LMFDB - Embedded newform 960.2.f.i.769.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(769,960)]

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, 0, 0, 1]))

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

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

chi := DirichletCharacter("960.769");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	960.f (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:	$7.66563859404$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.2.f.i.769.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(2.00000 - 1.00000i) q^{5} +2.00000i q^{7} -1.00000 q^{9} +2.00000 q^{11} -6.00000i q^{13} +(-1.00000 - 2.00000i) q^{15} -2.00000i q^{17} +2.00000 q^{21} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +1.00000i q^{27} +8.00000 q^{31} -2.00000i q^{33} +(2.00000 + 4.00000i) q^{35} +2.00000i q^{37} -6.00000 q^{39} +2.00000 q^{41} -4.00000i q^{43} +(-2.00000 + 1.00000i) q^{45} -8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -6.00000i q^{53} +(4.00000 - 2.00000i) q^{55} -10.0000 q^{59} -2.00000 q^{61} -2.00000i q^{63} +(-6.00000 - 12.0000i) q^{65} +8.00000i q^{67} +4.00000 q^{69} -12.0000 q^{71} -4.00000i q^{73} +(-4.00000 - 3.00000i) q^{75} +4.00000i q^{77} +1.00000 q^{81} -4.00000i q^{83} +(-2.00000 - 4.00000i) q^{85} +10.0000 q^{89} +12.0000 q^{91} -8.00000i q^{93} +8.00000i q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 4 q^{5} - 2 q^{9} + 4 q^{11} - 2 q^{15} + 4 q^{21} + 6 q^{25} + 16 q^{31} + 4 q^{35} - 12 q^{39} + 4 q^{41} - 4 q^{45} + 6 q^{49} - 4 q^{51} + 8 q^{55} - 20 q^{59} - 4 q^{61} - 12 q^{65} + 8 q^{69}+ \cdots - 4 q^{99}+O(q^{100})$ 2 * q + 4 * q^5 - 2 * q^9 + 4 * q^11 - 2 * q^15 + 4 * q^21 + 6 * q^25 + 16 * q^31 + 4 * q^35 - 12 * q^39 + 4 * q^41 - 4 * q^45 + 6 * q^49 - 4 * q^51 + 8 * q^55 - 20 * q^59 - 4 * q^61 - 12 * q^65 + 8 * q^69 - 24 * q^71 - 8 * q^75 + 2 * q^81 - 4 * q^85 + 20 * q^89 + 24 * q^91 - 4 * q^99

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
$3$		−	1.00000i		−	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
$7$			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
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$	−1.00000	−	2.00000i	−0.258199	−	0.516398i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		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
$33$		−	2.00000i		−	0.348155i
$34$		0			0
$35$	2.00000	+	4.00000i	0.338062	+	0.676123i
$36$		0			0
$37$			2.00000i			0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$			2.00000i			0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$		0			0
$39$	−6.00000			−0.960769
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−2.00000	+	1.00000i	−0.298142	+	0.149071i
$46$		0			0
$47$		−	8.00000i		−	1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$		−	8.00000i		−	1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		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
$55$	4.00000	−	2.00000i	0.539360	−	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.0000			−1.30189			−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$	−6.00000	−	12.0000i	−0.744208	−	1.48842i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$		0			0
$75$	−4.00000	−	3.00000i	−0.461880	−	0.346410i
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	4.00000i		−	0.439057i	−0.975606	−	0.219529i	$-0.929548\pi$
$83$		−	4.00000i		−	0.439057i	0.975606	−	0.219529i	$-0.0704519\pi$
$84$		0			0
$85$	−2.00000	−	4.00000i	−0.216930	−	0.433861i
$86$		0			0
$87$		0			0
$88$		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
$91$	12.0000			1.25794
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$	−2.00000			−0.201008
$100$		0			0
$101$	8.00000			0.796030			0.398015	−	0.917379i	$-0.369699\pi$
$101$	8.00000			0.796030			0.398015	+	0.917379i	$0.369699\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$	4.00000	−	2.00000i	0.390360	−	0.195180i
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	10.0000			0.957826			0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000			0.957826			0.478913	+	0.877862i	$0.341031\pi$
$110$		0			0
$111$	2.00000			0.189832
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	4.00000	+	8.00000i	0.373002	+	0.746004i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	2.00000i		−	0.180334i
$124$		0			0
$125$	2.00000	−	11.0000i	0.178885	−	0.983870i
$126$		0			0
$127$			2.00000i			0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$			2.00000i			0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1.00000	+	2.00000i	0.0860663	+	0.172133i
$136$		0			0
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	3.00000i		−	0.247436i
$148$		0			0
$149$	−20.0000			−1.63846			−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000			−1.63846			−0.819232	+	0.573462i	$0.805600\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	16.0000	−	8.00000i	1.28515	−	0.642575i
$156$		0			0
$157$			22.0000i			1.75579i	0.478852	+	0.877896i	$0.341053\pi$
$157$			22.0000i			1.75579i	−0.478852	+	0.877896i	$0.658947\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	−8.00000			−0.630488
$162$		0			0
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$		0			0
$165$	−2.00000	−	4.00000i	−0.155700	−	0.311400i
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$			14.0000i			1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$			14.0000i			1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$		0			0
$177$			10.0000i			0.751646i
$178$		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
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$			2.00000i			0.147844i
$184$		0			0
$185$	2.00000	+	4.00000i	0.147043	+	0.294086i
$186$		0			0
$187$		−	4.00000i		−	0.292509i
$188$		0			0
$189$	−2.00000			−0.145479
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$		0			0
$195$	−12.0000	+	6.00000i	−0.859338	+	0.429669i
$196$		0			0
$197$			22.0000i			1.56744i	0.621117	+	0.783718i	$0.286679\pi$
$197$			22.0000i			1.56744i	−0.621117	+	0.783718i	$0.713321\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$	8.00000			0.564276
$202$		0			0
$203$		0			0
$204$		0			0
$205$	4.00000	−	2.00000i	0.279372	−	0.139686i
$206$		0			0
$207$		−	4.00000i		−	0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$			12.0000i			0.822226i
$214$		0			0
$215$	−4.00000	−	8.00000i	−0.272798	−	0.545595i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	−4.00000			−0.270295
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$		−	26.0000i		−	1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$		−	26.0000i		−	1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$		0			0
$225$	−3.00000	+	4.00000i	−0.200000	+	0.266667i
$226$		0			0
$227$			28.0000i			1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$			28.0000i			1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$		0			0
$229$	−10.0000			−0.660819			−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000			−0.660819			−0.330409	+	0.943838i	$0.607187\pi$
$230$		0			0
$231$	4.00000			0.263181
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	22.0000			1.41714			0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000			1.41714			0.708572	+	0.705638i	$0.249340\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	6.00000	−	3.00000i	0.383326	−	0.191663i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$			8.00000i			0.502956i
$254$		0			0
$255$	−4.00000	+	2.00000i	−0.250490	+	0.125245i
$256$		0			0
$257$			18.0000i			1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$			18.0000i			1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$		0			0
$259$	−4.00000			−0.248548
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4.00000i			0.246651i	0.992366	+	0.123325i	$0.0393559\pi$
$263$			4.00000i			0.246651i	−0.992366	+	0.123325i	$0.960644\pi$
$264$		0			0
$265$	−6.00000	−	12.0000i	−0.368577	−	0.737154i
$266$		0			0
$267$		−	10.0000i		−	0.611990i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		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
$273$		−	12.0000i		−	0.726273i
$274$		0			0
$275$	6.00000	−	8.00000i	0.361814	−	0.482418i
$276$		0			0
$277$			2.00000i			0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$			2.00000i			0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$		0			0
$279$	−8.00000			−0.478947
$280$		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
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	8.00000			0.468968
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$	−20.0000	+	10.0000i	−1.16445	+	0.582223i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$	8.00000			0.461112
$302$		0			0
$303$		−	8.00000i		−	0.459588i
$304$		0			0
$305$	−4.00000	+	2.00000i	−0.229039	+	0.114520i
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	−12.0000			−0.680458			−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000			−0.680458			−0.340229	+	0.940343i	$0.610505\pi$
$312$		0			0
$313$		−	4.00000i		−	0.226093i	−0.993590	−	0.113047i	$-0.963939\pi$
$313$		−	4.00000i		−	0.226093i	0.993590	−	0.113047i	$-0.0360610\pi$
$314$		0			0
$315$	−2.00000	−	4.00000i	−0.112687	−	0.225374i
$316$		0			0
$317$			2.00000i			0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$			2.00000i			0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−24.0000	−	18.0000i	−1.33128	−	0.998460i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$		−	2.00000i		−	0.109599i
$334$		0			0
$335$	8.00000	+	16.0000i	0.437087	+	0.874173i
$336$		0			0
$337$			28.0000i			1.52526i	0.646837	+	0.762629i	$0.276092\pi$
$337$			28.0000i			1.52526i	−0.646837	+	0.762629i	$0.723908\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$	8.00000	−	4.00000i	0.430706	−	0.215353i
$346$		0			0
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	6.00000			0.320256
$352$		0			0
$353$		−	14.0000i		−	0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$		−	14.0000i		−	0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$		0			0
$355$	−24.0000	+	12.0000i	−1.27379	+	0.636894i
$356$		0			0
$357$		−	4.00000i		−	0.211702i
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$			7.00000i			0.367405i
$364$		0			0
$365$	−4.00000	−	8.00000i	−0.209370	−	0.418739i
$366$		0			0
$367$			2.00000i			0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$			2.00000i			0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$	12.0000			0.623009
$372$		0			0
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$		0			0
$375$	−11.0000	−	2.00000i	−0.568038	−	0.103280i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$	2.00000			0.102463
$382$		0			0
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$	4.00000	+	8.00000i	0.203859	+	0.407718i
$386$		0			0
$387$			4.00000i			0.203331i
$388$		0			0
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$			18.0000i			0.907980i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			2.00000i			0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$			2.00000i			0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	22.0000			1.09863			0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000			1.09863			0.549314	+	0.835616i	$0.314889\pi$
$402$		0			0
$403$		−	48.0000i		−	2.39105i
$404$		0			0
$405$	2.00000	−	1.00000i	0.0993808	−	0.0496904i
$406$		0			0
$407$			4.00000i			0.198273i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$	18.0000			0.887875
$412$		0			0
$413$		−	20.0000i		−	0.984136i
$414$		0			0
$415$	−4.00000	−	8.00000i	−0.196352	−	0.392705i
$416$		0			0
$417$		−	20.0000i		−	0.979404i
$418$		0			0
$419$	10.0000			0.488532			0.244266	−	0.969708i	$-0.421453\pi$
$419$	10.0000			0.488532			0.244266	+	0.969708i	$0.421453\pi$
$420$		0			0
$421$	−22.0000			−1.07221			−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000			−1.07221			−0.536107	+	0.844150i	$0.680106\pi$
$422$		0			0
$423$			8.00000i			0.388973i
$424$		0			0
$425$	−8.00000	−	6.00000i	−0.388057	−	0.291043i
$426$		0			0
$427$		−	4.00000i		−	0.193574i
$428$		0			0
$429$	−12.0000			−0.579365
$430$		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
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$			36.0000i			1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$			36.0000i			1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$		0			0
$445$	20.0000	−	10.0000i	0.948091	−	0.474045i
$446$		0			0
$447$			20.0000i			0.945968i
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$		0			0
$453$		−	8.00000i		−	0.375873i
$454$		0			0
$455$	24.0000	−	12.0000i	1.12514	−	0.562569i
$456$		0			0
$457$		−	32.0000i		−	1.49690i	−0.663193	−	0.748448i	$-0.730799\pi$
$457$		−	32.0000i		−	1.49690i	0.663193	−	0.748448i	$-0.269201\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$		−	6.00000i		−	0.278844i	−0.990233	−	0.139422i	$-0.955476\pi$
$463$		−	6.00000i		−	0.278844i	0.990233	−	0.139422i	$-0.0445244\pi$
$464$		0			0
$465$	−8.00000	−	16.0000i	−0.370991	−	0.741982i
$466$		0			0
$467$		−	12.0000i		−	0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$		−	12.0000i		−	0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$	22.0000			1.01371
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$			6.00000i			0.274721i
$478$		0			0
$479$	20.0000			0.913823			0.456912	−	0.889512i	$-0.348956\pi$
$479$	20.0000			0.913823			0.456912	+	0.889512i	$0.348956\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		0			0
$483$			8.00000i			0.364013i
$484$		0			0
$485$	8.00000	+	16.0000i	0.363261	+	0.726523i
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$	16.0000			0.723545
$490$		0			0
$491$	−18.0000			−0.812329			−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000			−0.812329			−0.406164	+	0.913800i	$0.633134\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−4.00000	+	2.00000i	−0.179787	+	0.0898933i
$496$		0			0
$497$		−	24.0000i		−	1.07655i
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$			24.0000i			1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$			24.0000i			1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$		0			0
$505$	16.0000	−	8.00000i	0.711991	−	0.355995i
$506$		0			0
$507$			23.0000i			1.02147i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14.0000	+	28.0000i	0.616914	+	1.23383i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$		0			0
$519$	14.0000			0.614532
$520$		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
$523$			16.0000i			0.699631i	0.936819	+	0.349816i	$0.113756\pi$
$523$			16.0000i			0.699631i	−0.936819	+	0.349816i	$0.886244\pi$
$524$		0			0
$525$	6.00000	−	8.00000i	0.261861	−	0.349149i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$	10.0000			0.433963
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−12.0000	−	24.0000i	−0.518805	−	1.03761i
$536$		0			0
$537$			10.0000i			0.431532i
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$			2.00000i			0.0858282i
$544$		0			0
$545$	20.0000	−	10.0000i	0.856706	−	0.428353i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$		0			0
$549$	2.00000			0.0853579
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.00000	−	2.00000i	0.169791	−	0.0848953i
$556$		0			0
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$		−	44.0000i		−	1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$		−	44.0000i		−	1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$		0			0
$565$	6.00000	+	12.0000i	0.252422	+	0.504844i
$566$		0			0
$567$			2.00000i			0.0839921i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	−8.00000			−0.334790			−0.167395	−	0.985890i	$-0.553535\pi$
$571$	−8.00000			−0.334790			−0.167395	+	0.985890i	$0.553535\pi$
$572$		0			0
$573$			12.0000i			0.501307i
$574$		0			0
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$		0			0
$579$	−4.00000			−0.166234
$580$		0			0
$581$	8.00000			0.331896
$582$		0			0
$583$		−	12.0000i		−	0.496989i
$584$		0			0
$585$	6.00000	+	12.0000i	0.248069	+	0.496139i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	22.0000			0.904959
$592$		0			0
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$	8.00000	−	4.00000i	0.327968	−	0.163984i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$		−	8.00000i		−	0.325785i
$604$		0			0
$605$	−14.0000	+	7.00000i	−0.569181	+	0.284590i
$606$		0			0
$607$			22.0000i			0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$			22.0000i			0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−48.0000			−1.94187
$612$		0			0
$613$		−	26.0000i		−	1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$		−	26.0000i		−	1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$		0			0
$615$	−2.00000	−	4.00000i	−0.0806478	−	0.161296i
$616$		0			0
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$	−4.00000			−0.160514
$622$		0			0
$623$			20.0000i			0.801283i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$		−	12.0000i		−	0.476957i
$634$		0			0
$635$	2.00000	+	4.00000i	0.0793676	+	0.158735i
$636$		0			0
$637$		−	18.0000i		−	0.713186i
$638$		0			0
$639$	12.0000			0.474713
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$		−	24.0000i		−	0.946468i	−0.880937	−	0.473234i	$-0.843087\pi$
$643$		−	24.0000i		−	0.946468i	0.880937	−	0.473234i	$-0.156913\pi$
$644$		0			0
$645$	−8.00000	+	4.00000i	−0.315000	+	0.157500i
$646$		0			0
$647$		−	48.0000i		−	1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$		−	48.0000i		−	1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$	16.0000			0.627089
$652$		0			0
$653$		−	26.0000i		−	1.01746i	−0.860927	−	0.508729i	$-0.830115\pi$
$653$		−	26.0000i		−	1.01746i	0.860927	−	0.508729i	$-0.169885\pi$
$654$		0			0
$655$	−36.0000	+	18.0000i	−1.40664	+	0.703318i
$656$		0			0
$657$			4.00000i			0.156055i
$658$		0			0
$659$	50.0000			1.94772			0.973862	−	0.227142i	$-0.0729380\pi$
$659$	50.0000			1.94772			0.973862	+	0.227142i	$0.0729380\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$			12.0000i			0.466041i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−26.0000			−1.00522
$670$		0			0
$671$	−4.00000			−0.154418
$672$		0			0
$673$			36.0000i			1.38770i	0.720121	+	0.693849i	$0.244086\pi$
$673$			36.0000i			1.38770i	−0.720121	+	0.693849i	$0.755914\pi$
$674$		0			0
$675$	4.00000	+	3.00000i	0.153960	+	0.115470i
$676$		0			0
$677$			2.00000i			0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$			2.00000i			0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$	28.0000			1.07296
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	18.0000	+	36.0000i	0.687745	+	1.37549i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−36.0000			−1.37149
$690$		0			0
$691$	−8.00000			−0.304334			−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000			−0.304334			−0.152167	+	0.988355i	$0.548625\pi$
$692$		0			0
$693$		−	4.00000i		−	0.151947i
$694$		0			0
$695$	40.0000	−	20.0000i	1.51729	−	0.758643i
$696$		0			0
$697$		−	4.00000i		−	0.151511i
$698$		0			0
$699$	−14.0000			−0.529529
$700$		0			0
$701$	−32.0000			−1.20862			−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000			−1.20862			−0.604312	+	0.796748i	$0.706552\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−16.0000	+	8.00000i	−0.602595	+	0.301297i
$706$		0			0
$707$			16.0000i			0.601742i
$708$		0			0
$709$	30.0000			1.12667			0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000			1.12667			0.563337	+	0.826227i	$0.309517\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			32.0000i			1.19841i
$714$		0			0
$715$	−12.0000	−	24.0000i	−0.448775	−	0.897549i
$716$		0			0
$717$		−	20.0000i		−	0.746914i
$718$		0			0
$719$	−40.0000			−1.49175			−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000			−1.49175			−0.745874	+	0.666087i	$0.767968\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$		0			0
$723$		−	22.0000i		−	0.818189i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−8.00000			−0.295891
$732$		0			0
$733$			14.0000i			0.517102i	0.965998	+	0.258551i	$0.0832450\pi$
$733$			14.0000i			0.517102i	−0.965998	+	0.258551i	$0.916755\pi$
$734$		0			0
$735$	−3.00000	−	6.00000i	−0.110657	−	0.221313i
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	−40.0000	+	20.0000i	−1.46549	+	0.732743i
$746$		0			0
$747$			4.00000i			0.146352i
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$		0			0
$753$			18.0000i			0.655956i
$754$		0			0
$755$	16.0000	−	8.00000i	0.582300	−	0.291150i
$756$		0			0
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		0			0
$759$	8.00000			0.290382
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$			20.0000i			0.724049i
$764$		0			0
$765$	2.00000	+	4.00000i	0.0723102	+	0.144620i
$766$		0			0
$767$			60.0000i			2.16647i
$768$		0			0
$769$	30.0000			1.08183			0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000			1.08183			0.540914	+	0.841078i	$0.318079\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$			54.0000i			1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$			54.0000i			1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$		0			0
$775$	24.0000	−	32.0000i	0.862105	−	1.14947i
$776$		0			0
$777$			4.00000i			0.143499i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−24.0000			−0.858788
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.0000	+	44.0000i	0.785214	+	1.57043i
$786$		0			0
$787$		−	32.0000i		−	1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$		−	32.0000i		−	1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$		0			0
$789$	4.00000			0.142404
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$			12.0000i			0.426132i
$794$		0			0
$795$	−12.0000	+	6.00000i	−0.425596	+	0.212798i
$796$		0			0
$797$			2.00000i			0.0708436i	0.999372	+	0.0354218i	$0.0112775\pi$
$797$			2.00000i			0.0708436i	−0.999372	+	0.0354218i	$0.988723\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$		−	8.00000i		−	0.282314i
$804$		0			0
$805$	−16.0000	+	8.00000i	−0.563926	+	0.281963i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	30.0000			1.05474			0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000			1.05474			0.527372	+	0.849635i	$0.323177\pi$
$810$		0			0
$811$	52.0000			1.82597			0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000			1.82597			0.912983	+	0.407997i	$0.133772\pi$
$812$		0			0
$813$		−	8.00000i		−	0.280572i
$814$		0			0
$815$	16.0000	+	32.0000i	0.560456	+	1.12091i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−12.0000			−0.419314
$820$		0			0
$821$	8.00000			0.279202			0.139601	−	0.990208i	$-0.455418\pi$
$821$	8.00000			0.279202			0.139601	+	0.990208i	$0.455418\pi$
$822$		0			0
$823$		−	6.00000i		−	0.209147i	−0.994517	−	0.104573i	$-0.966652\pi$
$823$		−	6.00000i		−	0.209147i	0.994517	−	0.104573i	$-0.0333477\pi$
$824$		0			0
$825$	−8.00000	−	6.00000i	−0.278524	−	0.208893i
$826$		0			0
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	2.00000			0.0693792
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	12.0000	+	24.0000i	0.415277	+	0.830554i
$836$		0			0
$837$			8.00000i			0.276520i
$838$		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
$841$	−29.0000			−1.00000
$842$		0			0
$843$			18.0000i			0.619953i
$844$		0			0
$845$	−46.0000	+	23.0000i	−1.58245	+	0.791224i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$		0			0
$849$	16.0000			0.549119
$850$		0			0
$851$	−8.00000			−0.274236
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	42.0000i		−	1.43469i	−0.696717	−	0.717346i	$-0.745357\pi$
$857$		−	42.0000i		−	1.43469i	0.696717	−	0.717346i	$-0.254643\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$	4.00000			0.136320
$862$		0			0
$863$		−	36.0000i		−	1.22545i	−0.790295	−	0.612727i	$-0.790072\pi$
$863$		−	36.0000i		−	1.22545i	0.790295	−	0.612727i	$-0.209928\pi$
$864$		0			0
$865$	14.0000	+	28.0000i	0.476014	+	0.952029i
$866$		0			0
$867$		−	13.0000i		−	0.441503i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$		−	8.00000i		−	0.270759i
$874$		0			0
$875$	22.0000	+	4.00000i	0.743736	+	0.135225i
$876$		0			0
$877$			22.0000i			0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$			22.0000i			0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$		0			0
$879$	−6.00000			−0.202375
$880$		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
$883$		−	24.0000i		−	0.807664i	−0.914833	−	0.403832i	$-0.867678\pi$
$883$		−	24.0000i		−	0.807664i	0.914833	−	0.403832i	$-0.132322\pi$
$884$		0			0
$885$	10.0000	+	20.0000i	0.336146	+	0.672293i
$886$		0			0
$887$			12.0000i			0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$			12.0000i			0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$		0			0
$889$	−4.00000			−0.134156
$890$		0			0
$891$	2.00000			0.0670025
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−20.0000	+	10.0000i	−0.668526	+	0.334263i
$896$		0			0
$897$		−	24.0000i		−	0.801337i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		−	8.00000i		−	0.266223i
$904$		0			0
$905$	−4.00000	+	2.00000i	−0.132964	+	0.0664822i
$906$		0			0
$907$		−	12.0000i		−	0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$		−	12.0000i		−	0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$		0			0
$909$	−8.00000			−0.265343
$910$		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
$913$		−	8.00000i		−	0.264761i
$914$		0			0
$915$	2.00000	+	4.00000i	0.0661180	+	0.132236i
$916$		0			0
$917$		−	36.0000i		−	1.18882i
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−12.0000			−0.395413
$922$		0			0
$923$			72.0000i			2.36991i
$924$		0			0
$925$	8.00000	+	6.00000i	0.263038	+	0.197279i
$926$		0			0
$927$		−	14.0000i		−	0.459820i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			12.0000i			0.392862i
$934$		0			0
$935$	−4.00000	−	8.00000i	−0.130814	−	0.261628i
$936$		0			0
$937$			8.00000i			0.261349i	0.991425	+	0.130674i	$0.0417142\pi$
$937$			8.00000i			0.261349i	−0.991425	+	0.130674i	$0.958286\pi$
$938$		0			0
$939$	−4.00000			−0.130535
$940$		0			0
$941$	28.0000			0.912774			0.456387	−	0.889781i	$-0.349143\pi$
$941$	28.0000			0.912774			0.456387	+	0.889781i	$0.349143\pi$
$942$		0			0
$943$			8.00000i			0.260516i
$944$		0			0
$945$	−4.00000	+	2.00000i	−0.130120	+	0.0650600i
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	2.00000			0.0648544
$952$		0			0
$953$			46.0000i			1.49009i	0.667016	+	0.745043i	$0.267571\pi$
$953$			46.0000i			1.49009i	−0.667016	+	0.745043i	$0.732429\pi$
$954$		0			0
$955$	−24.0000	+	12.0000i	−0.776622	+	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.0000			−1.16250
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			12.0000i			0.386695i
$964$		0			0
$965$	−4.00000	−	8.00000i	−0.128765	−	0.257529i
$966$		0			0
$967$		−	38.0000i		−	1.22200i	−0.791632	−	0.610999i	$-0.790768\pi$
$967$		−	38.0000i		−	1.22200i	0.791632	−	0.610999i	$-0.209232\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000			−0.577647			−0.288824	−	0.957382i	$-0.593264\pi$
$971$	−18.0000			−0.577647			−0.288824	+	0.957382i	$0.593264\pi$
$972$		0			0
$973$			40.0000i			1.28234i
$974$		0			0
$975$	−18.0000	+	24.0000i	−0.576461	+	0.768615i
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$	20.0000			0.639203
$980$		0			0
$981$	−10.0000			−0.319275
$982$		0			0
$983$		−	16.0000i		−	0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$		−	16.0000i		−	0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$		0			0
$985$	22.0000	+	44.0000i	0.700978	+	1.40196i
$986$		0			0
$987$		−	16.0000i		−	0.509286i
$988$		0			0
$989$	16.0000			0.508770
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	18.0000i		−	0.570066i	−0.958518	−	0.285033i	$-0.907995\pi$
$997$		−	18.0000i		−	0.570066i	0.958518	−	0.285033i	$-0.0920045\pi$
$998$		0			0
$999$	−2.00000			−0.0632772

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.f.i.769.1		2
3.2	odd	2		2880.2.f.c.1729.2		2
4.3	odd	2		960.2.f.h.769.2		2
5.2	odd	4		4800.2.a.m.1.1		1
5.3	odd	4		4800.2.a.cj.1.1		1
5.4	even	2	inner	960.2.f.i.769.2		2
8.3	odd	2		30.2.c.a.19.2	yes	2
8.5	even	2		240.2.f.a.49.2		2
12.11	even	2		2880.2.f.e.1729.2		2
15.14	odd	2		2880.2.f.c.1729.1		2
16.3	odd	4		3840.2.d.g.2689.1		2
16.5	even	4		3840.2.d.j.2689.2		2
16.11	odd	4		3840.2.d.y.2689.2		2
16.13	even	4		3840.2.d.x.2689.1		2
20.3	even	4		4800.2.a.l.1.1		1
20.7	even	4		4800.2.a.cg.1.1		1
20.19	odd	2		960.2.f.h.769.1		2
24.5	odd	2		720.2.f.f.289.1		2
24.11	even	2		90.2.c.a.19.1		2
40.3	even	4		150.2.a.c.1.1		1
40.13	odd	4		1200.2.a.g.1.1		1
40.19	odd	2		30.2.c.a.19.1	✓	2
40.27	even	4		150.2.a.a.1.1		1
40.29	even	2		240.2.f.a.49.1		2
40.37	odd	4		1200.2.a.m.1.1		1
56.3	even	6		1470.2.n.a.79.1		4
56.11	odd	6		1470.2.n.h.79.1		4
56.19	even	6		1470.2.n.a.949.2		4
56.27	even	2		1470.2.g.g.589.2		2
56.51	odd	6		1470.2.n.h.949.2		4
60.59	even	2		2880.2.f.e.1729.1		2
72.11	even	6		810.2.i.b.109.1		4
72.43	odd	6		810.2.i.e.109.2		4
72.59	even	6		810.2.i.b.379.2		4
72.67	odd	6		810.2.i.e.379.1		4
80.19	odd	4		3840.2.d.y.2689.1		2
80.29	even	4		3840.2.d.j.2689.1		2
80.59	odd	4		3840.2.d.g.2689.2		2
80.69	even	4		3840.2.d.x.2689.2		2
120.29	odd	2		720.2.f.f.289.2		2
120.53	even	4		3600.2.a.bg.1.1		1
120.59	even	2		90.2.c.a.19.2		2
120.77	even	4		3600.2.a.o.1.1		1
120.83	odd	4		450.2.a.b.1.1		1
120.107	odd	4		450.2.a.f.1.1		1
280.19	even	6		1470.2.n.a.949.1		4
280.27	odd	4		7350.2.a.bg.1.1		1
280.59	even	6		1470.2.n.a.79.2		4
280.83	odd	4		7350.2.a.cc.1.1		1
280.139	even	2		1470.2.g.g.589.1		2
280.179	odd	6		1470.2.n.h.79.2		4
280.219	odd	6		1470.2.n.h.949.1		4
360.59	even	6		810.2.i.b.379.1		4
360.139	odd	6		810.2.i.e.379.2		4
360.259	odd	6		810.2.i.e.109.1		4
360.299	even	6		810.2.i.b.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.c.a.19.1	✓	2	40.19	odd	2
30.2.c.a.19.2	yes	2	8.3	odd	2
90.2.c.a.19.1		2	24.11	even	2
90.2.c.a.19.2		2	120.59	even	2
150.2.a.a.1.1		1	40.27	even	4
150.2.a.c.1.1		1	40.3	even	4
240.2.f.a.49.1		2	40.29	even	2
240.2.f.a.49.2		2	8.5	even	2
450.2.a.b.1.1		1	120.83	odd	4
450.2.a.f.1.1		1	120.107	odd	4
720.2.f.f.289.1		2	24.5	odd	2
720.2.f.f.289.2		2	120.29	odd	2
810.2.i.b.109.1		4	72.11	even	6
810.2.i.b.109.2		4	360.299	even	6
810.2.i.b.379.1		4	360.59	even	6
810.2.i.b.379.2		4	72.59	even	6
810.2.i.e.109.1		4	360.259	odd	6
810.2.i.e.109.2		4	72.43	odd	6
810.2.i.e.379.1		4	72.67	odd	6
810.2.i.e.379.2		4	360.139	odd	6
960.2.f.h.769.1		2	20.19	odd	2
960.2.f.h.769.2		2	4.3	odd	2
960.2.f.i.769.1		2	1.1	even	1	trivial
960.2.f.i.769.2		2	5.4	even	2	inner
1200.2.a.g.1.1		1	40.13	odd	4
1200.2.a.m.1.1		1	40.37	odd	4
1470.2.g.g.589.1		2	280.139	even	2
1470.2.g.g.589.2		2	56.27	even	2
1470.2.n.a.79.1		4	56.3	even	6
1470.2.n.a.79.2		4	280.59	even	6
1470.2.n.a.949.1		4	280.19	even	6
1470.2.n.a.949.2		4	56.19	even	6
1470.2.n.h.79.1		4	56.11	odd	6
1470.2.n.h.79.2		4	280.179	odd	6
1470.2.n.h.949.1		4	280.219	odd	6
1470.2.n.h.949.2		4	56.51	odd	6
2880.2.f.c.1729.1		2	15.14	odd	2
2880.2.f.c.1729.2		2	3.2	odd	2
2880.2.f.e.1729.1		2	60.59	even	2
2880.2.f.e.1729.2		2	12.11	even	2
3600.2.a.o.1.1		1	120.77	even	4
3600.2.a.bg.1.1		1	120.53	even	4
3840.2.d.g.2689.1		2	16.3	odd	4
3840.2.d.g.2689.2		2	80.59	odd	4
3840.2.d.j.2689.1		2	80.29	even	4
3840.2.d.j.2689.2		2	16.5	even	4
3840.2.d.x.2689.1		2	16.13	even	4
3840.2.d.x.2689.2		2	80.69	even	4
3840.2.d.y.2689.1		2	80.19	odd	4
3840.2.d.y.2689.2		2	16.11	odd	4
4800.2.a.l.1.1		1	20.3	even	4
4800.2.a.m.1.1		1	5.2	odd	4
4800.2.a.cg.1.1		1	20.7	even	4
4800.2.a.cj.1.1		1	5.3	odd	4
7350.2.a.bg.1.1		1	280.27	odd	4
7350.2.a.cc.1.1		1	280.83	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	960.2.f.i.769.1

Level	$960$
Weight	$2$
Character	960.769
Analytic conductor	$7.666$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$