LMFDB - Embedded newform 2520.2.bi.b.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,2,Mod(361,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))

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

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

chi := DirichletCharacter("2520.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2520.bi (of order $3$, degree $2$, 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:	$20.1223013094$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2520.361
Dual form			2520.2.bi.b.1801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +O(q^{10})$	$q+(-0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{11} +1.00000 q^{13} +(3.50000 + 6.06218i) q^{19} +(-2.50000 - 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-3.00000 + 5.19615i) q^{31} +(2.50000 + 0.866025i) q^{35} +(-1.50000 - 2.59808i) q^{37} +3.00000 q^{41} +8.00000 q^{43} +(-0.500000 - 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(2.50000 - 4.33013i) q^{53} -3.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-0.500000 - 0.866025i) q^{65} +6.00000 q^{71} +(7.00000 - 12.1244i) q^{73} +(1.50000 + 7.79423i) q^{77} +(8.00000 + 13.8564i) q^{79} +16.0000 q^{83} +(3.00000 + 5.19615i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(3.50000 - 6.06218i) q^{95} +16.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - q^5 - 4 * q^7	$ 2 q - q^{5} - 4 q^{7} + 3 q^{11} + 2 q^{13} + 7 q^{19} - 5 q^{23} - q^{25} - 6 q^{31} + 5 q^{35} - 3 q^{37} + 6 q^{41} + 16 q^{43} - q^{47} + 2 q^{49} + 5 q^{53} - 6 q^{55} - 4 q^{59} + 8 q^{61} - q^{65} + 12 q^{71} + 14 q^{73} + 3 q^{77} + 16 q^{79} + 32 q^{83} + 6 q^{89} - 4 q^{91} + 7 q^{95} + 32 q^{97}+O(q^{100}) $ 2 * q - q^5 - 4 * q^7 + 3 * q^11 + 2 * q^13 + 7 * q^19 - 5 * q^23 - q^25 - 6 * q^31 + 5 * q^35 - 3 * q^37 + 6 * q^41 + 16 * q^43 - q^47 + 2 * q^49 + 5 * q^53 - 6 * q^55 - 4 * q^59 + 8 * q^61 - q^65 + 12 * q^71 + 14 * q^73 + 3 * q^77 + 16 * q^79 + 32 * q^83 + 6 * q^89 - 4 * q^91 + 7 * q^95 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.50000	+	6.06218i	0.802955	+	1.39076i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	−0.999694	−	0.0247559i	$-0.992119\pi$
$23$	−2.50000	−	4.33013i	−0.521286	−	0.902894i	0.478407	−	0.878138i	$-0.341214\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	0.460152	+	0.887840i	$0.347795\pi$
$31$	−3.00000	+	5.19615i	−0.538816	+	0.933257i	−0.998968	+	0.0454165i	$0.985539\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.50000	+	0.866025i	0.422577	+	0.146385i
$36$		0			0
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	0.715981	−	0.698119i	$-0.245980\pi$
$37$	−1.50000	−	2.59808i	−0.246598	−	0.427121i	−0.962580	+	0.270998i	$0.912646\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000			0.468521			0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000			0.468521			0.234261	+	0.972174i	$0.424733\pi$
$42$		0			0
$43$	8.00000			1.21999			0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000			1.21999			0.609994	+	0.792406i	$0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.500000	−	0.866025i	−0.0729325	−	0.126323i	0.827253	−	0.561830i	$-0.189902\pi$
$47$	−0.500000	−	0.866025i	−0.0729325	−	0.126323i	−0.900185	+	0.435507i	$0.856569\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.50000	−	4.33013i	0.343401	−	0.594789i	−0.641661	−	0.766989i	$-0.721754\pi$
$53$	2.50000	−	4.33013i	0.343401	−	0.594789i	0.985062	+	0.172200i	$0.0550875\pi$
$54$		0			0
$55$	−3.00000			−0.404520
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	−0.966342	−	0.257260i	$-0.917180\pi$
$59$	−2.00000	+	3.46410i	−0.260378	+	0.450988i	0.705965	+	0.708247i	$0.250514\pi$
$60$		0			0
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	0.999901	+	0.0140840i	$0.00448323\pi$
$61$	4.00000	+	6.92820i	0.512148	+	0.887066i	−0.487753	+	0.872982i	$0.662183\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.500000	−	0.866025i	−0.0620174	−	0.107417i
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$	7.00000	−	12.1244i	0.819288	−	1.41905i	0.906208	−	0.422833i	$-0.138964\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.50000	+	7.79423i	0.170941	+	0.888235i
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	16.0000			1.75623			0.878114	−	0.478451i	$-0.158802\pi$
$83$	16.0000			1.75623			0.878114	+	0.478451i	$0.158802\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	0.980071	−	0.198650i	$-0.0636557\pi$
$89$	3.00000	+	5.19615i	0.317999	+	0.550791i	−0.662071	+	0.749441i	$0.730322\pi$
$90$		0			0
$91$	−2.00000	+	1.73205i	−0.209657	+	0.181568i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.50000	−	6.06218i	0.359092	−	0.621966i
$96$		0			0
$97$	16.0000			1.62455			0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000			1.62455			0.812277	+	0.583272i	$0.198228\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	−0.595466	−	0.803380i	$-0.703033\pi$
$101$	4.00000	−	6.92820i	0.398015	−	0.689382i	0.993481	+	0.113998i	$0.0363659\pi$
$102$		0			0
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	0.598858	−	0.800855i	$-0.295621\pi$
$103$	−4.00000	−	6.92820i	−0.394132	−	0.682656i	−0.992990	+	0.118199i	$0.962288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.00000	+	12.1244i	0.676716	+	1.17211i	0.975964	+	0.217931i	$0.0699306\pi$
$107$	7.00000	+	12.1244i	0.676716	+	1.17211i	−0.299249	+	0.954175i	$0.596736\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−2.50000	+	4.33013i	−0.233126	+	0.403786i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	11.0000			0.976092			0.488046	−	0.872818i	$-0.337710\pi$
$127$	11.0000			0.976092			0.488046	+	0.872818i	$0.337710\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.50000	+	6.06218i	0.305796	+	0.529655i	0.977438	−	0.211221i	$-0.0677440\pi$
$131$	3.50000	+	6.06218i	0.305796	+	0.529655i	−0.671642	+	0.740876i	$0.734411\pi$
$132$		0			0
$133$	−17.5000	−	6.06218i	−1.51744	−	0.525657i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.50000	−	2.59808i	0.125436	−	0.217262i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	0.426948	+	0.904276i	$0.359589\pi$
$151$	−7.00000	+	12.1244i	−0.569652	+	0.986666i	−0.996600	+	0.0823900i	$0.973745\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.00000			0.481932
$156$		0			0
$157$	7.50000	−	12.9904i	0.598565	−	1.03675i	−0.394468	−	0.918910i	$-0.629071\pi$
$157$	7.50000	−	12.9904i	0.598565	−	1.03675i	0.993033	−	0.117836i	$-0.0375956\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	12.5000	+	4.33013i	0.985138	+	0.341262i
$162$		0			0
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	0.665771	−	0.746156i	$-0.268103\pi$
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	23.0000			1.77979			0.889897	−	0.456162i	$-0.150776\pi$
$167$	23.0000			1.77979			0.889897	+	0.456162i	$0.150776\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.50000	+	7.79423i	0.342129	+	0.592584i	0.984828	−	0.173534i	$-0.0555188\pi$
$173$	4.50000	+	7.79423i	0.342129	+	0.592584i	−0.642699	+	0.766119i	$0.722185\pi$
$174$		0			0
$175$	−0.500000	−	2.59808i	−0.0377964	−	0.196396i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−7.50000	+	12.9904i	−0.560576	+	0.970947i	0.436870	+	0.899525i	$0.356087\pi$
$179$	−7.50000	+	12.9904i	−0.560576	+	0.970947i	−0.997446	+	0.0714220i	$0.977246\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.50000	+	2.59808i	−0.110282	+	0.191014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.0000	−	19.0526i	−0.795932	−	1.37859i	−0.922246	−	0.386604i	$-0.873648\pi$
$191$	−11.0000	−	19.0526i	−0.795932	−	1.37859i	0.126314	−	0.991990i	$-0.459685\pi$
$192$		0			0
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	1.00000	−	1.73205i	0.0719816	−	0.124676i	0.899770	+	0.436365i	$0.143734\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−15.0000			−1.06871			−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000			−1.06871			−0.534353	+	0.845262i	$0.679445\pi$
$198$		0			0
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	−0.0299585	−	0.999551i	$-0.509538\pi$
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	0.880616	−	0.473831i	$-0.157129\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.50000	−	2.59808i	−0.104765	−	0.181458i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	21.0000			1.45260
$210$		0			0
$211$	9.00000			0.619586			0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000			0.619586			0.309793	+	0.950804i	$0.399740\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	−	6.92820i	−0.272798	−	0.472500i
$216$		0			0
$217$	−3.00000	−	15.5885i	−0.203653	−	1.05821i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0000	+	20.7846i	−0.796468	+	1.37952i	0.125435	+	0.992102i	$0.459967\pi$
$227$	−12.0000	+	20.7846i	−0.796468	+	1.37952i	−0.921903	+	0.387421i	$0.873366\pi$
$228$		0			0
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	0.231287	−	0.972886i	$-0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−11.0000	−	19.0526i	−0.720634	−	1.24817i	−0.960746	−	0.277429i	$-0.910518\pi$
$233$	−11.0000	−	19.0526i	−0.720634	−	1.24817i	0.240112	−	0.970745i	$-0.422816\pi$
$234$		0			0
$235$	−0.500000	+	0.866025i	−0.0326164	+	0.0564933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000			0.129369			0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000			0.129369			0.0646846	+	0.997906i	$0.479396\pi$
$240$		0			0
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	−0.956456	−	0.291877i	$-0.905720\pi$
$241$	−3.50000	+	6.06218i	−0.225455	+	0.390499i	0.731001	+	0.682376i	$0.239053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.50000	+	2.59808i	−0.415270	+	0.165985i
$246$		0			0
$247$	3.50000	+	6.06218i	0.222700	+	0.385727i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	31.0000			1.95670			0.978351	−	0.206951i	$-0.0663540\pi$
$251$	31.0000			1.95670			0.978351	+	0.206951i	$0.0663540\pi$
$252$		0			0
$253$	−15.0000			−0.943042
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	0.615948	−	0.787787i	$-0.288773\pi$
$257$	−6.00000	−	10.3923i	−0.374270	−	0.648254i	−0.990217	+	0.139533i	$0.955440\pi$
$258$		0			0
$259$	7.50000	+	2.59808i	0.466027	+	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.0000	+	27.7128i	−0.986602	+	1.70885i	−0.352014	+	0.935995i	$0.614503\pi$
$263$	−16.0000	+	27.7128i	−0.986602	+	1.70885i	−0.634588	+	0.772851i	$0.718830\pi$
$264$		0			0
$265$	−5.00000			−0.307148
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−6.00000	+	10.3923i	−0.365826	+	0.633630i	−0.988908	−	0.148527i	$-0.952547\pi$
$269$	−6.00000	+	10.3923i	−0.365826	+	0.633630i	0.623082	+	0.782157i	$0.285880\pi$
$270$		0			0
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	0.957328	+	0.289003i	$0.0933238\pi$
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	−0.228380	+	0.973572i	$0.573343\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.50000	+	2.59808i	0.0904534	+	0.156670i
$276$		0			0
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−5.00000	+	8.66025i	−0.300421	+	0.520344i	0.675810	+	0.737075i	$0.263794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	17.0000			1.01413			0.507067	−	0.861906i	$-0.330729\pi$
$281$	17.0000			1.01413			0.507067	+	0.861906i	$0.330729\pi$
$282$		0			0
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	−0.579437	−	0.815017i	$-0.696728\pi$
$283$	7.00000	−	12.1244i	0.416107	−	0.720718i	0.995544	+	0.0942988i	$0.0300609\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.00000	+	5.19615i	−0.354169	+	0.306719i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.00000			−0.175262			−0.0876309	−	0.996153i	$-0.527930\pi$
$293$	−3.00000			−0.175262			−0.0876309	+	0.996153i	$0.527930\pi$
$294$		0			0
$295$	4.00000			0.232889
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.50000	−	4.33013i	−0.144579	−	0.250418i
$300$		0			0
$301$	−16.0000	+	13.8564i	−0.922225	+	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000	−	6.92820i	0.229039	−	0.396708i
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	−0.917143	−	0.398559i	$-0.869511\pi$
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	0.803733	+	0.594990i	$0.202844\pi$
$312$		0			0
$313$	−6.00000	−	10.3923i	−0.339140	−	0.587408i	0.645131	−	0.764072i	$-0.276803\pi$
$313$	−6.00000	−	10.3923i	−0.339140	−	0.587408i	−0.984271	+	0.176664i	$0.943469\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.500000	+	0.866025i	−0.0277350	+	0.0480384i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.50000	+	0.866025i	0.137829	+	0.0477455i
$330$		0			0
$331$	8.50000	+	14.7224i	0.467202	+	0.809218i	0.999298	−	0.0374662i	$-0.0119287\pi$
$331$	8.50000	+	14.7224i	0.467202	+	0.809218i	−0.532096	+	0.846684i	$0.678595\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	36.0000			1.96104			0.980522	−	0.196407i	$-0.0629273\pi$
$337$	36.0000			1.96104			0.980522	+	0.196407i	$0.0629273\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	9.00000	+	15.5885i	0.487377	+	0.844162i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.0000	+	29.4449i	−0.912608	+	1.58068i	−0.102241	+	0.994760i	$0.532601\pi$
$347$	−17.0000	+	29.4449i	−0.912608	+	1.58068i	−0.810366	+	0.585923i	$0.800732\pi$
$348$		0			0
$349$	−4.00000			−0.214115			−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000			−0.214115			−0.107058	+	0.994253i	$0.534143\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	−0.999476	−	0.0323801i	$-0.989691\pi$
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	0.471696	−	0.881761i	$-0.343642\pi$
$360$		0			0
$361$	−15.0000	+	25.9808i	−0.789474	+	1.36741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−14.0000			−0.732793
$366$		0			0
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	−0.923869	−	0.382709i	$-0.874991\pi$
$367$	−2.50000	+	4.33013i	−0.130499	+	0.226031i	0.793370	+	0.608740i	$0.208325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	2.50000	+	12.9904i	0.129794	+	0.674427i
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.00000			0.0513665			0.0256833	−	0.999670i	$-0.491824\pi$
$379$	1.00000			0.0513665			0.0256833	+	0.999670i	$0.491824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.500000	−	0.866025i	−0.0255488	−	0.0442518i	0.852968	−	0.521963i	$-0.174800\pi$
$383$	−0.500000	−	0.866025i	−0.0255488	−	0.0442518i	−0.878517	+	0.477711i	$0.841467\pi$
$384$		0			0
$385$	6.00000	−	5.19615i	0.305788	−	0.264820i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	−0.321716	−	0.946836i	$-0.604260\pi$
$389$	13.0000	−	22.5167i	0.659126	−	1.14164i	0.980842	−	0.194804i	$-0.0624070\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.00000	−	13.8564i	0.402524	−	0.697191i
$396$		0			0
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	0.998492	−	0.0549046i	$-0.0174855\pi$
$397$	9.00000	+	15.5885i	0.451697	+	0.782362i	−0.546795	+	0.837267i	$0.684152\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.5000	−	25.1147i	−0.724095	−	1.25417i	−0.959345	−	0.282235i	$-0.908924\pi$
$401$	−14.5000	−	25.1147i	−0.724095	−	1.25417i	0.235250	−	0.971935i	$-0.424409\pi$
$402$		0			0
$403$	−3.00000	+	5.19615i	−0.149441	+	0.258839i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.00000			−0.446113
$408$		0			0
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	−0.639430	−	0.768849i	$-0.720830\pi$
$409$	7.00000	−	12.1244i	0.346128	−	0.599511i	0.985558	+	0.169338i	$0.0541630\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.00000	−	10.3923i	−0.0984136	−	0.511372i
$414$		0			0
$415$	−8.00000	−	13.8564i	−0.392705	−	0.680184i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.0000			−0.732798			−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000			−0.732798			−0.366399	+	0.930458i	$0.619409\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−20.0000	−	6.92820i	−0.967868	−	0.335279i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	8.00000	−	13.8564i	0.385346	−	0.667440i	−0.606471	−	0.795106i	$-0.707415\pi$
$431$	8.00000	−	13.8564i	0.385346	−	0.667440i	0.991817	+	0.127666i	$0.0407486\pi$
$432$		0			0
$433$	8.00000			0.384455			0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000			0.384455			0.192228	+	0.981350i	$0.438429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	17.5000	−	30.3109i	0.837139	−	1.44997i
$438$		0			0
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	0.940963	+	0.338508i	$0.109922\pi$
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	−0.177325	+	0.984152i	$0.556744\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.0000	−	34.6410i	−0.950229	−	1.64584i	−0.744927	−	0.667146i	$-0.767516\pi$
$443$	−20.0000	−	34.6410i	−0.950229	−	1.64584i	−0.205301	−	0.978699i	$-0.565817\pi$
$444$		0			0
$445$	3.00000	−	5.19615i	0.142214	−	0.246321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	21.0000			0.991051			0.495526	−	0.868593i	$-0.334975\pi$
$449$	21.0000			0.991051			0.495526	+	0.868593i	$0.334975\pi$
$450$		0			0
$451$	4.50000	−	7.79423i	0.211897	−	0.367016i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.50000	+	0.866025i	0.117202	+	0.0405999i
$456$		0			0
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	0.996038	−	0.0889312i	$-0.0283451\pi$
$457$	9.00000	+	15.5885i	0.421002	+	0.729197i	−0.575036	+	0.818128i	$0.695012\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−36.0000			−1.67669			−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000			−1.67669			−0.838344	+	0.545142i	$0.816476\pi$
$462$		0			0
$463$	−9.00000			−0.418265			−0.209133	−	0.977887i	$-0.567064\pi$
$463$	−9.00000			−0.418265			−0.209133	+	0.977887i	$0.567064\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.00000	−	6.92820i	−0.185098	−	0.320599i	0.758512	−	0.651660i	$-0.225927\pi$
$467$	−4.00000	−	6.92820i	−0.185098	−	0.320599i	−0.943610	+	0.331061i	$0.892594\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	12.0000	−	20.7846i	0.551761	−	0.955677i
$474$		0			0
$475$	−7.00000			−0.321182
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.0000	+	29.4449i	−0.776750	+	1.34537i	0.157056	+	0.987590i	$0.449800\pi$
$479$	−17.0000	+	29.4449i	−0.776750	+	1.34537i	−0.933806	+	0.357780i	$0.883534\pi$
$480$		0			0
$481$	−1.50000	−	2.59808i	−0.0683941	−	0.118462i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000	−	13.8564i	−0.363261	−	0.629187i
$486$		0			0
$487$	−20.0000	+	34.6410i	−0.906287	+	1.56973i	−0.0871056	+	0.996199i	$0.527762\pi$
$487$	−20.0000	+	34.6410i	−0.906287	+	1.56973i	−0.819181	+	0.573535i	$0.805572\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−28.0000			−1.26362			−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000			−1.26362			−0.631811	+	0.775122i	$0.717688\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	+	10.3923i	−0.538274	+	0.466159i
$498$		0			0
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	−0.999054	−	0.0434940i	$-0.986151\pi$
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	0.461860	−	0.886953i	$-0.347182\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−28.0000			−1.24846			−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000			−1.24846			−0.624229	+	0.781241i	$0.714587\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.00000	−	8.66025i	−0.221621	−	0.383859i	0.733679	−	0.679496i	$-0.237801\pi$
$509$	−5.00000	−	8.66025i	−0.221621	−	0.383859i	−0.955300	+	0.295637i	$0.904468\pi$
$510$		0			0
$511$	7.00000	+	36.3731i	0.309662	+	1.60905i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−4.00000	+	6.92820i	−0.176261	+	0.305293i
$516$		0			0
$517$	−3.00000			−0.131940
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−16.5000	+	28.5788i	−0.722878	+	1.25206i	0.236963	+	0.971519i	$0.423848\pi$
$521$	−16.5000	+	28.5788i	−0.722878	+	1.25206i	−0.959841	+	0.280543i	$0.909485\pi$
$522$		0			0
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	−0.996719	−	0.0809336i	$-0.974210\pi$
$523$	−13.0000	−	22.5167i	−0.568450	−	0.984585i	0.428269	−	0.903651i	$-0.359124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.00000	+	1.73205i	−0.0434783	+	0.0753066i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.00000			0.129944
$534$		0			0
$535$	7.00000	−	12.1244i	0.302636	−	0.524182i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16.5000	−	12.9904i	−0.710705	−	0.559535i
$540$		0			0
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	0.886721	−	0.462304i	$-0.152977\pi$
$541$	1.00000	+	1.73205i	0.0429934	+	0.0744667i	−0.843728	+	0.536771i	$0.819644\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.00000			−0.0856706
$546$		0			0
$547$	24.0000			1.02617			0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000			1.02617			0.513083	+	0.858339i	$0.328503\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−40.0000	−	13.8564i	−1.70097	−	0.589234i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	−0.945473	−	0.325701i	$-0.894400\pi$
$557$	−4.50000	+	7.79423i	−0.190671	+	0.330252i	0.754802	+	0.655953i	$0.227733\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	9.00000	−	15.5885i	0.379305	−	0.656975i	−0.611656	−	0.791123i	$-0.709497\pi$
$563$	9.00000	−	15.5885i	0.379305	−	0.656975i	0.990961	+	0.134148i	$0.0428299\pi$
$564$		0			0
$565$	3.00000	+	5.19615i	0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.5000	−	28.5788i	−0.691716	−	1.19809i	−0.971275	−	0.237959i	$-0.923522\pi$
$569$	−16.5000	−	28.5788i	−0.691716	−	1.19809i	0.279559	−	0.960128i	$-0.409812\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5.00000			0.208514
$576$		0			0
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	−0.886090	−	0.463513i	$-0.846589\pi$
$577$	−1.00000	+	1.73205i	−0.0416305	+	0.0721062i	0.844459	+	0.535620i	$0.179922\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−32.0000	+	27.7128i	−1.32758	+	1.14972i
$582$		0			0
$583$	−7.50000	−	12.9904i	−0.310618	−	0.538007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.0000			0.742940			0.371470	−	0.928445i	$-0.378854\pi$
$587$	18.0000			0.742940			0.371470	+	0.928445i	$0.378854\pi$
$588$		0			0
$589$	−42.0000			−1.73058
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.00000	+	6.92820i	0.164260	+	0.284507i	0.936392	−	0.350955i	$-0.114143\pi$
$593$	4.00000	+	6.92820i	0.164260	+	0.284507i	−0.772132	+	0.635462i	$0.780810\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.0000	+	19.0526i	−0.449448	+	0.778466i	−0.998350	−	0.0574201i	$-0.981713\pi$
$599$	−11.0000	+	19.0526i	−0.449448	+	0.778466i	0.548902	+	0.835887i	$0.315046\pi$
$600$		0			0
$601$	38.0000			1.55005			0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000			1.55005			0.775026	+	0.631929i	$0.217737\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	0.786212	−	0.617957i	$-0.212039\pi$
$607$	−3.50000	−	6.06218i	−0.142061	−	0.246056i	−0.928272	+	0.371901i	$0.878706\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.500000	−	0.866025i	−0.0202278	−	0.0350356i
$612$		0			0
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	−0.760724	−	0.649075i	$-0.775156\pi$
$613$	4.50000	−	7.79423i	0.181753	−	0.314806i	0.942478	+	0.334269i	$0.108489\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.00000			−0.161034			−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000			−0.161034			−0.0805170	+	0.996753i	$0.525657\pi$
$618$		0			0
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	−0.365927	−	0.930644i	$-0.619248\pi$
$619$	15.5000	−	26.8468i	0.622998	−	1.07906i	0.988924	−	0.148420i	$-0.0474187\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−15.0000	−	5.19615i	−0.600962	−	0.208179i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	14.0000			0.557331			0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000			0.557331			0.278666	+	0.960388i	$0.410108\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.50000	−	9.52628i	−0.218261	−	0.378039i
$636$		0			0
$637$	1.00000	−	6.92820i	0.0396214	−	0.274505i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.5000	+	30.3109i	−0.691208	+	1.19721i	0.280234	+	0.959932i	$0.409588\pi$
$641$	−17.5000	+	30.3109i	−0.691208	+	1.19721i	−0.971442	+	0.237276i	$0.923745\pi$
$642$		0			0
$643$	−34.0000			−1.34083			−0.670415	−	0.741987i	$-0.733884\pi$
$643$	−34.0000			−1.34083			−0.670415	+	0.741987i	$0.733884\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.5000	−	37.2391i	0.845252	−	1.46402i	−0.0401498	−	0.999194i	$-0.512784\pi$
$647$	21.5000	−	37.2391i	0.845252	−	1.46402i	0.885402	−	0.464826i	$-0.153883\pi$
$648$		0			0
$649$	6.00000	+	10.3923i	0.235521	+	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4.50000	−	7.79423i	−0.176099	−	0.305012i	0.764442	−	0.644692i	$-0.223014\pi$
$653$	−4.50000	−	7.79423i	−0.176099	−	0.305012i	−0.940541	+	0.339680i	$0.889681\pi$
$654$		0			0
$655$	3.50000	−	6.06218i	0.136756	−	0.236869i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	0.155235	+	0.987878i	$0.450387\pi$
$661$	−20.0000	+	34.6410i	−0.777910	+	1.34738i	−0.933144	+	0.359502i	$0.882947\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.50000	+	18.1865i	0.135724	+	0.705244i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$	12.0000			0.462566			0.231283	−	0.972887i	$-0.425708\pi$
$673$	12.0000			0.462566			0.231283	+	0.972887i	$0.425708\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	17.5000	+	30.3109i	0.672580	+	1.16494i	0.977170	+	0.212459i	$0.0681471\pi$
$677$	17.5000	+	30.3109i	0.672580	+	1.16494i	−0.304590	+	0.952483i	$0.598520\pi$
$678$		0			0
$679$	−32.0000	+	27.7128i	−1.22805	+	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	−0.901750	−	0.432259i	$-0.857717\pi$
$683$	−2.00000	+	3.46410i	−0.0765279	+	0.132550i	0.825222	+	0.564809i	$0.191050\pi$
$684$		0			0
$685$	2.00000			0.0764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.50000	−	4.33013i	0.0952424	−	0.164965i
$690$		0			0
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	0.729039	−	0.684472i	$-0.239967\pi$
$691$	−6.00000	−	10.3923i	−0.228251	−	0.395342i	−0.957290	+	0.289130i	$0.906634\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2.00000	+	3.46410i	0.0758643	+	0.131401i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$	10.5000	−	18.1865i	0.396015	−	0.685918i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.00000	+	20.7846i	0.150435	+	0.781686i
$708$		0			0
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	0.826023	−	0.563636i	$-0.190598\pi$
$709$	−2.00000	−	3.46410i	−0.0751116	−	0.130097i	−0.901135	+	0.433539i	$0.857265\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	30.0000			1.12351
$714$		0			0
$715$	−3.00000			−0.112194
$716$		0			0
$717$		0			0
$718$		0			0
$719$	1.00000	+	1.73205i	0.0372937	+	0.0645946i	0.884070	−	0.467355i	$-0.154793\pi$
$719$	1.00000	+	1.73205i	0.0372937	+	0.0645946i	−0.846776	+	0.531949i	$0.821460\pi$
$720$		0			0
$721$	20.0000	+	6.92820i	0.744839	+	0.258020i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−33.0000			−1.22390			−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000			−1.22390			−0.611951	+	0.790896i	$0.709615\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	0.635515	−	0.772088i	$-0.280788\pi$
$733$	−9.50000	−	16.4545i	−0.350891	−	0.607760i	−0.986406	+	0.164328i	$0.947454\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	0.191714	+	0.981451i	$0.438596\pi$
$739$	−20.5000	+	35.5070i	−0.754105	+	1.30615i	−0.945818	+	0.324697i	$0.894738\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.00000			−0.183432			−0.0917161	−	0.995785i	$-0.529235\pi$
$743$	−5.00000			−0.183432			−0.0917161	+	0.995785i	$0.529235\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−35.0000	−	12.1244i	−1.27887	−	0.443014i
$750$		0			0
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	−0.989423	−	0.145062i	$-0.953662\pi$
$751$	−17.0000	−	29.4449i	−0.620339	−	1.07446i	0.369084	−	0.929396i	$-0.379672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.0000			0.509512
$756$		0			0
$757$	−14.0000			−0.508839			−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000			−0.508839			−0.254419	+	0.967094i	$0.581884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21.5000	+	37.2391i	0.779374	+	1.34992i	0.932303	+	0.361679i	$0.117796\pi$
$761$	21.5000	+	37.2391i	0.779374	+	1.34992i	−0.152928	+	0.988237i	$0.548870\pi$
$762$		0			0
$763$	1.00000	+	5.19615i	0.0362024	+	0.188113i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.00000	+	3.46410i	−0.0722158	+	0.125081i
$768$		0			0
$769$	35.0000			1.26213			0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000			1.26213			0.631066	+	0.775729i	$0.282618\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−15.5000	+	26.8468i	−0.557496	+	0.965612i	0.440208	+	0.897896i	$0.354905\pi$
$773$	−15.5000	+	26.8468i	−0.557496	+	0.965612i	−0.997705	+	0.0677162i	$0.978429\pi$
$774$		0			0
$775$	−3.00000	−	5.19615i	−0.107763	−	0.186651i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	10.5000	+	18.1865i	0.376202	+	0.651600i
$780$		0			0
$781$	9.00000	−	15.5885i	0.322045	−	0.557799i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−15.0000			−0.535373
$786$		0			0
$787$	−19.0000	+	32.9090i	−0.677277	+	1.17308i	0.298521	+	0.954403i	$0.403507\pi$
$787$	−19.0000	+	32.9090i	−0.677277	+	1.17308i	−0.975798	+	0.218675i	$0.929827\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000	−	10.3923i	0.426671	−	0.369508i
$792$		0			0
$793$	4.00000	+	6.92820i	0.142044	+	0.246028i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−21.0000	−	36.3731i	−0.741074	−	1.28358i
$804$		0			0
$805$	−2.50000	−	12.9904i	−0.0881134	−	0.457851i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27.5000	+	47.6314i	−0.966849	+	1.67463i	−0.262284	+	0.964991i	$0.584476\pi$
$809$	−27.5000	+	47.6314i	−0.966849	+	1.67463i	−0.704564	+	0.709640i	$0.748858\pi$
$810$		0			0
$811$	5.00000			0.175574			0.0877869	−	0.996139i	$-0.472021\pi$
$811$	5.00000			0.175574			0.0877869	+	0.996139i	$0.472021\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	+	6.92820i	−0.140114	+	0.242684i
$816$		0			0
$817$	28.0000	+	48.4974i	0.979596	+	1.69671i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	0.955636	+	0.294549i	$0.0951694\pi$
$821$	21.0000	+	36.3731i	0.732905	+	1.26943i	−0.222731	+	0.974880i	$0.571497\pi$
$822$		0			0
$823$	24.0000	−	41.5692i	0.836587	−	1.44901i	−0.0561440	−	0.998423i	$-0.517881\pi$
$823$	24.0000	−	41.5692i	0.836587	−	1.44901i	0.892731	−	0.450589i	$-0.148786\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.0000			−1.04320			−0.521601	−	0.853189i	$-0.674665\pi$
$827$	−30.0000			−1.04320			−0.521601	+	0.853189i	$0.674665\pi$
$828$		0			0
$829$	−12.0000	+	20.7846i	−0.416777	+	0.721879i	−0.995613	−	0.0935647i	$-0.970174\pi$
$829$	−12.0000	+	20.7846i	−0.416777	+	0.721879i	0.578836	+	0.815444i	$0.303507\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−11.5000	−	19.9186i	−0.397974	−	0.689311i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.00000	+	10.3923i	0.206406	+	0.357506i
$846$		0			0
$847$	−5.00000	−	1.73205i	−0.171802	−	0.0595140i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.50000	+	12.9904i	−0.257097	+	0.445305i
$852$		0			0
$853$	23.0000			0.787505			0.393753	−	0.919216i	$-0.371177\pi$
$853$	23.0000			0.787505			0.393753	+	0.919216i	$0.371177\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	−0.244701	−	0.969599i	$-0.578690\pi$
$857$	21.0000	−	36.3731i	0.717346	−	1.24248i	0.962048	−	0.272882i	$-0.0879768\pi$
$858$		0			0
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	0.829887	−	0.557931i	$-0.188405\pi$
$859$	−2.00000	−	3.46410i	−0.0682391	−	0.118194i	−0.898126	+	0.439738i	$0.855071\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.50000	−	7.79423i	−0.153182	−	0.265319i	0.779214	−	0.626758i	$-0.215619\pi$
$863$	−4.50000	−	7.79423i	−0.153182	−	0.265319i	−0.932395	+	0.361440i	$0.882285\pi$
$864$		0			0
$865$	4.50000	−	7.79423i	0.153005	−	0.265012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	48.0000			1.62829
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.00000	+	1.73205i	−0.0676123	+	0.0585540i
$876$		0			0
$877$	−28.5000	−	49.3634i	−0.962377	−	1.66689i	−0.716504	−	0.697583i	$-0.754259\pi$
$877$	−28.5000	−	49.3634i	−0.962377	−	1.66689i	−0.245873	−	0.969302i	$-0.579075\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.00000			−0.101073			−0.0505363	−	0.998722i	$-0.516093\pi$
$881$	−3.00000			−0.101073			−0.0505363	+	0.998722i	$0.516093\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	0.747539	−	0.664218i	$-0.231235\pi$
$887$	−6.00000	−	10.3923i	−0.201460	−	0.348939i	−0.948999	+	0.315279i	$0.897902\pi$
$888$		0			0
$889$	−22.0000	+	19.0526i	−0.737856	+	0.639002i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.50000	−	6.06218i	0.117123	−	0.202863i
$894$		0			0
$895$	15.0000			0.501395
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	−0.988816	−	0.149140i	$-0.952349\pi$
$907$	−11.0000	+	19.0526i	−0.365249	+	0.632630i	0.623567	+	0.781770i	$0.285683\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	42.0000			1.39152			0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000			1.39152			0.695761	+	0.718273i	$0.255067\pi$
$912$		0			0
$913$	24.0000	−	41.5692i	0.794284	−	1.37574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.5000	−	6.06218i	−0.577901	−	0.200191i
$918$		0			0
$919$	−10.0000	−	17.3205i	−0.329870	−	0.571351i	0.652616	−	0.757689i	$-0.273671\pi$
$919$	−10.0000	−	17.3205i	−0.329870	−	0.571351i	−0.982486	+	0.186338i	$0.940338\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.00000			0.197492
$924$		0			0
$925$	3.00000			0.0986394
$926$		0			0
$927$		0			0
$928$		0			0
$929$	13.5000	+	23.3827i	0.442921	+	0.767161i	0.997905	−	0.0646999i	$-0.0206090\pi$
$929$	13.5000	+	23.3827i	0.442921	+	0.767161i	−0.554984	+	0.831861i	$0.687276\pi$
$930$		0			0
$931$	45.5000	−	18.1865i	1.49120	−	0.596040i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−20.0000			−0.653372			−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000			−0.653372			−0.326686	+	0.945133i	$0.605932\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9.00000	+	15.5885i	−0.293392	+	0.508169i	−0.974609	−	0.223912i	$-0.928117\pi$
$941$	−9.00000	+	15.5885i	−0.293392	+	0.508169i	0.681218	+	0.732081i	$0.261451\pi$
$942$		0			0
$943$	−7.50000	−	12.9904i	−0.244234	−	0.423025i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	19.0000	+	32.9090i	0.617417	+	1.06940i	0.989955	+	0.141381i	$0.0451542\pi$
$947$	19.0000	+	32.9090i	0.617417	+	1.06940i	−0.372538	+	0.928017i	$0.621512\pi$
$948$		0			0
$949$	7.00000	−	12.1244i	0.227230	−	0.393573i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−4.00000			−0.129573			−0.0647864	−	0.997899i	$-0.520637\pi$
$953$	−4.00000			−0.129573			−0.0647864	+	0.997899i	$0.520637\pi$
$954$		0			0
$955$	−11.0000	+	19.0526i	−0.355952	+	0.616526i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−1.00000	−	5.19615i	−0.0322917	−	0.167793i
$960$		0			0
$961$	−2.50000	−	4.33013i	−0.0806452	−	0.139682i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.00000			−0.0643823
$966$		0			0
$967$	4.00000			0.128631			0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000			0.128631			0.0643157	+	0.997930i	$0.479514\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.50000	−	7.79423i	−0.144412	−	0.250129i	0.784741	−	0.619823i	$-0.212796\pi$
$971$	−4.50000	−	7.79423i	−0.144412	−	0.250129i	−0.929153	+	0.369694i	$0.879462\pi$
$972$		0			0
$973$	8.00000	−	6.92820i	0.256468	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.0000	−	46.7654i	0.863807	−	1.49616i	−0.00442082	−	0.999990i	$-0.501407\pi$
$977$	27.0000	−	46.7654i	0.863807	−	1.49616i	0.868227	−	0.496167i	$-0.165259\pi$
$978$		0			0
$979$	18.0000			0.575282
$980$		0			0
$981$		0			0
$982$		0			0
$983$	10.5000	−	18.1865i	0.334898	−	0.580060i	−0.648567	−	0.761157i	$-0.724631\pi$
$983$	10.5000	−	18.1865i	0.334898	−	0.580060i	0.983465	+	0.181097i	$0.0579648\pi$
$984$		0			0
$985$	7.50000	+	12.9904i	0.238970	+	0.413908i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20.0000	−	34.6410i	−0.635963	−	1.10152i
$990$		0			0
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	−0.849705	−	0.527258i	$-0.823220\pi$
$991$	1.00000	−	1.73205i	0.0317660	−	0.0550204i	0.881471	+	0.472237i	$0.156554\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−24.0000			−0.760851
$996$		0			0
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	0.229135	+	0.973395i	$0.426410\pi$
$997$	−23.0000	+	39.8372i	−0.728417	+	1.26166i	−0.957552	+	0.288261i	$0.906923\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.bi.b.361.1		2
3.2	odd	2		840.2.bg.d.361.1	yes	2
7.2	even	3	inner	2520.2.bi.b.1801.1		2
12.11	even	2		1680.2.bg.i.1201.1		2
21.2	odd	6		840.2.bg.d.121.1	✓	2
21.11	odd	6		5880.2.a.f.1.1		1
21.17	even	6		5880.2.a.bh.1.1		1
84.23	even	6		1680.2.bg.i.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.bg.d.121.1	✓	2	21.2	odd	6
840.2.bg.d.361.1	yes	2	3.2	odd	2
1680.2.bg.i.961.1		2	84.23	even	6
1680.2.bg.i.1201.1		2	12.11	even	2
2520.2.bi.b.361.1		2	1.1	even	1	trivial
2520.2.bi.b.1801.1		2	7.2	even	3	inner
5880.2.a.f.1.1		1	21.11	odd	6
5880.2.a.bh.1.1		1	21.17	even	6

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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2520.bi (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-2.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{11} +1.00000 q^{13} +(3.50000 + 6.06218i) q^{19} +(-2.50000 - 4.33013i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-3.00000 + 5.19615i) q^{31} +(2.50000 + 0.866025i) q^{35} +(-1.50000 - 2.59808i) q^{37} +3.00000 q^{41} +8.00000 q^{43} +(-0.500000 - 0.866025i) q^{47} +(1.00000 - 6.92820i) q^{49} +(2.50000 - 4.33013i) q^{53} -3.00000 q^{55} +(-2.00000 + 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-0.500000 - 0.866025i) q^{65} +6.00000 q^{71} +(7.00000 - 12.1244i) q^{73} +(1.50000 + 7.79423i) q^{77} +(8.00000 + 13.8564i) q^{79} +16.0000 q^{83} +(3.00000 + 5.19615i) q^{89} +(-2.00000 + 1.73205i) q^{91} +(3.50000 - 6.06218i) q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - q^5 - 4 * q^7	\( 2 q - q^{5} - 4 q^{7} + 3 q^{11} + 2 q^{13} + 7 q^{19} - 5 q^{23} - q^{25} - 6 q^{31} + 5 q^{35} - 3 q^{37} + 6 q^{41} + 16 q^{43} - q^{47} + 2 q^{49} + 5 q^{53} - 6 q^{55} - 4 q^{59} + 8 q^{61} - q^{65} + 12 q^{71} + 14 q^{73} + 3 q^{77} + 16 q^{79} + 32 q^{83} + 6 q^{89} - 4 q^{91} + 7 q^{95} + 32 q^{97}+O(q^{100}) \) 2 * q - q^5 - 4 * q^7 + 3 * q^11 + 2 * q^13 + 7 * q^19 - 5 * q^23 - q^25 - 6 * q^31 + 5 * q^35 - 3 * q^37 + 6 * q^41 + 16 * q^43 - q^47 + 2 * q^49 + 5 * q^53 - 6 * q^55 - 4 * q^59 + 8 * q^61 - q^65 + 12 * q^71 + 14 * q^73 + 3 * q^77 + 16 * q^79 + 32 * q^83 + 6 * q^89 - 4 * q^91 + 7 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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