LMFDB - Embedded newform 880.2.b.d.529.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [880,2,Mod(529,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.529"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,0,0,-2,0,-2,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$7.02683537787$
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:	$ 2 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.2.b.d.529.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} -2.00000i q^{17} +4.00000 q^{19} +8.00000 q^{21} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} +2.00000i q^{33} +(8.00000 + 4.00000i) q^{35} +8.00000i q^{37} -12.0000 q^{39} +6.00000 q^{41} -12.0000i q^{43} +(-1.00000 + 2.00000i) q^{45} +10.0000i q^{47} -9.00000 q^{49} -4.00000 q^{51} +(-1.00000 + 2.00000i) q^{55} -8.00000i q^{57} -4.00000 q^{59} -10.0000 q^{61} -4.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} +2.00000i q^{67} -12.0000 q^{69} +8.00000 q^{71} +2.00000i q^{73} +(-8.00000 + 6.00000i) q^{75} -4.00000i q^{77} +4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +(-4.00000 - 2.00000i) q^{85} -4.00000i q^{87} +14.0000 q^{89} +24.0000 q^{91} +16.0000i q^{93} +(4.00000 - 8.00000i) q^{95} +4.00000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} + 8 q^{19} + 16 q^{21} - 6 q^{25} + 4 q^{29} - 16 q^{31} + 16 q^{35} - 24 q^{39} + 12 q^{41} - 2 q^{45} - 18 q^{49} - 8 q^{51} - 2 q^{55} - 8 q^{59} - 20 q^{61}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^11 - 8 * q^15 + 8 * q^19 + 16 * q^21 - 6 * q^25 + 4 * q^29 - 16 * q^31 + 16 * q^35 - 24 * q^39 + 12 * q^41 - 2 * q^45 - 18 * q^49 - 8 * q^51 - 2 * q^55 - 8 * q^59 - 20 * q^61 - 24 * q^65 - 24 * q^69 + 16 * q^71 - 16 * q^75 + 8 * q^79 - 22 * q^81 - 8 * q^85 + 28 * q^89 + 48 * q^91 + 8 * q^95 + 2 * q^99

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
$3$		−	2.00000i		−	1.15470i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$		−	2.00000i		−	1.15470i	0.816497	−	0.577350i	$-0.195913\pi$
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$	−4.00000	−	2.00000i	−1.03280	−	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
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	8.00000			1.74574
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		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
$35$	8.00000	+	4.00000i	1.35225	+	0.676123i
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	−12.0000			−1.92154
$40$		0			0
$41$	6.00000			0.937043			0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000			0.937043			0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$		−	12.0000i		−	1.82998i	−0.403473	−	0.914991i	$-0.632197\pi$
$43$		−	12.0000i		−	1.82998i	0.403473	−	0.914991i	$-0.367803\pi$
$44$		0			0
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$			10.0000i			1.45865i	0.684167	+	0.729325i	$0.260166\pi$
$47$			10.0000i			1.45865i	−0.684167	+	0.729325i	$0.739834\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$	−4.00000			−0.560112
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	−1.00000	+	2.00000i	−0.134840	+	0.269680i
$56$		0			0
$57$		−	8.00000i		−	1.05963i
$58$		0			0
$59$	−4.00000			−0.520756			−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000			−0.520756			−0.260378	+	0.965507i	$0.583847\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		−	4.00000i		−	0.503953i
$64$		0			0
$65$	−12.0000	−	6.00000i	−1.48842	−	0.744208i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	−12.0000			−1.44463
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$	−8.00000	+	6.00000i	−0.923760	+	0.692820i
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$	−4.00000	−	2.00000i	−0.433861	−	0.216930i
$86$		0			0
$87$		−	4.00000i		−	0.428845i
$88$		0			0
$89$	14.0000			1.48400			0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000			1.48400			0.741999	+	0.670402i	$0.233878\pi$
$90$		0			0
$91$	24.0000			2.51588
$92$		0			0
$93$			16.0000i			1.65912i
$94$		0			0
$95$	4.00000	−	8.00000i	0.410391	−	0.820783i
$96$		0			0
$97$			4.00000i			0.406138i	0.979164	+	0.203069i	$0.0650917\pi$
$97$			4.00000i			0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		−	2.00000i		−	0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$		−	2.00000i		−	0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$		0			0
$105$	8.00000	−	16.0000i	0.780720	−	1.56144i
$106$		0			0
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$	16.0000			1.51865
$112$		0			0
$113$			4.00000i			0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$			4.00000i			0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$		0			0
$115$	−12.0000	−	6.00000i	−1.11901	−	0.559503i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	12.0000i		−	1.08200i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			8.00000i			0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$			8.00000i			0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$		0			0
$129$	−24.0000			−2.11308
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$			16.0000i			1.38738i
$134$		0			0
$135$	−8.00000	−	4.00000i	−0.688530	−	0.344265i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	20.0000			1.68430
$142$		0			0
$143$			6.00000i			0.501745i
$144$		0			0
$145$	2.00000	−	4.00000i	0.166091	−	0.332182i
$146$		0			0
$147$			18.0000i			1.48461i
$148$		0			0
$149$	14.0000			1.14692			0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000			1.14692			0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	−8.00000	+	16.0000i	−0.642575	+	1.28515i
$156$		0			0
$157$			20.0000i			1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$			20.0000i			1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	24.0000			1.89146
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$		0			0
$165$	4.00000	+	2.00000i	0.311400	+	0.155700i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			22.0000i			1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$			22.0000i			1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$		0			0
$175$	16.0000	−	12.0000i	1.20949	−	0.907115i
$176$		0			0
$177$			8.00000i			0.601317i
$178$		0			0
$179$	4.00000			0.298974			0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000			0.298974			0.149487	+	0.988764i	$0.452238\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$			20.0000i			1.47844i
$184$		0			0
$185$	16.0000	+	8.00000i	1.17634	+	0.588172i
$186$		0			0
$187$			2.00000i			0.146254i
$188$		0			0
$189$	16.0000			1.16383
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		0			0
$195$	−12.0000	+	24.0000i	−0.859338	+	1.71868i
$196$		0			0
$197$		−	6.00000i		−	0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$		−	6.00000i		−	0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$	4.00000			0.282138
$202$		0			0
$203$			8.00000i			0.561490i
$204$		0			0
$205$	6.00000	−	12.0000i	0.419058	−	0.838116i
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		−	16.0000i		−	1.09630i
$214$		0			0
$215$	−24.0000	−	12.0000i	−1.63679	−	0.818393i
$216$		0			0
$217$		−	32.0000i		−	2.17230i
$218$		0			0
$219$	4.00000			0.270295
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$			10.0000i			0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$			10.0000i			0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$	−8.00000			−0.526361
$232$		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
$235$	20.0000	+	10.0000i	1.30466	+	0.652328i
$236$		0			0
$237$		−	8.00000i		−	0.519656i
$238$		0			0
$239$	4.00000			0.258738			0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000			0.258738			0.129369	+	0.991596i	$0.458705\pi$
$240$		0			0
$241$	6.00000			0.386494			0.193247	−	0.981150i	$-0.438098\pi$
$241$	6.00000			0.386494			0.193247	+	0.981150i	$0.438098\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	−9.00000	+	18.0000i	−0.574989	+	1.14998i
$246$		0			0
$247$		−	24.0000i		−	1.52708i
$248$		0			0
$249$	8.00000			0.506979
$250$		0			0
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$		0			0
$253$			6.00000i			0.377217i
$254$		0			0
$255$	−4.00000	+	8.00000i	−0.250490	+	0.500979i
$256$		0			0
$257$			24.0000i			1.49708i	0.663090	+	0.748539i	$0.269245\pi$
$257$			24.0000i			1.49708i	−0.663090	+	0.748539i	$0.730755\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$		0			0
$261$	−2.00000			−0.123797
$262$		0			0
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	28.0000i		−	1.71357i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		−	48.0000i		−	2.90509i
$274$		0			0
$275$	3.00000	+	4.00000i	0.180907	+	0.241209i
$276$		0			0
$277$			14.0000i			0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$			14.0000i			0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$		0			0
$279$	8.00000			0.478947
$280$		0			0
$281$	22.0000			1.31241			0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000			1.31241			0.656205	+	0.754583i	$0.272161\pi$
$282$		0			0
$283$		−	28.0000i		−	1.66443i	−0.554455	−	0.832214i	$-0.687073\pi$
$283$		−	28.0000i		−	1.66443i	0.554455	−	0.832214i	$-0.312927\pi$
$284$		0			0
$285$	−16.0000	−	8.00000i	−0.947758	−	0.473879i
$286$		0			0
$287$			24.0000i			1.41668i
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	8.00000			0.468968
$292$		0			0
$293$			26.0000i			1.51894i	0.650545	+	0.759468i	$0.274541\pi$
$293$			26.0000i			1.51894i	−0.650545	+	0.759468i	$0.725459\pi$
$294$		0			0
$295$	−4.00000	+	8.00000i	−0.232889	+	0.465778i
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	−36.0000			−2.08193
$300$		0			0
$301$	48.0000			2.76667
$302$		0			0
$303$		−	20.0000i		−	1.14897i
$304$		0			0
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$		0			0
$307$			8.00000i			0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$			8.00000i			0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$		0			0
$309$	−4.00000			−0.227552
$310$		0			0
$311$	8.00000			0.453638			0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000			0.453638			0.226819	+	0.973937i	$0.427167\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−8.00000	−	4.00000i	−0.450749	−	0.225374i
$316$		0			0
$317$		−	28.0000i		−	1.57264i	−0.617822	−	0.786318i	$-0.711985\pi$
$317$		−	28.0000i		−	1.57264i	0.617822	−	0.786318i	$-0.288015\pi$
$318$		0			0
$319$	−2.00000			−0.111979
$320$		0			0
$321$	−16.0000			−0.893033
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	−24.0000	+	18.0000i	−1.33128	+	0.998460i
$326$		0			0
$327$		−	28.0000i		−	1.54840i
$328$		0			0
$329$	−40.0000			−2.20527
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		−	8.00000i		−	0.438397i
$334$		0			0
$335$	4.00000	+	2.00000i	0.218543	+	0.109272i
$336$		0			0
$337$		−	2.00000i		−	0.108947i	−0.998515	−	0.0544735i	$-0.982652\pi$
$337$		−	2.00000i		−	0.108947i	0.998515	−	0.0544735i	$-0.0173480\pi$
$338$		0			0
$339$	8.00000			0.434500
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$	−12.0000	+	24.0000i	−0.646058	+	1.29212i
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−22.0000			−1.17763			−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000			−1.17763			−0.588817	+	0.808267i	$0.700406\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$			8.00000i			0.425797i	0.977074	+	0.212899i	$0.0682904\pi$
$353$			8.00000i			0.425797i	−0.977074	+	0.212899i	$0.931710\pi$
$354$		0			0
$355$	8.00000	−	16.0000i	0.424596	−	0.849192i
$356$		0			0
$357$		−	16.0000i		−	0.846810i
$358$		0			0
$359$	4.00000			0.211112			0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000			0.211112			0.105556	+	0.994413i	$0.466338\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	4.00000	+	2.00000i	0.209370	+	0.104685i
$366$		0			0
$367$		−	30.0000i		−	1.56599i	−0.622030	−	0.782994i	$-0.713692\pi$
$367$		−	30.0000i		−	1.56599i	0.622030	−	0.782994i	$-0.286308\pi$
$368$		0			0
$369$	−6.00000			−0.312348
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	26.0000i		−	1.34623i	−0.739538	−	0.673114i	$-0.764956\pi$
$373$		−	26.0000i		−	1.34623i	0.739538	−	0.673114i	$-0.235044\pi$
$374$		0			0
$375$	4.00000	+	22.0000i	0.206559	+	1.13608i
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$	16.0000			0.819705
$382$		0			0
$383$			30.0000i			1.53293i	0.642287	+	0.766464i	$0.277986\pi$
$383$			30.0000i			1.53293i	−0.642287	+	0.766464i	$0.722014\pi$
$384$		0			0
$385$	−8.00000	−	4.00000i	−0.407718	−	0.203859i
$386$		0			0
$387$			12.0000i			0.609994i
$388$		0			0
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	−12.0000			−0.606866
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.00000	−	8.00000i	0.201262	−	0.402524i
$396$		0			0
$397$			20.0000i			1.00377i	0.864934	+	0.501886i	$0.167360\pi$
$397$			20.0000i			1.00377i	−0.864934	+	0.501886i	$0.832640\pi$
$398$		0			0
$399$	32.0000			1.60200
$400$		0			0
$401$	−34.0000			−1.69788			−0.848939	−	0.528490i	$-0.822758\pi$
$401$	−34.0000			−1.69788			−0.848939	+	0.528490i	$0.822758\pi$
$402$		0			0
$403$			48.0000i			2.39105i
$404$		0			0
$405$	−11.0000	+	22.0000i	−0.546594	+	1.09319i
$406$		0			0
$407$		−	8.00000i		−	0.396545i
$408$		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
$411$	−24.0000			−1.18383
$412$		0			0
$413$		−	16.0000i		−	0.787309i
$414$		0			0
$415$	8.00000	+	4.00000i	0.392705	+	0.196352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.0000			1.36789			0.683945	−	0.729534i	$-0.260263\pi$
$419$	28.0000			1.36789			0.683945	+	0.729534i	$0.260263\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		−	10.0000i		−	0.486217i
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$		−	40.0000i		−	1.93574i
$428$		0			0
$429$	12.0000			0.579365
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$		−	28.0000i		−	1.34559i	−0.739827	−	0.672797i	$-0.765093\pi$
$433$		−	28.0000i		−	1.34559i	0.739827	−	0.672797i	$-0.234907\pi$
$434$		0			0
$435$	−8.00000	−	4.00000i	−0.383571	−	0.191785i
$436$		0			0
$437$		−	24.0000i		−	1.14808i
$438$		0			0
$439$	−16.0000			−0.763638			−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000			−0.763638			−0.381819	+	0.924237i	$0.624702\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$			14.0000i			0.665160i	0.943075	+	0.332580i	$0.107919\pi$
$443$			14.0000i			0.665160i	−0.943075	+	0.332580i	$0.892081\pi$
$444$		0			0
$445$	14.0000	−	28.0000i	0.663664	−	1.32733i
$446$		0			0
$447$		−	28.0000i		−	1.32435i
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$		−	16.0000i		−	0.751746i
$454$		0			0
$455$	24.0000	−	48.0000i	1.12514	−	2.25027i
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$		0			0
$459$	−8.00000			−0.373408
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		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
$465$	32.0000	+	16.0000i	1.48396	+	0.741982i
$466$		0			0
$467$			2.00000i			0.0925490i	0.998929	+	0.0462745i	$0.0147349\pi$
$467$			2.00000i			0.0925490i	−0.998929	+	0.0462745i	$0.985265\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$	40.0000			1.84310
$472$		0			0
$473$			12.0000i			0.551761i
$474$		0			0
$475$	−12.0000	−	16.0000i	−0.550598	−	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.00000			0.182765			0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000			0.182765			0.0913823	+	0.995816i	$0.470871\pi$
$480$		0			0
$481$	48.0000			2.18861
$482$		0			0
$483$		−	48.0000i		−	2.18408i
$484$		0			0
$485$	8.00000	+	4.00000i	0.363261	+	0.181631i
$486$		0			0
$487$		−	6.00000i		−	0.271886i	−0.990717	−	0.135943i	$-0.956594\pi$
$487$		−	6.00000i		−	0.271886i	0.990717	−	0.135943i	$-0.0434064\pi$
$488$		0			0
$489$	−20.0000			−0.904431
$490$		0			0
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$		−	4.00000i		−	0.180151i
$494$		0			0
$495$	1.00000	−	2.00000i	0.0449467	−	0.0898933i
$496$		0			0
$497$			32.0000i			1.43540i
$498$		0			0
$499$	28.0000			1.25345			0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000			1.25345			0.626726	+	0.779240i	$0.284395\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12.0000i		−	0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$		−	12.0000i		−	0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$		0			0
$507$			46.0000i			2.04293i
$508$		0			0
$509$	14.0000			0.620539			0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000			0.620539			0.310270	+	0.950649i	$0.399581\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$		−	16.0000i		−	0.706417i
$514$		0			0
$515$	−4.00000	−	2.00000i	−0.176261	−	0.0881305i
$516$		0			0
$517$		−	10.0000i		−	0.439799i
$518$		0			0
$519$	44.0000			1.93139
$520$		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
$525$	−24.0000	−	32.0000i	−1.04745	−	1.39659i
$526$		0			0
$527$			16.0000i			0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	4.00000			0.173585
$532$		0			0
$533$		−	36.0000i		−	1.55933i
$534$		0			0
$535$	−16.0000	−	8.00000i	−0.691740	−	0.345870i
$536$		0			0
$537$		−	8.00000i		−	0.345225i
$538$		0			0
$539$	9.00000			0.387657
$540$		0			0
$541$	−30.0000			−1.28980			−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000			−1.28980			−0.644900	+	0.764267i	$0.723101\pi$
$542$		0			0
$543$			44.0000i			1.88822i
$544$		0			0
$545$	14.0000	−	28.0000i	0.599694	−	1.19939i
$546$		0			0
$547$		−	12.0000i		−	0.513083i	−0.966533	−	0.256541i	$-0.917417\pi$
$547$		−	12.0000i		−	0.513083i	0.966533	−	0.256541i	$-0.0825830\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	16.0000	−	32.0000i	0.679162	−	1.35832i
$556$		0			0
$557$			2.00000i			0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$			2.00000i			0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$		0			0
$559$	−72.0000			−3.04528
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$		−	24.0000i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$		−	24.0000i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$		0			0
$565$	8.00000	+	4.00000i	0.336563	+	0.168281i
$566$		0			0
$567$		−	44.0000i		−	1.84783i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.0000	+	18.0000i	−1.00087	+	0.750652i
$576$		0			0
$577$		−	4.00000i		−	0.166522i	−0.996528	−	0.0832611i	$-0.973466\pi$
$577$		−	4.00000i		−	0.166522i	0.996528	−	0.0832611i	$-0.0265335\pi$
$578$		0			0
$579$	−28.0000			−1.16364
$580$		0			0
$581$	−16.0000			−0.663792
$582$		0			0
$583$		0			0
$584$		0			0
$585$	12.0000	+	6.00000i	0.496139	+	0.248069i
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$	−12.0000			−0.493614
$592$		0			0
$593$			42.0000i			1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$			42.0000i			1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$		0			0
$595$	8.00000	−	16.0000i	0.327968	−	0.655936i
$596$		0			0
$597$			32.0000i			1.30967i
$598$		0			0
$599$	−40.0000			−1.63436			−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000			−1.63436			−0.817178	+	0.576386i	$0.804463\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$		0			0
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$	1.00000	−	2.00000i	0.0406558	−	0.0813116i
$606$		0			0
$607$			48.0000i			1.94826i	0.225989	+	0.974130i	$0.427439\pi$
$607$			48.0000i			1.94826i	−0.225989	+	0.974130i	$0.572561\pi$
$608$		0			0
$609$	16.0000			0.648353
$610$		0			0
$611$	60.0000			2.42734
$612$		0			0
$613$			14.0000i			0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$			14.0000i			0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$		0			0
$615$	−24.0000	−	12.0000i	−0.967773	−	0.483887i
$616$		0			0
$617$			4.00000i			0.161034i	0.996753	+	0.0805170i	$0.0256571\pi$
$617$			4.00000i			0.161034i	−0.996753	+	0.0805170i	$0.974343\pi$
$618$		0			0
$619$	−36.0000			−1.44696			−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000			−1.44696			−0.723481	+	0.690344i	$0.757459\pi$
$620$		0			0
$621$	−24.0000			−0.963087
$622$		0			0
$623$			56.0000i			2.24359i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$			8.00000i			0.319489i
$628$		0			0
$629$	16.0000			0.637962
$630$		0			0
$631$	−48.0000			−1.91085			−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000			−1.91085			−0.955425	+	0.295234i	$0.904602\pi$
$632$		0			0
$633$		−	32.0000i		−	1.27189i
$634$		0			0
$635$	16.0000	+	8.00000i	0.634941	+	0.317470i
$636$		0			0
$637$			54.0000i			2.13956i
$638$		0			0
$639$	−8.00000			−0.316475
$640$		0			0
$641$	−2.00000			−0.0789953			−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000			−0.0789953			−0.0394976	+	0.999220i	$0.512576\pi$
$642$		0			0
$643$			22.0000i			0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$			22.0000i			0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$		0			0
$645$	−24.0000	+	48.0000i	−0.944999	+	1.89000i
$646$		0			0
$647$		−	18.0000i		−	0.707653i	−0.935311	−	0.353827i	$-0.884880\pi$
$647$		−	18.0000i		−	0.707653i	0.935311	−	0.353827i	$-0.115120\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$	−64.0000			−2.50836
$652$		0			0
$653$		−	24.0000i		−	0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$		−	24.0000i		−	0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	2.00000i		−	0.0780274i
$658$		0			0
$659$	24.0000			0.934907			0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000			0.934907			0.467454	+	0.884018i	$0.345171\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$		0			0
$663$			24.0000i			0.932083i
$664$		0			0
$665$	32.0000	+	16.0000i	1.24091	+	0.620453i
$666$		0			0
$667$		−	12.0000i		−	0.464642i
$668$		0			0
$669$	20.0000			0.773245
$670$		0			0
$671$	10.0000			0.386046
$672$		0			0
$673$		−	14.0000i		−	0.539660i	−0.962908	−	0.269830i	$-0.913032\pi$
$673$		−	14.0000i		−	0.539660i	0.962908	−	0.269830i	$-0.0869676\pi$
$674$		0			0
$675$	−16.0000	+	12.0000i	−0.615840	+	0.461880i
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$	−16.0000			−0.614024
$680$		0			0
$681$	24.0000			0.919682
$682$		0			0
$683$		−	30.0000i		−	1.14792i	−0.818884	−	0.573959i	$-0.805407\pi$
$683$		−	30.0000i		−	1.14792i	0.818884	−	0.573959i	$-0.194593\pi$
$684$		0			0
$685$	−24.0000	−	12.0000i	−0.916993	−	0.458496i
$686$		0			0
$687$		−	12.0000i		−	0.457829i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.00000			−0.152167			−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000			−0.152167			−0.0760836	+	0.997101i	$0.524242\pi$
$692$		0			0
$693$			4.00000i			0.151947i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	12.0000i		−	0.454532i
$698$		0			0
$699$	28.0000			1.05906
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$			32.0000i			1.20690i
$704$		0			0
$705$	20.0000	−	40.0000i	0.753244	−	1.50649i
$706$		0			0
$707$			40.0000i			1.50435i
$708$		0			0
$709$	42.0000			1.57734			0.788672	−	0.614815i	$-0.210769\pi$
$709$	42.0000			1.57734			0.788672	+	0.614815i	$0.210769\pi$
$710$		0			0
$711$	−4.00000			−0.150012
$712$		0			0
$713$			48.0000i			1.79761i
$714$		0			0
$715$	12.0000	+	6.00000i	0.448775	+	0.224387i
$716$		0			0
$717$		−	8.00000i		−	0.298765i
$718$		0			0
$719$	8.00000			0.298350			0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000			0.298350			0.149175	+	0.988811i	$0.452338\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$		−	12.0000i		−	0.446285i
$724$		0			0
$725$	−6.00000	−	8.00000i	−0.222834	−	0.297113i
$726$		0			0
$727$			30.0000i			1.11264i	0.830969	+	0.556319i	$0.187787\pi$
$727$			30.0000i			1.11264i	−0.830969	+	0.556319i	$0.812213\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$			6.00000i			0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$			6.00000i			0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$		0			0
$735$	36.0000	+	18.0000i	1.32788	+	0.663940i
$736$		0			0
$737$		−	2.00000i		−	0.0736709i
$738$		0			0
$739$	8.00000			0.294285			0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000			0.294285			0.147142	+	0.989115i	$0.452992\pi$
$740$		0			0
$741$	−48.0000			−1.76332
$742$		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
$745$	14.0000	−	28.0000i	0.512920	−	1.02584i
$746$		0			0
$747$		−	4.00000i		−	0.146352i
$748$		0			0
$749$	32.0000			1.16925
$750$		0			0
$751$	8.00000			0.291924			0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000			0.291924			0.145962	+	0.989290i	$0.453372\pi$
$752$		0			0
$753$			24.0000i			0.874609i
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	28.0000i		−	1.01768i	−0.860862	−	0.508839i	$-0.830075\pi$
$757$		−	28.0000i		−	1.01768i	0.860862	−	0.508839i	$-0.169925\pi$
$758$		0			0
$759$	12.0000			0.435572
$760$		0			0
$761$	−22.0000			−0.797499			−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000			−0.797499			−0.398750	+	0.917060i	$0.630556\pi$
$762$		0			0
$763$			56.0000i			2.02734i
$764$		0			0
$765$	4.00000	+	2.00000i	0.144620	+	0.0723102i
$766$		0			0
$767$			24.0000i			0.866590i
$768$		0			0
$769$	−14.0000			−0.504853			−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000			−0.504853			−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$	48.0000			1.72868
$772$		0			0
$773$		−	48.0000i		−	1.72644i	−0.504828	−	0.863220i	$-0.668444\pi$
$773$		−	48.0000i		−	1.72644i	0.504828	−	0.863220i	$-0.331556\pi$
$774$		0			0
$775$	24.0000	+	32.0000i	0.862105	+	1.14947i
$776$		0			0
$777$			64.0000i			2.29599i
$778$		0			0
$779$	24.0000			0.859889
$780$		0			0
$781$	−8.00000			−0.286263
$782$		0			0
$783$		−	8.00000i		−	0.285897i
$784$		0			0
$785$	40.0000	+	20.0000i	1.42766	+	0.713831i
$786$		0			0
$787$			24.0000i			0.855508i	0.903895	+	0.427754i	$0.140695\pi$
$787$			24.0000i			0.855508i	−0.903895	+	0.427754i	$0.859305\pi$
$788$		0			0
$789$	48.0000			1.70885
$790$		0			0
$791$	−16.0000			−0.568895
$792$		0			0
$793$			60.0000i			2.13066i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$	20.0000			0.707549
$800$		0			0
$801$	−14.0000			−0.494666
$802$		0			0
$803$		−	2.00000i		−	0.0705785i
$804$		0			0
$805$	24.0000	−	48.0000i	0.845889	−	1.69178i
$806$		0			0
$807$		−	28.0000i		−	0.985647i
$808$		0			0
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$	20.0000			0.702295			0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000			0.702295			0.351147	+	0.936320i	$0.385792\pi$
$812$		0			0
$813$		−	48.0000i		−	1.68343i
$814$		0			0
$815$	−20.0000	−	10.0000i	−0.700569	−	0.350285i
$816$		0			0
$817$		−	48.0000i		−	1.67931i
$818$		0			0
$819$	−24.0000			−0.838628
$820$		0			0
$821$	−18.0000			−0.628204			−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000			−0.628204			−0.314102	+	0.949389i	$0.601703\pi$
$822$		0			0
$823$		−	2.00000i		−	0.0697156i	−0.999392	−	0.0348578i	$-0.988902\pi$
$823$		−	2.00000i		−	0.0697156i	0.999392	−	0.0348578i	$-0.0110978\pi$
$824$		0			0
$825$	8.00000	−	6.00000i	0.278524	−	0.208893i
$826$		0			0
$827$			8.00000i			0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$			8.00000i			0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$		0			0
$829$	26.0000			0.903017			0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000			0.903017			0.451509	+	0.892267i	$0.350886\pi$
$830$		0			0
$831$	28.0000			0.971309
$832$		0			0
$833$			18.0000i			0.623663i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			32.0000i			1.10608i
$838$		0			0
$839$	−48.0000			−1.65714			−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000			−1.65714			−0.828572	+	0.559883i	$0.810846\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		−	44.0000i		−	1.51544i
$844$		0			0
$845$	−23.0000	+	46.0000i	−0.791224	+	1.58245i
$846$		0			0
$847$			4.00000i			0.137442i
$848$		0			0
$849$	−56.0000			−1.92192
$850$		0			0
$851$	48.0000			1.64542
$852$		0			0
$853$			14.0000i			0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$			14.0000i			0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$		0			0
$855$	−4.00000	+	8.00000i	−0.136797	+	0.273594i
$856$		0			0
$857$		−	10.0000i		−	0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$		−	10.0000i		−	0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$		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
$861$	48.0000			1.63584
$862$		0			0
$863$		−	26.0000i		−	0.885050i	−0.896756	−	0.442525i	$-0.854083\pi$
$863$		−	26.0000i		−	0.885050i	0.896756	−	0.442525i	$-0.145917\pi$
$864$		0			0
$865$	44.0000	+	22.0000i	1.49604	+	0.748022i
$866$		0			0
$867$		−	26.0000i		−	0.883006i
$868$		0			0
$869$	−4.00000			−0.135691
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$		−	4.00000i		−	0.135379i
$874$		0			0
$875$	−8.00000	−	44.0000i	−0.270449	−	1.48747i
$876$		0			0
$877$			2.00000i			0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$			2.00000i			0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$		0			0
$879$	52.0000			1.75392
$880$		0			0
$881$	34.0000			1.14549			0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000			1.14549			0.572745	+	0.819734i	$0.305879\pi$
$882$		0			0
$883$			46.0000i			1.54802i	0.633171	+	0.774012i	$0.281753\pi$
$883$			46.0000i			1.54802i	−0.633171	+	0.774012i	$0.718247\pi$
$884$		0			0
$885$	16.0000	+	8.00000i	0.537834	+	0.268917i
$886$		0			0
$887$			48.0000i			1.61168i	0.592132	+	0.805841i	$0.298286\pi$
$887$			48.0000i			1.61168i	−0.592132	+	0.805841i	$0.701714\pi$
$888$		0			0
$889$	−32.0000			−1.07325
$890$		0			0
$891$	11.0000			0.368514
$892$		0			0
$893$			40.0000i			1.33855i
$894$		0			0
$895$	4.00000	−	8.00000i	0.133705	−	0.267411i
$896$		0			0
$897$			72.0000i			2.40401i
$898$		0			0
$899$	−16.0000			−0.533630
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	96.0000i		−	3.19468i
$904$		0			0
$905$	−22.0000	+	44.0000i	−0.731305	+	1.46261i
$906$		0			0
$907$		−	38.0000i		−	1.26177i	−0.775877	−	0.630885i	$-0.782692\pi$
$907$		−	38.0000i		−	1.26177i	0.775877	−	0.630885i	$-0.217308\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		−	4.00000i		−	0.132381i
$914$		0			0
$915$	40.0000	+	20.0000i	1.32236	+	0.661180i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$	16.0000			0.527218
$922$		0			0
$923$		−	48.0000i		−	1.57994i
$924$		0			0
$925$	32.0000	−	24.0000i	1.05215	−	0.789115i
$926$		0			0
$927$			2.00000i			0.0656886i
$928$		0			0
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$	−36.0000			−1.17985
$932$		0			0
$933$		−	16.0000i		−	0.523816i
$934$		0			0
$935$	4.00000	+	2.00000i	0.130814	+	0.0654070i
$936$		0			0
$937$			26.0000i			0.849383i	0.905338	+	0.424691i	$0.139617\pi$
$937$			26.0000i			0.849383i	−0.905338	+	0.424691i	$0.860383\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−30.0000			−0.977972			−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000			−0.977972			−0.488986	+	0.872292i	$0.662633\pi$
$942$		0			0
$943$		−	36.0000i		−	1.17232i
$944$		0			0
$945$	16.0000	−	32.0000i	0.520480	−	1.04096i
$946$		0			0
$947$			2.00000i			0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$			2.00000i			0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$	−56.0000			−1.81592
$952$		0			0
$953$		−	42.0000i		−	1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$		−	42.0000i		−	1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			4.00000i			0.129302i
$958$		0			0
$959$	48.0000			1.55000
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$			8.00000i			0.257796i
$964$		0			0
$965$	−28.0000	−	14.0000i	−0.901352	−	0.450676i
$966$		0			0
$967$			32.0000i			1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$			32.0000i			1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$		0			0
$969$	−16.0000			−0.513994
$970$		0			0
$971$	−36.0000			−1.15529			−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	36.0000	+	48.0000i	1.15292	+	1.53723i
$976$		0			0
$977$			60.0000i			1.91957i	0.280736	+	0.959785i	$0.409421\pi$
$977$			60.0000i			1.91957i	−0.280736	+	0.959785i	$0.590579\pi$
$978$		0			0
$979$	−14.0000			−0.447442
$980$		0			0
$981$	−14.0000			−0.446986
$982$		0			0
$983$		−	42.0000i		−	1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$		−	42.0000i		−	1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$		0			0
$985$	−12.0000	−	6.00000i	−0.382352	−	0.191176i
$986$		0			0
$987$			80.0000i			2.54643i
$988$		0			0
$989$	−72.0000			−2.28947
$990$		0			0
$991$	−24.0000			−0.762385			−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000			−0.762385			−0.381193	+	0.924496i	$0.624487\pi$
$992$		0			0
$993$		−	8.00000i		−	0.253872i
$994$		0			0
$995$	−16.0000	+	32.0000i	−0.507234	+	1.01447i
$996$		0			0
$997$		−	46.0000i		−	1.45683i	−0.685134	−	0.728417i	$-0.740256\pi$
$997$		−	46.0000i		−	1.45683i	0.685134	−	0.728417i	$-0.259744\pi$
$998$		0			0
$999$	32.0000			1.01244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.2.b.d.529.1		2
4.3	odd	2		440.2.b.b.89.2	yes	2
5.2	odd	4		4400.2.a.c.1.1		1
5.3	odd	4		4400.2.a.bb.1.1		1
5.4	even	2	inner	880.2.b.d.529.2		2
12.11	even	2		3960.2.d.b.3169.2		2
20.3	even	4		2200.2.a.b.1.1		1
20.7	even	4		2200.2.a.j.1.1		1
20.19	odd	2		440.2.b.b.89.1	✓	2
60.59	even	2		3960.2.d.b.3169.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.b.89.1	✓	2	20.19	odd	2
440.2.b.b.89.2	yes	2	4.3	odd	2
880.2.b.d.529.1		2	1.1	even	1	trivial
880.2.b.d.529.2		2	5.4	even	2	inner
2200.2.a.b.1.1		1	20.3	even	4
2200.2.a.j.1.1		1	20.7	even	4
3960.2.d.b.3169.1		2	60.59	even	2
3960.2.d.b.3169.2		2	12.11	even	2
4400.2.a.c.1.1		1	5.2	odd	4
4400.2.a.bb.1.1		1	5.3	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

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +(1.00000 - 2.00000i) q^{5} +4.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} -6.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} -2.00000i q^{17} +4.00000 q^{19} +8.00000 q^{21} -6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -4.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} +2.00000i q^{33} +(8.00000 + 4.00000i) q^{35} +8.00000i q^{37} -12.0000 q^{39} +6.00000 q^{41} -12.0000i q^{43} +(-1.00000 + 2.00000i) q^{45} +10.0000i q^{47} -9.00000 q^{49} -4.00000 q^{51} +(-1.00000 + 2.00000i) q^{55} -8.00000i q^{57} -4.00000 q^{59} -10.0000 q^{61} -4.00000i q^{63} +(-12.0000 - 6.00000i) q^{65} +2.00000i q^{67} -12.0000 q^{69} +8.00000 q^{71} +2.00000i q^{73} +(-8.00000 + 6.00000i) q^{75} -4.00000i q^{77} +4.00000 q^{79} -11.0000 q^{81} +4.00000i q^{83} +(-4.00000 - 2.00000i) q^{85} -4.00000i q^{87} +14.0000 q^{89} +24.0000 q^{91} +16.0000i q^{93} +(4.00000 - 8.00000i) q^{95} +4.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9} - 2 q^{11} - 8 q^{15} + 8 q^{19} + 16 q^{21} - 6 q^{25} + 4 q^{29} - 16 q^{31} + 16 q^{35} - 24 q^{39} + 12 q^{41} - 2 q^{45} - 18 q^{49} - 8 q^{51} - 2 q^{55} - 8 q^{59} - 20 q^{61}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^11 - 8 * q^15 + 8 * q^19 + 16 * q^21 - 6 * q^25 + 4 * q^29 - 16 * q^31 + 16 * q^35 - 24 * q^39 + 12 * q^41 - 2 * q^45 - 18 * q^49 - 8 * q^51 - 2 * q^55 - 8 * q^59 - 20 * q^61 - 24 * q^65 - 24 * q^69 + 16 * q^71 - 16 * q^75 + 8 * q^79 - 22 * q^81 - 8 * q^85 + 28 * q^89 + 48 * q^91 + 8 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more