LMFDB - Embedded newform 2736.2.s.c.1873.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(577,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2736.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.s (of order $3$, degree $2$, not 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:	$21.8470699930$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1873.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2736.1873
Dual form			2736.2.s.c.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{7} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{7} +2.00000 q^{11} +(3.50000 + 6.06218i) q^{13} +(4.00000 - 1.73205i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(2.00000 + 3.46410i) q^{29} -1.00000 q^{31} +(-6.00000 + 10.3923i) q^{35} +7.00000 q^{37} +(2.00000 - 3.46410i) q^{41} +(3.50000 - 6.06218i) q^{43} +(-1.00000 - 1.73205i) q^{47} +2.00000 q^{49} +(-2.00000 - 3.46410i) q^{53} +(-4.00000 + 6.92820i) q^{55} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} -28.0000 q^{65} +(1.50000 + 2.59808i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(1.50000 - 2.59808i) q^{73} +6.00000 q^{77} +(2.50000 - 4.33013i) q^{79} -12.0000 q^{83} +(9.00000 + 15.5885i) q^{89} +(10.5000 + 18.1865i) q^{91} +(-2.00000 + 17.3205i) q^{95} +(-5.00000 + 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 + 6 * q^7	$ 2 q - 4 q^{5} + 6 q^{7} + 4 q^{11} + 7 q^{13} + 8 q^{19} + 4 q^{23} - 11 q^{25} + 4 q^{29} - 2 q^{31} - 12 q^{35} + 14 q^{37} + 4 q^{41} + 7 q^{43} - 2 q^{47} + 4 q^{49} - 4 q^{53} - 8 q^{55} + 6 q^{59} + q^{61} - 56 q^{65} + 3 q^{67} - 2 q^{71} + 3 q^{73} + 12 q^{77} + 5 q^{79} - 24 q^{83} + 18 q^{89} + 21 q^{91} - 4 q^{95} - 10 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 6 * q^7 + 4 * q^11 + 7 * q^13 + 8 * q^19 + 4 * q^23 - 11 * q^25 + 4 * q^29 - 2 * q^31 - 12 * q^35 + 14 * q^37 + 4 * q^41 + 7 * q^43 - 2 * q^47 + 4 * q^49 - 4 * q^53 - 8 * q^55 + 6 * q^59 + q^61 - 56 * q^65 + 3 * q^67 - 2 * q^71 + 3 * q^73 + 12 * q^77 + 5 * q^79 - 24 * q^83 + 18 * q^89 + 21 * q^91 - 4 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	3.00000			1.13389			0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000			1.13389			0.566947	+	0.823754i	$0.308125\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	3.50000	+	6.06218i	0.970725	+	1.68135i	0.693375	+	0.720577i	$0.256123\pi$
$13$	3.50000	+	6.06218i	0.970725	+	1.68135i	0.277350	+	0.960769i	$0.410544\pi$
$14$		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
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$19$	4.00000	−	1.73205i	0.917663	−	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.00000	+	3.46410i	0.371391	+	0.643268i	0.989780	−	0.142605i	$-0.0455477\pi$
$29$	2.00000	+	3.46410i	0.371391	+	0.643268i	−0.618389	+	0.785872i	$0.712214\pi$
$30$		0			0
$31$	−1.00000			−0.179605			−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000			−0.179605			−0.0898027	+	0.995960i	$0.528624\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000	+	10.3923i	−1.01419	+	1.75662i
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	−0.666523	−	0.745485i	$-0.732218\pi$
$41$	2.00000	−	3.46410i	0.312348	−	0.541002i	0.978870	+	0.204483i	$0.0655513\pi$
$42$		0			0
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	−0.465478	−	0.885059i	$-0.654118\pi$
$43$	3.50000	−	6.06218i	0.533745	−	0.924473i	0.999223	−	0.0394140i	$-0.0125491\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	0.783830	−	0.620975i	$-0.213263\pi$
$47$	−1.00000	−	1.73205i	−0.145865	−	0.252646i	−0.929695	+	0.368329i	$0.879930\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$	−4.00000	+	6.92820i	−0.539360	+	0.934199i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−28.0000			−3.47297
$66$		0			0
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	0.942987	−	0.332830i	$-0.108004\pi$
$67$	1.50000	+	2.59808i	0.183254	+	0.317406i	−0.759733	+	0.650236i	$0.774670\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	−0.919244	−	0.393688i	$-0.871199\pi$
$71$	−1.00000	+	1.73205i	−0.118678	+	0.205557i	0.800566	+	0.599245i	$0.204532\pi$
$72$		0			0
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	−0.764794	−	0.644275i	$-0.777159\pi$
$73$	1.50000	−	2.59808i	0.175562	−	0.304082i	0.940356	+	0.340193i	$0.110493\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.00000			0.683763
$78$		0			0
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	−0.690426	−	0.723403i	$-0.742577\pi$
$79$	2.50000	−	4.33013i	0.281272	−	0.487177i	0.971698	+	0.236225i	$0.0759104\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.736644	+	0.676280i	$0.236409\pi$
$89$	9.00000	+	15.5885i	0.953998	+	1.65237i	0.217354	+	0.976093i	$0.430258\pi$
$90$		0			0
$91$	10.5000	+	18.1865i	1.10070	+	1.90647i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.00000	+	17.3205i	−0.205196	+	1.77705i
$96$		0			0
$97$	−5.00000	+	8.66025i	−0.507673	+	0.879316i	0.492287	+	0.870433i	$0.336161\pi$
$97$	−5.00000	+	8.66025i	−0.507673	+	0.879316i	−0.999961	+	0.00888289i	$0.997172\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	0.911479	−	0.411346i	$-0.134941\pi$
$101$	1.00000	+	1.73205i	0.0995037	+	0.172345i	−0.811976	+	0.583691i	$0.801608\pi$
$102$		0			0
$103$	−9.00000			−0.886796			−0.443398	−	0.896325i	$-0.646227\pi$
$103$	−9.00000			−0.886796			−0.443398	+	0.896325i	$0.646227\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$	−3.00000	+	5.19615i	−0.287348	+	0.497701i	−0.973176	−	0.230063i	$-0.926107\pi$
$109$	−3.00000	+	5.19615i	−0.287348	+	0.497701i	0.685828	+	0.727764i	$0.259440\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$	−16.0000			−1.49201
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$127$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	−0.141865	−	0.989886i	$-0.545310\pi$
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	0.928199	−	0.372084i	$-0.121357\pi$
$132$		0			0
$133$	12.0000	−	5.19615i	1.04053	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	0.820141	−	0.572161i	$-0.193895\pi$
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	−0.905577	+	0.424182i	$0.860562\pi$
$138$		0			0
$139$	10.5000	+	18.1865i	0.890598	+	1.54256i	0.839159	+	0.543885i	$0.183047\pi$
$139$	10.5000	+	18.1865i	0.890598	+	1.54256i	0.0514389	+	0.998676i	$0.483619\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.00000	+	12.1244i	0.585369	+	1.01389i
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	0.216394	+	0.976306i	$0.430570\pi$
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	−0.953703	+	0.300750i	$0.902763\pi$
$150$		0			0
$151$	−20.0000			−1.62758			−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000			−1.62758			−0.813788	+	0.581161i	$0.802599\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	−	3.46410i	0.160644	−	0.278243i
$156$		0			0
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	−0.971219	−	0.238190i	$-0.923446\pi$
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	0.691888	+	0.722005i	$0.256779\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000	+	10.3923i	0.472866	+	0.819028i
$162$		0			0
$163$	−11.0000			−0.861586			−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000			−0.861586			−0.430793	+	0.902451i	$0.641766\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	0.726293	−	0.687386i	$-0.241242\pi$
$167$	−3.00000	−	5.19615i	−0.232147	−	0.402090i	−0.958440	+	0.285295i	$0.907908\pi$
$168$		0			0
$169$	−18.0000	+	31.1769i	−1.38462	+	2.39822i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	−0.542583	−	0.840002i	$-0.682554\pi$
$173$	6.00000	−	10.3923i	0.456172	−	0.790112i	0.998755	+	0.0498898i	$0.0158870\pi$
$174$		0			0
$175$	−16.5000	−	28.5788i	−1.24728	−	2.16036i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	1.00000	+	1.73205i	0.0743294	+	0.128742i	0.900794	−	0.434246i	$-0.142985\pi$
$181$	1.00000	+	1.73205i	0.0743294	+	0.128742i	−0.826465	+	0.562988i	$0.809652\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.0000	+	24.2487i	−1.02930	+	1.78280i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−20.0000			−1.44715			−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000			−1.44715			−0.723575	+	0.690246i	$0.757502\pi$
$192$		0			0
$193$	10.5000	−	18.1865i	0.755807	−	1.30910i	−0.189166	−	0.981945i	$-0.560578\pi$
$193$	10.5000	−	18.1865i	0.755807	−	1.30910i	0.944972	−	0.327150i	$-0.106088\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	10.0000			0.712470			0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000			0.712470			0.356235	+	0.934396i	$0.384060\pi$
$198$		0			0
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	0.714893	−	0.699234i	$-0.246476\pi$
$199$	−3.50000	−	6.06218i	−0.248108	−	0.429736i	−0.963001	+	0.269498i	$0.913142\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.00000	+	10.3923i	0.421117	+	0.729397i
$204$		0			0
$205$	8.00000	+	13.8564i	0.558744	+	0.967773i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	8.00000	−	3.46410i	0.553372	−	0.239617i
$210$		0			0
$211$	4.50000	−	7.79423i	0.309793	−	0.536577i	−0.668524	−	0.743690i	$-0.733074\pi$
$211$	4.50000	−	7.79423i	0.309793	−	0.536577i	0.978317	+	0.207114i	$0.0664070\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.0000	+	24.2487i	0.954792	+	1.65375i
$216$		0			0
$217$	−3.00000			−0.203653
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.50000	−	4.33013i	0.167412	−	0.289967i	−0.770097	−	0.637927i	$-0.779792\pi$
$223$	2.50000	−	4.33013i	0.167412	−	0.289967i	0.937509	+	0.347960i	$0.113126\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.00000			0.265489			0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000			0.265489			0.132745	+	0.991150i	$0.457621\pi$
$228$		0			0
$229$	−7.00000			−0.462573			−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000			−0.462573			−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	−0.750867	−	0.660454i	$-0.770364\pi$
$233$	3.00000	−	5.19615i	0.196537	−	0.340411i	0.947403	+	0.320043i	$0.103697\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.00000	+	6.92820i	−0.255551	+	0.442627i
$246$		0			0
$247$	24.5000	+	18.1865i	1.55890	+	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−7.00000	−	12.1244i	−0.441836	−	0.765283i	0.555990	−	0.831189i	$-0.312339\pi$
$251$	−7.00000	−	12.1244i	−0.441836	−	0.765283i	−0.997826	+	0.0659066i	$0.979006\pi$
$252$		0			0
$253$	4.00000	+	6.92820i	0.251478	+	0.435572i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−14.0000	−	24.2487i	−0.873296	−	1.51259i	−0.858567	−	0.512702i	$-0.828645\pi$
$257$	−14.0000	−	24.2487i	−0.873296	−	1.51259i	−0.0147291	−	0.999892i	$-0.504689\pi$
$258$		0			0
$259$	21.0000			1.30488
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	−0.449622	−	0.893219i	$-0.648441\pi$
$269$	9.00000	−	15.5885i	0.548740	−	0.950445i	0.998361	−	0.0572259i	$-0.0182255\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.0000	−	19.0526i	−0.663325	−	1.14891i
$276$		0			0
$277$	−26.0000			−1.56219			−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−26.0000			−1.56219			−0.781094	+	0.624413i	$0.785338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.00000	+	3.46410i	0.119310	+	0.206651i	0.919494	−	0.393103i	$-0.128598\pi$
$281$	2.00000	+	3.46410i	0.119310	+	0.206651i	−0.800184	+	0.599754i	$0.795265\pi$
$282$		0			0
$283$	−6.00000	+	10.3923i	−0.356663	+	0.617758i	−0.987401	−	0.158237i	$-0.949419\pi$
$283$	−6.00000	+	10.3923i	−0.356663	+	0.617758i	0.630738	+	0.775996i	$0.282752\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00000	−	10.3923i	0.354169	−	0.613438i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$	12.0000	+	20.7846i	0.698667	+	1.21013i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.0000	+	24.2487i	−0.809641	+	1.40234i
$300$		0			0
$301$	10.5000	−	18.1865i	0.605210	−	1.04825i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	−0.642447	−	0.766330i	$-0.722081\pi$
$307$	6.00000	−	10.3923i	0.342438	−	0.593120i	0.984885	+	0.173210i	$0.0554140\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	34.0000			1.92796			0.963982	−	0.265969i	$-0.0856919\pi$
$311$	34.0000			1.92796			0.963982	+	0.265969i	$0.0856919\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	−0.976764	−	0.214318i	$-0.931247\pi$
$317$	−12.0000	−	20.7846i	−0.673987	−	1.16738i	0.302777	−	0.953062i	$-0.402086\pi$
$318$		0			0
$319$	4.00000	+	6.92820i	0.223957	+	0.387905i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	38.5000	−	66.6840i	2.13560	−	3.69896i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.00000	−	5.19615i	−0.165395	−	0.286473i
$330$		0			0
$331$	−23.0000			−1.26419			−0.632097	−	0.774889i	$-0.717806\pi$
$331$	−23.0000			−1.26419			−0.632097	+	0.774889i	$0.717806\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000			−0.655630
$336$		0			0
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	−0.986960	−	0.160968i	$-0.948538\pi$
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	0.632882	+	0.774248i	$0.281872\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.00000			−0.108306
$342$		0			0
$343$	−15.0000			−0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−14.0000	+	24.2487i	−0.751559	+	1.30174i	0.195507	+	0.980702i	$0.437365\pi$
$347$	−14.0000	+	24.2487i	−0.751559	+	1.30174i	−0.947067	+	0.321037i	$0.895969\pi$
$348$		0			0
$349$	31.0000			1.65939			0.829696	−	0.558216i	$-0.188514\pi$
$349$	31.0000			1.65939			0.829696	+	0.558216i	$0.188514\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$	−4.00000	−	6.92820i	−0.212298	−	0.367711i
$356$		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
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.00000	+	10.3923i	0.314054	+	0.543958i
$366$		0			0
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	0.999989	−	0.00473247i	$-0.00150640\pi$
$367$	9.50000	+	16.4545i	0.495896	+	0.858917i	−0.504093	+	0.863649i	$0.668173\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−14.0000	+	24.2487i	−0.721037	+	1.24887i
$378$		0			0
$379$	21.0000			1.07870			0.539349	−	0.842082i	$-0.318670\pi$
$379$	21.0000			1.07870			0.539349	+	0.842082i	$0.318670\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	−0.629890	−	0.776684i	$-0.716900\pi$
$383$	7.00000	−	12.1244i	0.357683	−	0.619526i	0.987573	+	0.157159i	$0.0502334\pi$
$384$		0			0
$385$	−12.0000	+	20.7846i	−0.611577	+	1.05928i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	−0.991500	−	0.130105i	$-0.958469\pi$
$389$	−12.0000	−	20.7846i	−0.608424	−	1.05382i	0.383076	−	0.923717i	$-0.374865\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	10.0000	+	17.3205i	0.503155	+	0.871489i
$396$		0			0
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	−0.655534	−	0.755166i	$-0.727556\pi$
$397$	6.50000	−	11.2583i	0.326226	−	0.565039i	0.981760	+	0.190126i	$0.0608897\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	−0.713751	−	0.700399i	$-0.753005\pi$
$401$	5.00000	−	8.66025i	0.249688	−	0.432472i	0.963439	+	0.267927i	$0.0863386\pi$
$402$		0			0
$403$	−3.50000	−	6.06218i	−0.174347	−	0.301979i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	14.0000			0.693954
$408$		0			0
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	9.00000	−	15.5885i	0.442861	−	0.767058i
$414$		0			0
$415$	24.0000	−	41.5692i	1.17811	−	2.04055i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	30.0000			1.46560			0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000			1.46560			0.732798	+	0.680446i	$0.238214\pi$
$420$		0			0
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	0.353233	+	0.935536i	$0.385082\pi$
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	−0.986814	+	0.161859i	$0.948251\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.50000	+	2.59808i	0.0725901	+	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	0.784683	−	0.619897i	$-0.212826\pi$
$431$	−3.00000	−	5.19615i	−0.144505	−	0.250290i	−0.929188	+	0.369607i	$0.879492\pi$
$432$		0			0
$433$	−11.5000	−	19.9186i	−0.552655	−	0.957226i	−0.998082	−	0.0619079i	$-0.980282\pi$
$433$	−11.5000	−	19.9186i	−0.552655	−	0.957226i	0.445427	−	0.895318i	$-0.353052\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.0000	+	10.3923i	0.669711	+	0.497131i
$438$		0			0
$439$	11.5000	−	19.9186i	0.548865	−	0.950662i	−0.449488	−	0.893287i	$-0.648393\pi$
$439$	11.5000	−	19.9186i	0.548865	−	0.950662i	0.998353	−	0.0573756i	$-0.0182733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	−72.0000			−3.41313
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.0000			−0.755087			−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000			−0.755087			−0.377543	+	0.925992i	$0.623231\pi$
$450$		0			0
$451$	4.00000	−	6.92820i	0.188353	−	0.326236i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−84.0000			−3.93798
$456$		0			0
$457$	29.0000			1.35656			0.678281	−	0.734802i	$-0.262725\pi$
$457$	29.0000			1.35656			0.678281	+	0.734802i	$0.262725\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−17.0000	+	29.4449i	−0.791769	+	1.37138i	0.133102	+	0.991102i	$0.457506\pi$
$461$	−17.0000	+	29.4449i	−0.791769	+	1.37138i	−0.924871	+	0.380282i	$0.875827\pi$
$462$		0			0
$463$	−19.0000			−0.883005			−0.441502	−	0.897260i	$-0.645554\pi$
$463$	−19.0000			−0.883005			−0.441502	+	0.897260i	$0.645554\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	4.50000	+	7.79423i	0.207791	+	0.359904i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	7.00000	−	12.1244i	0.321860	−	0.557478i
$474$		0			0
$475$	−38.5000	−	28.5788i	−1.76650	−	1.31129i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−17.0000	−	29.4449i	−0.776750	−	1.34537i	−0.933806	−	0.357780i	$-0.883534\pi$
$479$	−17.0000	−	29.4449i	−0.776750	−	1.34537i	0.157056	−	0.987590i	$-0.449800\pi$
$480$		0			0
$481$	24.5000	+	42.4352i	1.11710	+	1.93488i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.0000	−	34.6410i	−0.908153	−	1.57297i
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−13.0000	+	22.5167i	−0.586682	+	1.01616i	0.407982	+	0.912990i	$0.366233\pi$
$491$	−13.0000	+	22.5167i	−0.586682	+	1.01616i	−0.994663	+	0.103173i	$0.967101\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.00000	+	5.19615i	−0.134568	+	0.233079i
$498$		0			0
$499$	−5.50000	+	9.52628i	−0.246214	+	0.426455i	−0.962472	−	0.271380i	$-0.912520\pi$
$499$	−5.50000	+	9.52628i	−0.246214	+	0.426455i	0.716258	+	0.697835i	$0.245853\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	0.968223	−	0.250090i	$-0.0804603\pi$
$503$	6.00000	+	10.3923i	0.267527	+	0.463370i	−0.700696	+	0.713460i	$0.747127\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	0.791849	−	0.610718i	$-0.209119\pi$
$509$	−3.00000	−	5.19615i	−0.132973	−	0.230315i	−0.924821	+	0.380402i	$0.875786\pi$
$510$		0			0
$511$	4.50000	−	7.79423i	0.199068	−	0.344796i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.0000	−	31.1769i	0.793175	−	1.37382i
$516$		0			0
$517$	−2.00000	−	3.46410i	−0.0879599	−	0.152351i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−38.0000			−1.66481			−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000			−1.66481			−0.832405	+	0.554168i	$0.813037\pi$
$522$		0			0
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	0.986718	+	0.162446i	$0.0519382\pi$
$523$	14.5000	+	25.1147i	0.634041	+	1.09819i	−0.352677	+	0.935745i	$0.614728\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	28.0000			1.21281
$534$		0			0
$535$	−4.00000	+	6.92820i	−0.172935	+	0.299532i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.00000			0.172292
$540$		0			0
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	0.994715	−	0.102677i	$-0.0327407\pi$
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	−0.586278	+	0.810110i	$0.699407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−12.0000	−	20.7846i	−0.514024	−	0.890315i
$546$		0			0
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	−0.989490	−	0.144604i	$-0.953809\pi$
$547$	−14.5000	−	25.1147i	−0.619975	−	1.07383i	0.369514	−	0.929225i	$-0.379524\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.0000	+	10.3923i	0.596420	+	0.442727i
$552$		0			0
$553$	7.50000	−	12.9904i	0.318932	−	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.0000	+	19.0526i	0.466085	+	0.807283i	0.999250	−	0.0387286i	$-0.0123308\pi$
$557$	11.0000	+	19.0526i	0.466085	+	0.807283i	−0.533165	+	0.846011i	$0.678997\pi$
$558$		0			0
$559$	49.0000			2.07248
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−42.0000			−1.77009			−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000			−1.77009			−0.885044	+	0.465506i	$0.845872\pi$
$564$		0			0
$565$	4.00000	−	6.92820i	0.168281	−	0.291472i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.0000			−0.670755			−0.335377	−	0.942084i	$-0.608864\pi$
$569$	−16.0000			−0.670755			−0.335377	+	0.942084i	$0.608864\pi$
$570$		0			0
$571$	7.00000			0.292941			0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000			0.292941			0.146470	+	0.989215i	$0.453209\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	22.0000	−	38.1051i	0.917463	−	1.58909i
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−36.0000			−1.49353
$582$		0			0
$583$	−4.00000	−	6.92820i	−0.165663	−	0.286937i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9.00000	−	15.5885i	0.371470	−	0.643404i	−0.618322	−	0.785925i	$-0.712187\pi$
$587$	9.00000	−	15.5885i	0.371470	−	0.643404i	0.989792	+	0.142520i	$0.0455206\pi$
$588$		0			0
$589$	−4.00000	+	1.73205i	−0.164817	+	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	0.969122	+	0.246581i	$0.0793071\pi$
$593$	17.0000	+	29.4449i	0.698106	+	1.20916i	−0.271016	+	0.962575i	$0.587360\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	−0.990752	−	0.135686i	$-0.956676\pi$
$599$	−15.0000	−	25.9808i	−0.612883	−	1.06155i	0.377869	−	0.925859i	$-0.376657\pi$
$600$		0			0
$601$	19.0000			0.775026			0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000			0.775026			0.387513	+	0.921864i	$0.373334\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0000	−	24.2487i	0.569181	−	0.985850i
$606$		0			0
$607$	−41.0000			−1.66414			−0.832069	−	0.554672i	$-0.812844\pi$
$607$	−41.0000			−1.66414			−0.832069	+	0.554672i	$0.812844\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	7.00000	−	12.1244i	0.283190	−	0.490499i
$612$		0			0
$613$	19.0000	−	32.9090i	0.767403	−	1.32918i	−0.171564	−	0.985173i	$-0.554882\pi$
$613$	19.0000	−	32.9090i	0.767403	−	1.32918i	0.938967	−	0.344008i	$-0.111785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	0.988343	−	0.152242i	$-0.0486493\pi$
$617$	9.00000	+	15.5885i	0.362326	+	0.627568i	−0.626017	+	0.779809i	$0.715316\pi$
$618$		0			0
$619$	23.0000			0.924448			0.462224	−	0.886763i	$-0.347052\pi$
$619$	23.0000			0.924448			0.462224	+	0.886763i	$0.347052\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	27.0000	+	46.7654i	1.08173	+	1.87362i
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−9.50000	−	16.4545i	−0.378189	−	0.655043i	0.612610	−	0.790386i	$-0.290120\pi$
$631$	−9.50000	−	16.4545i	−0.378189	−	0.655043i	−0.990799	+	0.135343i	$0.956786\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.00000	+	12.1244i	0.277350	+	0.480384i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	−0.631721	−	0.775196i	$-0.717651\pi$
$641$	9.00000	−	15.5885i	0.355479	−	0.615707i	0.987200	+	0.159489i	$0.0509845\pi$
$642$		0			0
$643$	6.50000	−	11.2583i	0.256335	−	0.443985i	−0.708922	−	0.705287i	$-0.750818\pi$
$643$	6.50000	−	11.2583i	0.256335	−	0.443985i	0.965257	+	0.261301i	$0.0841516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−28.0000			−1.10079			−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000			−1.10079			−0.550397	+	0.834903i	$0.685524\pi$
$648$		0			0
$649$	6.00000	−	10.3923i	0.235521	−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	30.0000			1.17399			0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000			1.17399			0.586995	+	0.809590i	$0.300311\pi$
$654$		0			0
$655$	36.0000	+	62.3538i	1.40664	+	2.43637i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.00000	+	8.66025i	0.194772	+	0.337356i	0.946826	−	0.321746i	$-0.104270\pi$
$659$	5.00000	+	8.66025i	0.194772	+	0.337356i	−0.752054	+	0.659102i	$0.770937\pi$
$660$		0			0
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	0.752252	−	0.658876i	$-0.228968\pi$
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	−0.946729	+	0.322031i	$0.895634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	+	51.9615i	−0.232670	+	2.01498i
$666$		0			0
$667$	−8.00000	+	13.8564i	−0.309761	+	0.536522i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.00000	+	1.73205i	0.0386046	+	0.0668651i
$672$		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
$677$	38.0000			1.46046			0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000			1.46046			0.730229	+	0.683202i	$0.239413\pi$
$678$		0			0
$679$	−15.0000	+	25.9808i	−0.575647	+	0.997050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	40.0000			1.53056			0.765279	−	0.643699i	$-0.222601\pi$
$683$	40.0000			1.53056			0.765279	+	0.643699i	$0.222601\pi$
$684$		0			0
$685$	8.00000			0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.0000	−	24.2487i	0.533358	−	0.923802i
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−84.0000			−3.18630
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4.00000	−	6.92820i	0.151078	−	0.261675i	−0.780546	−	0.625098i	$-0.785059\pi$
$701$	4.00000	−	6.92820i	0.151078	−	0.261675i	0.931624	+	0.363424i	$0.118392\pi$
$702$		0			0
$703$	28.0000	−	12.1244i	1.05604	−	0.457279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.00000	+	5.19615i	0.112827	+	0.195421i
$708$		0			0
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	0.892817	−	0.450420i	$-0.148726\pi$
$709$	1.50000	+	2.59808i	0.0563337	+	0.0975728i	−0.836483	+	0.547992i	$0.815392\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.00000	−	3.46410i	−0.0749006	−	0.129732i
$714$		0			0
$715$	−56.0000			−2.09428
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	−0.975759	−	0.218850i	$-0.929769\pi$
$719$	−8.00000	+	13.8564i	−0.298350	+	0.516757i	0.677409	+	0.735607i	$0.263103\pi$
$720$		0			0
$721$	−27.0000			−1.00553
$722$		0			0
$723$		0			0
$724$		0			0
$725$	22.0000	−	38.1051i	0.817059	−	1.41519i
$726$		0			0
$727$	18.5000	−	32.0429i	0.686127	−	1.18841i	−0.286954	−	0.957944i	$-0.592643\pi$
$727$	18.5000	−	32.0429i	0.686127	−	1.18841i	0.973081	−	0.230463i	$-0.0740239\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	30.0000			1.10808			0.554038	−	0.832492i	$-0.313086\pi$
$733$	30.0000			1.10808			0.554038	+	0.832492i	$0.313086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.00000	+	5.19615i	0.110506	+	0.191403i
$738$		0			0
$739$	17.5000	−	30.3109i	0.643748	−	1.11500i	−0.340841	−	0.940121i	$-0.610712\pi$
$739$	17.5000	−	30.3109i	0.643748	−	1.11500i	0.984589	−	0.174883i	$-0.0559548\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−13.0000	+	22.5167i	−0.476924	+	0.826056i	−0.999650	−	0.0264443i	$-0.991582\pi$
$743$	−13.0000	+	22.5167i	−0.476924	+	0.826056i	0.522727	+	0.852500i	$0.324915\pi$
$744$		0			0
$745$	−36.0000	−	62.3538i	−1.31894	−	2.28447i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.00000			0.219235
$750$		0			0
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	−0.996993	−	0.0774878i	$-0.975310\pi$
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	0.431390	−	0.902165i	$-0.358023\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	40.0000	−	69.2820i	1.45575	−	2.52143i
$756$		0			0
$757$	11.5000	−	19.9186i	0.417975	−	0.723953i	−0.577761	−	0.816206i	$-0.696073\pi$
$757$	11.5000	−	19.9186i	0.417975	−	0.723953i	0.995736	+	0.0922527i	$0.0294068\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	42.0000			1.52250			0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000			1.52250			0.761249	+	0.648459i	$0.224586\pi$
$762$		0			0
$763$	−9.00000	+	15.5885i	−0.325822	+	0.564340i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	42.0000			1.51653
$768$		0			0
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	0.907575	−	0.419890i	$-0.137931\pi$
$769$	2.50000	+	4.33013i	0.0901523	+	0.156148i	−0.817423	+	0.576038i	$0.804598\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.00000	+	5.19615i	0.107903	+	0.186893i	0.914920	−	0.403634i	$-0.132253\pi$
$773$	3.00000	+	5.19615i	0.107903	+	0.186893i	−0.807018	+	0.590527i	$0.798920\pi$
$774$		0			0
$775$	5.50000	+	9.52628i	0.197566	+	0.342194i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.00000	−	17.3205i	0.0716574	−	0.620572i
$780$		0			0
$781$	−2.00000	+	3.46410i	−0.0715656	+	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−14.0000	−	24.2487i	−0.499681	−	0.865474i
$786$		0			0
$787$	−49.0000			−1.74666			−0.873331	−	0.487128i	$-0.838045\pi$
$787$	−49.0000			−1.74666			−0.873331	+	0.487128i	$0.838045\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.00000			−0.213335
$792$		0			0
$793$	−3.50000	+	6.06218i	−0.124289	+	0.215274i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6.00000			−0.212531			−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000			−0.212531			−0.106265	+	0.994338i	$0.533889\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$		0			0
$805$	−48.0000			−1.69178
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−36.0000			−1.26569			−0.632846	−	0.774277i	$-0.718114\pi$
$809$	−36.0000			−1.26569			−0.632846	+	0.774277i	$0.718114\pi$
$810$		0			0
$811$	18.0000	+	31.1769i	0.632065	+	1.09477i	0.987129	+	0.159927i	$0.0511260\pi$
$811$	18.0000	+	31.1769i	0.632065	+	1.09477i	−0.355063	+	0.934842i	$0.615541\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	22.0000	−	38.1051i	0.770626	−	1.33476i
$816$		0			0
$817$	3.50000	−	30.3109i	0.122449	−	1.06044i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.00000	+	10.3923i	0.209401	+	0.362694i	0.951526	−	0.307568i	$-0.0995151\pi$
$821$	6.00000	+	10.3923i	0.209401	+	0.362694i	−0.742125	+	0.670262i	$0.766182\pi$
$822$		0			0
$823$	22.0000	+	38.1051i	0.766872	+	1.32826i	0.939251	+	0.343230i	$0.111521\pi$
$823$	22.0000	+	38.1051i	0.766872	+	1.32826i	−0.172379	+	0.985031i	$0.555146\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.00000	−	6.92820i	−0.139094	−	0.240917i	0.788060	−	0.615598i	$-0.211086\pi$
$827$	−4.00000	−	6.92820i	−0.139094	−	0.240917i	−0.927154	+	0.374681i	$0.877752\pi$
$828$		0			0
$829$	45.0000			1.56291			0.781457	−	0.623959i	$-0.214477\pi$
$829$	45.0000			1.56291			0.781457	+	0.623959i	$0.214477\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	24.0000			0.830554
$836$		0			0
$837$		0			0
$838$		0			0
$839$	20.0000	−	34.6410i	0.690477	−	1.19594i	−0.281205	−	0.959648i	$-0.590734\pi$
$839$	20.0000	−	34.6410i	0.690477	−	1.19594i	0.971682	−	0.236293i	$-0.0759325\pi$
$840$		0			0
$841$	6.50000	−	11.2583i	0.224138	−	0.388218i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−72.0000	−	124.708i	−2.47688	−	4.29007i
$846$		0			0
$847$	−21.0000			−0.721569
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14.0000	+	24.2487i	0.479914	+	0.831235i
$852$		0			0
$853$	4.50000	−	7.79423i	0.154077	−	0.266869i	−0.778646	−	0.627464i	$-0.784093\pi$
$853$	4.50000	−	7.79423i	0.154077	−	0.266869i	0.932723	+	0.360595i	$0.117426\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	−0.912705	−	0.408619i	$-0.866010\pi$
$857$	−3.00000	+	5.19615i	−0.102478	+	0.177497i	0.810227	+	0.586116i	$0.199344\pi$
$858$		0			0
$859$	3.50000	+	6.06218i	0.119418	+	0.206839i	0.919537	−	0.393003i	$-0.128564\pi$
$859$	3.50000	+	6.06218i	0.119418	+	0.206839i	−0.800119	+	0.599841i	$0.795230\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.00000			−0.204242			−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000			−0.204242			−0.102121	+	0.994772i	$0.532563\pi$
$864$		0			0
$865$	24.0000	+	41.5692i	0.816024	+	1.41340i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5.00000	−	8.66025i	0.169613	−	0.293779i
$870$		0			0
$871$	−10.5000	+	18.1865i	−0.355779	+	0.616227i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	72.0000			2.43404
$876$		0			0
$877$	−9.50000	+	16.4545i	−0.320792	+	0.555628i	−0.980652	−	0.195761i	$-0.937282\pi$
$877$	−9.50000	+	16.4545i	−0.320792	+	0.555628i	0.659860	+	0.751389i	$0.270616\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	24.0000			0.808581			0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000			0.808581			0.404290	+	0.914631i	$0.367519\pi$
$882$		0			0
$883$	−20.5000	−	35.5070i	−0.689880	−	1.19491i	−0.971876	−	0.235492i	$-0.924330\pi$
$883$	−20.5000	−	35.5070i	−0.689880	−	1.19491i	0.281996	−	0.959415i	$-0.409003\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.00000	−	8.66025i	−0.167884	−	0.290783i	0.769792	−	0.638295i	$-0.220360\pi$
$887$	−5.00000	−	8.66025i	−0.167884	−	0.290783i	−0.937676	+	0.347512i	$0.887027\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.00000	−	5.19615i	−0.234246	−	0.173883i
$894$		0			0
$895$	24.0000	−	41.5692i	0.802232	−	1.38951i
$896$		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
$905$	−8.00000			−0.265929
$906$		0			0
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	−0.650868	−	0.759191i	$-0.725595\pi$
$907$	10.0000	−	17.3205i	0.332045	−	0.575118i	0.982913	+	0.184073i	$0.0589282\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$	−24.0000			−0.794284
$914$		0			0
$915$		0			0
$916$		0			0
$917$	27.0000	−	46.7654i	0.891619	−	1.54433i
$918$		0			0
$919$	−11.0000			−0.362857			−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000			−0.362857			−0.181428	+	0.983404i	$0.558072\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.0000			−0.460816
$924$		0			0
$925$	−38.5000	−	66.6840i	−1.26587	−	2.19255i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4.00000	−	6.92820i	0.131236	−	0.227307i	−0.792917	−	0.609329i	$-0.791439\pi$
$929$	4.00000	−	6.92820i	0.131236	−	0.227307i	0.924153	+	0.382022i	$0.124772\pi$
$930$		0			0
$931$	8.00000	−	3.46410i	0.262189	−	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	12.5000	+	21.6506i	0.408357	+	0.707295i	0.994706	−	0.102763i	$-0.0327685\pi$
$937$	12.5000	+	21.6506i	0.408357	+	0.707295i	−0.586349	+	0.810059i	$0.699435\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.00000	+	13.8564i	0.260793	+	0.451706i	0.966453	−	0.256844i	$-0.0826828\pi$
$941$	8.00000	+	13.8564i	0.260793	+	0.451706i	−0.705660	+	0.708550i	$0.749349\pi$
$942$		0			0
$943$	16.0000			0.521032
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.0000	+	46.7654i	−0.877382	+	1.51967i	−0.0231788	+	0.999731i	$0.507379\pi$
$947$	−27.0000	+	46.7654i	−0.877382	+	1.51967i	−0.854203	+	0.519939i	$0.825955\pi$
$948$		0			0
$949$	21.0000			0.681689
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−5.00000	+	8.66025i	−0.161966	+	0.280533i	−0.935574	−	0.353132i	$-0.885117\pi$
$953$	−5.00000	+	8.66025i	−0.161966	+	0.280533i	0.773608	+	0.633665i	$0.218450\pi$
$954$		0			0
$955$	40.0000	−	69.2820i	1.29437	−	2.24191i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−3.00000	−	5.19615i	−0.0968751	−	0.167793i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	42.0000	+	72.7461i	1.35203	+	2.34178i
$966$		0			0
$967$	−4.50000	+	7.79423i	−0.144710	+	0.250645i	−0.929265	−	0.369414i	$-0.879558\pi$
$967$	−4.50000	+	7.79423i	−0.144710	+	0.250645i	0.784555	+	0.620060i	$0.212892\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.00000	+	12.1244i	−0.224641	+	0.389089i	−0.956212	−	0.292676i	$-0.905454\pi$
$971$	−7.00000	+	12.1244i	−0.224641	+	0.389089i	0.731571	+	0.681765i	$0.238788\pi$
$972$		0			0
$973$	31.5000	+	54.5596i	1.00984	+	1.74910i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.0000			−0.447900			−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000			−0.447900			−0.223950	+	0.974601i	$0.571895\pi$
$978$		0			0
$979$	18.0000	+	31.1769i	0.575282	+	0.996419i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	−0.909894	−	0.414840i	$-0.863838\pi$
$983$	−3.00000	+	5.19615i	−0.0956851	+	0.165732i	0.814209	+	0.580572i	$0.197171\pi$
$984$		0			0
$985$	−20.0000	+	34.6410i	−0.637253	+	1.10375i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$	−14.5000	+	25.1147i	−0.460608	+	0.797796i	−0.998991	−	0.0449040i	$-0.985702\pi$
$991$	−14.5000	+	25.1147i	−0.460608	+	0.797796i	0.538384	+	0.842700i	$0.319035\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	28.0000			0.887660
$996$		0			0
$997$	26.5000	+	45.8993i	0.839263	+	1.45365i	0.890511	+	0.454961i	$0.150347\pi$
$997$	26.5000	+	45.8993i	0.839263	+	1.45365i	−0.0512480	+	0.998686i	$0.516320\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.s.c.1873.1		2
3.2	odd	2		912.2.q.d.49.1		2
4.3	odd	2		342.2.g.d.163.1		2
12.11	even	2		114.2.e.a.49.1	yes	2
19.7	even	3	inner	2736.2.s.c.577.1		2
57.26	odd	6		912.2.q.d.577.1		2
76.7	odd	6		342.2.g.d.235.1		2
76.11	odd	6		6498.2.a.l.1.1		1
76.27	even	6		6498.2.a.x.1.1		1
228.11	even	6		2166.2.a.f.1.1		1
228.83	even	6		114.2.e.a.7.1	✓	2
228.179	odd	6		2166.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.e.a.7.1	✓	2	228.83	even	6
114.2.e.a.49.1	yes	2	12.11	even	2
342.2.g.d.163.1		2	4.3	odd	2
342.2.g.d.235.1		2	76.7	odd	6
912.2.q.d.49.1		2	3.2	odd	2
912.2.q.d.577.1		2	57.26	odd	6
2166.2.a.c.1.1		1	228.179	odd	6
2166.2.a.f.1.1		1	228.11	even	6
2736.2.s.c.577.1		2	19.7	even	3	inner
2736.2.s.c.1873.1		2	1.1	even	1	trivial
6498.2.a.l.1.1		1	76.11	odd	6
6498.2.a.x.1.1		1	76.27	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{7} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{5} +3.00000 q^{7} +2.00000 q^{11} +(3.50000 + 6.06218i) q^{13} +(4.00000 - 1.73205i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(2.00000 + 3.46410i) q^{29} -1.00000 q^{31} +(-6.00000 + 10.3923i) q^{35} +7.00000 q^{37} +(2.00000 - 3.46410i) q^{41} +(3.50000 - 6.06218i) q^{43} +(-1.00000 - 1.73205i) q^{47} +2.00000 q^{49} +(-2.00000 - 3.46410i) q^{53} +(-4.00000 + 6.92820i) q^{55} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} -28.0000 q^{65} +(1.50000 + 2.59808i) q^{67} +(-1.00000 + 1.73205i) q^{71} +(1.50000 - 2.59808i) q^{73} +6.00000 q^{77} +(2.50000 - 4.33013i) q^{79} -12.0000 q^{83} +(9.00000 + 15.5885i) q^{89} +(10.5000 + 18.1865i) q^{91} +(-2.00000 + 17.3205i) q^{95} +(-5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 + 6 * q^7	\( 2 q - 4 q^{5} + 6 q^{7} + 4 q^{11} + 7 q^{13} + 8 q^{19} + 4 q^{23} - 11 q^{25} + 4 q^{29} - 2 q^{31} - 12 q^{35} + 14 q^{37} + 4 q^{41} + 7 q^{43} - 2 q^{47} + 4 q^{49} - 4 q^{53} - 8 q^{55} + 6 q^{59} + q^{61} - 56 q^{65} + 3 q^{67} - 2 q^{71} + 3 q^{73} + 12 q^{77} + 5 q^{79} - 24 q^{83} + 18 q^{89} + 21 q^{91} - 4 q^{95} - 10 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 6 * q^7 + 4 * q^11 + 7 * q^13 + 8 * q^19 + 4 * q^23 - 11 * q^25 + 4 * q^29 - 2 * q^31 - 12 * q^35 + 14 * q^37 + 4 * q^41 + 7 * q^43 - 2 * q^47 + 4 * q^49 - 4 * q^53 - 8 * q^55 + 6 * q^59 + q^61 - 56 * q^65 + 3 * q^67 - 2 * q^71 + 3 * q^73 + 12 * q^77 + 5 * q^79 - 24 * q^83 + 18 * q^89 + 21 * q^91 - 4 * q^95 - 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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