LMFDB - Embedded newform 2520.2.bi.f.1801.1

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			1801.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2520.1801
Dual form			2520.2.bi.f.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.500000 + 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} -6.00000 q^{41} -1.00000 q^{43} +(5.00000 - 8.66025i) q^{47} +(5.50000 - 4.33013i) q^{49} -2.00000 q^{55} +(3.00000 - 5.19615i) q^{61} +(-2.50000 + 4.33013i) q^{65} +(7.50000 + 12.9904i) q^{67} +6.00000 q^{71} +(1.50000 + 2.59808i) q^{73} +(4.00000 + 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} -6.00000 q^{83} -4.00000 q^{85} +(12.5000 - 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q - q^5 + 5 * q^7	$ 2 q - q^{5} + 5 q^{7} + 2 q^{11} + 10 q^{13} + 4 q^{17} - 3 q^{19} - 4 q^{23} - q^{25} - 8 q^{29} + q^{31} - q^{35} + q^{37} - 12 q^{41} - 2 q^{43} + 10 q^{47} + 11 q^{49} - 4 q^{55} + 6 q^{61} - 5 q^{65} + 15 q^{67} + 12 q^{71} + 3 q^{73} + 8 q^{77} - 9 q^{79} - 12 q^{83} - 8 q^{85} + 25 q^{91} - 3 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - q^5 + 5 * q^7 + 2 * q^11 + 10 * q^13 + 4 * q^17 - 3 * q^19 - 4 * q^23 - q^25 - 8 * q^29 + q^31 - q^35 + q^37 - 12 * q^41 - 2 * q^43 + 10 * q^47 + 11 * q^49 - 4 * q^55 + 6 * q^61 - 5 * q^65 + 15 * q^67 + 12 * q^71 + 3 * q^73 + 8 * q^77 - 9 * q^79 - 12 * q^83 - 8 * q^85 + 25 * q^91 - 3 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$7$	2.50000	−	0.866025i	0.944911	−	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	0.999853	−	0.0171533i	$-0.00546033\pi$
$17$	2.00000	+	3.46410i	0.485071	+	0.840168i	−0.514782	+	0.857321i	$0.672127\pi$
$18$		0			0
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	$-0.945157\pi$
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	0.641071	+	0.767482i	$0.278491\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		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
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	0.907428	−	0.420208i	$-0.138043\pi$
$31$	0.500000	+	0.866025i	0.0898027	+	0.155543i	−0.817625	+	0.575751i	$0.804710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.500000	+	2.59808i	−0.0845154	+	0.439155i
$36$		0			0
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	0.500000	−	0.866025i	0.0821995	−	0.142374i	0.904194	+	0.427121i	$0.140472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.00000	−	8.66025i	0.729325	−	1.26323i	−0.227844	−	0.973698i	$-0.573168\pi$
$47$	5.00000	−	8.66025i	0.729325	−	1.26323i	0.957169	−	0.289530i	$-0.0934991\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$53$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	−0.607535	−	0.794293i	$-0.707841\pi$
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	0.991645	+	0.128994i	$0.0411748\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.50000	+	4.33013i	−0.310087	+	0.537086i
$66$		0			0
$67$	7.50000	+	12.9904i	0.916271	+	1.58703i	0.805030	+	0.593234i	$0.202149\pi$
$67$	7.50000	+	12.9904i	0.916271	+	1.58703i	0.111241	+	0.993793i	$0.464517\pi$
$68$		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
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	0.940356	−	0.340193i	$-0.110493\pi$
$73$	1.50000	+	2.59808i	0.175562	+	0.304082i	−0.764794	+	0.644275i	$0.777159\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000	+	3.46410i	0.455842	+	0.394771i
$78$		0			0
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	0.493684	+	0.869641i	$0.335650\pi$
$79$	−4.50000	+	7.79423i	−0.506290	+	0.876919i	−0.999974	+	0.00727784i	$0.997683\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000			−0.658586			−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000			−0.658586			−0.329293	+	0.944228i	$0.606810\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	12.5000	−	4.33013i	1.31036	−	0.453921i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.50000	−	2.59808i	−0.153897	−	0.266557i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.833143	+	0.553058i	$0.186539\pi$
$101$	9.00000	+	15.5885i	0.895533	+	1.55111i	0.0623905	+	0.998052i	$0.480128\pi$
$102$		0			0
$103$	−3.50000	+	6.06218i	−0.344865	+	0.597324i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	−3.50000	+	6.06218i	−0.344865	+	0.597324i	0.640464	+	0.767988i	$0.278742\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	−0.753010	−	0.658009i	$-0.771399\pi$
$107$	2.00000	−	3.46410i	0.193347	−	0.334887i	0.946357	+	0.323122i	$0.104732\pi$
$108$		0			0
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	0.999512	+	0.0312328i	$0.00994332\pi$
$109$	5.50000	+	9.52628i	0.526804	+	0.912452i	−0.472708	+	0.881219i	$0.656723\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$	−2.00000	−	3.46410i	−0.186501	−	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	8.00000	+	6.92820i	0.733359	+	0.635107i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−17.0000			−1.50851			−0.754253	−	0.656584i	$-0.772001\pi$
$127$	−17.0000			−1.50851			−0.754253	+	0.656584i	$0.772001\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.00000	+	15.5885i	−0.786334	+	1.36197i	0.141865	+	0.989886i	$0.454690\pi$
$131$	−9.00000	+	15.5885i	−0.786334	+	1.36197i	−0.928199	+	0.372084i	$0.878643\pi$
$132$		0			0
$133$	−1.50000	+	7.79423i	−0.130066	+	0.675845i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	−17.0000			−1.44192			−0.720961	−	0.692976i	$-0.756299\pi$
$139$	−17.0000			−1.44192			−0.720961	+	0.692976i	$0.756299\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	5.00000	+	8.66025i	0.418121	+	0.724207i
$144$		0			0
$145$	2.00000	−	3.46410i	0.166091	−	0.287678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	−0.0870170	−	0.996207i	$-0.527733\pi$
$149$	10.0000	−	17.3205i	0.819232	−	1.41895i	0.906249	−	0.422744i	$-0.138933\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.00000			−0.0803219
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000	+	10.3923i	−0.157622	+	0.819028i
$162$		0			0
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	0.146772	+	0.989170i	$0.453112\pi$
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	−0.930033	+	0.367477i	$0.880222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000			−0.154765			−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000			−0.154765			−0.0773823	+	0.997001i	$0.524656\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	−0.931984	−	0.362500i	$-0.881923\pi$
$173$	−2.00000	+	3.46410i	−0.152057	+	0.263371i	0.779926	+	0.625871i	$0.215256\pi$
$174$		0			0
$175$	−2.00000	−	1.73205i	−0.151186	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	0.900975	−	0.433872i	$-0.142853\pi$
$179$	1.00000	+	1.73205i	0.0747435	+	0.129460i	−0.826231	+	0.563331i	$0.809520\pi$
$180$		0			0
$181$	25.0000			1.85824			0.929118	−	0.369784i	$-0.120568\pi$
$181$	25.0000			1.85824			0.929118	+	0.369784i	$0.120568\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.500000	+	0.866025i	0.0367607	+	0.0636715i
$186$		0			0
$187$	−4.00000	+	6.92820i	−0.292509	+	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		0			0
$193$	−5.50000	−	9.52628i	−0.395899	−	0.685717i	0.597317	−	0.802005i	$-0.296234\pi$
$193$	−5.50000	−	9.52628i	−0.395899	−	0.685717i	−0.993215	+	0.116289i	$0.962900\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	−0.996850	−	0.0793045i	$-0.974730\pi$
$199$	−8.00000	−	13.8564i	−0.567105	−	0.982255i	0.429745	−	0.902950i	$-0.358603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−10.0000	+	3.46410i	−0.701862	+	0.243132i
$204$		0			0
$205$	3.00000	−	5.19615i	0.209529	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	12.0000			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	0.500000	−	0.866025i	0.0340997	−	0.0590624i
$216$		0			0
$217$	2.00000	+	1.73205i	0.135769	+	0.117579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.0000	+	17.3205i	0.672673	+	1.16510i
$222$		0			0
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−5.00000	−	8.66025i	−0.331862	−	0.574801i	0.651015	−	0.759065i	$-0.274343\pi$
$227$	−5.00000	−	8.66025i	−0.331862	−	0.574801i	−0.982877	+	0.184263i	$0.941010\pi$
$228$		0			0
$229$	−12.5000	+	21.6506i	−0.826023	+	1.43071i	0.0751115	+	0.997175i	$0.476069\pi$
$229$	−12.5000	+	21.6506i	−0.826023	+	1.43071i	−0.901135	+	0.433539i	$0.857265\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	−0.404674	−	0.914461i	$-0.632615\pi$
$233$	9.00000	−	15.5885i	0.589610	−	1.02123i	0.994283	−	0.106773i	$-0.0340517\pi$
$234$		0			0
$235$	5.00000	+	8.66025i	0.326164	+	0.564933i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.0000			0.905585			0.452792	−	0.891616i	$-0.350428\pi$
$239$	14.0000			0.905585			0.452792	+	0.891616i	$0.350428\pi$
$240$		0			0
$241$	1.00000	+	1.73205i	0.0644157	+	0.111571i	0.896435	−	0.443176i	$-0.146148\pi$
$241$	1.00000	+	1.73205i	0.0644157	+	0.111571i	−0.832019	+	0.554747i	$0.812815\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.00000	+	6.92820i	0.0638877	+	0.442627i
$246$		0			0
$247$	−7.50000	+	12.9904i	−0.477214	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−8.00000			−0.502956
$254$		0			0
$255$		0			0
$256$		0			0
$257$	15.0000	−	25.9808i	0.935674	−	1.62064i	0.162247	−	0.986750i	$-0.448126\pi$
$257$	15.0000	−	25.9808i	0.935674	−	1.62064i	0.773427	−	0.633885i	$-0.218541\pi$
$258$		0			0
$259$	0.500000	−	2.59808i	0.0310685	−	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.00000	+	3.46410i	0.123325	+	0.213606i	0.921077	−	0.389380i	$-0.127311\pi$
$263$	2.00000	+	3.46410i	0.123325	+	0.213606i	−0.797752	+	0.602986i	$0.793977\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.00000	+	12.1244i	0.426798	+	0.739235i	0.996586	−	0.0825561i	$-0.0263084\pi$
$269$	7.00000	+	12.1244i	0.426798	+	0.739235i	−0.569789	+	0.821791i	$0.692975\pi$
$270$		0			0
$271$	4.00000	−	6.92820i	0.242983	−	0.420858i	−0.718580	−	0.695444i	$-0.755208\pi$
$271$	4.00000	−	6.92820i	0.242983	−	0.420858i	0.961563	+	0.274586i	$0.0885408\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.00000	−	1.73205i	0.0603023	−	0.104447i
$276$		0			0
$277$	11.5000	+	19.9186i	0.690968	+	1.19679i	0.971521	+	0.236953i	$0.0761488\pi$
$277$	11.5000	+	19.9186i	0.690968	+	1.19679i	−0.280553	+	0.959839i	$0.590518\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.00000			−0.477240			−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000			−0.477240			−0.238620	+	0.971113i	$0.576695\pi$
$282$		0			0
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.797283	−	0.603606i	$-0.793730\pi$
$283$	−15.5000	−	26.8468i	−0.921379	−	1.59588i	−0.124096	−	0.992270i	$-0.539603\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−15.0000	+	5.19615i	−0.885422	+	0.306719i
$288$		0			0
$289$	0.500000	−	0.866025i	0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−32.0000			−1.86946			−0.934730	−	0.355359i	$-0.884359\pi$
$293$	−32.0000			−1.86946			−0.934730	+	0.355359i	$0.884359\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	+	17.3205i	−0.578315	+	1.00167i
$300$		0			0
$301$	−2.50000	+	0.866025i	−0.144098	+	0.0499169i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000	+	5.19615i	0.171780	+	0.297531i
$306$		0			0
$307$	21.0000			1.19853			0.599267	−	0.800549i	$-0.295459\pi$
$307$	21.0000			1.19853			0.599267	+	0.800549i	$0.295459\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	0.938460	−	0.345389i	$-0.112253\pi$
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	−0.768345	+	0.640036i	$0.778920\pi$
$312$		0			0
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	−0.710365	−	0.703833i	$-0.751470\pi$
$313$	4.50000	−	7.79423i	0.254355	−	0.440556i	0.964720	+	0.263278i	$0.0848035\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	−0.302777	−	0.953062i	$-0.597914\pi$
$317$	12.0000	−	20.7846i	0.673987	−	1.16738i	0.976764	−	0.214318i	$-0.0687530\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	−2.50000	−	4.33013i	−0.138675	−	0.240192i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	5.00000	−	25.9808i	0.275659	−	1.43237i
$330$		0			0
$331$	17.5000	−	30.3109i	0.961887	−	1.66604i	0.244131	−	0.969742i	$-0.421497\pi$
$331$	17.5000	−	30.3109i	0.961887	−	1.66604i	0.717756	−	0.696295i	$-0.245169\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.0000			−0.819538
$336$		0			0
$337$	−27.0000			−1.47078			−0.735392	−	0.677642i	$-0.763002\pi$
$337$	−27.0000			−1.47078			−0.735392	+	0.677642i	$0.763002\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.00000	+	1.73205i	−0.0541530	+	0.0937958i
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	0.658824	−	0.752297i	$-0.271054\pi$
$347$	−6.00000	−	10.3923i	−0.322097	−	0.557888i	−0.980921	+	0.194409i	$0.937721\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	−0.971208	−	0.238233i	$-0.923432\pi$
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	0.279288	−	0.960207i	$-0.409902\pi$
$354$		0			0
$355$	−3.00000	+	5.19615i	−0.159223	+	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	−0.663123	−	0.748511i	$-0.730769\pi$
$359$	6.00000	−	10.3923i	0.316668	−	0.548485i	0.979791	+	0.200026i	$0.0641026\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000			−0.157027
$366$		0			0
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	0.992776	+	0.119982i	$0.0382835\pi$
$367$	11.5000	+	19.9186i	0.600295	+	1.03974i	−0.392481	+	0.919760i	$0.628383\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−13.5000	+	23.3827i	−0.699004	+	1.21071i	0.269809	+	0.962914i	$0.413039\pi$
$373$	−13.5000	+	23.3827i	−0.699004	+	1.21071i	−0.968812	+	0.247796i	$0.920294\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0000			−1.03005
$378$		0			0
$379$	25.0000			1.28416			0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000			1.28416			0.642082	+	0.766636i	$0.278071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	0.377531	+	0.925997i	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$		0			0
$385$	−5.00000	+	1.73205i	−0.254824	+	0.0882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$	−16.0000			−0.809155
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.50000	−	7.79423i	−0.226420	−	0.392170i
$396$		0			0
$397$	14.5000	−	25.1147i	0.727734	−	1.26047i	−0.230105	−	0.973166i	$-0.573907\pi$
$397$	14.5000	−	25.1147i	0.727734	−	1.26047i	0.957839	−	0.287307i	$-0.0927599\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4.00000	−	6.92820i	0.199750	−	0.345978i	−0.748697	−	0.662912i	$-0.769320\pi$
$401$	4.00000	−	6.92820i	0.199750	−	0.345978i	0.948447	+	0.316934i	$0.102654\pi$
$402$		0			0
$403$	2.50000	+	4.33013i	0.124534	+	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.00000			0.0991363
$408$		0			0
$409$	−14.5000	−	25.1147i	−0.716979	−	1.24184i	−0.962191	−	0.272374i	$-0.912191\pi$
$409$	−14.5000	−	25.1147i	−0.716979	−	1.24184i	0.245212	−	0.969469i	$-0.421142\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3.00000	−	5.19615i	0.147264	−	0.255069i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−26.0000			−1.27018			−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000			−1.27018			−0.635092	+	0.772437i	$0.719038\pi$
$420$		0			0
$421$	13.0000			0.633581			0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000			0.633581			0.316791	+	0.948495i	$0.397395\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	−	3.46410i	0.0970143	−	0.168034i
$426$		0			0
$427$	3.00000	−	15.5885i	0.145180	−	0.754378i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−13.0000	−	22.5167i	−0.626188	−	1.08459i	−0.988310	−	0.152459i	$-0.951281\pi$
$431$	−13.0000	−	22.5167i	−0.626188	−	1.08459i	0.362122	−	0.932131i	$-0.382052\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.00000	−	10.3923i	−0.287019	−	0.497131i
$438$		0			0
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	−0.754642	−	0.656136i	$-0.772190\pi$
$439$	4.00000	−	6.92820i	0.190910	−	0.330665i	0.945552	+	0.325471i	$0.105523\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	−0.426414	−	0.904528i	$-0.640223\pi$
$443$	12.0000	−	20.7846i	0.570137	−	0.987507i	0.996551	−	0.0829786i	$-0.0264433\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.0000			−1.60456			−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000			−1.60456			−0.802280	+	0.596948i	$0.796380\pi$
$450$		0			0
$451$	−6.00000	−	10.3923i	−0.282529	−	0.489355i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.50000	+	12.9904i	−0.117202	+	0.608998i
$456$		0			0
$457$	13.5000	−	23.3827i	0.631503	−	1.09380i	−0.355741	−	0.934585i	$-0.615772\pi$
$457$	13.5000	−	23.3827i	0.631503	−	1.09380i	0.987245	−	0.159211i	$-0.0508951\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−12.0000			−0.558896			−0.279448	−	0.960161i	$-0.590151\pi$
$461$	−12.0000			−0.558896			−0.279448	+	0.960161i	$0.590151\pi$
$462$		0			0
$463$	−31.0000			−1.44069			−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000			−1.44069			−0.720346	+	0.693615i	$0.756017\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.0000	+	36.3731i	−0.971764	+	1.68314i	−0.281539	+	0.959550i	$0.590845\pi$
$467$	−21.0000	+	36.3731i	−0.971764	+	1.68314i	−0.690225	+	0.723595i	$0.742488\pi$
$468$		0			0
$469$	30.0000	+	25.9808i	1.38527	+	1.19968i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.00000	−	1.73205i	−0.0459800	−	0.0796398i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	0.998392	+	0.0566937i	$0.0180558\pi$
$479$	12.0000	+	20.7846i	0.548294	+	0.949673i	−0.450098	+	0.892979i	$0.648611\pi$
$480$		0			0
$481$	2.50000	−	4.33013i	0.113990	−	0.197437i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.00000	+	1.73205i	−0.0454077	+	0.0786484i
$486$		0			0
$487$	11.5000	+	19.9186i	0.521115	+	0.902597i	0.999698	+	0.0245553i	$0.00781698\pi$
$487$	11.5000	+	19.9186i	0.521115	+	0.902597i	−0.478584	+	0.878042i	$0.658850\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	−8.00000	−	13.8564i	−0.360302	−	0.624061i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	15.0000	−	5.19615i	0.672842	−	0.233079i
$498$		0			0
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	−0.933670	−	0.358134i	$-0.883413\pi$
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	0.776989	+	0.629515i	$0.216746\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−2.00000			−0.0891756			−0.0445878	−	0.999005i	$-0.514197\pi$
$503$	−2.00000			−0.0891756			−0.0445878	+	0.999005i	$0.514197\pi$
$504$		0			0
$505$	−18.0000			−0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	6.00000	+	5.19615i	0.265424	+	0.229864i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.50000	−	6.06218i	−0.154228	−	0.267131i
$516$		0			0
$517$	20.0000			0.879599
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	0.704137	−	0.710064i	$-0.251334\pi$
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	−0.967002	+	0.254769i	$0.918001\pi$
$522$		0			0
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	−0.972421	−	0.233233i	$-0.925070\pi$
$523$	−6.50000	+	11.2583i	−0.284225	+	0.492292i	0.688196	+	0.725525i	$0.258403\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−2.00000	+	3.46410i	−0.0871214	+	0.150899i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−30.0000			−1.29944
$534$		0			0
$535$	2.00000	+	3.46410i	0.0864675	+	0.149766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	13.0000	+	5.19615i	0.559950	+	0.223814i
$540$		0			0
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	0.623404	+	0.781900i	$0.285749\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.0000			−0.471188
$546$		0			0
$547$	20.0000			0.855138			0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000			0.855138			0.427569	+	0.903983i	$0.359370\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.00000	−	10.3923i	0.255609	−	0.442727i
$552$		0			0
$553$	−4.50000	+	23.3827i	−0.191359	+	0.994333i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.00000	−	8.66025i	−0.210725	−	0.364986i	0.741217	−	0.671266i	$-0.234249\pi$
$563$	−5.00000	−	8.66025i	−0.210725	−	0.364986i	−0.951942	+	0.306280i	$0.900916\pi$
$564$		0			0
$565$	−9.00000	+	15.5885i	−0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	0.500000	+	0.866025i	0.0209243	+	0.0362420i	0.876298	−	0.481770i	$-0.160006\pi$
$571$	0.500000	+	0.866025i	0.0209243	+	0.0362420i	−0.855374	+	0.518012i	$0.826672\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.00000			0.166812
$576$		0			0
$577$	−23.5000	−	40.7032i	−0.978318	−	1.69450i	−0.668521	−	0.743693i	$-0.733072\pi$
$577$	−23.5000	−	40.7032i	−0.978318	−	1.69450i	−0.309797	−	0.950803i	$-0.600261\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−15.0000	+	5.19615i	−0.622305	+	0.215573i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−8.00000			−0.330195			−0.165098	−	0.986277i	$-0.552794\pi$
$587$	−8.00000			−0.330195			−0.165098	+	0.986277i	$0.552794\pi$
$588$		0			0
$589$	−3.00000			−0.123613
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−11.0000	+	19.0526i	−0.451716	+	0.782395i	−0.998493	−	0.0548835i	$-0.982521\pi$
$593$	−11.0000	+	19.0526i	−0.451716	+	0.782395i	0.546777	+	0.837278i	$0.315855\pi$
$594$		0			0
$595$	−10.0000	+	3.46410i	−0.409960	+	0.142014i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2.00000	+	3.46410i	0.0817178	+	0.141539i	0.903988	−	0.427558i	$-0.140626\pi$
$599$	2.00000	+	3.46410i	0.0817178	+	0.141539i	−0.822270	+	0.569097i	$0.807293\pi$
$600$		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
$605$	3.50000	+	6.06218i	0.142295	+	0.246463i
$606$		0			0
$607$	−8.50000	+	14.7224i	−0.345004	+	0.597565i	−0.985355	−	0.170518i	$-0.945456\pi$
$607$	−8.50000	+	14.7224i	−0.345004	+	0.597565i	0.640350	+	0.768083i	$0.278789\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	25.0000	−	43.3013i	1.01139	−	1.75178i
$612$		0			0
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	0.747208	−	0.664590i	$-0.231394\pi$
$613$	−5.00000	−	8.66025i	−0.201948	−	0.349784i	−0.949156	+	0.314806i	$0.898061\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.0000			0.885687			0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000			0.885687			0.442843	+	0.896599i	$0.353970\pi$
$618$		0			0
$619$	−0.500000	−	0.866025i	−0.0200967	−	0.0348085i	0.855802	−	0.517303i	$-0.173064\pi$
$619$	−0.500000	−	0.866025i	−0.0200967	−	0.0348085i	−0.875899	+	0.482495i	$0.839731\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.00000			0.159490
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.50000	−	14.7224i	0.337312	−	0.584242i
$636$		0			0
$637$	27.5000	−	21.6506i	1.08959	−	0.857829i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.00000	−	3.46410i	−0.0789953	−	0.136824i	0.823821	−	0.566849i	$-0.191838\pi$
$641$	−2.00000	−	3.46410i	−0.0789953	−	0.136824i	−0.902817	+	0.430026i	$0.858505\pi$
$642$		0			0
$643$	−49.0000			−1.93237			−0.966186	−	0.257847i	$-0.916987\pi$
$643$	−49.0000			−1.93237			−0.966186	+	0.257847i	$0.916987\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	−0.901464	−	0.432855i	$-0.857506\pi$
$647$	−21.0000	−	36.3731i	−0.825595	−	1.42997i	0.0758684	−	0.997118i	$-0.475827\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7.00000	+	12.1244i	−0.273931	+	0.474463i	−0.969865	−	0.243643i	$-0.921657\pi$
$653$	−7.00000	+	12.1244i	−0.273931	+	0.474463i	0.695934	+	0.718106i	$0.254991\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$	6.50000	+	11.2583i	0.252821	+	0.437898i	0.964301	−	0.264807i	$-0.0853084\pi$
$661$	6.50000	+	11.2583i	0.252821	+	0.437898i	−0.711481	+	0.702706i	$0.751975\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	−	5.19615i	−0.232670	−	0.201498i
$666$		0			0
$667$	8.00000	−	13.8564i	0.309761	−	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	−0.279453	−	0.960159i	$-0.590153\pi$
$677$	18.0000	−	31.1769i	0.691796	−	1.19823i	0.971249	−	0.238067i	$-0.0765137\pi$
$678$		0			0
$679$	5.00000	−	1.73205i	0.191882	−	0.0664700i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.00000	−	13.8564i	−0.306111	−	0.530201i	0.671397	−	0.741098i	$-0.265695\pi$
$683$	−8.00000	−	13.8564i	−0.306111	−	0.530201i	−0.977508	+	0.210898i	$0.932361\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	7.50000	−	12.9904i	0.285313	−	0.494177i	−0.687372	−	0.726306i	$-0.741236\pi$
$691$	7.50000	−	12.9904i	0.285313	−	0.494177i	0.972685	+	0.232128i	$0.0745690\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.50000	−	14.7224i	0.322423	−	0.558454i
$696$		0			0
$697$	−12.0000	−	20.7846i	−0.454532	−	0.787273i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	36.0000			1.35970			0.679851	−	0.733351i	$-0.262045\pi$
$701$	36.0000			1.35970			0.679851	+	0.733351i	$0.262045\pi$
$702$		0			0
$703$	1.50000	+	2.59808i	0.0565736	+	0.0979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	36.0000	+	31.1769i	1.35392	+	1.17253i
$708$		0			0
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	−0.347325	−	0.937745i	$-0.612910\pi$
$709$	17.0000	−	29.4449i	0.638448	−	1.10583i	0.985773	−	0.168080i	$-0.0537568\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.00000			−0.149801
$714$		0			0
$715$	−10.0000			−0.373979
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	−0.647965	−	0.761670i	$-0.724380\pi$
$719$	9.00000	−	15.5885i	0.335643	−	0.581351i	0.983608	+	0.180319i	$0.0577130\pi$
$720$		0			0
$721$	−3.50000	+	18.1865i	−0.130347	+	0.677302i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2.00000	+	3.46410i	0.0742781	+	0.128654i
$726$		0			0
$727$	−11.0000			−0.407967			−0.203984	−	0.978974i	$-0.565389\pi$
$727$	−11.0000			−0.407967			−0.203984	+	0.978974i	$0.565389\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.00000	−	3.46410i	−0.0739727	−	0.128124i
$732$		0			0
$733$	−14.5000	+	25.1147i	−0.535570	+	0.927634i	0.463566	+	0.886062i	$0.346570\pi$
$733$	−14.5000	+	25.1147i	−0.535570	+	0.927634i	−0.999136	+	0.0415715i	$0.986764\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−15.0000	+	25.9808i	−0.552532	+	0.957014i
$738$		0			0
$739$	−17.5000	−	30.3109i	−0.643748	−	1.11500i	−0.984589	−	0.174883i	$-0.944045\pi$
$739$	−17.5000	−	30.3109i	−0.643748	−	1.11500i	0.340841	−	0.940121i	$-0.389288\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22.0000			−0.807102			−0.403551	−	0.914957i	$-0.632224\pi$
$743$	−22.0000			−0.807102			−0.403551	+	0.914957i	$0.632224\pi$
$744$		0			0
$745$	10.0000	+	17.3205i	0.366372	+	0.634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	2.00000	−	10.3923i	0.0730784	−	0.379727i
$750$		0			0
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	−0.722718	−	0.691143i	$-0.757107\pi$
$751$	6.50000	−	11.2583i	0.237188	−	0.410822i	0.959906	+	0.280321i	$0.0904408\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000			0.291150
$756$		0			0
$757$	−54.0000			−1.96266			−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−54.0000			−1.96266			−0.981332	+	0.192323i	$0.938398\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.00000	−	3.46410i	0.0724999	−	0.125574i	−0.827496	−	0.561471i	$-0.810236\pi$
$761$	2.00000	−	3.46410i	0.0724999	−	0.125574i	0.899996	+	0.435897i	$0.143569\pi$
$762$		0			0
$763$	22.0000	+	19.0526i	0.796453	+	0.689749i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−41.0000			−1.47850			−0.739249	−	0.673432i	$-0.764819\pi$
$769$	−41.0000			−1.47850			−0.739249	+	0.673432i	$0.764819\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	0.761987	−	0.647592i	$-0.224224\pi$
$773$	−5.00000	−	8.66025i	−0.179838	−	0.311488i	−0.941825	+	0.336104i	$0.890891\pi$
$774$		0			0
$775$	0.500000	−	0.866025i	0.0179605	−	0.0311086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.00000	−	15.5885i	0.322458	−	0.558514i
$780$		0			0
$781$	6.00000	+	10.3923i	0.214697	+	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.00000			0.0713831
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	45.0000	−	15.5885i	1.60002	−	0.554262i
$792$		0			0
$793$	15.0000	−	25.9808i	0.532666	−	0.922604i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.0000			−0.991811			−0.495905	−	0.868377i	$-0.665164\pi$
$797$	−28.0000			−0.991811			−0.495905	+	0.868377i	$0.665164\pi$
$798$		0			0
$799$	40.0000			1.41510
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	+	5.19615i	−0.105868	+	0.183368i
$804$		0			0
$805$	−8.00000	−	6.92820i	−0.281963	−	0.244187i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	0.605269	−	0.796021i	$-0.293065\pi$
$809$	−11.0000	−	19.0526i	−0.386739	−	0.669852i	−0.992009	+	0.126168i	$0.959732\pi$
$810$		0			0
$811$	−24.0000			−0.842754			−0.421377	−	0.906886i	$-0.638453\pi$
$811$	−24.0000			−0.842754			−0.421377	+	0.906886i	$0.638453\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10.0000	−	17.3205i	−0.350285	−	0.606711i
$816$		0			0
$817$	1.50000	−	2.59808i	0.0524784	−	0.0908952i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.0000	−	25.9808i	0.523504	−	0.906735i	−0.476122	−	0.879379i	$-0.657958\pi$
$821$	15.0000	−	25.9808i	0.523504	−	0.906735i	0.999626	−	0.0273557i	$-0.00870868\pi$
$822$		0			0
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	0.577474	−	0.816409i	$-0.304038\pi$
$823$	−12.0000	−	20.7846i	−0.418294	−	0.724506i	−0.995768	+	0.0919029i	$0.970705\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	−18.5000	−	32.0429i	−0.642532	−	1.11290i	−0.984866	−	0.173319i	$-0.944551\pi$
$829$	−18.5000	−	32.0429i	−0.642532	−	1.11290i	0.342334	−	0.939578i	$-0.388783\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	26.0000	+	10.3923i	0.900847	+	0.360072i
$834$		0			0
$835$	1.00000	−	1.73205i	0.0346064	−	0.0599401i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−20.0000			−0.690477			−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000			−0.690477			−0.345238	+	0.938515i	$0.612202\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−6.00000	+	10.3923i	−0.206406	+	0.357506i
$846$		0			0
$847$	3.50000	−	18.1865i	0.120261	−	0.624897i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000	+	3.46410i	0.0685591	+	0.118748i
$852$		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
$857$	−20.0000	−	34.6410i	−0.683187	−	1.18331i	−0.974003	−	0.226536i	$-0.927260\pi$
$857$	−20.0000	−	34.6410i	−0.683187	−	1.18331i	0.290816	−	0.956779i	$-0.406073\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.00000	−	12.1244i	0.238283	−	0.412718i	−0.721939	−	0.691957i	$-0.756749\pi$
$863$	7.00000	−	12.1244i	0.238283	−	0.412718i	0.960222	+	0.279239i	$0.0900822\pi$
$864$		0			0
$865$	−2.00000	−	3.46410i	−0.0680020	−	0.117783i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.0000			−0.610608
$870$		0			0
$871$	37.5000	+	64.9519i	1.27064	+	2.20081i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2.50000	−	0.866025i	0.0845154	−	0.0292770i
$876$		0			0
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	−0.723298	−	0.690536i	$-0.757375\pi$
$877$	7.00000	−	12.1244i	0.236373	−	0.409410i	0.959671	+	0.281126i	$0.0907079\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−12.0000			−0.404290			−0.202145	−	0.979356i	$-0.564791\pi$
$881$	−12.0000			−0.404290			−0.202145	+	0.979356i	$0.564791\pi$
$882$		0			0
$883$	−15.0000			−0.504790			−0.252395	−	0.967624i	$-0.581218\pi$
$883$	−15.0000			−0.504790			−0.252395	+	0.967624i	$0.581218\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	0.496340	+	0.868128i	$0.334677\pi$
$887$	−15.0000	+	25.9808i	−0.503651	+	0.872349i	−0.999991	+	0.00422062i	$0.998657\pi$
$888$		0			0
$889$	−42.5000	+	14.7224i	−1.42540	+	0.493775i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	15.0000	+	25.9808i	0.501956	+	0.869413i
$894$		0			0
$895$	−2.00000			−0.0668526
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.00000	−	3.46410i	−0.0667037	−	0.115534i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.5000	+	21.6506i	−0.415514	+	0.719691i
$906$		0			0
$907$	−8.50000	−	14.7224i	−0.282238	−	0.488850i	0.689698	−	0.724097i	$-0.257743\pi$
$907$	−8.50000	−	14.7224i	−0.282238	−	0.488850i	−0.971936	+	0.235247i	$0.924410\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$	−6.00000	−	10.3923i	−0.198571	−	0.343935i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.00000	+	46.7654i	−0.297206	+	1.54433i
$918$		0			0
$919$	−28.5000	+	49.3634i	−0.940128	+	1.62835i	−0.174905	+	0.984585i	$0.555962\pi$
$919$	−28.5000	+	49.3634i	−0.940128	+	1.62835i	−0.765224	+	0.643765i	$0.777372\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	30.0000			0.987462
$924$		0			0
$925$	−1.00000			−0.0328798
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.0000	+	32.9090i	−0.623370	+	1.07971i	0.365484	+	0.930818i	$0.380904\pi$
$929$	−19.0000	+	32.9090i	−0.623370	+	1.07971i	−0.988854	+	0.148890i	$0.952430\pi$
$930$		0			0
$931$	3.00000	+	20.7846i	0.0983210	+	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4.00000	−	6.92820i	−0.130814	−	0.226576i
$936$		0			0
$937$	39.0000			1.27407			0.637037	−	0.770833i	$-0.280160\pi$
$937$	39.0000			1.27407			0.637037	+	0.770833i	$0.280160\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$	12.0000	−	20.7846i	0.390774	−	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	−0.999896	−	0.0144491i	$-0.995401\pi$
$947$	−15.0000	+	25.9808i	−0.487435	+	0.844261i	0.512461	+	0.858710i	$0.328734\pi$
$948$		0			0
$949$	7.50000	+	12.9904i	0.243460	+	0.421686i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−48.0000			−1.55487			−0.777436	−	0.628962i	$-0.783480\pi$
$953$	−48.0000			−1.55487			−0.777436	+	0.628962i	$0.783480\pi$
$954$		0			0
$955$	9.00000	+	15.5885i	0.291233	+	0.504431i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24.0000	+	20.7846i	0.775000	+	0.671170i
$960$		0			0
$961$	15.0000	−	25.9808i	0.483871	−	0.838089i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0000			0.354103
$966$		0			0
$967$	5.00000			0.160789			0.0803946	−	0.996763i	$-0.474382\pi$
$967$	5.00000			0.160789			0.0803946	+	0.996763i	$0.474382\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16.0000	−	27.7128i	0.513464	−	0.889346i	−0.486414	−	0.873729i	$-0.661695\pi$
$971$	16.0000	−	27.7128i	0.513464	−	0.889346i	0.999878	−	0.0156178i	$-0.00497150\pi$
$972$		0			0
$973$	−42.5000	+	14.7224i	−1.36249	+	0.471979i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	0.986586	−	0.163242i	$-0.0521952\pi$
$977$	11.0000	+	19.0526i	0.351921	+	0.609545i	−0.634665	+	0.772787i	$0.718862\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	0.939992	+	0.341197i	$0.110832\pi$
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	−0.174511	+	0.984655i	$0.555834\pi$
$984$		0			0
$985$	−6.00000	+	10.3923i	−0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.00000	−	3.46410i	0.0635963	−	0.110152i
$990$		0			0
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	0.993364	−	0.115015i	$-0.0366917\pi$
$991$	12.5000	+	21.6506i	0.397076	+	0.687755i	−0.596288	+	0.802771i	$0.703358\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16.0000			0.507234
$996$		0			0
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	0.993213	−	0.116310i	$-0.0371066\pi$
$997$	12.5000	+	21.6506i	0.395879	+	0.685682i	−0.597334	+	0.801993i	$0.703773\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.2.bi.f.1801.1		2
3.2	odd	2		840.2.bg.f.121.1	✓	2
7.4	even	3	inner	2520.2.bi.f.361.1		2
12.11	even	2		1680.2.bg.f.961.1		2
21.2	odd	6		5880.2.a.e.1.1		1
21.5	even	6		5880.2.a.bg.1.1		1
21.11	odd	6		840.2.bg.f.361.1	yes	2
84.11	even	6		1680.2.bg.f.1201.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.bg.f.121.1	✓	2	3.2	odd	2
840.2.bg.f.361.1	yes	2	21.11	odd	6
1680.2.bg.f.961.1		2	12.11	even	2
1680.2.bg.f.1201.1		2	84.11	even	6
2520.2.bi.f.361.1		2	7.4	even	3	inner
2520.2.bi.f.1801.1		2	1.1	even	1	trivial
5880.2.a.e.1.1		1	21.2	odd	6
5880.2.a.bg.1.1		1	21.5	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)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(1.00000 + 1.73205i) q^{11} +5.00000 q^{13} +(2.00000 + 3.46410i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} -4.00000 q^{29} +(0.500000 + 0.866025i) q^{31} +(-0.500000 + 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} -6.00000 q^{41} -1.00000 q^{43} +(5.00000 - 8.66025i) q^{47} +(5.50000 - 4.33013i) q^{49} -2.00000 q^{55} +(3.00000 - 5.19615i) q^{61} +(-2.50000 + 4.33013i) q^{65} +(7.50000 + 12.9904i) q^{67} +6.00000 q^{71} +(1.50000 + 2.59808i) q^{73} +(4.00000 + 3.46410i) q^{77} +(-4.50000 + 7.79423i) q^{79} -6.00000 q^{83} -4.00000 q^{85} +(12.5000 - 4.33013i) q^{91} +(-1.50000 - 2.59808i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q - q^5 + 5 * q^7	\( 2 q - q^{5} + 5 q^{7} + 2 q^{11} + 10 q^{13} + 4 q^{17} - 3 q^{19} - 4 q^{23} - q^{25} - 8 q^{29} + q^{31} - q^{35} + q^{37} - 12 q^{41} - 2 q^{43} + 10 q^{47} + 11 q^{49} - 4 q^{55} + 6 q^{61} - 5 q^{65} + 15 q^{67} + 12 q^{71} + 3 q^{73} + 8 q^{77} - 9 q^{79} - 12 q^{83} - 8 q^{85} + 25 q^{91} - 3 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - q^5 + 5 * q^7 + 2 * q^11 + 10 * q^13 + 4 * q^17 - 3 * q^19 - 4 * q^23 - q^25 - 8 * q^29 + q^31 - q^35 + q^37 - 12 * q^41 - 2 * q^43 + 10 * q^47 + 11 * q^49 - 4 * q^55 + 6 * q^61 - 5 * q^65 + 15 * q^67 + 12 * q^71 + 3 * q^73 + 8 * q^77 - 9 * q^79 - 12 * q^83 - 8 * q^85 + 25 * q^91 - 3 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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