LMFDB - Embedded newform 80.2.c.a.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [80,2,Mod(49,80)] 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("80.49"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(80, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	80.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.638803216170$
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 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	80.49
Dual form			80.2.c.a.49.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -4.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} -4.00000 q^{19} +4.00000 q^{21} -2.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +4.00000i q^{27} -2.00000 q^{29} +8.00000i q^{33} +(4.00000 + 2.00000i) q^{35} +4.00000i q^{37} +8.00000 q^{39} +2.00000 q^{41} -6.00000i q^{43} +(1.00000 - 2.00000i) q^{45} +6.00000i q^{47} +3.00000 q^{49} +4.00000i q^{53} +(-4.00000 + 8.00000i) q^{55} -8.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +2.00000i q^{63} +(8.00000 + 4.00000i) q^{65} -14.0000i q^{67} +4.00000 q^{69} -8.00000 q^{71} -8.00000i q^{73} +(8.00000 - 6.00000i) q^{75} -8.00000i q^{77} +16.0000 q^{79} -11.0000 q^{81} +2.00000i q^{83} -4.00000i q^{87} -6.00000 q^{89} -8.00000 q^{91} +(4.00000 - 8.00000i) q^{95} +16.0000i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{9} + 8 q^{11} - 8 q^{15} - 8 q^{19} + 8 q^{21} - 6 q^{25} - 4 q^{29} + 8 q^{35} + 16 q^{39} + 4 q^{41} + 2 q^{45} + 6 q^{49} - 8 q^{55} - 24 q^{59} - 20 q^{61} + 16 q^{65} + 8 q^{69}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^9 + 8 * q^11 - 8 * q^15 - 8 * q^19 + 8 * q^21 - 6 * q^25 - 4 * q^29 + 8 * q^35 + 16 * q^39 + 4 * q^41 + 2 * q^45 + 6 * q^49 - 8 * q^55 - 24 * q^59 - 20 * q^61 + 16 * q^65 + 8 * q^69 - 16 * q^71 + 16 * q^75 + 32 * q^79 - 22 * q^81 - 12 * q^89 - 16 * q^91 + 8 * q^95 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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	$0.195913\pi$
$3$			2.00000i			1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$		0			0
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$		−	4.00000i		−	1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$		−	4.00000i		−	1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$		0			0
$15$	−4.00000	−	2.00000i	−1.03280	−	0.516398i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$	4.00000			0.872872
$22$		0			0
$23$		−	2.00000i		−	0.417029i	−0.978019	−	0.208514i	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		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	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$			8.00000i			1.39262i
$34$		0			0
$35$	4.00000	+	2.00000i	0.676123	+	0.338062i
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$	8.00000			1.28103
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$		−	6.00000i		−	0.914991i	−0.889212	−	0.457496i	$-0.848747\pi$
$43$		−	6.00000i		−	0.914991i	0.889212	−	0.457496i	$-0.151253\pi$
$44$		0			0
$45$	1.00000	−	2.00000i	0.149071	−	0.298142i
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$	−4.00000	+	8.00000i	−0.539360	+	1.07872i
$56$		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
$63$			2.00000i			0.251976i
$64$		0			0
$65$	8.00000	+	4.00000i	0.992278	+	0.496139i
$66$		0			0
$67$		−	14.0000i		−	1.71037i	−0.518321	−	0.855186i	$-0.673443\pi$
$67$		−	14.0000i		−	1.71037i	0.518321	−	0.855186i	$-0.326557\pi$
$68$		0			0
$69$	4.00000			0.481543
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$		0			0
$75$	8.00000	−	6.00000i	0.923760	−	0.692820i
$76$		0			0
$77$		−	8.00000i		−	0.911685i
$78$		0			0
$79$	16.0000			1.80014			0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000			1.80014			0.900070	+	0.435745i	$0.143515\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$			2.00000i			0.219529i	0.993958	+	0.109764i	$0.0350096\pi$
$83$			2.00000i			0.219529i	−0.993958	+	0.109764i	$0.964990\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	4.00000i		−	0.428845i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−8.00000			−0.838628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	8.00000i	0.410391	−	0.820783i
$96$		0			0
$97$			16.0000i			1.62455i	0.583272	+	0.812277i	$0.301772\pi$
$97$			16.0000i			1.62455i	−0.583272	+	0.812277i	$0.698228\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		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
$105$	−4.00000	+	8.00000i	−0.390360	+	0.780720i
$106$		0			0
$107$			10.0000i			0.966736i	0.875417	+	0.483368i	$0.160587\pi$
$107$			10.0000i			0.966736i	−0.875417	+	0.483368i	$0.839413\pi$
$108$		0			0
$109$	6.00000			0.574696			0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000			0.574696			0.287348	+	0.957826i	$0.407226\pi$
$110$		0			0
$111$	−8.00000			−0.759326
$112$		0			0
$113$		−	16.0000i		−	1.50515i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$		−	16.0000i		−	1.50515i	0.658505	−	0.752577i	$-0.271189\pi$
$114$		0			0
$115$	4.00000	+	2.00000i	0.373002	+	0.186501i
$116$		0			0
$117$			4.00000i			0.369800i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$			4.00000i			0.360668i
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$	12.0000			1.05654
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$			8.00000i			0.693688i
$134$		0			0
$135$	−8.00000	−	4.00000i	−0.688530	−	0.344265i
$136$		0			0
$137$			8.00000i			0.683486i	0.939793	+	0.341743i	$0.111017\pi$
$137$			8.00000i			0.683486i	−0.939793	+	0.341743i	$0.888983\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$	−12.0000			−1.01058
$142$		0			0
$143$		−	16.0000i		−	1.33799i
$144$		0			0
$145$	2.00000	−	4.00000i	0.166091	−	0.332182i
$146$		0			0
$147$			6.00000i			0.494872i
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		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
$153$		0			0
$154$		0			0
$155$		0			0
$156$		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
$159$	−8.00000			−0.634441
$160$		0			0
$161$	−4.00000			−0.315244
$162$		0			0
$163$			2.00000i			0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$			2.00000i			0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$		0			0
$165$	−16.0000	−	8.00000i	−1.24560	−	0.622799i
$166$		0			0
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$			12.0000i			0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$			12.0000i			0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$		0			0
$175$	−8.00000	+	6.00000i	−0.604743	+	0.453557i
$176$		0			0
$177$		−	24.0000i		−	1.80395i
$178$		0			0
$179$	−4.00000			−0.298974			−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000			−0.298974			−0.149487	+	0.988764i	$0.547762\pi$
$180$		0			0
$181$	22.0000			1.63525			0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$		−	20.0000i		−	1.47844i
$184$		0			0
$185$	−8.00000	−	4.00000i	−0.588172	−	0.294086i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.00000			0.581914
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			16.0000i			1.15171i	0.817554	+	0.575853i	$0.195330\pi$
$193$			16.0000i			1.15171i	−0.817554	+	0.575853i	$0.804670\pi$
$194$		0			0
$195$	−8.00000	+	16.0000i	−0.572892	+	1.14578i
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		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
$201$	28.0000			1.97497
$202$		0			0
$203$			4.00000i			0.280745i
$204$		0			0
$205$	−2.00000	+	4.00000i	−0.139686	+	0.279372i
$206$		0			0
$207$			2.00000i			0.139010i
$208$		0			0
$209$	−16.0000			−1.10674
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		−	16.0000i		−	1.09630i
$214$		0			0
$215$	12.0000	+	6.00000i	0.818393	+	0.409197i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	16.0000			1.08118
$220$		0			0
$221$		0			0
$222$		0			0
$223$			6.00000i			0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$			6.00000i			0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$		0			0
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$			2.00000i			0.132745i	0.997795	+	0.0663723i	$0.0211425\pi$
$227$			2.00000i			0.132745i	−0.997795	+	0.0663723i	$0.978857\pi$
$228$		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
$231$	16.0000			1.05272
$232$		0			0
$233$			24.0000i			1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$			24.0000i			1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$		0			0
$235$	−12.0000	−	6.00000i	−0.782794	−	0.391397i
$236$		0			0
$237$			32.0000i			2.07862i
$238$		0			0
$239$	−16.0000			−1.03495			−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000			−1.03495			−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$	−22.0000			−1.41714			−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000			−1.41714			−0.708572	+	0.705638i	$0.750660\pi$
$242$		0			0
$243$		−	10.0000i		−	0.641500i
$244$		0			0
$245$	−3.00000	+	6.00000i	−0.191663	+	0.383326i
$246$		0			0
$247$			16.0000i			1.01806i
$248$		0			0
$249$	−4.00000			−0.253490
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	2.00000			0.123797
$262$		0			0
$263$			30.0000i			1.84988i	0.380114	+	0.924940i	$0.375885\pi$
$263$			30.0000i			1.84988i	−0.380114	+	0.924940i	$0.624115\pi$
$264$		0			0
$265$	−8.00000	−	4.00000i	−0.491436	−	0.245718i
$266$		0			0
$267$		−	12.0000i		−	0.734388i
$268$		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
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		−	16.0000i		−	0.968364i
$274$		0			0
$275$	−12.0000	−	16.0000i	−0.723627	−	0.964836i
$276$		0			0
$277$		−	12.0000i		−	0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$		−	12.0000i		−	0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	18.0000			1.07379			0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000			1.07379			0.536895	+	0.843649i	$0.319597\pi$
$282$		0			0
$283$		−	6.00000i		−	0.356663i	−0.983970	−	0.178331i	$-0.942930\pi$
$283$		−	6.00000i		−	0.356663i	0.983970	−	0.178331i	$-0.0570699\pi$
$284$		0			0
$285$	16.0000	+	8.00000i	0.947758	+	0.473879i
$286$		0			0
$287$		−	4.00000i		−	0.236113i
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	−32.0000			−1.87587
$292$		0			0
$293$		−	28.0000i		−	1.63578i	−0.575376	−	0.817889i	$-0.695144\pi$
$293$		−	28.0000i		−	1.63578i	0.575376	−	0.817889i	$-0.304856\pi$
$294$		0			0
$295$	12.0000	−	24.0000i	0.698667	−	1.39733i
$296$		0			0
$297$			16.0000i			0.928414i
$298$		0			0
$299$	−8.00000			−0.462652
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$			12.0000i			0.689382i
$304$		0			0
$305$	10.0000	−	20.0000i	0.572598	−	1.14520i
$306$		0			0
$307$			2.00000i			0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$			2.00000i			0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$		0			0
$309$	−28.0000			−1.59286
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$		−	8.00000i		−	0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$		−	8.00000i		−	0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$		0			0
$315$	−4.00000	−	2.00000i	−0.225374	−	0.112687i
$316$		0			0
$317$			12.0000i			0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$			12.0000i			0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$		0			0
$321$	−20.0000			−1.11629
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−16.0000	+	12.0000i	−0.887520	+	0.665640i
$326$		0			0
$327$			12.0000i			0.663602i
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$		−	4.00000i		−	0.219199i
$334$		0			0
$335$	28.0000	+	14.0000i	1.52980	+	0.764902i
$336$		0			0
$337$		−	16.0000i		−	0.871576i	−0.900049	−	0.435788i	$-0.856470\pi$
$337$		−	16.0000i		−	0.871576i	0.900049	−	0.435788i	$-0.143530\pi$
$338$		0			0
$339$	32.0000			1.73800
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$	−4.00000	+	8.00000i	−0.215353	+	0.430706i
$346$		0			0
$347$		−	22.0000i		−	1.18102i	−0.807030	−	0.590511i	$-0.798926\pi$
$347$		−	22.0000i		−	1.18102i	0.807030	−	0.590511i	$-0.201074\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$	16.0000			0.854017
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	8.00000	−	16.0000i	0.424596	−	0.849192i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$			10.0000i			0.524864i
$364$		0			0
$365$	16.0000	+	8.00000i	0.837478	+	0.418739i
$366$		0			0
$367$		−	10.0000i		−	0.521996i	−0.965339	−	0.260998i	$-0.915948\pi$
$367$		−	10.0000i		−	0.521996i	0.965339	−	0.260998i	$-0.0840516\pi$
$368$		0			0
$369$	−2.00000			−0.104116
$370$		0			0
$371$	8.00000			0.415339
$372$		0			0
$373$			20.0000i			1.03556i	0.855514	+	0.517780i	$0.173242\pi$
$373$			20.0000i			1.03556i	−0.855514	+	0.517780i	$0.826758\pi$
$374$		0			0
$375$	4.00000	+	22.0000i	0.206559	+	1.13608i
$376$		0			0
$377$			8.00000i			0.412021i
$378$		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
$381$	−12.0000			−0.614779
$382$		0			0
$383$			6.00000i			0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$			6.00000i			0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$		0			0
$385$	16.0000	+	8.00000i	0.815436	+	0.407718i
$386$		0			0
$387$			6.00000i			0.304997i
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			24.0000i			1.21064i
$394$		0			0
$395$	−16.0000	+	32.0000i	−0.805047	+	1.61009i
$396$		0			0
$397$		−	36.0000i		−	1.80679i	−0.428811	−	0.903394i	$-0.641067\pi$
$397$		−	36.0000i		−	1.80679i	0.428811	−	0.903394i	$-0.358933\pi$
$398$		0			0
$399$	−16.0000			−0.801002
$400$		0			0
$401$	−14.0000			−0.699127			−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000			−0.699127			−0.349563	+	0.936913i	$0.613670\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	11.0000	−	22.0000i	0.546594	−	1.09319i
$406$		0			0
$407$			16.0000i			0.793091i
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$	−16.0000			−0.789222
$412$		0			0
$413$			24.0000i			1.18096i
$414$		0			0
$415$	−4.00000	−	2.00000i	−0.196352	−	0.0981761i
$416$		0			0
$417$			8.00000i			0.391762i
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		−	6.00000i		−	0.291730i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			20.0000i			0.967868i
$428$		0			0
$429$	32.0000			1.54497
$430$		0			0
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	8.00000	+	4.00000i	0.383571	+	0.191785i
$436$		0			0
$437$			8.00000i			0.382692i
$438$		0			0
$439$	−24.0000			−1.14546			−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000			−1.14546			−0.572729	+	0.819745i	$0.694115\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$		0			0
$445$	6.00000	−	12.0000i	0.284427	−	0.568855i
$446$		0			0
$447$		−	36.0000i		−	1.70274i
$448$		0			0
$449$	−6.00000			−0.283158			−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000			−0.283158			−0.141579	+	0.989927i	$0.545218\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$			16.0000i			0.751746i
$454$		0			0
$455$	8.00000	−	16.0000i	0.375046	−	0.750092i
$456$		0			0
$457$		−	24.0000i		−	1.12267i	−0.827588	−	0.561336i	$-0.810287\pi$
$457$		−	24.0000i		−	1.12267i	0.827588	−	0.561336i	$-0.189713\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$		−	26.0000i		−	1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$		−	26.0000i		−	1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			18.0000i			0.832941i	0.909149	+	0.416470i	$0.136733\pi$
$467$			18.0000i			0.832941i	−0.909149	+	0.416470i	$0.863267\pi$
$468$		0			0
$469$	−28.0000			−1.29292
$470$		0			0
$471$	8.00000			0.368621
$472$		0			0
$473$		−	24.0000i		−	1.10352i
$474$		0			0
$475$	12.0000	+	16.0000i	0.550598	+	0.734130i
$476$		0			0
$477$		−	4.00000i		−	0.183147i
$478$		0			0
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$		−	8.00000i		−	0.364013i
$484$		0			0
$485$	−32.0000	−	16.0000i	−1.45305	−	0.726523i
$486$		0			0
$487$		−	2.00000i		−	0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$		−	2.00000i		−	0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	4.00000			0.180517			0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000			0.180517			0.0902587	+	0.995918i	$0.471231\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	4.00000	−	8.00000i	0.179787	−	0.359573i
$496$		0			0
$497$			16.0000i			0.717698i
$498$		0			0
$499$	−4.00000			−0.179065			−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000			−0.179065			−0.0895323	+	0.995984i	$0.528537\pi$
$500$		0			0
$501$	36.0000			1.60836
$502$		0			0
$503$		−	34.0000i		−	1.51599i	−0.652263	−	0.757993i	$-0.726180\pi$
$503$		−	34.0000i		−	1.51599i	0.652263	−	0.757993i	$-0.273820\pi$
$504$		0			0
$505$	−6.00000	+	12.0000i	−0.266996	+	0.533993i
$506$		0			0
$507$		−	6.00000i		−	0.266469i
$508$		0			0
$509$	−34.0000			−1.50702			−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000			−1.50702			−0.753512	+	0.657434i	$0.771642\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		0			0
$513$		−	16.0000i		−	0.706417i
$514$		0			0
$515$	−28.0000	−	14.0000i	−1.23383	−	0.616914i
$516$		0			0
$517$			24.0000i			1.05552i
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$			10.0000i			0.437269i	0.975807	+	0.218635i	$0.0701603\pi$
$523$			10.0000i			0.437269i	−0.975807	+	0.218635i	$0.929840\pi$
$524$		0			0
$525$	−12.0000	−	16.0000i	−0.523723	−	0.698297i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	19.0000			0.826087
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$		−	8.00000i		−	0.346518i
$534$		0			0
$535$	−20.0000	−	10.0000i	−0.864675	−	0.432338i
$536$		0			0
$537$		−	8.00000i		−	0.345225i
$538$		0			0
$539$	12.0000			0.516877
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$			44.0000i			1.88822i
$544$		0			0
$545$	−6.00000	+	12.0000i	−0.257012	+	0.514024i
$546$		0			0
$547$			34.0000i			1.45374i	0.686778	+	0.726868i	$0.259025\pi$
$547$			34.0000i			1.45374i	−0.686778	+	0.726868i	$0.740975\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$	8.00000	−	16.0000i	0.339581	−	0.679162i
$556$		0			0
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$			18.0000i			0.758610i	0.925272	+	0.379305i	$0.123837\pi$
$563$			18.0000i			0.758610i	−0.925272	+	0.379305i	$0.876163\pi$
$564$		0			0
$565$	32.0000	+	16.0000i	1.34625	+	0.673125i
$566$		0			0
$567$			22.0000i			0.923913i
$568$		0			0
$569$	18.0000			0.754599			0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000			0.754599			0.377300	+	0.926091i	$0.376853\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−8.00000	+	6.00000i	−0.333623	+	0.250217i
$576$		0			0
$577$			32.0000i			1.33218i	0.745873	+	0.666089i	$0.232033\pi$
$577$			32.0000i			1.33218i	−0.745873	+	0.666089i	$0.767967\pi$
$578$		0			0
$579$	−32.0000			−1.32987
$580$		0			0
$581$	4.00000			0.165948
$582$		0			0
$583$			16.0000i			0.662652i
$584$		0			0
$585$	−8.00000	−	4.00000i	−0.330759	−	0.165380i
$586$		0			0
$587$		−	38.0000i		−	1.56843i	−0.620491	−	0.784214i	$-0.713066\pi$
$587$		−	38.0000i		−	1.56843i	0.620491	−	0.784214i	$-0.286934\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	24.0000			0.987228
$592$		0			0
$593$			16.0000i			0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$			16.0000i			0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	16.0000i		−	0.654836i
$598$		0			0
$599$	40.0000			1.63436			0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000			1.63436			0.817178	+	0.576386i	$0.195537\pi$
$600$		0			0
$601$	18.0000			0.734235			0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000			0.734235			0.367118	+	0.930175i	$0.380345\pi$
$602$		0			0
$603$			14.0000i			0.570124i
$604$		0			0
$605$	−5.00000	+	10.0000i	−0.203279	+	0.406558i
$606$		0			0
$607$		−	10.0000i		−	0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$		−	10.0000i		−	0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$		0			0
$609$	−8.00000			−0.324176
$610$		0			0
$611$	24.0000			0.970936
$612$		0			0
$613$			20.0000i			0.807792i	0.914805	+	0.403896i	$0.132344\pi$
$613$			20.0000i			0.807792i	−0.914805	+	0.403896i	$0.867656\pi$
$614$		0			0
$615$	−8.00000	−	4.00000i	−0.322591	−	0.161296i
$616$		0			0
$617$		−	8.00000i		−	0.322068i	−0.986949	−	0.161034i	$-0.948517\pi$
$617$		−	8.00000i		−	0.322068i	0.986949	−	0.161034i	$-0.0514829\pi$
$618$		0			0
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$		0			0
$621$	8.00000			0.321029
$622$		0			0
$623$			12.0000i			0.480770i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		−	32.0000i		−	1.27796i
$628$		0			0
$629$		0			0
$630$		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
$633$		−	8.00000i		−	0.317971i
$634$		0			0
$635$	−12.0000	−	6.00000i	−0.476205	−	0.238103i
$636$		0			0
$637$		−	12.0000i		−	0.475457i
$638$		0			0
$639$	8.00000			0.316475
$640$		0			0
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$			18.0000i			0.709851i	0.934895	+	0.354925i	$0.115494\pi$
$643$			18.0000i			0.709851i	−0.934895	+	0.354925i	$0.884506\pi$
$644$		0			0
$645$	−12.0000	+	24.0000i	−0.472500	+	0.944999i
$646$		0			0
$647$		−	2.00000i		−	0.0786281i	−0.999227	−	0.0393141i	$-0.987483\pi$
$647$		−	2.00000i		−	0.0786281i	0.999227	−	0.0393141i	$-0.0125173\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	36.0000i		−	1.40879i	−0.709809	−	0.704394i	$-0.751219\pi$
$653$		−	36.0000i		−	1.40879i	0.709809	−	0.704394i	$-0.248781\pi$
$654$		0			0
$655$	−12.0000	+	24.0000i	−0.468879	+	0.937758i
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	14.0000			0.544537			0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000			0.544537			0.272268	+	0.962221i	$0.412226\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−16.0000	−	8.00000i	−0.620453	−	0.310227i
$666$		0			0
$667$			4.00000i			0.154881i
$668$		0			0
$669$	−12.0000			−0.463947
$670$		0			0
$671$	−40.0000			−1.54418
$672$		0			0
$673$			16.0000i			0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$			16.0000i			0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$		0			0
$675$	16.0000	−	12.0000i	0.615840	−	0.461880i
$676$		0			0
$677$		−	12.0000i		−	0.461197i	−0.973049	−	0.230599i	$-0.925932\pi$
$677$		−	12.0000i		−	0.461197i	0.973049	−	0.230599i	$-0.0740685\pi$
$678$		0			0
$679$	32.0000			1.22805
$680$		0			0
$681$	−4.00000			−0.153280
$682$		0			0
$683$		−	38.0000i		−	1.45403i	−0.686622	−	0.727015i	$-0.740907\pi$
$683$		−	38.0000i		−	1.45403i	0.686622	−	0.727015i	$-0.259093\pi$
$684$		0			0
$685$	−16.0000	−	8.00000i	−0.611329	−	0.305664i
$686$		0			0
$687$			12.0000i			0.457829i
$688$		0			0
$689$	16.0000			0.609551
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$		0			0
$693$			8.00000i			0.303895i
$694$		0			0
$695$	−4.00000	+	8.00000i	−0.151729	+	0.303457i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−48.0000			−1.81553
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$		−	16.0000i		−	0.603451i
$704$		0			0
$705$	12.0000	−	24.0000i	0.451946	−	0.903892i
$706$		0			0
$707$		−	12.0000i		−	0.451306i
$708$		0			0
$709$	38.0000			1.42712			0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000			1.42712			0.713560	+	0.700594i	$0.247082\pi$
$710$		0			0
$711$	−16.0000			−0.600047
$712$		0			0
$713$		0			0
$714$		0			0
$715$	32.0000	+	16.0000i	1.19673	+	0.598366i
$716$		0			0
$717$		−	32.0000i		−	1.19506i
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	28.0000			1.04277
$722$		0			0
$723$		−	44.0000i		−	1.63638i
$724$		0			0
$725$	6.00000	+	8.00000i	0.222834	+	0.297113i
$726$		0			0
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$	−12.0000	−	6.00000i	−0.442627	−	0.221313i
$736$		0			0
$737$		−	56.0000i		−	2.06279i
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$	−32.0000			−1.17555
$742$		0			0
$743$			30.0000i			1.10059i	0.834969	+	0.550297i	$0.185485\pi$
$743$			30.0000i			1.10059i	−0.834969	+	0.550297i	$0.814515\pi$
$744$		0			0
$745$	18.0000	−	36.0000i	0.659469	−	1.31894i
$746$		0			0
$747$		−	2.00000i		−	0.0731762i
$748$		0			0
$749$	20.0000			0.730784
$750$		0			0
$751$	16.0000			0.583848			0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000			0.583848			0.291924	+	0.956441i	$0.405705\pi$
$752$		0			0
$753$			40.0000i			1.45768i
$754$		0			0
$755$	−8.00000	+	16.0000i	−0.291150	+	0.582300i
$756$		0			0
$757$			52.0000i			1.88997i	0.327111	+	0.944986i	$0.393925\pi$
$757$			52.0000i			1.88997i	−0.327111	+	0.944986i	$0.606075\pi$
$758$		0			0
$759$	16.0000			0.580763
$760$		0			0
$761$	−38.0000			−1.37750			−0.688749	−	0.724999i	$-0.741840\pi$
$761$	−38.0000			−1.37750			−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			48.0000i			1.73318i
$768$		0			0
$769$	2.00000			0.0721218			0.0360609	−	0.999350i	$-0.488519\pi$
$769$	2.00000			0.0721218			0.0360609	+	0.999350i	$0.488519\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			20.0000i			0.719350i	0.933078	+	0.359675i	$0.117112\pi$
$773$			20.0000i			0.719350i	−0.933078	+	0.359675i	$0.882888\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			16.0000i			0.573997i
$778$		0			0
$779$	−8.00000			−0.286630
$780$		0			0
$781$	−32.0000			−1.14505
$782$		0			0
$783$		−	8.00000i		−	0.285897i
$784$		0			0
$785$	8.00000	+	4.00000i	0.285532	+	0.142766i
$786$		0			0
$787$			2.00000i			0.0712923i	0.999364	+	0.0356462i	$0.0113489\pi$
$787$			2.00000i			0.0712923i	−0.999364	+	0.0356462i	$0.988651\pi$
$788$		0			0
$789$	−60.0000			−2.13606
$790$		0			0
$791$	−32.0000			−1.13779
$792$		0			0
$793$			40.0000i			1.42044i
$794$		0			0
$795$	8.00000	−	16.0000i	0.283731	−	0.567462i
$796$		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
$799$		0			0
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$		−	32.0000i		−	1.12926i
$804$		0			0
$805$	4.00000	−	8.00000i	0.140981	−	0.281963i
$806$		0			0
$807$			12.0000i			0.422420i
$808$		0			0
$809$	42.0000			1.47664			0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000			1.47664			0.738321	+	0.674450i	$0.235619\pi$
$810$		0			0
$811$	36.0000			1.26413			0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000			1.26413			0.632065	+	0.774915i	$0.282207\pi$
$812$		0			0
$813$			32.0000i			1.12229i
$814$		0			0
$815$	−4.00000	−	2.00000i	−0.140114	−	0.0700569i
$816$		0			0
$817$			24.0000i			0.839654i
$818$		0			0
$819$	8.00000			0.279543
$820$		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
$825$	32.0000	−	24.0000i	1.11410	−	0.835573i
$826$		0			0
$827$			42.0000i			1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$			42.0000i			1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$	24.0000			0.832551
$832$		0			0
$833$		0			0
$834$		0			0
$835$	36.0000	+	18.0000i	1.24583	+	0.622916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$			36.0000i			1.23991i
$844$		0			0
$845$	3.00000	−	6.00000i	0.103203	−	0.206406i
$846$		0			0
$847$		−	10.0000i		−	0.343604i
$848$		0			0
$849$	12.0000			0.411839
$850$		0			0
$851$	8.00000			0.274236
$852$		0			0
$853$		−	28.0000i		−	0.958702i	−0.877623	−	0.479351i	$-0.840872\pi$
$853$		−	28.0000i		−	0.958702i	0.877623	−	0.479351i	$-0.159128\pi$
$854$		0			0
$855$	−4.00000	+	8.00000i	−0.136797	+	0.273594i
$856$		0			0
$857$		−	8.00000i		−	0.273275i	−0.990621	−	0.136637i	$-0.956370\pi$
$857$		−	8.00000i		−	0.273275i	0.990621	−	0.136637i	$-0.0436295\pi$
$858$		0			0
$859$	−44.0000			−1.50126			−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000			−1.50126			−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$	8.00000			0.272639
$862$		0			0
$863$		−	42.0000i		−	1.42970i	−0.699280	−	0.714848i	$-0.746496\pi$
$863$		−	42.0000i		−	1.42970i	0.699280	−	0.714848i	$-0.253504\pi$
$864$		0			0
$865$	−24.0000	−	12.0000i	−0.816024	−	0.408012i
$866$		0			0
$867$			34.0000i			1.15470i
$868$		0			0
$869$	64.0000			2.17105
$870$		0			0
$871$	−56.0000			−1.89749
$872$		0			0
$873$		−	16.0000i		−	0.541518i
$874$		0			0
$875$	−4.00000	−	22.0000i	−0.135225	−	0.743736i
$876$		0			0
$877$			44.0000i			1.48577i	0.669417	+	0.742887i	$0.266544\pi$
$877$			44.0000i			1.48577i	−0.669417	+	0.742887i	$0.733456\pi$
$878$		0			0
$879$	56.0000			1.88883
$880$		0			0
$881$	10.0000			0.336909			0.168454	−	0.985709i	$-0.446122\pi$
$881$	10.0000			0.336909			0.168454	+	0.985709i	$0.446122\pi$
$882$		0			0
$883$			50.0000i			1.68263i	0.540542	+	0.841317i	$0.318219\pi$
$883$			50.0000i			1.68263i	−0.540542	+	0.841317i	$0.681781\pi$
$884$		0			0
$885$	48.0000	+	24.0000i	1.61350	+	0.806751i
$886$		0			0
$887$		−	18.0000i		−	0.604381i	−0.953248	−	0.302190i	$-0.902282\pi$
$887$		−	18.0000i		−	0.604381i	0.953248	−	0.302190i	$-0.0977178\pi$
$888$		0			0
$889$	12.0000			0.402467
$890$		0			0
$891$	−44.0000			−1.47406
$892$		0			0
$893$		−	24.0000i		−	0.803129i
$894$		0			0
$895$	4.00000	−	8.00000i	0.133705	−	0.267411i
$896$		0			0
$897$		−	16.0000i		−	0.534224i
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		−	24.0000i		−	0.798670i
$904$		0			0
$905$	−22.0000	+	44.0000i	−0.731305	+	1.46261i
$906$		0			0
$907$		−	22.0000i		−	0.730498i	−0.930910	−	0.365249i	$-0.880984\pi$
$907$		−	22.0000i		−	0.730498i	0.930910	−	0.365249i	$-0.119016\pi$
$908$		0			0
$909$	−6.00000			−0.199007
$910$		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
$913$			8.00000i			0.264761i
$914$		0			0
$915$	40.0000	+	20.0000i	1.32236	+	0.661180i
$916$		0			0
$917$		−	24.0000i		−	0.792550i
$918$		0			0
$919$	−24.0000			−0.791687			−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000			−0.791687			−0.395843	+	0.918318i	$0.629548\pi$
$920$		0			0
$921$	−4.00000			−0.131804
$922$		0			0
$923$			32.0000i			1.05329i
$924$		0			0
$925$	16.0000	−	12.0000i	0.526077	−	0.394558i
$926$		0			0
$927$		−	14.0000i		−	0.459820i
$928$		0			0
$929$	−38.0000			−1.24674			−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000			−1.24674			−0.623370	+	0.781927i	$0.714237\pi$
$930$		0			0
$931$	−12.0000			−0.393284
$932$		0			0
$933$		−	48.0000i		−	1.57145i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	40.0000i		−	1.30674i	−0.757037	−	0.653372i	$-0.773354\pi$
$937$		−	40.0000i		−	1.30674i	0.757037	−	0.653372i	$-0.226646\pi$
$938$		0			0
$939$	16.0000			0.522140
$940$		0			0
$941$	14.0000			0.456387			0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000			0.456387			0.228193	+	0.973616i	$0.426718\pi$
$942$		0			0
$943$		−	4.00000i		−	0.130258i
$944$		0			0
$945$	−8.00000	+	16.0000i	−0.260240	+	0.520480i
$946$		0			0
$947$		−	30.0000i		−	0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$		−	30.0000i		−	0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$		0			0
$949$	−32.0000			−1.03876
$950$		0			0
$951$	−24.0000			−0.778253
$952$		0			0
$953$			24.0000i			0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$953$			24.0000i			0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		−	16.0000i		−	0.517207i
$958$		0			0
$959$	16.0000			0.516667
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		−	10.0000i		−	0.322245i
$964$		0			0
$965$	−32.0000	−	16.0000i	−1.03012	−	0.515058i
$966$		0			0
$967$		−	50.0000i		−	1.60789i	−0.594703	−	0.803946i	$-0.702730\pi$
$967$		−	50.0000i		−	1.60789i	0.594703	−	0.803946i	$-0.297270\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$	−24.0000	−	32.0000i	−0.768615	−	1.02482i
$976$		0			0
$977$			32.0000i			1.02377i	0.859054	+	0.511885i	$0.171053\pi$
$977$			32.0000i			1.02377i	−0.859054	+	0.511885i	$0.828947\pi$
$978$		0			0
$979$	−24.0000			−0.767043
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$			46.0000i			1.46717i	0.679597	+	0.733586i	$0.262155\pi$
$983$			46.0000i			1.46717i	−0.679597	+	0.733586i	$0.737845\pi$
$984$		0			0
$985$	24.0000	+	12.0000i	0.764704	+	0.382352i
$986$		0			0
$987$			24.0000i			0.763928i
$988$		0			0
$989$	−12.0000			−0.381578
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$		−	56.0000i		−	1.77711i
$994$		0			0
$995$	8.00000	−	16.0000i	0.253617	−	0.507234i
$996$		0			0
$997$		−	12.0000i		−	0.380044i	−0.981780	−	0.190022i	$-0.939144\pi$
$997$		−	12.0000i		−	0.380044i	0.981780	−	0.190022i	$-0.0608559\pi$
$998$		0			0
$999$	−16.0000			−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.2.c.a.49.2		2
3.2	odd	2		720.2.f.e.289.1		2
4.3	odd	2		40.2.c.a.9.1	✓	2
5.2	odd	4		400.2.a.g.1.1		1
5.3	odd	4		400.2.a.b.1.1		1
5.4	even	2	inner	80.2.c.a.49.1		2
8.3	odd	2		320.2.c.c.129.2		2
8.5	even	2		320.2.c.b.129.1		2
12.11	even	2		360.2.f.c.289.1		2
15.2	even	4		3600.2.a.bb.1.1		1
15.8	even	4		3600.2.a.k.1.1		1
15.14	odd	2		720.2.f.e.289.2		2
16.3	odd	4		1280.2.f.f.129.2		2
16.5	even	4		1280.2.f.e.129.1		2
16.11	odd	4		1280.2.f.a.129.1		2
16.13	even	4		1280.2.f.b.129.2		2
20.3	even	4		200.2.a.d.1.1		1
20.7	even	4		200.2.a.b.1.1		1
20.19	odd	2		40.2.c.a.9.2	yes	2
24.5	odd	2		2880.2.f.i.1729.2		2
24.11	even	2		2880.2.f.h.1729.2		2
28.27	even	2		1960.2.g.b.1569.2		2
40.3	even	4		1600.2.a.f.1.1		1
40.13	odd	4		1600.2.a.u.1.1		1
40.19	odd	2		320.2.c.c.129.1		2
40.27	even	4		1600.2.a.v.1.1		1
40.29	even	2		320.2.c.b.129.2		2
40.37	odd	4		1600.2.a.d.1.1		1
60.23	odd	4		1800.2.a.s.1.1		1
60.47	odd	4		1800.2.a.j.1.1		1
60.59	even	2		360.2.f.c.289.2		2
80.19	odd	4		1280.2.f.a.129.2		2
80.29	even	4		1280.2.f.e.129.2		2
80.59	odd	4		1280.2.f.f.129.1		2
80.69	even	4		1280.2.f.b.129.1		2
120.29	odd	2		2880.2.f.i.1729.1		2
120.59	even	2		2880.2.f.h.1729.1		2
140.27	odd	4		9800.2.a.bf.1.1		1
140.83	odd	4		9800.2.a.d.1.1		1
140.139	even	2		1960.2.g.b.1569.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.2.c.a.9.1	✓	2	4.3	odd	2
40.2.c.a.9.2	yes	2	20.19	odd	2
80.2.c.a.49.1		2	5.4	even	2	inner
80.2.c.a.49.2		2	1.1	even	1	trivial
200.2.a.b.1.1		1	20.7	even	4
200.2.a.d.1.1		1	20.3	even	4
320.2.c.b.129.1		2	8.5	even	2
320.2.c.b.129.2		2	40.29	even	2
320.2.c.c.129.1		2	40.19	odd	2
320.2.c.c.129.2		2	8.3	odd	2
360.2.f.c.289.1		2	12.11	even	2
360.2.f.c.289.2		2	60.59	even	2
400.2.a.b.1.1		1	5.3	odd	4
400.2.a.g.1.1		1	5.2	odd	4
720.2.f.e.289.1		2	3.2	odd	2
720.2.f.e.289.2		2	15.14	odd	2
1280.2.f.a.129.1		2	16.11	odd	4
1280.2.f.a.129.2		2	80.19	odd	4
1280.2.f.b.129.1		2	80.69	even	4
1280.2.f.b.129.2		2	16.13	even	4
1280.2.f.e.129.1		2	16.5	even	4
1280.2.f.e.129.2		2	80.29	even	4
1280.2.f.f.129.1		2	80.59	odd	4
1280.2.f.f.129.2		2	16.3	odd	4
1600.2.a.d.1.1		1	40.37	odd	4
1600.2.a.f.1.1		1	40.3	even	4
1600.2.a.u.1.1		1	40.13	odd	4
1600.2.a.v.1.1		1	40.27	even	4
1800.2.a.j.1.1		1	60.47	odd	4
1800.2.a.s.1.1		1	60.23	odd	4
1960.2.g.b.1569.1		2	140.139	even	2
1960.2.g.b.1569.2		2	28.27	even	2
2880.2.f.h.1729.1		2	120.59	even	2
2880.2.f.h.1729.2		2	24.11	even	2
2880.2.f.i.1729.1		2	120.29	odd	2
2880.2.f.i.1729.2		2	24.5	odd	2
3600.2.a.k.1.1		1	15.8	even	4
3600.2.a.bb.1.1		1	15.2	even	4
9800.2.a.d.1.1		1	140.83	odd	4
9800.2.a.bf.1.1		1	140.27	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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	80.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +(-1.00000 + 2.00000i) q^{5} -2.00000i q^{7} -1.00000 q^{9} +4.00000 q^{11} -4.00000i q^{13} +(-4.00000 - 2.00000i) q^{15} -4.00000 q^{19} +4.00000 q^{21} -2.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +4.00000i q^{27} -2.00000 q^{29} +8.00000i q^{33} +(4.00000 + 2.00000i) q^{35} +4.00000i q^{37} +8.00000 q^{39} +2.00000 q^{41} -6.00000i q^{43} +(1.00000 - 2.00000i) q^{45} +6.00000i q^{47} +3.00000 q^{49} +4.00000i q^{53} +(-4.00000 + 8.00000i) q^{55} -8.00000i q^{57} -12.0000 q^{59} -10.0000 q^{61} +2.00000i q^{63} +(8.00000 + 4.00000i) q^{65} -14.0000i q^{67} +4.00000 q^{69} -8.00000 q^{71} -8.00000i q^{73} +(8.00000 - 6.00000i) q^{75} -8.00000i q^{77} +16.0000 q^{79} -11.0000 q^{81} +2.00000i q^{83} -4.00000i q^{87} -6.00000 q^{89} -8.00000 q^{91} +(4.00000 - 8.00000i) q^{95} +16.0000i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{9} + 8 q^{11} - 8 q^{15} - 8 q^{19} + 8 q^{21} - 6 q^{25} - 4 q^{29} + 8 q^{35} + 16 q^{39} + 4 q^{41} + 2 q^{45} + 6 q^{49} - 8 q^{55} - 24 q^{59} - 20 q^{61} + 16 q^{65} + 8 q^{69}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^9 + 8 * q^11 - 8 * q^15 - 8 * q^19 + 8 * q^21 - 6 * q^25 - 4 * q^29 + 8 * q^35 + 16 * q^39 + 4 * q^41 + 2 * q^45 + 6 * q^49 - 8 * q^55 - 24 * q^59 - 20 * q^61 + 16 * q^65 + 8 * q^69 - 16 * q^71 + 16 * q^75 + 32 * q^79 - 22 * q^81 - 12 * q^89 - 16 * q^91 + 8 * q^95 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more