LMFDB - Embedded newform 3120.2.l.g.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("3120.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 390)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +2.00000i q^{17} -4.00000 q^{19} +2.00000 q^{21} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -4.00000 q^{29} -8.00000 q^{31} -2.00000i q^{33} +(2.00000 - 4.00000i) q^{35} +6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -2.00000i q^{53} +(-4.00000 - 2.00000i) q^{55} -4.00000i q^{57} +10.0000 q^{59} -14.0000 q^{61} +2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} +16.0000i q^{67} +4.00000 q^{71} -8.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} +4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +(-2.00000 + 4.00000i) q^{85} -4.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} -8.00000i q^{93} +(-8.00000 - 4.00000i) q^{95} +12.0000i q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^5 - 2 * q^9	$ 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{19} + 4 q^{21} + 6 q^{25} - 8 q^{29} - 16 q^{31} + 4 q^{35} - 2 q^{39} - 12 q^{41} - 4 q^{45} + 6 q^{49} - 4 q^{51} - 8 q^{55} + 20 q^{59} - 28 q^{61} - 2 q^{65} + 8 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 4 q^{85} - 12 q^{89} + 4 q^{91} - 16 q^{95} + 4 q^{99}+O(q^{100}) $ 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^19 + 4 * q^21 + 6 * q^25 - 8 * q^29 - 16 * q^31 + 4 * q^35 - 2 * q^39 - 12 * q^41 - 4 * q^45 + 6 * q^49 - 4 * q^51 - 8 * q^55 + 20 * q^59 - 28 * q^61 - 2 * q^65 + 8 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 - 4 * q^85 - 12 * q^89 + 4 * q^91 - 16 * q^95 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$5$	2.00000	+	1.00000i	0.894427	+	0.447214i
$6$		0			0
$7$		−	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
$11$	−2.00000			−0.603023			−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000			−0.603023			−0.301511	+	0.953463i	$0.597491\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$	−1.00000	+	2.00000i	−0.258199	+	0.516398i
$16$		0			0
$17$			2.00000i			0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$			2.00000i			0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	2.00000			0.436436
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		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
$35$	2.00000	−	4.00000i	0.338062	−	0.676123i
$36$		0			0
$37$			6.00000i			0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$			6.00000i			0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	−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
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	−2.00000	−	1.00000i	−0.298142	−	0.149071i
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$	−2.00000			−0.280056
$52$		0			0
$53$		−	2.00000i		−	0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$		−	2.00000i		−	0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$		0			0
$55$	−4.00000	−	2.00000i	−0.539360	−	0.269680i
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	−14.0000			−1.79252			−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000			−1.79252			−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$			2.00000i			0.251976i
$64$		0			0
$65$	−1.00000	+	2.00000i	−0.124035	+	0.248069i
$66$		0			0
$67$			16.0000i			1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$			16.0000i			1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	4.00000			0.474713			0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000			0.474713			0.237356	+	0.971423i	$0.423719\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$		0			0
$75$	−4.00000	+	3.00000i	−0.461880	+	0.346410i
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$	−2.00000	+	4.00000i	−0.216930	+	0.433861i
$86$		0			0
$87$		−	4.00000i		−	0.428845i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	2.00000			0.209657
$92$		0			0
$93$		−	8.00000i		−	0.829561i
$94$		0			0
$95$	−8.00000	−	4.00000i	−0.820783	−	0.410391i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$	2.00000			0.201008
$100$		0			0
$101$	4.00000			0.398015			0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000			0.398015			0.199007	+	0.979998i	$0.436228\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$	4.00000	+	2.00000i	0.390360	+	0.195180i
$106$		0			0
$107$			4.00000i			0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$			4.00000i			0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$		0			0
$109$	18.0000			1.72409			0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000			1.72409			0.862044	+	0.506834i	$0.169184\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$		−	6.00000i		−	0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$		−	6.00000i		−	0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	1.00000i		−	0.0924500i
$118$		0			0
$119$	4.00000			0.366679
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$	2.00000	+	11.0000i	0.178885	+	0.983870i
$126$		0			0
$127$		−	6.00000i		−	0.532414i	−0.963916	−	0.266207i	$-0.914230\pi$
$127$		−	6.00000i		−	0.532414i	0.963916	−	0.266207i	$-0.0857705\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$	1.00000	−	2.00000i	0.0860663	−	0.172133i
$136$		0			0
$137$		−	14.0000i		−	1.19610i	−0.801459	−	0.598050i	$-0.795942\pi$
$137$		−	14.0000i		−	1.19610i	0.801459	−	0.598050i	$-0.204058\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	−8.00000			−0.673722
$142$		0			0
$143$		−	2.00000i		−	0.167248i
$144$		0			0
$145$	−8.00000	−	4.00000i	−0.664364	−	0.332182i
$146$		0			0
$147$			3.00000i			0.247436i
$148$		0			0
$149$	−12.0000			−0.983078			−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000			−0.983078			−0.491539	+	0.870855i	$0.663566\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$		0			0
$153$		−	2.00000i		−	0.161690i
$154$		0			0
$155$	−16.0000	−	8.00000i	−1.28515	−	0.642575i
$156$		0			0
$157$			10.0000i			0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$			10.0000i			0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$		0			0
$159$	2.00000			0.158610
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		0			0
$165$	2.00000	−	4.00000i	0.155700	−	0.311400i
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$		−	14.0000i		−	1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$		−	14.0000i		−	1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$		0			0
$175$	8.00000	−	6.00000i	0.604743	−	0.453557i
$176$		0			0
$177$			10.0000i			0.751646i
$178$		0			0
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		−	14.0000i		−	1.03491i
$184$		0			0
$185$	−6.00000	+	12.0000i	−0.441129	+	0.882258i
$186$		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	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	−2.00000	−	1.00000i	−0.143223	−	0.0716115i
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$		0			0
$201$	−16.0000			−1.12855
$202$		0			0
$203$			8.00000i			0.561490i
$204$		0			0
$205$	−12.0000	−	6.00000i	−0.838116	−	0.419058i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000			0.553372
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$			4.00000i			0.274075i
$214$		0			0
$215$	−4.00000	+	8.00000i	−0.272798	+	0.545595i
$216$		0			0
$217$			16.0000i			1.08615i
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	−2.00000			−0.134535
$222$		0			0
$223$		−	14.0000i		−	0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$		−	14.0000i		−	0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$		0			0
$225$	−3.00000	−	4.00000i	−0.200000	−	0.266667i
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		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
$235$	−8.00000	+	16.0000i	−0.521862	+	1.04372i
$236$		0			0
$237$		−	8.00000i		−	0.519656i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	6.00000	+	3.00000i	0.383326	+	0.191663i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$	−12.0000			−0.760469
$250$		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
$253$		0			0
$254$		0			0
$255$	−4.00000	−	2.00000i	−0.250490	−	0.125245i
$256$		0			0
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	12.0000			0.745644
$260$		0			0
$261$	4.00000			0.247594
$262$		0			0
$263$		−	8.00000i		−	0.493301i	−0.969104	−	0.246651i	$-0.920670\pi$
$263$		−	8.00000i		−	0.493301i	0.969104	−	0.246651i	$-0.0793300\pi$
$264$		0			0
$265$	2.00000	−	4.00000i	0.122859	−	0.245718i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$	20.0000			1.21942			0.609711	−	0.792624i	$-0.291286\pi$
$269$	20.0000			1.21942			0.609711	+	0.792624i	$0.291286\pi$
$270$		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
$273$			2.00000i			0.121046i
$274$		0			0
$275$	−6.00000	−	8.00000i	−0.361814	−	0.482418i
$276$		0			0
$277$		−	2.00000i		−	0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$		−	2.00000i		−	0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$		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
$285$	4.00000	−	8.00000i	0.236940	−	0.473879i
$286$		0			0
$287$			12.0000i			0.708338i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$		−	26.0000i		−	1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$		−	26.0000i		−	1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$		0			0
$295$	20.0000	+	10.0000i	1.16445	+	0.582223i
$296$		0			0
$297$			2.00000i			0.116052i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	8.00000			0.461112
$302$		0			0
$303$			4.00000i			0.229794i
$304$		0			0
$305$	−28.0000	−	14.0000i	−1.60328	−	0.801638i
$306$		0			0
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		0			0
$309$	−6.00000			−0.341328
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$			20.0000i			1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$			20.0000i			1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$		0			0
$315$	−2.00000	+	4.00000i	−0.112687	+	0.225374i
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$	8.00000			0.447914
$320$		0			0
$321$	−4.00000			−0.223258
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	−4.00000	+	3.00000i	−0.221880	+	0.166410i
$326$		0			0
$327$			18.0000i			0.995402i
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	12.0000			0.659580			0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000			0.659580			0.329790	+	0.944054i	$0.393022\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$	−16.0000	+	32.0000i	−0.874173	+	1.74835i
$336$		0			0
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		0			0
$346$		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
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	1.00000			0.0533761
$352$		0			0
$353$			34.0000i			1.80964i	0.425797	+	0.904819i	$0.359994\pi$
$353$			34.0000i			1.80964i	−0.425797	+	0.904819i	$0.640006\pi$
$354$		0			0
$355$	8.00000	+	4.00000i	0.424596	+	0.212298i
$356$		0			0
$357$			4.00000i			0.211702i
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	7.00000i		−	0.367405i
$364$		0			0
$365$	8.00000	−	16.0000i	0.418739	−	0.837478i
$366$		0			0
$367$			10.0000i			0.521996i	0.965339	+	0.260998i	$0.0840516\pi$
$367$			10.0000i			0.521996i	−0.965339	+	0.260998i	$0.915948\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$	−4.00000			−0.207670
$372$		0			0
$373$			22.0000i			1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$			22.0000i			1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$		0			0
$375$	−11.0000	+	2.00000i	−0.568038	+	0.103280i
$376$		0			0
$377$		−	4.00000i		−	0.206010i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$	6.00000			0.307389
$382$		0			0
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$	−4.00000	+	8.00000i	−0.203859	+	0.407718i
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		0			0
$389$	−16.0000			−0.811232			−0.405616	−	0.914044i	$-0.632943\pi$
$389$	−16.0000			−0.811232			−0.405616	+	0.914044i	$0.632943\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			6.00000i			0.302660i
$394$		0			0
$395$	−16.0000	−	8.00000i	−0.805047	−	0.402524i
$396$		0			0
$397$		−	18.0000i		−	0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$		−	18.0000i		−	0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$		0			0
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$		0			0
$405$	2.00000	+	1.00000i	0.0993808	+	0.0496904i
$406$		0			0
$407$		−	12.0000i		−	0.594818i
$408$		0			0
$409$	26.0000			1.28562			0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000			1.28562			0.642809	+	0.766027i	$0.277769\pi$
$410$		0			0
$411$	14.0000			0.690569
$412$		0			0
$413$		−	20.0000i		−	0.984136i
$414$		0			0
$415$	−12.0000	+	24.0000i	−0.589057	+	1.17811i
$416$		0			0
$417$			4.00000i			0.195881i
$418$		0			0
$419$	−14.0000			−0.683945			−0.341972	−	0.939710i	$-0.611095\pi$
$419$	−14.0000			−0.683945			−0.341972	+	0.939710i	$0.611095\pi$
$420$		0			0
$421$	34.0000			1.65706			0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000			1.65706			0.828529	+	0.559946i	$0.189178\pi$
$422$		0			0
$423$		−	8.00000i		−	0.388973i
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$			28.0000i			1.35501i
$428$		0			0
$429$	2.00000			0.0965609
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$		−	12.0000i		−	0.576683i	−0.957528	−	0.288342i	$-0.906896\pi$
$433$		−	12.0000i		−	0.576683i	0.957528	−	0.288342i	$-0.0931039\pi$
$434$		0			0
$435$	4.00000	−	8.00000i	0.191785	−	0.383571i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$			4.00000i			0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$			4.00000i			0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$		0			0
$445$	−12.0000	−	6.00000i	−0.568855	−	0.284427i
$446$		0			0
$447$		−	12.0000i		−	0.567581i
$448$		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
$451$	12.0000			0.565058
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$	4.00000	+	2.00000i	0.187523	+	0.0937614i
$456$		0			0
$457$		−	28.0000i		−	1.30978i	−0.755722	−	0.654892i	$-0.772714\pi$
$457$		−	28.0000i		−	1.30978i	0.755722	−	0.654892i	$-0.227286\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$			6.00000i			0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$			6.00000i			0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$		0			0
$465$	8.00000	−	16.0000i	0.370991	−	0.741982i
$466$		0			0
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	32.0000			1.47762
$470$		0			0
$471$	−10.0000			−0.460776
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$			2.00000i			0.0915737i
$478$		0			0
$479$	36.0000			1.64488			0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000			1.64488			0.822441	+	0.568850i	$0.192612\pi$
$480$		0			0
$481$	−6.00000			−0.273576
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−12.0000	+	24.0000i	−0.544892	+	1.08978i
$486$		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
$489$	8.00000			0.361773
$490$		0			0
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$		0			0
$493$		−	8.00000i		−	0.360302i
$494$		0			0
$495$	4.00000	+	2.00000i	0.179787	+	0.0898933i
$496$		0			0
$497$		−	8.00000i		−	0.358849i
$498$		0			0
$499$	36.0000			1.61158			0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000			1.61158			0.805791	+	0.592200i	$0.201741\pi$
$500$		0			0
$501$	12.0000			0.536120
$502$		0			0
$503$		−	36.0000i		−	1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$		−	36.0000i		−	1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$		0			0
$505$	8.00000	+	4.00000i	0.355995	+	0.177998i
$506$		0			0
$507$		−	1.00000i		−	0.0444116i
$508$		0			0
$509$	40.0000			1.77297			0.886484	−	0.462758i	$-0.153140\pi$
$509$	40.0000			1.77297			0.886484	+	0.462758i	$0.153140\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		0			0
$513$			4.00000i			0.176604i
$514$		0			0
$515$	−6.00000	+	12.0000i	−0.264392	+	0.528783i
$516$		0			0
$517$		−	16.0000i		−	0.703679i
$518$		0			0
$519$	14.0000			0.614532
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		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
$529$	23.0000			1.00000
$530$		0			0
$531$	−10.0000			−0.433963
$532$		0			0
$533$		−	6.00000i		−	0.259889i
$534$		0			0
$535$	−4.00000	+	8.00000i	−0.172935	+	0.345870i
$536$		0			0
$537$		−	18.0000i		−	0.776757i
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		−	6.00000i		−	0.257485i
$544$		0			0
$545$	36.0000	+	18.0000i	1.54207	+	0.771035i
$546$		0			0
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$		0			0
$549$	14.0000			0.597505
$550$		0			0
$551$	16.0000			0.681623
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	−12.0000	−	6.00000i	−0.509372	−	0.254686i
$556$		0			0
$557$		−	14.0000i		−	0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$		−	14.0000i		−	0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		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
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$			12.0000i			0.501307i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			28.0000i			1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$577$			28.0000i			1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$			4.00000i			0.165663i
$584$		0			0
$585$	1.00000	−	2.00000i	0.0413449	−	0.0826898i
$586$		0			0
$587$		−	36.0000i		−	1.48588i	−0.669359	−	0.742940i	$-0.733431\pi$
$587$		−	36.0000i		−	1.48588i	0.669359	−	0.742940i	$-0.266569\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$	−10.0000			−0.411345
$592$		0			0
$593$		−	34.0000i		−	1.39621i	−0.715994	−	0.698106i	$-0.754026\pi$
$593$		−	34.0000i		−	1.39621i	0.715994	−	0.698106i	$-0.245974\pi$
$594$		0			0
$595$	8.00000	+	4.00000i	0.327968	+	0.163984i
$596$		0			0
$597$		−	24.0000i		−	0.982255i
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$		−	16.0000i		−	0.651570i
$604$		0			0
$605$	−14.0000	−	7.00000i	−0.569181	−	0.284590i
$606$		0			0
$607$		−	18.0000i		−	0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$		−	18.0000i		−	0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$		0			0
$609$	−8.00000			−0.324176
$610$		0			0
$611$	−8.00000			−0.323645
$612$		0			0
$613$			2.00000i			0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$			2.00000i			0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$		0			0
$615$	6.00000	−	12.0000i	0.241943	−	0.483887i
$616$		0			0
$617$			46.0000i			1.85189i	0.377658	+	0.925945i	$0.376729\pi$
$617$			46.0000i			1.85189i	−0.377658	+	0.925945i	$0.623271\pi$
$618$		0			0
$619$	−24.0000			−0.964641			−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000			−0.964641			−0.482321	+	0.875995i	$0.660206\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			8.00000i			0.319489i
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$			4.00000i			0.158986i
$634$		0			0
$635$	6.00000	−	12.0000i	0.238103	−	0.476205i
$636$		0			0
$637$			3.00000i			0.118864i
$638$		0			0
$639$	−4.00000			−0.158238
$640$		0			0
$641$	−38.0000			−1.50091			−0.750455	−	0.660922i	$-0.770166\pi$
$641$	−38.0000			−1.50091			−0.750455	+	0.660922i	$0.770166\pi$
$642$		0			0
$643$			32.0000i			1.26196i	0.775800	+	0.630978i	$0.217346\pi$
$643$			32.0000i			1.26196i	−0.775800	+	0.630978i	$0.782654\pi$
$644$		0			0
$645$	−8.00000	−	4.00000i	−0.315000	−	0.157500i
$646$		0			0
$647$		−	36.0000i		−	1.41531i	−0.706560	−	0.707653i	$-0.749754\pi$
$647$		−	36.0000i		−	1.41531i	0.706560	−	0.707653i	$-0.250246\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$	−16.0000			−0.627089
$652$		0			0
$653$			34.0000i			1.33052i	0.746611	+	0.665261i	$0.231680\pi$
$653$			34.0000i			1.33052i	−0.746611	+	0.665261i	$0.768320\pi$
$654$		0			0
$655$	12.0000	+	6.00000i	0.468879	+	0.234439i
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$	10.0000			0.389545			0.194772	−	0.980848i	$-0.437603\pi$
$659$	10.0000			0.389545			0.194772	+	0.980848i	$0.437603\pi$
$660$		0			0
$661$	30.0000			1.16686			0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000			1.16686			0.583432	+	0.812162i	$0.301709\pi$
$662$		0			0
$663$		−	2.00000i		−	0.0776736i
$664$		0			0
$665$	−8.00000	+	16.0000i	−0.310227	+	0.620453i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	14.0000			0.541271
$670$		0			0
$671$	28.0000			1.08093
$672$		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
$677$		−	10.0000i		−	0.384331i	−0.981363	−	0.192166i	$-0.938449\pi$
$677$		−	10.0000i		−	0.384331i	0.981363	−	0.192166i	$-0.0615511\pi$
$678$		0			0
$679$	24.0000			0.921035
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		0			0
$685$	14.0000	−	28.0000i	0.534913	−	1.06983i
$686$		0			0
$687$			22.0000i			0.839352i
$688$		0			0
$689$	2.00000			0.0761939
$690$		0			0
$691$	28.0000			1.06517			0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000			1.06517			0.532585	+	0.846376i	$0.321221\pi$
$692$		0			0
$693$		−	4.00000i		−	0.151947i
$694$		0			0
$695$	8.00000	+	4.00000i	0.303457	+	0.151729i
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$	−14.0000			−0.529529
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$	−16.0000	−	8.00000i	−0.602595	−	0.301297i
$706$		0			0
$707$		−	8.00000i		−	0.300871i
$708$		0			0
$709$	−34.0000			−1.27690			−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000			−1.27690			−0.638448	+	0.769665i	$0.720423\pi$
$710$		0			0
$711$	8.00000			0.300023
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.00000	−	4.00000i	0.0747958	−	0.149592i
$716$		0			0
$717$		−	12.0000i		−	0.448148i
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	12.0000			0.446903
$722$		0			0
$723$		−	18.0000i		−	0.669427i
$724$		0			0
$725$	−12.0000	−	16.0000i	−0.445669	−	0.594225i
$726$		0			0
$727$		−	34.0000i		−	1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$		−	34.0000i		−	1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$		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	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		0			0
$735$	−3.00000	+	6.00000i	−0.110657	+	0.221313i
$736$		0			0
$737$		−	32.0000i		−	1.17874i
$738$		0			0
$739$	28.0000			1.03000			0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000			1.03000			0.514998	+	0.857191i	$0.327793\pi$
$740$		0			0
$741$	4.00000			0.146944
$742$		0			0
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	−24.0000	−	12.0000i	−0.879292	−	0.439646i
$746$		0			0
$747$		−	12.0000i		−	0.439057i
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		−	18.0000i		−	0.655956i
$754$		0			0
$755$	32.0000	+	16.0000i	1.16460	+	0.582300i
$756$		0			0
$757$		−	26.0000i		−	0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$		−	26.0000i		−	0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.0000			1.37750			0.688749	−	0.724999i	$-0.258160\pi$
$761$	38.0000			1.37750			0.688749	+	0.724999i	$0.258160\pi$
$762$		0			0
$763$		−	36.0000i		−	1.30329i
$764$		0			0
$765$	2.00000	−	4.00000i	0.0723102	−	0.144620i
$766$		0			0
$767$			10.0000i			0.361079i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$		−	22.0000i		−	0.791285i	−0.918405	−	0.395643i	$-0.870522\pi$
$773$		−	22.0000i		−	0.791285i	0.918405	−	0.395643i	$-0.129478\pi$
$774$		0			0
$775$	−24.0000	−	32.0000i	−0.862105	−	1.14947i
$776$		0			0
$777$			12.0000i			0.430498i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$			4.00000i			0.142948i
$784$		0			0
$785$	−10.0000	+	20.0000i	−0.356915	+	0.713831i
$786$		0			0
$787$			8.00000i			0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$			8.00000i			0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$		0			0
$789$	8.00000			0.284808
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$		−	14.0000i		−	0.497155i
$794$		0			0
$795$	4.00000	+	2.00000i	0.141865	+	0.0709327i
$796$		0			0
$797$			14.0000i			0.495905i	0.968772	+	0.247953i	$0.0797578\pi$
$797$			14.0000i			0.495905i	−0.968772	+	0.247953i	$0.920242\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$			16.0000i			0.564628i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			20.0000i			0.704033i
$808$		0			0
$809$	22.0000			0.773479			0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000			0.773479			0.386739	+	0.922189i	$0.373601\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$	8.00000	−	16.0000i	0.280228	−	0.560456i
$816$		0			0
$817$		−	16.0000i		−	0.559769i
$818$		0			0
$819$	−2.00000			−0.0698857
$820$		0			0
$821$	24.0000			0.837606			0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000			0.837606			0.418803	+	0.908077i	$0.362450\pi$
$822$		0			0
$823$			34.0000i			1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$			34.0000i			1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$		0			0
$825$	8.00000	−	6.00000i	0.278524	−	0.208893i
$826$		0			0
$827$			12.0000i			0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$			12.0000i			0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$		0			0
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$		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
$841$	−13.0000			−0.448276
$842$		0			0
$843$		−	18.0000i		−	0.619953i
$844$		0			0
$845$	−2.00000	−	1.00000i	−0.0688021	−	0.0344010i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		0			0
$849$	−16.0000			−0.549119
$850$		0			0
$851$		0			0
$852$		0			0
$853$			18.0000i			0.616308i	0.951336	+	0.308154i	$0.0997113\pi$
$853$			18.0000i			0.616308i	−0.951336	+	0.308154i	$0.900289\pi$
$854$		0			0
$855$	8.00000	+	4.00000i	0.273594	+	0.136797i
$856$		0			0
$857$			34.0000i			1.16142i	0.814111	+	0.580709i	$0.197225\pi$
$857$			34.0000i			1.16142i	−0.814111	+	0.580709i	$0.802775\pi$
$858$		0			0
$859$	20.0000			0.682391			0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000			0.682391			0.341196	+	0.939992i	$0.389168\pi$
$860$		0			0
$861$	−12.0000			−0.408959
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$	14.0000	−	28.0000i	0.476014	−	0.952029i
$866$		0			0
$867$			13.0000i			0.441503i
$868$		0			0
$869$	16.0000			0.542763
$870$		0			0
$871$	−16.0000			−0.542139
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$	22.0000	−	4.00000i	0.743736	−	0.135225i
$876$		0			0
$877$			42.0000i			1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$			42.0000i			1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$		0			0
$879$	26.0000			0.876958
$880$		0			0
$881$	−10.0000			−0.336909			−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000			−0.336909			−0.168454	+	0.985709i	$0.553878\pi$
$882$		0			0
$883$			24.0000i			0.807664i	0.914833	+	0.403832i	$0.132322\pi$
$883$			24.0000i			0.807664i	−0.914833	+	0.403832i	$0.867678\pi$
$884$		0			0
$885$	−10.0000	+	20.0000i	−0.336146	+	0.672293i
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−12.0000			−0.402467
$890$		0			0
$891$	−2.00000			−0.0670025
$892$		0			0
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$	−36.0000	−	18.0000i	−1.20335	−	0.601674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	32.0000			1.06726
$900$		0			0
$901$	4.00000			0.133259
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$	−12.0000	−	6.00000i	−0.398893	−	0.199447i
$906$		0			0
$907$			28.0000i			0.929725i	0.885383	+	0.464862i	$0.153896\pi$
$907$			28.0000i			0.929725i	−0.885383	+	0.464862i	$0.846104\pi$
$908$		0			0
$909$	−4.00000			−0.132672
$910$		0			0
$911$	−16.0000			−0.530104			−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000			−0.530104			−0.265052	+	0.964234i	$0.585389\pi$
$912$		0			0
$913$		−	24.0000i		−	0.794284i
$914$		0			0
$915$	14.0000	−	28.0000i	0.462826	−	0.925651i
$916$		0			0
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	−48.0000			−1.58337			−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000			−1.58337			−0.791687	+	0.610927i	$0.790797\pi$
$920$		0			0
$921$	−28.0000			−0.922631
$922$		0			0
$923$			4.00000i			0.131662i
$924$		0			0
$925$	−24.0000	+	18.0000i	−0.789115	+	0.591836i
$926$		0			0
$927$		−	6.00000i		−	0.197066i
$928$		0			0
$929$	50.0000			1.64045			0.820223	−	0.572043i	$-0.193849\pi$
$929$	50.0000			1.64045			0.820223	+	0.572043i	$0.193849\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		0			0
$933$		−	20.0000i		−	0.654771i
$934$		0			0
$935$	4.00000	−	8.00000i	0.130814	−	0.261628i
$936$		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
$939$	−20.0000			−0.652675
$940$		0			0
$941$	−44.0000			−1.43436			−0.717180	−	0.696888i	$-0.754567\pi$
$941$	−44.0000			−1.43436			−0.717180	+	0.696888i	$0.754567\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−4.00000	−	2.00000i	−0.130120	−	0.0650600i
$946$		0			0
$947$			36.0000i			1.16984i	0.811090	+	0.584921i	$0.198875\pi$
$947$			36.0000i			1.16984i	−0.811090	+	0.584921i	$0.801125\pi$
$948$		0			0
$949$	8.00000			0.259691
$950$		0			0
$951$	18.0000			0.583690
$952$		0			0
$953$		−	6.00000i		−	0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$		−	6.00000i		−	0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$		0			0
$955$	24.0000	+	12.0000i	0.776622	+	0.388311i
$956$		0			0
$957$			8.00000i			0.258603i
$958$		0			0
$959$	−28.0000			−0.904167
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	4.00000i		−	0.128898i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	58.0000i		−	1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$		−	58.0000i		−	1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$		0			0
$969$	8.00000			0.256997
$970$		0			0
$971$	−50.0000			−1.60458			−0.802288	−	0.596937i	$-0.796384\pi$
$971$	−50.0000			−1.60458			−0.802288	+	0.596937i	$0.796384\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$	−3.00000	−	4.00000i	−0.0960769	−	0.128103i
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	12.0000			0.383522
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$			56.0000i			1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$			56.0000i			1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$		0			0
$985$	−10.0000	+	20.0000i	−0.318626	+	0.637253i
$986$		0			0
$987$			16.0000i			0.509286i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$			12.0000i			0.380808i
$994$		0			0
$995$	−48.0000	−	24.0000i	−1.52170	−	0.760851i
$996$		0			0
$997$		−	14.0000i		−	0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$		−	14.0000i		−	0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$		0			0
$999$	6.00000			0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.e.c.79.1	✓	2	20.19	odd	2
390.2.e.c.79.2	yes	2	4.3	odd	2
1170.2.e.a.469.1		2	12.11	even	2
1170.2.e.a.469.2		2	60.59	even	2
1950.2.a.c.1.1		1	20.7	even	4
1950.2.a.z.1.1		1	20.3	even	4
3120.2.l.g.1249.1		2	5.4	even	2	inner
3120.2.l.g.1249.2		2	1.1	even	1	trivial
5850.2.a.t.1.1		1	60.23	odd	4
5850.2.a.bj.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(2.00000 + 1.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} -2.00000 q^{11} +1.00000i q^{13} +(-1.00000 + 2.00000i) q^{15} +2.00000i q^{17} -4.00000 q^{19} +2.00000 q^{21} +(3.00000 + 4.00000i) q^{25} -1.00000i q^{27} -4.00000 q^{29} -8.00000 q^{31} -2.00000i q^{33} +(2.00000 - 4.00000i) q^{35} +6.00000i q^{37} -1.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-2.00000 - 1.00000i) q^{45} +8.00000i q^{47} +3.00000 q^{49} -2.00000 q^{51} -2.00000i q^{53} +(-4.00000 - 2.00000i) q^{55} -4.00000i q^{57} +10.0000 q^{59} -14.0000 q^{61} +2.00000i q^{63} +(-1.00000 + 2.00000i) q^{65} +16.0000i q^{67} +4.00000 q^{71} -8.00000i q^{73} +(-4.00000 + 3.00000i) q^{75} +4.00000i q^{77} -8.00000 q^{79} +1.00000 q^{81} +12.0000i q^{83} +(-2.00000 + 4.00000i) q^{85} -4.00000i q^{87} -6.00000 q^{89} +2.00000 q^{91} -8.00000i q^{93} +(-8.00000 - 4.00000i) q^{95} +12.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^5 - 2 * q^9	\( 2 q + 4 q^{5} - 2 q^{9} - 4 q^{11} - 2 q^{15} - 8 q^{19} + 4 q^{21} + 6 q^{25} - 8 q^{29} - 16 q^{31} + 4 q^{35} - 2 q^{39} - 12 q^{41} - 4 q^{45} + 6 q^{49} - 4 q^{51} - 8 q^{55} + 20 q^{59} - 28 q^{61} - 2 q^{65} + 8 q^{71} - 8 q^{75} - 16 q^{79} + 2 q^{81} - 4 q^{85} - 12 q^{89} + 4 q^{91} - 16 q^{95} + 4 q^{99}+O(q^{100}) \) 2 * q + 4 * q^5 - 2 * q^9 - 4 * q^11 - 2 * q^15 - 8 * q^19 + 4 * q^21 + 6 * q^25 - 8 * q^29 - 16 * q^31 + 4 * q^35 - 2 * q^39 - 12 * q^41 - 4 * q^45 + 6 * q^49 - 4 * q^51 - 8 * q^55 + 20 * q^59 - 28 * q^61 - 2 * q^65 + 8 * q^71 - 8 * q^75 - 16 * q^79 + 2 * q^81 - 4 * q^85 - 12 * q^89 + 4 * q^91 - 16 * q^95 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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