LMFDB - Embedded newform 1600.2.l.c.401.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,2,Mod(401,1600)] 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("1600.401"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.l (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$12.7760643234$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			401.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.401
Dual form			1600.2.l.c.1201.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.00000i) q^{3} -1.00000i q^{9} +(3.00000 - 3.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} -4.00000 q^{17} +(-1.00000 - 1.00000i) q^{19} -8.00000i q^{23} +(4.00000 - 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{29} +6.00000 q^{33} +(-3.00000 + 3.00000i) q^{37} -6.00000i q^{39} +(-3.00000 + 3.00000i) q^{43} +2.00000 q^{47} +7.00000 q^{49} +(-4.00000 - 4.00000i) q^{51} +(-9.00000 + 9.00000i) q^{53} -2.00000i q^{57} +(9.00000 - 9.00000i) q^{59} +(-5.00000 - 5.00000i) q^{61} +(3.00000 + 3.00000i) q^{67} +(8.00000 - 8.00000i) q^{69} +6.00000i q^{71} -6.00000i q^{73} +8.00000 q^{79} +5.00000 q^{81} +(9.00000 + 9.00000i) q^{83} -6.00000i q^{87} -12.0000i q^{89} +12.0000 q^{97} +(-3.00000 - 3.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{11} - 6 q^{13} - 8 q^{17} - 2 q^{19} + 8 q^{27} - 6 q^{29} + 12 q^{33} - 6 q^{37} - 6 q^{43} + 4 q^{47} + 14 q^{49} - 8 q^{51} - 18 q^{53} + 18 q^{59} - 10 q^{61} + 6 q^{67} + 16 q^{69}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^11 - 6 * q^13 - 8 * q^17 - 2 * q^19 + 8 * q^27 - 6 * q^29 + 12 * q^33 - 6 * q^37 - 6 * q^43 + 4 * q^47 + 14 * q^49 - 8 * q^51 - 18 * q^53 + 18 * q^59 - 10 * q^61 + 6 * q^67 + 16 * q^69 + 16 * q^79 + 10 * q^81 + 18 * q^83 + 24 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	0.934172	−	0.356822i	$-0.116140\pi$
$3$	1.00000	+	1.00000i	0.577350	+	0.577350i	−0.356822	+	0.934172i	$0.616140\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	3.00000	−	3.00000i	0.904534	−	0.904534i	−0.0912903	−	0.995824i	$-0.529099\pi$
$11$	3.00000	−	3.00000i	0.904534	−	0.904534i	0.995824	+	0.0912903i	$0.0290991\pi$
$12$		0			0
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	0.155747	−	0.987797i	$-0.450222\pi$
$13$	−3.00000	−	3.00000i	−0.832050	−	0.832050i	−0.987797	+	0.155747i	$0.950222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$	−1.00000	−	1.00000i	−0.229416	−	0.229416i	0.583033	−	0.812449i	$-0.301866\pi$
$19$	−1.00000	−	1.00000i	−0.229416	−	0.229416i	−0.812449	+	0.583033i	$0.801866\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	8.00000i		−	1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$		−	8.00000i		−	1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$	6.00000			1.04447
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	−0.909312	−	0.416115i	$-0.863391\pi$
$37$	−3.00000	+	3.00000i	−0.493197	+	0.493197i	0.416115	+	0.909312i	$0.363391\pi$
$38$		0			0
$39$		−	6.00000i		−	0.960769i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	−0.897833	−	0.440337i	$-0.854859\pi$
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	0.440337	+	0.897833i	$0.354859\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.00000			0.291730			0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000			0.291730			0.145865	+	0.989305i	$0.453403\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−4.00000	−	4.00000i	−0.560112	−	0.560112i
$52$		0			0
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.274721	+	0.961524i	$0.588586\pi$
$53$	−9.00000	+	9.00000i	−1.23625	+	1.23625i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	2.00000i		−	0.264906i
$58$		0			0
$59$	9.00000	−	9.00000i	1.17170	−	1.17170i	0.189896	−	0.981804i	$-0.439185\pi$
$59$	9.00000	−	9.00000i	1.17170	−	1.17170i	0.981804	−	0.189896i	$-0.0608151\pi$
$60$		0			0
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	0.310416	−	0.950601i	$-0.399532\pi$
$61$	−5.00000	−	5.00000i	−0.640184	−	0.640184i	−0.950601	+	0.310416i	$0.899532\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	0.866202	−	0.499694i	$-0.166554\pi$
$67$	3.00000	+	3.00000i	0.366508	+	0.366508i	−0.499694	+	0.866202i	$0.666554\pi$
$68$		0			0
$69$	8.00000	−	8.00000i	0.963087	−	0.963087i
$70$		0			0
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	8.00000			0.900070			0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000			0.900070			0.450035	+	0.893011i	$0.351411\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$	9.00000	+	9.00000i	0.987878	+	0.987878i	0.999927	−	0.0120491i	$-0.00383543\pi$
$83$	9.00000	+	9.00000i	0.987878	+	0.987878i	−0.0120491	+	0.999927i	$0.503835\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		0			0
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000			1.21842			0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000			1.21842			0.609208	+	0.793011i	$0.291488\pi$
$98$		0			0
$99$	−3.00000	−	3.00000i	−0.301511	−	0.301511i
$100$		0			0
$101$	3.00000	−	3.00000i	0.298511	−	0.298511i	−0.541919	−	0.840431i	$-0.682302\pi$
$101$	3.00000	−	3.00000i	0.298511	−	0.298511i	0.840431	+	0.541919i	$0.182302\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.00000	+	9.00000i	−0.870063	+	0.870063i	−0.992479	−	0.122416i	$-0.960936\pi$
$107$	−9.00000	+	9.00000i	−0.870063	+	0.870063i	0.122416	+	0.992479i	$0.460936\pi$
$108$		0			0
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	0.753374	−	0.657592i	$-0.228425\pi$
$109$	1.00000	+	1.00000i	0.0957826	+	0.0957826i	−0.657592	+	0.753374i	$0.728425\pi$
$110$		0			0
$111$	−6.00000			−0.569495
$112$		0			0
$113$	8.00000			0.752577			0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000			0.752577			0.376288	+	0.926503i	$0.377200\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−3.00000	+	3.00000i	−0.277350	+	0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	7.00000i		−	0.636364i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−6.00000			−0.532414			−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000			−0.532414			−0.266207	+	0.963916i	$0.585770\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$	9.00000	+	9.00000i	0.786334	+	0.786334i	0.980891	−	0.194557i	$-0.0623271\pi$
$131$	9.00000	+	9.00000i	0.786334	+	0.786334i	−0.194557	+	0.980891i	$0.562327\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$			2.00000i			0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$			2.00000i			0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$		0			0
$139$	−7.00000	+	7.00000i	−0.593732	+	0.593732i	−0.938638	−	0.344905i	$-0.887911\pi$
$139$	−7.00000	+	7.00000i	−0.593732	+	0.593732i	0.344905	+	0.938638i	$0.387911\pi$
$140$		0			0
$141$	2.00000	+	2.00000i	0.168430	+	0.168430i
$142$		0			0
$143$	−18.0000			−1.50524
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	+	7.00000i	0.577350	+	0.577350i
$148$		0			0
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−3.00000	+	3.00000i	−0.245770	+	0.245770i	0.573462	+	0.819232i	$0.305600\pi$
$150$		0			0
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$		0			0
$153$			4.00000i			0.323381i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	0.249974	−	0.968252i	$-0.419578\pi$
$157$	−9.00000	−	9.00000i	−0.718278	−	0.718278i	−0.968252	+	0.249974i	$0.919578\pi$
$158$		0			0
$159$	−18.0000			−1.42749
$160$		0			0
$161$		0			0
$162$		0			0
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	0.965465	−	0.260531i	$-0.0838976\pi$
$163$	9.00000	+	9.00000i	0.704934	+	0.704934i	−0.260531	+	0.965465i	$0.583898\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$			5.00000i			0.384615i
$170$		0			0
$171$	−1.00000	+	1.00000i	−0.0764719	+	0.0764719i
$172$		0			0
$173$	9.00000	+	9.00000i	0.684257	+	0.684257i	0.960957	−	0.276699i	$-0.0892406\pi$
$173$	9.00000	+	9.00000i	0.684257	+	0.684257i	−0.276699	+	0.960957i	$0.589241\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	18.0000			1.35296
$178$		0			0
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	0.810277	−	0.586047i	$-0.199317\pi$
$179$	3.00000	+	3.00000i	0.224231	+	0.224231i	−0.586047	+	0.810277i	$0.699317\pi$
$180$		0			0
$181$	−1.00000	+	1.00000i	−0.0743294	+	0.0743294i	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−1.00000	+	1.00000i	−0.0743294	+	0.0743294i	0.668965	+	0.743294i	$0.266738\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.0000	+	12.0000i	−0.877527	+	0.877527i
$188$		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
$193$	12.0000			0.863779			0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000			0.863779			0.431889	+	0.901927i	$0.357847\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	−0.506188	−	0.862423i	$-0.668946\pi$
$197$	5.00000	−	5.00000i	0.356235	−	0.356235i	0.862423	+	0.506188i	$0.168946\pi$
$198$		0			0
$199$			2.00000i			0.141776i	0.997484	+	0.0708881i	$0.0225833\pi$
$199$			2.00000i			0.141776i	−0.997484	+	0.0708881i	$0.977417\pi$
$200$		0			0
$201$			6.00000i			0.423207i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.00000			−0.556038
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	−11.0000	−	11.0000i	−0.757271	−	0.757271i	0.218554	−	0.975825i	$-0.429866\pi$
$211$	−11.0000	−	11.0000i	−0.757271	−	0.757271i	−0.975825	+	0.218554i	$0.929866\pi$
$212$		0			0
$213$	−6.00000	+	6.00000i	−0.411113	+	0.411113i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	6.00000	−	6.00000i	0.405442	−	0.405442i
$220$		0			0
$221$	12.0000	+	12.0000i	0.807207	+	0.807207i
$222$		0			0
$223$	6.00000			0.401790			0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000			0.401790			0.200895	+	0.979613i	$0.435615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.00000	−	9.00000i	−0.597351	−	0.597351i	0.342256	−	0.939607i	$-0.388809\pi$
$227$	−9.00000	−	9.00000i	−0.597351	−	0.597351i	−0.939607	+	0.342256i	$0.888809\pi$
$228$		0			0
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	−0.899498	−	0.436925i	$-0.856068\pi$
$229$	−7.00000	+	7.00000i	−0.462573	+	0.462573i	0.436925	+	0.899498i	$0.356068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	22.0000i		−	1.44127i	−0.693316	−	0.720634i	$-0.743851\pi$
$233$		−	22.0000i		−	1.44127i	0.693316	−	0.720634i	$-0.256149\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.00000	+	8.00000i	0.519656	+	0.519656i
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−18.0000			−1.15948			−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000			−1.15948			−0.579741	+	0.814801i	$0.696846\pi$
$242$		0			0
$243$	−7.00000	−	7.00000i	−0.449050	−	0.449050i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			6.00000i			0.381771i
$248$		0			0
$249$			18.0000i			1.14070i
$250$		0			0
$251$	−9.00000	+	9.00000i	−0.568075	+	0.568075i	−0.931589	−	0.363514i	$-0.881577\pi$
$251$	−9.00000	+	9.00000i	−0.568075	+	0.568075i	0.363514	+	0.931589i	$0.381577\pi$
$252$		0			0
$253$	−24.0000	−	24.0000i	−1.50887	−	1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−8.00000			−0.499026			−0.249513	−	0.968371i	$-0.580271\pi$
$257$	−8.00000			−0.499026			−0.249513	+	0.968371i	$0.580271\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.00000	+	3.00000i	−0.185695	+	0.185695i
$262$		0			0
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.0000	−	12.0000i	0.734388	−	0.734388i
$268$		0			0
$269$	9.00000	+	9.00000i	0.548740	+	0.548740i	0.926076	−	0.377337i	$-0.123160\pi$
$269$	9.00000	+	9.00000i	0.548740	+	0.548740i	−0.377337	+	0.926076i	$0.623160\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.00000	+	3.00000i	−0.180253	+	0.180253i	−0.791466	−	0.611213i	$-0.790682\pi$
$277$	−3.00000	+	3.00000i	−0.180253	+	0.180253i	0.611213	+	0.791466i	$0.290682\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		0			0
$283$	−15.0000	+	15.0000i	−0.891657	+	0.891657i	−0.994679	−	0.103022i	$-0.967149\pi$
$283$	−15.0000	+	15.0000i	−0.891657	+	0.891657i	0.103022	+	0.994679i	$0.467149\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	12.0000	+	12.0000i	0.703452	+	0.703452i
$292$		0			0
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	−0.919313	−	0.393527i	$-0.871255\pi$
$293$	−9.00000	+	9.00000i	−0.525786	+	0.525786i	0.393527	+	0.919313i	$0.371255\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	24.0000i		−	1.39262i
$298$		0			0
$299$	−24.0000	+	24.0000i	−1.38796	+	1.38796i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	6.00000			0.344691
$304$		0			0
$305$		0			0
$306$		0			0
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	0.787515	−	0.616296i	$-0.211367\pi$
$307$	3.00000	+	3.00000i	0.171219	+	0.171219i	−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			6.00000i			0.340229i	0.985424	+	0.170114i	$0.0544137\pi$
$311$			6.00000i			0.340229i	−0.985424	+	0.170114i	$0.945586\pi$
$312$		0			0
$313$			6.00000i			0.339140i	0.985518	+	0.169570i	$0.0542379\pi$
$313$			6.00000i			0.339140i	−0.985518	+	0.169570i	$0.945762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	0.875812	−	0.482653i	$-0.160327\pi$
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	−0.482653	+	0.875812i	$0.660327\pi$
$318$		0			0
$319$	−18.0000			−1.00781
$320$		0			0
$321$	−18.0000			−1.00466
$322$		0			0
$323$	4.00000	+	4.00000i	0.222566	+	0.222566i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			2.00000i			0.110600i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.00000	+	5.00000i	−0.274825	+	0.274825i	−0.831039	−	0.556214i	$-0.812253\pi$
$331$	−5.00000	+	5.00000i	−0.274825	+	0.274825i	0.556214	+	0.831039i	$0.312253\pi$
$332$		0			0
$333$	3.00000	+	3.00000i	0.164399	+	0.164399i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−24.0000			−1.30736			−0.653682	−	0.756770i	$-0.726776\pi$
$337$	−24.0000			−1.30736			−0.653682	+	0.756770i	$0.726776\pi$
$338$		0			0
$339$	8.00000	+	8.00000i	0.434500	+	0.434500i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	19.0000	−	19.0000i	1.01997	−	1.01997i	0.0201770	−	0.999796i	$-0.493577\pi$
$347$	19.0000	−	19.0000i	1.01997	−	1.01997i	0.999796	−	0.0201770i	$-0.00642298\pi$
$348$		0			0
$349$	5.00000	+	5.00000i	0.267644	+	0.267644i	0.828150	−	0.560506i	$-0.189393\pi$
$349$	5.00000	+	5.00000i	0.267644	+	0.267644i	−0.560506	+	0.828150i	$0.689393\pi$
$350$		0			0
$351$	−24.0000			−1.28103
$352$		0			0
$353$	16.0000			0.851594			0.425797	−	0.904819i	$-0.359994\pi$
$353$	16.0000			0.851594			0.425797	+	0.904819i	$0.359994\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			18.0000i			0.950004i	0.879985	+	0.475002i	$0.157553\pi$
$359$			18.0000i			0.950004i	−0.879985	+	0.475002i	$0.842447\pi$
$360$		0			0
$361$		−	17.0000i		−	0.894737i
$362$		0			0
$363$	7.00000	−	7.00000i	0.367405	−	0.367405i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	18.0000			0.939592			0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000			0.939592			0.469796	+	0.882775i	$0.344327\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	−0.625161	−	0.780496i	$-0.714967\pi$
$373$	3.00000	−	3.00000i	0.155334	−	0.155334i	0.780496	+	0.625161i	$0.214967\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			18.0000i			0.927047i
$378$		0			0
$379$	1.00000	−	1.00000i	0.0513665	−	0.0513665i	−0.680957	−	0.732323i	$-0.738436\pi$
$379$	1.00000	−	1.00000i	0.0513665	−	0.0513665i	0.732323	+	0.680957i	$0.238436\pi$
$380$		0			0
$381$	−6.00000	−	6.00000i	−0.307389	−	0.307389i
$382$		0			0
$383$	−10.0000			−0.510976			−0.255488	−	0.966812i	$-0.582236\pi$
$383$	−10.0000			−0.510976			−0.255488	+	0.966812i	$0.582236\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.00000	+	3.00000i	0.152499	+	0.152499i
$388$		0			0
$389$	−15.0000	+	15.0000i	−0.760530	+	0.760530i	−0.976418	−	0.215888i	$-0.930735\pi$
$389$	−15.0000	+	15.0000i	−0.760530	+	0.760530i	0.215888	+	0.976418i	$0.430735\pi$
$390$		0			0
$391$			32.0000i			1.61831i
$392$		0			0
$393$			18.0000i			0.907980i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	0.444220	−	0.895918i	$-0.353481\pi$
$397$	−9.00000	−	9.00000i	−0.451697	−	0.451697i	−0.895918	+	0.444220i	$0.853481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	30.0000			1.49813			0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000			1.49813			0.749064	+	0.662497i	$0.230503\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			18.0000i			0.892227i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−2.00000	+	2.00000i	−0.0986527	+	0.0986527i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−14.0000			−0.685583
$418$		0			0
$419$	15.0000	+	15.0000i	0.732798	+	0.732798i	0.971173	−	0.238375i	$-0.0766148\pi$
$419$	15.0000	+	15.0000i	0.732798	+	0.732798i	−0.238375	+	0.971173i	$0.576615\pi$
$420$		0			0
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	−0.818373	−	0.574688i	$-0.805124\pi$
$421$	−5.00000	+	5.00000i	−0.243685	+	0.243685i	0.574688	+	0.818373i	$0.305124\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−18.0000	−	18.0000i	−0.869048	−	0.869048i
$430$		0			0
$431$	24.0000			1.15604			0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000			1.15604			0.578020	+	0.816023i	$0.303826\pi$
$432$		0			0
$433$	−36.0000			−1.73005			−0.865025	−	0.501729i	$-0.832697\pi$
$433$	−36.0000			−1.73005			−0.865025	+	0.501729i	$0.832697\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.00000	+	8.00000i	−0.382692	+	0.382692i
$438$		0			0
$439$			10.0000i			0.477274i	0.971109	+	0.238637i	$0.0767006\pi$
$439$			10.0000i			0.477274i	−0.971109	+	0.238637i	$0.923299\pi$
$440$		0			0
$441$		−	7.00000i		−	0.333333i
$442$		0			0
$443$	9.00000	−	9.00000i	0.427603	−	0.427603i	−0.460208	−	0.887811i	$-0.652225\pi$
$443$	9.00000	−	9.00000i	0.427603	−	0.427603i	0.887811	+	0.460208i	$0.152225\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−6.00000			−0.283790
$448$		0			0
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	18.0000	−	18.0000i	0.845714	−	0.845714i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$	−16.0000	+	16.0000i	−0.746816	+	0.746816i
$460$		0			0
$461$	3.00000	+	3.00000i	0.139724	+	0.139724i	0.773509	−	0.633785i	$-0.218500\pi$
$461$	3.00000	+	3.00000i	0.139724	+	0.139724i	−0.633785	+	0.773509i	$0.718500\pi$
$462$		0			0
$463$	30.0000			1.39422			0.697109	−	0.716965i	$-0.254469\pi$
$463$	30.0000			1.39422			0.697109	+	0.716965i	$0.254469\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.00000	−	5.00000i	−0.231372	−	0.231372i	0.581893	−	0.813265i	$-0.302312\pi$
$467$	−5.00000	−	5.00000i	−0.231372	−	0.231372i	−0.813265	+	0.581893i	$0.802312\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	18.0000i		−	0.829396i
$472$		0			0
$473$			18.0000i			0.827641i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	9.00000	+	9.00000i	0.412082	+	0.412082i
$478$		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
$481$	18.0000			0.820729
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		0			0
$489$			18.0000i			0.813988i
$490$		0			0
$491$	15.0000	−	15.0000i	0.676941	−	0.676941i	−0.282366	−	0.959307i	$-0.591119\pi$
$491$	15.0000	−	15.0000i	0.676941	−	0.676941i	0.959307	+	0.282366i	$0.0911193\pi$
$492$		0			0
$493$	12.0000	+	12.0000i	0.540453	+	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−29.0000	−	29.0000i	−1.29822	−	1.29822i	−0.929568	−	0.368650i	$-0.879820\pi$
$499$	−29.0000	−	29.0000i	−1.29822	−	1.29822i	−0.368650	−	0.929568i	$-0.620180\pi$
$500$		0			0
$501$	8.00000	−	8.00000i	0.357414	−	0.357414i
$502$		0			0
$503$			32.0000i			1.42681i	0.700752	+	0.713405i	$0.252848\pi$
$503$			32.0000i			1.42681i	−0.700752	+	0.713405i	$0.747152\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.00000	+	5.00000i	−0.222058	+	0.222058i
$508$		0			0
$509$	9.00000	+	9.00000i	0.398918	+	0.398918i	0.877851	−	0.478933i	$-0.158976\pi$
$509$	9.00000	+	9.00000i	0.398918	+	0.398918i	−0.478933	+	0.877851i	$0.658976\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.00000			−0.353209
$514$		0			0
$515$		0			0
$516$		0			0
$517$	6.00000	−	6.00000i	0.263880	−	0.263880i
$518$		0			0
$519$			18.0000i			0.790112i
$520$		0			0
$521$			24.0000i			1.05146i	0.850652	+	0.525730i	$0.176208\pi$
$521$			24.0000i			1.05146i	−0.850652	+	0.525730i	$0.823792\pi$
$522$		0			0
$523$	9.00000	−	9.00000i	0.393543	−	0.393543i	−0.482405	−	0.875948i	$-0.660237\pi$
$523$	9.00000	−	9.00000i	0.393543	−	0.393543i	0.875948	+	0.482405i	$0.160237\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	−9.00000	−	9.00000i	−0.390567	−	0.390567i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$			6.00000i			0.258919i
$538$		0			0
$539$	21.0000	−	21.0000i	0.904534	−	0.904534i
$540$		0			0
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	0.685283	−	0.728277i	$-0.259678\pi$
$541$	−1.00000	−	1.00000i	−0.0429934	−	0.0429934i	−0.728277	+	0.685283i	$0.759678\pi$
$542$		0			0
$543$	−2.00000			−0.0858282
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.00000	+	3.00000i	0.128271	+	0.128271i	0.768328	−	0.640057i	$-0.221089\pi$
$547$	3.00000	+	3.00000i	0.128271	+	0.128271i	−0.640057	+	0.768328i	$0.721089\pi$
$548$		0			0
$549$	−5.00000	+	5.00000i	−0.213395	+	0.213395i
$550$		0			0
$551$			6.00000i			0.255609i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.00000	−	9.00000i	−0.381342	−	0.381342i	0.490243	−	0.871586i	$-0.336908\pi$
$557$	−9.00000	−	9.00000i	−0.381342	−	0.381342i	−0.871586	+	0.490243i	$0.836908\pi$
$558$		0			0
$559$	18.0000			0.761319
$560$		0			0
$561$	−24.0000			−1.01328
$562$		0			0
$563$	−19.0000	−	19.0000i	−0.800755	−	0.800755i	0.182459	−	0.983213i	$-0.441594\pi$
$563$	−19.0000	−	19.0000i	−0.800755	−	0.800755i	−0.983213	+	0.182459i	$0.941594\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$		0			0
$571$	11.0000	−	11.0000i	0.460336	−	0.460336i	−0.438430	−	0.898765i	$-0.644465\pi$
$571$	11.0000	−	11.0000i	0.460336	−	0.460336i	0.898765	+	0.438430i	$0.144465\pi$
$572$		0			0
$573$	24.0000	+	24.0000i	1.00261	+	1.00261i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	24.0000			0.999133			0.499567	−	0.866276i	$-0.333493\pi$
$577$	24.0000			0.999133			0.499567	+	0.866276i	$0.333493\pi$
$578$		0			0
$579$	12.0000	+	12.0000i	0.498703	+	0.498703i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			54.0000i			2.23645i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.00000	+	9.00000i	−0.371470	+	0.371470i	−0.868012	−	0.496543i	$-0.834603\pi$
$587$	−9.00000	+	9.00000i	−0.371470	+	0.371470i	0.496543	+	0.868012i	$0.334603\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	10.0000			0.411345
$592$		0			0
$593$	−32.0000			−1.31408			−0.657041	−	0.753855i	$-0.728192\pi$
$593$	−32.0000			−1.31408			−0.657041	+	0.753855i	$0.728192\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.00000	+	2.00000i	−0.0818546	+	0.0818546i
$598$		0			0
$599$		−	30.0000i		−	1.22577i	−0.790173	−	0.612883i	$-0.790010\pi$
$599$		−	30.0000i		−	1.22577i	0.790173	−	0.612883i	$-0.209990\pi$
$600$		0			0
$601$			36.0000i			1.46847i	0.678895	+	0.734235i	$0.262459\pi$
$601$			36.0000i			1.46847i	−0.678895	+	0.734235i	$0.737541\pi$
$602$		0			0
$603$	3.00000	−	3.00000i	0.122169	−	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	42.0000			1.70473			0.852364	−	0.522949i	$-0.175168\pi$
$607$	42.0000			1.70473			0.852364	+	0.522949i	$0.175168\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.00000	−	6.00000i	−0.242734	−	0.242734i
$612$		0			0
$613$	27.0000	−	27.0000i	1.09052	−	1.09052i	0.0950469	−	0.995473i	$-0.469700\pi$
$613$	27.0000	−	27.0000i	1.09052	−	1.09052i	0.995473	−	0.0950469i	$-0.0303001\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10.0000i		−	0.402585i	−0.979531	−	0.201292i	$-0.935486\pi$
$617$		−	10.0000i		−	0.402585i	0.979531	−	0.201292i	$-0.0645141\pi$
$618$		0			0
$619$	13.0000	−	13.0000i	0.522514	−	0.522514i	−0.395816	−	0.918330i	$-0.629538\pi$
$619$	13.0000	−	13.0000i	0.522514	−	0.522514i	0.918330	+	0.395816i	$0.129538\pi$
$620$		0			0
$621$	−32.0000	−	32.0000i	−1.28412	−	1.28412i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−6.00000	−	6.00000i	−0.239617	−	0.239617i
$628$		0			0
$629$	12.0000	−	12.0000i	0.478471	−	0.478471i
$630$		0			0
$631$		−	2.00000i		−	0.0796187i	−0.999207	−	0.0398094i	$-0.987325\pi$
$631$		−	2.00000i		−	0.0796187i	0.999207	−	0.0398094i	$-0.0126751\pi$
$632$		0			0
$633$		−	22.0000i		−	0.874421i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−21.0000	−	21.0000i	−0.832050	−	0.832050i
$638$		0			0
$639$	6.00000			0.237356
$640$		0			0
$641$	30.0000			1.18493			0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000			1.18493			0.592464	+	0.805597i	$0.298155\pi$
$642$		0			0
$643$	−27.0000	−	27.0000i	−1.06478	−	1.06478i	−0.997751	−	0.0670247i	$-0.978649\pi$
$643$	−27.0000	−	27.0000i	−1.06478	−	1.06478i	−0.0670247	−	0.997751i	$-0.521351\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.0000i			1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$			32.0000i			1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$		0			0
$649$		−	54.0000i		−	2.11969i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	+	9.00000i	0.352197	+	0.352197i	0.860927	−	0.508729i	$-0.169885\pi$
$653$	9.00000	+	9.00000i	0.352197	+	0.352197i	−0.508729	+	0.860927i	$0.669885\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.00000			−0.234082
$658$		0			0
$659$	−21.0000	−	21.0000i	−0.818044	−	0.818044i	0.167781	−	0.985824i	$-0.446340\pi$
$659$	−21.0000	−	21.0000i	−0.818044	−	0.818044i	−0.985824	+	0.167781i	$0.946340\pi$
$660$		0			0
$661$	−29.0000	+	29.0000i	−1.12797	+	1.12797i	−0.137462	+	0.990507i	$0.543895\pi$
$661$	−29.0000	+	29.0000i	−1.12797	+	1.12797i	−0.990507	+	0.137462i	$0.956105\pi$
$662$		0			0
$663$			24.0000i			0.932083i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−24.0000	+	24.0000i	−0.929284	+	0.929284i
$668$		0			0
$669$	6.00000	+	6.00000i	0.231973	+	0.231973i
$670$		0			0
$671$	−30.0000			−1.15814
$672$		0			0
$673$	−12.0000			−0.462566			−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000			−0.462566			−0.231283	+	0.972887i	$0.574292\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.00000	−	9.00000i	0.345898	−	0.345898i	−0.512681	−	0.858579i	$-0.671348\pi$
$677$	9.00000	−	9.00000i	0.345898	−	0.345898i	0.858579	+	0.512681i	$0.171348\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	18.0000i		−	0.689761i
$682$		0			0
$683$	13.0000	−	13.0000i	0.497431	−	0.497431i	−0.413206	−	0.910637i	$-0.635591\pi$
$683$	13.0000	−	13.0000i	0.497431	−	0.497431i	0.910637	+	0.413206i	$0.135591\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−14.0000			−0.534133
$688$		0			0
$689$	54.0000			2.05724
$690$		0			0
$691$	5.00000	+	5.00000i	0.190209	+	0.190209i	0.795786	−	0.605577i	$-0.207058\pi$
$691$	5.00000	+	5.00000i	0.190209	+	0.190209i	−0.605577	+	0.795786i	$0.707058\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	22.0000	−	22.0000i	0.832116	−	0.832116i
$700$		0			0
$701$	3.00000	+	3.00000i	0.113308	+	0.113308i	0.761488	−	0.648179i	$-0.224469\pi$
$701$	3.00000	+	3.00000i	0.113308	+	0.113308i	−0.648179	+	0.761488i	$0.724469\pi$
$702$		0			0
$703$	6.00000			0.226294
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	13.0000	−	13.0000i	0.488225	−	0.488225i	−0.419521	−	0.907746i	$-0.637802\pi$
$709$	13.0000	−	13.0000i	0.488225	−	0.488225i	0.907746	+	0.419521i	$0.137802\pi$
$710$		0			0
$711$		−	8.00000i		−	0.300023i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	24.0000	+	24.0000i	0.896296	+	0.896296i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−18.0000	−	18.0000i	−0.669427	−	0.669427i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	24.0000i		−	0.890111i	−0.895503	−	0.445055i	$-0.853184\pi$
$727$		−	24.0000i		−	0.890111i	0.895503	−	0.445055i	$-0.146816\pi$
$728$		0			0
$729$		−	29.0000i		−	1.07407i
$730$		0			0
$731$	12.0000	−	12.0000i	0.443836	−	0.443836i
$732$		0			0
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	0.649529	−	0.760337i	$-0.274966\pi$
$733$	−3.00000	−	3.00000i	−0.110808	−	0.110808i	−0.760337	+	0.649529i	$0.774966\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.0000			0.663039
$738$		0			0
$739$	19.0000	+	19.0000i	0.698926	+	0.698926i	0.964179	−	0.265253i	$-0.0854554\pi$
$739$	19.0000	+	19.0000i	0.698926	+	0.698926i	−0.265253	+	0.964179i	$0.585455\pi$
$740$		0			0
$741$	−6.00000	+	6.00000i	−0.220416	+	0.220416i
$742$		0			0
$743$		−	8.00000i		−	0.293492i	−0.989174	−	0.146746i	$-0.953120\pi$
$743$		−	8.00000i		−	0.293492i	0.989174	−	0.146746i	$-0.0468799\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	9.00000	−	9.00000i	0.329293	−	0.329293i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$	−18.0000			−0.655956
$754$		0			0
$755$		0			0
$756$		0			0
$757$	33.0000	−	33.0000i	1.19941	−	1.19941i	0.225061	−	0.974345i	$-0.427742\pi$
$757$	33.0000	−	33.0000i	1.19941	−	1.19941i	0.974345	−	0.225061i	$-0.0722580\pi$
$758$		0			0
$759$		−	48.0000i		−	1.74229i
$760$		0			0
$761$		−	48.0000i		−	1.74000i	−0.493053	−	0.869999i	$-0.664119\pi$
$761$		−	48.0000i		−	1.74000i	0.493053	−	0.869999i	$-0.335881\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−54.0000			−1.94983
$768$		0			0
$769$	18.0000			0.649097			0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000			0.649097			0.324548	+	0.945869i	$0.394788\pi$
$770$		0			0
$771$	−8.00000	−	8.00000i	−0.288113	−	0.288113i
$772$		0			0
$773$	23.0000	−	23.0000i	0.827253	−	0.827253i	−0.159883	−	0.987136i	$-0.551112\pi$
$773$	23.0000	−	23.0000i	0.827253	−	0.827253i	0.987136	+	0.159883i	$0.0511118\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	18.0000	+	18.0000i	0.644091	+	0.644091i
$782$		0			0
$783$	−24.0000			−0.857690
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−33.0000	−	33.0000i	−1.17632	−	1.17632i	−0.980674	−	0.195649i	$-0.937319\pi$
$787$	−33.0000	−	33.0000i	−1.17632	−	1.17632i	−0.195649	−	0.980674i	$-0.562681\pi$
$788$		0			0
$789$	16.0000	−	16.0000i	0.569615	−	0.569615i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			30.0000i			1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	0.958410	−	0.285395i	$-0.0921249\pi$
$797$	19.0000	+	19.0000i	0.673015	+	0.673015i	−0.285395	+	0.958410i	$0.592125\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$	−12.0000			−0.423999
$802$		0			0
$803$	−18.0000	−	18.0000i	−0.635206	−	0.635206i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			18.0000i			0.633630i
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−37.0000	+	37.0000i	−1.29925	+	1.29925i	−0.370356	+	0.928890i	$0.620764\pi$
$811$	−37.0000	+	37.0000i	−1.29925	+	1.29925i	−0.928890	+	0.370356i	$0.879236\pi$
$812$		0			0
$813$	−16.0000	−	16.0000i	−0.561144	−	0.561144i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.00000			0.209913
$818$		0			0
$819$		0			0
$820$		0			0
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.488603	−	0.872506i	$-0.337507\pi$
$821$	39.0000	−	39.0000i	1.36111	−	1.36111i	0.872506	−	0.488603i	$-0.162493\pi$
$822$		0			0
$823$		−	24.0000i		−	0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$		−	24.0000i		−	0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	31.0000	−	31.0000i	1.07798	−	1.07798i	0.0812847	−	0.996691i	$-0.474098\pi$
$827$	31.0000	−	31.0000i	1.07798	−	1.07798i	0.996691	−	0.0812847i	$-0.0259023\pi$
$828$		0			0
$829$	−35.0000	−	35.0000i	−1.21560	−	1.21560i	−0.969157	−	0.246443i	$-0.920738\pi$
$829$	−35.0000	−	35.0000i	−1.21560	−	1.21560i	−0.246443	−	0.969157i	$-0.579262\pi$
$830$		0			0
$831$	−6.00000			−0.208138
$832$		0			0
$833$	−28.0000			−0.970143
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			42.0000i			1.45000i	0.688748	+	0.725001i	$0.258161\pi$
$839$			42.0000i			1.45000i	−0.688748	+	0.725001i	$0.741839\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$		0			0
$843$	−12.0000	+	12.0000i	−0.413302	+	0.413302i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−30.0000			−1.02960
$850$		0			0
$851$	24.0000	+	24.0000i	0.822709	+	0.822709i
$852$		0			0
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	−0.402034	−	0.915625i	$-0.631697\pi$
$853$	15.0000	−	15.0000i	0.513590	−	0.513590i	0.915625	+	0.402034i	$0.131697\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	38.0000i		−	1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$		−	38.0000i		−	1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$		0			0
$859$	−7.00000	+	7.00000i	−0.238837	+	0.238837i	−0.816368	−	0.577531i	$-0.804016\pi$
$859$	−7.00000	+	7.00000i	−0.238837	+	0.238837i	0.577531	+	0.816368i	$0.304016\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22.0000			0.748889			0.374444	−	0.927249i	$-0.377833\pi$
$863$	22.0000			0.748889			0.374444	+	0.927249i	$0.377833\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.00000	−	1.00000i	−0.0339618	−	0.0339618i
$868$		0			0
$869$	24.0000	−	24.0000i	0.814144	−	0.814144i
$870$		0			0
$871$		−	18.0000i		−	0.609907i
$872$		0			0
$873$		−	12.0000i		−	0.406138i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	0.755942	−	0.654639i	$-0.227179\pi$
$877$	3.00000	+	3.00000i	0.101303	+	0.101303i	−0.654639	+	0.755942i	$0.727179\pi$
$878$		0			0
$879$	−18.0000			−0.607125
$880$		0			0
$881$	−6.00000			−0.202145			−0.101073	−	0.994879i	$-0.532227\pi$
$881$	−6.00000			−0.202145			−0.101073	+	0.994879i	$0.532227\pi$
$882$		0			0
$883$	21.0000	+	21.0000i	0.706706	+	0.706706i	0.965841	−	0.259135i	$-0.0834374\pi$
$883$	21.0000	+	21.0000i	0.706706	+	0.706706i	−0.259135	+	0.965841i	$0.583437\pi$
$884$		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
$891$	15.0000	−	15.0000i	0.502519	−	0.502519i
$892$		0			0
$893$	−2.00000	−	2.00000i	−0.0669274	−	0.0669274i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−48.0000			−1.60267
$898$		0			0
$899$		0			0
$900$		0			0
$901$	36.0000	−	36.0000i	1.19933	−	1.19933i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−21.0000	+	21.0000i	−0.697294	+	0.697294i	−0.963826	−	0.266532i	$-0.914122\pi$
$907$	−21.0000	+	21.0000i	−0.697294	+	0.697294i	0.266532	+	0.963826i	$0.414122\pi$
$908$		0			0
$909$	−3.00000	−	3.00000i	−0.0995037	−	0.0995037i
$910$		0			0
$911$	48.0000			1.59031			0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000			1.59031			0.795155	+	0.606406i	$0.207389\pi$
$912$		0			0
$913$	54.0000			1.78714
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	54.0000i		−	1.78130i	−0.454694	−	0.890648i	$-0.650251\pi$
$919$		−	54.0000i		−	1.78130i	0.454694	−	0.890648i	$-0.349749\pi$
$920$		0			0
$921$			6.00000i			0.197707i
$922$		0			0
$923$	18.0000	−	18.0000i	0.592477	−	0.592477i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−30.0000			−0.984268			−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000			−0.984268			−0.492134	+	0.870519i	$0.663783\pi$
$930$		0			0
$931$	−7.00000	−	7.00000i	−0.229416	−	0.229416i
$932$		0			0
$933$	−6.00000	+	6.00000i	−0.196431	+	0.196431i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			54.0000i			1.76410i	0.471153	+	0.882052i	$0.343838\pi$
$937$			54.0000i			1.76410i	−0.471153	+	0.882052i	$0.656162\pi$
$938$		0			0
$939$	−6.00000	+	6.00000i	−0.195803	+	0.195803i
$940$		0			0
$941$	27.0000	+	27.0000i	0.880175	+	0.880175i	0.993552	−	0.113377i	$-0.0361668\pi$
$941$	27.0000	+	27.0000i	0.880175	+	0.880175i	−0.113377	+	0.993552i	$0.536167\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27.0000	+	27.0000i	0.877382	+	0.877382i	0.993263	−	0.115881i	$-0.0369691\pi$
$947$	27.0000	+	27.0000i	0.877382	+	0.877382i	−0.115881	+	0.993263i	$0.536969\pi$
$948$		0			0
$949$	−18.0000	+	18.0000i	−0.584305	+	0.584305i
$950$		0			0
$951$			14.0000i			0.453981i
$952$		0			0
$953$			22.0000i			0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$			22.0000i			0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−18.0000	−	18.0000i	−0.581857	−	0.581857i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$	9.00000	+	9.00000i	0.290021	+	0.290021i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			48.0000i			1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$			48.0000i			1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$		0			0
$969$			8.00000i			0.256997i
$970$		0			0
$971$	−9.00000	+	9.00000i	−0.288824	+	0.288824i	−0.836615	−	0.547791i	$-0.815469\pi$
$971$	−9.00000	+	9.00000i	−0.288824	+	0.288824i	0.547791	+	0.836615i	$0.315469\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−28.0000			−0.895799			−0.447900	−	0.894084i	$-0.647828\pi$
$977$	−28.0000			−0.895799			−0.447900	+	0.894084i	$0.647828\pi$
$978$		0			0
$979$	−36.0000	−	36.0000i	−1.15056	−	1.15056i
$980$		0			0
$981$	1.00000	−	1.00000i	0.0319275	−	0.0319275i
$982$		0			0
$983$			16.0000i			0.510321i	0.966899	+	0.255160i	$0.0821283\pi$
$983$			16.0000i			0.510321i	−0.966899	+	0.255160i	$0.917872\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.0000	+	24.0000i	0.763156	+	0.763156i
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	−10.0000			−0.317340
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	−0.550079	−	0.835112i	$-0.685403\pi$
$997$	9.00000	−	9.00000i	0.285033	−	0.285033i	0.835112	+	0.550079i	$0.185403\pi$
$998$		0			0
$999$			24.0000i			0.759326i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.2.l.c.401.1		2
4.3	odd	2		400.2.l.a.301.1		2
5.2	odd	4		320.2.q.b.209.1		2
5.3	odd	4		320.2.q.a.209.1		2
5.4	even	2		1600.2.l.b.401.1		2
16.5	even	4	inner	1600.2.l.c.1201.1		2
16.11	odd	4		400.2.l.a.101.1		2
20.3	even	4		80.2.q.a.29.1	✓	2
20.7	even	4		80.2.q.b.29.1	yes	2
20.19	odd	2		400.2.l.b.301.1		2
40.3	even	4		640.2.q.b.289.1		2
40.13	odd	4		640.2.q.d.289.1		2
40.27	even	4		640.2.q.c.289.1		2
40.37	odd	4		640.2.q.a.289.1		2
60.23	odd	4		720.2.bm.b.109.1		2
60.47	odd	4		720.2.bm.a.109.1		2
80.3	even	4		640.2.q.c.609.1		2
80.13	odd	4		640.2.q.a.609.1		2
80.27	even	4		80.2.q.a.69.1	yes	2
80.37	odd	4		320.2.q.a.49.1		2
80.43	even	4		80.2.q.b.69.1	yes	2
80.53	odd	4		320.2.q.b.49.1		2
80.59	odd	4		400.2.l.b.101.1		2
80.67	even	4		640.2.q.b.609.1		2
80.69	even	4		1600.2.l.b.1201.1		2
80.77	odd	4		640.2.q.d.609.1		2
240.107	odd	4		720.2.bm.b.469.1		2
240.203	odd	4		720.2.bm.a.469.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
80.2.q.a.29.1	✓	2	20.3	even	4
80.2.q.a.69.1	yes	2	80.27	even	4
80.2.q.b.29.1	yes	2	20.7	even	4
80.2.q.b.69.1	yes	2	80.43	even	4
320.2.q.a.49.1		2	80.37	odd	4
320.2.q.a.209.1		2	5.3	odd	4
320.2.q.b.49.1		2	80.53	odd	4
320.2.q.b.209.1		2	5.2	odd	4
400.2.l.a.101.1		2	16.11	odd	4
400.2.l.a.301.1		2	4.3	odd	2
400.2.l.b.101.1		2	80.59	odd	4
400.2.l.b.301.1		2	20.19	odd	2
640.2.q.a.289.1		2	40.37	odd	4
640.2.q.a.609.1		2	80.13	odd	4
640.2.q.b.289.1		2	40.3	even	4
640.2.q.b.609.1		2	80.67	even	4
640.2.q.c.289.1		2	40.27	even	4
640.2.q.c.609.1		2	80.3	even	4
640.2.q.d.289.1		2	40.13	odd	4
640.2.q.d.609.1		2	80.77	odd	4
720.2.bm.a.109.1		2	60.47	odd	4
720.2.bm.a.469.1		2	240.203	odd	4
720.2.bm.b.109.1		2	60.23	odd	4
720.2.bm.b.469.1		2	240.107	odd	4
1600.2.l.b.401.1		2	5.4	even	2
1600.2.l.b.1201.1		2	80.69	even	4
1600.2.l.c.401.1		2	1.1	even	1	trivial
1600.2.l.c.1201.1		2	16.5	even	4	inner

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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1600.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.00000i) q^{3} -1.00000i q^{9} +(3.00000 - 3.00000i) q^{11} +(-3.00000 - 3.00000i) q^{13} -4.00000 q^{17} +(-1.00000 - 1.00000i) q^{19} -8.00000i q^{23} +(4.00000 - 4.00000i) q^{27} +(-3.00000 - 3.00000i) q^{29} +6.00000 q^{33} +(-3.00000 + 3.00000i) q^{37} -6.00000i q^{39} +(-3.00000 + 3.00000i) q^{43} +2.00000 q^{47} +7.00000 q^{49} +(-4.00000 - 4.00000i) q^{51} +(-9.00000 + 9.00000i) q^{53} -2.00000i q^{57} +(9.00000 - 9.00000i) q^{59} +(-5.00000 - 5.00000i) q^{61} +(3.00000 + 3.00000i) q^{67} +(8.00000 - 8.00000i) q^{69} +6.00000i q^{71} -6.00000i q^{73} +8.00000 q^{79} +5.00000 q^{81} +(9.00000 + 9.00000i) q^{83} -6.00000i q^{87} -12.0000i q^{89} +12.0000 q^{97} +(-3.00000 - 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{11} - 6 q^{13} - 8 q^{17} - 2 q^{19} + 8 q^{27} - 6 q^{29} + 12 q^{33} - 6 q^{37} - 6 q^{43} + 4 q^{47} + 14 q^{49} - 8 q^{51} - 18 q^{53} + 18 q^{59} - 10 q^{61} + 6 q^{67} + 16 q^{69}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^11 - 6 * q^13 - 8 * q^17 - 2 * q^19 + 8 * q^27 - 6 * q^29 + 12 * q^33 - 6 * q^37 - 6 * q^43 + 4 * q^47 + 14 * q^49 - 8 * q^51 - 18 * q^53 + 18 * q^59 - 10 * q^61 + 6 * q^67 + 16 * q^69 + 16 * q^79 + 10 * q^81 + 18 * q^83 + 24 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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