LMFDB - Embedded newform 403.2.c.a.311.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [403,2,Mod(311,403)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(403, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("403.311");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.c (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:	$3.21797120146$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			311.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	403.311
Dual form			403.2.c.a.311.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{3} +2.00000 q^{4} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{3} +2.00000 q^{4} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +1.00000i q^{11} +4.00000 q^{12} +(-2.00000 - 3.00000i) q^{13} +8.00000i q^{15} +4.00000 q^{16} -2.00000 q^{17} -2.00000i q^{19} +8.00000i q^{20} -4.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -4.00000 q^{27} -4.00000i q^{28} +8.00000 q^{29} -1.00000i q^{31} +2.00000i q^{33} +8.00000 q^{35} +2.00000 q^{36} -3.00000i q^{37} +(-4.00000 - 6.00000i) q^{39} -4.00000i q^{41} +4.00000 q^{43} +2.00000i q^{44} +4.00000i q^{45} -2.00000i q^{47} +8.00000 q^{48} +3.00000 q^{49} -4.00000 q^{51} +(-4.00000 - 6.00000i) q^{52} -4.00000 q^{53} -4.00000 q^{55} -4.00000i q^{57} +6.00000i q^{59} +16.0000i q^{60} +10.0000 q^{61} -2.00000i q^{63} +8.00000 q^{64} +(12.0000 - 8.00000i) q^{65} +16.0000i q^{67} -4.00000 q^{68} -8.00000 q^{69} -6.00000i q^{71} -7.00000i q^{73} -22.0000 q^{75} -4.00000i q^{76} +2.00000 q^{77} -16.0000 q^{79} +16.0000i q^{80} -11.0000 q^{81} +12.0000i q^{83} -8.00000i q^{84} -8.00000i q^{85} +16.0000 q^{87} -15.0000i q^{89} +(-6.00000 + 4.00000i) q^{91} -8.00000 q^{92} -2.00000i q^{93} +8.00000 q^{95} +2.00000i q^{97} +1.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9	$ 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9} + 8 q^{12} - 4 q^{13} + 8 q^{16} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 8 q^{27} + 16 q^{29} + 16 q^{35} + 4 q^{36} - 8 q^{39} + 8 q^{43} + 16 q^{48} + 6 q^{49} - 8 q^{51} - 8 q^{52} - 8 q^{53} - 8 q^{55} + 20 q^{61} + 16 q^{64} + 24 q^{65} - 8 q^{68} - 16 q^{69} - 44 q^{75} + 4 q^{77} - 32 q^{79} - 22 q^{81} + 32 q^{87} - 12 q^{91} - 16 q^{92} + 16 q^{95}+O(q^{100}) $ 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9 + 8 * q^12 - 4 * q^13 + 8 * q^16 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 8 * q^27 + 16 * q^29 + 16 * q^35 + 4 * q^36 - 8 * q^39 + 8 * q^43 + 16 * q^48 + 6 * q^49 - 8 * q^51 - 8 * q^52 - 8 * q^53 - 8 * q^55 + 20 * q^61 + 16 * q^64 + 24 * q^65 - 8 * q^68 - 16 * q^69 - 44 * q^75 + 4 * q^77 - 32 * q^79 - 22 * q^81 + 32 * q^87 - 12 * q^91 - 16 * q^92 + 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/403\mathbb{Z}\right)^\times$.

$n$	$249$	$313$
$\chi(n)$	$-1$	$1$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$	2.00000			1.15470			0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000			1.15470			0.577350	+	0.816497i	$0.304087\pi$
$4$	2.00000			1.00000
$5$			4.00000i			1.78885i	0.447214	+	0.894427i	$0.352416\pi$
$5$			4.00000i			1.78885i	−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.00000i			0.301511i	0.988571	+	0.150756i	$0.0481707\pi$
$11$			1.00000i			0.301511i	−0.988571	+	0.150756i	$0.951829\pi$
$12$	4.00000			1.15470
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$13$	−2.00000	−	3.00000i	−0.554700	−	0.832050i
$14$		0			0
$15$			8.00000i			2.06559i
$16$	4.00000			1.00000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$			8.00000i			1.78885i
$21$		−	4.00000i		−	0.872872i
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−11.0000			−2.20000
$26$		0			0
$27$	−4.00000			−0.769800
$28$		−	4.00000i		−	0.755929i
$29$	8.00000			1.48556			0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000			1.48556			0.742781	+	0.669534i	$0.233506\pi$
$30$		0			0
$31$		−	1.00000i		−	0.179605i
$31$		−	1.00000i		−	0.179605i
$32$		0			0
$33$			2.00000i			0.348155i
$34$		0			0
$35$	8.00000			1.35225
$36$	2.00000			0.333333
$37$		−	3.00000i		−	0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$		−	3.00000i		−	0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$		0			0
$39$	−4.00000	−	6.00000i	−0.640513	−	0.960769i
$40$		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
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$			2.00000i			0.301511i
$45$			4.00000i			0.596285i
$46$		0			0
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	8.00000			1.15470
$49$	3.00000			0.428571
$50$		0			0
$51$	−4.00000			−0.560112
$52$	−4.00000	−	6.00000i	−0.554700	−	0.832050i
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$		−	4.00000i		−	0.529813i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$			16.0000i			2.06559i
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		−	2.00000i		−	0.251976i
$64$	8.00000			1.00000
$65$	12.0000	−	8.00000i	1.48842	−	0.992278i
$66$		0			0
$67$			16.0000i			1.95471i	0.211604	+	0.977356i	$0.432131\pi$
$67$			16.0000i			1.95471i	−0.211604	+	0.977356i	$0.567869\pi$
$68$	−4.00000			−0.485071
$69$	−8.00000			−0.963087
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$		−	7.00000i		−	0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$		−	7.00000i		−	0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$		0			0
$75$	−22.0000			−2.54034
$76$		−	4.00000i		−	0.458831i
$77$	2.00000			0.227921
$78$		0			0
$79$	−16.0000			−1.80014			−0.900070	−	0.435745i	$-0.856485\pi$
$79$	−16.0000			−1.80014			−0.900070	+	0.435745i	$0.856485\pi$
$80$			16.0000i			1.78885i
$81$	−11.0000			−1.22222
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		−	8.00000i		−	0.872872i
$85$		−	8.00000i		−	0.867722i
$86$		0			0
$87$	16.0000			1.71538
$88$		0			0
$89$		−	15.0000i		−	1.59000i	−0.606612	−	0.794998i	$-0.707472\pi$
$89$		−	15.0000i		−	1.59000i	0.606612	−	0.794998i	$-0.292528\pi$
$90$		0			0
$91$	−6.00000	+	4.00000i	−0.628971	+	0.419314i
$92$	−8.00000			−0.834058
$93$		−	2.00000i		−	0.207390i
$94$		0			0
$95$	8.00000			0.820783
$96$		0			0
$97$			2.00000i			0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$			2.00000i			0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$	−22.0000			−2.20000
$101$	−3.00000			−0.298511			−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000			−0.298511			−0.149256	+	0.988799i	$0.547688\pi$
$102$		0			0
$103$	−15.0000			−1.47799			−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000			−1.47799			−0.738997	+	0.673709i	$0.764700\pi$
$104$		0			0
$105$	16.0000			1.56144
$106$		0			0
$107$	9.00000			0.870063			0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000			0.870063			0.435031	+	0.900415i	$0.356737\pi$
$108$	−8.00000			−0.769800
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$		−	6.00000i		−	0.569495i
$112$		−	8.00000i		−	0.755929i
$113$	−17.0000			−1.59923			−0.799613	−	0.600516i	$-0.794962\pi$
$113$	−17.0000			−1.59923			−0.799613	+	0.600516i	$0.794962\pi$
$114$		0			0
$115$		−	16.0000i		−	1.49201i
$116$	16.0000			1.48556
$117$	−2.00000	−	3.00000i	−0.184900	−	0.277350i
$118$		0			0
$119$			4.00000i			0.366679i
$120$		0			0
$121$	10.0000			0.909091
$122$		0			0
$123$		−	8.00000i		−	0.721336i
$124$		−	2.00000i		−	0.179605i
$125$		−	24.0000i		−	2.14663i
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$		0			0
$129$	8.00000			0.704361
$130$		0			0
$131$	1.00000			0.0873704			0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000			0.0873704			0.0436852	+	0.999045i	$0.486090\pi$
$132$			4.00000i			0.348155i
$133$	−4.00000			−0.346844
$134$		0			0
$135$		−	16.0000i		−	1.37706i
$136$		0			0
$137$			6.00000i			0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$			6.00000i			0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$	16.0000			1.35225
$141$		−	4.00000i		−	0.336861i
$142$		0			0
$143$	3.00000	−	2.00000i	0.250873	−	0.167248i
$144$	4.00000			0.333333
$145$			32.0000i			2.65746i
$146$		0			0
$147$	6.00000			0.494872
$148$		−	6.00000i		−	0.493197i
$149$			22.0000i			1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$			22.0000i			1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$		0			0
$151$			17.0000i			1.38344i	0.722166	+	0.691720i	$0.243147\pi$
$151$			17.0000i			1.38344i	−0.722166	+	0.691720i	$0.756853\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	4.00000			0.321288
$156$	−8.00000	−	12.0000i	−0.640513	−	0.960769i
$157$	13.0000			1.03751			0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000			1.03751			0.518756	+	0.854922i	$0.326395\pi$
$158$		0			0
$159$	−8.00000			−0.634441
$160$		0			0
$161$			8.00000i			0.630488i
$162$		0			0
$163$			12.0000i			0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$			12.0000i			0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$		−	8.00000i		−	0.624695i
$165$	−8.00000			−0.622799
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$		−	2.00000i		−	0.152944i
$172$	8.00000			0.609994
$173$	−9.00000			−0.684257			−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000			−0.684257			−0.342129	+	0.939653i	$0.611148\pi$
$174$		0			0
$175$			22.0000i			1.66304i
$176$			4.00000i			0.301511i
$177$			12.0000i			0.901975i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$			8.00000i			0.596285i
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$	20.0000			1.47844
$184$		0			0
$185$	12.0000			0.882258
$186$		0			0
$187$		−	2.00000i		−	0.146254i
$188$		−	4.00000i		−	0.291730i
$189$			8.00000i			0.581914i
$190$		0			0
$191$	−5.00000			−0.361787			−0.180894	−	0.983503i	$-0.557899\pi$
$191$	−5.00000			−0.361787			−0.180894	+	0.983503i	$0.557899\pi$
$192$	16.0000			1.15470
$193$			14.0000i			1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$			14.0000i			1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$		0			0
$195$	24.0000	−	16.0000i	1.71868	−	1.14578i
$196$	6.00000			0.428571
$197$		−	15.0000i		−	1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$		−	15.0000i		−	1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$		0			0
$199$	10.0000			0.708881			0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000			0.708881			0.354441	+	0.935079i	$0.384671\pi$
$200$		0			0
$201$			32.0000i			2.25711i
$202$		0			0
$203$		−	16.0000i		−	1.12298i
$204$	−8.00000			−0.560112
$205$	16.0000			1.11749
$206$		0			0
$207$	−4.00000			−0.278019
$208$	−8.00000	−	12.0000i	−0.554700	−	0.832050i
$209$	2.00000			0.138343
$210$		0			0
$211$	19.0000			1.30801			0.654007	−	0.756489i	$-0.273087\pi$
$211$	19.0000			1.30801			0.654007	+	0.756489i	$0.273087\pi$
$212$	−8.00000			−0.549442
$213$		−	12.0000i		−	0.822226i
$214$		0			0
$215$			16.0000i			1.09119i
$216$		0			0
$217$	−2.00000			−0.135769
$218$		0			0
$219$		−	14.0000i		−	0.946032i
$220$	−8.00000			−0.539360
$221$	4.00000	+	6.00000i	0.269069	+	0.403604i
$222$		0			0
$223$		−	19.0000i		−	1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$		0			0
$225$	−11.0000			−0.733333
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		−	8.00000i		−	0.529813i
$229$		−	29.0000i		−	1.91637i	−0.286143	−	0.958187i	$-0.592373\pi$
$229$		−	29.0000i		−	1.91637i	0.286143	−	0.958187i	$-0.407627\pi$
$230$		0			0
$231$	4.00000			0.263181
$232$		0			0
$233$	13.0000			0.851658			0.425829	−	0.904804i	$-0.359982\pi$
$233$	13.0000			0.851658			0.425829	+	0.904804i	$0.359982\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$			12.0000i			0.781133i
$237$	−32.0000			−2.07862
$238$		0			0
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$			32.0000i			2.06559i
$241$			14.0000i			0.901819i	0.892570	+	0.450910i	$0.148900\pi$
$241$			14.0000i			0.901819i	−0.892570	+	0.450910i	$0.851100\pi$
$242$		0			0
$243$	−10.0000			−0.641500
$244$	20.0000			1.28037
$245$			12.0000i			0.766652i
$246$		0			0
$247$	−6.00000	+	4.00000i	−0.381771	+	0.254514i
$248$		0			0
$249$			24.0000i			1.52094i
$250$		0			0
$251$	22.0000			1.38863			0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000			1.38863			0.694314	+	0.719672i	$0.255708\pi$
$252$		−	4.00000i		−	0.251976i
$253$		−	4.00000i		−	0.251478i
$254$		0			0
$255$		−	16.0000i		−	1.00196i
$256$	16.0000			1.00000
$257$	21.0000			1.30994			0.654972	−	0.755653i	$-0.272680\pi$
$257$	21.0000			1.30994			0.654972	+	0.755653i	$0.272680\pi$
$258$		0			0
$259$	−6.00000			−0.372822
$260$	24.0000	−	16.0000i	1.48842	−	0.992278i
$261$	8.00000			0.495188
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		−	16.0000i		−	0.982872i
$266$		0			0
$267$		−	30.0000i		−	1.83597i
$268$			32.0000i			1.95471i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$		−	15.0000i		−	0.911185i	−0.890188	−	0.455593i	$-0.849427\pi$
$271$		−	15.0000i		−	0.911185i	0.890188	−	0.455593i	$-0.150573\pi$
$272$	−8.00000			−0.485071
$273$	−12.0000	+	8.00000i	−0.726273	+	0.484182i
$274$		0			0
$275$		−	11.0000i		−	0.663325i
$276$	−16.0000			−0.963087
$277$	−16.0000			−0.961347			−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000			−0.961347			−0.480673	+	0.876900i	$0.659608\pi$
$278$		0			0
$279$		−	1.00000i		−	0.0598684i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$		0			0
$283$	13.0000			0.772770			0.386385	−	0.922338i	$-0.373724\pi$
$283$	13.0000			0.772770			0.386385	+	0.922338i	$0.373724\pi$
$284$		−	12.0000i		−	0.712069i
$285$	16.0000			0.947758
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$			4.00000i			0.234484i
$292$		−	14.0000i		−	0.819288i
$293$		−	18.0000i		−	1.05157i	−0.850617	−	0.525786i	$-0.823771\pi$
$293$		−	18.0000i		−	1.05157i	0.850617	−	0.525786i	$-0.176229\pi$
$294$		0			0
$295$	−24.0000			−1.39733
$296$		0			0
$297$		−	4.00000i		−	0.232104i
$298$		0			0
$299$	8.00000	+	12.0000i	0.462652	+	0.693978i
$300$	−44.0000			−2.54034
$301$		−	8.00000i		−	0.461112i
$302$		0			0
$303$	−6.00000			−0.344691
$304$		−	8.00000i		−	0.458831i
$305$			40.0000i			2.29039i
$306$		0			0
$307$		−	14.0000i		−	0.799022i	−0.916728	−	0.399511i	$-0.869180\pi$
$307$		−	14.0000i		−	0.799022i	0.916728	−	0.399511i	$-0.130820\pi$
$308$	4.00000			0.227921
$309$	−30.0000			−1.70664
$310$		0			0
$311$	15.0000			0.850572			0.425286	−	0.905059i	$-0.360174\pi$
$311$	15.0000			0.850572			0.425286	+	0.905059i	$0.360174\pi$
$312$		0			0
$313$	16.0000			0.904373			0.452187	−	0.891923i	$-0.350644\pi$
$313$	16.0000			0.904373			0.452187	+	0.891923i	$0.350644\pi$
$314$		0			0
$315$	8.00000			0.450749
$316$	−32.0000			−1.80014
$317$			24.0000i			1.34797i	0.738743	+	0.673987i	$0.235420\pi$
$317$			24.0000i			1.34797i	−0.738743	+	0.673987i	$0.764580\pi$
$318$		0			0
$319$			8.00000i			0.447914i
$320$			32.0000i			1.78885i
$321$	18.0000			1.00466
$322$		0			0
$323$			4.00000i			0.222566i
$324$	−22.0000			−1.22222
$325$	22.0000	+	33.0000i	1.22034	+	1.83051i
$326$		0			0
$327$		−	4.00000i		−	0.221201i
$328$		0			0
$329$	−4.00000			−0.220527
$330$		0			0
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$			24.0000i			1.31717i
$333$		−	3.00000i		−	0.164399i
$334$		0			0
$335$	−64.0000			−3.49669
$336$		−	16.0000i		−	0.872872i
$337$	8.00000			0.435788			0.217894	−	0.975972i	$-0.430081\pi$
$337$	8.00000			0.435788			0.217894	+	0.975972i	$0.430081\pi$
$338$		0			0
$339$	−34.0000			−1.84663
$340$		−	16.0000i		−	0.867722i
$341$	1.00000			0.0541530
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		−	32.0000i		−	1.72282i
$346$		0			0
$347$	−2.00000			−0.107366			−0.0536828	−	0.998558i	$-0.517096\pi$
$347$	−2.00000			−0.107366			−0.0536828	+	0.998558i	$0.517096\pi$
$348$	32.0000			1.71538
$349$		−	36.0000i		−	1.92704i	−0.267644	−	0.963518i	$-0.586245\pi$
$349$		−	36.0000i		−	1.92704i	0.267644	−	0.963518i	$-0.413755\pi$
$350$		0			0
$351$	8.00000	+	12.0000i	0.427008	+	0.640513i
$352$		0			0
$353$			15.0000i			0.798369i	0.916871	+	0.399185i	$0.130707\pi$
$353$			15.0000i			0.798369i	−0.916871	+	0.399185i	$0.869293\pi$
$354$		0			0
$355$	24.0000			1.27379
$356$		−	30.0000i		−	1.59000i
$357$			8.00000i			0.423405i
$358$		0			0
$359$		−	20.0000i		−	1.05556i	−0.849381	−	0.527780i	$-0.823025\pi$
$359$		−	20.0000i		−	1.05556i	0.849381	−	0.527780i	$-0.176975\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$	20.0000			1.04973
$364$	−12.0000	+	8.00000i	−0.628971	+	0.419314i
$365$	28.0000			1.46559
$366$		0			0
$367$	4.00000			0.208798			0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000			0.208798			0.104399	+	0.994535i	$0.466708\pi$
$368$	−16.0000			−0.834058
$369$		−	4.00000i		−	0.208232i
$370$		0			0
$371$			8.00000i			0.415339i
$372$		−	4.00000i		−	0.207390i
$373$	9.00000			0.466002			0.233001	−	0.972476i	$-0.425145\pi$
$373$	9.00000			0.466002			0.233001	+	0.972476i	$0.425145\pi$
$374$		0			0
$375$		−	48.0000i		−	2.47871i
$376$		0			0
$377$	−16.0000	−	24.0000i	−0.824042	−	1.23606i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	16.0000			0.820783
$381$	−8.00000			−0.409852
$382$		0			0
$383$			13.0000i			0.664269i	0.943232	+	0.332134i	$0.107769\pi$
$383$			13.0000i			0.664269i	−0.943232	+	0.332134i	$0.892231\pi$
$384$		0			0
$385$			8.00000i			0.407718i
$386$		0			0
$387$	4.00000			0.203331
$388$			4.00000i			0.203069i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$	2.00000			0.100887
$394$		0			0
$395$		−	64.0000i		−	3.22019i
$396$			2.00000i			0.100504i
$397$		−	6.00000i		−	0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$		−	6.00000i		−	0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$		0			0
$399$	−8.00000			−0.400501
$400$	−44.0000			−2.20000
$401$			29.0000i			1.44819i	0.689700	+	0.724095i	$0.257743\pi$
$401$			29.0000i			1.44819i	−0.689700	+	0.724095i	$0.742257\pi$
$402$		0			0
$403$	−3.00000	+	2.00000i	−0.149441	+	0.0996271i
$404$	−6.00000			−0.298511
$405$		−	44.0000i		−	2.18638i
$406$		0			0
$407$	3.00000			0.148704
$408$		0			0
$409$		−	3.00000i		−	0.148340i	−0.997246	−	0.0741702i	$-0.976369\pi$
$409$		−	3.00000i		−	0.148340i	0.997246	−	0.0741702i	$-0.0236308\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$	−30.0000			−1.47799
$413$	12.0000			0.590481
$414$		0			0
$415$	−48.0000			−2.35623
$416$		0			0
$417$	−28.0000			−1.37117
$418$		0			0
$419$	−40.0000			−1.95413			−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000			−1.95413			−0.977064	+	0.212946i	$0.931694\pi$
$420$	32.0000			1.56144
$421$			26.0000i			1.26716i	0.773676	+	0.633581i	$0.218416\pi$
$421$			26.0000i			1.26716i	−0.773676	+	0.633581i	$0.781584\pi$
$422$		0			0
$423$		−	2.00000i		−	0.0972433i
$424$		0			0
$425$	22.0000			1.06716
$426$		0			0
$427$		−	20.0000i		−	0.967868i
$428$	18.0000			0.870063
$429$	6.00000	−	4.00000i	0.289683	−	0.193122i
$430$		0			0
$431$			18.0000i			0.867029i	0.901146	+	0.433515i	$0.142727\pi$
$431$			18.0000i			0.867029i	−0.901146	+	0.433515i	$0.857273\pi$
$432$	−16.0000			−0.769800
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$			64.0000i			3.06857i
$436$		−	4.00000i		−	0.191565i
$437$			8.00000i			0.382692i
$438$		0			0
$439$	−4.00000			−0.190910			−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000			−0.190910			−0.0954548	+	0.995434i	$0.530431\pi$
$440$		0			0
$441$	3.00000			0.142857
$442$		0			0
$443$	−5.00000			−0.237557			−0.118779	−	0.992921i	$-0.537898\pi$
$443$	−5.00000			−0.237557			−0.118779	+	0.992921i	$0.537898\pi$
$444$		−	12.0000i		−	0.569495i
$445$	60.0000			2.84427
$446$		0			0
$447$			44.0000i			2.08113i
$448$		−	16.0000i		−	0.755929i
$449$		−	6.00000i		−	0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$		−	6.00000i		−	0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$		0			0
$451$	4.00000			0.188353
$452$	−34.0000			−1.59923
$453$			34.0000i			1.59746i
$454$		0			0
$455$	−16.0000	−	24.0000i	−0.750092	−	1.12514i
$456$		0			0
$457$		−	31.0000i		−	1.45012i	−0.688686	−	0.725059i	$-0.741812\pi$
$457$		−	31.0000i		−	1.45012i	0.688686	−	0.725059i	$-0.258188\pi$
$458$		0			0
$459$	8.00000			0.373408
$460$		−	32.0000i		−	1.49201i
$461$		−	11.0000i		−	0.512321i	−0.966634	−	0.256161i	$-0.917542\pi$
$461$		−	11.0000i		−	0.512321i	0.966634	−	0.256161i	$-0.0824576\pi$
$462$		0			0
$463$			5.00000i			0.232370i	0.993228	+	0.116185i	$0.0370665\pi$
$463$			5.00000i			0.232370i	−0.993228	+	0.116185i	$0.962933\pi$
$464$	32.0000			1.48556
$465$	8.00000			0.370991
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$	−4.00000	−	6.00000i	−0.184900	−	0.277350i
$469$	32.0000			1.47762
$470$		0			0
$471$	26.0000			1.19802
$472$		0			0
$473$			4.00000i			0.183920i
$474$		0			0
$475$			22.0000i			1.00943i
$476$			8.00000i			0.366679i
$477$	−4.00000			−0.183147
$478$		0			0
$479$			30.0000i			1.37073i	0.728197	+	0.685367i	$0.240358\pi$
$479$			30.0000i			1.37073i	−0.728197	+	0.685367i	$0.759642\pi$
$480$		0			0
$481$	−9.00000	+	6.00000i	−0.410365	+	0.273576i
$482$		0			0
$483$			16.0000i			0.728025i
$484$	20.0000			0.909091
$485$	−8.00000			−0.363261
$486$		0			0
$487$			16.0000i			0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$			16.0000i			0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$		0			0
$489$			24.0000i			1.08532i
$490$		0			0
$491$	20.0000			0.902587			0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000			0.902587			0.451294	+	0.892375i	$0.350963\pi$
$492$		−	16.0000i		−	0.721336i
$493$	−16.0000			−0.720604
$494$		0			0
$495$	−4.00000			−0.179787
$496$		−	4.00000i		−	0.179605i
$497$	−12.0000			−0.538274
$498$		0			0
$499$		−	24.0000i		−	1.07439i	−0.843459	−	0.537194i	$-0.819484\pi$
$499$		−	24.0000i		−	1.07439i	0.843459	−	0.537194i	$-0.180516\pi$
$500$		−	48.0000i		−	2.14663i
$501$			6.00000i			0.268060i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		−	12.0000i		−	0.533993i
$506$		0			0
$507$	−10.0000	+	24.0000i	−0.444116	+	1.06588i
$508$	−8.00000			−0.354943
$509$			30.0000i			1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$			30.0000i			1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$		0			0
$511$	−14.0000			−0.619324
$512$		0			0
$513$			8.00000i			0.353209i
$514$		0			0
$515$		−	60.0000i		−	2.64392i
$516$	16.0000			0.704361
$517$	2.00000			0.0879599
$518$		0			0
$519$	−18.0000			−0.790112
$520$		0			0
$521$	−15.0000			−0.657162			−0.328581	−	0.944476i	$-0.606570\pi$
$521$	−15.0000			−0.657162			−0.328581	+	0.944476i	$0.606570\pi$
$522$		0			0
$523$	22.0000			0.961993			0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000			0.961993			0.480996	+	0.876723i	$0.340275\pi$
$524$	2.00000			0.0873704
$525$			44.0000i			1.92032i
$526$		0			0
$527$			2.00000i			0.0871214i
$528$			8.00000i			0.348155i
$529$	−7.00000			−0.304348
$530$		0			0
$531$			6.00000i			0.260378i
$532$	−8.00000			−0.346844
$533$	−12.0000	+	8.00000i	−0.519778	+	0.346518i
$534$		0			0
$535$			36.0000i			1.55642i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			3.00000i			0.129219i
$540$		−	32.0000i		−	1.37706i
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	−12.0000			−0.514969
$544$		0			0
$545$	8.00000			0.342682
$546$		0			0
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$			12.0000i			0.512615i
$549$	10.0000			0.426790
$550$		0			0
$551$		−	16.0000i		−	0.681623i
$552$		0			0
$553$			32.0000i			1.36078i
$554$		0			0
$555$	24.0000			1.01874
$556$	−28.0000			−1.18746
$557$		−	9.00000i		−	0.381342i	−0.981654	−	0.190671i	$-0.938934\pi$
$557$		−	9.00000i		−	0.381342i	0.981654	−	0.190671i	$-0.0610664\pi$
$558$		0			0
$559$	−8.00000	−	12.0000i	−0.338364	−	0.507546i
$560$	32.0000			1.35225
$561$		−	4.00000i		−	0.168880i
$562$		0			0
$563$	−31.0000			−1.30649			−0.653247	−	0.757145i	$-0.726594\pi$
$563$	−31.0000			−1.30649			−0.653247	+	0.757145i	$0.726594\pi$
$564$		−	8.00000i		−	0.336861i
$565$		−	68.0000i		−	2.86078i
$566$		0			0
$567$			22.0000i			0.923913i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		0			0
$571$	18.0000			0.753277			0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000			0.753277			0.376638	+	0.926360i	$0.377080\pi$
$572$	6.00000	−	4.00000i	0.250873	−	0.167248i
$573$	−10.0000			−0.417756
$574$		0			0
$575$	44.0000			1.83493
$576$	8.00000			0.333333
$577$			16.0000i			0.666089i	0.942911	+	0.333044i	$0.108076\pi$
$577$			16.0000i			0.666089i	−0.942911	+	0.333044i	$0.891924\pi$
$578$		0			0
$579$			28.0000i			1.16364i
$580$			64.0000i			2.65746i
$581$	24.0000			0.995688
$582$		0			0
$583$		−	4.00000i		−	0.165663i
$584$		0			0
$585$	12.0000	−	8.00000i	0.496139	−	0.330759i
$586$		0			0
$587$		−	31.0000i		−	1.27951i	−0.768580	−	0.639753i	$-0.779036\pi$
$587$		−	31.0000i		−	1.27951i	0.768580	−	0.639753i	$-0.220964\pi$
$588$	12.0000			0.494872
$589$	−2.00000			−0.0824086
$590$		0			0
$591$		−	30.0000i		−	1.23404i
$592$		−	12.0000i		−	0.493197i
$593$		−	42.0000i		−	1.72473i	−0.506284	−	0.862367i	$-0.668981\pi$
$593$		−	42.0000i		−	1.72473i	0.506284	−	0.862367i	$-0.331019\pi$
$594$		0			0
$595$	−16.0000			−0.655936
$596$			44.0000i			1.80231i
$597$	20.0000			0.818546
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−24.0000			−0.978980			−0.489490	−	0.872009i	$-0.662817\pi$
$601$	−24.0000			−0.978980			−0.489490	+	0.872009i	$0.662817\pi$
$602$		0			0
$603$			16.0000i			0.651570i
$604$			34.0000i			1.38344i
$605$			40.0000i			1.62623i
$606$		0			0
$607$	37.0000			1.50178			0.750892	−	0.660425i	$-0.229624\pi$
$607$	37.0000			1.50178			0.750892	+	0.660425i	$0.229624\pi$
$608$		0			0
$609$		−	32.0000i		−	1.29671i
$610$		0			0
$611$	−6.00000	+	4.00000i	−0.242734	+	0.161823i
$612$	−4.00000			−0.161690
$613$		−	30.0000i		−	1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$		−	30.0000i		−	1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$		0			0
$615$	32.0000			1.29036
$616$		0			0
$617$		−	22.0000i		−	0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$		−	22.0000i		−	0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$		0			0
$619$			15.0000i			0.602901i	0.953482	+	0.301450i	$0.0974708\pi$
$619$			15.0000i			0.602901i	−0.953482	+	0.301450i	$0.902529\pi$
$620$	8.00000			0.321288
$621$	16.0000			0.642058
$622$		0			0
$623$	−30.0000			−1.20192
$624$	−16.0000	−	24.0000i	−0.640513	−	0.960769i
$625$	41.0000			1.64000
$626$		0			0
$627$	4.00000			0.159745
$628$	26.0000			1.03751
$629$			6.00000i			0.239236i
$630$		0			0
$631$		−	35.0000i		−	1.39333i	−0.717398	−	0.696664i	$-0.754667\pi$
$631$		−	35.0000i		−	1.39333i	0.717398	−	0.696664i	$-0.245333\pi$
$632$		0			0
$633$	38.0000			1.51036
$634$		0			0
$635$		−	16.0000i		−	0.634941i
$636$	−16.0000			−0.634441
$637$	−6.00000	−	9.00000i	−0.237729	−	0.356593i
$638$		0			0
$639$		−	6.00000i		−	0.237356i
$640$		0			0
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$			49.0000i			1.93237i	0.257847	+	0.966186i	$0.416987\pi$
$643$			49.0000i			1.93237i	−0.257847	+	0.966186i	$0.583013\pi$
$644$			16.0000i			0.630488i
$645$			32.0000i			1.26000i
$646$		0			0
$647$	−6.00000			−0.235884			−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000			−0.235884			−0.117942	+	0.993020i	$0.537630\pi$
$648$		0			0
$649$	−6.00000			−0.235521
$650$		0			0
$651$	−4.00000			−0.156772
$652$			24.0000i			0.939913i
$653$	−21.0000			−0.821794			−0.410897	−	0.911682i	$-0.634784\pi$
$653$	−21.0000			−0.821794			−0.410897	+	0.911682i	$0.634784\pi$
$654$		0			0
$655$			4.00000i			0.156293i
$656$		−	16.0000i		−	0.624695i
$657$		−	7.00000i		−	0.273096i
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$	−16.0000			−0.622799
$661$			42.0000i			1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$			42.0000i			1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$		0			0
$663$	8.00000	+	12.0000i	0.310694	+	0.466041i
$664$		0			0
$665$		−	16.0000i		−	0.620453i
$666$		0			0
$667$	−32.0000			−1.23904
$668$			6.00000i			0.232147i
$669$		−	38.0000i		−	1.46916i
$670$		0			0
$671$			10.0000i			0.386046i
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$	44.0000			1.69356
$676$	−10.0000	+	24.0000i	−0.384615	+	0.923077i
$677$	−16.0000			−0.614930			−0.307465	−	0.951559i	$-0.599481\pi$
$677$	−16.0000			−0.614930			−0.307465	+	0.951559i	$0.599481\pi$
$678$		0			0
$679$	4.00000			0.153506
$680$		0			0
$681$			16.0000i			0.613121i
$682$		0			0
$683$			32.0000i			1.22445i	0.790685	+	0.612223i	$0.209725\pi$
$683$			32.0000i			1.22445i	−0.790685	+	0.612223i	$0.790275\pi$
$684$		−	4.00000i		−	0.152944i
$685$	−24.0000			−0.916993
$686$		0			0
$687$		−	58.0000i		−	2.21284i
$688$	16.0000			0.609994
$689$	8.00000	+	12.0000i	0.304776	+	0.457164i
$690$		0			0
$691$			12.0000i			0.456502i	0.973602	+	0.228251i	$0.0733006\pi$
$691$			12.0000i			0.456502i	−0.973602	+	0.228251i	$0.926699\pi$
$692$	−18.0000			−0.684257
$693$	2.00000			0.0759737
$694$		0			0
$695$		−	56.0000i		−	2.12420i
$696$		0			0
$697$			8.00000i			0.303022i
$698$		0			0
$699$	26.0000			0.983410
$700$			44.0000i			1.66304i
$701$	41.0000			1.54855			0.774274	−	0.632850i	$-0.218115\pi$
$701$	41.0000			1.54855			0.774274	+	0.632850i	$0.218115\pi$
$702$		0			0
$703$	−6.00000			−0.226294
$704$			8.00000i			0.301511i
$705$	16.0000			0.602595
$706$		0			0
$707$			6.00000i			0.225653i
$708$			24.0000i			0.901975i
$709$			25.0000i			0.938895i	0.882960	+	0.469447i	$0.155547\pi$
$709$			25.0000i			0.938895i	−0.882960	+	0.469447i	$0.844453\pi$
$710$		0			0
$711$	−16.0000			−0.600047
$712$		0			0
$713$			4.00000i			0.149801i
$714$		0			0
$715$	8.00000	+	12.0000i	0.299183	+	0.448775i
$716$		0			0
$717$		−	32.0000i		−	1.19506i
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$			16.0000i			0.596285i
$721$			30.0000i			1.11726i
$722$		0			0
$723$			28.0000i			1.04133i
$724$	−12.0000			−0.445976
$725$	−88.0000			−3.26824
$726$		0			0
$727$	20.0000			0.741759			0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000			0.741759			0.370879	+	0.928681i	$0.379056\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−8.00000			−0.295891
$732$	40.0000			1.47844
$733$		−	8.00000i		−	0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$		−	8.00000i		−	0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$		0			0
$735$			24.0000i			0.885253i
$736$		0			0
$737$	−16.0000			−0.589368
$738$		0			0
$739$		−	23.0000i		−	0.846069i	−0.906114	−	0.423034i	$-0.860965\pi$
$739$		−	23.0000i		−	0.846069i	0.906114	−	0.423034i	$-0.139035\pi$
$740$	24.0000			0.882258
$741$	−12.0000	+	8.00000i	−0.440831	+	0.293887i
$742$		0			0
$743$		−	33.0000i		−	1.21065i	−0.795977	−	0.605326i	$-0.793043\pi$
$743$		−	33.0000i		−	1.21065i	0.795977	−	0.605326i	$-0.206957\pi$
$744$		0			0
$745$	−88.0000			−3.22407
$746$		0			0
$747$			12.0000i			0.439057i
$748$		−	4.00000i		−	0.146254i
$749$		−	18.0000i		−	0.657706i
$750$		0			0
$751$	35.0000			1.27717			0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000			1.27717			0.638584	+	0.769552i	$0.279520\pi$
$752$		−	8.00000i		−	0.291730i
$753$	44.0000			1.60345
$754$		0			0
$755$	−68.0000			−2.47477
$756$			16.0000i			0.581914i
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		0			0
$759$		−	8.00000i		−	0.290382i
$760$		0			0
$761$		−	22.0000i		−	0.797499i	−0.917060	−	0.398750i	$-0.869444\pi$
$761$		−	22.0000i		−	0.797499i	0.917060	−	0.398750i	$-0.130556\pi$
$762$		0			0
$763$	−4.00000			−0.144810
$764$	−10.0000			−0.361787
$765$		−	8.00000i		−	0.289241i
$766$		0			0
$767$	18.0000	−	12.0000i	0.649942	−	0.433295i
$768$	32.0000			1.15470
$769$		−	2.00000i		−	0.0721218i	−0.999350	−	0.0360609i	$-0.988519\pi$
$769$		−	2.00000i		−	0.0721218i	0.999350	−	0.0360609i	$-0.0114810\pi$
$770$		0			0
$771$	42.0000			1.51259
$772$			28.0000i			1.00774i
$773$		−	47.0000i		−	1.69047i	−0.534393	−	0.845236i	$-0.679460\pi$
$773$		−	47.0000i		−	1.69047i	0.534393	−	0.845236i	$-0.320540\pi$
$774$		0			0
$775$			11.0000i			0.395132i
$776$		0			0
$777$	−12.0000			−0.430498
$778$		0			0
$779$	−8.00000			−0.286630
$780$	48.0000	−	32.0000i	1.71868	−	1.14578i
$781$	6.00000			0.214697
$782$		0			0
$783$	−32.0000			−1.14359
$784$	12.0000			0.428571
$785$			52.0000i			1.85596i
$786$		0			0
$787$			3.00000i			0.106938i	0.998569	+	0.0534692i	$0.0170279\pi$
$787$			3.00000i			0.106938i	−0.998569	+	0.0534692i	$0.982972\pi$
$788$		−	30.0000i		−	1.06871i
$789$	−48.0000			−1.70885
$790$		0			0
$791$			34.0000i			1.20890i
$792$		0			0
$793$	−20.0000	−	30.0000i	−0.710221	−	1.06533i
$794$		0			0
$795$		−	32.0000i		−	1.13492i
$796$	20.0000			0.708881
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$			4.00000i			0.141510i
$800$		0			0
$801$		−	15.0000i		−	0.529999i
$802$		0			0
$803$	7.00000			0.247025
$804$			64.0000i			2.25711i
$805$	−32.0000			−1.12785
$806$		0			0
$807$	20.0000			0.704033
$808$		0			0
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$		−	22.0000i		−	0.772524i	−0.922389	−	0.386262i	$-0.873766\pi$
$811$		−	22.0000i		−	0.772524i	0.922389	−	0.386262i	$-0.126234\pi$
$812$		−	32.0000i		−	1.12298i
$813$		−	30.0000i		−	1.05215i
$814$		0			0
$815$	−48.0000			−1.68137
$816$	−16.0000			−0.560112
$817$		−	8.00000i		−	0.279885i
$818$		0			0
$819$	−6.00000	+	4.00000i	−0.209657	+	0.139771i
$820$	32.0000			1.11749
$821$			10.0000i			0.349002i	0.984657	+	0.174501i	$0.0558313\pi$
$821$			10.0000i			0.349002i	−0.984657	+	0.174501i	$0.944169\pi$
$822$		0			0
$823$	−10.0000			−0.348578			−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000			−0.348578			−0.174289	+	0.984695i	$0.555763\pi$
$824$		0			0
$825$		−	22.0000i		−	0.765942i
$826$		0			0
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	−8.00000			−0.278019
$829$	−18.0000			−0.625166			−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000			−0.625166			−0.312583	+	0.949890i	$0.601194\pi$
$830$		0			0
$831$	−32.0000			−1.11007
$832$	−16.0000	−	24.0000i	−0.554700	−	0.832050i
$833$	−6.00000			−0.207888
$834$		0			0
$835$	−12.0000			−0.415277
$836$	4.00000			0.138343
$837$			4.00000i			0.138260i
$838$		0			0
$839$		−	18.0000i		−	0.621429i	−0.950503	−	0.310715i	$-0.899432\pi$
$839$		−	18.0000i		−	0.621429i	0.950503	−	0.310715i	$-0.100568\pi$
$840$		0			0
$841$	35.0000			1.20690
$842$		0			0
$843$			48.0000i			1.65321i
$844$	38.0000			1.30801
$845$	−48.0000	−	20.0000i	−1.65125	−	0.688021i
$846$		0			0
$847$		−	20.0000i		−	0.687208i
$848$	−16.0000			−0.549442
$849$	26.0000			0.892318
$850$		0			0
$851$			12.0000i			0.411355i
$852$		−	24.0000i		−	0.822226i
$853$			16.0000i			0.547830i	0.961754	+	0.273915i	$0.0883186\pi$
$853$			16.0000i			0.547830i	−0.961754	+	0.273915i	$0.911681\pi$
$854$		0			0
$855$	8.00000			0.273594
$856$		0			0
$857$	−19.0000			−0.649028			−0.324514	−	0.945881i	$-0.605201\pi$
$857$	−19.0000			−0.649028			−0.324514	+	0.945881i	$0.605201\pi$
$858$		0			0
$859$	−32.0000			−1.09183			−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000			−1.09183			−0.545913	+	0.837842i	$0.683817\pi$
$860$			32.0000i			1.09119i
$861$	−16.0000			−0.545279
$862$		0			0
$863$		−	8.00000i		−	0.272323i	−0.990687	−	0.136162i	$-0.956523\pi$
$863$		−	8.00000i		−	0.272323i	0.990687	−	0.136162i	$-0.0434766\pi$
$864$		0			0
$865$		−	36.0000i		−	1.22404i
$866$		0			0
$867$	−26.0000			−0.883006
$868$	−4.00000			−0.135769
$869$		−	16.0000i		−	0.542763i
$870$		0			0
$871$	48.0000	−	32.0000i	1.62642	−	1.08428i
$872$		0			0
$873$			2.00000i			0.0676897i
$874$		0			0
$875$	−48.0000			−1.62270
$876$		−	28.0000i		−	0.946032i
$877$		−	34.0000i		−	1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$		−	34.0000i		−	1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$		0			0
$879$		−	36.0000i		−	1.21425i
$880$	−16.0000			−0.539360
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	6.00000			0.201916			0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000			0.201916			0.100958	+	0.994891i	$0.467809\pi$
$884$	8.00000	+	12.0000i	0.269069	+	0.403604i
$885$	−48.0000			−1.61350
$886$		0			0
$887$	−32.0000			−1.07445			−0.537227	−	0.843437i	$-0.680528\pi$
$887$	−32.0000			−1.07445			−0.537227	+	0.843437i	$0.680528\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$		−	11.0000i		−	0.368514i
$892$		−	38.0000i		−	1.27233i
$893$	−4.00000			−0.133855
$894$		0			0
$895$		0			0
$896$		0			0
$897$	16.0000	+	24.0000i	0.534224	+	0.801337i
$898$		0			0
$899$		−	8.00000i		−	0.266815i
$900$	−22.0000			−0.733333
$901$	8.00000			0.266519
$902$		0			0
$903$		−	16.0000i		−	0.532447i
$904$		0			0
$905$		−	24.0000i		−	0.797787i
$906$		0			0
$907$	37.0000			1.22856			0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000			1.22856			0.614282	+	0.789086i	$0.289446\pi$
$908$			16.0000i			0.530979i
$909$	−3.00000			−0.0995037
$910$		0			0
$911$	22.0000			0.728893			0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000			0.728893			0.364446	+	0.931224i	$0.381258\pi$
$912$		−	16.0000i		−	0.529813i
$913$	−12.0000			−0.397142
$914$		0			0
$915$			80.0000i			2.64472i
$916$		−	58.0000i		−	1.91637i
$917$		−	2.00000i		−	0.0660458i
$918$		0			0
$919$	11.0000			0.362857			0.181428	−	0.983404i	$-0.441928\pi$
$919$	11.0000			0.362857			0.181428	+	0.983404i	$0.441928\pi$
$920$		0			0
$921$		−	28.0000i		−	0.922631i
$922$		0			0
$923$	−18.0000	+	12.0000i	−0.592477	+	0.394985i
$924$	8.00000			0.263181
$925$			33.0000i			1.08503i
$926$		0			0
$927$	−15.0000			−0.492665
$928$		0			0
$929$		−	57.0000i		−	1.87011i	−0.354504	−	0.935055i	$-0.615350\pi$
$929$		−	57.0000i		−	1.87011i	0.354504	−	0.935055i	$-0.384650\pi$
$930$		0			0
$931$		−	6.00000i		−	0.196642i
$932$	26.0000			0.851658
$933$	30.0000			0.982156
$934$		0			0
$935$	8.00000			0.261628
$936$		0			0
$937$	−7.00000			−0.228680			−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000			−0.228680			−0.114340	+	0.993442i	$0.536475\pi$
$938$		0			0
$939$	32.0000			1.04428
$940$	16.0000			0.521862
$941$			21.0000i			0.684580i	0.939594	+	0.342290i	$0.111203\pi$
$941$			21.0000i			0.684580i	−0.939594	+	0.342290i	$0.888797\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$			24.0000i			0.781133i
$945$	−32.0000			−1.04096
$946$		0			0
$947$		−	29.0000i		−	0.942373i	−0.882034	−	0.471187i	$-0.843826\pi$
$947$		−	29.0000i		−	0.942373i	0.882034	−	0.471187i	$-0.156174\pi$
$948$	−64.0000			−2.07862
$949$	−21.0000	+	14.0000i	−0.681689	+	0.454459i
$950$		0			0
$951$			48.0000i			1.55651i
$952$		0			0
$953$	8.00000			0.259145			0.129573	−	0.991570i	$-0.458639\pi$
$953$	8.00000			0.259145			0.129573	+	0.991570i	$0.458639\pi$
$954$		0			0
$955$		−	20.0000i		−	0.647185i
$956$		−	32.0000i		−	1.03495i
$957$			16.0000i			0.517207i
$958$		0			0
$959$	12.0000			0.387500
$960$			64.0000i			2.06559i
$961$	−1.00000			−0.0322581
$962$		0			0
$963$	9.00000			0.290021
$964$			28.0000i			0.901819i
$965$	−56.0000			−1.80270
$966$		0			0
$967$			37.0000i			1.18984i	0.803785	+	0.594920i	$0.202816\pi$
$967$			37.0000i			1.18984i	−0.803785	+	0.594920i	$0.797184\pi$
$968$		0			0
$969$			8.00000i			0.256997i
$970$		0			0
$971$	−57.0000			−1.82922			−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000			−1.82922			−0.914609	+	0.404341i	$0.867501\pi$
$972$	−20.0000			−0.641500
$973$			28.0000i			0.897639i
$974$		0			0
$975$	44.0000	+	66.0000i	1.40913	+	2.11369i
$976$	40.0000			1.28037
$977$			44.0000i			1.40768i	0.710356	+	0.703842i	$0.248534\pi$
$977$			44.0000i			1.40768i	−0.710356	+	0.703842i	$0.751466\pi$
$978$		0			0
$979$	15.0000			0.479402
$980$			24.0000i			0.766652i
$981$		−	2.00000i		−	0.0638551i
$982$		0			0
$983$		−	48.0000i		−	1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$		−	48.0000i		−	1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$		0			0
$985$	60.0000			1.91176
$986$		0			0
$987$	−8.00000			−0.254643
$988$	−12.0000	+	8.00000i	−0.381771	+	0.254514i
$989$	−16.0000			−0.508770
$990$		0			0
$991$	−46.0000			−1.46124			−0.730619	−	0.682785i	$-0.760768\pi$
$991$	−46.0000			−1.46124			−0.730619	+	0.682785i	$0.760768\pi$
$992$		0			0
$993$			8.00000i			0.253872i
$994$		0			0
$995$			40.0000i			1.26809i
$996$			48.0000i			1.52094i
$997$	51.0000			1.61519			0.807593	−	0.589740i	$-0.200770\pi$
$997$	51.0000			1.61519			0.807593	+	0.589740i	$0.200770\pi$
$998$		0			0
$999$			12.0000i			0.379663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.c.a.311.2	yes	2
13.5	odd	4		5239.2.a.c.1.1		1
13.8	odd	4		5239.2.a.b.1.1		1
13.12	even	2	inner	403.2.c.a.311.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.c.a.311.1	✓	2	13.12	even	2	inner
403.2.c.a.311.2	yes	2	1.1	even	1	trivial
5239.2.a.b.1.1		1	13.8	odd	4
5239.2.a.c.1.1		1	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.00000 q^{3} +2.00000 q^{4} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{3} +2.00000 q^{4} +4.00000i q^{5} -2.00000i q^{7} +1.00000 q^{9} +1.00000i q^{11} +4.00000 q^{12} +(-2.00000 - 3.00000i) q^{13} +8.00000i q^{15} +4.00000 q^{16} -2.00000 q^{17} -2.00000i q^{19} +8.00000i q^{20} -4.00000i q^{21} -4.00000 q^{23} -11.0000 q^{25} -4.00000 q^{27} -4.00000i q^{28} +8.00000 q^{29} -1.00000i q^{31} +2.00000i q^{33} +8.00000 q^{35} +2.00000 q^{36} -3.00000i q^{37} +(-4.00000 - 6.00000i) q^{39} -4.00000i q^{41} +4.00000 q^{43} +2.00000i q^{44} +4.00000i q^{45} -2.00000i q^{47} +8.00000 q^{48} +3.00000 q^{49} -4.00000 q^{51} +(-4.00000 - 6.00000i) q^{52} -4.00000 q^{53} -4.00000 q^{55} -4.00000i q^{57} +6.00000i q^{59} +16.0000i q^{60} +10.0000 q^{61} -2.00000i q^{63} +8.00000 q^{64} +(12.0000 - 8.00000i) q^{65} +16.0000i q^{67} -4.00000 q^{68} -8.00000 q^{69} -6.00000i q^{71} -7.00000i q^{73} -22.0000 q^{75} -4.00000i q^{76} +2.00000 q^{77} -16.0000 q^{79} +16.0000i q^{80} -11.0000 q^{81} +12.0000i q^{83} -8.00000i q^{84} -8.00000i q^{85} +16.0000 q^{87} -15.0000i q^{89} +(-6.00000 + 4.00000i) q^{91} -8.00000 q^{92} -2.00000i q^{93} +8.00000 q^{95} +2.00000i q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9	\( 2 q + 4 q^{3} + 4 q^{4} + 2 q^{9} + 8 q^{12} - 4 q^{13} + 8 q^{16} - 4 q^{17} - 8 q^{23} - 22 q^{25} - 8 q^{27} + 16 q^{29} + 16 q^{35} + 4 q^{36} - 8 q^{39} + 8 q^{43} + 16 q^{48} + 6 q^{49} - 8 q^{51} - 8 q^{52} - 8 q^{53} - 8 q^{55} + 20 q^{61} + 16 q^{64} + 24 q^{65} - 8 q^{68} - 16 q^{69} - 44 q^{75} + 4 q^{77} - 32 q^{79} - 22 q^{81} + 32 q^{87} - 12 q^{91} - 16 q^{92} + 16 q^{95}+O(q^{100}) \) 2 * q + 4 * q^3 + 4 * q^4 + 2 * q^9 + 8 * q^12 - 4 * q^13 + 8 * q^16 - 4 * q^17 - 8 * q^23 - 22 * q^25 - 8 * q^27 + 16 * q^29 + 16 * q^35 + 4 * q^36 - 8 * q^39 + 8 * q^43 + 16 * q^48 + 6 * q^49 - 8 * q^51 - 8 * q^52 - 8 * q^53 - 8 * q^55 + 20 * q^61 + 16 * q^64 + 24 * q^65 - 8 * q^68 - 16 * q^69 - 44 * q^75 + 4 * q^77 - 32 * q^79 - 22 * q^81 + 32 * q^87 - 12 * q^91 - 16 * q^92 + 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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