LMFDB - Embedded newform 1520.2.d.b.609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not 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:	$12.1372611072$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.b.609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +4.00000 q^{31} +(-4.00000 - 2.00000i) q^{35} +10.0000i q^{37} -10.0000 q^{41} -2.00000i q^{43} +(-3.00000 + 6.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} +10.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +2.00000 q^{61} +6.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +8.00000i q^{67} -4.00000 q^{71} +4.00000i q^{73} +8.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +18.0000i q^{83} +(8.00000 + 4.00000i) q^{85} +2.00000 q^{89} +4.00000 q^{91} +(-1.00000 + 2.00000i) q^{95} -6.00000i q^{97} +12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{19} - 6 q^{25} + 12 q^{29} + 8 q^{31} - 8 q^{35} - 20 q^{41} - 6 q^{45} + 6 q^{49} - 8 q^{55} + 4 q^{61} + 8 q^{65} - 8 q^{71} + 8 q^{79} + 18 q^{81} + 16 q^{85}+ \cdots + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^9 + 8 * q^11 + 2 * q^19 - 6 * q^25 + 12 * q^29 + 8 * q^31 - 8 * q^35 - 20 * q^41 - 6 * q^45 + 6 * q^49 - 8 * q^55 + 4 * q^61 + 8 * q^65 - 8 * q^71 + 8 * q^79 + 18 * q^81 + 16 * q^85 + 4 * q^89 + 8 * q^91 - 2 * q^95 + 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0		1.00000			$0$
$3$		0			0		−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
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$		0			0
$9$	3.00000			1.00000
$10$		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
$13$		−	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
$15$		0			0
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$		0			0
$22$		0			0
$23$			6.00000i			1.25109i	0.780189	+	0.625543i	$0.215123\pi$
$23$			6.00000i			1.25109i	−0.780189	+	0.625543i	$0.784877\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.00000	−	2.00000i	−0.676123	−	0.338062i
$36$		0			0
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$		0			0
$45$	−3.00000	+	6.00000i	−0.447214	+	0.894427i
$46$		0			0
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$			10.0000i			1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$			10.0000i			1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$		0			0
$55$	−4.00000	+	8.00000i	−0.539360	+	1.07872i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$			6.00000i			0.755929i
$64$		0			0
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			8.00000i			0.911685i
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$			18.0000i			1.97576i	0.155230	+	0.987878i	$0.450388\pi$
$83$			18.0000i			1.97576i	−0.155230	+	0.987878i	$0.549612\pi$
$84$		0			0
$85$	8.00000	+	4.00000i	0.867722	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	2.00000			0.212000			0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000			0.212000			0.106000	+	0.994366i	$0.466196\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.00000	+	2.00000i	−0.102598	+	0.205196i
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		0			0
$99$	12.0000			1.20605
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		−	16.0000i		−	1.57653i	−0.615338	−	0.788263i	$-0.710980\pi$
$103$		−	16.0000i		−	1.57653i	0.615338	−	0.788263i	$-0.289020\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	4.00000i		−	0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$		−	4.00000i		−	0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$		0			0
$109$	−6.00000			−0.574696			−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000			−0.574696			−0.287348	+	0.957826i	$0.592774\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	−12.0000	−	6.00000i	−1.11901	−	0.559503i
$116$		0			0
$117$		−	6.00000i		−	0.554700i
$118$		0			0
$119$	8.00000			0.733359
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		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
$133$			2.00000i			0.173422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	8.00000i		−	0.668994i
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$	−24.0000			−1.95309			−0.976546	−	0.215308i	$-0.930924\pi$
$151$	−24.0000			−1.95309			−0.976546	+	0.215308i	$0.930924\pi$
$152$		0			0
$153$		−	12.0000i		−	0.970143i
$154$		0			0
$155$	−4.00000	+	8.00000i	−0.321288	+	0.642575i
$156$		0			0
$157$			8.00000i			0.638470i	0.947676	+	0.319235i	$0.103426\pi$
$157$			8.00000i			0.638470i	−0.947676	+	0.319235i	$0.896574\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−12.0000			−0.945732
$162$		0			0
$163$		−	10.0000i		−	0.783260i	−0.920123	−	0.391630i	$-0.871911\pi$
$163$		−	10.0000i		−	0.783260i	0.920123	−	0.391630i	$-0.128089\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.0000i			0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$			12.0000i			0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	3.00000			0.229416
$172$		0			0
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$	8.00000	−	6.00000i	0.604743	−	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−20.0000	−	10.0000i	−1.47043	−	0.735215i
$186$		0			0
$187$		−	16.0000i		−	1.17004i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	24.0000			1.73658			0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000			1.73658			0.868290	+	0.496058i	$0.165220\pi$
$192$		0			0
$193$		−	26.0000i		−	1.87152i	−0.352636	−	0.935760i	$-0.614715\pi$
$193$		−	26.0000i		−	1.87152i	0.352636	−	0.935760i	$-0.385285\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			12.0000i			0.842235i
$204$		0			0
$205$	10.0000	−	20.0000i	0.698430	−	1.39686i
$206$		0			0
$207$			18.0000i			1.25109i
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	+	2.00000i	0.272798	+	0.136399i
$216$		0			0
$217$			8.00000i			0.543075i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.00000			−0.538138
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−9.00000	−	12.0000i	−0.600000	−	0.800000i
$226$		0			0
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			16.0000i			1.04819i	0.851658	+	0.524097i	$0.175597\pi$
$233$			16.0000i			1.04819i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	12.0000	+	6.00000i	0.782794	+	0.391397i
$236$		0			0
$237$		0			0
$238$		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
$241$	−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
$243$		0			0
$244$		0			0
$245$	−3.00000	+	6.00000i	−0.191663	+	0.383326i
$246$		0			0
$247$		−	2.00000i		−	0.127257i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.00000			−0.252478			−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000			−0.252478			−0.126239	+	0.992000i	$0.540291\pi$
$252$		0			0
$253$			24.0000i			1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−20.0000			−1.24274
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$			6.00000i			0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$			6.00000i			0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$		0			0
$265$	−20.0000	−	10.0000i	−1.22859	−	0.614295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−12.0000	−	16.0000i	−0.723627	−	0.964836i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		0			0
$279$	12.0000			0.718421
$280$		0			0
$281$	−26.0000			−1.55103			−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000			−1.55103			−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		−	14.0000i		−	0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$		−	14.0000i		−	0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	20.0000i		−	1.18056i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	14.0000i		−	0.817889i	−0.912559	−	0.408944i	$-0.865897\pi$
$293$		−	14.0000i		−	0.817889i	0.912559	−	0.408944i	$-0.134103\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000			0.693978
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000	+	4.00000i	−0.114520	+	0.229039i
$306$		0			0
$307$		−	20.0000i		−	1.14146i	−0.821138	−	0.570730i	$-0.806660\pi$
$307$		−	20.0000i		−	1.14146i	0.821138	−	0.570730i	$-0.193340\pi$
$308$		0			0
$309$		0			0
$310$		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
$313$		−	24.0000i		−	1.35656i	−0.734803	−	0.678280i	$-0.762726\pi$
$313$		−	24.0000i		−	1.35656i	0.734803	−	0.678280i	$-0.237274\pi$
$314$		0			0
$315$	−12.0000	−	6.00000i	−0.676123	−	0.338062i
$316$		0			0
$317$		−	22.0000i		−	1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$		−	22.0000i		−	1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.00000i		−	0.222566i
$324$		0			0
$325$	−8.00000	+	6.00000i	−0.443760	+	0.332820i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.0000			0.661581
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$			30.0000i			1.64399i
$334$		0			0
$335$	−16.0000	−	8.00000i	−0.874173	−	0.437087i
$336$		0			0
$337$		−	22.0000i		−	1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$		−	22.0000i		−	1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$			20.0000i			1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	6.00000i		−	0.322097i	−0.986947	−	0.161048i	$-0.948512\pi$
$347$		−	6.00000i		−	0.322097i	0.986947	−	0.161048i	$-0.0514875\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$			4.00000i			0.212899i	0.994318	+	0.106449i	$0.0339482\pi$
$353$			4.00000i			0.212899i	−0.994318	+	0.106449i	$0.966052\pi$
$354$		0			0
$355$	4.00000	−	8.00000i	0.212298	−	0.424596i
$356$		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
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.00000	−	4.00000i	−0.418739	−	0.209370i
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		0			0
$369$	−30.0000			−1.56174
$370$		0			0
$371$	−20.0000			−1.03835
$372$		0			0
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	12.0000i		−	0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$		−	12.0000i		−	0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$		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
$391$	24.0000			1.21373
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000	+	8.00000i	−0.201262	+	0.402524i
$396$		0			0
$397$			8.00000i			0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$			8.00000i			0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	18.0000			0.898877			0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000			0.898877			0.449439	+	0.893311i	$0.351624\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$		0			0
$405$	−9.00000	+	18.0000i	−0.447214	+	0.894427i
$406$		0			0
$407$			40.0000i			1.98273i
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−36.0000	−	18.0000i	−1.76717	−	0.883585i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	36.0000			1.75872			0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000			1.75872			0.879358	+	0.476162i	$0.157972\pi$
$420$		0			0
$421$	−30.0000			−1.46211			−0.731055	−	0.682318i	$-0.760972\pi$
$421$	−30.0000			−1.46211			−0.731055	+	0.682318i	$0.760972\pi$
$422$		0			0
$423$		−	18.0000i		−	0.875190i
$424$		0			0
$425$	−16.0000	+	12.0000i	−0.776114	+	0.582086i
$426$		0			0
$427$			4.00000i			0.193574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		−	34.0000i		−	1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$		−	34.0000i		−	1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			6.00000i			0.287019i
$438$		0			0
$439$	20.0000			0.954548			0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000			0.954548			0.477274	+	0.878755i	$0.341625\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$		0			0
$443$			2.00000i			0.0950229i	0.998871	+	0.0475114i	$0.0151291\pi$
$443$			2.00000i			0.0950229i	−0.998871	+	0.0475114i	$0.984871\pi$
$444$		0			0
$445$	−2.00000	+	4.00000i	−0.0948091	+	0.189618i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−10.0000			−0.471929			−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000			−0.471929			−0.235965	+	0.971762i	$0.575825\pi$
$450$		0			0
$451$	−40.0000			−1.88353
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.00000	+	8.00000i	−0.187523	+	0.375046i
$456$		0			0
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.0000			0.652045			0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000			0.652045			0.326023	+	0.945362i	$0.394291\pi$
$462$		0			0
$463$		−	2.00000i		−	0.0929479i	−0.998920	−	0.0464739i	$-0.985202\pi$
$463$		−	2.00000i		−	0.0929479i	0.998920	−	0.0464739i	$-0.0147984\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	38.0000i		−	1.75843i	−0.476425	−	0.879215i	$-0.658068\pi$
$467$		−	38.0000i		−	1.75843i	0.476425	−	0.879215i	$-0.341932\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	8.00000i		−	0.367840i
$474$		0			0
$475$	−3.00000	−	4.00000i	−0.137649	−	0.183533i
$476$		0			0
$477$			30.0000i			1.37361i
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	+	6.00000i	0.544892	+	0.272446i
$486$		0			0
$487$		−	20.0000i		−	0.906287i	−0.891438	−	0.453143i	$-0.850303\pi$
$487$		−	20.0000i		−	0.906287i	0.891438	−	0.453143i	$-0.149697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		0			0
$493$		−	24.0000i		−	1.08091i
$494$		0			0
$495$	−12.0000	+	24.0000i	−0.539360	+	1.07872i
$496$		0			0
$497$		−	8.00000i		−	0.358849i
$498$		0			0
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	10.0000i		−	0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$		−	10.0000i		−	0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$		0			0
$505$	6.00000	−	12.0000i	0.266996	−	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$		0			0
$511$	−8.00000			−0.353899
$512$		0			0
$513$		0			0
$514$		0			0
$515$	32.0000	+	16.0000i	1.41009	+	0.705044i
$516$		0			0
$517$		−	24.0000i		−	1.05552i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$			44.0000i			1.92399i	0.273075	+	0.961993i	$0.411959\pi$
$523$			44.0000i			1.92399i	−0.273075	+	0.961993i	$0.588041\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	16.0000i		−	0.696971i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$			20.0000i			0.866296i
$534$		0			0
$535$	8.00000	+	4.00000i	0.345870	+	0.172935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12.0000			0.516877
$540$		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
$543$		0			0
$544$		0			0
$545$	6.00000	−	12.0000i	0.257012	−	0.514024i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$		0			0
$549$	6.00000			0.256074
$550$		0			0
$551$	6.00000			0.255609
$552$		0			0
$553$			8.00000i			0.340195i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			16.0000i			0.677942i	0.940797	+	0.338971i	$0.110079\pi$
$557$			16.0000i			0.677942i	−0.940797	+	0.338971i	$0.889921\pi$
$558$		0			0
$559$	−4.00000			−0.169182
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	12.0000i		−	0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$		−	12.0000i		−	0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$		0			0
$565$	−12.0000	−	6.00000i	−0.504844	−	0.252422i
$566$		0			0
$567$			18.0000i			0.755929i
$568$		0			0
$569$	−42.0000			−1.76073			−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000			−1.76073			−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000	−	18.0000i	1.00087	−	0.750652i
$576$		0			0
$577$		−	24.0000i		−	0.999133i	−0.866276	−	0.499567i	$-0.833493\pi$
$577$		−	24.0000i		−	0.999133i	0.866276	−	0.499567i	$-0.166507\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−36.0000			−1.49353
$582$		0			0
$583$			40.0000i			1.65663i
$584$		0			0
$585$	12.0000	+	6.00000i	0.496139	+	0.248069i
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	8.00000i		−	0.328521i	−0.986417	−	0.164260i	$-0.947476\pi$
$593$		−	8.00000i		−	0.328521i	0.986417	−	0.164260i	$-0.0525237\pi$
$594$		0			0
$595$	−8.00000	+	16.0000i	−0.327968	+	0.655936i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$		0			0
$605$	−5.00000	+	10.0000i	−0.203279	+	0.406558i
$606$		0			0
$607$		−	44.0000i		−	1.78590i	−0.450151	−	0.892952i	$-0.648630\pi$
$607$		−	44.0000i		−	1.78590i	0.450151	−	0.892952i	$-0.351370\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$		−	24.0000i		−	0.969351i	−0.874694	−	0.484675i	$-0.838938\pi$
$613$		−	24.0000i		−	0.969351i	0.874694	−	0.484675i	$-0.161062\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	12.0000i		−	0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$		−	12.0000i		−	0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.00000i			0.160257i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	40.0000			1.59490
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	6.00000i		−	0.237729i
$638$		0			0
$639$	−12.0000			−0.474713
$640$		0			0
$641$	6.00000			0.236986			0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000			0.236986			0.118493	+	0.992955i	$0.462194\pi$
$642$		0			0
$643$		−	38.0000i		−	1.49857i	−0.662246	−	0.749287i	$-0.730396\pi$
$643$		−	38.0000i		−	1.49857i	0.662246	−	0.749287i	$-0.269604\pi$
$644$		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
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			16.0000i			0.626128i	0.949732	+	0.313064i	$0.101356\pi$
$653$			16.0000i			0.626128i	−0.949732	+	0.313064i	$0.898644\pi$
$654$		0			0
$655$	−12.0000	+	24.0000i	−0.468879	+	0.937758i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$	−16.0000			−0.623272			−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000			−0.623272			−0.311636	+	0.950202i	$0.600877\pi$
$660$		0			0
$661$	−30.0000			−1.16686			−0.583432	−	0.812162i	$-0.698291\pi$
$661$	−30.0000			−1.16686			−0.583432	+	0.812162i	$0.698291\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−4.00000	−	2.00000i	−0.155113	−	0.0775567i
$666$		0			0
$667$			36.0000i			1.39393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8.00000			0.308837
$672$		0			0
$673$			34.0000i			1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$			34.0000i			1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	18.0000i		−	0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$		−	18.0000i		−	0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	12.0000i		−	0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$		−	12.0000i		−	0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.0000			0.761939
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$			24.0000i			0.911685i
$694$		0			0
$695$	4.00000	−	8.00000i	0.151729	−	0.303457i
$696$		0			0
$697$			40.0000i			1.51511i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$			10.0000i			0.377157i
$704$		0			0
$705$		0			0
$706$		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
$711$	12.0000			0.450035
$712$		0			0
$713$			24.0000i			0.898807i
$714$		0			0
$715$	16.0000	+	8.00000i	0.598366	+	0.299183i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	32.0000			1.19174
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	−	24.0000i	−0.668503	−	0.891338i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−8.00000			−0.295891
$732$		0			0
$733$		−	12.0000i		−	0.443230i	−0.975134	−	0.221615i	$-0.928867\pi$
$733$		−	12.0000i		−	0.443230i	0.975134	−	0.221615i	$-0.0711328\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			32.0000i			1.17874i
$738$		0			0
$739$	4.00000			0.147142			0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000			0.147142			0.0735712	+	0.997290i	$0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4.00000i			0.146746i	0.997305	+	0.0733729i	$0.0233763\pi$
$743$			4.00000i			0.146746i	−0.997305	+	0.0733729i	$0.976624\pi$
$744$		0			0
$745$	−10.0000	+	20.0000i	−0.366372	+	0.732743i
$746$		0			0
$747$			54.0000i			1.97576i
$748$		0			0
$749$	8.00000			0.292314
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.0000	−	48.0000i	0.873449	−	1.74690i
$756$		0			0
$757$			4.00000i			0.145382i	0.997354	+	0.0726912i	$0.0231588\pi$
$757$			4.00000i			0.145382i	−0.997354	+	0.0726912i	$0.976841\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6.00000			0.217500			0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000			0.217500			0.108750	+	0.994069i	$0.465315\pi$
$762$		0			0
$763$		−	12.0000i		−	0.434429i
$764$		0			0
$765$	24.0000	+	12.0000i	0.867722	+	0.433861i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−26.0000			−0.937584			−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000			−0.937584			−0.468792	+	0.883309i	$0.655311\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			46.0000i			1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$			46.0000i			1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$		0			0
$775$	−12.0000	−	16.0000i	−0.431053	−	0.574737i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−10.0000			−0.358287
$780$		0			0
$781$	−16.0000			−0.572525
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−16.0000	−	8.00000i	−0.571064	−	0.285532i
$786$		0			0
$787$		−	44.0000i		−	1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$		−	44.0000i		−	1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$		−	4.00000i		−	0.142044i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.00000i			0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$			6.00000i			0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$		0			0
$799$	−24.0000			−0.849059
$800$		0			0
$801$	6.00000			0.212000
$802$		0			0
$803$			16.0000i			0.564628i
$804$		0			0
$805$	12.0000	−	24.0000i	0.422944	−	0.845889i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		0			0
$811$	24.0000			0.842754			0.421377	−	0.906886i	$-0.361547\pi$
$811$	24.0000			0.842754			0.421377	+	0.906886i	$0.361547\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	20.0000	+	10.0000i	0.700569	+	0.350285i
$816$		0			0
$817$		−	2.00000i		−	0.0699711i
$818$		0			0
$819$	12.0000			0.419314
$820$		0			0
$821$	−42.0000			−1.46581			−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000			−1.46581			−0.732905	+	0.680331i	$0.761836\pi$
$822$		0			0
$823$			42.0000i			1.46403i	0.681290	+	0.732014i	$0.261419\pi$
$823$			42.0000i			1.46403i	−0.681290	+	0.732014i	$0.738581\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.0000i			0.834562i	0.908778	+	0.417281i	$0.137017\pi$
$827$			24.0000i			0.834562i	−0.908778	+	0.417281i	$0.862983\pi$
$828$		0			0
$829$	−14.0000			−0.486240			−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000			−0.486240			−0.243120	+	0.969996i	$0.578171\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	12.0000i		−	0.415775i
$834$		0			0
$835$	−24.0000	−	12.0000i	−0.830554	−	0.415277i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−12.0000			−0.414286			−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000			−0.414286			−0.207143	+	0.978311i	$0.566417\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−9.00000	+	18.0000i	−0.309609	+	0.619219i
$846$		0			0
$847$			10.0000i			0.343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−60.0000			−2.05677
$852$		0			0
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$		0			0
$855$	−3.00000	+	6.00000i	−0.102598	+	0.205196i
$856$		0			0
$857$			6.00000i			0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$			6.00000i			0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			28.0000i			0.953131i	0.879139	+	0.476566i	$0.158119\pi$
$863$			28.0000i			0.953131i	−0.879139	+	0.476566i	$0.841881\pi$
$864$		0			0
$865$	12.0000	+	6.00000i	0.408012	+	0.204006i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	16.0000			0.542763
$870$		0			0
$871$	16.0000			0.542139
$872$		0			0
$873$		−	18.0000i		−	0.609208i
$874$		0			0
$875$	4.00000	+	22.0000i	0.135225	+	0.743736i
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		0			0
$883$		−	34.0000i		−	1.14419i	−0.820187	−	0.572096i	$-0.806131\pi$
$883$		−	34.0000i		−	1.14419i	0.820187	−	0.572096i	$-0.193869\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			8.00000i			0.268614i	0.990940	+	0.134307i	$0.0428808\pi$
$887$			8.00000i			0.268614i	−0.990940	+	0.134307i	$0.957119\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	36.0000			1.20605
$892$		0			0
$893$		−	6.00000i		−	0.200782i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$	40.0000			1.33259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	18.0000	−	36.0000i	0.598340	−	1.19668i
$906$		0			0
$907$			12.0000i			0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$			12.0000i			0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$		0			0
$909$	−18.0000			−0.597022
$910$		0			0
$911$	−8.00000			−0.265052			−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000			−0.265052			−0.132526	+	0.991180i	$0.542309\pi$
$912$		0			0
$913$			72.0000i			2.38285i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			24.0000i			0.792550i
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			8.00000i			0.263323i
$924$		0			0
$925$	40.0000	−	30.0000i	1.31519	−	0.986394i
$926$		0			0
$927$		−	48.0000i		−	1.57653i
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	3.00000			0.0983210
$932$		0			0
$933$		0			0
$934$		0			0
$935$	32.0000	+	16.0000i	1.04651	+	0.523256i
$936$		0			0
$937$			20.0000i			0.653372i	0.945133	+	0.326686i	$0.105932\pi$
$937$			20.0000i			0.653372i	−0.945133	+	0.326686i	$0.894068\pi$
$938$		0			0
$939$		0			0
$940$		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
$943$		−	60.0000i		−	1.95387i
$944$		0			0
$945$		0			0
$946$		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
$949$	8.00000			0.259691
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	30.0000i		−	0.971795i	−0.874016	−	0.485898i	$-0.838493\pi$
$953$		−	30.0000i		−	0.971795i	0.874016	−	0.485898i	$-0.161507\pi$
$954$		0			0
$955$	−24.0000	+	48.0000i	−0.776622	+	1.55324i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000			0.775000
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$		−	12.0000i		−	0.386695i
$964$		0			0
$965$	52.0000	+	26.0000i	1.67394	+	0.836970i
$966$		0			0
$967$		−	2.00000i		−	0.0643157i	−0.999483	−	0.0321578i	$-0.989762\pi$
$967$		−	2.00000i		−	0.0643157i	0.999483	−	0.0321578i	$-0.0102379\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−24.0000			−0.770197			−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000			−0.770197			−0.385098	+	0.922876i	$0.625832\pi$
$972$		0			0
$973$		−	8.00000i		−	0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			14.0000i			0.447900i	0.974601	+	0.223950i	$0.0718952\pi$
$977$			14.0000i			0.447900i	−0.974601	+	0.223950i	$0.928105\pi$
$978$		0			0
$979$	8.00000			0.255681
$980$		0			0
$981$	−18.0000			−0.574696
$982$		0			0
$983$			12.0000i			0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$			12.0000i			0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$		0			0
$985$	−24.0000	−	12.0000i	−0.764704	−	0.382352i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12.0000			0.381578
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000	+	32.0000i	−0.507234	+	1.01447i
$996$		0			0
$997$		−	32.0000i		−	1.01345i	−0.862108	−	0.506725i	$-0.830856\pi$
$997$		−	32.0000i		−	1.01345i	0.862108	−	0.506725i	$-0.169144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.b.609.2		2
4.3	odd	2		95.2.b.a.39.2	yes	2
5.2	odd	4		7600.2.a.i.1.1		1
5.3	odd	4		7600.2.a.l.1.1		1
5.4	even	2	inner	1520.2.d.b.609.1		2
12.11	even	2		855.2.c.b.514.1		2
20.3	even	4		475.2.a.c.1.1		1
20.7	even	4		475.2.a.a.1.1		1
20.19	odd	2		95.2.b.a.39.1	✓	2
60.23	odd	4		4275.2.a.e.1.1		1
60.47	odd	4		4275.2.a.p.1.1		1
60.59	even	2		855.2.c.b.514.2		2
76.75	even	2		1805.2.b.c.1084.1		2
380.227	odd	4		9025.2.a.h.1.1		1
380.303	odd	4		9025.2.a.c.1.1		1
380.379	even	2		1805.2.b.c.1084.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.b.a.39.1	✓	2	20.19	odd	2
95.2.b.a.39.2	yes	2	4.3	odd	2
475.2.a.a.1.1		1	20.7	even	4
475.2.a.c.1.1		1	20.3	even	4
855.2.c.b.514.1		2	12.11	even	2
855.2.c.b.514.2		2	60.59	even	2
1520.2.d.b.609.1		2	5.4	even	2	inner
1520.2.d.b.609.2		2	1.1	even	1	trivial
1805.2.b.c.1084.1		2	76.75	even	2
1805.2.b.c.1084.2		2	380.379	even	2
4275.2.a.e.1.1		1	60.23	odd	4
4275.2.a.p.1.1		1	60.47	odd	4
7600.2.a.i.1.1		1	5.2	odd	4
7600.2.a.l.1.1		1	5.3	odd	4
9025.2.a.c.1.1		1	380.303	odd	4
9025.2.a.h.1.1		1	380.227	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 2.00000i) q^{5} +2.00000i q^{7} +3.00000 q^{9} +4.00000 q^{11} -2.00000i q^{13} -4.00000i q^{17} +1.00000 q^{19} +6.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +4.00000 q^{31} +(-4.00000 - 2.00000i) q^{35} +10.0000i q^{37} -10.0000 q^{41} -2.00000i q^{43} +(-3.00000 + 6.00000i) q^{45} -6.00000i q^{47} +3.00000 q^{49} +10.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +2.00000 q^{61} +6.00000i q^{63} +(4.00000 + 2.00000i) q^{65} +8.00000i q^{67} -4.00000 q^{71} +4.00000i q^{73} +8.00000i q^{77} +4.00000 q^{79} +9.00000 q^{81} +18.0000i q^{83} +(8.00000 + 4.00000i) q^{85} +2.00000 q^{89} +4.00000 q^{91} +(-1.00000 + 2.00000i) q^{95} -6.00000i q^{97} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 6 q^{9} + 8 q^{11} + 2 q^{19} - 6 q^{25} + 12 q^{29} + 8 q^{31} - 8 q^{35} - 20 q^{41} - 6 q^{45} + 6 q^{49} - 8 q^{55} + 4 q^{61} + 8 q^{65} - 8 q^{71} + 8 q^{79} + 18 q^{81} + 16 q^{85}+ \cdots + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^9 + 8 * q^11 + 2 * q^19 - 6 * q^25 + 12 * q^29 + 8 * q^31 - 8 * q^35 - 20 * q^41 - 6 * q^45 + 6 * q^49 - 8 * q^55 + 4 * q^61 + 8 * q^65 - 8 * q^71 + 8 * q^79 + 18 * q^81 + 16 * q^85 + 4 * q^89 + 8 * q^91 - 2 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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