LMFDB - Embedded newform 1600.2.s.a.943.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(207,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("1600.207");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1600.s (of order $4$, degree $2$, 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:	$12.7760643234$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 80)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			943.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1600.943
Dual form			1600.2.s.a.207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} +(-3.00000 + 3.00000i) q^{7} +1.00000 q^{9} +(1.00000 + 1.00000i) q^{11} +2.00000i q^{13} +(-1.00000 + 1.00000i) q^{17} +(3.00000 + 3.00000i) q^{19} +(6.00000 - 6.00000i) q^{21} +(-1.00000 - 1.00000i) q^{23} +4.00000 q^{27} +(-7.00000 + 7.00000i) q^{29} -2.00000i q^{31} +(-2.00000 - 2.00000i) q^{33} -6.00000i q^{37} -4.00000i q^{39} -4.00000i q^{41} +4.00000i q^{43} +(-7.00000 - 7.00000i) q^{47} -11.0000i q^{49} +(2.00000 - 2.00000i) q^{51} +8.00000 q^{53} +(-6.00000 - 6.00000i) q^{57} +(-3.00000 + 3.00000i) q^{59} +(-1.00000 - 1.00000i) q^{61} +(-3.00000 + 3.00000i) q^{63} -4.00000i q^{67} +(2.00000 + 2.00000i) q^{69} +(-3.00000 + 3.00000i) q^{73} -6.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -2.00000 q^{83} +(14.0000 - 14.0000i) q^{87} -6.00000 q^{89} +(-6.00000 - 6.00000i) q^{91} +4.00000i q^{93} +(11.0000 - 11.0000i) q^{97} +(1.00000 + 1.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 2 q^{23} + 8 q^{27} - 14 q^{29} - 4 q^{33} - 14 q^{47} + 4 q^{51} + 16 q^{53} - 12 q^{57} - 6 q^{59} - 2 q^{61} - 6 q^{63}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 6 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 2 * q^23 + 8 * q^27 - 14 * q^29 - 4 * q^33 - 14 * q^47 + 4 * q^51 + 16 * q^53 - 12 * q^57 - 6 * q^59 - 2 * q^61 - 6 * q^63 + 4 * q^69 - 6 * q^73 - 12 * q^77 - 16 * q^79 - 22 * q^81 - 4 * q^83 + 28 * q^87 - 12 * q^89 - 12 * q^91 + 22 * q^97 + 2 * q^99

Character values

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

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

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
$3$	−2.00000			−1.15470			−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000			−1.15470			−0.577350	+	0.816497i	$0.695913\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	0.841605	−	0.540094i	$-0.181611\pi$
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	−0.540094	+	0.841605i	$0.681611\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	−0.817898	−	0.575363i	$-0.804861\pi$
$17$	−1.00000	+	1.00000i	−0.242536	+	0.242536i	0.575363	+	0.817898i	$0.304861\pi$
$18$		0			0
$19$	3.00000	+	3.00000i	0.688247	+	0.688247i	0.961844	−	0.273597i	$-0.0882135\pi$
$19$	3.00000	+	3.00000i	0.688247	+	0.688247i	−0.273597	+	0.961844i	$0.588214\pi$
$20$		0			0
$21$	6.00000	−	6.00000i	1.30931	−	1.30931i
$22$		0			0
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	0.595121	−	0.803636i	$-0.297104\pi$
$23$	−1.00000	−	1.00000i	−0.208514	−	0.208514i	−0.803636	+	0.595121i	$0.797104\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	4.00000			0.769800
$28$		0			0
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.371391	+	0.928477i	$0.621119\pi$
$29$	−7.00000	+	7.00000i	−1.29987	+	1.29987i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$	−2.00000	−	2.00000i	−0.348155	−	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	6.00000i		−	0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$		−	6.00000i		−	0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$		0			0
$39$		−	4.00000i		−	0.640513i
$40$		0			0
$41$		−	4.00000i		−	0.624695i	−0.949968	−	0.312348i	$-0.898885\pi$
$41$		−	4.00000i		−	0.624695i	0.949968	−	0.312348i	$-0.101115\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.00000	−	7.00000i	−1.02105	−	1.02105i	−0.999774	−	0.0212814i	$-0.993225\pi$
$47$	−7.00000	−	7.00000i	−1.02105	−	1.02105i	−0.0212814	−	0.999774i	$-0.506775\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$	2.00000	−	2.00000i	0.280056	−	0.280056i
$52$		0			0
$53$	8.00000			1.09888			0.549442	−	0.835532i	$-0.314840\pi$
$53$	8.00000			1.09888			0.549442	+	0.835532i	$0.314840\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.00000	−	6.00000i	−0.794719	−	0.794719i
$58$		0			0
$59$	−3.00000	+	3.00000i	−0.390567	+	0.390567i	−0.874889	−	0.484323i	$-0.839066\pi$
$59$	−3.00000	+	3.00000i	−0.390567	+	0.390567i	0.484323	+	0.874889i	$0.339066\pi$
$60$		0			0
$61$	−1.00000	−	1.00000i	−0.128037	−	0.128037i	0.640184	−	0.768221i	$-0.278858\pi$
$61$	−1.00000	−	1.00000i	−0.128037	−	0.128037i	−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$	−3.00000	+	3.00000i	−0.377964	+	0.377964i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$	2.00000	+	2.00000i	0.240772	+	0.240772i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−6.00000			−0.683763
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$		0			0
$83$	−2.00000			−0.219529			−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000			−0.219529			−0.109764	+	0.993958i	$0.535010\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	14.0000	−	14.0000i	1.50096	−	1.50096i
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−6.00000	−	6.00000i	−0.628971	−	0.628971i
$92$		0			0
$93$			4.00000i			0.414781i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.124684	−	0.992196i	$-0.460208\pi$
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.992196	−	0.124684i	$-0.0397918\pi$
$98$		0			0
$99$	1.00000	+	1.00000i	0.100504	+	0.100504i
$100$		0			0
$101$	−5.00000	+	5.00000i	−0.497519	+	0.497519i	−0.910665	−	0.413146i	$-0.864430\pi$
$101$	−5.00000	+	5.00000i	−0.497519	+	0.497519i	0.413146	+	0.910665i	$0.364430\pi$
$102$		0			0
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	0.416480	−	0.909145i	$-0.363264\pi$
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	−0.909145	+	0.416480i	$0.863264\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$	5.00000	−	5.00000i	0.478913	−	0.478913i	−0.425871	−	0.904784i	$-0.640032\pi$
$109$	5.00000	−	5.00000i	0.478913	−	0.478913i	0.904784	+	0.425871i	$0.140032\pi$
$110$		0			0
$111$			12.0000i			1.13899i
$112$		0			0
$113$	−13.0000	−	13.0000i	−1.22294	−	1.22294i	−0.966583	−	0.256354i	$-0.917479\pi$
$113$	−13.0000	−	13.0000i	−1.22294	−	1.22294i	−0.256354	−	0.966583i	$-0.582521\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$		−	6.00000i		−	0.550019i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$			8.00000i			0.721336i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−7.00000	−	7.00000i	−0.621150	−	0.621150i	0.324676	−	0.945825i	$-0.394745\pi$
$127$	−7.00000	−	7.00000i	−0.621150	−	0.621150i	−0.945825	+	0.324676i	$0.894745\pi$
$128$		0			0
$129$		−	8.00000i		−	0.704361i
$130$		0			0
$131$	7.00000	−	7.00000i	0.611593	−	0.611593i	−0.331768	−	0.943361i	$-0.607645\pi$
$131$	7.00000	−	7.00000i	0.611593	−	0.611593i	0.943361	+	0.331768i	$0.107645\pi$
$132$		0			0
$133$	−18.0000			−1.56080
$134$		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
$139$	9.00000	−	9.00000i	0.763370	−	0.763370i	−0.213560	−	0.976930i	$-0.568506\pi$
$139$	9.00000	−	9.00000i	0.763370	−	0.763370i	0.976930	+	0.213560i	$0.0685059\pi$
$140$		0			0
$141$	14.0000	+	14.0000i	1.17901	+	1.17901i
$142$		0			0
$143$	−2.00000	+	2.00000i	−0.167248	+	0.167248i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			22.0000i			1.81453i
$148$		0			0
$149$	1.00000	+	1.00000i	0.0819232	+	0.0819232i	0.746881	−	0.664958i	$-0.231550\pi$
$149$	1.00000	+	1.00000i	0.0819232	+	0.0819232i	−0.664958	+	0.746881i	$0.731550\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$	−1.00000	+	1.00000i	−0.0808452	+	0.0808452i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−20.0000			−1.59617			−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000			−1.59617			−0.798087	+	0.602542i	$0.794154\pi$
$158$		0			0
$159$	−16.0000			−1.26888
$160$		0			0
$161$	6.00000			0.472866
$162$		0			0
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	−0.813588	−	0.581441i	$-0.802489\pi$
$167$	−3.00000	+	3.00000i	−0.232147	+	0.232147i	0.581441	+	0.813588i	$0.302489\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	3.00000	+	3.00000i	0.229416	+	0.229416i
$172$		0			0
$173$			6.00000i			0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$			6.00000i			0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	6.00000	−	6.00000i	0.450988	−	0.450988i
$178$		0			0
$179$	−5.00000	−	5.00000i	−0.373718	−	0.373718i	0.495112	−	0.868829i	$-0.335127\pi$
$179$	−5.00000	−	5.00000i	−0.373718	−	0.373718i	−0.868829	+	0.495112i	$0.835127\pi$
$180$		0			0
$181$	3.00000	−	3.00000i	0.222988	−	0.222988i	−0.586767	−	0.809756i	$-0.699600\pi$
$181$	3.00000	−	3.00000i	0.222988	−	0.222988i	0.809756	+	0.586767i	$0.199600\pi$
$182$		0			0
$183$	2.00000	+	2.00000i	0.147844	+	0.147844i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.00000			−0.146254
$188$		0			0
$189$	−12.0000	+	12.0000i	−0.872872	+	0.872872i
$190$		0			0
$191$		−	18.0000i		−	1.30243i	−0.758891	−	0.651217i	$-0.774259\pi$
$191$		−	18.0000i		−	1.30243i	0.758891	−	0.651217i	$-0.225741\pi$
$192$		0			0
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.996534	+	0.0831899i	$0.0265108\pi$
$193$	15.0000	+	15.0000i	1.07972	+	1.07972i	0.0831899	+	0.996534i	$0.473489\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			6.00000i			0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$			6.00000i			0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$		0			0
$199$			10.0000i			0.708881i	0.935079	+	0.354441i	$0.115329\pi$
$199$			10.0000i			0.708881i	−0.935079	+	0.354441i	$0.884671\pi$
$200$		0			0
$201$			8.00000i			0.564276i
$202$		0			0
$203$		−	42.0000i		−	2.94782i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.00000	−	1.00000i	−0.0695048	−	0.0695048i
$208$		0			0
$209$			6.00000i			0.415029i
$210$		0			0
$211$	19.0000	−	19.0000i	1.30801	−	1.30801i	0.385167	−	0.922847i	$-0.374144\pi$
$211$	19.0000	−	19.0000i	1.30801	−	1.30801i	0.922847	−	0.385167i	$-0.125856\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.00000	+	6.00000i	0.407307	+	0.407307i
$218$		0			0
$219$	6.00000	−	6.00000i	0.405442	−	0.405442i
$220$		0			0
$221$	−2.00000	−	2.00000i	−0.134535	−	0.134535i
$222$		0			0
$223$	−9.00000	+	9.00000i	−0.602685	+	0.602685i	−0.941024	−	0.338340i	$-0.890135\pi$
$223$	−9.00000	+	9.00000i	−0.602685	+	0.602685i	0.338340	+	0.941024i	$0.390135\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	12.0000i		−	0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$		−	12.0000i		−	0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$		0			0
$229$	1.00000	+	1.00000i	0.0660819	+	0.0660819i	0.739375	−	0.673293i	$-0.235121\pi$
$229$	1.00000	+	1.00000i	0.0660819	+	0.0660819i	−0.673293	+	0.739375i	$0.735121\pi$
$230$		0			0
$231$	12.0000			0.789542
$232$		0			0
$233$	9.00000	−	9.00000i	0.589610	−	0.589610i	−0.347916	−	0.937526i	$-0.613111\pi$
$233$	9.00000	−	9.00000i	0.589610	−	0.589610i	0.937526	+	0.347916i	$0.113111\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	16.0000			1.03931
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$		0			0
$243$	10.0000			0.641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.00000	+	6.00000i	−0.381771	+	0.381771i
$248$		0			0
$249$	4.00000			0.253490
$250$		0			0
$251$	−11.0000	−	11.0000i	−0.694314	−	0.694314i	0.268864	−	0.963178i	$-0.413352\pi$
$251$	−11.0000	−	11.0000i	−0.694314	−	0.694314i	−0.963178	+	0.268864i	$0.913352\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.0000	+	13.0000i	−0.810918	+	0.810918i	−0.984771	−	0.173854i	$-0.944378\pi$
$257$	−13.0000	+	13.0000i	−0.810918	+	0.810918i	0.173854	+	0.984771i	$0.444378\pi$
$258$		0			0
$259$	18.0000	+	18.0000i	1.11847	+	1.11847i
$260$		0			0
$261$	−7.00000	+	7.00000i	−0.433289	+	0.433289i
$262$		0			0
$263$	7.00000	+	7.00000i	0.431638	+	0.431638i	0.889185	−	0.457547i	$-0.151272\pi$
$263$	7.00000	+	7.00000i	0.431638	+	0.431638i	−0.457547	+	0.889185i	$0.651272\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.0000			0.734388
$268$		0			0
$269$	1.00000	−	1.00000i	0.0609711	−	0.0609711i	−0.675964	−	0.736935i	$-0.736272\pi$
$269$	1.00000	−	1.00000i	0.0609711	−	0.0609711i	0.736935	+	0.675964i	$0.236272\pi$
$270$		0			0
$271$			30.0000i			1.82237i	0.411997	+	0.911185i	$0.364831\pi$
$271$			30.0000i			1.82237i	−0.411997	+	0.911185i	$0.635169\pi$
$272$		0			0
$273$	12.0000	+	12.0000i	0.726273	+	0.726273i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		0			0
$279$		−	2.00000i		−	0.119737i
$280$		0			0
$281$			16.0000i			0.954480i	0.878773	+	0.477240i	$0.158363\pi$
$281$			16.0000i			0.954480i	−0.878773	+	0.477240i	$0.841637\pi$
$282$		0			0
$283$			12.0000i			0.713326i	0.934233	+	0.356663i	$0.116086\pi$
$283$			12.0000i			0.713326i	−0.934233	+	0.356663i	$0.883914\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	+	12.0000i	0.708338	+	0.708338i
$288$		0			0
$289$			15.0000i			0.882353i
$290$		0			0
$291$	−22.0000	+	22.0000i	−1.28966	+	1.28966i
$292$		0			0
$293$	12.0000			0.701047			0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000			0.701047			0.350524	+	0.936554i	$0.386004\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	4.00000	+	4.00000i	0.232104	+	0.232104i
$298$		0			0
$299$	2.00000	−	2.00000i	0.115663	−	0.115663i
$300$		0			0
$301$	−12.0000	−	12.0000i	−0.691669	−	0.691669i
$302$		0			0
$303$	10.0000	−	10.0000i	0.574485	−	0.574485i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		0			0
$309$	10.0000	+	10.0000i	0.568880	+	0.568880i
$310$		0			0
$311$	−16.0000			−0.907277			−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000			−0.907277			−0.453638	+	0.891186i	$0.649874\pi$
$312$		0			0
$313$	13.0000	−	13.0000i	0.734803	−	0.734803i	−0.236764	−	0.971567i	$-0.576087\pi$
$313$	13.0000	−	13.0000i	0.734803	−	0.734803i	0.971567	+	0.236764i	$0.0760868\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.00000			−0.449325			−0.224662	−	0.974437i	$-0.572128\pi$
$317$	−8.00000			−0.449325			−0.224662	+	0.974437i	$0.572128\pi$
$318$		0			0
$319$	−14.0000			−0.783850
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	−6.00000			−0.333849
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−10.0000	+	10.0000i	−0.553001	+	0.553001i
$328$		0			0
$329$	42.0000			2.31553
$330$		0			0
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.985689	+	0.168576i	$0.0539168\pi$
$331$	21.0000	+	21.0000i	1.15426	+	1.15426i	0.168576	+	0.985689i	$0.446083\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	11.0000	−	11.0000i	0.599208	−	0.599208i	−0.340894	−	0.940102i	$-0.610730\pi$
$337$	11.0000	−	11.0000i	0.599208	−	0.599208i	0.940102	+	0.340894i	$0.110730\pi$
$338$		0			0
$339$	26.0000	+	26.0000i	1.41213	+	1.41213i
$340$		0			0
$341$	2.00000	−	2.00000i	0.108306	−	0.108306i
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.00000			0.107366			0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000			0.107366			0.0536828	+	0.998558i	$0.482904\pi$
$348$		0			0
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	−0.782826	−	0.622240i	$-0.786223\pi$
$349$	−3.00000	+	3.00000i	−0.160586	+	0.160586i	0.622240	+	0.782826i	$0.286223\pi$
$350$		0			0
$351$			8.00000i			0.427008i
$352$		0			0
$353$	−13.0000	−	13.0000i	−0.691920	−	0.691920i	0.270734	−	0.962654i	$-0.412734\pi$
$353$	−13.0000	−	13.0000i	−0.691920	−	0.691920i	−0.962654	+	0.270734i	$0.912734\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			12.0000i			0.635107i
$358$		0			0
$359$		−	14.0000i		−	0.738892i	−0.929252	−	0.369446i	$-0.879548\pi$
$359$		−	14.0000i		−	0.738892i	0.929252	−	0.369446i	$-0.120452\pi$
$360$		0			0
$361$		−	1.00000i		−	0.0526316i
$362$		0			0
$363$			18.0000i			0.944755i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	21.0000	+	21.0000i	1.09619	+	1.09619i	0.994852	+	0.101339i	$0.0323127\pi$
$367$	21.0000	+	21.0000i	1.09619	+	1.09619i	0.101339	+	0.994852i	$0.467687\pi$
$368$		0			0
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$	−24.0000	+	24.0000i	−1.24602	+	1.24602i
$372$		0			0
$373$	4.00000			0.207112			0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000			0.207112			0.103556	+	0.994624i	$0.466978\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.0000	−	14.0000i	−0.721037	−	0.721037i
$378$		0			0
$379$	−15.0000	+	15.0000i	−0.770498	+	0.770498i	−0.978194	−	0.207695i	$-0.933404\pi$
$379$	−15.0000	+	15.0000i	−0.770498	+	0.770498i	0.207695	+	0.978194i	$0.433404\pi$
$380$		0			0
$381$	14.0000	+	14.0000i	0.717242	+	0.717242i
$382$		0			0
$383$	−5.00000	+	5.00000i	−0.255488	+	0.255488i	−0.823216	−	0.567728i	$-0.807823\pi$
$383$	−5.00000	+	5.00000i	−0.255488	+	0.255488i	0.567728	+	0.823216i	$0.307823\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			4.00000i			0.203331i
$388$		0			0
$389$	−23.0000	−	23.0000i	−1.16615	−	1.16615i	−0.983105	−	0.183041i	$-0.941406\pi$
$389$	−23.0000	−	23.0000i	−1.16615	−	1.16615i	−0.183041	−	0.983105i	$-0.558594\pi$
$390$		0			0
$391$	2.00000			0.101144
$392$		0			0
$393$	−14.0000	+	14.0000i	−0.706207	+	0.706207i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	32.0000			1.60603			0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000			1.60603			0.803017	+	0.595956i	$0.203227\pi$
$398$		0			0
$399$	36.0000			1.80225
$400$		0			0
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$	4.00000			0.199254
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	−	6.00000i	0.297409	−	0.297409i
$408$		0			0
$409$	−38.0000			−1.87898			−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000			−1.87898			−0.939490	+	0.342578i	$0.888700\pi$
$410$		0			0
$411$	−18.0000	−	18.0000i	−0.887875	−	0.887875i
$412$		0			0
$413$		−	18.0000i		−	0.885722i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−18.0000	+	18.0000i	−0.881464	+	0.881464i
$418$		0			0
$419$	−17.0000	−	17.0000i	−0.830504	−	0.830504i	0.157081	−	0.987586i	$-0.449792\pi$
$419$	−17.0000	−	17.0000i	−0.830504	−	0.830504i	−0.987586	+	0.157081i	$0.949792\pi$
$420$		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
$423$	−7.00000	−	7.00000i	−0.340352	−	0.340352i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	6.00000			0.290360
$428$		0			0
$429$	4.00000	−	4.00000i	0.193122	−	0.193122i
$430$		0			0
$431$		−	2.00000i		−	0.0963366i	−0.998839	−	0.0481683i	$-0.984662\pi$
$431$		−	2.00000i		−	0.0963366i	0.998839	−	0.0481683i	$-0.0153384\pi$
$432$		0			0
$433$	−5.00000	−	5.00000i	−0.240285	−	0.240285i	0.576683	−	0.816968i	$-0.304347\pi$
$433$	−5.00000	−	5.00000i	−0.240285	−	0.240285i	−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	6.00000i		−	0.287019i
$438$		0			0
$439$			26.0000i			1.24091i	0.784241	+	0.620456i	$0.213053\pi$
$439$			26.0000i			1.24091i	−0.784241	+	0.620456i	$0.786947\pi$
$440$		0			0
$441$		−	11.0000i		−	0.523810i
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−2.00000	−	2.00000i	−0.0945968	−	0.0945968i
$448$		0			0
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		0			0
$451$	4.00000	−	4.00000i	0.188353	−	0.188353i
$452$		0			0
$453$	16.0000			0.751746
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.00000	−	7.00000i	−0.327446	−	0.327446i	0.524168	−	0.851615i	$-0.324376\pi$
$457$	−7.00000	−	7.00000i	−0.327446	−	0.327446i	−0.851615	+	0.524168i	$0.824376\pi$
$458$		0			0
$459$	−4.00000	+	4.00000i	−0.186704	+	0.186704i
$460$		0			0
$461$	−21.0000	−	21.0000i	−0.978068	−	0.978068i	0.0216971	−	0.999765i	$-0.493093\pi$
$461$	−21.0000	−	21.0000i	−0.978068	−	0.978068i	−0.999765	+	0.0216971i	$0.993093\pi$
$462$		0			0
$463$	19.0000	−	19.0000i	0.883005	−	0.883005i	−0.110834	−	0.993839i	$-0.535352\pi$
$463$	19.0000	−	19.0000i	0.883005	−	0.883005i	0.993839	+	0.110834i	$0.0353522\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			28.0000i			1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$			28.0000i			1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$		0			0
$469$	12.0000	+	12.0000i	0.554109	+	0.554109i
$470$		0			0
$471$	40.0000			1.84310
$472$		0			0
$473$	−4.00000	+	4.00000i	−0.183920	+	0.183920i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	8.00000			0.366295
$478$		0			0
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	12.0000			0.547153
$482$		0			0
$483$	−12.0000			−0.546019
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.0000	+	15.0000i	−0.679715	+	0.679715i	−0.959936	−	0.280221i	$-0.909592\pi$
$487$	−15.0000	+	15.0000i	−0.679715	+	0.679715i	0.280221	+	0.959936i	$0.409592\pi$
$488$		0			0
$489$	−28.0000			−1.26620
$490$		0			0
$491$	9.00000	+	9.00000i	0.406164	+	0.406164i	0.880399	−	0.474234i	$-0.157275\pi$
$491$	9.00000	+	9.00000i	0.406164	+	0.406164i	−0.474234	+	0.880399i	$0.657275\pi$
$492$		0			0
$493$		−	14.0000i		−	0.630528i
$494$		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
$501$	6.00000	−	6.00000i	0.268060	−	0.268060i
$502$		0			0
$503$	−29.0000	−	29.0000i	−1.29305	−	1.29305i	−0.932893	−	0.360153i	$-0.882725\pi$
$503$	−29.0000	−	29.0000i	−1.29305	−	1.29305i	−0.360153	−	0.932893i	$-0.617275\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−18.0000			−0.799408
$508$		0			0
$509$	17.0000	−	17.0000i	0.753512	−	0.753512i	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	17.0000	−	17.0000i	0.753512	−	0.753512i	0.975133	+	0.221621i	$0.0711348\pi$
$510$		0			0
$511$		−	18.0000i		−	0.796273i
$512$		0			0
$513$	12.0000	+	12.0000i	0.529813	+	0.529813i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	14.0000i		−	0.615719i
$518$		0			0
$519$		−	12.0000i		−	0.526742i
$520$		0			0
$521$		−	16.0000i		−	0.700973i	−0.936568	−	0.350486i	$-0.886016\pi$
$521$		−	16.0000i		−	0.700973i	0.936568	−	0.350486i	$-0.113984\pi$
$522$		0			0
$523$			20.0000i			0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$			20.0000i			0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.00000	+	2.00000i	0.0871214	+	0.0871214i
$528$		0			0
$529$		−	21.0000i		−	0.913043i
$530$		0			0
$531$	−3.00000	+	3.00000i	−0.130189	+	0.130189i
$532$		0			0
$533$	8.00000			0.346518
$534$		0			0
$535$		0			0
$536$		0			0
$537$	10.0000	+	10.0000i	0.431532	+	0.431532i
$538$		0			0
$539$	11.0000	−	11.0000i	0.473804	−	0.473804i
$540$		0			0
$541$	15.0000	+	15.0000i	0.644900	+	0.644900i	0.951756	−	0.306856i	$-0.0992769\pi$
$541$	15.0000	+	15.0000i	0.644900	+	0.644900i	−0.306856	+	0.951756i	$0.599277\pi$
$542$		0			0
$543$	−6.00000	+	6.00000i	−0.257485	+	0.257485i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	28.0000i		−	1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$		−	28.0000i		−	1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$		0			0
$549$	−1.00000	−	1.00000i	−0.0426790	−	0.0426790i
$550$		0			0
$551$	−42.0000			−1.78926
$552$		0			0
$553$	24.0000	−	24.0000i	1.02058	−	1.02058i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−28.0000			−1.18640			−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000			−1.18640			−0.593199	+	0.805056i	$0.702135\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$	4.00000			0.168880
$562$		0			0
$563$	−18.0000			−0.758610			−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000			−0.758610			−0.379305	+	0.925272i	$0.623837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	33.0000	−	33.0000i	1.38587	−	1.38587i
$568$		0			0
$569$	−6.00000			−0.251533			−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000			−0.251533			−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	25.0000	+	25.0000i	1.04622	+	1.04622i	0.998879	+	0.0473385i	$0.0150740\pi$
$571$	25.0000	+	25.0000i	1.04622	+	1.04622i	0.0473385	+	0.998879i	$0.484926\pi$
$572$		0			0
$573$			36.0000i			1.50392i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.00000	+	9.00000i	−0.374675	+	0.374675i	−0.869177	−	0.494502i	$-0.835351\pi$
$577$	−9.00000	+	9.00000i	−0.374675	+	0.374675i	0.494502	+	0.869177i	$0.335351\pi$
$578$		0			0
$579$	−30.0000	−	30.0000i	−1.24676	−	1.24676i
$580$		0			0
$581$	6.00000	−	6.00000i	0.248922	−	0.248922i
$582$		0			0
$583$	8.00000	+	8.00000i	0.331326	+	0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	2.00000			0.0825488			0.0412744	−	0.999148i	$-0.486858\pi$
$587$	2.00000			0.0825488			0.0412744	+	0.999148i	$0.486858\pi$
$588$		0			0
$589$	6.00000	−	6.00000i	0.247226	−	0.247226i
$590$		0			0
$591$		−	12.0000i		−	0.493614i
$592$		0			0
$593$	−17.0000	−	17.0000i	−0.698106	−	0.698106i	0.265896	−	0.964002i	$-0.414332\pi$
$593$	−17.0000	−	17.0000i	−0.698106	−	0.698106i	−0.964002	+	0.265896i	$0.914332\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	20.0000i		−	0.818546i
$598$		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
$601$			16.0000i			0.652654i	0.945257	+	0.326327i	$0.105811\pi$
$601$			16.0000i			0.652654i	−0.945257	+	0.326327i	$0.894189\pi$
$602$		0			0
$603$		−	4.00000i		−	0.162893i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−23.0000	−	23.0000i	−0.933541	−	0.933541i	0.0643840	−	0.997925i	$-0.479492\pi$
$607$	−23.0000	−	23.0000i	−0.933541	−	0.933541i	−0.997925	+	0.0643840i	$0.979492\pi$
$608$		0			0
$609$			84.0000i			3.40385i
$610$		0			0
$611$	14.0000	−	14.0000i	0.566379	−	0.566379i
$612$		0			0
$613$	−8.00000			−0.323117			−0.161558	−	0.986863i	$-0.551652\pi$
$613$	−8.00000			−0.323117			−0.161558	+	0.986863i	$0.551652\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.0000	+	25.0000i	1.00646	+	1.00646i	0.999979	+	0.00648312i	$0.00206366\pi$
$617$	25.0000	+	25.0000i	1.00646	+	1.00646i	0.00648312	+	0.999979i	$0.497936\pi$
$618$		0			0
$619$	−7.00000	+	7.00000i	−0.281354	+	0.281354i	−0.833649	−	0.552295i	$-0.813752\pi$
$619$	−7.00000	+	7.00000i	−0.281354	+	0.281354i	0.552295	+	0.833649i	$0.313752\pi$
$620$		0			0
$621$	−4.00000	−	4.00000i	−0.160514	−	0.160514i
$622$		0			0
$623$	18.0000	−	18.0000i	0.721155	−	0.721155i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	12.0000i		−	0.479234i
$628$		0			0
$629$	6.00000	+	6.00000i	0.239236	+	0.239236i
$630$		0			0
$631$	−24.0000			−0.955425			−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000			−0.955425			−0.477712	+	0.878516i	$0.658534\pi$
$632$		0			0
$633$	−38.0000	+	38.0000i	−1.51036	+	1.51036i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	22.0000			0.871672
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−26.0000			−1.02534			−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000			−1.02534			−0.512670	+	0.858586i	$0.671344\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−15.0000	+	15.0000i	−0.589711	+	0.589711i	−0.937553	−	0.347842i	$-0.886914\pi$
$647$	−15.0000	+	15.0000i	−0.589711	+	0.589711i	0.347842	+	0.937553i	$0.386914\pi$
$648$		0			0
$649$	−6.00000			−0.235521
$650$		0			0
$651$	−12.0000	−	12.0000i	−0.470317	−	0.470317i
$652$		0			0
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.00000	+	3.00000i	−0.117041	+	0.117041i
$658$		0			0
$659$	11.0000	+	11.0000i	0.428499	+	0.428499i	0.888117	−	0.459618i	$-0.152014\pi$
$659$	11.0000	+	11.0000i	0.428499	+	0.428499i	−0.459618	+	0.888117i	$0.652014\pi$
$660$		0			0
$661$	−25.0000	+	25.0000i	−0.972387	+	0.972387i	−0.999629	−	0.0272416i	$-0.991328\pi$
$661$	−25.0000	+	25.0000i	−0.972387	+	0.972387i	0.0272416	+	0.999629i	$0.491328\pi$
$662$		0			0
$663$	4.00000	+	4.00000i	0.155347	+	0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	14.0000			0.542082
$668$		0			0
$669$	18.0000	−	18.0000i	0.695920	−	0.695920i
$670$		0			0
$671$		−	2.00000i		−	0.0772091i
$672$		0			0
$673$	−1.00000	−	1.00000i	−0.0385472	−	0.0385472i	0.687570	−	0.726118i	$-0.258677\pi$
$673$	−1.00000	−	1.00000i	−0.0385472	−	0.0385472i	−0.726118	+	0.687570i	$0.758677\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$			66.0000i			2.53285i
$680$		0			0
$681$			24.0000i			0.919682i
$682$		0			0
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−2.00000	−	2.00000i	−0.0763048	−	0.0763048i
$688$		0			0
$689$			16.0000i			0.609551i
$690$		0			0
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	−0.982919	−	0.184041i	$-0.941082\pi$
$691$	−21.0000	+	21.0000i	−0.798878	+	0.798878i	0.184041	+	0.982919i	$0.441082\pi$
$692$		0			0
$693$	−6.00000			−0.227921
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.00000	+	4.00000i	0.151511	+	0.151511i
$698$		0			0
$699$	−18.0000	+	18.0000i	−0.680823	+	0.680823i
$700$		0			0
$701$	−13.0000	−	13.0000i	−0.491003	−	0.491003i	0.417619	−	0.908622i	$-0.362865\pi$
$701$	−13.0000	−	13.0000i	−0.491003	−	0.491003i	−0.908622	+	0.417619i	$0.862865\pi$
$702$		0			0
$703$	18.0000	−	18.0000i	0.678883	−	0.678883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	30.0000i		−	1.12827i
$708$		0			0
$709$	1.00000	+	1.00000i	0.0375558	+	0.0375558i	0.725635	−	0.688080i	$-0.241546\pi$
$709$	1.00000	+	1.00000i	0.0375558	+	0.0375558i	−0.688080	+	0.725635i	$0.741546\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$	−2.00000	+	2.00000i	−0.0749006	+	0.0749006i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	32.0000			1.19340			0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000			1.19340			0.596699	+	0.802465i	$0.296479\pi$
$720$		0			0
$721$	30.0000			1.11726
$722$		0			0
$723$	28.0000			1.04133
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−7.00000	+	7.00000i	−0.259616	+	0.259616i	−0.824898	−	0.565282i	$-0.808767\pi$
$727$	−7.00000	+	7.00000i	−0.259616	+	0.259616i	0.565282	+	0.824898i	$0.308767\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−4.00000	−	4.00000i	−0.147945	−	0.147945i
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000	−	4.00000i	0.147342	−	0.147342i
$738$		0			0
$739$	−21.0000	−	21.0000i	−0.772497	−	0.772497i	0.206045	−	0.978543i	$-0.433941\pi$
$739$	−21.0000	−	21.0000i	−0.772497	−	0.772497i	−0.978543	+	0.206045i	$0.933941\pi$
$740$		0			0
$741$	12.0000	−	12.0000i	0.440831	−	0.440831i
$742$		0			0
$743$	31.0000	+	31.0000i	1.13728	+	1.13728i	0.988936	+	0.148344i	$0.0473942\pi$
$743$	31.0000	+	31.0000i	1.13728	+	1.13728i	0.148344	+	0.988936i	$0.452606\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−2.00000			−0.0731762
$748$		0			0
$749$	18.0000	−	18.0000i	0.657706	−	0.657706i
$750$		0			0
$751$		−	50.0000i		−	1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$		−	50.0000i		−	1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$		0			0
$753$	22.0000	+	22.0000i	0.801725	+	0.801725i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			2.00000i			0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$			2.00000i			0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$		0			0
$759$			4.00000i			0.145191i
$760$		0			0
$761$			40.0000i			1.45000i	0.688749	+	0.724999i	$0.258160\pi$
$761$			40.0000i			1.45000i	−0.688749	+	0.724999i	$0.741840\pi$
$762$		0			0
$763$			30.0000i			1.08607i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.00000	−	6.00000i	−0.216647	−	0.216647i
$768$		0			0
$769$		−	4.00000i		−	0.144244i	−0.997396	−	0.0721218i	$-0.977023\pi$
$769$		−	4.00000i		−	0.144244i	0.997396	−	0.0721218i	$-0.0229770\pi$
$770$		0			0
$771$	26.0000	−	26.0000i	0.936367	−	0.936367i
$772$		0			0
$773$	−48.0000			−1.72644			−0.863220	−	0.504828i	$-0.831556\pi$
$773$	−48.0000			−1.72644			−0.863220	+	0.504828i	$0.831556\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−36.0000	−	36.0000i	−1.29149	−	1.29149i
$778$		0			0
$779$	12.0000	−	12.0000i	0.429945	−	0.429945i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−28.0000	+	28.0000i	−1.00064	+	1.00064i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			4.00000i			0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$			4.00000i			0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$		0			0
$789$	−14.0000	−	14.0000i	−0.498413	−	0.498413i
$790$		0			0
$791$	78.0000			2.77336
$792$		0			0
$793$	2.00000	−	2.00000i	0.0710221	−	0.0710221i
$794$		0			0
$795$

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

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} +(-3.00000 + 3.00000i) q^{7} +1.00000 q^{9} +(1.00000 + 1.00000i) q^{11} +2.00000i q^{13} +(-1.00000 + 1.00000i) q^{17} +(3.00000 + 3.00000i) q^{19} +(6.00000 - 6.00000i) q^{21} +(-1.00000 - 1.00000i) q^{23} +4.00000 q^{27} +(-7.00000 + 7.00000i) q^{29} -2.00000i q^{31} +(-2.00000 - 2.00000i) q^{33} -6.00000i q^{37} -4.00000i q^{39} -4.00000i q^{41} +4.00000i q^{43} +(-7.00000 - 7.00000i) q^{47} -11.0000i q^{49} +(2.00000 - 2.00000i) q^{51} +8.00000 q^{53} +(-6.00000 - 6.00000i) q^{57} +(-3.00000 + 3.00000i) q^{59} +(-1.00000 - 1.00000i) q^{61} +(-3.00000 + 3.00000i) q^{63} -4.00000i q^{67} +(2.00000 + 2.00000i) q^{69} +(-3.00000 + 3.00000i) q^{73} -6.00000 q^{77} -8.00000 q^{79} -11.0000 q^{81} -2.00000 q^{83} +(14.0000 - 14.0000i) q^{87} -6.00000 q^{89} +(-6.00000 - 6.00000i) q^{91} +4.00000i q^{93} +(11.0000 - 11.0000i) q^{97} +(1.00000 + 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 6 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{17} + 6 q^{19} + 12 q^{21} - 2 q^{23} + 8 q^{27} - 14 q^{29} - 4 q^{33} - 14 q^{47} + 4 q^{51} + 16 q^{53} - 12 q^{57} - 6 q^{59} - 2 q^{61} - 6 q^{63}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 6 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^17 + 6 * q^19 + 12 * q^21 - 2 * q^23 + 8 * q^27 - 14 * q^29 - 4 * q^33 - 14 * q^47 + 4 * q^51 + 16 * q^53 - 12 * q^57 - 6 * q^59 - 2 * q^61 - 6 * q^63 + 4 * q^69 - 6 * q^73 - 12 * q^77 - 16 * q^79 - 22 * q^81 - 4 * q^83 + 28 * q^87 - 12 * q^89 - 12 * q^91 + 22 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more