749$			40.0000i			1.46157i
$750$		0			0
$751$	−8.00000			−0.291924			−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000			−0.291924			−0.145962	+	0.989290i	$0.546628\pi$
$752$		0			0
$753$			26.0000i			0.947493i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$		0			0
$759$	−12.0000			−0.435572
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$	32.0000			1.15848
$764$		0			0
$765$	8.00000	−	4.00000i	0.289241	−	0.144620i
$766$		0			0
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$		−	28.0000i		−	1.00840i
$772$		0			0
$773$	−6.00000			−0.215805			−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000			−0.215805			−0.107903	+	0.994161i	$0.534413\pi$
$774$		0			0
$775$	24.0000	−	32.0000i	0.862105	−	1.14947i
$776$		0			0
$777$			20.0000i			0.717496i
$778$		0			0
$779$			12.0000i			0.429945i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			4.00000i			0.142948i
$784$		0			0
$785$	−14.0000	−	28.0000i	−0.499681	−	0.999363i
$786$		0			0
$787$	−36.0000			−1.28326			−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000			−1.28326			−0.641631	+	0.767014i	$0.721742\pi$
$788$		0			0
$789$		−	10.0000i		−	0.356009i
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$		−	16.0000i		−	0.568177i
$794$		0			0
$795$	10.0000	+	20.0000i	0.354663	+	0.709327i
$796$		0			0
$797$	2.00000			0.0708436			0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000			0.0708436			0.0354218	+	0.999372i	$0.488723\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$	10.0000			0.353333
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−12.0000	−	24.0000i	−0.422944	−	0.845889i
$806$		0			0
$807$		−	12.0000i		−	0.422420i
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$			14.0000i			0.491606i	0.969320	+	0.245803i	$0.0790517\pi$
$811$			14.0000i			0.491606i	−0.969320	+	0.245803i	$0.920948\pi$
$812$		0			0
$813$	−16.0000			−0.561144
$814$		0			0
$815$	−4.00000	−	8.00000i	−0.140114	−	0.280228i
$816$		0			0
$817$			8.00000i			0.279885i
$818$		0			0
$819$			4.00000i			0.139771i
$820$		0			0
$821$			12.0000i			0.418803i	0.977830	+	0.209401i	$0.0671515\pi$
$821$			12.0000i			0.418803i	−0.977830	+	0.209401i	$0.932848\pi$
$822$		0			0
$823$			2.00000i			0.0697156i	0.999392	+	0.0348578i	$0.0110978\pi$
$823$			2.00000i			0.0697156i	−0.999392	+	0.0348578i	$0.988902\pi$
$824$		0			0
$825$	−8.00000	−	6.00000i	−0.278524	−	0.208893i
$826$		0			0
$827$	−52.0000			−1.80822			−0.904109	−	0.427303i	$-0.859464\pi$
$827$	−52.0000			−1.80822			−0.904109	+	0.427303i	$0.859464\pi$
$828$		0			0
$829$			32.0000i			1.11141i	0.831381	+	0.555703i	$0.187551\pi$
$829$			32.0000i			1.11141i	−0.831381	+	0.555703i	$0.812449\pi$
$830$		0			0
$831$	10.0000			0.346896
$832$		0			0
$833$		−	12.0000i		−	0.415775i
$834$		0			0
$835$	20.0000	−	10.0000i	0.692129	−	0.346064i
$836$		0			0
$837$	−8.00000			−0.276520
$838$		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
$841$	13.0000			0.448276
$842$		0			0
$843$	−22.0000			−0.757720
$844$		0			0
$845$	−9.00000	−	18.0000i	−0.309609	−	0.619219i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$			60.0000i			2.05677i
$852$		0			0
$853$	−54.0000			−1.84892			−0.924462	−	0.381273i	$-0.875486\pi$
$853$	−54.0000			−1.84892			−0.924462	+	0.381273i	$0.875486\pi$
$854$		0			0
$855$	4.00000	−	2.00000i	0.136797	−	0.0683986i
$856$		0			0
$857$			28.0000i			0.956462i	0.878234	+	0.478231i	$0.158722\pi$
$857$			28.0000i			0.956462i	−0.878234	+	0.478231i	$0.841278\pi$
$858$		0			0
$859$			14.0000i			0.477674i	0.971060	+	0.238837i	$0.0767661\pi$
$859$			14.0000i			0.477674i	−0.971060	+	0.238837i	$0.923234\pi$
$860$		0			0
$861$		−	12.0000i		−	0.408959i
$862$		0			0
$863$		−	34.0000i		−	1.15737i	−0.815550	−	0.578687i	$-0.803565\pi$
$863$		−	34.0000i		−	1.15737i	0.815550	−	0.578687i	$-0.196435\pi$
$864$		0			0
$865$	−22.0000	−	44.0000i	−0.748022	−	1.49604i
$866$		0			0
$867$	1.00000			0.0339618
$868$		0			0
$869$		−	32.0000i		−	1.08553i
$870$		0			0
$871$	24.0000			0.813209
$872$		0			0
$873$		−	16.0000i		−	0.541518i
$874$		0			0
$875$	4.00000	−	22.0000i	0.135225	−	0.743736i
$876$		0			0
$877$	2.00000			0.0675352			0.0337676	−	0.999430i	$-0.489249\pi$
$877$	2.00000			0.0675352			0.0337676	+	0.999430i	$0.489249\pi$
$878$		0			0
$879$	2.00000			0.0674583
$880$		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
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$	12.0000	−	6.00000i	0.403376	−	0.201688i
$886$		0			0
$887$		−	42.0000i		−	1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$		−	42.0000i		−	1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$		0			0
$889$	28.0000			0.939090
$890$		0			0
$891$			2.00000i			0.0670025i
$892$		0			0
$893$	12.0000			0.401565
$894$		0			0
$895$	12.0000	−	6.00000i	0.401116	−	0.200558i
$896$		0			0
$897$			12.0000i			0.400668i
$898$		0			0
$899$		−	32.0000i		−	1.06726i
$900$		0			0
$901$		−	40.0000i		−	1.33259i
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	16.0000	−	8.00000i	0.531858	−	0.265929i
$906$		0			0
$907$	28.0000			0.929725			0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000			0.929725			0.464862	+	0.885383i	$0.346104\pi$
$908$		0			0
$909$			12.0000i			0.398015i
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$		−	8.00000i		−	0.264761i
$914$		0			0
$915$	16.0000	−	8.00000i	0.528944	−	0.264472i
$916$		0			0
$917$	−20.0000			−0.660458
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	20.0000			0.659022
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−30.0000	+	40.0000i	−0.986394	+	1.31519i
$926$		0			0
$927$		−	6.00000i		−	0.197066i
$928$		0			0
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$		−	6.00000i		−	0.196642i
$932$		0			0
$933$	−16.0000			−0.523816
$934$		0			0
$935$	8.00000	+	16.0000i	0.261628	+	0.523256i
$936$		0			0
$937$			56.0000i			1.82944i	0.404088	+	0.914720i	$0.367589\pi$
$937$			56.0000i			1.82944i	−0.404088	+	0.914720i	$0.632411\pi$
$938$		0			0
$939$		−	16.0000i		−	0.522140i
$940$		0			0
$941$			12.0000i			0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$			12.0000i			0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$		0			0
$943$		−	36.0000i		−	1.17232i
$944$		0			0
$945$	−4.00000	+	2.00000i	−0.130120	+	0.0650600i
$946$		0			0
$947$	−20.0000			−0.649913			−0.324956	−	0.945729i	$-0.605350\pi$
$947$	−20.0000			−0.649913			−0.324956	+	0.945729i	$0.605350\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	2.00000			0.0648544
$952$		0			0
$953$		−	36.0000i		−	1.16615i	−0.812417	−	0.583077i	$-0.801849\pi$
$953$		−	36.0000i		−	1.16615i	0.812417	−	0.583077i	$-0.198151\pi$
$954$		0			0
$955$	24.0000	+	48.0000i	0.776622	+	1.55324i
$956$		0			0
$957$	−8.00000			−0.258603
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	20.0000			0.644491
$964$		0			0
$965$	−48.0000	+	24.0000i	−1.54517	+	0.772587i
$966$		0			0
$967$		−	30.0000i		−	0.964735i	−0.875969	−	0.482367i	$-0.839777\pi$
$967$		−	30.0000i		−	0.964735i	0.875969	−	0.482367i	$-0.160223\pi$
$968$		0			0
$969$	−8.00000			−0.256997
$970$		0			0
$971$		−	6.00000i		−	0.192549i	−0.995355	−	0.0962746i	$-0.969307\pi$
$971$		−	6.00000i		−	0.192549i	0.995355	−	0.0962746i	$-0.0306927\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$		0			0
$975$	−6.00000	+	8.00000i	−0.192154	+	0.256205i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$			20.0000i			0.639203i
$980$		0			0
$981$		−	16.0000i		−	0.510841i
$982$		0			0
$983$			46.0000i			1.46717i	0.679597	+	0.733586i	$0.262155\pi$
$983$			46.0000i			1.46717i	−0.679597	+	0.733586i	$0.737845\pi$
$984$		0			0
$985$	−6.00000	−	12.0000i	−0.191176	−	0.382352i
$986$		0			0
$987$	−12.0000			−0.381964
$988$		0			0
$989$		−	24.0000i		−	0.763156i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$	8.00000	+	16.0000i	0.253617	+	0.507234i
$996$		0			0
$997$	10.0000			0.316703			0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000			0.316703			0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	10.0000			0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.d.d.289.2	yes	2
3.2	odd	2		2880.2.d.a.289.1		2
4.3	odd	2		960.2.d.b.289.2	yes	2
5.2	odd	4		4800.2.k.c.2401.1		2
5.3	odd	4		4800.2.k.f.2401.2		2
5.4	even	2		960.2.d.a.289.2	yes	2
8.3	odd	2		960.2.d.c.289.1	yes	2
8.5	even	2		960.2.d.a.289.1	✓	2
12.11	even	2		2880.2.d.b.289.1		2
15.14	odd	2		2880.2.d.c.289.1		2
16.3	odd	4		3840.2.f.c.769.1		2
16.5	even	4		3840.2.f.a.769.1		2
16.11	odd	4		3840.2.f.b.769.2		2
16.13	even	4		3840.2.f.d.769.2		2
20.3	even	4		4800.2.k.b.2401.1		2
20.7	even	4		4800.2.k.g.2401.2		2
20.19	odd	2		960.2.d.c.289.2	yes	2
24.5	odd	2		2880.2.d.c.289.2		2
24.11	even	2		2880.2.d.d.289.2		2
40.3	even	4		4800.2.k.b.2401.2		2
40.13	odd	4		4800.2.k.f.2401.1		2
40.19	odd	2		960.2.d.b.289.1	yes	2
40.27	even	4		4800.2.k.g.2401.1		2
40.29	even	2	inner	960.2.d.d.289.1	yes	2
40.37	odd	4		4800.2.k.c.2401.2		2
60.59	even	2		2880.2.d.d.289.1		2
80.19	odd	4		3840.2.f.c.769.2		2
80.29	even	4		3840.2.f.d.769.1		2
80.59	odd	4		3840.2.f.b.769.1		2
80.69	even	4		3840.2.f.a.769.2		2
120.29	odd	2		2880.2.d.a.289.2		2
120.59	even	2		2880.2.d.b.289.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.d.a.289.1	✓	2	8.5	even	2
960.2.d.a.289.2	yes	2	5.4	even	2
960.2.d.b.289.1	yes	2	40.19	odd	2
960.2.d.b.289.2	yes	2	4.3	odd	2
960.2.d.c.289.1	yes	2	8.3	odd	2
960.2.d.c.289.2	yes	2	20.19	odd	2
960.2.d.d.289.1	yes	2	40.29	even	2	inner
960.2.d.d.289.2	yes	2	1.1	even	1	trivial
2880.2.d.a.289.1		2	3.2	odd	2
2880.2.d.a.289.2		2	120.29	odd	2
2880.2.d.b.289.1		2	12.11	even	2
2880.2.d.b.289.2		2	120.59	even	2
2880.2.d.c.289.1		2	15.14	odd	2
2880.2.d.c.289.2		2	24.5	odd	2
2880.2.d.d.289.1		2	60.59	even	2
2880.2.d.d.289.2		2	24.11	even	2
3840.2.f.a.769.1		2	16.5	even	4
3840.2.f.a.769.2		2	80.69	even	4
3840.2.f.b.769.1		2	80.59	odd	4
3840.2.f.b.769.2		2	16.11	odd	4
3840.2.f.c.769.1		2	16.3	odd	4
3840.2.f.c.769.2		2	80.19	odd	4
3840.2.f.d.769.1		2	80.29	even	4
3840.2.f.d.769.2		2	16.13	even	4
4800.2.k.b.2401.1		2	20.3	even	4
4800.2.k.b.2401.2		2	40.3	even	4
4800.2.k.c.2401.1		2	5.2	odd	4
4800.2.k.c.2401.2		2	40.37	odd	4
4800.2.k.f.2401.1		2	40.13	odd	4
4800.2.k.f.2401.2		2	5.3	odd	4
4800.2.k.g.2401.1		2	40.27	even	4
4800.2.k.g.2401.2		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +(1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +1.00000 q^{9} +2.00000i q^{11} +2.00000 q^{13} +(1.00000 + 2.00000i) q^{15} -4.00000i q^{17} -2.00000i q^{19} +2.00000i q^{21} +6.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} +1.00000 q^{27} +4.00000i q^{29} -8.00000 q^{31} +2.00000i q^{33} +(-4.00000 + 2.00000i) q^{35} +10.0000 q^{37} +2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{43} +(1.00000 + 2.00000i) q^{45} +6.00000i q^{47} +3.00000 q^{49} -4.00000i q^{51} +10.0000 q^{53} +(-4.00000 + 2.00000i) q^{55} -2.00000i q^{57} -6.00000i q^{59} -8.00000i q^{61} +2.00000i q^{63} +(2.00000 + 4.00000i) q^{65} +12.0000 q^{67} +6.00000i q^{69} +(-3.00000 + 4.00000i) q^{75} -4.00000 q^{77} -16.0000 q^{79} +1.00000 q^{81} -4.00000 q^{83} +(8.00000 - 4.00000i) q^{85} +4.00000i q^{87} +10.0000 q^{89} +4.00000i q^{91} -8.00000 q^{93} +(4.00000 - 2.00000i) q^{95} -16.0000i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{25} + 2 q^{27} - 16 q^{31} - 8 q^{35} + 20 q^{37} + 4 q^{39} - 12 q^{41} - 8 q^{43} + 2 q^{45} + 6 q^{49} + 20 q^{53} - 8 q^{55} + 4 q^{65} + 24 q^{67} - 6 q^{75} - 8 q^{77} - 32 q^{79} + 2 q^{81} - 8 q^{83} + 16 q^{85} + 20 q^{89} - 16 q^{93} + 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^25 + 2 * q^27 - 16 * q^31 - 8 * q^35 + 20 * q^37 + 4 * q^39 - 12 * q^41 - 8 * q^43 + 2 * q^45 + 6 * q^49 + 20 * q^53 - 8 * q^55 + 4 * q^65 + 24 * q^67 - 6 * q^75 - 8 * q^77 - 32 * q^79 + 2 * q^81 - 8 * q^83 + 16 * q^85 + 20 * q^89 - 16 * q^93 + 8 * q^95

Introduction

L-functions

Modular forms

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Database

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