LMFDB - Embedded newform 3840.2.d.x.2689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3840,2,Mod(2689,3840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3840, 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("3840.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3840 = 2^{8} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3840.d (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:	$30.6625543762$
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:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3840.2689
Dual form			3840.2.d.x.2689.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3840\mathbb{Z}\right)^\times$ .

$n$	$511$	$1537$	$2561$	$2821$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		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		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		−	2.00000i		−	0.436436i
$22$		0			0
$23$		−	4.00000i		−	0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$		−	4.00000i		−	0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		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
$35$	−4.00000	+	2.00000i	−0.676123	+	0.338062i
$36$		0			0
$37$	2.00000			0.328798			0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000			0.328798			0.164399	+	0.986394i	$0.447432\pi$
$38$		0			0
$39$	6.00000			0.960769
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		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
$47$		−	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
$51$		−	2.00000i		−	0.280056i
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			10.0000i			1.30189i	0.759125	+	0.650945i	$0.225627\pi$
$59$			10.0000i			1.30189i	−0.759125	+	0.650945i	$0.774373\pi$
$60$		0			0
$61$		−	2.00000i		−	0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$		−	2.00000i		−	0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$		0			0
$65$	−6.00000	−	12.0000i	−0.744208	−	1.48842i
$66$		0			0
$67$	−8.00000			−0.977356			−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000			−0.977356			−0.488678	+	0.872464i	$0.662521\pi$
$68$		0			0
$69$		−	4.00000i		−	0.481543i
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$	−3.00000	+	4.00000i	−0.346410	+	0.461880i
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$		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
$83$	4.00000			0.439057			0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000			0.439057			0.219529	+	0.975606i	$0.429548\pi$
$84$		0			0
$85$	−4.00000	+	2.00000i	−0.433861	+	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.0000			−1.06000			−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000			−1.06000			−0.529999	+	0.847998i	$0.677808\pi$
$90$		0			0
$91$		−	12.0000i		−	1.25794i
$92$		0			0
$93$	8.00000			0.829561
$94$		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
$99$		−	2.00000i		−	0.201008i
$100$		0			0
$101$		−	8.00000i		−	0.796030i	−0.917379	−	0.398015i	$-0.869699\pi$
$101$		−	8.00000i		−	0.796030i	0.917379	−	0.398015i	$-0.130301\pi$
$102$		0			0
$103$		−	14.0000i		−	1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$		−	14.0000i		−	1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$		0			0
$105$	−4.00000	+	2.00000i	−0.390360	+	0.195180i
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		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
$115$	−8.00000	+	4.00000i	−0.746004	+	0.373002i
$116$		0			0
$117$	6.00000			0.554700
$118$		0			0
$119$	−4.00000			−0.366679
$120$		0			0
$121$	7.00000			0.636364
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		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
$131$		−	18.0000i		−	1.57267i	−0.617802	−	0.786334i	$-0.711977\pi$
$131$		−	18.0000i		−	1.57267i	0.617802	−	0.786334i	$-0.288023\pi$
$132$		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	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$		−	20.0000i		−	1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$		−	20.0000i		−	1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$		0			0
$141$		−	8.00000i		−	0.673722i
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	3.00000			0.247436
$148$		0			0
$149$			20.0000i			1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$149$			20.0000i			1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$	−22.0000			−1.75579			−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000			−1.75579			−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$	−8.00000			−0.630488
$162$		0			0
$163$	−16.0000			−1.25322			−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000			−1.25322			−0.626608	+	0.779334i	$0.715557\pi$
$164$		0			0
$165$	−4.00000	+	2.00000i	−0.311400	+	0.155700i
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−14.0000			−1.06440			−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000			−1.06440			−0.532200	+	0.846619i	$0.678635\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$		0			0
$177$			10.0000i			0.751646i
$178$		0			0
$179$		−	10.0000i		−	0.747435i	−0.927543	−	0.373718i	$-0.878083\pi$
$179$		−	10.0000i		−	0.747435i	0.927543	−	0.373718i	$-0.121917\pi$
$180$		0			0
$181$			2.00000i			0.148659i	0.997234	+	0.0743294i	$0.0236816\pi$
$181$			2.00000i			0.148659i	−0.997234	+	0.0743294i	$0.976318\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$		0			0
$185$	−2.00000	−	4.00000i	−0.147043	−	0.294086i
$186$		0			0
$187$	−4.00000			−0.292509
$188$		0			0
$189$		−	2.00000i		−	0.145479i
$190$		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
$195$	−6.00000	−	12.0000i	−0.429669	−	0.859338i
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		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
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$		0			0
$207$		−	4.00000i		−	0.278019i
$208$		0			0
$209$		0			0
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		0			0
$213$	12.0000			0.822226
$214$		0			0
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$		−	16.0000i		−	1.08615i
$218$		0			0
$219$			4.00000i			0.270295i
$220$		0			0
$221$		−	12.0000i		−	0.807207i
$222$		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
$227$	−28.0000			−1.85843			−0.929213	−	0.369546i	$-0.879513\pi$
$227$	−28.0000			−1.85843			−0.929213	+	0.369546i	$0.879513\pi$
$228$		0			0
$229$			10.0000i			0.660819i	0.943838	+	0.330409i	$0.107187\pi$
$229$			10.0000i			0.660819i	−0.943838	+	0.330409i	$0.892813\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$			14.0000i			0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$			14.0000i			0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$		0			0
$235$	−16.0000	+	8.00000i	−1.04372	+	0.521862i
$236$		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
$243$	1.00000			0.0641500
$244$		0			0
$245$	−3.00000	−	6.00000i	−0.191663	−	0.383326i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$			18.0000i			1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$			18.0000i			1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		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
$259$		−	4.00000i		−	0.248548i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	4.00000i		−	0.246651i	−0.992366	−	0.123325i	$-0.960644\pi$
$263$		−	4.00000i		−	0.246651i	0.992366	−	0.123325i	$-0.0393559\pi$
$264$		0			0
$265$	6.00000	+	12.0000i	0.368577	+	0.737154i
$266$		0			0
$267$	−10.0000			−0.611990
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		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
$275$	8.00000	+	6.00000i	0.482418	+	0.361814i
$276$		0			0
$277$	2.00000			0.120168			0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000			0.120168			0.0600842	+	0.998193i	$0.480863\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$	16.0000			0.951101			0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000			0.951101			0.475551	+	0.879688i	$0.342249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$			8.00000i			0.468968i
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$	20.0000	−	10.0000i	1.16445	−	0.582223i
$296$		0			0
$297$		−	2.00000i		−	0.116052i
$298$		0			0
$299$		−	24.0000i		−	1.38796i
$300$		0			0
$301$			8.00000i			0.461112i
$302$		0			0
$303$		−	8.00000i		−	0.459588i
$304$		0			0
$305$	−4.00000	+	2.00000i	−0.229039	+	0.114520i
$306$		0			0
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		−	14.0000i		−	0.796432i
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$			4.00000i			0.226093i	0.993590	+	0.113047i	$0.0360610\pi$
$313$			4.00000i			0.226093i	−0.993590	+	0.113047i	$0.963939\pi$
$314$		0			0
$315$	−4.00000	+	2.00000i	−0.225374	+	0.112687i
$316$		0			0
$317$	−2.00000			−0.112331			−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000			−0.112331			−0.0561656	+	0.998421i	$0.517887\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−18.0000	+	24.0000i	−0.998460	+	1.33128i
$326$		0			0
$327$			10.0000i			0.553001i
$328$		0			0
$329$	−16.0000			−0.882109
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		0			0
$333$	2.00000			0.109599
$334$		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
$339$			6.00000i			0.325875i
$340$		0			0
$341$		−	16.0000i		−	0.866449i
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	−8.00000	+	4.00000i	−0.430706	+	0.215353i
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$			10.0000i			0.535288i	0.963518	+	0.267644i	$0.0862451\pi$
$349$			10.0000i			0.535288i	−0.963518	+	0.267644i	$0.913755\pi$
$350$		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
$355$	−12.0000	−	24.0000i	−0.636894	−	1.27379i
$356$		0			0
$357$	−4.00000			−0.211702
$358$		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
$363$	7.00000			0.367405
$364$		0			0
$365$	8.00000	−	4.00000i	0.418739	−	0.209370i
$366$		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
$371$			12.0000i			0.623009i
$372$		0			0
$373$	−6.00000			−0.310668			−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000			−0.310668			−0.155334	+	0.987862i	$0.549645\pi$
$374$		0			0
$375$	11.0000	+	2.00000i	0.568038	+	0.103280i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$			2.00000i			0.102463i
$382$		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
$387$	−4.00000			−0.203331
$388$		0			0
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$		0			0
$393$		−	18.0000i		−	0.907980i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		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
$403$	48.0000			2.39105
$404$		0			0
$405$	−1.00000	−	2.00000i	−0.0496904	−	0.0993808i
$406$		0			0
$407$		−	4.00000i		−	0.198273i
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		−	18.0000i		−	0.887875i
$412$		0			0
$413$	20.0000			0.984136
$414$		0			0
$415$	−4.00000	−	8.00000i	−0.196352	−	0.392705i
$416$		0			0
$417$		−	20.0000i		−	0.979404i
$418$		0			0
$419$			10.0000i			0.488532i	0.969708	+	0.244266i	$0.0785470\pi$
$419$			10.0000i			0.488532i	−0.969708	+	0.244266i	$0.921453\pi$
$420$		0			0
$421$			22.0000i			1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$			22.0000i			1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	8.00000	+	6.00000i	0.388057	+	0.291043i
$426$		0			0
$427$	−4.00000			−0.193574
$428$		0			0
$429$		−	12.0000i		−	0.579365i
$430$		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
$443$	36.0000			1.71041			0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000			1.71041			0.855206	+	0.518289i	$0.173431\pi$
$444$		0			0
$445$	10.0000	+	20.0000i	0.474045	+	0.948091i
$446$		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
$451$			4.00000i			0.188353i
$452$		0			0
$453$	−8.00000			−0.375873
$454$		0			0
$455$	−24.0000	+	12.0000i	−1.12514	+	0.562569i
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$		0			0
$459$		−	2.00000i		−	0.0933520i
$460$		0			0
$461$		−	12.0000i		−	0.558896i	−0.960161	−	0.279448i	$-0.909849\pi$
$461$		−	12.0000i		−	0.558896i	0.960161	−	0.279448i	$-0.0901514\pi$
$462$		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
$467$	12.0000			0.555294			0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000			0.555294			0.277647	+	0.960683i	$0.410445\pi$
$468$		0			0
$469$			16.0000i			0.738811i
$470$		0			0
$471$	−22.0000			−1.01371
$472$		0			0
$473$			8.00000i			0.367840i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−6.00000			−0.274721
$478$		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
$483$	−8.00000			−0.364013
$484$		0			0
$485$	16.0000	−	8.00000i	0.726523	−	0.363261i
$486$		0			0
$487$			18.0000i			0.815658i	0.913058	+	0.407829i	$0.133714\pi$
$487$			18.0000i			0.815658i	−0.913058	+	0.407829i	$0.866286\pi$
$488$		0			0
$489$	−16.0000			−0.723545
$490$		0			0
$491$			18.0000i			0.812329i	0.913800	+	0.406164i	$0.133134\pi$
$491$			18.0000i			0.812329i	−0.913800	+	0.406164i	$0.866866\pi$
$492$		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		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$		−	12.0000i		−	0.536120i
$502$		0			0
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$		0			0
$505$	−16.0000	+	8.00000i	−0.711991	+	0.355995i
$506$		0			0
$507$	23.0000			1.02147
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	8.00000			0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.0000	+	14.0000i	−1.23383	+	0.616914i
$516$		0			0
$517$	−16.0000			−0.703679
$518$		0			0
$519$	−14.0000			−0.614532
$520$		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
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$	8.00000	+	6.00000i	0.349149	+	0.261861i
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	7.00000			0.304348
$530$		0			0
$531$			10.0000i			0.433963i
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	12.0000	+	24.0000i	0.518805	+	1.03761i
$536$		0			0
$537$		−	10.0000i		−	0.431532i
$538$		0			0
$539$		−	6.00000i		−	0.258438i
$540$		0			0
$541$			38.0000i			1.63375i	0.576816	+	0.816874i	$0.304295\pi$
$541$			38.0000i			1.63375i	−0.576816	+	0.816874i	$0.695705\pi$
$542$		0			0
$543$			2.00000i			0.0858282i
$544$		0			0
$545$	20.0000	−	10.0000i	0.856706	−	0.428353i
$546$		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
$549$		−	2.00000i		−	0.0853579i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.00000	−	4.00000i	−0.0848953	−	0.169791i
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$	−4.00000			−0.168880
$562$		0			0
$563$	44.0000			1.85438			0.927189	−	0.374593i	$-0.122217\pi$
$563$	44.0000			1.85438			0.927189	+	0.374593i	$0.122217\pi$
$564$		0			0
$565$	12.0000	−	6.00000i	0.504844	−	0.252422i
$566$		0			0
$567$		−	2.00000i		−	0.0839921i
$568$		0			0
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$			8.00000i			0.334790i	0.985890	+	0.167395i	$0.0535355\pi$
$571$			8.00000i			0.334790i	−0.985890	+	0.167395i	$0.946465\pi$
$572$		0			0
$573$	−12.0000			−0.501307
$574$		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
$579$		−	4.00000i		−	0.166234i
$580$		0			0
$581$		−	8.00000i		−	0.331896i
$582$		0			0
$583$			12.0000i			0.496989i
$584$		0			0
$585$	−6.00000	−	12.0000i	−0.248069	−	0.496139i
$586$		0			0
$587$	−12.0000			−0.495293			−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000			−0.495293			−0.247647	+	0.968850i	$0.579657\pi$
$588$		0			0
$589$		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
$595$	4.00000	+	8.00000i	0.163984	+	0.327968i
$596$		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	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$	−8.00000			−0.325785
$604$		0			0
$605$	−7.00000	−	14.0000i	−0.284590	−	0.569181i
$606$		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
$611$		−	48.0000i		−	1.94187i
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$		0			0
$615$	2.00000	+	4.00000i	0.0806478	+	0.161296i
$616$		0			0
$617$			2.00000i			0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		0			0
$619$			20.0000i			0.803868i	0.915669	+	0.401934i	$0.131662\pi$
$619$			20.0000i			0.803868i	−0.915669	+	0.401934i	$0.868338\pi$
$620$		0			0
$621$		−	4.00000i		−	0.160514i
$622$		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
$629$		−	4.00000i		−	0.159490i
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$			12.0000i			0.476957i
$634$		0			0
$635$	4.00000	−	2.00000i	0.158735	−	0.0793676i
$636$		0			0
$637$	18.0000			0.713186
$638$		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
$643$	24.0000			0.946468			0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000			0.946468			0.473234	+	0.880937i	$0.343087\pi$
$644$		0			0
$645$	4.00000	+	8.00000i	0.157500	+	0.315000i
$646$		0			0
$647$			48.0000i			1.88707i	0.331266	+	0.943537i	$0.392524\pi$
$647$			48.0000i			1.88707i	−0.331266	+	0.943537i	$0.607476\pi$
$648$		0			0
$649$	20.0000			0.785069
$650$		0			0
$651$		−	16.0000i		−	0.627089i
$652$		0			0
$653$	26.0000			1.01746			0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000			1.01746			0.508729	+	0.860927i	$0.330115\pi$
$654$		0			0
$655$	−36.0000	+	18.0000i	−1.40664	+	0.703318i
$656$		0			0
$657$			4.00000i			0.156055i
$658$		0			0
$659$			50.0000i			1.94772i	0.227142	+	0.973862i	$0.427062\pi$
$659$			50.0000i			1.94772i	−0.227142	+	0.973862i	$0.572938\pi$
$660$		0			0
$661$			2.00000i			0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$			2.00000i			0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$		0			0
$663$		−	12.0000i		−	0.466041i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		−	26.0000i		−	1.00522i
$670$		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
$675$	−3.00000	+	4.00000i	−0.115470	+	0.153960i
$676$		0			0
$677$	2.00000			0.0768662			0.0384331	−	0.999261i	$-0.487763\pi$
$677$	2.00000			0.0768662			0.0384331	+	0.999261i	$0.487763\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$		0			0
$681$	−28.0000			−1.07296
$682$		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
$685$	−36.0000	+	18.0000i	−1.37549	+	0.687745i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	−36.0000			−1.37149
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		0			0
$693$	−4.00000			−0.151947
$694$		0			0
$695$	−40.0000	+	20.0000i	−1.51729	+	0.758643i
$696$		0			0
$697$			4.00000i			0.151511i
$698$		0			0
$699$			14.0000i			0.529529i
$700$		0			0
$701$		−	32.0000i		−	1.20862i	−0.796748	−	0.604312i	$-0.793448\pi$
$701$		−	32.0000i		−	1.20862i	0.796748	−	0.604312i	$-0.206552\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−16.0000	+	8.00000i	−0.602595	+	0.301297i
$706$		0			0
$707$	−16.0000			−0.601742
$708$		0			0
$709$		−	30.0000i		−	1.12667i	−0.826227	−	0.563337i	$-0.809517\pi$
$709$		−	30.0000i		−	1.12667i	0.826227	−	0.563337i	$-0.190483\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	32.0000i		−	1.19841i
$714$		0			0
$715$	−24.0000	+	12.0000i	−0.897549	+	0.448775i
$716$		0			0
$717$	20.0000			0.746914
$718$		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
$723$	22.0000			0.818189
$724$		0			0
$725$		0			0
$726$		0			0
$727$			18.0000i			0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$			18.0000i			0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			8.00000i			0.295891i
$732$		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
$737$			16.0000i			0.589368i
$738$		0			0
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	24.0000i		−	0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$		−	24.0000i		−	0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$		0			0
$745$	40.0000	−	20.0000i	1.46549	−	0.732743i
$746$		0			0
$747$	4.00000			0.146352
$748$		0			0
$749$			24.0000i			0.876941i
$750$		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
$755$	8.00000	+	16.0000i	0.291150	+	0.582300i
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	−8.00000			−0.290382
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	20.0000			0.724049
$764$		0			0
$765$	−4.00000	+	2.00000i	−0.144620	+	0.0723102i
$766$		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
$771$			18.0000i			0.648254i
$772$		0			0
$773$	54.0000			1.94225			0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000			1.94225			0.971123	+	0.238581i	$0.0766824\pi$
$774$		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
$781$		−	24.0000i		−	0.858788i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	22.0000	+	44.0000i	0.785214	+	1.57043i
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$		0			0
$789$		−	4.00000i		−	0.142404i
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$		−	12.0000i		−	0.426132i
$794$		0			0
$795$	6.00000	+	12.0000i	0.212798	+	0.425596i
$796$		0			0
$797$	−2.00000			−0.0708436			−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000			−0.0708436			−0.0354218	+	0.999372i	$0.511277\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$	−10.0000			−0.353333
$802$		0			0
$803$	8.00000			0.282314
$804$		0			0
$805$	8.00000	+	16.0000i	0.281963	+	0.563926i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	52.0000i		−	1.82597i	−0.407997	−	0.912983i	$-0.633772\pi$
$811$		−	52.0000i		−	1.82597i	0.407997	−	0.912983i	$-0.366228\pi$
$812$		0			0
$813$	8.00000			0.280572
$814$		0			0
$815$	16.0000	+	32.0000i	0.560456	+	1.12091i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	12.0000i		−	0.419314i
$820$		0			0
$821$		−	8.00000i		−	0.279202i	−0.990208	−	0.139601i	$-0.955418\pi$
$821$		−	8.00000i		−	0.279202i	0.990208	−	0.139601i	$-0.0445820\pi$
$822$		0			0
$823$			6.00000i			0.209147i	0.994517	+	0.104573i	$0.0333477\pi$
$823$			6.00000i			0.209147i	−0.994517	+	0.104573i	$0.966652\pi$
$824$		0			0
$825$	8.00000	+	6.00000i	0.278524	+	0.208893i
$826$		0			0
$827$	28.0000			0.973655			0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000			0.973655			0.486828	+	0.873498i	$0.338154\pi$
$828$		0			0
$829$		−	30.0000i		−	1.04194i	−0.853574	−	0.520972i	$-0.825570\pi$
$829$		−	30.0000i		−	1.04194i	0.853574	−	0.520972i	$-0.174430\pi$
$830$		0			0
$831$	2.00000			0.0693792
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	−24.0000	+	12.0000i	−0.830554	+	0.415277i
$836$		0			0
$837$	8.00000			0.276520
$838$		0			0
$839$	−40.0000			−1.38095			−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000			−1.38095			−0.690477	+	0.723355i	$0.742599\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	18.0000			0.619953
$844$		0			0
$845$	−23.0000	−	46.0000i	−0.791224	−	1.58245i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$		0			0
$849$	16.0000			0.549119
$850$		0			0
$851$		−	8.00000i		−	0.274236i
$852$		0			0
$853$	14.0000			0.479351			0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000			0.479351			0.239675	+	0.970853i	$0.422959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			42.0000i			1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$			42.0000i			1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$		0			0
$859$			20.0000i			0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$			20.0000i			0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$		0			0
$861$			4.00000i			0.136320i
$862$		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
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−48.0000			−1.62642
$872$		0			0
$873$			8.00000i			0.270759i
$874$		0			0
$875$	4.00000	−	22.0000i	0.135225	−	0.743736i
$876$		0			0
$877$	−22.0000			−0.742887			−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000			−0.742887			−0.371444	+	0.928456i	$0.621137\pi$
$878$		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
$883$	24.0000			0.807664			0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000			0.807664			0.403832	+	0.914833i	$0.367678\pi$
$884$		0			0
$885$	20.0000	−	10.0000i	0.672293	−	0.336146i
$886$		0			0
$887$		−	12.0000i		−	0.402921i	−0.979497	−	0.201460i	$-0.935431\pi$
$887$		−	12.0000i		−	0.402921i	0.979497	−	0.201460i	$-0.0645687\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$		0			0
$891$		−	2.00000i		−	0.0670025i
$892$		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
$901$			12.0000i			0.399778i
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$	4.00000	−	2.00000i	0.132964	−	0.0664822i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$		0			0
$909$		−	8.00000i		−	0.265343i
$910$		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
$915$	−4.00000	+	2.00000i	−0.132236	+	0.0661180i
$916$		0			0
$917$	−36.0000			−1.18882
$918$		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
$923$	72.0000			2.36991
$924$		0			0
$925$	−6.00000	+	8.00000i	−0.197279	+	0.263038i
$926$		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
$933$	12.0000			0.392862
$934$		0			0
$935$	4.00000	+	8.00000i	0.130814	+	0.261628i
$936$		0			0
$937$		−	8.00000i		−	0.261349i	−0.991425	−	0.130674i	$-0.958286\pi$
$937$		−	8.00000i		−	0.261349i	0.991425	−	0.130674i	$-0.0417142\pi$
$938$		0			0
$939$			4.00000i			0.130535i
$940$		0			0
$941$			28.0000i			0.912774i	0.889781	+	0.456387i	$0.150857\pi$
$941$			28.0000i			0.912774i	−0.889781	+	0.456387i	$0.849143\pi$
$942$		0			0
$943$			8.00000i			0.260516i
$944$		0			0
$945$	−4.00000	+	2.00000i	−0.130120	+	0.0650600i
$946$		0			0
$947$	12.0000			0.389948			0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000			0.389948			0.194974	+	0.980808i	$0.437538\pi$
$948$		0			0
$949$			24.0000i			0.779073i
$950$		0			0
$951$	−2.00000			−0.0648544
$952$		0			0
$953$		−	46.0000i		−	1.49009i	−0.667016	−	0.745043i	$-0.732429\pi$
$953$		−	46.0000i		−	1.49009i	0.667016	−	0.745043i	$-0.267571\pi$
$954$		0			0
$955$	12.0000	+	24.0000i	0.388311	+	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.0000			−1.16250
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$	−12.0000			−0.386695
$964$		0			0
$965$	−8.00000	+	4.00000i	−0.257529	+	0.128765i
$966$		0			0
$967$			38.0000i			1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$			38.0000i			1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			18.0000i			0.577647i	0.957382	+	0.288824i	$0.0932642\pi$
$971$			18.0000i			0.577647i	−0.957382	+	0.288824i	$0.906736\pi$
$972$		0			0
$973$	−40.0000			−1.28234
$974$		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
$979$			20.0000i			0.639203i
$980$		0			0
$981$			10.0000i			0.319275i
$982$		0			0
$983$			16.0000i			0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$			16.0000i			0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$		0			0
$985$	−22.0000	−	44.0000i	−0.700978	−	1.40196i
$986$		0			0
$987$	−16.0000			−0.509286
$988$		0			0
$989$			16.0000i			0.508770i
$990$		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
$997$	−18.0000			−0.570066			−0.285033	−	0.958518i	$-0.592005\pi$
$997$	−18.0000			−0.570066			−0.285033	+	0.958518i	$0.592005\pi$
$998$		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	3840.2.d.x.2689.1		2
4.3	odd	2		3840.2.d.g.2689.1		2
5.4	even	2		3840.2.d.j.2689.1		2
8.3	odd	2		3840.2.d.y.2689.2		2
8.5	even	2		3840.2.d.j.2689.2		2
16.3	odd	4		30.2.c.a.19.2	yes	2
16.5	even	4		960.2.f.i.769.1		2
16.11	odd	4		960.2.f.h.769.2		2
16.13	even	4		240.2.f.a.49.2		2
20.19	odd	2		3840.2.d.y.2689.1		2
40.19	odd	2		3840.2.d.g.2689.2		2
40.29	even	2	inner	3840.2.d.x.2689.2		2
48.5	odd	4		2880.2.f.c.1729.2		2
48.11	even	4		2880.2.f.e.1729.2		2
48.29	odd	4		720.2.f.f.289.1		2
48.35	even	4		90.2.c.a.19.1		2
80.3	even	4		150.2.a.c.1.1		1
80.13	odd	4		1200.2.a.g.1.1		1
80.19	odd	4		30.2.c.a.19.1	✓	2
80.27	even	4		4800.2.a.cg.1.1		1
80.29	even	4		240.2.f.a.49.1		2
80.37	odd	4		4800.2.a.m.1.1		1
80.43	even	4		4800.2.a.l.1.1		1
80.53	odd	4		4800.2.a.cj.1.1		1
80.59	odd	4		960.2.f.h.769.1		2
80.67	even	4		150.2.a.a.1.1		1
80.69	even	4		960.2.f.i.769.2		2
80.77	odd	4		1200.2.a.m.1.1		1
112.3	even	12		1470.2.n.a.79.1		4
112.19	even	12		1470.2.n.a.949.2		4
112.51	odd	12		1470.2.n.h.949.2		4
112.67	odd	12		1470.2.n.h.79.1		4
112.83	even	4		1470.2.g.g.589.2		2
144.67	odd	12		810.2.i.e.379.1		4
144.83	even	12		810.2.i.b.109.1		4
144.115	odd	12		810.2.i.e.109.2		4
144.131	even	12		810.2.i.b.379.2		4
240.29	odd	4		720.2.f.f.289.2		2
240.59	even	4		2880.2.f.e.1729.1		2
240.77	even	4		3600.2.a.o.1.1		1
240.83	odd	4		450.2.a.b.1.1		1
240.149	odd	4		2880.2.f.c.1729.1		2
240.173	even	4		3600.2.a.bg.1.1		1
240.179	even	4		90.2.c.a.19.2		2
240.227	odd	4		450.2.a.f.1.1		1
560.19	even	12		1470.2.n.a.949.1		4
560.83	odd	4		7350.2.a.cc.1.1		1
560.179	odd	12		1470.2.n.h.79.2		4
560.307	odd	4		7350.2.a.bg.1.1		1
560.339	even	12		1470.2.n.a.79.2		4
560.419	even	4		1470.2.g.g.589.1		2
560.499	odd	12		1470.2.n.h.949.1		4
720.259	odd	12		810.2.i.e.109.1		4
720.419	even	12		810.2.i.b.379.1		4
720.499	odd	12		810.2.i.e.379.2		4
720.659	even	12		810.2.i.b.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.c.a.19.1	✓	2	80.19	odd	4
30.2.c.a.19.2	yes	2	16.3	odd	4
90.2.c.a.19.1		2	48.35	even	4
90.2.c.a.19.2		2	240.179	even	4
150.2.a.a.1.1		1	80.67	even	4
150.2.a.c.1.1		1	80.3	even	4
240.2.f.a.49.1		2	80.29	even	4
240.2.f.a.49.2		2	16.13	even	4
450.2.a.b.1.1		1	240.83	odd	4
450.2.a.f.1.1		1	240.227	odd	4
720.2.f.f.289.1		2	48.29	odd	4
720.2.f.f.289.2		2	240.29	odd	4
810.2.i.b.109.1		4	144.83	even	12
810.2.i.b.109.2		4	720.659	even	12
810.2.i.b.379.1		4	720.419	even	12
810.2.i.b.379.2		4	144.131	even	12
810.2.i.e.109.1		4	720.259	odd	12
810.2.i.e.109.2		4	144.115	odd	12
810.2.i.e.379.1		4	144.67	odd	12
810.2.i.e.379.2		4	720.499	odd	12
960.2.f.h.769.1		2	80.59	odd	4
960.2.f.h.769.2		2	16.11	odd	4
960.2.f.i.769.1		2	16.5	even	4
960.2.f.i.769.2		2	80.69	even	4
1200.2.a.g.1.1		1	80.13	odd	4
1200.2.a.m.1.1		1	80.77	odd	4
1470.2.g.g.589.1		2	560.419	even	4
1470.2.g.g.589.2		2	112.83	even	4
1470.2.n.a.79.1		4	112.3	even	12
1470.2.n.a.79.2		4	560.339	even	12
1470.2.n.a.949.1		4	560.19	even	12
1470.2.n.a.949.2		4	112.19	even	12
1470.2.n.h.79.1		4	112.67	odd	12
1470.2.n.h.79.2		4	560.179	odd	12
1470.2.n.h.949.1		4	560.499	odd	12
1470.2.n.h.949.2		4	112.51	odd	12
2880.2.f.c.1729.1		2	240.149	odd	4
2880.2.f.c.1729.2		2	48.5	odd	4
2880.2.f.e.1729.1		2	240.59	even	4
2880.2.f.e.1729.2		2	48.11	even	4
3600.2.a.o.1.1		1	240.77	even	4
3600.2.a.bg.1.1		1	240.173	even	4
3840.2.d.g.2689.1		2	4.3	odd	2
3840.2.d.g.2689.2		2	40.19	odd	2
3840.2.d.j.2689.1		2	5.4	even	2
3840.2.d.j.2689.2		2	8.5	even	2
3840.2.d.x.2689.1		2	1.1	even	1	trivial
3840.2.d.x.2689.2		2	40.29	even	2	inner
3840.2.d.y.2689.1		2	20.19	odd	2
3840.2.d.y.2689.2		2	8.3	odd	2
4800.2.a.l.1.1		1	80.43	even	4
4800.2.a.m.1.1		1	80.37	odd	4
4800.2.a.cg.1.1		1	80.27	even	4
4800.2.a.cj.1.1		1	80.53	odd	4
7350.2.a.bg.1.1		1	560.307	odd	4
7350.2.a.cc.1.1		1	560.83	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	3840.2.d.x.2689.1

Level	$3840$
Weight	$2$
Character	3840.2689
Analytic conductor	$30.663$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$