LMFDB - Embedded newform 4680.2.l.c.2809.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (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:	$37.3699881460$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.c.2809.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 2.00000i) q^{5} -5.00000i q^{7} +O(q^{10})$	$q+(1.00000 + 2.00000i) q^{5} -5.00000i q^{7} -5.00000 q^{11} -1.00000i q^{13} +3.00000i q^{17} +4.00000 q^{19} +5.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{29} +(10.0000 - 5.00000i) q^{35} -7.00000i q^{37} -11.0000 q^{41} +12.0000i q^{43} -6.00000i q^{47} -18.0000 q^{49} +1.00000i q^{53} +(-5.00000 - 10.0000i) q^{55} +12.0000 q^{59} -7.00000 q^{61} +(2.00000 - 1.00000i) q^{65} +4.00000i q^{67} +7.00000 q^{71} +14.0000i q^{73} +25.0000i q^{77} +5.00000 q^{79} -2.00000i q^{83} +(-6.00000 + 3.00000i) q^{85} -3.00000 q^{89} -5.00000 q^{91} +(4.00000 + 8.00000i) q^{95} -1.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 10 q^{11} + 8 q^{19} - 6 q^{25} - 8 q^{29} + 20 q^{35} - 22 q^{41} - 36 q^{49} - 10 q^{55} + 24 q^{59} - 14 q^{61} + 4 q^{65} + 14 q^{71} + 10 q^{79} - 12 q^{85} - 6 q^{89} - 10 q^{91} + 8 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 10 * q^11 + 8 * q^19 - 6 * q^25 - 8 * q^29 + 20 * q^35 - 22 * q^41 - 36 * q^49 - 10 * q^55 + 24 * q^59 - 14 * q^61 + 4 * q^65 + 14 * q^71 + 10 * q^79 - 12 * q^85 - 6 * q^89 - 10 * q^91 + 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$5$	1.00000	+	2.00000i	0.447214	+	0.894427i
$6$		0			0
$7$		−	5.00000i		−	1.88982i	−0.327327	−	0.944911i	$-0.606148\pi$
$7$		−	5.00000i		−	1.88982i	0.327327	−	0.944911i	$-0.393852\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.00000			−1.50756			−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000			−1.50756			−0.753778	+	0.657129i	$0.771771\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.00000i			1.04257i	0.853382	+	0.521286i	$0.174548\pi$
$23$			5.00000i			1.04257i	−0.853382	+	0.521286i	$0.825452\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000			−0.742781			−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000			−0.742781			−0.371391	+	0.928477i	$0.621119\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	10.0000	−	5.00000i	1.69031	−	0.845154i
$36$		0			0
$37$		−	7.00000i		−	1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$		−	7.00000i		−	1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−11.0000			−1.71791			−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000			−1.71791			−0.858956	+	0.512050i	$0.828886\pi$
$42$		0			0
$43$			12.0000i			1.82998i	0.403473	+	0.914991i	$0.367803\pi$
$43$			12.0000i			1.82998i	−0.403473	+	0.914991i	$0.632197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	−18.0000			−2.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$			1.00000i			0.137361i	0.997639	+	0.0686803i	$0.0218788\pi$
$53$			1.00000i			0.137361i	−0.997639	+	0.0686803i	$0.978121\pi$
$54$		0			0
$55$	−5.00000	−	10.0000i	−0.674200	−	1.34840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	−7.00000			−0.896258			−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000			−0.896258			−0.448129	+	0.893969i	$0.647910\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	−	1.00000i	0.248069	−	0.124035i
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.00000			0.830747			0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000			0.830747			0.415374	+	0.909651i	$0.363651\pi$
$72$		0			0
$73$			14.0000i			1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$			14.0000i			1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			25.0000i			2.84901i
$78$		0			0
$79$	5.00000			0.562544			0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000			0.562544			0.281272	+	0.959628i	$0.409244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	2.00000i		−	0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$		−	2.00000i		−	0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$		0			0
$85$	−6.00000	+	3.00000i	−0.650791	+	0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	−5.00000			−0.524142
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	8.00000i	0.410391	+	0.820783i
$96$		0			0
$97$		−	1.00000i		−	0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$		−	1.00000i		−	0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$			14.0000i			1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$			14.0000i			1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			11.0000i			1.06341i	0.846930	+	0.531705i	$0.178449\pi$
$107$			11.0000i			1.06341i	−0.846930	+	0.531705i	$0.821551\pi$
$108$		0			0
$109$	12.0000			1.14939			0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000			1.14939			0.574696	+	0.818367i	$0.305120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			14.0000i			1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$			14.0000i			1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$		0			0
$115$	−10.0000	+	5.00000i	−0.932505	+	0.466252i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	15.0000			1.37505
$120$		0			0
$121$	14.0000			1.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−11.0000	−	2.00000i	−0.983870	−	0.178885i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		−	20.0000i		−	1.73422i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			14.0000i			1.19610i	0.801459	+	0.598050i	$0.204058\pi$
$137$			14.0000i			1.19610i	−0.801459	+	0.598050i	$0.795942\pi$
$138$		0			0
$139$	11.0000			0.933008			0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000			0.933008			0.466504	+	0.884519i	$0.345513\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			5.00000i			0.418121i
$144$		0			0
$145$	−4.00000	−	8.00000i	−0.332182	−	0.664364i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−15.0000			−1.22885			−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000			−1.22885			−0.614424	+	0.788976i	$0.710612\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			4.00000i			0.319235i	0.987179	+	0.159617i	$0.0510260\pi$
$157$			4.00000i			0.319235i	−0.987179	+	0.159617i	$0.948974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.0000			1.97028
$162$		0			0
$163$		−	7.00000i		−	0.548282i	−0.961689	−	0.274141i	$-0.911606\pi$
$163$		−	7.00000i		−	0.548282i	0.961689	−	0.274141i	$-0.0883936\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		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
$175$	20.0000	+	15.0000i	1.51186	+	1.13389i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−10.0000			−0.747435			−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000			−0.747435			−0.373718	+	0.927543i	$0.621917\pi$
$180$		0			0
$181$	−23.0000			−1.70958			−0.854788	−	0.518977i	$-0.826313\pi$
$181$	−23.0000			−1.70958			−0.854788	+	0.518977i	$0.826313\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	14.0000	−	7.00000i	1.02930	−	0.514650i
$186$		0			0
$187$		−	15.0000i		−	1.09691i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.00000			−0.434145			−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000			−0.434145			−0.217072	+	0.976156i	$0.569651\pi$
$192$		0			0
$193$			23.0000i			1.65558i	0.561041	+	0.827788i	$0.310401\pi$
$193$			23.0000i			1.65558i	−0.561041	+	0.827788i	$0.689599\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			20.0000i			1.40372i
$204$		0			0
$205$	−11.0000	−	22.0000i	−0.768273	−	1.53655i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−20.0000			−1.38343
$210$		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
$213$		0			0
$214$		0			0
$215$	−24.0000	+	12.0000i	−1.63679	+	0.818393i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3.00000			0.201802
$222$		0			0
$223$			24.0000i			1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$			24.0000i			1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	6.00000			0.396491			0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000			0.396491			0.198246	+	0.980152i	$0.436476\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.00000i		−	0.589610i	−0.955557	−	0.294805i	$-0.904745\pi$
$233$		−	9.00000i		−	0.589610i	0.955557	−	0.294805i	$-0.0952546\pi$
$234$		0			0
$235$	12.0000	−	6.00000i	0.782794	−	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.00000			0.323423			0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000			0.323423			0.161712	+	0.986838i	$0.448299\pi$
$240$		0			0
$241$	−4.00000			−0.257663			−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000			−0.257663			−0.128831	+	0.991667i	$0.541123\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−18.0000	−	36.0000i	−1.14998	−	2.29996i
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$		−	25.0000i		−	1.57174i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			6.00000i			0.374270i	0.982334	+	0.187135i	$0.0599201\pi$
$257$			6.00000i			0.374270i	−0.982334	+	0.187135i	$0.940080\pi$
$258$		0			0
$259$	−35.0000			−2.17479
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$		0			0
$265$	−2.00000	+	1.00000i	−0.122859	+	0.0614295i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000			−0.365826			−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000			−0.365826			−0.182913	+	0.983129i	$0.558553\pi$
$270$		0			0
$271$	2.00000			0.121491			0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000			0.121491			0.0607457	+	0.998153i	$0.480652\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.0000	−	20.0000i	0.904534	−	1.20605i
$276$		0			0
$277$		−	30.0000i		−	1.80253i	−0.433273	−	0.901263i	$-0.642641\pi$
$277$		−	30.0000i		−	1.80253i	0.433273	−	0.901263i	$-0.357359\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			18.0000i			1.06999i	0.844856	+	0.534994i	$0.179686\pi$
$283$			18.0000i			1.06999i	−0.844856	+	0.534994i	$0.820314\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			55.0000i			3.24655i
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$			20.0000i			1.16841i	0.811605	+	0.584206i	$0.198594\pi$
$293$			20.0000i			1.16841i	−0.811605	+	0.584206i	$0.801406\pi$
$294$		0			0
$295$	12.0000	+	24.0000i	0.698667	+	1.39733i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.00000			0.289157
$300$		0			0
$301$	60.0000			3.45834
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.00000	−	14.0000i	−0.400819	−	0.801638i
$306$		0			0
$307$		−	25.0000i		−	1.42683i	−0.700744	−	0.713413i	$-0.747149\pi$
$307$		−	25.0000i		−	1.42683i	0.700744	−	0.713413i	$-0.252851\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.0000			−1.36092			−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000			−1.36092			−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$			20.0000i			1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$			20.0000i			1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	24.0000i		−	1.34797i	−0.738743	−	0.673987i	$-0.764580\pi$
$317$		−	24.0000i		−	1.34797i	0.738743	−	0.673987i	$-0.235420\pi$
$318$		0			0
$319$	20.0000			1.11979
$320$		0			0
$321$		0			0
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$	4.00000	+	3.00000i	0.221880	+	0.166410i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−30.0000			−1.65395
$330$		0			0
$331$	4.00000			0.219860			0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000			0.219860			0.109930	+	0.993939i	$0.464937\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	+	4.00000i	−0.437087	+	0.218543i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			55.0000i			2.96972i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			7.00000i			0.375780i	0.982190	+	0.187890i	$0.0601648\pi$
$347$			7.00000i			0.375780i	−0.982190	+	0.187890i	$0.939835\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			30.0000i			1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$			30.0000i			1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$		0			0
$355$	7.00000	+	14.0000i	0.371521	+	0.743043i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−32.0000			−1.68890			−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000			−1.68890			−0.844448	+	0.535638i	$0.820071\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.0000	+	14.0000i	−1.46559	+	0.732793i
$366$		0			0
$367$		−	22.0000i		−	1.14839i	−0.818718	−	0.574195i	$-0.805315\pi$
$367$		−	22.0000i		−	1.14839i	0.818718	−	0.574195i	$-0.194685\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5.00000			0.259587
$372$		0			0
$373$		−	14.0000i		−	0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$		−	14.0000i		−	0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			4.00000i			0.206010i
$378$		0			0
$379$	−32.0000			−1.64373			−0.821865	−	0.569683i	$-0.807066\pi$
$379$	−32.0000			−1.64373			−0.821865	+	0.569683i	$0.807066\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	6.00000i		−	0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$		−	6.00000i		−	0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$		0			0
$385$	−50.0000	+	25.0000i	−2.54824	+	1.27412i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−24.0000			−1.21685			−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000			−1.21685			−0.608424	+	0.793612i	$0.708198\pi$
$390$		0			0
$391$	−15.0000			−0.758583
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.00000	+	10.0000i	0.251577	+	0.503155i
$396$		0			0
$397$		−	3.00000i		−	0.150566i	−0.997162	−	0.0752828i	$-0.976014\pi$
$397$		−	3.00000i		−	0.150566i	0.997162	−	0.0752828i	$-0.0239860\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.0000			0.699127			0.349563	−	0.936913i	$-0.386330\pi$
$401$	14.0000			0.699127			0.349563	+	0.936913i	$0.386330\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			35.0000i			1.73489i
$408$		0			0
$409$	−20.0000			−0.988936			−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000			−0.988936			−0.494468	+	0.869196i	$0.664637\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	60.0000i		−	2.95241i
$414$		0			0
$415$	4.00000	−	2.00000i	0.196352	−	0.0981761i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	−14.0000			−0.682318			−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000			−0.682318			−0.341159	+	0.940006i	$0.610819\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000	−	9.00000i	−0.582086	−	0.436564i
$426$		0			0
$427$			35.0000i			1.69377i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.00000			−0.192673			−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000			−0.192673			−0.0963366	+	0.995349i	$0.530713\pi$
$432$		0			0
$433$			18.0000i			0.865025i	0.901628	+	0.432512i	$0.142373\pi$
$433$			18.0000i			0.865025i	−0.901628	+	0.432512i	$0.857627\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			20.0000i			0.956730i
$438$		0			0
$439$	27.0000			1.28864			0.644320	−	0.764756i	$-0.277141\pi$
$439$	27.0000			1.28864			0.644320	+	0.764756i	$0.277141\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			21.0000i			0.997740i	0.866677	+	0.498870i	$0.166252\pi$
$443$			21.0000i			0.997740i	−0.866677	+	0.498870i	$0.833748\pi$
$444$		0			0
$445$	−3.00000	−	6.00000i	−0.142214	−	0.284427i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−9.00000			−0.424736			−0.212368	−	0.977190i	$-0.568118\pi$
$449$	−9.00000			−0.424736			−0.212368	+	0.977190i	$0.568118\pi$
$450$		0			0
$451$	55.0000			2.58985
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5.00000	−	10.0000i	−0.234404	−	0.468807i
$456$		0			0
$457$			39.0000i			1.82434i	0.409809	+	0.912172i	$0.365595\pi$
$457$			39.0000i			1.82434i	−0.409809	+	0.912172i	$0.634405\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	33.0000			1.53696			0.768482	−	0.639872i	$-0.221013\pi$
$461$	33.0000			1.53696			0.768482	+	0.639872i	$0.221013\pi$
$462$		0			0
$463$			1.00000i			0.0464739i	0.999730	+	0.0232370i	$0.00739722\pi$
$463$			1.00000i			0.0464739i	−0.999730	+	0.0232370i	$0.992603\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			15.0000i			0.694117i	0.937843	+	0.347059i	$0.112820\pi$
$467$			15.0000i			0.694117i	−0.937843	+	0.347059i	$0.887180\pi$
$468$		0			0
$469$	20.0000			0.923514
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	60.0000i		−	2.75880i
$474$		0			0
$475$	−12.0000	+	16.0000i	−0.550598	+	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11.0000			0.502603			0.251301	−	0.967909i	$-0.419141\pi$
$479$	11.0000			0.502603			0.251301	+	0.967909i	$0.419141\pi$
$480$		0			0
$481$	−7.00000			−0.319173
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000	−	1.00000i	0.0908153	−	0.0454077i
$486$		0			0
$487$		−	23.0000i		−	1.04223i	−0.853487	−	0.521115i	$-0.825516\pi$
$487$		−	23.0000i		−	1.04223i	0.853487	−	0.521115i	$-0.174484\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$		0			0
$493$		−	12.0000i		−	0.540453i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	35.0000i		−	1.56996i
$498$		0			0
$499$	−14.0000			−0.626726			−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000			−0.626726			−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.00000i		−	0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$		−	8.00000i		−	0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$		0			0
$505$	6.00000	+	12.0000i	0.266996	+	0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13.0000			0.576215			0.288107	−	0.957598i	$-0.406974\pi$
$509$	13.0000			0.576215			0.288107	+	0.957598i	$0.406974\pi$
$510$		0			0
$511$	70.0000			3.09662
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.0000	+	14.0000i	−1.23383	+	0.616914i
$516$		0			0
$517$			30.0000i			1.31940i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.0000			0.613351			0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000			0.613351			0.306676	+	0.951814i	$0.400783\pi$
$522$		0			0
$523$		−	38.0000i		−	1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$		−	38.0000i		−	1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.00000			−0.0869565
$530$		0			0
$531$		0			0
$532$		0			0
$533$			11.0000i			0.476463i
$534$		0			0
$535$	−22.0000	+	11.0000i	−0.951143	+	0.475571i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	90.0000			3.87657
$540$		0			0
$541$	16.0000			0.687894			0.343947	−	0.938989i	$-0.388236\pi$
$541$	16.0000			0.687894			0.343947	+	0.938989i	$0.388236\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	12.0000	+	24.0000i	0.514024	+	1.02805i
$546$		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
$549$		0			0
$550$		0			0
$551$	−16.0000			−0.681623
$552$		0			0
$553$		−	25.0000i		−	1.06311i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	39.0000i		−	1.64365i	−0.569737	−	0.821827i	$-0.692955\pi$
$563$		−	39.0000i		−	1.64365i	0.569737	−	0.821827i	$-0.307045\pi$
$564$		0			0
$565$	−28.0000	+	14.0000i	−1.17797	+	0.588984i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	43.0000			1.79949			0.899747	−	0.436412i	$-0.143751\pi$
$571$	43.0000			1.79949			0.899747	+	0.436412i	$0.143751\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.0000	−	15.0000i	−0.834058	−	0.625543i
$576$		0			0
$577$		−	17.0000i		−	0.707719i	−0.935299	−	0.353860i	$-0.884869\pi$
$577$		−	17.0000i		−	0.707719i	0.935299	−	0.353860i	$-0.115131\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−10.0000			−0.414870
$582$		0			0
$583$		−	5.00000i		−	0.207079i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	34.0000i		−	1.40333i	−0.712507	−	0.701665i	$-0.752440\pi$
$587$		−	34.0000i		−	1.40333i	0.712507	−	0.701665i	$-0.247560\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			16.0000i			0.657041i	0.944497	+	0.328521i	$0.106550\pi$
$593$			16.0000i			0.657041i	−0.944497	+	0.328521i	$0.893450\pi$
$594$		0			0
$595$	15.0000	+	30.0000i	0.614940	+	1.22988i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	34.0000			1.38920			0.694601	−	0.719395i	$-0.255581\pi$
$599$	34.0000			1.38920			0.694601	+	0.719395i	$0.255581\pi$
$600$		0			0
$601$	15.0000			0.611863			0.305931	−	0.952054i	$-0.401032\pi$
$601$	15.0000			0.611863			0.305931	+	0.952054i	$0.401032\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0000	+	28.0000i	0.569181	+	1.13836i
$606$		0			0
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.00000			−0.242734
$612$		0			0
$613$		−	11.0000i		−	0.444286i	−0.975014	−	0.222143i	$-0.928695\pi$
$613$		−	11.0000i		−	0.444286i	0.975014	−	0.222143i	$-0.0713052\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			4.00000i			0.161034i	0.996753	+	0.0805170i	$0.0256571\pi$
$617$			4.00000i			0.161034i	−0.996753	+	0.0805170i	$0.974343\pi$
$618$		0			0
$619$	−10.0000			−0.401934			−0.200967	−	0.979598i	$-0.564408\pi$
$619$	−10.0000			−0.401934			−0.200967	+	0.979598i	$0.564408\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			15.0000i			0.600962i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	21.0000			0.837325
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000	+	6.00000i	−0.476205	+	0.238103i
$636$		0			0
$637$			18.0000i			0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	36.0000			1.42191			0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000			1.42191			0.710957	+	0.703235i	$0.248262\pi$
$642$		0			0
$643$		−	39.0000i		−	1.53801i	−0.639243	−	0.769005i	$-0.720752\pi$
$643$		−	39.0000i		−	1.53801i	0.639243	−	0.769005i	$-0.279248\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			29.0000i			1.14011i	0.821607	+	0.570054i	$0.193078\pi$
$647$			29.0000i			1.14011i	−0.821607	+	0.570054i	$0.806922\pi$
$648$		0			0
$649$	−60.0000			−2.35521
$650$		0			0
$651$		0			0
$652$		0			0
$653$			18.0000i			0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$			18.0000i			0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$		0			0
$655$	−12.0000	−	24.0000i	−0.468879	−	0.937758i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$	−46.0000			−1.78919			−0.894596	−	0.446875i	$-0.852537\pi$
$661$	−46.0000			−1.78919			−0.894596	+	0.446875i	$0.852537\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	40.0000	−	20.0000i	1.55113	−	0.775567i
$666$		0			0
$667$		−	20.0000i		−	0.774403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	35.0000			1.35116
$672$		0			0
$673$		−	10.0000i		−	0.385472i	−0.981251	−	0.192736i	$-0.938264\pi$
$673$		−	10.0000i		−	0.385472i	0.981251	−	0.192736i	$-0.0617360\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15.0000i			0.576497i	0.957556	+	0.288248i	$0.0930729\pi$
$677$			15.0000i			0.576497i	−0.957556	+	0.288248i	$0.906927\pi$
$678$		0			0
$679$	−5.00000			−0.191882
$680$		0			0
$681$		0			0
$682$		0			0
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$	−28.0000	+	14.0000i	−1.06983	+	0.534913i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.00000			0.0380970
$690$		0			0
$691$	−18.0000			−0.684752			−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000			−0.684752			−0.342376	+	0.939563i	$0.611232\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	11.0000	+	22.0000i	0.417254	+	0.834508i
$696$		0			0
$697$		−	33.0000i		−	1.24996i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−4.00000			−0.151078			−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000			−0.151078			−0.0755390	+	0.997143i	$0.524068\pi$
$702$		0			0
$703$		−	28.0000i		−	1.05604i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	30.0000i		−	1.12827i
$708$		0			0
$709$	20.0000			0.751116			0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000			0.751116			0.375558	+	0.926799i	$0.377451\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−10.0000	+	5.00000i	−0.373979	+	0.186989i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	22.0000			0.820462			0.410231	−	0.911982i	$-0.365448\pi$
$719$	22.0000			0.820462			0.410231	+	0.911982i	$0.365448\pi$
$720$		0			0
$721$	70.0000			2.60694
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.0000	−	16.0000i	0.445669	−	0.594225i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.0000			−1.33151
$732$		0			0
$733$			3.00000i			0.110808i	0.998464	+	0.0554038i	$0.0176446\pi$
$733$			3.00000i			0.110808i	−0.998464	+	0.0554038i	$0.982355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	20.0000i		−	0.736709i
$738$		0			0
$739$	−24.0000			−0.882854			−0.441427	−	0.897297i	$-0.645528\pi$
$739$	−24.0000			−0.882854			−0.441427	+	0.897297i	$0.645528\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			36.0000i			1.32071i	0.750953	+	0.660356i	$0.229595\pi$
$743$			36.0000i			1.32071i	−0.750953	+	0.660356i	$0.770405\pi$
$744$		0			0
$745$	−15.0000	−	30.0000i	−0.549557	−	1.09911i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	55.0000			2.00966
$750$		0			0
$751$	17.0000			0.620339			0.310169	−	0.950681i	$-0.399614\pi$
$751$	17.0000			0.620339			0.310169	+	0.950681i	$0.399614\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.00000	−	4.00000i	−0.0727875	−	0.145575i
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−10.0000			−0.362500			−0.181250	−	0.983437i	$-0.558014\pi$
$761$	−10.0000			−0.362500			−0.181250	+	0.983437i	$0.558014\pi$
$762$		0			0
$763$		−	60.0000i		−	2.17215i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	12.0000i		−	0.433295i
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			2.00000i			0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$			2.00000i			0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−44.0000			−1.57646
$780$		0			0
$781$	−35.0000			−1.25240
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−8.00000	+	4.00000i	−0.285532	+	0.142766i
$786$		0			0
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	70.0000			2.48891
$792$		0			0
$793$			7.00000i			0.248577i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			45.0000i			1.59398i	0.603991	+	0.796991i	$0.293576\pi$
$797$			45.0000i			1.59398i	−0.603991	+	0.796991i	$0.706424\pi$
$798$		0			0
$799$	18.0000			0.636794
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	70.0000i		−	2.47025i
$804$		0			0
$805$	25.0000	+	50.0000i	0.881134	+	1.76227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.0000			0.843795			0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000			0.843795			0.421898	+	0.906644i	$0.361364\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.0000	−	7.00000i	0.490399	−	0.245199i
$816$		0			0
$817$			48.0000i			1.67931i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−3.00000			−0.104701			−0.0523504	−	0.998629i	$-0.516671\pi$
$821$	−3.00000			−0.104701			−0.0523504	+	0.998629i	$0.516671\pi$
$822$		0			0
$823$		−	26.0000i		−	0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$		−	26.0000i		−	0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	54.0000i		−	1.87099i
$834$		0			0
$835$	36.0000	−	18.0000i	1.24583	−	0.622916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−3.00000			−0.103572			−0.0517858	−	0.998658i	$-0.516491\pi$
$839$	−3.00000			−0.103572			−0.0517858	+	0.998658i	$0.516491\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.00000	−	2.00000i	−0.0344010	−	0.0688021i
$846$		0			0
$847$		−	70.0000i		−	2.40523i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	35.0000			1.19978
$852$		0			0
$853$			49.0000i			1.67773i	0.544341	+	0.838864i	$0.316780\pi$
$853$			49.0000i			1.67773i	−0.544341	+	0.838864i	$0.683220\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	11.0000i		−	0.375753i	−0.982193	−	0.187876i	$-0.939840\pi$
$857$		−	11.0000i		−	0.375753i	0.982193	−	0.187876i	$-0.0601604\pi$
$858$		0			0
$859$	−1.00000			−0.0341196			−0.0170598	−	0.999854i	$-0.505431\pi$
$859$	−1.00000			−0.0341196			−0.0170598	+	0.999854i	$0.505431\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	24.0000i		−	0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$		−	24.0000i		−	0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$		0			0
$865$	−12.0000	+	6.00000i	−0.408012	+	0.204006i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−25.0000			−0.848067
$870$		0			0
$871$	4.00000			0.135535
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10.0000	+	55.0000i	−0.338062	+	1.85934i
$876$		0			0
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−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
$883$		−	22.0000i		−	0.740359i	−0.928960	−	0.370179i	$-0.879296\pi$
$883$		−	22.0000i		−	0.740359i	0.928960	−	0.370179i	$-0.120704\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			9.00000i			0.302190i	0.988519	+	0.151095i	$0.0482800\pi$
$887$			9.00000i			0.302190i	−0.988519	+	0.151095i	$0.951720\pi$
$888$		0			0
$889$	30.0000			1.00617
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	24.0000i		−	0.803129i
$894$		0			0
$895$	−10.0000	−	20.0000i	−0.334263	−	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−3.00000			−0.0999445
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−23.0000	−	46.0000i	−0.764546	−	1.52909i
$906$		0			0
$907$		−	22.0000i		−	0.730498i	−0.930910	−	0.365249i	$-0.880984\pi$
$907$		−	22.0000i		−	0.730498i	0.930910	−	0.365249i	$-0.119016\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.00000			0.198789			0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000			0.198789			0.0993944	+	0.995048i	$0.468309\pi$
$912$		0			0
$913$			10.0000i			0.330952i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			60.0000i			1.98137i
$918$		0			0
$919$	−51.0000			−1.68233			−0.841167	−	0.540775i	$-0.818131\pi$
$919$	−51.0000			−1.68233			−0.841167	+	0.540775i	$0.818131\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	7.00000i		−	0.230408i
$924$		0			0
$925$	28.0000	+	21.0000i	0.920634	+	0.690476i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−31.0000			−1.01708			−0.508539	−	0.861039i	$-0.669814\pi$
$929$	−31.0000			−1.01708			−0.508539	+	0.861039i	$0.669814\pi$
$930$		0			0
$931$	−72.0000			−2.35970
$932$		0			0
$933$		0			0
$934$		0			0
$935$	30.0000	−	15.0000i	0.981105	−	0.490552i
$936$		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
$939$		0			0
$940$		0			0
$941$	7.00000			0.228193			0.114097	−	0.993470i	$-0.463603\pi$
$941$	7.00000			0.228193			0.114097	+	0.993470i	$0.463603\pi$
$942$		0			0
$943$		−	55.0000i		−	1.79105i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	30.0000i		−	0.974869i	−0.873160	−	0.487435i	$-0.837933\pi$
$947$		−	30.0000i		−	0.974869i	0.873160	−	0.487435i	$-0.162067\pi$
$948$		0			0
$949$	14.0000			0.454459
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	3.00000i		−	0.0971795i	−0.998819	−	0.0485898i	$-0.984527\pi$
$953$		−	3.00000i		−	0.0971795i	0.998819	−	0.0485898i	$-0.0154727\pi$
$954$		0			0
$955$	−6.00000	−	12.0000i	−0.194155	−	0.388311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	70.0000			2.26042
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−46.0000	+	23.0000i	−1.48079	+	0.740396i
$966$		0			0
$967$			40.0000i			1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$			40.0000i			1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−54.0000			−1.73294			−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000			−1.73294			−0.866471	+	0.499227i	$0.833617\pi$
$972$		0			0
$973$		−	55.0000i		−	1.76322i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	46.0000i		−	1.47167i	−0.677161	−	0.735835i	$-0.736790\pi$
$977$		−	46.0000i		−	1.47167i	0.677161	−	0.735835i	$-0.263210\pi$
$978$		0			0
$979$	15.0000			0.479402
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	40.0000i		−	1.27580i	−0.770118	−	0.637901i	$-0.779803\pi$
$983$		−	40.0000i		−	1.27580i	0.770118	−	0.637901i	$-0.220197\pi$
$984$		0			0
$985$	−36.0000	+	18.0000i	−1.14706	+	0.573528i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−60.0000			−1.90789
$990$		0			0
$991$	1.00000			0.0317660			0.0158830	−	0.999874i	$-0.494944\pi$
$991$	1.00000			0.0317660			0.0158830	+	0.999874i	$0.494944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000	−	32.0000i	−0.507234	−	1.01447i
$996$		0			0
$997$		−	42.0000i		−	1.33015i	−0.746775	−	0.665077i	$-0.768399\pi$
$997$		−	42.0000i		−	1.33015i	0.746775	−	0.665077i	$-0.231601\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.c.2809.2		2
3.2	odd	2		1560.2.l.a.1249.1	✓	2
5.4	even	2	inner	4680.2.l.c.2809.1		2
12.11	even	2		3120.2.l.b.1249.2		2
15.2	even	4		7800.2.a.l.1.1		1
15.8	even	4		7800.2.a.m.1.1		1
15.14	odd	2		1560.2.l.a.1249.2	yes	2
60.59	even	2		3120.2.l.b.1249.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.a.1249.1	✓	2	3.2	odd	2
1560.2.l.a.1249.2	yes	2	15.14	odd	2
3120.2.l.b.1249.1		2	60.59	even	2
3120.2.l.b.1249.2		2	12.11	even	2
4680.2.l.c.2809.1		2	5.4	even	2	inner
4680.2.l.c.2809.2		2	1.1	even	1	trivial
7800.2.a.l.1.1		1	15.2	even	4
7800.2.a.m.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 2.00000i) q^{5} -5.00000i q^{7} +O(q^{10})\)	\(q+(1.00000 + 2.00000i) q^{5} -5.00000i q^{7} -5.00000 q^{11} -1.00000i q^{13} +3.00000i q^{17} +4.00000 q^{19} +5.00000i q^{23} +(-3.00000 + 4.00000i) q^{25} -4.00000 q^{29} +(10.0000 - 5.00000i) q^{35} -7.00000i q^{37} -11.0000 q^{41} +12.0000i q^{43} -6.00000i q^{47} -18.0000 q^{49} +1.00000i q^{53} +(-5.00000 - 10.0000i) q^{55} +12.0000 q^{59} -7.00000 q^{61} +(2.00000 - 1.00000i) q^{65} +4.00000i q^{67} +7.00000 q^{71} +14.0000i q^{73} +25.0000i q^{77} +5.00000 q^{79} -2.00000i q^{83} +(-6.00000 + 3.00000i) q^{85} -3.00000 q^{89} -5.00000 q^{91} +(4.00000 + 8.00000i) q^{95} -1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 10 q^{11} + 8 q^{19} - 6 q^{25} - 8 q^{29} + 20 q^{35} - 22 q^{41} - 36 q^{49} - 10 q^{55} + 24 q^{59} - 14 q^{61} + 4 q^{65} + 14 q^{71} + 10 q^{79} - 12 q^{85} - 6 q^{89} - 10 q^{91} + 8 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 10 * q^11 + 8 * q^19 - 6 * q^25 - 8 * q^29 + 20 * q^35 - 22 * q^41 - 36 * q^49 - 10 * q^55 + 24 * q^59 - 14 * q^61 + 4 * q^65 + 14 * q^71 + 10 * q^79 - 12 * q^85 - 6 * q^89 - 10 * q^91 + 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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