LMFDB - Embedded newform 80.2.n.b.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,2,Mod(47,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("80.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	80.n (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$0.638803216170$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			47.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	80.47
Dual form			80.2.n.b.63.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.73205 - 1.73205i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +O(q^{10})$	\(q+(-1.73205 - 1.73205i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +3.46410i q^{11} +(1.00000 - 1.00000i) q^{13} +(-1.73205 + 5.19615i) q^{15} +(1.00000 + 1.00000i) q^{17} +6.92820 q^{19} -6.00000 q^{21} +(-1.73205 - 1.73205i) q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{29} -3.46410i q^{31} +(6.00000 - 6.00000i) q^{33} +(-5.19615 - 1.73205i) q^{35} +(5.00000 + 5.00000i) q^{37} -3.46410 q^{39} +2.00000 q^{41} +(-1.73205 - 1.73205i) q^{43} +(6.00000 - 3.00000i) q^{45} +(1.73205 - 1.73205i) q^{47} +1.00000i q^{49} -3.46410i q^{51} +(-7.00000 + 7.00000i) q^{53} +(6.92820 - 3.46410i) q^{55} +(-12.0000 - 12.0000i) q^{57} -6.92820 q^{59} +6.00000 q^{61} +(5.19615 + 5.19615i) q^{63} +(-3.00000 - 1.00000i) q^{65} +(-5.19615 + 5.19615i) q^{67} +6.00000i q^{69} +10.3923i q^{71} +(-7.00000 + 7.00000i) q^{73} +(12.1244 - 1.73205i) q^{75} +(6.00000 + 6.00000i) q^{77} +9.00000 q^{81} +(12.1244 + 12.1244i) q^{83} +(1.00000 - 3.00000i) q^{85} +(-6.92820 + 6.92820i) q^{87} -8.00000i q^{89} -3.46410i q^{91} +(-6.00000 + 6.00000i) q^{93} +(-6.92820 - 13.8564i) q^{95} +(-7.00000 - 7.00000i) q^{97} -10.3923 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} + 4 q^{13} + 4 q^{17} - 24 q^{21} - 12 q^{25} + 24 q^{33} + 20 q^{37} + 8 q^{41} + 24 q^{45} - 28 q^{53} - 48 q^{57} + 24 q^{61} - 12 q^{65} - 28 q^{73} + 24 q^{77} + 36 q^{81} + 4 q^{85} - 24 q^{93} - 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 4 * q^13 + 4 * q^17 - 24 * q^21 - 12 * q^25 + 24 * q^33 + 20 * q^37 + 8 * q^41 + 24 * q^45 - 28 * q^53 - 48 * q^57 + 24 * q^61 - 12 * q^65 - 28 * q^73 + 24 * q^77 + 36 * q^81 + 4 * q^85 - 24 * q^93 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$17$	$21$	$31$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.73205	−	1.73205i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$3$	−1.73205	−	1.73205i	−1.00000	−	1.00000i	−1.00000			$\pi$
$4$		0			0
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$	1.73205	−	1.73205i	0.654654	−	0.654654i	−0.299456	−	0.954110i	$-0.596805\pi$
$7$	1.73205	−	1.73205i	0.654654	−	0.654654i	0.954110	+	0.299456i	$0.0968053\pi$
$8$		0			0
$9$			3.00000i			1.00000i
$10$		0			0
$11$			3.46410i			1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$			3.46410i			1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$	−1.73205	+	5.19615i	−0.447214	+	1.34164i
$16$		0			0
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	0.817898	−	0.575363i	$-0.195139\pi$
$17$	1.00000	+	1.00000i	0.242536	+	0.242536i	−0.575363	+	0.817898i	$0.695139\pi$
$18$		0			0
$19$	6.92820			1.58944			0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820			1.58944			0.794719	+	0.606977i	$0.207618\pi$
$20$		0			0
$21$	−6.00000			−1.30931
$22$		0			0
$23$	−1.73205	−	1.73205i	−0.361158	−	0.361158i	0.503081	−	0.864239i	$-0.332200\pi$
$23$	−1.73205	−	1.73205i	−0.361158	−	0.361158i	−0.864239	+	0.503081i	$0.832200\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	4.00000i		−	0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$		−	4.00000i		−	0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	6.00000	−	6.00000i	1.04447	−	1.04447i
$34$		0			0
$35$	−5.19615	−	1.73205i	−0.878310	−	0.292770i
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$	−3.46410			−0.554700
$40$		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
$43$	−1.73205	−	1.73205i	−0.264135	−	0.264135i	0.562596	−	0.826732i	$-0.309803\pi$
$43$	−1.73205	−	1.73205i	−0.264135	−	0.264135i	−0.826732	+	0.562596i	$0.809803\pi$
$44$		0			0
$45$	6.00000	−	3.00000i	0.894427	−	0.447214i
$46$		0			0
$47$	1.73205	−	1.73205i	0.252646	−	0.252646i	−0.569409	−	0.822054i	$-0.692828\pi$
$47$	1.73205	−	1.73205i	0.252646	−	0.252646i	0.822054	+	0.569409i	$0.192828\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$		−	3.46410i		−	0.485071i
$52$		0			0
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	−0.999287	−	0.0377628i	$-0.987977\pi$
$53$	−7.00000	+	7.00000i	−0.961524	+	0.961524i	0.0377628	+	0.999287i	$0.487977\pi$
$54$		0			0
$55$	6.92820	−	3.46410i	0.934199	−	0.467099i
$56$		0			0
$57$	−12.0000	−	12.0000i	−1.58944	−	1.58944i
$58$		0			0
$59$	−6.92820			−0.901975			−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820			−0.901975			−0.450988	+	0.892530i	$0.648928\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$	5.19615	+	5.19615i	0.654654	+	0.654654i
$64$		0			0
$65$	−3.00000	−	1.00000i	−0.372104	−	0.124035i
$66$		0			0
$67$	−5.19615	+	5.19615i	−0.634811	+	0.634811i	−0.949271	−	0.314460i	$-0.898177\pi$
$67$	−5.19615	+	5.19615i	−0.634811	+	0.634811i	0.314460	+	0.949271i	$0.398177\pi$
$68$		0			0
$69$			6.00000i			0.722315i
$70$		0			0
$71$			10.3923i			1.23334i	0.787222	+	0.616670i	$0.211519\pi$
$71$			10.3923i			1.23334i	−0.787222	+	0.616670i	$0.788481\pi$
$72$		0			0
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	−0.986005	−	0.166717i	$-0.946683\pi$
$73$	−7.00000	+	7.00000i	−0.819288	+	0.819288i	0.166717	+	0.986005i	$0.446683\pi$
$74$		0			0
$75$	12.1244	−	1.73205i	1.40000	−	0.200000i
$76$		0			0
$77$	6.00000	+	6.00000i	0.683763	+	0.683763i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	12.1244	+	12.1244i	1.33082	+	1.33082i	0.904636	+	0.426185i	$0.140143\pi$
$83$	12.1244	+	12.1244i	1.33082	+	1.33082i	0.426185	+	0.904636i	$0.359857\pi$
$84$		0			0
$85$	1.00000	−	3.00000i	0.108465	−	0.325396i
$86$		0			0
$87$	−6.92820	+	6.92820i	−0.742781	+	0.742781i
$88$		0			0
$89$		−	8.00000i		−	0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$89$		−	8.00000i		−	0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$90$		0			0
$91$		−	3.46410i		−	0.363137i
$92$		0			0
$93$	−6.00000	+	6.00000i	−0.622171	+	0.622171i
$94$		0			0
$95$	−6.92820	−	13.8564i	−0.710819	−	1.42164i
$96$		0			0
$97$	−7.00000	−	7.00000i	−0.710742	−	0.710742i	0.255948	−	0.966691i	$-0.417612\pi$
$97$	−7.00000	−	7.00000i	−0.710742	−	0.710742i	−0.966691	+	0.255948i	$0.917612\pi$
$98$		0			0
$99$	−10.3923			−1.04447
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$	−1.73205	−	1.73205i	−0.170664	−	0.170664i	0.616607	−	0.787271i	$-0.288507\pi$
$103$	−1.73205	−	1.73205i	−0.170664	−	0.170664i	−0.787271	+	0.616607i	$0.788507\pi$
$104$		0			0
$105$	6.00000	+	12.0000i	0.585540	+	1.17108i
$106$		0			0
$107$	8.66025	−	8.66025i	0.837218	−	0.837218i	−0.151274	−	0.988492i	$-0.548337\pi$
$107$	8.66025	−	8.66025i	0.837218	−	0.837218i	0.988492	+	0.151274i	$0.0483374\pi$
$108$		0			0
$109$		−	12.0000i		−	1.14939i	−0.818367	−	0.574696i	$-0.805120\pi$
$109$		−	12.0000i		−	1.14939i	0.818367	−	0.574696i	$-0.194880\pi$
$110$		0			0
$111$		−	17.3205i		−	1.64399i
$112$		0			0
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	−0.143065	−	0.989713i	$-0.545696\pi$
$113$	9.00000	−	9.00000i	0.846649	−	0.846649i	0.989713	+	0.143065i	$0.0456957\pi$
$114$		0			0
$115$	−1.73205	+	5.19615i	−0.161515	+	0.484544i
$116$		0			0
$117$	3.00000	+	3.00000i	0.277350	+	0.277350i
$118$		0			0
$119$	3.46410			0.317554
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	−3.46410	−	3.46410i	−0.312348	−	0.312348i
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$	−12.1244	+	12.1244i	−1.07586	+	1.07586i	−0.0789869	+	0.996876i	$0.525169\pi$
$127$	−12.1244	+	12.1244i	−1.07586	+	1.07586i	−0.996876	+	0.0789869i	$0.974831\pi$
$128$		0			0
$129$			6.00000i			0.528271i
$130$		0			0
$131$		−	10.3923i		−	0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$		−	10.3923i		−	0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$		0			0
$133$	12.0000	−	12.0000i	1.04053	−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.00000	+	9.00000i	0.768922	+	0.768922i	0.977917	−	0.208995i	$-0.0670192\pi$
$137$	9.00000	+	9.00000i	0.768922	+	0.768922i	−0.208995	+	0.977917i	$0.567019\pi$
$138$		0			0
$139$	−6.92820			−0.587643			−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820			−0.587643			−0.293821	+	0.955860i	$0.594927\pi$
$140$		0			0
$141$	−6.00000			−0.505291
$142$		0			0
$143$	3.46410	+	3.46410i	0.289683	+	0.289683i
$144$		0			0
$145$	−8.00000	+	4.00000i	−0.664364	+	0.332182i
$146$		0			0
$147$	1.73205	−	1.73205i	0.142857	−	0.142857i
$148$		0			0
$149$		−	4.00000i		−	0.327693i	−0.986486	−	0.163846i	$-0.947610\pi$
$149$		−	4.00000i		−	0.327693i	0.986486	−	0.163846i	$-0.0523901\pi$
$150$		0			0
$151$			10.3923i			0.845714i	0.906196	+	0.422857i	$0.138973\pi$
$151$			10.3923i			0.845714i	−0.906196	+	0.422857i	$0.861027\pi$
$152$		0			0
$153$	−3.00000	+	3.00000i	−0.242536	+	0.242536i
$154$		0			0
$155$	−6.92820	+	3.46410i	−0.556487	+	0.278243i
$156$		0			0
$157$	−11.0000	−	11.0000i	−0.877896	−	0.877896i	0.115421	−	0.993317i	$-0.463178\pi$
$157$	−11.0000	−	11.0000i	−0.877896	−	0.877896i	−0.993317	+	0.115421i	$0.963178\pi$
$158$		0			0
$159$	24.2487			1.92305
$160$		0			0
$161$	−6.00000			−0.472866
$162$		0			0
$163$	−1.73205	−	1.73205i	−0.135665	−	0.135665i	0.636013	−	0.771678i	$-0.280582\pi$
$163$	−1.73205	−	1.73205i	−0.135665	−	0.135665i	−0.771678	+	0.636013i	$0.780582\pi$
$164$		0			0
$165$	−18.0000	−	6.00000i	−1.40130	−	0.467099i
$166$		0			0
$167$	15.5885	−	15.5885i	1.20627	−	1.20627i	0.234045	−	0.972226i	$-0.424804\pi$
$167$	15.5885	−	15.5885i	1.20627	−	1.20627i	0.972226	−	0.234045i	$-0.0751964\pi$
$168$		0			0
$169$			11.0000i			0.846154i
$170$		0			0
$171$			20.7846i			1.58944i
$172$		0			0
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.152057	+	0.988372i	$0.548590\pi$
$173$	−15.0000	+	15.0000i	−1.14043	+	1.14043i	−0.988372	+	0.152057i	$0.951410\pi$
$174$		0			0
$175$	1.73205	+	12.1244i	0.130931	+	0.916515i
$176$		0			0
$177$	12.0000	+	12.0000i	0.901975	+	0.901975i
$178$		0			0
$179$	6.92820			0.517838			0.258919	−	0.965899i	$-0.416634\pi$
$179$	6.92820			0.517838			0.258919	+	0.965899i	$0.416634\pi$
$180$		0			0
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$	−10.3923	−	10.3923i	−0.768221	−	0.768221i
$184$		0			0
$185$	5.00000	−	15.0000i	0.367607	−	1.10282i
$186$		0			0
$187$	−3.46410	+	3.46410i	−0.253320	+	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			24.2487i			1.75458i	0.479965	+	0.877288i	$0.340649\pi$
$191$			24.2487i			1.75458i	−0.479965	+	0.877288i	$0.659351\pi$
$192$		0			0
$193$	1.00000	−	1.00000i	0.0719816	−	0.0719816i	−0.670199	−	0.742181i	$-0.733791\pi$
$193$	1.00000	−	1.00000i	0.0719816	−	0.0719816i	0.742181	+	0.670199i	$0.233791\pi$
$194$		0			0
$195$	3.46410	+	6.92820i	0.248069	+	0.496139i
$196$		0			0
$197$	−3.00000	−	3.00000i	−0.213741	−	0.213741i	0.592113	−	0.805855i	$-0.298294\pi$
$197$	−3.00000	−	3.00000i	−0.213741	−	0.213741i	−0.805855	+	0.592113i	$0.798294\pi$
$198$		0			0
$199$	−13.8564			−0.982255			−0.491127	−	0.871088i	$-0.663415\pi$
$199$	−13.8564			−0.982255			−0.491127	+	0.871088i	$0.663415\pi$
$200$		0			0
$201$	18.0000			1.26962
$202$		0			0
$203$	−6.92820	−	6.92820i	−0.486265	−	0.486265i
$204$		0			0
$205$	−2.00000	−	4.00000i	−0.139686	−	0.279372i
$206$		0			0
$207$	5.19615	−	5.19615i	0.361158	−	0.361158i
$208$		0			0
$209$			24.0000i			1.66011i
$210$		0			0
$211$		−	10.3923i		−	0.715436i	−0.933830	−	0.357718i	$-0.883555\pi$
$211$		−	10.3923i		−	0.715436i	0.933830	−	0.357718i	$-0.116445\pi$
$212$		0			0
$213$	18.0000	−	18.0000i	1.23334	−	1.23334i
$214$		0			0
$215$	−1.73205	+	5.19615i	−0.118125	+	0.354375i
$216$		0			0
$217$	−6.00000	−	6.00000i	−0.407307	−	0.407307i
$218$		0			0
$219$	24.2487			1.63858
$220$		0			0
$221$	2.00000			0.134535
$222$		0			0
$223$	−15.5885	−	15.5885i	−1.04388	−	1.04388i	−0.998992	−	0.0448883i	$-0.985707\pi$
$223$	−15.5885	−	15.5885i	−1.04388	−	1.04388i	−0.0448883	−	0.998992i	$-0.514293\pi$
$224$		0			0
$225$	−12.0000	−	9.00000i	−0.800000	−	0.600000i
$226$		0			0
$227$	8.66025	−	8.66025i	0.574801	−	0.574801i	−0.358665	−	0.933466i	$-0.616768\pi$
$227$	8.66025	−	8.66025i	0.574801	−	0.574801i	0.933466	+	0.358665i	$0.116768\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$		−	20.7846i		−	1.36753i
$232$		0			0
$233$	1.00000	−	1.00000i	0.0655122	−	0.0655122i	−0.673592	−	0.739104i	$-0.735249\pi$
$233$	1.00000	−	1.00000i	0.0655122	−	0.0655122i	0.739104	+	0.673592i	$0.235249\pi$
$234$		0			0
$235$	−5.19615	−	1.73205i	−0.338960	−	0.112987i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−27.7128			−1.79259			−0.896296	−	0.443455i	$-0.853752\pi$
$239$	−27.7128			−1.79259			−0.896296	+	0.443455i	$0.853752\pi$
$240$		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
$243$	−15.5885	−	15.5885i	−1.00000	−	1.00000i
$244$		0			0
$245$	2.00000	−	1.00000i	0.127775	−	0.0638877i
$246$		0			0
$247$	6.92820	−	6.92820i	0.440831	−	0.440831i
$248$		0			0
$249$		−	42.0000i		−	2.66164i
$250$		0			0
$251$			3.46410i			0.218652i	0.994006	+	0.109326i	$0.0348693\pi$
$251$			3.46410i			0.218652i	−0.994006	+	0.109326i	$0.965131\pi$
$252$		0			0
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$		0			0
$255$	−6.92820	+	3.46410i	−0.433861	+	0.216930i
$256$		0			0
$257$	9.00000	+	9.00000i	0.561405	+	0.561405i	0.929706	−	0.368302i	$-0.120061\pi$
$257$	9.00000	+	9.00000i	0.561405	+	0.561405i	−0.368302	+	0.929706i	$0.620061\pi$
$258$		0			0
$259$	17.3205			1.07624
$260$		0			0
$261$	12.0000			0.742781
$262$		0			0
$263$	12.1244	+	12.1244i	0.747620	+	0.747620i	0.974032	−	0.226412i	$-0.0726995\pi$
$263$	12.1244	+	12.1244i	0.747620	+	0.747620i	−0.226412	+	0.974032i	$0.572700\pi$
$264$		0			0
$265$	21.0000	+	7.00000i	1.29002	+	0.430007i
$266$		0			0
$267$	−13.8564	+	13.8564i	−0.847998	+	0.847998i
$268$		0			0
$269$			4.00000i			0.243884i	0.992537	+	0.121942i	$0.0389122\pi$
$269$			4.00000i			0.243884i	−0.992537	+	0.121942i	$0.961088\pi$
$270$		0			0
$271$		−	3.46410i		−	0.210429i	−0.994450	−	0.105215i	$-0.966447\pi$
$271$		−	3.46410i		−	0.210429i	0.994450	−	0.105215i	$-0.0335529\pi$
$272$		0			0
$273$	−6.00000	+	6.00000i	−0.363137	+	0.363137i
$274$		0			0
$275$	−13.8564	−	10.3923i	−0.835573	−	0.626680i
$276$		0			0
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	0.980015	−	0.198921i	$-0.0637438\pi$
$277$	13.0000	+	13.0000i	0.781094	+	0.781094i	−0.198921	+	0.980015i	$0.563744\pi$
$278$		0			0
$279$	10.3923			0.622171
$280$		0			0
$281$	−14.0000			−0.835170			−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000			−0.835170			−0.417585	+	0.908638i	$0.637123\pi$
$282$		0			0
$283$	12.1244	+	12.1244i	0.720718	+	0.720718i	0.968751	−	0.248033i	$-0.0797843\pi$
$283$	12.1244	+	12.1244i	0.720718	+	0.720718i	−0.248033	+	0.968751i	$0.579784\pi$
$284$		0			0
$285$	−12.0000	+	36.0000i	−0.710819	+	2.13246i
$286$		0			0
$287$	3.46410	−	3.46410i	0.204479	−	0.204479i
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$			24.2487i			1.42148i
$292$		0			0
$293$	9.00000	−	9.00000i	0.525786	−	0.525786i	−0.393527	−	0.919313i	$-0.628745\pi$
$293$	9.00000	−	9.00000i	0.525786	−	0.525786i	0.919313	+	0.393527i	$0.128745\pi$
$294$		0			0
$295$	6.92820	+	13.8564i	0.403376	+	0.806751i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.46410			−0.200334
$300$		0			0
$301$	−6.00000			−0.345834
$302$		0			0
$303$	17.3205	+	17.3205i	0.995037	+	0.995037i
$304$		0			0
$305$	−6.00000	−	12.0000i	−0.343559	−	0.687118i
$306$		0			0
$307$	−5.19615	+	5.19615i	−0.296560	+	0.296560i	−0.839665	−	0.543105i	$-0.817249\pi$
$307$	−5.19615	+	5.19615i	−0.296560	+	0.296560i	0.543105	+	0.839665i	$0.317249\pi$
$308$		0			0
$309$			6.00000i			0.341328i
$310$		0			0
$311$		−	17.3205i		−	0.982156i	−0.871116	−	0.491078i	$-0.836603\pi$
$311$		−	17.3205i		−	0.982156i	0.871116	−	0.491078i	$-0.163397\pi$
$312$		0			0
$313$	−7.00000	+	7.00000i	−0.395663	+	0.395663i	−0.876700	−	0.481037i	$-0.840260\pi$
$313$	−7.00000	+	7.00000i	−0.395663	+	0.395663i	0.481037	+	0.876700i	$0.340260\pi$
$314$		0			0
$315$	5.19615	−	15.5885i	0.292770	−	0.878310i
$316$		0			0
$317$	−19.0000	−	19.0000i	−1.06715	−	1.06715i	−0.997577	−	0.0695692i	$-0.977838\pi$
$317$	−19.0000	−	19.0000i	−1.06715	−	1.06715i	−0.0695692	−	0.997577i	$-0.522162\pi$
$318$		0			0
$319$	13.8564			0.775810
$320$		0			0
$321$	−30.0000			−1.67444
$322$		0			0
$323$	6.92820	+	6.92820i	0.385496	+	0.385496i
$324$		0			0
$325$	1.00000	+	7.00000i	0.0554700	+	0.388290i
$326$		0			0
$327$	−20.7846	+	20.7846i	−1.14939	+	1.14939i
$328$		0			0
$329$		−	6.00000i		−	0.330791i
$330$		0			0
$331$		−	24.2487i		−	1.33283i	−0.745581	−	0.666415i	$-0.767828\pi$
$331$		−	24.2487i		−	1.33283i	0.745581	−	0.666415i	$-0.232172\pi$
$332$		0			0
$333$	−15.0000	+	15.0000i	−0.821995	+	0.821995i
$334$		0			0
$335$	15.5885	+	5.19615i	0.851688	+	0.283896i
$336$		0			0
$337$	1.00000	+	1.00000i	0.0544735	+	0.0544735i	0.733819	−	0.679345i	$-0.237736\pi$
$337$	1.00000	+	1.00000i	0.0544735	+	0.0544735i	−0.679345	+	0.733819i	$0.737736\pi$
$338$		0			0
$339$	−31.1769			−1.69330
$340$		0			0
$341$	12.0000			0.649836
$342$		0			0
$343$	13.8564	+	13.8564i	0.748176	+	0.748176i
$344$		0			0
$345$	12.0000	−	6.00000i	0.646058	−	0.323029i
$346$		0			0
$347$	−5.19615	+	5.19615i	−0.278944	+	0.278944i	−0.832687	−	0.553743i	$-0.813199\pi$
$347$	−5.19615	+	5.19615i	−0.278944	+	0.278944i	0.553743	+	0.832687i	$0.313199\pi$
$348$		0			0
$349$			12.0000i			0.642345i	0.947021	+	0.321173i	$0.104077\pi$
$349$			12.0000i			0.642345i	−0.947021	+	0.321173i	$0.895923\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	−	1.00000i	0.0532246	−	0.0532246i	−0.679994	−	0.733218i	$-0.738017\pi$
$353$	1.00000	−	1.00000i	0.0532246	−	0.0532246i	0.733218	+	0.679994i	$0.238017\pi$
$354$		0			0
$355$	20.7846	−	10.3923i	1.10313	−	0.551566i
$356$		0			0
$357$	−6.00000	−	6.00000i	−0.317554	−	0.317554i
$358$		0			0
$359$	−13.8564			−0.731313			−0.365657	−	0.930750i	$-0.619156\pi$
$359$	−13.8564			−0.731313			−0.365657	+	0.930750i	$0.619156\pi$
$360$		0			0
$361$	29.0000			1.52632
$362$		0			0
$363$	1.73205	+	1.73205i	0.0909091	+	0.0909091i
$364$		0			0
$365$	21.0000	+	7.00000i	1.09919	+	0.366397i
$366$		0			0
$367$	15.5885	−	15.5885i	0.813711	−	0.813711i	−0.171477	−	0.985188i	$-0.554854\pi$
$367$	15.5885	−	15.5885i	0.813711	−	0.813711i	0.985188	+	0.171477i	$0.0548540\pi$
$368$		0			0
$369$			6.00000i			0.312348i
$370$		0			0
$371$			24.2487i			1.25893i
$372$		0			0
$373$	1.00000	−	1.00000i	0.0517780	−	0.0517780i	−0.680744	−	0.732522i	$-0.738343\pi$
$373$	1.00000	−	1.00000i	0.0517780	−	0.0517780i	0.732522	+	0.680744i	$0.238343\pi$
$374$		0			0
$375$	−15.5885	−	22.5167i	−0.804984	−	1.16276i
$376$		0			0
$377$	−4.00000	−	4.00000i	−0.206010	−	0.206010i
$378$		0			0
$379$	−6.92820			−0.355878			−0.177939	−	0.984042i	$-0.556943\pi$
$379$	−6.92820			−0.355878			−0.177939	+	0.984042i	$0.556943\pi$
$380$		0			0
$381$	42.0000			2.15173
$382$		0			0
$383$	−15.5885	−	15.5885i	−0.796533	−	0.796533i	0.186014	−	0.982547i	$-0.440443\pi$
$383$	−15.5885	−	15.5885i	−0.796533	−	0.796533i	−0.982547	+	0.186014i	$0.940443\pi$
$384$		0			0
$385$	6.00000	−	18.0000i	0.305788	−	0.917365i
$386$		0			0
$387$	5.19615	−	5.19615i	0.264135	−	0.264135i
$388$		0			0
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$		−	3.46410i		−	0.175187i
$392$		0			0
$393$	−18.0000	+	18.0000i	−0.907980	+	0.907980i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−11.0000	−	11.0000i	−0.552074	−	0.552074i	0.374965	−	0.927039i	$-0.377655\pi$
$397$	−11.0000	−	11.0000i	−0.552074	−	0.552074i	−0.927039	+	0.374965i	$0.877655\pi$
$398$		0			0
$399$	−41.5692			−2.08106
$400$		0			0
$401$	2.00000			0.0998752			0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000			0.0998752			0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$	−3.46410	−	3.46410i	−0.172559	−	0.172559i
$404$		0			0
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$		0			0
$407$	−17.3205	+	17.3205i	−0.858546	+	0.858546i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		−	31.1769i		−	1.53784i
$412$		0			0
$413$	−12.0000	+	12.0000i	−0.590481	+	0.590481i
$414$		0			0
$415$	12.1244	−	36.3731i	0.595161	−	1.78548i
$416$		0			0
$417$	12.0000	+	12.0000i	0.587643	+	0.587643i
$418$		0			0
$419$	6.92820			0.338465			0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820			0.338465			0.169232	+	0.985576i	$0.445871\pi$
$420$		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
$423$	5.19615	+	5.19615i	0.252646	+	0.252646i
$424$		0			0
$425$	−7.00000	+	1.00000i	−0.339550	+	0.0485071i
$426$		0			0
$427$	10.3923	−	10.3923i	0.502919	−	0.502919i
$428$		0			0
$429$		−	12.0000i		−	0.579365i
$430$		0			0
$431$		−	3.46410i		−	0.166860i	−0.996514	−	0.0834300i	$-0.973413\pi$
$431$		−	3.46410i		−	0.166860i	0.996514	−	0.0834300i	$-0.0265875\pi$
$432$		0			0
$433$	−7.00000	+	7.00000i	−0.336399	+	0.336399i	−0.855010	−	0.518611i	$-0.826449\pi$
$433$	−7.00000	+	7.00000i	−0.336399	+	0.336399i	0.518611	+	0.855010i	$0.326449\pi$
$434$		0			0
$435$	20.7846	+	6.92820i	0.996546	+	0.332182i
$436$		0			0
$437$	−12.0000	−	12.0000i	−0.574038	−	0.574038i
$438$		0			0
$439$	13.8564			0.661330			0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564			0.661330			0.330665	+	0.943748i	$0.392727\pi$
$440$		0			0
$441$	−3.00000			−0.142857
$442$		0			0
$443$	12.1244	+	12.1244i	0.576046	+	0.576046i	0.933811	−	0.357766i	$-0.116461\pi$
$443$	12.1244	+	12.1244i	0.576046	+	0.576046i	−0.357766	+	0.933811i	$0.616461\pi$
$444$		0			0
$445$	−16.0000	+	8.00000i	−0.758473	+	0.379236i
$446$		0			0
$447$	−6.92820	+	6.92820i	−0.327693	+	0.327693i
$448$		0			0
$449$			8.00000i			0.377543i	0.982021	+	0.188772i	$0.0604506\pi$
$449$			8.00000i			0.377543i	−0.982021	+	0.188772i	$0.939549\pi$
$450$		0			0
$451$			6.92820i			0.326236i
$452$		0			0
$453$	18.0000	−	18.0000i	0.845714	−	0.845714i
$454$		0			0
$455$	−6.92820	+	3.46410i	−0.324799	+	0.162400i
$456$		0			0
$457$	17.0000	+	17.0000i	0.795226	+	0.795226i	0.982339	−	0.187112i	$-0.0599128\pi$
$457$	17.0000	+	17.0000i	0.795226	+	0.795226i	−0.187112	+	0.982339i	$0.559913\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.00000			−0.0931493			−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000			−0.0931493			−0.0465746	+	0.998915i	$0.514831\pi$
$462$		0			0
$463$	−1.73205	−	1.73205i	−0.0804952	−	0.0804952i	0.665713	−	0.746208i	$-0.268128\pi$
$463$	−1.73205	−	1.73205i	−0.0804952	−	0.0804952i	−0.746208	+	0.665713i	$0.768128\pi$
$464$		0			0
$465$	18.0000	+	6.00000i	0.834730	+	0.278243i
$466$		0			0
$467$	−19.0526	+	19.0526i	−0.881647	+	0.881647i	−0.993702	−	0.112055i	$-0.964257\pi$
$467$	−19.0526	+	19.0526i	−0.881647	+	0.881647i	0.112055	+	0.993702i	$0.464257\pi$
$468$		0			0
$469$			18.0000i			0.831163i
$470$		0			0
$471$			38.1051i			1.75579i
$472$		0			0
$473$	6.00000	−	6.00000i	0.275880	−	0.275880i
$474$		0			0
$475$	−20.7846	+	27.7128i	−0.953663	+	1.27155i
$476$		0			0
$477$	−21.0000	−	21.0000i	−0.961524	−	0.961524i
$478$		0			0
$479$	27.7128			1.26623			0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128			1.26623			0.633115	+	0.774057i	$0.281776\pi$
$480$		0			0
$481$	10.0000			0.455961
$482$		0			0
$483$	10.3923	+	10.3923i	0.472866	+	0.472866i
$484$		0			0
$485$	−7.00000	+	21.0000i	−0.317854	+	0.953561i
$486$		0			0
$487$	1.73205	−	1.73205i	0.0784867	−	0.0784867i	−0.666774	−	0.745260i	$-0.732325\pi$
$487$	1.73205	−	1.73205i	0.0784867	−	0.0784867i	0.745260	+	0.666774i	$0.232325\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$		−	24.2487i		−	1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$		−	24.2487i		−	1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$		0			0
$493$	4.00000	−	4.00000i	0.180151	−	0.180151i
$494$		0			0
$495$	10.3923	+	20.7846i	0.467099	+	0.934199i
$496$		0			0
$497$	18.0000	+	18.0000i	0.807410	+	0.807410i
$498$		0			0
$499$	−20.7846			−0.930447			−0.465223	−	0.885193i	$-0.654026\pi$
$499$	−20.7846			−0.930447			−0.465223	+	0.885193i	$0.654026\pi$
$500$		0			0
$501$	−54.0000			−2.41254
$502$		0			0
$503$	−1.73205	−	1.73205i	−0.0772283	−	0.0772283i	0.667437	−	0.744666i	$-0.267391\pi$
$503$	−1.73205	−	1.73205i	−0.0772283	−	0.0772283i	−0.744666	+	0.667437i	$0.767391\pi$
$504$		0			0
$505$	10.0000	+	20.0000i	0.444994	+	0.889988i
$506$		0			0
$507$	19.0526	−	19.0526i	0.846154	−	0.846154i
$508$		0			0
$509$		−	20.0000i		−	0.886484i	−0.896402	−	0.443242i	$-0.853828\pi$
$509$		−	20.0000i		−	0.886484i	0.896402	−	0.443242i	$-0.146172\pi$
$510$		0			0
$511$			24.2487i			1.07270i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.73205	+	5.19615i	−0.0763233	+	0.228970i
$516$		0			0
$517$	6.00000	+	6.00000i	0.263880	+	0.263880i
$518$		0			0
$519$	51.9615			2.28086
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$	12.1244	+	12.1244i	0.530161	+	0.530161i	0.920620	−	0.390459i	$-0.127684\pi$
$523$	12.1244	+	12.1244i	0.530161	+	0.530161i	−0.390459	+	0.920620i	$0.627684\pi$
$524$		0			0
$525$	18.0000	−	24.0000i	0.785584	−	1.04745i
$526$		0			0
$527$	3.46410	−	3.46410i	0.150899	−	0.150899i
$528$		0			0
$529$		−	17.0000i		−	0.739130i
$530$		0			0
$531$		−	20.7846i		−	0.901975i
$532$		0			0
$533$	2.00000	−	2.00000i	0.0866296	−	0.0866296i
$534$		0			0
$535$	−25.9808	−	8.66025i	−1.12325	−	0.374415i
$536$		0			0
$537$	−12.0000	−	12.0000i	−0.517838	−	0.517838i
$538$		0			0
$539$	−3.46410			−0.149209
$540$		0			0
$541$	−18.0000			−0.773880			−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000			−0.773880			−0.386940	+	0.922105i	$0.626468\pi$
$542$		0			0
$543$	−10.3923	−	10.3923i	−0.445976	−	0.445976i
$544$		0			0
$545$	−24.0000	+	12.0000i	−1.02805	+	0.514024i
$546$		0			0
$547$	8.66025	−	8.66025i	0.370286	−	0.370286i	−0.497296	−	0.867581i	$-0.665674\pi$
$547$	8.66025	−	8.66025i	0.370286	−	0.370286i	0.867581	+	0.497296i	$0.165674\pi$
$548$		0			0
$549$			18.0000i			0.768221i
$550$		0			0
$551$		−	27.7128i		−	1.18061i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−34.6410	+	17.3205i	−1.47043	+	0.735215i
$556$		0			0
$557$	29.0000	+	29.0000i	1.22877	+	1.22877i	0.964430	+	0.264340i	$0.0851541\pi$
$557$	29.0000	+	29.0000i	1.22877	+	1.22877i	0.264340	+	0.964430i	$0.414846\pi$
$558$		0			0
$559$	−3.46410			−0.146516
$560$		0			0
$561$	12.0000			0.506640
$562$		0			0
$563$	−29.4449	−	29.4449i	−1.24095	−	1.24095i	−0.959606	−	0.281347i	$-0.909219\pi$
$563$	−29.4449	−	29.4449i	−1.24095	−	1.24095i	−0.281347	−	0.959606i	$-0.590781\pi$
$564$		0			0
$565$	−27.0000	−	9.00000i	−1.13590	−	0.378633i
$566$		0			0
$567$	15.5885	−	15.5885i	0.654654	−	0.654654i
$568$		0			0
$569$			16.0000i			0.670755i	0.942084	+	0.335377i	$0.108864\pi$
$569$			16.0000i			0.670755i	−0.942084	+	0.335377i	$0.891136\pi$
$570$		0			0
$571$			31.1769i			1.30471i	0.757912	+	0.652357i	$0.226220\pi$
$571$			31.1769i			1.30471i	−0.757912	+	0.652357i	$0.773780\pi$
$572$		0			0
$573$	42.0000	−	42.0000i	1.75458	−	1.75458i
$574$		0			0
$575$	12.1244	−	1.73205i	0.505621	−	0.0722315i
$576$		0			0
$577$	−7.00000	−	7.00000i	−0.291414	−	0.291414i	0.546225	−	0.837639i	$-0.316064\pi$
$577$	−7.00000	−	7.00000i	−0.291414	−	0.291414i	−0.837639	+	0.546225i	$0.816064\pi$
$578$		0			0
$579$	−3.46410			−0.143963
$580$		0			0
$581$	42.0000			1.74245
$582$		0			0
$583$	−24.2487	−	24.2487i	−1.00428	−	1.00428i
$584$		0			0
$585$	3.00000	−	9.00000i	0.124035	−	0.372104i
$586$		0			0
$587$	−5.19615	+	5.19615i	−0.214468	+	0.214468i	−0.806162	−	0.591694i	$-0.798459\pi$
$587$	−5.19615	+	5.19615i	−0.214468	+	0.214468i	0.591694	+	0.806162i	$0.298459\pi$
$588$		0			0
$589$		−	24.0000i		−	0.988903i
$590$		0			0
$591$			10.3923i			0.427482i
$592$		0			0
$593$	33.0000	−	33.0000i	1.35515	−	1.35515i	0.475352	−	0.879796i	$-0.342321\pi$
$593$	33.0000	−	33.0000i	1.35515	−	1.35515i	0.879796	−	0.475352i	$-0.157679\pi$
$594$		0			0
$595$	−3.46410	−	6.92820i	−0.142014	−	0.284029i
$596$		0			0
$597$	24.0000	+	24.0000i	0.982255	+	0.982255i
$598$		0			0
$599$	13.8564			0.566157			0.283079	−	0.959097i	$-0.408644\pi$
$599$	13.8564			0.566157			0.283079	+	0.959097i	$0.408644\pi$
$600$		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
$603$	−15.5885	−	15.5885i	−0.634811	−	0.634811i
$604$		0			0
$605$	1.00000	+	2.00000i	0.0406558	+	0.0813116i
$606$		0			0
$607$	−25.9808	+	25.9808i	−1.05453	+	1.05453i	−0.0561015	+	0.998425i	$0.517867\pi$
$607$	−25.9808	+	25.9808i	−1.05453	+	1.05453i	−0.998425	+	0.0561015i	$0.982133\pi$
$608$		0			0
$609$			24.0000i			0.972529i
$610$		0			0
$611$		−	3.46410i		−	0.140143i
$612$		0			0
$613$	−31.0000	+	31.0000i	−1.25208	+	1.25208i	−0.297291	+	0.954787i	$0.596083\pi$
$613$	−31.0000	+	31.0000i	−1.25208	+	1.25208i	−0.954787	+	0.297291i	$0.903917\pi$
$614$		0			0
$615$	−3.46410	+	10.3923i	−0.139686	+	0.419058i
$616$		0			0
$617$	−7.00000	−	7.00000i	−0.281809	−	0.281809i	0.552021	−	0.833830i	$-0.313857\pi$
$617$	−7.00000	−	7.00000i	−0.281809	−	0.281809i	−0.833830	+	0.552021i	$0.813857\pi$
$618$		0			0
$619$	20.7846			0.835404			0.417702	−	0.908584i	$-0.362836\pi$
$619$	20.7846			0.835404			0.417702	+	0.908584i	$0.362836\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.8564	−	13.8564i	−0.555145	−	0.555145i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$	41.5692	−	41.5692i	1.66011	−	1.66011i
$628$		0			0
$629$			10.0000i			0.398726i
$630$		0			0
$631$		−	17.3205i		−	0.689519i	−0.938691	−	0.344759i	$-0.887961\pi$
$631$		−	17.3205i		−	0.689519i	0.938691	−	0.344759i	$-0.112039\pi$
$632$		0			0
$633$	−18.0000	+	18.0000i	−0.715436	+	0.715436i
$634$		0			0
$635$	36.3731	+	12.1244i	1.44342	+	0.481140i
$636$		0			0
$637$	1.00000	+	1.00000i	0.0396214	+	0.0396214i
$638$		0			0
$639$	−31.1769			−1.23334
$640$		0			0
$641$	26.0000			1.02694			0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000			1.02694			0.513469	+	0.858108i	$0.328360\pi$
$642$		0			0
$643$	−15.5885	−	15.5885i	−0.614749	−	0.614749i	0.329431	−	0.944180i	$-0.393143\pi$
$643$	−15.5885	−	15.5885i	−0.614749	−	0.614749i	−0.944180	+	0.329431i	$0.893143\pi$
$644$		0			0
$645$	12.0000	−	6.00000i	0.472500	−	0.236250i
$646$		0			0
$647$	1.73205	−	1.73205i	0.0680939	−	0.0680939i	−0.672240	−	0.740334i	$-0.734668\pi$
$647$	1.73205	−	1.73205i	0.0680939	−	0.0680939i	0.740334	+	0.672240i	$0.234668\pi$
$648$		0			0
$649$		−	24.0000i		−	0.942082i
$650$		0			0
$651$			20.7846i			0.814613i
$652$		0			0
$653$	−23.0000	+	23.0000i	−0.900060	+	0.900060i	−0.995441	−	0.0953813i	$-0.969593\pi$
$653$	−23.0000	+	23.0000i	−0.900060	+	0.900060i	0.0953813	+	0.995441i	$0.469593\pi$
$654$		0			0
$655$	−20.7846	+	10.3923i	−0.812122	+	0.406061i
$656$		0			0
$657$	−21.0000	−	21.0000i	−0.819288	−	0.819288i
$658$		0			0
$659$	34.6410			1.34942			0.674711	−	0.738082i	$-0.264268\pi$
$659$	34.6410			1.34942			0.674711	+	0.738082i	$0.264268\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$	−3.46410	−	3.46410i	−0.134535	−	0.134535i
$664$		0			0
$665$	−36.0000	−	12.0000i	−1.39602	−	0.465340i
$666$		0			0
$667$	−6.92820	+	6.92820i	−0.268261	+	0.268261i
$668$		0			0
$669$			54.0000i			2.08776i
$670$		0			0
$671$			20.7846i			0.802381i
$672$		0			0
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	−0.0356839	−	0.999363i	$-0.511361\pi$
$673$	25.0000	−	25.0000i	0.963679	−	0.963679i	0.999363	+	0.0356839i	$0.0113610\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−35.0000	−	35.0000i	−1.34516	−	1.34516i	−0.890838	−	0.454321i	$-0.849882\pi$
$677$	−35.0000	−	35.0000i	−1.34516	−	1.34516i	−0.454321	−	0.890838i	$-0.650118\pi$
$678$		0			0
$679$	−24.2487			−0.930580
$680$		0			0
$681$	−30.0000			−1.14960
$682$		0			0
$683$	−1.73205	−	1.73205i	−0.0662751	−	0.0662751i	0.673192	−	0.739467i	$-0.264923\pi$
$683$	−1.73205	−	1.73205i	−0.0662751	−	0.0662751i	−0.739467	+	0.673192i	$0.764923\pi$
$684$		0			0
$685$	9.00000	−	27.0000i	0.343872	−	1.03162i
$686$		0			0
$687$	34.6410	−	34.6410i	1.32164	−	1.32164i
$688$		0			0
$689$			14.0000i			0.533358i
$690$		0			0
$691$			45.0333i			1.71315i	0.516024	+	0.856574i	$0.327412\pi$
$691$			45.0333i			1.71315i	−0.516024	+	0.856574i	$0.672588\pi$
$692$		0			0
$693$	−18.0000	+	18.0000i	−0.683763	+	0.683763i
$694$		0			0
$695$	6.92820	+	13.8564i	0.262802	+	0.525603i
$696$		0			0
$697$	2.00000	+	2.00000i	0.0757554	+	0.0757554i
$698$		0			0
$699$	−3.46410			−0.131024
$700$		0			0
$701$	38.0000			1.43524			0.717620	−	0.696435i	$-0.245231\pi$
$701$	38.0000			1.43524			0.717620	+	0.696435i	$0.245231\pi$
$702$		0			0
$703$	34.6410	+	34.6410i	1.30651	+	1.30651i
$704$		0			0
$705$	6.00000	+	12.0000i	0.225973	+	0.451946i
$706$		0			0
$707$	−17.3205	+	17.3205i	−0.651405	+	0.651405i
$708$		0			0
$709$			4.00000i			0.150223i	0.997175	+	0.0751116i	$0.0239313\pi$
$709$			4.00000i			0.150223i	−0.997175	+	0.0751116i	$0.976069\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.00000	+	6.00000i	−0.224702	+	0.224702i
$714$		0			0
$715$	3.46410	−	10.3923i	0.129550	−	0.388650i
$716$		0			0
$717$	48.0000	+	48.0000i	1.79259	+	1.79259i
$718$		0			0
$719$	27.7128			1.03351			0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128			1.03351			0.516757	+	0.856132i	$0.327139\pi$
$720$		0			0
$721$	−6.00000			−0.223452
$722$		0			0
$723$	38.1051	+	38.1051i	1.41714	+	1.41714i
$724$		0			0
$725$	16.0000	+	12.0000i	0.594225	+	0.445669i
$726$		0			0
$727$	1.73205	−	1.73205i	0.0642382	−	0.0642382i	−0.674258	−	0.738496i	$-0.735536\pi$
$727$	1.73205	−	1.73205i	0.0642382	−	0.0642382i	0.738496	+	0.674258i	$0.235536\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		−	3.46410i		−	0.128124i
$732$		0			0
$733$	1.00000	−	1.00000i	0.0369358	−	0.0369358i	−0.688398	−	0.725333i	$-0.741686\pi$
$733$	1.00000	−	1.00000i	0.0369358	−	0.0369358i	0.725333	+	0.688398i	$0.241686\pi$
$734$		0			0
$735$	−5.19615	−	1.73205i	−0.191663	−	0.0638877i
$736$		0			0
$737$	−18.0000	−	18.0000i	−0.663039	−	0.663039i
$738$		0			0
$739$	−20.7846			−0.764574			−0.382287	−	0.924044i	$-0.624863\pi$
$739$	−20.7846			−0.764574			−0.382287	+	0.924044i	$0.624863\pi$
$740$		0			0
$741$	−24.0000			−0.881662
$742$		0			0
$743$	12.1244	+	12.1244i	0.444799	+	0.444799i	0.893621	−	0.448822i	$-0.148156\pi$
$743$	12.1244	+	12.1244i	0.444799	+	0.444799i	−0.448822	+	0.893621i	$0.648156\pi$
$744$		0			0
$745$	−8.00000	+	4.00000i	−0.293097	+	0.146549i
$746$		0			0
$747$	−36.3731	+	36.3731i	−1.33082	+	1.33082i
$748$		0			0
$749$		−	30.0000i		−	1.09618i
$750$		0			0
$751$		−	3.46410i		−	0.126407i	−0.998001	−	0.0632034i	$-0.979868\pi$
$751$		−	3.46410i		−	0.126407i	0.998001	−	0.0632034i	$-0.0201317\pi$
$752$		0			0
$753$	6.00000	−	6.00000i	0.218652	−	0.218652i
$754$		0			0
$755$	20.7846	−	10.3923i	0.756429	−	0.378215i
$756$		0			0
$757$	29.0000	+	29.0000i	1.05402	+	1.05402i	0.998455	+	0.0555680i	$0.0176970\pi$
$757$	29.0000	+	29.0000i	1.05402	+	1.05402i	0.0555680	+	0.998455i	$0.482303\pi$
$758$		0			0
$759$	−20.7846			−0.754434
$760$		0			0
$761$	10.0000			0.362500			0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000			0.362500			0.181250	+	0.983437i	$0.441986\pi$
$762$		0			0
$763$	−20.7846	−	20.7846i	−0.752453	−	0.752453i
$764$		0			0
$765$	9.00000	+	3.00000i	0.325396	+	0.108465i
$766$		0			0
$767$	−6.92820	+	6.92820i	−0.250163	+	0.250163i
$768$		0			0
$769$		−	48.0000i		−	1.73092i	−0.500974	−	0.865462i	$-0.667025\pi$
$769$		−	48.0000i		−	1.73092i	0.500974	−	0.865462i	$-0.332975\pi$
$770$		0			0
$771$		−	31.1769i		−	1.12281i
$772$		0			0
$773$	1.00000	−	1.00000i	0.0359675	−	0.0359675i	−0.688894	−	0.724862i	$-0.741904\pi$
$773$	1.00000	−	1.00000i	0.0359675	−	0.0359675i	0.724862	+	0.688894i	$0.241904\pi$
$774$		0			0
$775$	13.8564	+	10.3923i	0.497737	+	0.373303i
$776$		0			0
$777$	−30.0000	−	30.0000i	−1.07624	−	1.07624i
$778$		0			0
$779$	13.8564			0.496457
$780$		0			0
$781$	−36.0000			−1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.0000	+	33.0000i	−0.392607	+	1.17782i
$786$		0			0
$787$	−5.19615	+	5.19615i	−0.185223	+	0.185223i	−0.793627	−	0.608404i	$-0.791810\pi$
$787$	−5.19615	+	5.19615i	−0.185223	+	0.185223i	0.608404	+	0.793627i	$0.291810\pi$
$788$		0			0
$789$		−	42.0000i		−	1.49524i
$790$		0			0
$791$		−	31.1769i		−	1.10852i

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	80.n (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.73205 - 1.73205i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.73205 - 1.73205i) q^{3} +(-1.00000 - 2.00000i) q^{5} +(1.73205 - 1.73205i) q^{7} +3.00000i q^{9} +3.46410i q^{11} +(1.00000 - 1.00000i) q^{13} +(-1.73205 + 5.19615i) q^{15} +(1.00000 + 1.00000i) q^{17} +6.92820 q^{19} -6.00000 q^{21} +(-1.73205 - 1.73205i) q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000i q^{29} -3.46410i q^{31} +(6.00000 - 6.00000i) q^{33} +(-5.19615 - 1.73205i) q^{35} +(5.00000 + 5.00000i) q^{37} -3.46410 q^{39} +2.00000 q^{41} +(-1.73205 - 1.73205i) q^{43} +(6.00000 - 3.00000i) q^{45} +(1.73205 - 1.73205i) q^{47} +1.00000i q^{49} -3.46410i q^{51} +(-7.00000 + 7.00000i) q^{53} +(6.92820 - 3.46410i) q^{55} +(-12.0000 - 12.0000i) q^{57} -6.92820 q^{59} +6.00000 q^{61} +(5.19615 + 5.19615i) q^{63} +(-3.00000 - 1.00000i) q^{65} +(-5.19615 + 5.19615i) q^{67} +6.00000i q^{69} +10.3923i q^{71} +(-7.00000 + 7.00000i) q^{73} +(12.1244 - 1.73205i) q^{75} +(6.00000 + 6.00000i) q^{77} +9.00000 q^{81} +(12.1244 + 12.1244i) q^{83} +(1.00000 - 3.00000i) q^{85} +(-6.92820 + 6.92820i) q^{87} -8.00000i q^{89} -3.46410i q^{91} +(-6.00000 + 6.00000i) q^{93} +(-6.92820 - 13.8564i) q^{95} +(-7.00000 - 7.00000i) q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} + 4 q^{13} + 4 q^{17} - 24 q^{21} - 12 q^{25} + 24 q^{33} + 20 q^{37} + 8 q^{41} + 24 q^{45} - 28 q^{53} - 48 q^{57} + 24 q^{61} - 12 q^{65} - 28 q^{73} + 24 q^{77} + 36 q^{81} + 4 q^{85} - 24 q^{93} - 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 4 * q^13 + 4 * q^17 - 24 * q^21 - 12 * q^25 + 24 * q^33 + 20 * q^37 + 8 * q^41 + 24 * q^45 - 28 * q^53 - 48 * q^57 + 24 * q^61 - 12 * q^65 - 28 * q^73 + 24 * q^77 + 36 * q^81 + 4 * q^85 - 24 * q^93 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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