LMFDB - Embedded newform 880.2.b.e.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)

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")

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);

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 110)
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.e.529.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +(1.00000 + 2.00000i) q^{5} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} +(4.00000 - 2.00000i) q^{15} -6.00000i q^{17} +4.00000 q^{19} +2.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} +10.0000 q^{29} +8.00000 q^{31} +2.00000i q^{33} -8.00000i q^{37} -4.00000 q^{39} -2.00000 q^{41} +(-1.00000 - 2.00000i) q^{45} +2.00000i q^{47} +7.00000 q^{49} -12.0000 q^{51} +(-1.00000 - 2.00000i) q^{55} -8.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +(4.00000 - 2.00000i) q^{65} -6.00000i q^{67} +4.00000 q^{69} +6.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +12.0000 q^{79} -11.0000 q^{81} -16.0000i q^{83} +(12.0000 - 6.00000i) q^{85} -20.0000i q^{87} -18.0000 q^{89} -16.0000i q^{93} +(4.00000 + 8.00000i) q^{95} +12.0000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9} - 2 q^{11} + 8 q^{15} + 8 q^{19} - 6 q^{25} + 20 q^{29} + 16 q^{31} - 8 q^{39} - 4 q^{41} - 2 q^{45} + 14 q^{49} - 24 q^{51} - 2 q^{55} - 24 q^{59} - 20 q^{61} + 8 q^{65} + 8 q^{69}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^11 + 8 * q^15 + 8 * q^19 - 6 * q^25 + 20 * q^29 + 16 * q^31 - 8 * q^39 - 4 * q^41 - 2 * q^45 + 14 * q^49 - 24 * q^51 - 2 * q^55 - 24 * q^59 - 20 * q^61 + 8 * q^65 + 8 * q^69 + 16 * q^75 + 24 * q^79 - 22 * q^81 + 24 * q^85 - 36 * q^89 + 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
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$	4.00000	−	2.00000i	1.03280	−	0.516398i
$16$		0			0
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$		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
$21$		0			0
$22$		0			0
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		−	4.00000i		−	0.769800i
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	8.00000			1.43684			0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000			1.43684			0.718421	+	0.695608i	$0.244865\pi$
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	−4.00000			−0.640513
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1.00000	−	2.00000i	−0.149071	−	0.298142i
$46$		0			0
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−12.0000			−1.68034
$52$		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
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		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
$63$		0			0
$64$		0			0
$65$	4.00000	−	2.00000i	0.496139	−	0.248069i
$66$		0			0
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	8.00000	+	6.00000i	0.923760	+	0.692820i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$		−	16.0000i		−	1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$		−	16.0000i		−	1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$		0			0
$85$	12.0000	−	6.00000i	1.30158	−	0.650791i
$86$		0			0
$87$		−	20.0000i		−	2.14423i
$88$		0			0
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	16.0000i		−	1.65912i
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		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
$99$	1.00000			0.100504
$100$		0			0
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	−16.0000			−1.51865
$112$		0			0
$113$			12.0000i			1.12887i	0.825479	+	0.564433i	$0.190905\pi$
$113$			12.0000i			1.12887i	−0.825479	+	0.564433i	$0.809095\pi$
$114$		0			0
$115$	−4.00000	+	2.00000i	−0.373002	+	0.186501i
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$		−	4.00000i		−	0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$		−	4.00000i		−	0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.00000			0.698963			0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000			0.698963			0.349482	+	0.936943i	$0.386358\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	8.00000	−	4.00000i	0.688530	−	0.344265i
$136$		0			0
$137$			12.0000i			1.02523i	0.858619	+	0.512615i	$0.171323\pi$
$137$			12.0000i			1.02523i	−0.858619	+	0.512615i	$0.828677\pi$
$138$		0			0
$139$	8.00000			0.678551			0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000			0.678551			0.339276	+	0.940687i	$0.389818\pi$
$140$		0			0
$141$	4.00000			0.336861
$142$		0			0
$143$			2.00000i			0.167248i
$144$		0			0
$145$	10.0000	+	20.0000i	0.830455	+	1.66091i
$146$		0			0
$147$		−	14.0000i		−	1.15470i
$148$		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	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$	8.00000	+	16.0000i	0.642575	+	1.28515i
$156$		0			0
$157$		−	4.00000i		−	0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$		−	4.00000i		−	0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$		0			0
$165$	−4.00000	+	2.00000i	−0.311400	+	0.155700i
$166$		0			0
$167$			20.0000i			1.54765i	0.633402	+	0.773823i	$0.281658\pi$
$167$			20.0000i			1.54765i	−0.633402	+	0.773823i	$0.718342\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	−4.00000			−0.305888
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			24.0000i			1.80395i
$178$		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
$181$	−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
$183$			20.0000i			1.47844i
$184$		0			0
$185$	16.0000	−	8.00000i	1.17634	−	0.588172i
$186$		0			0
$187$			6.00000i			0.438763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−16.0000			−1.15772			−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000			−1.15772			−0.578860	+	0.815427i	$0.696502\pi$
$192$		0			0
$193$		−	10.0000i		−	0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$		−	10.0000i		−	0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$		0			0
$195$	−4.00000	−	8.00000i	−0.286446	−	0.572892i
$196$		0			0
$197$		−	2.00000i		−	0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$		−	2.00000i		−	0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$		0			0
$201$	−12.0000			−0.846415
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2.00000	−	4.00000i	−0.139686	−	0.279372i
$206$		0			0
$207$		−	2.00000i		−	0.139010i
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	12.0000			0.810885
$220$		0			0
$221$	−12.0000			−0.807207
$222$		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
$227$			24.0000i			1.59294i	0.604681	+	0.796468i	$0.293301\pi$
$227$			24.0000i			1.59294i	−0.604681	+	0.796468i	$0.706699\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	−4.00000	+	2.00000i	−0.260931	+	0.130466i
$236$		0			0
$237$		−	24.0000i		−	1.55897i
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$			10.0000i			0.641500i
$244$		0			0
$245$	7.00000	+	14.0000i	0.447214	+	0.894427i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$	−32.0000			−2.02792
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$	−12.0000	−	24.0000i	−0.751469	−	1.50294i
$256$		0			0
$257$			8.00000i			0.499026i	0.968371	+	0.249513i	$0.0802706\pi$
$257$			8.00000i			0.499026i	−0.968371	+	0.249513i	$0.919729\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−10.0000			−0.618984
$262$		0			0
$263$			12.0000i			0.739952i	0.929041	+	0.369976i	$0.120634\pi$
$263$			12.0000i			0.739952i	−0.929041	+	0.369976i	$0.879366\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$			36.0000i			2.20316i
$268$		0			0
$269$	−18.0000			−1.09748			−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000			−1.09748			−0.548740	+	0.835993i	$0.684892\pi$
$270$		0			0
$271$	8.00000			0.485965			0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000			0.485965			0.242983	+	0.970031i	$0.421874\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	−	4.00000i	0.180907	−	0.241209i
$276$		0			0
$277$		−	22.0000i		−	1.32185i	−0.750451	−	0.660926i	$-0.770164\pi$
$277$		−	22.0000i		−	1.32185i	0.750451	−	0.660926i	$-0.229836\pi$
$278$		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
$283$		−	32.0000i		−	1.90220i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$		−	32.0000i		−	1.90220i	0.308879	−	0.951101i	$-0.400046\pi$
$284$		0			0
$285$	16.0000	−	8.00000i	0.947758	−	0.473879i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	24.0000			1.40690
$292$		0			0
$293$			14.0000i			0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$			14.0000i			0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$		0			0
$295$	−12.0000	−	24.0000i	−0.698667	−	1.39733i
$296$		0			0
$297$			4.00000i			0.232104i
$298$		0			0
$299$	4.00000			0.231326
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	4.00000i		−	0.229794i
$304$		0			0
$305$	−10.0000	−	20.0000i	−0.572598	−	1.14520i
$306$		0			0
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$	28.0000			1.59286
$310$		0			0
$311$	−32.0000			−1.81455			−0.907277	−	0.420534i	$-0.861843\pi$
$311$	−32.0000			−1.81455			−0.907277	+	0.420534i	$0.861843\pi$
$312$		0			0
$313$		−	16.0000i		−	0.904373i	−0.891923	−	0.452187i	$-0.850644\pi$
$313$		−	16.0000i		−	0.904373i	0.891923	−	0.452187i	$-0.149356\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.0000i			1.57264i	0.617822	+	0.786318i	$0.288015\pi$
$317$			28.0000i			1.57264i	−0.617822	+	0.786318i	$0.711985\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$	8.00000	+	6.00000i	0.443760	+	0.332820i
$326$		0			0
$327$			4.00000i			0.221201i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.0000			−0.659580			−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000			−0.659580			−0.329790	+	0.944054i	$0.606978\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	12.0000	−	6.00000i	0.655630	−	0.327815i
$336$		0			0
$337$			10.0000i			0.544735i	0.962193	+	0.272367i	$0.0878066\pi$
$337$			10.0000i			0.544735i	−0.962193	+	0.272367i	$0.912193\pi$
$338$		0			0
$339$	24.0000			1.30350
$340$		0			0
$341$	−8.00000			−0.433224
$342$		0			0
$343$		0			0
$344$		0			0
$345$	4.00000	+	8.00000i	0.215353	+	0.430706i
$346$		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
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$	−8.00000			−0.427008
$352$		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
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		−	2.00000i		−	0.104973i
$364$		0			0
$365$	−12.0000	+	6.00000i	−0.628109	+	0.314054i
$366$		0			0
$367$			26.0000i			1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$			26.0000i			1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$		0			0
$369$	2.00000			0.104116
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$	−4.00000	+	22.0000i	−0.206559	+	1.13608i
$376$		0			0
$377$		−	20.0000i		−	1.03005i
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$	−8.00000			−0.409852
$382$		0			0
$383$		−	18.0000i		−	0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$		−	18.0000i		−	0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		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
$393$		−	16.0000i		−	0.807093i
$394$		0			0
$395$	12.0000	+	24.0000i	0.603786	+	1.20757i
$396$		0			0
$397$			28.0000i			1.40528i	0.711546	+	0.702640i	$0.247995\pi$
$397$			28.0000i			1.40528i	−0.711546	+	0.702640i	$0.752005\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		−	16.0000i		−	0.797017i
$404$		0			0
$405$	−11.0000	−	22.0000i	−0.546594	−	1.09319i
$406$		0			0
$407$			8.00000i			0.396545i
$408$		0			0
$409$	−2.00000			−0.0988936			−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000			−0.0988936			−0.0494468	+	0.998777i	$0.515746\pi$
$410$		0			0
$411$	24.0000			1.18383
$412$		0			0
$413$		0			0
$414$		0			0
$415$	32.0000	−	16.0000i	1.57082	−	0.785409i
$416$		0			0
$417$		−	16.0000i		−	0.783523i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	24.0000	+	18.0000i	1.16417	+	0.873128i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	28.0000			1.34871			0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000			1.34871			0.674356	+	0.738406i	$0.264421\pi$
$432$		0			0
$433$		−	4.00000i		−	0.192228i	−0.995370	−	0.0961139i	$-0.969359\pi$
$433$		−	4.00000i		−	0.192228i	0.995370	−	0.0961139i	$-0.0306413\pi$
$434$		0			0
$435$	40.0000	−	20.0000i	1.91785	−	0.958927i
$436$		0			0
$437$			8.00000i			0.382692i
$438$		0			0
$439$	16.0000			0.763638			0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000			0.763638			0.381819	+	0.924237i	$0.375298\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		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
$445$	−18.0000	−	36.0000i	−0.853282	−	1.70656i
$446$		0			0
$447$		−	28.0000i		−	1.32435i
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$	2.00000			0.0941763
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	18.0000i		−	0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$		−	18.0000i		−	0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$		0			0
$459$	−24.0000			−1.12022
$460$		0			0
$461$	2.00000			0.0931493			0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000			0.0931493			0.0465746	+	0.998915i	$0.485169\pi$
$462$		0			0
$463$			34.0000i			1.58011i	0.613033	+	0.790057i	$0.289949\pi$
$463$			34.0000i			1.58011i	−0.613033	+	0.790057i	$0.710051\pi$
$464$		0			0
$465$	32.0000	−	16.0000i	1.48396	−	0.741982i
$466$		0			0
$467$		−	6.00000i		−	0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$		−	6.00000i		−	0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−8.00000			−0.368621
$472$		0			0
$473$		0			0
$474$		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	$-0.529129\pi$
$479$	−4.00000			−0.182765			−0.0913823	+	0.995816i	$0.529129\pi$
$480$		0			0
$481$	−16.0000			−0.729537
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−24.0000	+	12.0000i	−1.08978	+	0.544892i
$486$		0			0
$487$			2.00000i			0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$			2.00000i			0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$		0			0
$489$	12.0000			0.542659
$490$		0			0
$491$	−36.0000			−1.62466			−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000			−1.62466			−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$		−	60.0000i		−	2.70226i
$494$		0			0
$495$	1.00000	+	2.00000i	0.0449467	+	0.0898933i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	20.0000			0.895323			0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000			0.895323			0.447661	+	0.894203i	$0.352257\pi$
$500$		0			0
$501$	40.0000			1.78707
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	2.00000	+	4.00000i	0.0889988	+	0.177998i
$506$		0			0
$507$		−	18.0000i		−	0.799408i
$508$		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
$511$		0			0
$512$		0			0
$513$		−	16.0000i		−	0.706417i
$514$		0			0
$515$	−28.0000	+	14.0000i	−1.23383	+	0.616914i
$516$		0			0
$517$		−	2.00000i		−	0.0879599i
$518$		0			0
$519$	4.00000			0.175581
$520$		0			0
$521$	22.0000			0.963837			0.481919	−	0.876216i	$-0.339940\pi$
$521$	22.0000			0.963837			0.481919	+	0.876216i	$0.339940\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	48.0000i		−	2.09091i
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$			4.00000i			0.173259i
$534$		0			0
$535$	−24.0000	+	12.0000i	−1.03761	+	0.518805i
$536$		0			0
$537$		−	8.00000i		−	0.345225i
$538$		0			0
$539$	−7.00000			−0.301511
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		0			0
$543$			12.0000i			0.514969i
$544$		0			0
$545$	−2.00000	−	4.00000i	−0.0856706	−	0.171341i
$546$		0			0
$547$		−	16.0000i		−	0.684111i	−0.939680	−	0.342055i	$-0.888877\pi$
$547$		−	16.0000i		−	0.684111i	0.939680	−	0.342055i	$-0.111123\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	40.0000			1.70406
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−16.0000	−	32.0000i	−0.679162	−	1.35832i
$556$		0			0
$557$			38.0000i			1.61011i	0.593199	+	0.805056i	$0.297865\pi$
$557$			38.0000i			1.61011i	−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	−24.0000	+	12.0000i	−1.00969	+	0.504844i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−26.0000			−1.08998			−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000			−1.08998			−0.544988	+	0.838444i	$0.683466\pi$
$570$		0			0
$571$	−12.0000			−0.502184			−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000			−0.502184			−0.251092	+	0.967963i	$0.580790\pi$
$572$		0			0
$573$			32.0000i			1.33682i
$574$		0			0
$575$	−8.00000	−	6.00000i	−0.333623	−	0.250217i
$576$		0			0
$577$			4.00000i			0.166522i	0.996528	+	0.0832611i	$0.0265335\pi$
$577$			4.00000i			0.166522i	−0.996528	+	0.0832611i	$0.973466\pi$
$578$		0			0
$579$	−20.0000			−0.831172
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−4.00000	+	2.00000i	−0.165380	+	0.0826898i
$586$		0			0
$587$		−	22.0000i		−	0.908037i	−0.890992	−	0.454019i	$-0.849990\pi$
$587$		−	22.0000i		−	0.908037i	0.890992	−	0.454019i	$-0.150010\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$	−4.00000			−0.164538
$592$		0			0
$593$		−	18.0000i		−	0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$		−	18.0000i		−	0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			16.0000i			0.654836i
$598$		0			0
$599$	−8.00000			−0.326871			−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000			−0.326871			−0.163436	+	0.986554i	$0.552258\pi$
$600$		0			0
$601$	−34.0000			−1.38689			−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000			−1.38689			−0.693444	+	0.720510i	$0.743908\pi$
$602$		0			0
$603$			6.00000i			0.244339i
$604$		0			0
$605$	1.00000	+	2.00000i	0.0406558	+	0.0813116i
$606$		0			0
$607$		−	28.0000i		−	1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$		−	28.0000i		−	1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.00000			0.161823
$612$		0			0
$613$		−	6.00000i		−	0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$		−	6.00000i		−	0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$		0			0
$615$	−8.00000	+	4.00000i	−0.322591	+	0.161296i
$616$		0			0
$617$		−	36.0000i		−	1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$		−	36.0000i		−	1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$			8.00000i			0.319489i
$628$		0			0
$629$	−48.0000			−1.91389
$630$		0			0
$631$	40.0000			1.59237			0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000			1.59237			0.796187	+	0.605050i	$0.206847\pi$
$632$		0			0
$633$			16.0000i			0.635943i
$634$		0			0
$635$	8.00000	−	4.00000i	0.317470	−	0.158735i
$636$		0			0
$637$		−	14.0000i		−	0.554700i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$		−	26.0000i		−	1.02534i	−0.858586	−	0.512670i	$-0.828656\pi$
$643$		−	26.0000i		−	1.02534i	0.858586	−	0.512670i	$-0.171344\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.0000i			1.17942i	0.807614	+	0.589711i	$0.200758\pi$
$647$			30.0000i			1.17942i	−0.807614	+	0.589711i	$0.799242\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$		0			0
$652$		0			0
$653$			24.0000i			0.939193i	0.882881	+	0.469596i	$0.155601\pi$
$653$			24.0000i			0.939193i	−0.882881	+	0.469596i	$0.844399\pi$
$654$		0			0
$655$	8.00000	+	16.0000i	0.312586	+	0.625172i
$656$		0			0
$657$		−	6.00000i		−	0.234082i
$658$		0			0
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$	−42.0000			−1.63361			−0.816805	−	0.576913i	$-0.804257\pi$
$661$	−42.0000			−1.63361			−0.816805	+	0.576913i	$0.804257\pi$
$662$		0			0
$663$			24.0000i			0.932083i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			20.0000i			0.774403i
$668$		0			0
$669$	−28.0000			−1.08254
$670$		0			0
$671$	10.0000			0.386046
$672$		0			0
$673$		−	10.0000i		−	0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$		−	10.0000i		−	0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$		0			0
$675$	16.0000	+	12.0000i	0.615840	+	0.461880i
$676$		0			0
$677$		−	14.0000i		−	0.538064i	−0.963131	−	0.269032i	$-0.913296\pi$
$677$		−	14.0000i		−	0.538064i	0.963131	−	0.269032i	$-0.0867037\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	48.0000			1.83936
$682$		0			0
$683$			26.0000i			0.994862i	0.867503	+	0.497431i	$0.165723\pi$
$683$			26.0000i			0.994862i	−0.867503	+	0.497431i	$0.834277\pi$
$684$		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
$691$	20.0000			0.760836			0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000			0.760836			0.380418	+	0.924815i	$0.375780\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	+	16.0000i	0.303457	+	0.606915i
$696$		0			0
$697$			12.0000i			0.454532i
$698$		0			0
$699$	20.0000			0.756469
$700$		0			0
$701$	6.00000			0.226617			0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000			0.226617			0.113308	+	0.993560i	$0.463855\pi$
$702$		0			0
$703$		−	32.0000i		−	1.20690i
$704$		0			0
$705$	4.00000	+	8.00000i	0.150649	+	0.301297i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$	−4.00000	+	2.00000i	−0.149592	+	0.0747958i
$716$		0			0
$717$		−	24.0000i		−	0.896296i
$718$		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
$721$		0			0
$722$		0			0
$723$			4.00000i			0.148762i
$724$		0			0
$725$	−30.0000	+	40.0000i	−1.11417	+	1.48556i
$726$		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
$731$		0			0
$732$		0			0
$733$			50.0000i			1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$			50.0000i			1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$		0			0
$735$	28.0000	−	14.0000i	1.03280	−	0.516398i
$736$		0			0
$737$			6.00000i			0.221013i
$738$		0			0
$739$	−32.0000			−1.17714			−0.588570	−	0.808447i	$-0.700309\pi$
$739$	−32.0000			−1.17714			−0.588570	+	0.808447i	$0.700309\pi$
$740$		0			0
$741$	−16.0000			−0.587775
$742$		0			0
$743$			28.0000i			1.02722i	0.858024	+	0.513610i	$0.171692\pi$
$743$			28.0000i			1.02722i	−0.858024	+	0.513610i	$0.828308\pi$
$744$		0			0
$745$	14.0000	+	28.0000i	0.512920	+	1.02584i
$746$		0			0
$747$			16.0000i			0.585409i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$			40.0000i			1.45768i
$754$		0			0
$755$	−8.00000	−	16.0000i	−0.291150	−	0.582300i
$756$		0			0
$757$			12.0000i			0.436147i	0.975932	+	0.218074i	$0.0699773\pi$
$757$			12.0000i			0.436147i	−0.975932	+	0.218074i	$0.930023\pi$
$758$		0			0
$759$	−4.00000			−0.145191
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−12.0000	+	6.00000i	−0.433861	+	0.216930i
$766$		0			0
$767$			24.0000i			0.866590i
$768$		0			0
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	16.0000			0.576226
$772$		0			0
$773$		−	32.0000i		−	1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$		−	32.0000i		−	1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$		0			0
$775$	−24.0000	+	32.0000i	−0.862105	+	1.14947i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	40.0000i		−	1.42948i
$784$		0			0
$785$	8.00000	−	4.00000i	0.285532	−	0.142766i
$786$		0			0
$787$		−	52.0000i		−	1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$		−	52.0000i		−	1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$		0			0
$792$		0			0
$793$			20.0000i			0.710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$	18.0000			0.635999
$802$		0			0
$803$		−	6.00000i		−	0.211735i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			36.0000i			1.26726i
$808$		0			0
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		−	16.0000i		−	0.561144i
$814$		0			0
$815$	−12.0000	+	6.00000i	−0.420342	+	0.210171i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−2.00000			−0.0698005			−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000			−0.0698005			−0.0349002	+	0.999391i	$0.511111\pi$
$822$		0			0
$823$		−	34.0000i		−	1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$		−	34.0000i		−	1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$		0			0
$825$	−8.00000	−	6.00000i	−0.278524	−	0.208893i
$826$		0			0
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$		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
$831$	−44.0000			−1.52634
$832$		0			0
$833$		−	42.0000i		−	1.45521i
$834$		0			0
$835$	−40.0000	+	20.0000i	−1.38426	+	0.692129i
$836$		0			0
$837$		−	32.0000i		−	1.10608i
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$			36.0000i			1.23991i
$844$		0			0
$845$	9.00000	+	18.0000i	0.309609	+	0.619219i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−64.0000			−2.19647
$850$		0			0
$851$	16.0000			0.548473
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$		0			0
$855$	−4.00000	−	8.00000i	−0.136797	−	0.273594i
$856$		0			0
$857$		−	46.0000i		−	1.57133i	−0.618652	−	0.785665i	$-0.712321\pi$
$857$		−	46.0000i		−	1.57133i	0.618652	−	0.785665i	$-0.287679\pi$
$858$		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
$861$		0			0
$862$		0			0
$863$			6.00000i			0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$			6.00000i			0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$		0			0
$865$	−4.00000	+	2.00000i	−0.136004	+	0.0680020i
$866$		0			0
$867$			38.0000i			1.29055i
$868$		0			0
$869$	−12.0000			−0.407072
$870$		0			0
$871$	−12.0000			−0.406604
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			38.0000i			1.28317i	0.767052	+	0.641584i	$0.221723\pi$
$877$			38.0000i			1.28317i	−0.767052	+	0.641584i	$0.778277\pi$
$878$		0			0
$879$	28.0000			0.944417
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$			30.0000i			1.00958i	0.863242	+	0.504790i	$0.168430\pi$
$883$			30.0000i			1.00958i	−0.863242	+	0.504790i	$0.831570\pi$
$884$		0			0
$885$	−48.0000	+	24.0000i	−1.61350	+	0.806751i
$886$		0			0
$887$			4.00000i			0.134307i	0.997743	+	0.0671534i	$0.0213917\pi$
$887$			4.00000i			0.134307i	−0.997743	+	0.0671534i	$0.978608\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	11.0000			0.368514
$892$		0			0
$893$			8.00000i			0.267710i
$894$		0			0
$895$	4.00000	+	8.00000i	0.133705	+	0.267411i
$896$		0			0
$897$		−	8.00000i		−	0.267112i
$898$		0			0
$899$	80.0000			2.66815
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.00000	−	12.0000i	−0.199447	−	0.398893i
$906$		0			0
$907$			50.0000i			1.66022i	0.557598	+	0.830111i	$0.311723\pi$
$907$			50.0000i			1.66022i	−0.557598	+	0.830111i	$0.688277\pi$
$908$		0			0
$909$	−2.00000			−0.0663358
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$			16.0000i			0.529523i
$914$		0			0
$915$	−40.0000	+	20.0000i	−1.32236	+	0.661180i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$	−8.00000			−0.263609
$922$		0			0
$923$		0			0
$924$		0			0
$925$	32.0000	+	24.0000i	1.05215	+	0.789115i
$926$		0			0
$927$		−	14.0000i		−	0.459820i
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	28.0000			0.917663
$932$		0			0
$933$			64.0000i			2.09527i
$934$		0			0
$935$	−12.0000	+	6.00000i	−0.392442	+	0.196221i
$936$		0			0
$937$		−	18.0000i		−	0.588034i	−0.955800	−	0.294017i	$-0.905008\pi$
$937$		−	18.0000i		−	0.588034i	0.955800	−	0.294017i	$-0.0949923\pi$
$938$		0			0
$939$	−32.0000			−1.04428
$940$		0			0
$941$	−22.0000			−0.717180			−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000			−0.717180			−0.358590	+	0.933495i	$0.616742\pi$
$942$		0			0
$943$		−	4.00000i		−	0.130258i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	22.0000i		−	0.714904i	−0.933932	−	0.357452i	$-0.883646\pi$
$947$		−	22.0000i		−	0.714904i	0.933932	−	0.357452i	$-0.116354\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$	56.0000			1.81592
$952$		0			0
$953$		−	46.0000i		−	1.49009i	−0.667016	−	0.745043i	$-0.732429\pi$
$953$		−	46.0000i		−	1.49009i	0.667016	−	0.745043i	$-0.267571\pi$
$954$		0			0
$955$	−16.0000	−	32.0000i	−0.517748	−	1.03550i
$956$		0			0
$957$			20.0000i			0.646508i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	33.0000			1.06452
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$	20.0000	−	10.0000i	0.643823	−	0.321911i
$966$		0			0
$967$			4.00000i			0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$			4.00000i			0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$		0			0
$969$	−48.0000			−1.54198
$970$		0			0
$971$	−20.0000			−0.641831			−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000			−0.641831			−0.320915	+	0.947108i	$0.603990\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	12.0000	−	16.0000i	0.384308	−	0.512410i
$976$		0			0
$977$			20.0000i			0.639857i	0.947442	+	0.319928i	$0.103659\pi$
$977$			20.0000i			0.639857i	−0.947442	+	0.319928i	$0.896341\pi$
$978$		0			0
$979$	18.0000			0.575282
$980$		0			0
$981$	2.00000			0.0638551
$982$		0			0
$983$			22.0000i			0.701691i	0.936433	+	0.350846i	$0.114106\pi$
$983$			22.0000i			0.701691i	−0.936433	+	0.350846i	$0.885894\pi$
$984$		0			0
$985$	4.00000	−	2.00000i	0.127451	−	0.0637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−40.0000			−1.27064			−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000			−1.27064			−0.635321	+	0.772248i	$0.719132\pi$
$992$		0			0
$993$			24.0000i			0.761617i
$994$		0			0
$995$	−8.00000	−	16.0000i	−0.253617	−	0.507234i
$996$		0			0
$997$		−	58.0000i		−	1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$		−	58.0000i		−	1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$		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.e.529.1		2
4.3	odd	2		110.2.b.b.89.1	✓	2
5.2	odd	4		4400.2.a.f.1.1		1
5.3	odd	4		4400.2.a.ba.1.1		1
5.4	even	2	inner	880.2.b.e.529.2		2
12.11	even	2		990.2.c.c.199.2		2
20.3	even	4		550.2.a.c.1.1		1
20.7	even	4		550.2.a.k.1.1		1
20.19	odd	2		110.2.b.b.89.2	yes	2
44.43	even	2		1210.2.b.d.969.2		2
60.23	odd	4		4950.2.a.bj.1.1		1
60.47	odd	4		4950.2.a.j.1.1		1
60.59	even	2		990.2.c.c.199.1		2
220.43	odd	4		6050.2.a.x.1.1		1
220.87	odd	4		6050.2.a.q.1.1		1
220.219	even	2		1210.2.b.d.969.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.b.b.89.1	✓	2	4.3	odd	2
110.2.b.b.89.2	yes	2	20.19	odd	2
550.2.a.c.1.1		1	20.3	even	4
550.2.a.k.1.1		1	20.7	even	4
880.2.b.e.529.1		2	1.1	even	1	trivial
880.2.b.e.529.2		2	5.4	even	2	inner
990.2.c.c.199.1		2	60.59	even	2
990.2.c.c.199.2		2	12.11	even	2
1210.2.b.d.969.1		2	220.219	even	2
1210.2.b.d.969.2		2	44.43	even	2
4400.2.a.f.1.1		1	5.2	odd	4
4400.2.a.ba.1.1		1	5.3	odd	4
4950.2.a.j.1.1		1	60.47	odd	4
4950.2.a.bj.1.1		1	60.23	odd	4
6050.2.a.q.1.1		1	220.87	odd	4
6050.2.a.x.1.1		1	220.43	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)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +(1.00000 + 2.00000i) q^{5} -1.00000 q^{9} -1.00000 q^{11} -2.00000i q^{13} +(4.00000 - 2.00000i) q^{15} -6.00000i q^{17} +4.00000 q^{19} +2.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{27} +10.0000 q^{29} +8.00000 q^{31} +2.00000i q^{33} -8.00000i q^{37} -4.00000 q^{39} -2.00000 q^{41} +(-1.00000 - 2.00000i) q^{45} +2.00000i q^{47} +7.00000 q^{49} -12.0000 q^{51} +(-1.00000 - 2.00000i) q^{55} -8.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +(4.00000 - 2.00000i) q^{65} -6.00000i q^{67} +4.00000 q^{69} +6.00000i q^{73} +(8.00000 + 6.00000i) q^{75} +12.0000 q^{79} -11.0000 q^{81} -16.0000i q^{83} +(12.0000 - 6.00000i) q^{85} -20.0000i q^{87} -18.0000 q^{89} -16.0000i q^{93} +(4.00000 + 8.00000i) q^{95} +12.0000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9} - 2 q^{11} + 8 q^{15} + 8 q^{19} - 6 q^{25} + 20 q^{29} + 16 q^{31} - 8 q^{39} - 4 q^{41} - 2 q^{45} + 14 q^{49} - 24 q^{51} - 2 q^{55} - 24 q^{59} - 20 q^{61} + 8 q^{65} + 8 q^{69}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 - 2 * q^11 + 8 * q^15 + 8 * q^19 - 6 * q^25 + 20 * q^29 + 16 * q^31 - 8 * q^39 - 4 * q^41 - 2 * q^45 + 14 * q^49 - 24 * q^51 - 2 * q^55 - 24 * q^59 - 20 * q^61 + 8 * q^65 + 8 * q^69 + 16 * q^75 + 24 * q^79 - 22 * q^81 + 24 * q^85 - 36 * q^89 + 8 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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