LMFDB - Embedded newform 4020.2.g.a.1609.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, 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("4020.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.g (of order $2$, degree $1$, 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:	$32.0998616126$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.a.1609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +2.00000i q^{13} +(2.00000 + 1.00000i) q^{15} -2.00000i q^{17} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} -2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} +7.00000 q^{49} +2.00000 q^{51} -4.00000i q^{53} -2.00000 q^{59} -10.0000 q^{61} +(4.00000 + 2.00000i) q^{65} -1.00000i q^{67} +8.00000 q^{69} -10.0000 q^{71} -8.00000i q^{73} +(4.00000 - 3.00000i) q^{75} +14.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +(-4.00000 - 2.00000i) q^{85} -6.00000i q^{87} +2.00000 q^{89} -2.00000i q^{93} +14.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^9	$ 2 q + 2 q^{5} - 2 q^{9} + 4 q^{15} - 6 q^{25} - 12 q^{29} - 4 q^{31} - 4 q^{39} - 12 q^{41} - 2 q^{45} + 14 q^{49} + 4 q^{51} - 4 q^{59} - 20 q^{61} + 8 q^{65} + 16 q^{69} - 20 q^{71} + 8 q^{75} + 28 q^{79} + 2 q^{81} - 8 q^{85} + 4 q^{89}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^9 + 4 * q^15 - 6 * q^25 - 12 * q^29 - 4 * q^31 - 4 * q^39 - 12 * q^41 - 2 * q^45 + 14 * q^49 + 4 * q^51 - 4 * q^59 - 20 * q^61 + 8 * q^65 + 16 * q^69 - 20 * q^71 + 8 * q^75 + 28 * q^79 + 2 * q^81 - 8 * q^85 + 4 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$5$	1.00000	−	2.00000i	0.447214	−	0.894427i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		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
$15$	2.00000	+	1.00000i	0.516398	+	0.258199i
$16$		0			0
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		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
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	−2.00000			−0.320256
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		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
$45$	−1.00000	+	2.00000i	−0.149071	+	0.298142i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$		−	4.00000i		−	0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$		−	4.00000i		−	0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000			−0.260378			−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000			−0.260378			−0.130189	+	0.991489i	$0.541558\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.00000	+	2.00000i	0.496139	+	0.248069i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$	−10.0000			−1.18678			−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000			−1.18678			−0.593391	+	0.804914i	$0.702211\pi$
$72$		0			0
$73$		−	8.00000i		−	0.936329i	−0.883641	−	0.468165i	$-0.844915\pi$
$73$		−	8.00000i		−	0.936329i	0.883641	−	0.468165i	$-0.155085\pi$
$74$		0			0
$75$	4.00000	−	3.00000i	0.461880	−	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	14.0000			1.57512			0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000			1.57512			0.787562	+	0.616236i	$0.211343\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−4.00000	−	2.00000i	−0.433861	−	0.216930i
$86$		0			0
$87$		−	6.00000i		−	0.643268i
$88$		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
$91$		0			0
$92$		0			0
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			14.0000i			1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$			14.0000i			1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	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
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−16.0000	−	8.00000i	−1.49201	−	0.746004i
$116$		0			0
$117$		−	2.00000i		−	0.184900i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		−	6.00000i		−	0.541002i
$124$		0			0
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$		−	20.0000i		−	1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$		−	20.0000i		−	1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$		0			0
$129$	−4.00000			−0.352180
$130$		0			0
$131$	−2.00000			−0.174741			−0.0873704	−	0.996176i	$-0.527846\pi$
$131$	−2.00000			−0.174741			−0.0873704	+	0.996176i	$0.527846\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−2.00000	−	1.00000i	−0.172133	−	0.0860663i
$136$		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
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−6.00000	+	12.0000i	−0.498273	+	0.996546i
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$			2.00000i			0.161690i
$154$		0			0
$155$	−2.00000	+	4.00000i	−0.160644	+	0.321288i
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	4.00000			0.317221
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	8.00000i		−	0.626608i	−0.949653	−	0.313304i	$-0.898564\pi$
$163$		−	8.00000i		−	0.626608i	0.949653	−	0.313304i	$-0.101436\pi$
$164$		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
$169$	9.00000			0.692308
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	2.00000i		−	0.150329i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		−	10.0000i		−	0.739221i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			4.00000i			0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$			4.00000i			0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$		0			0
$195$	−2.00000	+	4.00000i	−0.143223	+	0.286446i
$196$		0			0
$197$			24.0000i			1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$			24.0000i			1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−6.00000	+	12.0000i	−0.419058	+	0.838116i
$206$		0			0
$207$			8.00000i			0.556038i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000			0.275371			0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000			0.275371			0.137686	+	0.990476i	$0.456034\pi$
$212$		0			0
$213$		−	10.0000i		−	0.685189i
$214$		0			0
$215$	8.00000	+	4.00000i	0.545595	+	0.272798i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	8.00000			0.540590
$220$		0			0
$221$	4.00000			0.269069
$222$		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
$225$	3.00000	+	4.00000i	0.200000	+	0.266667i
$226$		0			0
$227$		−	28.0000i		−	1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$		−	28.0000i		−	1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$		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
$233$		−	28.0000i		−	1.83434i	−0.398495	−	0.917170i	$-0.630467\pi$
$233$		−	28.0000i		−	1.83434i	0.398495	−	0.917170i	$-0.369533\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			14.0000i			0.909398i
$238$		0			0
$239$	20.0000			1.29369			0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000			1.29369			0.646846	+	0.762620i	$0.276088\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	7.00000	−	14.0000i	0.447214	−	0.894427i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	12.0000			0.760469
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	2.00000	−	4.00000i	0.125245	−	0.250490i
$256$		0			0
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.00000			0.371391
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−8.00000	−	4.00000i	−0.491436	−	0.245718i
$266$		0			0
$267$			2.00000i			0.122398i
$268$		0			0
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	−22.0000			−1.33640			−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000			−1.33640			−0.668202	+	0.743980i	$0.732936\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		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
$279$	2.00000			0.119737
$280$		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
$283$		−	12.0000i		−	0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$		−	12.0000i		−	0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	13.0000			0.764706
$290$		0			0
$291$	−14.0000			−0.820695
$292$		0			0
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	−2.00000	+	4.00000i	−0.116445	+	0.232889i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	16.0000			0.925304
$300$		0			0
$301$		0			0
$302$		0			0
$303$			10.0000i			0.574485i
$304$		0			0
$305$	−10.0000	+	20.0000i	−0.572598	+	1.14520i
$306$		0			0
$307$		−	28.0000i		−	1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$		−	28.0000i		−	1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$		0			0
$313$		−	18.0000i		−	1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$		−	18.0000i		−	1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	10.0000i		−	0.561656i	−0.959758	−	0.280828i	$-0.909391\pi$
$317$		−	10.0000i		−	0.561656i	0.959758	−	0.280828i	$-0.0906090\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	4.00000			0.223258
$322$		0			0
$323$		0			0
$324$		0			0
$325$	8.00000	−	6.00000i	0.443760	−	0.332820i
$326$		0			0
$327$		−	10.0000i		−	0.553001i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	22.0000			1.20923			0.604615	−	0.796518i	$-0.293327\pi$
$331$	22.0000			1.20923			0.604615	+	0.796518i	$0.293327\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.00000	−	1.00000i	−0.109272	−	0.0546358i
$336$		0			0
$337$			26.0000i			1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$			26.0000i			1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$	8.00000	−	16.0000i	0.430706	−	0.861411i
$346$		0			0
$347$		−	28.0000i		−	1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$		−	28.0000i		−	1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$	−10.0000	+	20.0000i	−0.530745	+	1.06149i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000			−0.527780			−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000			−0.527780			−0.263890	+	0.964553i	$0.585006\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		−	11.0000i		−	0.577350i
$364$		0			0
$365$	−16.0000	−	8.00000i	−0.837478	−	0.418739i
$366$		0			0
$367$			24.0000i			1.25279i	0.779506	+	0.626395i	$0.215470\pi$
$367$			24.0000i			1.25279i	−0.779506	+	0.626395i	$0.784530\pi$
$368$		0			0
$369$	6.00000			0.312348
$370$		0			0
$371$		0			0
$372$		0			0
$373$			22.0000i			1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$			22.0000i			1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$		0			0
$375$	−2.00000	−	11.0000i	−0.103280	−	0.568038i
$376$		0			0
$377$		−	12.0000i		−	0.618031i
$378$		0			0
$379$	−14.0000			−0.719132			−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000			−0.719132			−0.359566	+	0.933120i	$0.617075\pi$
$380$		0			0
$381$	20.0000			1.02463
$382$		0			0
$383$		−	4.00000i		−	0.204390i	−0.994764	−	0.102195i	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.00000i		−	0.203331i
$388$		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
$391$	−16.0000			−0.809155
$392$		0			0
$393$		−	2.00000i		−	0.100887i
$394$		0			0
$395$	14.0000	−	28.0000i	0.704416	−	1.40883i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$		−	4.00000i		−	0.199254i
$404$		0			0
$405$	1.00000	−	2.00000i	0.0496904	−	0.0993808i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	6.00000			0.296681			0.148340	−	0.988936i	$-0.452607\pi$
$409$	6.00000			0.296681			0.148340	+	0.988936i	$0.452607\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−24.0000	−	12.0000i	−1.17811	−	0.589057i
$416$		0			0
$417$		−	10.0000i		−	0.489702i
$418$		0			0
$419$	−6.00000			−0.293119			−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000			−0.293119			−0.146560	+	0.989202i	$0.546820\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−8.00000	+	6.00000i	−0.388057	+	0.291043i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$			34.0000i			1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$			34.0000i			1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$		0			0
$435$	−12.0000	−	6.00000i	−0.575356	−	0.287678i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−20.0000			−0.954548			−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000			−0.954548			−0.477274	+	0.878755i	$0.658375\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$			12.0000i			0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		0			0
$445$	2.00000	−	4.00000i	0.0948091	−	0.189618i
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$	38.0000			1.79333			0.896665	−	0.442709i	$-0.145982\pi$
$449$	38.0000			1.79333			0.896665	+	0.442709i	$0.145982\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			8.00000i			0.375873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			32.0000i			1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$			32.0000i			1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$		0			0
$459$	−2.00000			−0.0933520
$460$		0			0
$461$	−6.00000			−0.279448			−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000			−0.279448			−0.139724	+	0.990190i	$0.544622\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$	−4.00000	−	2.00000i	−0.185496	−	0.0927478i
$466$		0			0
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−12.0000			−0.552931
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$			4.00000i			0.183147i
$478$		0			0
$479$	18.0000			0.822441			0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000			0.822441			0.411220	+	0.911536i	$0.365103\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	28.0000	+	14.0000i	1.27141	+	0.635707i
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$		0			0
$489$	8.00000			0.361773
$490$		0			0
$491$	2.00000			0.0902587			0.0451294	−	0.998981i	$-0.485630\pi$
$491$	2.00000			0.0902587			0.0451294	+	0.998981i	$0.485630\pi$
$492$		0			0
$493$			12.0000i			0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		0			0
$503$			16.0000i			0.713405i	0.934218	+	0.356702i	$0.116099\pi$
$503$			16.0000i			0.713405i	−0.934218	+	0.356702i	$0.883901\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$		0			0
$507$			9.00000i			0.399704i
$508$		0			0
$509$	2.00000			0.0886484			0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000			0.0886484			0.0443242	+	0.999017i	$0.485887\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−2.00000			−0.0877903
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$		−	16.0000i		−	0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			4.00000i			0.174243i
$528$		0			0
$529$	−41.0000			−1.78261
$530$		0			0
$531$	2.00000			0.0867926
$532$		0			0
$533$		−	12.0000i		−	0.519778i
$534$		0			0
$535$	−8.00000	−	4.00000i	−0.345870	−	0.172935i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.0000			−0.601907			−0.300954	−	0.953639i	$-0.597305\pi$
$541$	−14.0000			−0.601907			−0.300954	+	0.953639i	$0.597305\pi$
$542$		0			0
$543$		−	22.0000i		−	0.944110i
$544$		0			0
$545$	−10.0000	+	20.0000i	−0.428353	+	0.856706i
$546$		0			0
$547$		−	20.0000i		−	0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$		−	20.0000i		−	0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$		0			0
$549$	10.0000			0.426790
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	42.0000i		−	1.77960i	−0.456354	−	0.889799i	$-0.650845\pi$
$557$		−	42.0000i		−	1.77960i	0.456354	−	0.889799i	$-0.349155\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$			24.0000i			1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$			24.0000i			1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	38.0000			1.59304			0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000			1.59304			0.796521	+	0.604610i	$0.206671\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−32.0000	+	24.0000i	−1.33449	+	1.00087i
$576$		0			0
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$		0			0
$579$	−4.00000			−0.166234
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−4.00000	−	2.00000i	−0.165380	−	0.0826898i
$586$		0			0
$587$			24.0000i			0.990586i	0.868726	+	0.495293i	$0.164939\pi$
$587$			24.0000i			0.990586i	−0.868726	+	0.495293i	$0.835061\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$	−24.0000			−0.987228
$592$		0			0
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	20.0000i		−	0.818546i
$598$		0			0
$599$	20.0000			0.817178			0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000			0.817178			0.408589	+	0.912719i	$0.366021\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	−11.0000	+	22.0000i	−0.447214	+	0.894427i
$606$		0			0
$607$		−	20.0000i		−	0.811775i	−0.913923	−	0.405887i	$-0.866962\pi$
$607$		−	20.0000i		−	0.811775i	0.913923	−	0.405887i	$-0.133038\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			40.0000i			1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$			40.0000i			1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$		0			0
$615$	−12.0000	−	6.00000i	−0.483887	−	0.241943i
$616$		0			0
$617$			6.00000i			0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$			6.00000i			0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	−8.00000			−0.321029
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−18.0000			−0.716569			−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000			−0.716569			−0.358284	+	0.933613i	$0.616638\pi$
$632$		0			0
$633$			4.00000i			0.158986i
$634$		0			0
$635$	−40.0000	−	20.0000i	−1.58735	−	0.793676i
$636$		0			0
$637$			14.0000i			0.554700i
$638$		0			0
$639$	10.0000			0.395594
$640$		0			0
$641$	2.00000			0.0789953			0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000			0.0789953			0.0394976	+	0.999220i	$0.487424\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$		0			0
$645$	−4.00000	+	8.00000i	−0.157500	+	0.315000i
$646$		0			0
$647$			24.0000i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$			24.0000i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			48.0000i			1.87839i	0.343391	+	0.939193i	$0.388424\pi$
$653$			48.0000i			1.87839i	−0.343391	+	0.939193i	$0.611576\pi$
$654$		0			0
$655$	−2.00000	+	4.00000i	−0.0781465	+	0.156293i
$656$		0			0
$657$			8.00000i			0.312110i
$658$		0			0
$659$	34.0000			1.32445			0.662226	−	0.749304i	$-0.269612\pi$
$659$	34.0000			1.32445			0.662226	+	0.749304i	$0.269612\pi$
$660$		0			0
$661$	−2.00000			−0.0777910			−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000			−0.0777910			−0.0388955	+	0.999243i	$0.512384\pi$
$662$		0			0
$663$			4.00000i			0.155347i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			48.0000i			1.85857i
$668$		0			0
$669$	16.0000			0.618596
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	34.0000i		−	1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$		−	34.0000i		−	1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$		0			0
$675$	−4.00000	+	3.00000i	−0.153960	+	0.115470i
$676$		0			0
$677$		−	8.00000i		−	0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$677$		−	8.00000i		−	0.307465i	0.988113	−	0.153732i	$-0.0491294\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	28.0000			1.07296
$682$		0			0
$683$		−	8.00000i		−	0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$		−	8.00000i		−	0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$		0			0
$685$	−24.0000	−	12.0000i	−0.916993	−	0.458496i
$686$		0			0
$687$			10.0000i			0.381524i
$688$		0			0
$689$	8.00000			0.304776
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.0000	+	20.0000i	−0.379322	+	0.758643i
$696$		0			0
$697$			12.0000i			0.454532i
$698$		0			0
$699$	28.0000			1.05906
$700$		0			0
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−6.00000			−0.225335			−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000			−0.225335			−0.112667	+	0.993633i	$0.535939\pi$
$710$		0			0
$711$	−14.0000			−0.525041
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			20.0000i			0.746914i
$718$		0			0
$719$	−42.0000			−1.56634			−0.783168	−	0.621810i	$-0.786397\pi$
$719$	−42.0000			−1.56634			−0.783168	+	0.621810i	$0.786397\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			10.0000i			0.371904i
$724$		0			0
$725$	18.0000	+	24.0000i	0.668503	+	0.891338i
$726$		0			0
$727$			32.0000i			1.18681i	0.804902	+	0.593407i	$0.202218\pi$
$727$			32.0000i			1.18681i	−0.804902	+	0.593407i	$0.797782\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	8.00000			0.295891
$732$		0			0
$733$			18.0000i			0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$			18.0000i			0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$		0			0
$735$	14.0000	+	7.00000i	0.516398	+	0.258199i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−10.0000			−0.367856			−0.183928	−	0.982940i	$-0.558881\pi$
$739$	−10.0000			−0.367856			−0.183928	+	0.982940i	$0.558881\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	−18.0000	+	36.0000i	−0.659469	+	1.31894i
$746$		0			0
$747$			12.0000i			0.439057i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$			4.00000i			0.145768i
$754$		0			0
$755$	8.00000	−	16.0000i	0.291150	−	0.582300i
$756$		0			0
$757$		−	2.00000i		−	0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$		−	2.00000i		−	0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	4.00000	+	2.00000i	0.144620	+	0.0723102i
$766$		0			0
$767$		−	4.00000i		−	0.144432i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	6.00000			0.216085
$772$		0			0
$773$			18.0000i			0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		0			0
$775$	6.00000	+	8.00000i	0.215526	+	0.287368i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			6.00000i			0.214423i
$784$		0			0
$785$	24.0000	+	12.0000i	0.856597	+	0.428298i
$786$		0			0
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	20.0000i		−	0.710221i
$794$		0			0
$795$	4.00000	−	8.00000i	0.141865	−	0.283731i
$796$		0			0
$797$		−	2.00000i		−	0.0708436i	−0.999372	−	0.0354218i	$-0.988723\pi$
$797$		−	2.00000i		−	0.0708436i	0.999372	−	0.0354218i	$-0.0112775\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−2.00000			−0.0706665
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$			14.0000i			0.492823i
$808$		0			0
$809$	26.0000			0.914111			0.457056	−	0.889438i	$-0.348904\pi$
$809$	26.0000			0.914111			0.457056	+	0.889438i	$0.348904\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		−	22.0000i		−	0.771574i
$814$		0			0
$815$	−16.0000	−	8.00000i	−0.560456	−	0.280228i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.0000			−0.349002			−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000			−0.349002			−0.174501	+	0.984657i	$0.555831\pi$
$822$		0			0
$823$		−	8.00000i		−	0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$		−	8.00000i		−	0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			4.00000i			0.139094i	0.997579	+	0.0695468i	$0.0221553\pi$
$827$			4.00000i			0.139094i	−0.997579	+	0.0695468i	$0.977845\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$		0			0
$833$		−	14.0000i		−	0.485071i
$834$		0			0
$835$	−16.0000	−	8.00000i	−0.553703	−	0.276851i
$836$		0			0
$837$			2.00000i			0.0691301i
$838$		0			0
$839$	−10.0000			−0.345238			−0.172619	−	0.984989i	$-0.555223\pi$
$839$	−10.0000			−0.345238			−0.172619	+	0.984989i	$0.555223\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		−	26.0000i		−	0.895488i
$844$		0			0
$845$	9.00000	−	18.0000i	0.309609	−	0.619219i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	12.0000			0.411839
$850$		0			0
$851$		0			0
$852$		0			0
$853$			4.00000i			0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$			4.00000i			0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	16.0000			0.545913			0.272956	−	0.962026i	$-0.411998\pi$
$859$	16.0000			0.545913			0.272956	+	0.962026i	$0.411998\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			24.0000i			0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$			24.0000i			0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$		0			0
$865$	4.00000	+	2.00000i	0.136004	+	0.0680020i
$866$		0			0
$867$			13.0000i			0.441503i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	2.00000			0.0677674
$872$		0			0
$873$		−	14.0000i		−	0.473828i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	4.00000i		−	0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$877$		−	4.00000i		−	0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$878$		0			0
$879$	−18.0000			−0.607125
$880$		0			0
$881$	−34.0000			−1.14549			−0.572745	−	0.819734i	$-0.694121\pi$
$881$	−34.0000			−1.14549			−0.572745	+	0.819734i	$0.694121\pi$
$882$		0			0
$883$			12.0000i			0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$			12.0000i			0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$		0			0
$885$	−4.00000	−	2.00000i	−0.134459	−	0.0672293i
$886$		0			0
$887$		−	24.0000i		−	0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$		−	24.0000i		−	0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$			16.0000i			0.534224i
$898$		0			0
$899$	12.0000			0.400222
$900$		0			0
$901$	−8.00000			−0.266519
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−22.0000	+	44.0000i	−0.731305	+	1.46261i
$906$		0			0
$907$			48.0000i			1.59381i	0.604102	+	0.796907i	$0.293532\pi$
$907$			48.0000i			1.59381i	−0.604102	+	0.796907i	$0.706468\pi$
$908$		0			0
$909$	−10.0000			−0.331679
$910$		0			0
$911$	−54.0000			−1.78910			−0.894550	−	0.446968i	$-0.852504\pi$
$911$	−54.0000			−1.78910			−0.894550	+	0.446968i	$0.852504\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−20.0000	−	10.0000i	−0.661180	−	0.330590i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−6.00000			−0.197922			−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000			−0.197922			−0.0989609	+	0.995091i	$0.531552\pi$
$920$		0			0
$921$	28.0000			0.922631
$922$		0			0
$923$		−	20.0000i		−	0.658308i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.00000			−0.0656179			−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000			−0.0656179			−0.0328089	+	0.999462i	$0.510445\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			12.0000i			0.392862i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	14.0000i		−	0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$		−	14.0000i		−	0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$		0			0
$939$	18.0000			0.587408
$940$		0			0
$941$	18.0000			0.586783			0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000			0.586783			0.293392	+	0.955992i	$0.405216\pi$
$942$		0			0
$943$			48.0000i			1.56310i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			12.0000i			0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$			12.0000i			0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$		0			0
$949$	16.0000			0.519382
$950$		0			0
$951$	10.0000			0.324272
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$			4.00000i			0.128898i
$964$		0			0
$965$	8.00000	+	4.00000i	0.257529	+	0.128765i
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−14.0000			−0.449281			−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000			−0.449281			−0.224641	+	0.974442i	$0.572121\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	6.00000	+	8.00000i	0.192154	+	0.256205i
$976$		0			0
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	10.0000			0.319275
$982$		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
$985$	48.0000	+	24.0000i	1.52941	+	0.764704i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000			1.01754
$990$		0			0
$991$	−6.00000			−0.190596			−0.0952981	−	0.995449i	$-0.530380\pi$
$991$	−6.00000			−0.190596			−0.0952981	+	0.995449i	$0.530380\pi$
$992$		0			0
$993$			22.0000i			0.698149i
$994$		0			0
$995$	−20.0000	+	40.0000i	−0.634043	+	1.26809i
$996$		0			0
$997$		−	36.0000i		−	1.14013i	−0.821599	−	0.570066i	$-0.806918\pi$
$997$		−	36.0000i		−	1.14013i	0.821599	−	0.570066i	$-0.193082\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.a.1609.2	yes	2
5.4	even	2	inner	4020.2.g.a.1609.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.a.1609.1	✓	2	5.4	even	2	inner
4020.2.g.a.1609.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(1.00000 - 2.00000i) q^{5} -1.00000 q^{9} +2.00000i q^{13} +(2.00000 + 1.00000i) q^{15} -2.00000i q^{17} -8.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{27} -6.00000 q^{29} -2.00000 q^{31} -2.00000 q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-1.00000 + 2.00000i) q^{45} +7.00000 q^{49} +2.00000 q^{51} -4.00000i q^{53} -2.00000 q^{59} -10.0000 q^{61} +(4.00000 + 2.00000i) q^{65} -1.00000i q^{67} +8.00000 q^{69} -10.0000 q^{71} -8.00000i q^{73} +(4.00000 - 3.00000i) q^{75} +14.0000 q^{79} +1.00000 q^{81} -12.0000i q^{83} +(-4.00000 - 2.00000i) q^{85} -6.00000i q^{87} +2.00000 q^{89} -2.00000i q^{93} +14.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^9	\( 2 q + 2 q^{5} - 2 q^{9} + 4 q^{15} - 6 q^{25} - 12 q^{29} - 4 q^{31} - 4 q^{39} - 12 q^{41} - 2 q^{45} + 14 q^{49} + 4 q^{51} - 4 q^{59} - 20 q^{61} + 8 q^{65} + 16 q^{69} - 20 q^{71} + 8 q^{75} + 28 q^{79} + 2 q^{81} - 8 q^{85} + 4 q^{89}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^9 + 4 * q^15 - 6 * q^25 - 12 * q^29 - 4 * q^31 - 4 * q^39 - 12 * q^41 - 2 * q^45 + 14 * q^49 + 4 * q^51 - 4 * q^59 - 20 * q^61 + 8 * q^65 + 16 * q^69 - 20 * q^71 + 8 * q^75 + 28 * q^79 + 2 * q^81 - 8 * q^85 + 4 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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