LMFDB - Embedded newform 880.2.f.d.351.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(351,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,-8,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$7.02683537787$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.170772624.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.6
Root			$1.41203 - 0.0786378i$ of defining polynomial
Character	$\chi$	$=$	880.351
Dual form			880.2.f.d.351.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287i q^{3} -1.00000 q^{5} +4.70285 q^{7} +2.37228 q^{9} +(-2.15121 - 2.52434i) q^{11} +1.01082i q^{13} -0.792287i q^{15} -2.71519i q^{17} +2.15121 q^{19} +3.72601i q^{21} -5.04868i q^{23} +1.00000 q^{25} +4.25639i q^{27} +9.15640i q^{29} -9.30506i q^{31} +(2.00000 - 1.70438i) q^{33} -4.70285 q^{35} +5.37228 q^{37} -0.800857 q^{39} +10.8608i q^{41} +2.55164 q^{43} -2.37228 q^{45} -1.87953i q^{47} +15.1168 q^{49} +2.15121 q^{51} +4.11684 q^{53} +(2.15121 + 2.52434i) q^{55} +1.70438i q^{57} +1.87953i q^{59} -7.13477i q^{61} +11.1565 q^{63} -1.01082i q^{65} +3.46410i q^{67} +4.00000 q^{69} +6.13592i q^{71} +11.8716i q^{73} +0.792287i q^{75} +(-10.1168 - 11.8716i) q^{77} +8.60485 q^{79} +3.74456 q^{81} -1.75079 q^{83} +2.71519i q^{85} -7.25450 q^{87} -4.11684 q^{89} +4.75372i q^{91} +7.37228 q^{93} -2.15121 q^{95} -14.0000 q^{97} +(-5.10328 - 5.98844i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5} - 4 q^{9} + 8 q^{25} + 16 q^{33} + 20 q^{37} + 4 q^{45} + 52 q^{49} - 36 q^{53} + 32 q^{69} - 12 q^{77} - 16 q^{81} + 36 q^{89} + 36 q^{93} - 112 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 4 * q^9 + 8 * q^25 + 16 * q^33 + 20 * q^37 + 4 * q^45 + 52 * q^49 - 36 * q^53 + 32 * q^69 - 12 * q^77 - 16 * q^81 + 36 * q^89 + 36 * q^93 - 112 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$			0.792287i			0.457427i	0.973494	+	0.228714i	$0.0734519\pi$
$3$			0.792287i			0.457427i	−0.973494	+	0.228714i	$0.926548\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	4.70285			1.77751			0.888756	−	0.458381i	$-0.151570\pi$
$7$	4.70285			1.77751			0.888756	+	0.458381i	$0.151570\pi$
$8$		0			0
$9$	2.37228			0.790760
$10$		0			0
$11$	−2.15121	−	2.52434i	−0.648615	−	0.761116i
$11$	−2.15121	−	2.52434i	−0.648615	−	0.761116i
$12$		0			0
$13$			1.01082i			0.280350i	0.990127	+	0.140175i	$0.0447665\pi$
$13$			1.01082i			0.280350i	−0.990127	+	0.140175i	$0.955233\pi$
$14$		0			0
$15$		−	0.792287i		−	0.204568i
$16$		0			0
$17$		−	2.71519i		−	0.658531i	−0.944237	−	0.329266i	$-0.893199\pi$
$17$		−	2.71519i		−	0.658531i	0.944237	−	0.329266i	$-0.106801\pi$
$18$		0			0
$19$	2.15121			0.493522			0.246761	−	0.969076i	$-0.420634\pi$
$19$	2.15121			0.493522			0.246761	+	0.969076i	$0.420634\pi$
$20$		0			0
$21$			3.72601i			0.813082i
$22$		0			0
$23$		−	5.04868i		−	1.05272i	−0.850261	−	0.526361i	$-0.823556\pi$
$23$		−	5.04868i		−	1.05272i	0.850261	−	0.526361i	$-0.176444\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			4.25639i			0.819142i
$28$		0			0
$29$			9.15640i			1.70030i	0.526540	+	0.850150i	$0.323489\pi$
$29$			9.15640i			1.70030i	−0.526540	+	0.850150i	$0.676511\pi$
$30$		0			0
$31$		−	9.30506i		−	1.67124i	−0.549309	−	0.835619i	$-0.685109\pi$
$31$		−	9.30506i		−	1.67124i	0.549309	−	0.835619i	$-0.314891\pi$
$32$		0			0
$33$	2.00000	−	1.70438i	0.348155	−	0.296694i
$34$		0			0
$35$	−4.70285			−0.794928
$36$		0			0
$37$	5.37228			0.883198			0.441599	−	0.897213i	$-0.354411\pi$
$37$	5.37228			0.883198			0.441599	+	0.897213i	$0.354411\pi$
$38$		0			0
$39$	−0.800857			−0.128240
$40$		0			0
$41$			10.8608i			1.69617i	0.529861	+	0.848084i	$0.322244\pi$
$41$			10.8608i			1.69617i	−0.529861	+	0.848084i	$0.677756\pi$
$42$		0			0
$43$	2.55164			0.389122			0.194561	−	0.980890i	$-0.437672\pi$
$43$	2.55164			0.389122			0.194561	+	0.980890i	$0.437672\pi$
$44$		0			0
$45$	−2.37228			−0.353639
$46$		0			0
$47$		−	1.87953i		−	0.274157i	−0.990560	−	0.137079i	$-0.956229\pi$
$47$		−	1.87953i		−	0.274157i	0.990560	−	0.137079i	$-0.0437713\pi$
$48$		0			0
$49$	15.1168			2.15955
$50$		0			0
$51$	2.15121			0.301230
$52$		0			0
$53$	4.11684			0.565492			0.282746	−	0.959195i	$-0.408755\pi$
$53$	4.11684			0.565492			0.282746	+	0.959195i	$0.408755\pi$
$54$		0			0
$55$	2.15121	+	2.52434i	0.290070	+	0.340382i
$56$		0			0
$57$			1.70438i			0.225750i
$58$		0			0
$59$			1.87953i			0.244694i	0.992487	+	0.122347i	$0.0390420\pi$
$59$			1.87953i			0.244694i	−0.992487	+	0.122347i	$0.960958\pi$
$60$		0			0
$61$		−	7.13477i		−	0.913513i	−0.889592	−	0.456757i	$-0.849011\pi$
$61$		−	7.13477i		−	0.913513i	0.889592	−	0.456757i	$-0.150989\pi$
$62$		0			0
$63$	11.1565			1.40559
$64$		0			0
$65$		−	1.01082i		−	0.125376i
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$			6.13592i			0.728199i	0.931360	+	0.364100i	$0.118623\pi$
$71$			6.13592i			0.728199i	−0.931360	+	0.364100i	$0.881377\pi$
$72$		0			0
$73$			11.8716i			1.38947i	0.719268	+	0.694733i	$0.244477\pi$
$73$			11.8716i			1.38947i	−0.719268	+	0.694733i	$0.755523\pi$
$74$		0			0
$75$			0.792287i			0.0914854i
$76$		0			0
$77$	−10.1168	−	11.8716i	−1.15292	−	1.35289i
$78$		0			0
$79$	8.60485			0.968122			0.484061	−	0.875034i	$-0.339161\pi$
$79$	8.60485			0.968122			0.484061	+	0.875034i	$0.339161\pi$
$80$		0			0
$81$	3.74456			0.416063
$82$		0			0
$83$	−1.75079			−0.192174			−0.0960868	−	0.995373i	$-0.530633\pi$
$83$	−1.75079			−0.192174			−0.0960868	+	0.995373i	$0.530633\pi$
$84$		0			0
$85$			2.71519i			0.294504i
$86$		0			0
$87$	−7.25450			−0.777764
$88$		0			0
$89$	−4.11684			−0.436385			−0.218192	−	0.975906i	$-0.570016\pi$
$89$	−4.11684			−0.436385			−0.218192	+	0.975906i	$0.570016\pi$
$90$		0			0
$91$			4.75372i			0.498325i
$92$		0			0
$93$	7.37228			0.764470
$94$		0			0
$95$	−2.15121			−0.220710
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	−5.10328	−	5.98844i	−0.512899	−	0.601861i
$100$		0			0
$101$		−	16.2912i		−	1.62103i	−0.585717	−	0.810516i	$-0.699187\pi$
$101$		−	16.2912i		−	1.62103i	0.585717	−	0.810516i	$-0.300813\pi$
$102$		0			0
$103$		−	8.21782i		−	0.809726i	−0.914377	−	0.404863i	$-0.867319\pi$
$103$		−	8.21782i		−	0.809726i	0.914377	−	0.404863i	$-0.132681\pi$
$104$		0			0
$105$		−	3.72601i		−	0.363621i
$106$		0			0
$107$	−15.4589			−1.49447			−0.747235	−	0.664560i	$-0.768619\pi$
$107$	−15.4589			−1.49447			−0.747235	+	0.664560i	$0.768619\pi$
$108$		0			0
$109$			7.45202i			0.713774i	0.934148	+	0.356887i	$0.116162\pi$
$109$			7.45202i			0.713774i	−0.934148	+	0.356887i	$0.883838\pi$
$110$		0			0
$111$			4.25639i			0.403999i
$112$		0			0
$113$	−11.4891			−1.08081			−0.540403	−	0.841406i	$-0.681728\pi$
$113$	−11.4891			−1.08081			−0.540403	+	0.841406i	$0.681728\pi$
$114$		0			0
$115$			5.04868i			0.470791i
$116$		0			0
$117$			2.39794i			0.221690i
$118$		0			0
$119$		−	12.7692i		−	1.17055i
$120$		0			0
$121$	−1.74456	+	10.8608i	−0.158597	+	0.987343i
$122$		0			0
$123$	−8.60485			−0.775873
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−15.4589			−1.37176			−0.685879	−	0.727716i	$-0.740582\pi$
$127$	−15.4589			−1.37176			−0.685879	+	0.727716i	$0.740582\pi$
$128$		0			0
$129$			2.02163i			0.177995i
$130$		0			0
$131$	11.5569			1.00973			0.504867	−	0.863197i	$-0.331542\pi$
$131$	11.5569			1.00973			0.504867	+	0.863197i	$0.331542\pi$
$132$		0			0
$133$	10.1168			0.877242
$134$		0			0
$135$		−	4.25639i		−	0.366332i
$136$		0			0
$137$	−11.4891			−0.981582			−0.490791	−	0.871277i	$-0.663292\pi$
$137$	−11.4891			−0.981582			−0.490791	+	0.871277i	$0.663292\pi$
$138$		0			0
$139$	0.800857			0.0679278			0.0339639	−	0.999423i	$-0.489187\pi$
$139$	0.800857			0.0679278			0.0339639	+	0.999423i	$0.489187\pi$
$140$		0			0
$141$	1.48913			0.125407
$142$		0			0
$143$	2.55164	−	2.17448i	0.213379	−	0.181839i
$144$		0			0
$145$		−	9.15640i		−	0.760398i
$146$		0			0
$147$			11.9769i			0.987836i
$148$		0			0
$149$			1.70438i			0.139628i	0.997560	+	0.0698141i	$0.0222406\pi$
$149$			1.70438i			0.139628i	−0.997560	+	0.0698141i	$0.977759\pi$
$150$		0			0
$151$	9.40571			0.765426			0.382713	−	0.923867i	$-0.374990\pi$
$151$	9.40571			0.765426			0.382713	+	0.923867i	$0.374990\pi$
$152$		0			0
$153$		−	6.44121i		−	0.520741i
$154$		0			0
$155$			9.30506i			0.747401i
$156$		0			0
$157$	−22.8614			−1.82454			−0.912269	−	0.409591i	$-0.865672\pi$
$157$	−22.8614			−1.82454			−0.912269	+	0.409591i	$0.865672\pi$
$158$		0			0
$159$			3.26172i			0.258671i
$160$		0			0
$161$		−	23.7432i		−	1.87123i
$162$		0			0
$163$		−	16.2333i		−	1.27149i	−0.771900	−	0.635744i	$-0.780693\pi$
$163$		−	16.2333i		−	1.27149i	0.771900	−	0.635744i	$-0.219307\pi$
$164$		0			0
$165$	−2.00000	+	1.70438i	−0.155700	+	0.132686i
$166$		0			0
$167$	14.9094			1.15373			0.576863	−	0.816841i	$-0.304277\pi$
$167$	14.9094			1.15373			0.576863	+	0.816841i	$0.304277\pi$
$168$		0			0
$169$	11.9783			0.921404
$170$		0			0
$171$	5.10328			0.390258
$172$		0			0
$173$		−	6.44121i		−	0.489716i	−0.969559	−	0.244858i	$-0.921259\pi$
$173$		−	6.44121i		−	0.489716i	0.969559	−	0.244858i	$-0.0787413\pi$
$174$		0			0
$175$	4.70285			0.355502
$176$		0			0
$177$	−1.48913			−0.111930
$178$		0			0
$179$		−	1.87953i		−	0.140482i	−0.997530	−	0.0702412i	$-0.977623\pi$
$179$		−	1.87953i		−	0.140482i	0.997530	−	0.0702412i	$-0.0223769\pi$
$180$		0			0
$181$	−3.48913			−0.259345			−0.129672	−	0.991557i	$-0.541393\pi$
$181$	−3.48913			−0.259345			−0.129672	+	0.991557i	$0.541393\pi$
$182$		0			0
$183$	5.65278			0.417866
$184$		0			0
$185$	−5.37228			−0.394978
$186$		0			0
$187$	−6.85407	+	5.84096i	−0.501219	+	0.427133i
$188$		0			0
$189$			20.0172i			1.45604i
$190$		0			0
$191$			16.7306i			1.21058i	0.796004	+	0.605292i	$0.206944\pi$
$191$			16.7306i			1.21058i	−0.796004	+	0.605292i	$0.793056\pi$
$192$		0			0
$193$			6.75846i			0.486485i	0.969966	+	0.243242i	$0.0782110\pi$
$193$			6.75846i			0.486485i	−0.969966	+	0.243242i	$0.921789\pi$
$194$		0			0
$195$	0.800857			0.0573505
$196$		0			0
$197$		−	26.7756i		−	1.90769i	−0.300307	−	0.953843i	$-0.597089\pi$
$197$		−	26.7756i		−	1.90769i	0.300307	−	0.953843i	$-0.402911\pi$
$198$		0			0
$199$			7.13058i			0.505474i	0.967535	+	0.252737i	$0.0813307\pi$
$199$			7.13058i			0.505474i	−0.967535	+	0.252737i	$0.918669\pi$
$200$		0			0
$201$	−2.74456			−0.193587
$202$		0			0
$203$			43.0612i			3.02231i
$204$		0			0
$205$		−	10.8608i		−	0.758550i
$206$		0			0
$207$		−	11.9769i		−	0.832451i
$208$		0			0
$209$	−4.62772	−	5.43039i	−0.320106	−	0.375628i
$210$		0			0
$211$	−26.0659			−1.79445			−0.897226	−	0.441572i	$-0.854421\pi$
$211$	−26.0659			−1.79445			−0.897226	+	0.441572i	$0.854421\pi$
$212$		0			0
$213$	−4.86141			−0.333098
$214$		0			0
$215$	−2.55164			−0.174021
$216$		0			0
$217$		−	43.7604i		−	2.97065i
$218$		0			0
$219$	−9.40571			−0.635579
$220$		0			0
$221$	2.74456			0.184619
$222$		0			0
$223$		−	19.8997i		−	1.33259i	−0.745690	−	0.666293i	$-0.767880\pi$
$223$		−	19.8997i		−	1.33259i	0.745690	−	0.666293i	$-0.232120\pi$
$224$		0			0
$225$	2.37228			0.158152
$226$		0			0
$227$	−6.85407			−0.454920			−0.227460	−	0.973787i	$-0.573042\pi$
$227$	−6.85407			−0.454920			−0.227460	+	0.973787i	$0.573042\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$	9.40571	−	8.01544i	0.618850	−	0.527377i
$232$		0			0
$233$			6.12395i			0.401193i	0.979674	+	0.200597i	$0.0642881\pi$
$233$			6.12395i			0.401193i	−0.979674	+	0.200597i	$0.935712\pi$
$234$		0			0
$235$			1.87953i			0.122607i
$236$		0			0
$237$			6.81751i			0.442845i
$238$		0			0
$239$	−8.60485			−0.556602			−0.278301	−	0.960494i	$-0.589771\pi$
$239$	−8.60485			−0.556602			−0.278301	+	0.960494i	$0.589771\pi$
$240$		0			0
$241$			14.2695i			0.919182i	0.888131	+	0.459591i	$0.152004\pi$
$241$			14.2695i			0.919182i	−0.888131	+	0.459591i	$0.847996\pi$
$242$		0			0
$243$			15.7359i			1.00946i
$244$		0			0
$245$	−15.1168			−0.965780
$246$		0			0
$247$			2.17448i			0.138359i
$248$		0			0
$249$		−	1.38712i		−	0.0879054i
$250$		0			0
$251$		−	6.63325i		−	0.418687i	−0.977842	−	0.209344i	$-0.932867\pi$
$251$		−	6.63325i		−	0.418687i	0.977842	−	0.209344i	$-0.0671327\pi$
$252$		0			0
$253$	−12.7446	+	10.8608i	−0.801244	+	0.682811i
$254$		0			0
$255$	−2.15121			−0.134714
$256$		0			0
$257$	−20.7446			−1.29401			−0.647005	−	0.762486i	$-0.723979\pi$
$257$	−20.7446			−1.29401			−0.647005	+	0.762486i	$0.723979\pi$
$258$		0			0
$259$	25.2651			1.56989
$260$		0			0
$261$			21.7216i			1.34453i
$262$		0			0
$263$	−14.9094			−0.919354			−0.459677	−	0.888086i	$-0.652035\pi$
$263$	−14.9094			−0.919354			−0.459677	+	0.888086i	$0.652035\pi$
$264$		0			0
$265$	−4.11684			−0.252896
$266$		0			0
$267$		−	3.26172i		−	0.199614i
$268$		0			0
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	−28.2171			−1.71407			−0.857034	−	0.515259i	$-0.827696\pi$
$271$	−28.2171			−1.71407			−0.857034	+	0.515259i	$0.827696\pi$
$272$		0			0
$273$	−3.76631			−0.227948
$274$		0			0
$275$	−2.15121	−	2.52434i	−0.129723	−	0.152223i
$276$		0			0
$277$			20.7107i			1.24439i	0.782863	+	0.622194i	$0.213758\pi$
$277$			20.7107i			1.24439i	−0.782863	+	0.622194i	$0.786242\pi$
$278$		0			0
$279$		−	22.0742i		−	1.32155i
$280$		0			0
$281$			6.81751i			0.406699i	0.979106	+	0.203349i	$0.0651827\pi$
$281$			6.81751i			0.406699i	−0.979106	+	0.203349i	$0.934817\pi$
$282$		0			0
$283$	−30.7688			−1.82901			−0.914507	−	0.404571i	$-0.867421\pi$
$283$	−30.7688			−1.82901			−0.914507	+	0.404571i	$0.867421\pi$
$284$		0			0
$285$		−	1.70438i		−	0.100959i
$286$		0			0
$287$			51.0767i			3.01496i
$288$		0			0
$289$	9.62772			0.566336
$290$		0			0
$291$		−	11.0920i		−	0.650226i
$292$		0			0
$293$		−	24.7540i		−	1.44614i	−0.690772	−	0.723072i	$-0.742729\pi$
$293$		−	24.7540i		−	1.44614i	0.690772	−	0.723072i	$-0.257271\pi$
$294$		0			0
$295$		−	1.87953i		−	0.109430i
$296$		0			0
$297$	10.7446	−	9.15640i	0.623463	−	0.531308i
$298$		0			0
$299$	5.10328			0.295130
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$	12.9073			0.741504
$304$		0			0
$305$			7.13477i			0.408536i
$306$		0			0
$307$	3.35250			0.191337			0.0956686	−	0.995413i	$-0.469501\pi$
$307$	3.35250			0.191337			0.0956686	+	0.995413i	$0.469501\pi$
$308$		0			0
$309$	6.51087			0.370391
$310$		0			0
$311$		−	10.8896i		−	0.617495i	−0.951144	−	0.308747i	$-0.900090\pi$
$311$		−	10.8896i		−	0.617495i	0.951144	−	0.308747i	$-0.0999098\pi$
$312$		0			0
$313$	24.7446			1.39865			0.699323	−	0.714806i	$-0.253485\pi$
$313$	24.7446			1.39865			0.699323	+	0.714806i	$0.253485\pi$
$314$		0			0
$315$	−11.1565			−0.628597
$316$		0			0
$317$	−1.37228			−0.0770750			−0.0385375	−	0.999257i	$-0.512270\pi$
$317$	−1.37228			−0.0770750			−0.0385375	+	0.999257i	$0.512270\pi$
$318$		0			0
$319$	23.1138	−	19.6974i	1.29413	−	1.10284i
$320$		0			0
$321$		−	12.2479i		−	0.683611i
$322$		0			0
$323$		−	5.84096i		−	0.325000i
$324$		0			0
$325$			1.01082i			0.0560700i
$326$		0			0
$327$	−5.90414			−0.326500
$328$		0			0
$329$		−	8.83915i		−	0.487318i
$330$		0			0
$331$			5.63858i			0.309925i	0.987920	+	0.154962i	$0.0495256\pi$
$331$			5.63858i			0.309925i	−0.987920	+	0.154962i	$0.950474\pi$
$332$		0			0
$333$	12.7446			0.698398
$334$		0			0
$335$		−	3.46410i		−	0.189264i
$336$		0			0
$337$			8.14558i			0.443718i	0.975079	+	0.221859i	$0.0712125\pi$
$337$			8.14558i			0.443718i	−0.975079	+	0.221859i	$0.928787\pi$
$338$		0			0
$339$		−	9.10268i		−	0.494390i
$340$		0			0
$341$	−23.4891	+	20.0172i	−1.27201	+	1.08399i
$342$		0			0
$343$	38.1723			2.06111
$344$		0			0
$345$	−4.00000			−0.215353
$346$		0			0
$347$	20.5622			1.10384			0.551918	−	0.833898i	$-0.313896\pi$
$347$	20.5622			1.10384			0.551918	+	0.833898i	$0.313896\pi$
$348$		0			0
$349$		−	25.1303i		−	1.34519i	−0.740009	−	0.672597i	$-0.765179\pi$
$349$		−	25.1303i		−	1.34519i	0.740009	−	0.672597i	$-0.234821\pi$
$350$		0			0
$351$	−4.30243			−0.229647
$352$		0			0
$353$	14.2337			0.757583			0.378791	−	0.925482i	$-0.376340\pi$
$353$	14.2337			0.757583			0.378791	+	0.925482i	$0.376340\pi$
$354$		0			0
$355$		−	6.13592i		−	0.325661i
$356$		0			0
$357$	10.1168			0.535440
$358$		0			0
$359$	14.5090			0.765755			0.382878	−	0.923799i	$-0.374933\pi$
$359$	14.5090			0.765755			0.382878	+	0.923799i	$0.374933\pi$
$360$		0			0
$361$	−14.3723			−0.756436
$362$		0			0
$363$	−8.60485	−	1.38219i	−0.451638	−	0.0725464i
$364$		0			0
$365$		−	11.8716i		−	0.621388i
$366$		0			0
$367$			26.8280i			1.40041i	0.713943	+	0.700204i	$0.246908\pi$
$367$			26.8280i			1.40041i	−0.713943	+	0.700204i	$0.753092\pi$
$368$		0			0
$369$			25.7648i			1.34126i
$370$		0			0
$371$	19.3609			1.00517
$372$		0			0
$373$			26.7756i			1.38639i	0.720750	+	0.693195i	$0.243798\pi$
$373$			26.7756i			1.38639i	−0.720750	+	0.693195i	$0.756202\pi$
$374$		0			0
$375$		−	0.792287i		−	0.0409135i
$376$		0			0
$377$	−9.25544			−0.476679
$378$		0			0
$379$			35.9306i			1.84563i	0.385240	+	0.922816i	$0.374119\pi$
$379$			35.9306i			1.84563i	−0.385240	+	0.922816i	$0.625881\pi$
$380$		0			0
$381$		−	12.2479i		−	0.627479i
$382$		0			0
$383$		−	5.04868i		−	0.257975i	−0.991646	−	0.128988i	$-0.958827\pi$
$383$		−	5.04868i		−	0.257975i	0.991646	−	0.128988i	$-0.0411728\pi$
$384$		0			0
$385$	10.1168	+	11.8716i	0.515602	+	0.605032i
$386$		0			0
$387$	6.05321			0.307702
$388$		0			0
$389$	31.7228			1.60841			0.804205	−	0.594352i	$-0.202591\pi$
$389$	31.7228			1.60841			0.804205	+	0.594352i	$0.202591\pi$
$390$		0			0
$391$	−13.7081			−0.693250
$392$		0			0
$393$			9.15640i			0.461879i
$394$		0			0
$395$	−8.60485			−0.432957
$396$		0			0
$397$	15.4891			0.777377			0.388688	−	0.921369i	$-0.372928\pi$
$397$	15.4891			0.777377			0.388688	+	0.921369i	$0.372928\pi$
$398$		0			0
$399$			8.01544i			0.401274i
$400$		0			0
$401$	25.3723			1.26703			0.633516	−	0.773730i	$-0.281611\pi$
$401$	25.3723			1.26703			0.633516	+	0.773730i	$0.281611\pi$
$402$		0			0
$403$	9.40571			0.468532
$404$		0			0
$405$	−3.74456			−0.186069
$406$		0			0
$407$	−11.5569	−	13.5615i	−0.572855	−	0.672216i
$408$		0			0
$409$			12.8824i			0.636994i	0.947924	+	0.318497i	$0.103178\pi$
$409$			12.8824i			0.636994i	−0.947924	+	0.318497i	$0.896822\pi$
$410$		0			0
$411$		−	9.10268i		−	0.449002i
$412$		0			0
$413$			8.83915i			0.434946i
$414$		0			0
$415$	1.75079			0.0859427
$416$		0			0
$417$			0.634508i			0.0310720i
$418$		0			0
$419$		−	13.5615i		−	0.662520i	−0.943539	−	0.331260i	$-0.892526\pi$
$419$		−	13.5615i		−	0.662520i	0.943539	−	0.331260i	$-0.107474\pi$
$420$		0			0
$421$	−1.76631			−0.0860848			−0.0430424	−	0.999073i	$-0.513705\pi$
$421$	−1.76631			−0.0860848			−0.0430424	+	0.999073i	$0.513705\pi$
$422$		0			0
$423$		−	4.45877i		−	0.216793i
$424$		0			0
$425$		−	2.71519i		−	0.131706i
$426$		0			0
$427$		−	33.5538i		−	1.62378i
$428$		0			0
$429$	1.72281	+	2.02163i	0.0831782	+	0.0976053i
$430$		0			0
$431$	−18.0106			−0.867538			−0.433769	−	0.901024i	$-0.642817\pi$
$431$	−18.0106			−0.867538			−0.433769	+	0.901024i	$0.642817\pi$
$432$		0			0
$433$	−10.2337			−0.491800			−0.245900	−	0.969295i	$-0.579083\pi$
$433$	−10.2337			−0.491800			−0.245900	+	0.969295i	$0.579083\pi$
$434$		0			0
$435$	7.25450			0.347826
$436$		0			0
$437$		−	10.8608i		−	0.519541i
$438$		0			0
$439$	18.8114			0.897820			0.448910	−	0.893577i	$-0.351812\pi$
$439$	18.8114			0.897820			0.448910	+	0.893577i	$0.351812\pi$
$440$		0			0
$441$	35.8614			1.70769
$442$		0			0
$443$			6.63325i			0.315155i	0.987507	+	0.157578i	$0.0503684\pi$
$443$			6.63325i			0.315155i	−0.987507	+	0.157578i	$0.949632\pi$
$444$		0			0
$445$	4.11684			0.195157
$446$		0			0
$447$	−1.35036			−0.0638697
$448$		0			0
$449$	−34.4674			−1.62662			−0.813308	−	0.581833i	$-0.802336\pi$
$449$	−34.4674			−1.62662			−0.813308	+	0.581833i	$0.802336\pi$
$450$		0			0
$451$	27.4163	−	23.3639i	1.29098	−	1.10016i
$452$		0			0
$453$			7.45202i			0.350127i
$454$		0			0
$455$		−	4.75372i		−	0.222858i
$456$		0			0
$457$		−	6.12395i		−	0.286466i	−0.989689	−	0.143233i	$-0.954250\pi$
$457$		−	6.12395i		−	0.286466i	0.989689	−	0.143233i	$-0.0457499\pi$
$458$		0			0
$459$	11.5569			0.539431
$460$		0			0
$461$			3.72601i			0.173538i	0.996228	+	0.0867688i	$0.0276541\pi$
$461$			3.72601i			0.173538i	−0.996228	+	0.0867688i	$0.972346\pi$
$462$		0			0
$463$		−	19.8997i		−	0.924820i	−0.886666	−	0.462410i	$-0.846985\pi$
$463$		−	19.8997i		−	0.924820i	0.886666	−	0.462410i	$-0.153015\pi$
$464$		0			0
$465$	−7.37228			−0.341881
$466$		0			0
$467$		−	1.38219i		−	0.0639603i	−0.999489	−	0.0319802i	$-0.989819\pi$
$467$		−	1.38219i		−	0.0639603i	0.999489	−	0.0319802i	$-0.0101813\pi$
$468$		0			0
$469$			16.2912i			0.752256i
$470$		0			0
$471$		−	18.1128i		−	0.834594i
$472$		0			0
$473$	−5.48913	−	6.44121i	−0.252390	−	0.296167i
$474$		0			0
$475$	2.15121			0.0987044
$476$		0			0
$477$	9.76631			0.447169
$478$		0			0
$479$	−31.7187			−1.44926			−0.724632	−	0.689136i	$-0.757990\pi$
$479$	−31.7187			−1.44926			−0.724632	+	0.689136i	$0.757990\pi$
$480$		0			0
$481$			5.43039i			0.247604i
$482$		0			0
$483$	18.8114			0.855949
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$		−	35.9306i		−	1.62817i	−0.580744	−	0.814086i	$-0.697238\pi$
$487$		−	35.9306i		−	1.62817i	0.580744	−	0.814086i	$-0.302762\pi$
$488$		0			0
$489$	12.8614			0.581613
$490$		0			0
$491$	−2.95207			−0.133225			−0.0666125	−	0.997779i	$-0.521219\pi$
$491$	−2.95207			−0.133225			−0.0666125	+	0.997779i	$0.521219\pi$
$492$		0			0
$493$	24.8614			1.11970
$494$		0			0
$495$	5.10328	+	5.98844i	0.229376	+	0.269160i
$496$		0			0
$497$			28.8563i			1.29438i
$498$		0			0
$499$		−	24.6535i		−	1.10364i	−0.833963	−	0.551820i	$-0.813933\pi$
$499$		−	24.6535i		−	1.10364i	0.833963	−	0.551820i	$-0.186067\pi$
$500$		0			0
$501$			11.8125i			0.527745i
$502$		0			0
$503$	17.0606			0.760696			0.380348	−	0.924843i	$-0.375804\pi$
$503$	17.0606			0.760696			0.380348	+	0.924843i	$0.375804\pi$
$504$		0			0
$505$			16.2912i			0.724947i
$506$		0			0
$507$			9.49021i			0.421475i
$508$		0			0
$509$	−3.25544			−0.144295			−0.0721474	−	0.997394i	$-0.522985\pi$
$509$	−3.25544			−0.144295			−0.0721474	+	0.997394i	$0.522985\pi$
$510$		0			0
$511$			55.8304i			2.46979i
$512$		0			0
$513$			9.15640i			0.404265i
$514$		0			0
$515$			8.21782i			0.362121i
$516$		0			0
$517$	−4.74456	+	4.04326i	−0.208666	+	0.177823i
$518$		0			0
$519$	5.10328			0.224009
$520$		0			0
$521$	−23.4891			−1.02908			−0.514539	−	0.857467i	$-0.672037\pi$
$521$	−23.4891			−1.02908			−0.514539	+	0.857467i	$0.672037\pi$
$522$		0			0
$523$	−24.8646			−1.08725			−0.543627	−	0.839327i	$-0.682949\pi$
$523$	−24.8646			−1.08725			−0.543627	+	0.839327i	$0.682949\pi$
$524$		0			0
$525$			3.72601i			0.162616i
$526$		0			0
$527$	−25.2651			−1.10056
$528$		0			0
$529$	−2.48913			−0.108223
$530$		0			0
$531$			4.45877i			0.193494i
$532$		0			0
$533$	−10.9783			−0.475521
$534$		0			0
$535$	15.4589			0.668347
$536$		0			0
$537$	1.48913			0.0642605
$538$		0			0
$539$	−32.5196	−	38.1600i	−1.40072	−	1.64367i
$540$		0			0
$541$		−	36.9429i		−	1.58830i	−0.607724	−	0.794149i	$-0.707917\pi$
$541$		−	36.9429i		−	1.58830i	0.607724	−	0.794149i	$-0.292083\pi$
$542$		0			0
$543$		−	2.76439i		−	0.118631i
$544$		0			0
$545$		−	7.45202i		−	0.319210i
$546$		0			0
$547$	10.3556			0.442775			0.221388	−	0.975186i	$-0.428941\pi$
$547$	10.3556			0.442775			0.221388	+	0.975186i	$0.428941\pi$
$548$		0			0
$549$		−	16.9257i		−	0.722370i
$550$		0			0
$551$			19.6974i			0.839136i
$552$		0			0
$553$	40.4674			1.72085
$554$		0			0
$555$		−	4.25639i		−	0.180674i
$556$		0			0
$557$			21.3452i			0.904427i	0.891910	+	0.452214i	$0.149366\pi$
$557$			21.3452i			0.904427i	−0.891910	+	0.452214i	$0.850634\pi$
$558$		0			0
$559$			2.57924i			0.109090i
$560$		0			0
$561$	−4.62772	−	5.43039i	−0.195382	−	0.229271i
$562$		0			0
$563$	−16.2598			−0.685268			−0.342634	−	0.939469i	$-0.611319\pi$
$563$	−16.2598			−0.685268			−0.342634	+	0.939469i	$0.611319\pi$
$564$		0			0
$565$	11.4891			0.483351
$566$		0			0
$567$	17.6101			0.739556
$568$		0			0
$569$			27.1519i			1.13827i	0.822245	+	0.569134i	$0.192722\pi$
$569$			27.1519i			1.13827i	−0.822245	+	0.569134i	$0.807278\pi$
$570$		0			0
$571$	24.4642			1.02380			0.511898	−	0.859047i	$-0.328943\pi$
$571$	24.4642			1.02380			0.511898	+	0.859047i	$0.328943\pi$
$572$		0			0
$573$	−13.2554			−0.553754
$574$		0			0
$575$		−	5.04868i		−	0.210544i
$576$		0			0
$577$	−3.48913			−0.145254			−0.0726271	−	0.997359i	$-0.523138\pi$
$577$	−3.48913			−0.145254			−0.0726271	+	0.997359i	$0.523138\pi$
$578$		0			0
$579$	−5.35464			−0.222531
$580$		0			0
$581$	−8.23369			−0.341591
$582$		0			0
$583$	−8.85621	−	10.3923i	−0.366787	−	0.430405i
$584$		0			0
$585$		−	2.39794i		−	0.0991426i
$586$		0			0
$587$		−	8.71516i		−	0.359713i	−0.983693	−	0.179857i	$-0.942437\pi$
$587$		−	8.71516i		−	0.359713i	0.983693	−	0.179857i	$-0.0575633\pi$
$588$		0			0
$589$		−	20.0172i		−	0.824793i
$590$		0			0
$591$	21.2140			0.872627
$592$		0			0
$593$		−	24.7540i		−	1.01653i	−0.861202	−	0.508263i	$-0.830288\pi$
$593$		−	24.7540i		−	1.01653i	0.861202	−	0.508263i	$-0.169712\pi$
$594$		0			0
$595$			12.7692i			0.523485i
$596$		0			0
$597$	−5.64947			−0.231217
$598$		0			0
$599$			35.4333i			1.44777i	0.689923	+	0.723883i	$0.257644\pi$
$599$			35.4333i			1.44777i	−0.689923	+	0.723883i	$0.742356\pi$
$600$		0			0
$601$			14.2695i			0.582066i	0.956713	+	0.291033i	$0.0939990\pi$
$601$			14.2695i			0.582066i	−0.956713	+	0.291033i	$0.906001\pi$
$602$		0			0
$603$			8.21782i			0.334656i
$604$		0			0
$605$	1.74456	−	10.8608i	0.0709266	−	0.441553i
$606$		0			0
$607$	−27.0158			−1.09654			−0.548270	−	0.836302i	$-0.684713\pi$
$607$	−27.0158			−1.09654			−0.548270	+	0.836302i	$0.684713\pi$
$608$		0			0
$609$	−34.1168			−1.38248
$610$		0			0
$611$	1.89986			0.0768600
$612$		0			0
$613$			15.9149i			0.642795i	0.946944	+	0.321398i	$0.104153\pi$
$613$			15.9149i			0.642795i	−0.946944	+	0.321398i	$0.895847\pi$
$614$		0			0
$615$	8.60485			0.346981
$616$		0			0
$617$	32.7446			1.31825			0.659123	−	0.752035i	$-0.270928\pi$
$617$	32.7446			1.31825			0.659123	+	0.752035i	$0.270928\pi$
$618$		0			0
$619$			29.0024i			1.16571i	0.812578	+	0.582853i	$0.198064\pi$
$619$			29.0024i			1.16571i	−0.812578	+	0.582853i	$0.801936\pi$
$620$		0			0
$621$	21.4891			0.862329
$622$		0			0
$623$	−19.3609			−0.775679
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	4.30243	−	3.66648i	0.171822	−	0.146425i
$628$		0			0
$629$		−	14.5868i		−	0.581613i
$630$		0			0
$631$			23.1615i			0.922044i	0.887389	+	0.461022i	$0.152517\pi$
$631$			23.1615i			0.922044i	−0.887389	+	0.461022i	$0.847483\pi$
$632$		0			0
$633$		−	20.6517i		−	0.820831i
$634$		0			0
$635$	15.4589			0.613469
$636$		0			0
$637$			15.2804i			0.605430i
$638$		0			0
$639$			14.5561i			0.575831i
$640$		0			0
$641$	36.3505			1.43576			0.717880	−	0.696167i	$-0.245113\pi$
$641$	36.3505			1.43576			0.717880	+	0.696167i	$0.245113\pi$
$642$		0			0
$643$			20.9870i			0.827646i	0.910357	+	0.413823i	$0.135807\pi$
$643$			20.9870i			0.827646i	−0.910357	+	0.413823i	$0.864193\pi$
$644$		0			0
$645$		−	2.02163i		−	0.0796017i
$646$		0			0
$647$			6.63325i			0.260780i	0.991463	+	0.130390i	$0.0416229\pi$
$647$			6.63325i			0.260780i	−0.991463	+	0.130390i	$0.958377\pi$
$648$		0			0
$649$	4.74456	−	4.04326i	0.186240	−	0.158712i
$650$		0			0
$651$	34.6708			1.35885
$652$		0			0
$653$	−27.0951			−1.06031			−0.530156	−	0.847900i	$-0.677867\pi$
$653$	−27.0951			−1.06031			−0.530156	+	0.847900i	$0.677867\pi$
$654$		0			0
$655$	−11.5569			−0.451566
$656$		0			0
$657$			28.1628i			1.09873i
$658$		0			0
$659$	0.549500			0.0214055			0.0107027	−	0.999943i	$-0.496593\pi$
$659$	0.549500			0.0214055			0.0107027	+	0.999943i	$0.496593\pi$
$660$		0			0
$661$	−19.2554			−0.748950			−0.374475	−	0.927237i	$-0.622177\pi$
$661$	−19.2554			−0.748950			−0.374475	+	0.927237i	$0.622177\pi$
$662$		0			0
$663$			2.17448i			0.0844499i
$664$		0			0
$665$	−10.1168			−0.392314
$666$		0			0
$667$	46.2277			1.78994
$668$		0			0
$669$	15.7663			0.609561
$670$		0			0
$671$	−18.0106	+	15.3484i	−0.695290	+	0.592519i
$672$		0			0
$673$		−	14.9631i		−	0.576785i	−0.957512	−	0.288392i	$-0.906879\pi$
$673$		−	14.9631i		−	0.576785i	0.957512	−	0.288392i	$-0.0931208\pi$
$674$		0			0
$675$			4.25639i			0.163828i
$676$		0			0
$677$		−	7.82833i		−	0.300867i	−0.988620	−	0.150434i	$-0.951933\pi$
$677$		−	7.82833i		−	0.300867i	0.988620	−	0.150434i	$-0.0480670\pi$
$678$		0			0
$679$	−65.8400			−2.52671
$680$		0			0
$681$		−	5.43039i		−	0.208093i
$682$		0			0
$683$		−	3.37153i		−	0.129008i	−0.997917	−	0.0645040i	$-0.979453\pi$
$683$		−	3.37153i		−	0.129008i	0.997917	−	0.0645040i	$-0.0205465\pi$
$684$		0			0
$685$	11.4891			0.438977
$686$		0			0
$687$		−	1.58457i		−	0.0604553i
$688$		0			0
$689$			4.16137i			0.158536i
$690$		0			0
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$		0			0
$693$	−24.0000	−	28.1628i	−0.911685	−	1.06981i
$694$		0			0
$695$	−0.800857			−0.0303782
$696$		0			0
$697$	29.4891			1.11698
$698$		0			0
$699$	−4.85193			−0.183517
$700$		0			0
$701$			10.5435i			0.398223i	0.979977	+	0.199112i	$0.0638057\pi$
$701$			10.5435i			0.398223i	−0.979977	+	0.199112i	$0.936194\pi$
$702$		0			0
$703$	11.5569			0.435878
$704$		0			0
$705$	−1.48913			−0.0560837
$706$		0			0
$707$		−	76.6150i		−	2.88140i
$708$		0			0
$709$	19.2554			0.723153			0.361577	−	0.932342i	$-0.382239\pi$
$709$	19.2554			0.723153			0.361577	+	0.932342i	$0.382239\pi$
$710$		0			0
$711$	20.4131			0.765552
$712$		0			0
$713$	−46.9783			−1.75935
$714$		0			0
$715$	−2.55164	+	2.17448i	−0.0954260	+	0.0813210i
$716$		0			0
$717$		−	6.81751i		−	0.254605i
$718$		0			0
$719$			22.5716i			0.841777i	0.907112	+	0.420889i	$0.138282\pi$
$719$			22.5716i			0.841777i	−0.907112	+	0.420889i	$0.861718\pi$
$720$		0			0
$721$		−	38.6472i		−	1.43930i
$722$		0			0
$723$	−11.3056			−0.420459
$724$		0			0
$725$			9.15640i			0.340060i
$726$		0			0
$727$			15.1460i			0.561735i	0.959747	+	0.280868i	$0.0906222\pi$
$727$			15.1460i			0.561735i	−0.959747	+	0.280868i	$0.909378\pi$
$728$		0			0
$729$	−1.23369			−0.0456921
$730$		0			0
$731$		−	6.92820i		−	0.256249i
$732$		0			0
$733$			41.0452i			1.51604i	0.652232	+	0.758019i	$0.273833\pi$
$733$			41.0452i			1.51604i	−0.652232	+	0.758019i	$0.726167\pi$
$734$		0			0
$735$		−	11.9769i		−	0.441774i
$736$		0			0
$737$	8.74456	−	7.45202i	0.322110	−	0.274499i
$738$		0			0
$739$	−9.40571			−0.345995			−0.172997	−	0.984922i	$-0.555345\pi$
$739$	−9.40571			−0.345995			−0.172997	+	0.984922i	$0.555345\pi$
$740$		0			0
$741$	−1.72281			−0.0632891
$742$		0			0
$743$	24.3151			0.892036			0.446018	−	0.895024i	$-0.352842\pi$
$743$	24.3151			0.892036			0.446018	+	0.895024i	$0.352842\pi$
$744$		0			0
$745$		−	1.70438i		−	0.0624436i
$746$		0			0
$747$	−4.15335			−0.151963
$748$		0			0
$749$	−72.7011			−2.65644
$750$		0			0
$751$			1.97210i			0.0719630i	0.999352	+	0.0359815i	$0.0114557\pi$
$751$			1.97210i			0.0719630i	−0.999352	+	0.0359815i	$0.988544\pi$
$752$		0			0
$753$	5.25544			0.191519
$754$		0			0
$755$	−9.40571			−0.342309
$756$		0			0
$757$	−38.4674			−1.39812			−0.699060	−	0.715063i	$-0.746398\pi$
$757$	−38.4674			−1.39812			−0.699060	+	0.715063i	$0.746398\pi$
$758$		0			0
$759$	−8.60485	−	10.0974i	−0.312336	−	0.366511i
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			35.0458i			1.26874i
$764$		0			0
$765$			6.44121i			0.232882i
$766$		0			0
$767$	−1.89986			−0.0685999
$768$		0			0
$769$			8.83915i			0.318748i	0.987218	+	0.159374i	$0.0509475\pi$
$769$			8.83915i			0.318748i	−0.987218	+	0.159374i	$0.949052\pi$
$770$		0			0
$771$		−	16.4356i		−	0.591915i
$772$		0			0
$773$	−49.3723			−1.77580			−0.887899	−	0.460039i	$-0.847835\pi$
$773$	−49.3723			−1.77580			−0.887899	+	0.460039i	$0.847835\pi$
$774$		0			0
$775$		−	9.30506i		−	0.334248i
$776$		0			0
$777$			20.0172i			0.718112i
$778$		0			0
$779$			23.3639i			0.837097i
$780$		0			0
$781$	15.4891	−	13.1997i	0.554244	−	0.472321i
$782$		0			0
$783$	−38.9732			−1.39279
$784$		0			0
$785$	22.8614			0.815959
$786$		0			0
$787$	−12.7582			−0.454781			−0.227390	−	0.973804i	$-0.573019\pi$
$787$	−12.7582			−0.454781			−0.227390	+	0.973804i	$0.573019\pi$
$788$		0			0
$789$		−	11.8125i		−	0.420538i
$790$		0			0
$791$	−54.0317			−1.92115
$792$		0			0
$793$	7.21194			0.256103
$794$		0			0
$795$		−	3.26172i		−	0.115681i
$796$		0			0
$797$	−0.510875			−0.0180961			−0.00904806	−	0.999959i	$-0.502880\pi$
$797$	−0.510875			−0.0180961			−0.00904806	+	0.999959i	$0.502880\pi$
$798$		0			0
$799$	−5.10328			−0.180541
$800$		0			0
$801$	−9.76631			−0.345076
$802$		0			0
$803$	29.9679	−	25.5383i	1.05754	−	0.901228i
$804$		0			0
$805$			23.7432i			0.836837i
$806$		0			0
$807$			4.75372i			0.167339i
$808$		0			0
$809$		−	48.8735i		−	1.71830i	−0.511723	−	0.859150i	$-0.670993\pi$
$809$		−	48.8735i		−	1.71830i	0.511723	−	0.859150i	$-0.329007\pi$
$810$		0			0
$811$	−37.3715			−1.31229			−0.656145	−	0.754635i	$-0.727814\pi$
$811$	−37.3715			−1.31229			−0.656145	+	0.754635i	$0.727814\pi$
$812$		0			0
$813$		−	22.3561i		−	0.784061i
$814$		0			0
$815$			16.2333i			0.568627i
$816$		0			0
$817$	5.48913			0.192040
$818$		0			0
$819$			11.2772i			0.394056i
$820$		0			0
$821$			9.47365i			0.330633i	0.986241	+	0.165316i	$0.0528645\pi$
$821$			9.47365i			0.330633i	−0.986241	+	0.165316i	$0.947135\pi$
$822$		0			0
$823$		−	47.6126i		−	1.65967i	−0.558009	−	0.829835i	$-0.688435\pi$
$823$		−	47.6126i		−	1.65967i	0.558009	−	0.829835i	$-0.311565\pi$
$824$		0			0
$825$	2.00000	−	1.70438i	0.0696311	−	0.0593388i
$826$		0			0
$827$	24.8646			0.864628			0.432314	−	0.901723i	$-0.357697\pi$
$827$	24.8646			0.864628			0.432314	+	0.901723i	$0.357697\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$	−16.4088			−0.569217
$832$		0			0
$833$		−	41.0452i		−	1.42213i
$834$		0			0
$835$	−14.9094			−0.515962
$836$		0			0
$837$	39.6060			1.36898
$838$		0			0
$839$			48.2025i			1.66413i	0.554675	+	0.832067i	$0.312843\pi$
$839$			48.2025i			1.66413i	−0.554675	+	0.832067i	$0.687157\pi$
$840$		0			0
$841$	−54.8397			−1.89102
$842$		0			0
$843$	−5.40143			−0.186035
$844$		0			0
$845$	−11.9783			−0.412064
$846$		0			0
$847$	−8.20442	+	51.0767i	−0.281907	+	1.75501i
$848$		0			0
$849$		−	24.3777i		−	0.836640i
$850$		0			0
$851$		−	27.1229i		−	0.929761i
$852$		0			0
$853$		−	39.0235i		−	1.33614i	−0.744098	−	0.668070i	$-0.767121\pi$
$853$		−	39.0235i		−	1.33614i	0.744098	−	0.668070i	$-0.232879\pi$
$854$		0			0
$855$	−5.10328			−0.174529
$856$		0			0
$857$			15.5976i			0.532804i	0.963862	+	0.266402i	$0.0858349\pi$
$857$			15.5976i			0.532804i	−0.963862	+	0.266402i	$0.914165\pi$
$858$		0			0
$859$		−	26.4232i		−	0.901548i	−0.892638	−	0.450774i	$-0.851148\pi$
$859$		−	26.4232i		−	0.901548i	0.892638	−	0.450774i	$-0.148852\pi$
$860$		0			0
$861$	−40.4674			−1.37912
$862$		0			0
$863$			34.3461i			1.16915i	0.811338	+	0.584577i	$0.198739\pi$
$863$			34.3461i			1.16915i	−0.811338	+	0.584577i	$0.801261\pi$
$864$		0			0
$865$			6.44121i			0.219008i
$866$		0			0
$867$			7.62792i			0.259058i
$868$		0			0
$869$	−18.5109	−	21.7216i	−0.627938	−	0.736853i
$870$		0			0
$871$	−3.50157			−0.118646
$872$		0			0
$873$	−33.2119			−1.12405
$874$		0			0
$875$	−4.70285			−0.158986
$876$		0			0
$877$			21.3452i			0.720778i	0.932802	+	0.360389i	$0.117356\pi$
$877$			21.3452i			0.720778i	−0.932802	+	0.360389i	$0.882644\pi$
$878$		0			0
$879$	19.6123			0.661506
$880$		0			0
$881$	−0.510875			−0.0172118			−0.00860590	−	0.999963i	$-0.502739\pi$
$881$	−0.510875			−0.0172118			−0.00860590	+	0.999963i	$0.502739\pi$
$882$		0			0
$883$			34.8434i			1.17257i	0.810104	+	0.586287i	$0.199411\pi$
$883$			34.8434i			1.17257i	−0.810104	+	0.586287i	$0.800589\pi$
$884$		0			0
$885$	1.48913			0.0500564
$886$		0			0
$887$	0.149072			0.00500535			0.00250267	−	0.999997i	$-0.499203\pi$
$887$	0.149072			0.00500535			0.00250267	+	0.999997i	$0.499203\pi$
$888$		0			0
$889$	−72.7011			−2.43832
$890$		0			0
$891$	−8.05535	−	9.45254i	−0.269864	−	0.316672i
$892$		0			0
$893$		−	4.04326i		−	0.135303i
$894$		0			0
$895$			1.87953i			0.0628257i
$896$		0			0
$897$			4.04326i			0.135001i
$898$		0			0
$899$	85.2009			2.84161
$900$		0			0
$901$		−	11.1780i		−	0.372394i
$902$		0			0
$903$			9.50744i			0.316388i
$904$		0			0
$905$	3.48913			0.115982
$906$		0			0
$907$		−	20.5822i		−	0.683422i	−0.939805	−	0.341711i	$-0.888994\pi$
$907$		−	20.5822i		−	0.683422i	0.939805	−	0.341711i	$-0.111006\pi$
$908$		0			0
$909$		−	38.6472i		−	1.28185i
$910$		0			0
$911$			48.1099i			1.59395i	0.604011	+	0.796976i	$0.293568\pi$
$911$			48.1099i			1.59395i	−0.604011	+	0.796976i	$0.706432\pi$
$912$		0			0
$913$	3.76631	+	4.41957i	0.124647	+	0.146267i
$914$		0			0
$915$	−5.65278			−0.186875
$916$		0			0
$917$	54.3505			1.79481
$918$		0			0
$919$	−8.60485			−0.283848			−0.141924	−	0.989878i	$-0.545329\pi$
$919$	−8.60485			−0.283848			−0.141924	+	0.989878i	$0.545329\pi$
$920$		0			0
$921$			2.65614i			0.0875228i
$922$		0			0
$923$	−6.20228			−0.204151
$924$		0			0
$925$	5.37228			0.176640
$926$		0			0
$927$		−	19.4950i		−	0.640299i
$928$		0			0
$929$	33.6060			1.10258			0.551288	−	0.834315i	$-0.314137\pi$
$929$	33.6060			1.10258			0.551288	+	0.834315i	$0.314137\pi$
$930$		0			0
$931$	32.5196			1.06579
$932$		0			0
$933$	8.62772			0.282459
$934$		0			0
$935$	6.85407	−	5.84096i	0.224152	−	0.191020i
$936$		0			0
$937$		−	33.5932i		−	1.09744i	−0.836006	−	0.548720i	$-0.815115\pi$
$937$		−	33.5932i		−	1.09744i	0.836006	−	0.548720i	$-0.184885\pi$
$938$		0			0
$939$			19.6048i			0.639778i
$940$		0			0
$941$			21.4043i			0.697760i	0.937167	+	0.348880i	$0.113438\pi$
$941$			21.4043i			0.697760i	−0.937167	+	0.348880i	$0.886562\pi$
$942$		0			0
$943$	54.8325			1.78559
$944$		0			0
$945$		−	20.0172i		−	0.651159i
$946$		0			0
$947$			19.9923i			0.649663i	0.945772	+	0.324832i	$0.105308\pi$
$947$			19.9923i			0.649663i	−0.945772	+	0.324832i	$0.894692\pi$
$948$		0			0
$949$	−12.0000			−0.389536
$950$		0			0
$951$		−	1.08724i		−	0.0352562i
$952$		0			0
$953$		−	48.1799i		−	1.56070i	−0.625342	−	0.780351i	$-0.715041\pi$
$953$		−	48.1799i		−	1.56070i	0.625342	−	0.780351i	$-0.284959\pi$
$954$		0			0
$955$		−	16.7306i		−	0.541390i
$956$		0			0
$957$	15.6060	+	18.3128i	0.504469	+	0.591969i
$958$		0			0
$959$	−54.0317			−1.74477
$960$		0			0
$961$	−55.5842			−1.79304
$962$		0			0
$963$	−36.6729			−1.18177
$964$		0			0
$965$		−	6.75846i		−	0.217563i
$966$		0			0
$967$	52.5323			1.68932			0.844662	−	0.535300i	$-0.179802\pi$
$967$	52.5323			1.68932			0.844662	+	0.535300i	$0.179802\pi$
$968$		0			0
$969$	4.62772			0.148664
$970$		0			0
$971$		−	28.4125i		−	0.911801i	−0.890031	−	0.455901i	$-0.849317\pi$
$971$		−	28.4125i		−	0.911801i	0.890031	−	0.455901i	$-0.150683\pi$
$972$		0			0
$973$	3.76631			0.120742
$974$		0			0
$975$	−0.800857			−0.0256479
$976$		0			0
$977$	34.4674			1.10271			0.551355	−	0.834271i	$-0.314111\pi$
$977$	34.4674			1.10271			0.551355	+	0.834271i	$0.314111\pi$
$978$		0			0
$979$	8.85621	+	10.3923i	0.283046	+	0.332140i
$980$		0			0
$981$			17.6783i			0.564424i
$982$		0			0
$983$			16.1407i			0.514808i	0.966304	+	0.257404i	$0.0828671\pi$
$983$			16.1407i			0.514808i	−0.966304	+	0.257404i	$0.917133\pi$
$984$		0			0
$985$			26.7756i			0.853143i
$986$		0			0
$987$	7.00314			0.222912
$988$		0			0
$989$		−	12.8824i		−	0.409637i
$990$		0			0
$991$		−	17.3205i		−	0.550204i	−0.961415	−	0.275102i	$-0.911288\pi$
$991$		−	17.3205i		−	0.550204i	0.961415	−	0.275102i	$-0.0887116\pi$
$992$		0			0
$993$	−4.46738			−0.141768
$994$		0			0
$995$		−	7.13058i		−	0.226055i
$996$		0			0
$997$		−	26.1411i		−	0.827898i	−0.910300	−	0.413949i	$-0.864149\pi$
$997$		−	26.1411i		−	0.827898i	0.910300	−	0.413949i	$-0.135851\pi$
$998$		0			0
$999$			22.8665i			0.723465i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.f.d.351.6	yes	8
4.3	odd	2	inner	880.2.f.d.351.3	✓	8
8.3	odd	2		3520.2.f.j.2111.5		8
8.5	even	2		3520.2.f.j.2111.4		8
11.10	odd	2	inner	880.2.f.d.351.5	yes	8
44.43	even	2	inner	880.2.f.d.351.4	yes	8
88.21	odd	2		3520.2.f.j.2111.3		8
88.43	even	2		3520.2.f.j.2111.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
880.2.f.d.351.3	✓	8	4.3	odd	2	inner
880.2.f.d.351.4	yes	8	44.43	even	2	inner
880.2.f.d.351.5	yes	8	11.10	odd	2	inner
880.2.f.d.351.6	yes	8	1.1	even	1	trivial
3520.2.f.j.2111.3		8	88.21	odd	2
3520.2.f.j.2111.4		8	8.5	even	2
3520.2.f.j.2111.5		8	8.3	odd	2
3520.2.f.j.2111.6		8	88.43	even	2

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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.792287i q^{3} -1.00000 q^{5} +4.70285 q^{7} +2.37228 q^{9} +(-2.15121 - 2.52434i) q^{11} +1.01082i q^{13} -0.792287i q^{15} -2.71519i q^{17} +2.15121 q^{19} +3.72601i q^{21} -5.04868i q^{23} +1.00000 q^{25} +4.25639i q^{27} +9.15640i q^{29} -9.30506i q^{31} +(2.00000 - 1.70438i) q^{33} -4.70285 q^{35} +5.37228 q^{37} -0.800857 q^{39} +10.8608i q^{41} +2.55164 q^{43} -2.37228 q^{45} -1.87953i q^{47} +15.1168 q^{49} +2.15121 q^{51} +4.11684 q^{53} +(2.15121 + 2.52434i) q^{55} +1.70438i q^{57} +1.87953i q^{59} -7.13477i q^{61} +11.1565 q^{63} -1.01082i q^{65} +3.46410i q^{67} +4.00000 q^{69} +6.13592i q^{71} +11.8716i q^{73} +0.792287i q^{75} +(-10.1168 - 11.8716i) q^{77} +8.60485 q^{79} +3.74456 q^{81} -1.75079 q^{83} +2.71519i q^{85} -7.25450 q^{87} -4.11684 q^{89} +4.75372i q^{91} +7.37228 q^{93} -2.15121 q^{95} -14.0000 q^{97} +(-5.10328 - 5.98844i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5} - 4 q^{9} + 8 q^{25} + 16 q^{33} + 20 q^{37} + 4 q^{45} + 52 q^{49} - 36 q^{53} + 32 q^{69} - 12 q^{77} - 16 q^{81} + 36 q^{89} + 36 q^{93} - 112 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 4 * q^9 + 8 * q^25 + 16 * q^33 + 20 * q^37 + 4 * q^45 + 52 * q^49 - 36 * q^53 + 32 * q^69 - 12 * q^77 - 16 * q^81 + 36 * q^89 + 36 * q^93 - 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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