LMFDB - Embedded newform 1900.2.s.a.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(49,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

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

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

chi := DirichletCharacter("1900.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.s (of order $6$, 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:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 76)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			349.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1900.349
Dual form			1900.2.s.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{3} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})$	$q+(0.866025 - 0.500000i) q^{3} +(-1.00000 + 1.73205i) q^{9} -4.00000 q^{11} +(-0.866025 - 0.500000i) q^{13} +(2.59808 - 1.50000i) q^{17} +(4.00000 + 1.73205i) q^{19} +(4.33013 + 2.50000i) q^{23} +5.00000i q^{27} +(3.50000 - 6.06218i) q^{29} +4.00000 q^{31} +(-3.46410 + 2.00000i) q^{33} +10.0000i q^{37} -1.00000 q^{39} +(2.50000 + 4.33013i) q^{41} +(4.33013 - 2.50000i) q^{43} +(6.06218 + 3.50000i) q^{47} +7.00000 q^{49} +(1.50000 - 2.59808i) q^{51} +(9.52628 + 5.50000i) q^{53} +(4.33013 - 0.500000i) q^{57} +(1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(2.59808 + 1.50000i) q^{67} +5.00000 q^{69} +(-5.50000 - 9.52628i) q^{71} +(-12.9904 + 7.50000i) q^{73} +(-6.50000 - 11.2583i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.00000i q^{87} +(1.50000 - 2.59808i) q^{89} +(3.46410 - 2.00000i) q^{93} +(-4.33013 + 2.50000i) q^{97} +(4.00000 - 6.92820i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 16 q^{11} + 16 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{39} + 10 q^{41} + 28 q^{49} + 6 q^{51} + 6 q^{59} - 22 q^{61} + 20 q^{69} - 22 q^{71} - 26 q^{79} - 2 q^{81} + 6 q^{89} + 16 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 16 * q^11 + 16 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^39 + 10 * q^41 + 28 * q^49 + 6 * q^51 + 6 * q^59 - 22 * q^61 + 20 * q^69 - 22 * q^71 - 26 * q^79 - 2 * q^81 + 6 * q^89 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$1$

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
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	−0.228714	−	0.973494i	$-0.573452\pi$
$3$	0.866025	−	0.500000i	0.500000	−	0.288675i	0.728714	+	0.684819i	$0.240119\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000	+	1.73205i	−0.333333	+	0.577350i
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	−0.866025	−	0.500000i	−0.240192	−	0.138675i	0.375073	−	0.926995i	$-0.377618\pi$
$13$	−0.866025	−	0.500000i	−0.240192	−	0.138675i	−0.615265	+	0.788320i	$0.710951\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.59808	−	1.50000i	0.630126	−	0.363803i	−0.150675	−	0.988583i	$-0.548145\pi$
$17$	2.59808	−	1.50000i	0.630126	−	0.363803i	0.780801	+	0.624780i	$0.214811\pi$
$18$		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
$23$	4.33013	+	2.50000i	0.902894	+	0.521286i	0.878138	−	0.478407i	$-0.158786\pi$
$23$	4.33013	+	2.50000i	0.902894	+	0.521286i	0.0247559	+	0.999694i	$0.492119\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	−0.333205	−	0.942855i	$-0.608130\pi$
$29$	3.50000	−	6.06218i	0.649934	−	1.12572i	0.983138	−	0.182864i	$-0.0585367\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$	−3.46410	+	2.00000i	−0.603023	+	0.348155i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.0000i			1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$			10.0000i			1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$		0			0
$39$	−1.00000			−0.160128
$40$		0			0
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	0.992507	−	0.122189i	$-0.0389915\pi$
$41$	2.50000	+	4.33013i	0.390434	+	0.676252i	−0.602072	+	0.798441i	$0.705658\pi$
$42$		0			0
$43$	4.33013	−	2.50000i	0.660338	−	0.381246i	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	4.33013	−	2.50000i	0.660338	−	0.381246i	0.792406	+	0.609994i	$0.208828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.06218	+	3.50000i	0.884260	+	0.510527i	0.872060	−	0.489398i	$-0.162783\pi$
$47$	6.06218	+	3.50000i	0.884260	+	0.510527i	0.0121990	+	0.999926i	$0.496117\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	1.50000	−	2.59808i	0.210042	−	0.363803i
$52$		0			0
$53$	9.52628	+	5.50000i	1.30854	+	0.755483i	0.981852	−	0.189651i	$-0.0607356\pi$
$53$	9.52628	+	5.50000i	1.30854	+	0.755483i	0.326683	+	0.945134i	$0.394069\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.33013	−	0.500000i	0.573539	−	0.0662266i
$58$		0			0
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	0.946993	−	0.321253i	$-0.104104\pi$
$59$	1.50000	+	2.59808i	0.195283	+	0.338241i	−0.751710	+	0.659494i	$0.770771\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	0.650236	−	0.759733i	$-0.274670\pi$
$67$	2.59808	+	1.50000i	0.317406	+	0.183254i	−0.332830	+	0.942987i	$0.608004\pi$
$68$		0			0
$69$	5.00000			0.601929
$70$		0			0
$71$	−5.50000	−	9.52628i	−0.652730	−	1.13056i	−0.982458	−	0.186485i	$-0.940290\pi$
$71$	−5.50000	−	9.52628i	−0.652730	−	1.13056i	0.329728	−	0.944076i	$-0.393043\pi$
$72$		0			0
$73$	−12.9904	+	7.50000i	−1.52041	+	0.877809i	−0.520699	+	0.853740i	$0.674329\pi$
$73$	−12.9904	+	7.50000i	−1.52041	+	0.877809i	−0.999710	+	0.0240681i	$0.992338\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−6.50000	−	11.2583i	−0.731307	−	1.26666i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	7.00000i		−	0.750479i
$88$		0			0
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	−0.775509	−	0.631337i	$-0.782506\pi$
$89$	1.50000	−	2.59808i	0.159000	−	0.275396i	0.934508	+	0.355942i	$0.115840\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.46410	−	2.00000i	0.359211	−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.33013	+	2.50000i	−0.439658	+	0.253837i	−0.703452	−	0.710742i	$-0.748359\pi$
$97$	−4.33013	+	2.50000i	−0.439658	+	0.253837i	0.263795	+	0.964579i	$0.415026\pi$
$98$		0			0
$99$	4.00000	−	6.92820i	0.402015	−	0.696311i
$100$		0			0
$101$	0.500000	−	0.866025i	0.0497519	−	0.0861727i	−0.840077	−	0.542467i	$-0.817490\pi$
$101$	0.500000	−	0.866025i	0.0497519	−	0.0861727i	0.889829	+	0.456294i	$0.150824\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			20.0000i			1.93347i	0.255774	+	0.966736i	$0.417670\pi$
$107$			20.0000i			1.93347i	−0.255774	+	0.966736i	$0.582330\pi$
$108$		0			0
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	0.928877	−	0.370387i	$-0.120775\pi$
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	−0.785203	+	0.619238i	$0.787442\pi$
$110$		0			0
$111$	5.00000	+	8.66025i	0.474579	+	0.821995i
$112$		0			0
$113$		−	14.0000i		−	1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$		−	14.0000i		−	1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.73205	−	1.00000i	0.160128	−	0.0924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	4.33013	+	2.50000i	0.390434	+	0.225417i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.59808	+	1.50000i	0.230542	+	0.133103i	0.610822	−	0.791768i	$-0.290839\pi$
$127$	2.59808	+	1.50000i	0.230542	+	0.133103i	−0.380280	+	0.924871i	$0.624172\pi$
$128$		0			0
$129$	2.50000	−	4.33013i	0.220113	−	0.381246i
$130$		0			0
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	−0.981824	−	0.189794i	$-0.939218\pi$
$131$	−7.50000	−	12.9904i	−0.655278	−	1.13497i	0.326546	−	0.945181i	$-0.394115\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.33013	+	2.50000i	0.369948	+	0.213589i	0.673436	−	0.739246i	$-0.264818\pi$
$137$	4.33013	+	2.50000i	0.369948	+	0.213589i	−0.303488	+	0.952835i	$0.598151\pi$
$138$		0			0
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	−0.609618	−	0.792695i	$-0.708677\pi$
$139$	4.50000	−	7.79423i	0.381685	−	0.661098i	0.991303	+	0.131597i	$0.0420106\pi$
$140$		0			0
$141$	7.00000			0.589506
$142$		0			0
$143$	3.46410	+	2.00000i	0.289683	+	0.167248i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	6.06218	−	3.50000i	0.500000	−	0.288675i
$148$		0			0
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	0.920904	−	0.389789i	$-0.127452\pi$
$149$	1.50000	+	2.59808i	0.122885	+	0.212843i	−0.798019	+	0.602632i	$0.794119\pi$
$150$		0			0
$151$	−16.0000			−1.30206			−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000			−1.30206			−0.651031	+	0.759051i	$0.725663\pi$
$152$		0			0
$153$			6.00000i			0.485071i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.06218	−	3.50000i	0.483814	−	0.279330i	−0.238190	−	0.971219i	$-0.576554\pi$
$157$	6.06218	−	3.50000i	0.483814	−	0.279330i	0.722005	+	0.691888i	$0.243221\pi$
$158$		0			0
$159$	11.0000			0.872357
$160$		0			0
$161$		0			0
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.9904	+	7.50000i	1.00523	+	0.580367i	0.909790	−	0.415068i	$-0.136242\pi$
$167$	12.9904	+	7.50000i	1.00523	+	0.580367i	0.0954356	+	0.995436i	$0.469576\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$	−7.00000	+	5.19615i	−0.535303	+	0.397360i
$172$		0			0
$173$	−12.9904	+	7.50000i	−0.987640	+	0.570214i	−0.904568	−	0.426329i	$-0.859807\pi$
$173$	−12.9904	+	7.50000i	−0.987640	+	0.570214i	−0.0830722	+	0.996544i	$0.526473\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2.59808	+	1.50000i	0.195283	+	0.112747i
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	2.50000	−	4.33013i	0.185824	−	0.321856i	−0.758030	−	0.652219i	$-0.773838\pi$
$181$	2.50000	−	4.33013i	0.185824	−	0.321856i	0.943854	+	0.330364i	$0.107171\pi$
$182$		0			0
$183$			11.0000i			0.813143i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−10.3923	+	6.00000i	−0.759961	+	0.438763i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$	−12.9904	+	7.50000i	−0.935068	+	0.539862i	−0.888411	−	0.459049i	$-0.848190\pi$
$193$	−12.9904	+	7.50000i	−0.935068	+	0.539862i	−0.0466572	+	0.998911i	$0.514857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			2.00000i			0.142494i	0.997459	+	0.0712470i	$0.0226979\pi$
$197$			2.00000i			0.142494i	−0.997459	+	0.0712470i	$0.977302\pi$
$198$		0			0
$199$	−3.50000	+	6.06218i	−0.248108	+	0.429736i	−0.963001	−	0.269498i	$-0.913142\pi$
$199$	−3.50000	+	6.06218i	−0.248108	+	0.429736i	0.714893	+	0.699234i	$0.246476\pi$
$200$		0			0
$201$	3.00000			0.211604
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.66025	+	5.00000i	−0.601929	+	0.347524i
$208$		0			0
$209$	−16.0000	−	6.92820i	−1.10674	−	0.479234i
$210$		0			0
$211$	4.50000	+	7.79423i	0.309793	+	0.536577i	0.978317	−	0.207114i	$-0.0664070\pi$
$211$	4.50000	+	7.79423i	0.309793	+	0.536577i	−0.668524	+	0.743690i	$0.733074\pi$
$212$		0			0
$213$	−9.52628	−	5.50000i	−0.652730	−	0.376854i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−7.50000	+	12.9904i	−0.506803	+	0.877809i
$220$		0			0
$221$	−3.00000			−0.201802
$222$		0			0
$223$	21.6506	−	12.5000i	1.44983	−	0.837062i	0.451363	−	0.892341i	$-0.350938\pi$
$223$	21.6506	−	12.5000i	1.44983	−	0.837062i	0.998471	+	0.0552786i	$0.0176047\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.1865	−	10.5000i	1.19144	−	0.687878i	0.232806	−	0.972523i	$-0.425209\pi$
$233$	18.1865	−	10.5000i	1.19144	−	0.687878i	0.958633	+	0.284645i	$0.0918758\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−11.2583	−	6.50000i	−0.731307	−	0.422220i
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	0.378963	+	0.925412i	$0.376281\pi$
$241$	−9.50000	+	16.4545i	−0.611949	+	1.05993i	−0.990912	+	0.134515i	$0.957053\pi$
$242$		0			0
$243$	−13.8564	−	8.00000i	−0.888889	−	0.513200i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.59808	−	3.50000i	−0.165312	−	0.222700i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.5000	−	26.8468i	0.978351	−	1.69455i	0.309951	−	0.950753i	$-0.399687\pi$
$251$	15.5000	−	26.8468i	0.978351	−	1.69455i	0.668400	−	0.743802i	$-0.266979\pi$
$252$		0			0
$253$	−17.3205	−	10.0000i	−1.08893	−	0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−19.9186	−	11.5000i	−1.24249	−	0.717350i	−0.272887	−	0.962046i	$-0.587979\pi$
$257$	−19.9186	−	11.5000i	−1.24249	−	0.717350i	−0.969600	+	0.244696i	$0.921312\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	7.00000	+	12.1244i	0.433289	+	0.750479i
$262$		0			0
$263$	7.79423	−	4.50000i	0.480613	−	0.277482i	−0.240059	−	0.970758i	$-0.577167\pi$
$263$	7.79423	−	4.50000i	0.480613	−	0.277482i	0.720672	+	0.693276i	$0.243833\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	3.00000i		−	0.183597i
$268$		0			0
$269$	13.5000	+	23.3827i	0.823110	+	1.42567i	0.903356	+	0.428892i	$0.141096\pi$
$269$	13.5000	+	23.3827i	0.823110	+	1.42567i	−0.0802460	+	0.996775i	$0.525571\pi$
$270$		0			0
$271$	−15.5000	−	26.8468i	−0.941558	−	1.63083i	−0.762501	−	0.646988i	$-0.776029\pi$
$271$	−15.5000	−	26.8468i	−0.941558	−	1.63083i	−0.179057	−	0.983839i	$-0.557305\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			10.0000i			0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$			10.0000i			0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$		0			0
$279$	−4.00000	+	6.92820i	−0.239474	+	0.414781i
$280$		0			0
$281$	−3.50000	+	6.06218i	−0.208792	+	0.361639i	−0.951334	−	0.308160i	$-0.900287\pi$
$281$	−3.50000	+	6.06218i	−0.208792	+	0.361639i	0.742542	+	0.669800i	$0.233620\pi$
$282$		0			0
$283$	7.79423	−	4.50000i	0.463319	−	0.267497i	−0.250120	−	0.968215i	$-0.580470\pi$
$283$	7.79423	−	4.50000i	0.463319	−	0.267497i	0.713439	+	0.700718i	$0.247137\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.00000	+	6.92820i	−0.235294	+	0.407541i
$290$		0			0
$291$	−2.50000	+	4.33013i	−0.146553	+	0.253837i
$292$		0			0
$293$			30.0000i			1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$			30.0000i			1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	20.0000i		−	1.16052i
$298$		0			0
$299$	−2.50000	−	4.33013i	−0.144579	−	0.250418i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		−	1.00000i		−	0.0574485i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	23.3827	−	13.5000i	1.33452	−	0.770486i	0.348532	−	0.937297i	$-0.386680\pi$
$307$	23.3827	−	13.5000i	1.33452	−	0.770486i	0.985989	+	0.166811i	$0.0533471\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−20.0000			−1.13410			−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000			−1.13410			−0.567048	+	0.823685i	$0.691915\pi$
$312$		0			0
$313$	9.52628	+	5.50000i	0.538457	+	0.310878i	0.744453	−	0.667674i	$-0.232710\pi$
$313$	9.52628	+	5.50000i	0.538457	+	0.310878i	−0.205996	+	0.978553i	$0.566043\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.9904	−	7.50000i	−0.729612	−	0.421242i	0.0886679	−	0.996061i	$-0.471739\pi$
$317$	−12.9904	−	7.50000i	−0.729612	−	0.421242i	−0.818280	+	0.574819i	$0.805072\pi$
$318$		0			0
$319$	−14.0000	+	24.2487i	−0.783850	+	1.35767i
$320$		0			0
$321$	10.0000	+	17.3205i	0.558146	+	0.966736i
$322$		0			0
$323$	12.9904	−	1.50000i	0.722804	−	0.0834622i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	2.59808	+	1.50000i	0.143674	+	0.0829502i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.0000			1.09930			0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000			1.09930			0.549650	+	0.835395i	$0.314761\pi$
$332$		0			0
$333$	−17.3205	−	10.0000i	−0.949158	−	0.547997i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−4.33013	+	2.50000i	−0.235877	+	0.136184i	−0.613280	−	0.789865i	$-0.710150\pi$
$337$	−4.33013	+	2.50000i	−0.235877	+	0.136184i	0.377403	+	0.926049i	$0.376817\pi$
$338$		0			0
$339$	−7.00000	−	12.1244i	−0.380188	−	0.658505i
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4.33013	+	2.50000i	−0.232453	+	0.134207i	−0.611703	−	0.791087i	$-0.709515\pi$
$347$	−4.33013	+	2.50000i	−0.232453	+	0.134207i	0.379250	+	0.925294i	$0.376182\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$	2.50000	−	4.33013i	0.133440	−	0.231125i
$352$		0			0
$353$		−	30.0000i		−	1.59674i	−0.602168	−	0.798369i	$-0.705696\pi$
$353$		−	30.0000i		−	1.59674i	0.602168	−	0.798369i	$-0.294304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.50000	+	12.9904i	0.395835	+	0.685606i	0.993207	−	0.116358i	$-0.0371219\pi$
$359$	7.50000	+	12.9904i	0.395835	+	0.685606i	−0.597372	+	0.801964i	$0.703789\pi$
$360$		0			0
$361$	13.0000	+	13.8564i	0.684211	+	0.729285i
$362$		0			0
$363$	4.33013	−	2.50000i	0.227273	−	0.131216i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−21.6506	−	12.5000i	−1.13015	−	0.652495i	−0.186180	−	0.982516i	$-0.559611\pi$
$367$	−21.6506	−	12.5000i	−1.13015	−	0.652495i	−0.943974	+	0.330021i	$0.892944\pi$
$368$		0			0
$369$	−10.0000			−0.520579
$370$		0			0
$371$		0			0
$372$		0			0
$373$			10.0000i			0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$			10.0000i			0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.06218	+	3.50000i	−0.312218	+	0.180259i
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$	3.00000			0.153695
$382$		0			0
$383$	25.1147	−	14.5000i	1.28330	−	0.740915i	0.305852	−	0.952079i	$-0.401059\pi$
$383$	25.1147	−	14.5000i	1.28330	−	0.740915i	0.977451	+	0.211164i	$0.0677253\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			10.0000i			0.508329i
$388$		0			0
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	−0.825491	−	0.564416i	$-0.809102\pi$
$389$	1.50000	−	2.59808i	0.0760530	−	0.131728i	0.901544	+	0.432688i	$0.142435\pi$
$390$		0			0
$391$	15.0000			0.758583
$392$		0			0
$393$	−12.9904	−	7.50000i	−0.655278	−	0.378325i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−21.6506	+	12.5000i	−1.08661	+	0.627357i	−0.932673	−	0.360723i	$-0.882530\pi$
$397$	−21.6506	+	12.5000i	−1.08661	+	0.627357i	−0.153941	+	0.988080i	$0.549197\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.50000	−	16.4545i	−0.474407	−	0.821698i	0.525163	−	0.851002i	$-0.324004\pi$
$401$	−9.50000	−	16.4545i	−0.474407	−	0.821698i	−0.999571	+	0.0293039i	$0.990671\pi$
$402$		0			0
$403$	−3.46410	−	2.00000i	−0.172559	−	0.0996271i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	40.0000i		−	1.98273i
$408$		0			0
$409$	−8.50000	+	14.7224i	−0.420298	+	0.727977i	−0.995968	−	0.0897044i	$-0.971408\pi$
$409$	−8.50000	+	14.7224i	−0.420298	+	0.727977i	0.575670	+	0.817682i	$0.304741\pi$
$410$		0			0
$411$	5.00000			0.246632
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	9.00000i		−	0.440732i
$418$		0			0
$419$	−12.0000			−0.586238			−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000			−0.586238			−0.293119	+	0.956076i	$0.594693\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	0.877952	−	0.478748i	$-0.158909\pi$
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	−0.853584	+	0.520955i	$0.825576\pi$
$422$		0			0
$423$	−12.1244	+	7.00000i	−0.589506	+	0.340352i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	0.494211	+	0.869342i	$0.335457\pi$
$431$	−10.5000	+	18.1865i	−0.505767	+	0.876014i	−0.999978	+	0.00667224i	$0.997876\pi$
$432$		0			0
$433$	−21.6506	−	12.5000i	−1.04046	−	0.600712i	−0.120499	−	0.992713i	$-0.538449\pi$
$433$	−21.6506	−	12.5000i	−1.04046	−	0.600712i	−0.919964	+	0.392002i	$0.871783\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.9904	+	17.5000i	0.621414	+	0.837139i
$438$		0			0
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	0.668184	−	0.743996i	$-0.267072\pi$
$439$	−6.50000	−	11.2583i	−0.310228	−	0.537331i	−0.978412	+	0.206666i	$0.933739\pi$
$440$		0			0
$441$	−7.00000	+	12.1244i	−0.333333	+	0.577350i
$442$		0			0
$443$	21.6506	+	12.5000i	1.02865	+	0.593893i	0.916598	−	0.399809i	$-0.130924\pi$
$443$	21.6506	+	12.5000i	1.02865	+	0.593893i	0.112054	+	0.993702i	$0.464257\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	2.59808	+	1.50000i	0.122885	+	0.0709476i
$448$		0			0
$449$	−22.0000			−1.03824			−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000			−1.03824			−0.519122	+	0.854700i	$0.673741\pi$
$450$		0			0
$451$	−10.0000	−	17.3205i	−0.470882	−	0.815591i
$452$		0			0
$453$	−13.8564	+	8.00000i	−0.651031	+	0.375873i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	22.0000i		−	1.02912i	−0.857455	−	0.514558i	$-0.827956\pi$
$457$		−	22.0000i		−	1.02912i	0.857455	−	0.514558i	$-0.172044\pi$
$458$		0			0
$459$	7.50000	+	12.9904i	0.350070	+	0.606339i
$460$		0			0
$461$	−5.50000	−	9.52628i	−0.256161	−	0.443683i	0.709050	−	0.705159i	$-0.249124\pi$
$461$	−5.50000	−	9.52628i	−0.256161	−	0.443683i	−0.965210	+	0.261476i	$0.915791\pi$
$462$		0			0
$463$			20.0000i			0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$			20.0000i			0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20.0000i		−	0.925490i	−0.886492	−	0.462745i	$-0.846865\pi$
$467$		−	20.0000i		−	0.925490i	0.886492	−	0.462745i	$-0.153135\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	3.50000	−	6.06218i	0.161271	−	0.279330i
$472$		0			0
$473$	−17.3205	+	10.0000i	−0.796398	+	0.459800i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−19.0526	+	11.0000i	−0.872357	+	0.503655i
$478$		0			0
$479$	−11.5000	+	19.9186i	−0.525448	+	0.910103i	0.474112	+	0.880464i	$0.342769\pi$
$479$	−11.5000	+	19.9186i	−0.525448	+	0.910103i	−0.999561	+	0.0296389i	$0.990564\pi$
$480$		0			0
$481$	5.00000	−	8.66025i	0.227980	−	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$		0			0
$489$	2.00000	+	3.46410i	0.0904431	+	0.156652i
$490$		0			0
$491$	0.500000	+	0.866025i	0.0225647	+	0.0390832i	0.877087	−	0.480331i	$-0.159483\pi$
$491$	0.500000	+	0.866025i	0.0225647	+	0.0390832i	−0.854523	+	0.519414i	$0.826150\pi$
$492$		0			0
$493$		−	21.0000i		−	0.945792i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	0.804627	−	0.593780i	$-0.202365\pi$
$499$	−2.50000	−	4.33013i	−0.111915	−	0.193843i	−0.916542	+	0.399937i	$0.869032\pi$
$500$		0			0
$501$	15.0000			0.670151
$502$		0			0
$503$	18.1865	+	10.5000i	0.810897	+	0.468172i	0.847267	−	0.531167i	$-0.178246\pi$
$503$	18.1865	+	10.5000i	0.810897	+	0.468172i	−0.0363700	+	0.999338i	$0.511579\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−10.3923	−	6.00000i	−0.461538	−	0.266469i
$508$		0			0
$509$	7.50000	−	12.9904i	0.332432	−	0.575789i	−0.650556	−	0.759458i	$-0.725464\pi$
$509$	7.50000	−	12.9904i	0.332432	−	0.575789i	0.982988	+	0.183669i	$0.0587976\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.66025	+	20.0000i	−0.382360	+	0.883022i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−24.2487	−	14.0000i	−1.06646	−	0.615719i
$518$		0			0
$519$	−7.50000	+	12.9904i	−0.329213	+	0.570214i
$520$		0			0
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$	−16.4545	−	9.50000i	−0.719504	−	0.415406i	0.0950659	−	0.995471i	$-0.469694\pi$
$523$	−16.4545	−	9.50000i	−0.719504	−	0.415406i	−0.814570	+	0.580065i	$0.803027\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	10.3923	−	6.00000i	0.452696	−	0.261364i
$528$		0			0
$529$	1.00000	+	1.73205i	0.0434783	+	0.0753066i
$530$		0			0
$531$	−6.00000			−0.260378
$532$		0			0
$533$		−	5.00000i		−	0.216574i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	10.3923	−	6.00000i	0.448461	−	0.258919i
$538$		0			0
$539$	−28.0000			−1.20605
$540$		0			0
$541$	0.500000	−	0.866025i	0.0214967	−	0.0372333i	−0.855077	−	0.518501i	$-0.826490\pi$
$541$	0.500000	−	0.866025i	0.0214967	−	0.0372333i	0.876574	+	0.481268i	$0.159824\pi$
$542$		0			0
$543$		−	5.00000i		−	0.214571i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−21.6506	−	12.5000i	−0.925714	−	0.534461i	−0.0402607	−	0.999189i	$-0.512819\pi$
$547$	−21.6506	−	12.5000i	−0.925714	−	0.534461i	−0.885454	+	0.464728i	$0.846152\pi$
$548$		0			0
$549$	−11.0000	−	19.0526i	−0.469469	−	0.813143i
$550$		0			0
$551$	24.5000	−	18.1865i	1.04374	−	0.774772i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.33013	+	2.50000i	0.183473	+	0.105928i	0.588924	−	0.808189i	$-0.299552\pi$
$557$	4.33013	+	2.50000i	0.183473	+	0.105928i	−0.405450	+	0.914117i	$0.632885\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$	−6.00000	+	10.3923i	−0.253320	+	0.438763i
$562$		0			0
$563$		−	36.0000i		−	1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$		−	36.0000i		−	1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−22.0000			−0.922288			−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000			−0.922288			−0.461144	+	0.887325i	$0.652561\pi$
$570$		0			0
$571$	−4.00000			−0.167395			−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000			−0.167395			−0.0836974	+	0.996491i	$0.526673\pi$
$572$		0			0
$573$	13.8564	−	8.00000i	0.578860	−	0.334205i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	10.0000i		−	0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$		−	10.0000i		−	0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$		0			0
$579$	−7.50000	+	12.9904i	−0.311689	+	0.539862i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−38.1051	−	22.0000i	−1.57815	−	0.911147i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.9904	−	7.50000i	0.536170	−	0.309558i	−0.207355	−	0.978266i	$-0.566486\pi$
$587$	12.9904	−	7.50000i	0.536170	−	0.309558i	0.743525	+	0.668708i	$0.233152\pi$
$588$		0			0
$589$	16.0000	+	6.92820i	0.659269	+	0.285472i
$590$		0			0
$591$	1.00000	+	1.73205i	0.0411345	+	0.0712470i
$592$		0			0
$593$	−4.33013	−	2.50000i	−0.177817	−	0.102663i	0.408450	−	0.912781i	$-0.366070\pi$
$593$	−4.33013	−	2.50000i	−0.177817	−	0.102663i	−0.586267	+	0.810118i	$0.699403\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			7.00000i			0.286491i
$598$		0			0
$599$	22.5000	−	38.9711i	0.919325	−	1.59232i	0.118882	−	0.992908i	$-0.462069\pi$
$599$	22.5000	−	38.9711i	0.919325	−	1.59232i	0.800443	−	0.599409i	$-0.204598\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$	−5.19615	+	3.00000i	−0.211604	+	0.122169i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			32.0000i			1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$			32.0000i			1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−3.50000	−	6.06218i	−0.141595	−	0.245249i
$612$		0			0
$613$	25.1147	−	14.5000i	1.01437	−	0.585649i	0.101905	−	0.994794i	$-0.467506\pi$
$613$	25.1147	−	14.5000i	1.01437	−	0.585649i	0.912470	+	0.409145i	$0.134173\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	38.9711	+	22.5000i	1.56892	+	0.905816i	0.996295	+	0.0859976i	$0.0274078\pi$
$617$	38.9711	+	22.5000i	1.56892	+	0.905816i	0.572624	+	0.819818i	$0.305926\pi$
$618$		0			0
$619$	20.0000			0.803868			0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000			0.803868			0.401934	+	0.915669i	$0.368338\pi$
$620$		0			0
$621$	−12.5000	+	21.6506i	−0.501608	+	0.868810i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−17.3205	+	2.00000i	−0.691714	+	0.0798723i
$628$		0			0
$629$	15.0000	+	25.9808i	0.598089	+	1.03592i
$630$		0			0
$631$	−20.5000	+	35.5070i	−0.816092	+	1.41351i	0.0924489	+	0.995717i	$0.470531\pi$
$631$	−20.5000	+	35.5070i	−0.816092	+	1.41351i	−0.908541	+	0.417796i	$0.862803\pi$
$632$		0			0
$633$	7.79423	+	4.50000i	0.309793	+	0.178859i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−6.06218	−	3.50000i	−0.240192	−	0.138675i
$638$		0			0
$639$	22.0000			0.870307
$640$		0			0
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	−0.937451	−	0.348117i	$-0.886821\pi$
$641$	−19.5000	−	33.7750i	−0.770204	−	1.33403i	0.167247	−	0.985915i	$-0.446512\pi$
$642$		0			0
$643$	−16.4545	+	9.50000i	−0.648901	+	0.374643i	−0.788035	−	0.615630i	$-0.788902\pi$
$643$	−16.4545	+	9.50000i	−0.648901	+	0.374643i	0.139134	+	0.990274i	$0.455568\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.0000i			1.25805i	0.777385	+	0.629025i	$0.216546\pi$
$647$			32.0000i			1.25805i	−0.777385	+	0.629025i	$0.783454\pi$
$648$		0			0
$649$	−6.00000	−	10.3923i	−0.235521	−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			30.0000i			1.17399i	0.809590	+	0.586995i	$0.199689\pi$
$653$			30.0000i			1.17399i	−0.809590	+	0.586995i	$0.800311\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	30.0000i		−	1.17041i
$658$		0			0
$659$	2.50000	−	4.33013i	0.0973862	−	0.168678i	−0.813216	−	0.581962i	$-0.802285\pi$
$659$	2.50000	−	4.33013i	0.0973862	−	0.168678i	0.910602	+	0.413284i	$0.135618\pi$
$660$		0			0
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	0.389503	+	0.921025i	$0.372647\pi$
$661$	−15.5000	+	26.8468i	−0.602880	+	1.04422i	−0.992383	+	0.123194i	$0.960686\pi$
$662$		0			0
$663$	−2.59808	+	1.50000i	−0.100901	+	0.0582552i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	30.3109	−	17.5000i	1.17364	−	0.677603i
$668$		0			0
$669$	12.5000	−	21.6506i	0.483278	−	0.837062i
$670$		0			0
$671$	22.0000	−	38.1051i	0.849301	−	1.47103i
$672$		0			0
$673$			10.0000i			0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$			10.0000i			0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			10.0000i			0.384331i	0.981363	+	0.192166i	$0.0615511\pi$
$677$			10.0000i			0.384331i	−0.981363	+	0.192166i	$0.938449\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−10.0000	−	17.3205i	−0.383201	−	0.663723i
$682$		0			0
$683$		−	16.0000i		−	0.612223i	−0.951996	−	0.306111i	$-0.900972\pi$
$683$		−	16.0000i		−	0.612223i	0.951996	−	0.306111i	$-0.0990280\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−1.73205	+	1.00000i	−0.0660819	+	0.0381524i
$688$		0			0
$689$	−5.50000	−	9.52628i	−0.209533	−	0.362922i
$690$		0			0
$691$	36.0000			1.36950			0.684752	−	0.728776i	$-0.259910\pi$
$691$	36.0000			1.36950			0.684752	+	0.728776i	$0.259910\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.9904	+	7.50000i	0.492046	+	0.284083i
$698$		0			0
$699$	10.5000	−	18.1865i	0.397146	−	0.687878i
$700$		0			0
$701$	−17.5000	−	30.3109i	−0.660966	−	1.14483i	−0.980362	−	0.197205i	$-0.936813\pi$
$701$	−17.5000	−	30.3109i	−0.660966	−	1.14483i	0.319396	−	0.947621i	$-0.396520\pi$
$702$		0			0
$703$	−17.3205	+	40.0000i	−0.653255	+	1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.50000	−	2.59808i	0.0563337	−	0.0975728i	−0.836483	−	0.547992i	$-0.815392\pi$
$709$	1.50000	−	2.59808i	0.0563337	−	0.0975728i	0.892817	+	0.450420i	$0.148726\pi$
$710$		0			0
$711$	26.0000			0.975076
$712$		0			0
$713$	17.3205	+	10.0000i	0.648658	+	0.374503i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−10.3923	+	6.00000i	−0.388108	+	0.224074i
$718$		0			0
$719$	3.50000	+	6.06218i	0.130528	+	0.226081i	0.923880	−	0.382682i	$-0.124999\pi$
$719$	3.50000	+	6.06218i	0.130528	+	0.226081i	−0.793352	+	0.608763i	$0.791666\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			19.0000i			0.706618i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	6.06218	−	3.50000i	0.224834	−	0.129808i	−0.383353	−	0.923602i	$-0.625231\pi$
$727$	6.06218	−	3.50000i	0.224834	−	0.129808i	0.608186	+	0.793794i	$0.291897\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	7.50000	−	12.9904i	0.277398	−	0.480467i
$732$		0			0
$733$			30.0000i			1.10808i	0.832492	+	0.554038i	$0.186914\pi$
$733$			30.0000i			1.10808i	−0.832492	+	0.554038i	$0.813086\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−10.3923	−	6.00000i	−0.382805	−	0.221013i
$738$		0			0
$739$	−6.50000	−	11.2583i	−0.239106	−	0.414144i	0.721352	−	0.692569i	$-0.243521\pi$
$739$	−6.50000	−	11.2583i	−0.239106	−	0.414144i	−0.960458	+	0.278425i	$0.910188\pi$
$740$		0			0
$741$	−4.00000	−	1.73205i	−0.146944	−	0.0636285i
$742$		0			0
$743$	21.6506	−	12.5000i	0.794285	−	0.458581i	−0.0471840	−	0.998886i	$-0.515025\pi$
$743$	21.6506	−	12.5000i	0.794285	−	0.458581i	0.841469	+	0.540306i	$0.181691\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−14.5000	+	25.1147i	−0.529113	+	0.916450i	0.470311	+	0.882501i	$0.344142\pi$
$751$	−14.5000	+	25.1147i	−0.529113	+	0.916450i	−0.999424	+	0.0339490i	$0.989192\pi$
$752$		0			0
$753$		−	31.0000i		−	1.12970i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−14.7224	+	8.50000i	−0.535096	+	0.308938i	−0.743089	−	0.669193i	$-0.766640\pi$
$757$	−14.7224	+	8.50000i	−0.535096	+	0.308938i	0.207993	+	0.978130i	$0.433307\pi$
$758$		0			0
$759$	−20.0000			−0.725954
$760$		0			0
$761$	30.0000			1.08750			0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000			1.08750			0.543750	+	0.839248i	$0.317004\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.00000i		−	0.108324i
$768$		0			0
$769$	3.50000	−	6.06218i	0.126213	−	0.218608i	−0.795993	−	0.605305i	$-0.793051\pi$
$769$	3.50000	−	6.06218i	0.126213	−	0.218608i	0.922207	+	0.386698i	$0.126384\pi$
$770$		0			0
$771$	−23.0000			−0.828325
$772$		0			0
$773$	−18.1865	−	10.5000i	−0.654124	−	0.377659i	0.135910	−	0.990721i	$-0.456604\pi$
$773$	−18.1865	−	10.5000i	−0.654124	−	0.377659i	−0.790034	+	0.613062i	$0.789937\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.50000	+	21.6506i	0.0895718	+	0.775715i
$780$		0			0
$781$	22.0000	+	38.1051i	0.787222	+	1.36351i
$782$		0			0
$783$	30.3109	+	17.5000i	1.08322	+	0.625399i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			52.0000i			1.85360i	0.375555	+	0.926800i	$0.377452\pi$
$787$			52.0000i			1.85360i	−0.375555	+	0.926800i	$0.622548\pi$
$788$		0			0
$789$	4.50000	−	7.79423i	0.160204	−	0.277482i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	9.52628	−	5.50000i	0.338288	−	0.195311i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.0000i			0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$			18.0000i			0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$		0			0
$799$	21.0000			0.742927
$800$		0			0
$801$	3.00000	+	5.19615i	0.106000	+	0.183597i
$802$		0			0
$803$	51.9615	−	30.0000i	1.83368	−	1.05868i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	23.3827	+	13.5000i	0.823110	+	0.475223i
$808$		0			0
$809$	18.0000			0.632846			0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000			0.632846			0.316423	+	0.948618i	$0.397518\pi$
$810$		0			0
$811$	−10.5000	+	18.1865i	−0.368705	+	0.638616i	−0.989363	−	0.145465i	$-0.953532\pi$
$811$	−10.5000	+	18.1865i	−0.368705	+	0.638616i	0.620658	+	0.784081i	$0.286865\pi$
$812$		0			0
$813$	−26.8468	−	15.5000i	−0.941558	−	0.543609i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	21.6506	−	2.50000i	0.757460	−	0.0874639i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	−0.776753	−	0.629805i	$-0.783135\pi$
$821$	4.50000	−	7.79423i	0.157051	−	0.272020i	0.933804	+	0.357785i	$0.116468\pi$
$822$		0			0
$823$	−26.8468	−	15.5000i	−0.935820	−	0.540296i	−0.0471726	−	0.998887i	$-0.515021\pi$
$823$	−26.8468	−	15.5000i	−0.935820	−	0.540296i	−0.888648	+	0.458591i	$0.848354\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.06218	+	3.50000i	0.210803	+	0.121707i	0.601684	−	0.798734i	$-0.294497\pi$
$827$	6.06218	+	3.50000i	0.210803	+	0.121707i	−0.390882	+	0.920441i	$0.627830\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$	5.00000	+	8.66025i	0.173448	+	0.300421i
$832$		0			0
$833$	18.1865	−	10.5000i	0.630126	−	0.363803i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			20.0000i			0.691301i
$838$		0			0
$839$	−12.5000	−	21.6506i	−0.431548	−	0.747463i	0.565459	−	0.824776i	$-0.308699\pi$
$839$	−12.5000	−	21.6506i	−0.431548	−	0.747463i	−0.997007	+	0.0773135i	$0.975366\pi$
$840$		0			0
$841$	−10.0000	−	17.3205i	−0.344828	−	0.597259i
$842$		0			0
$843$			7.00000i			0.241093i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	4.50000	−	7.79423i	0.154440	−	0.267497i
$850$		0			0
$851$	−25.0000	+	43.3013i	−0.856989	+	1.48435i
$852$		0			0
$853$	−33.7750	+	19.5000i	−1.15643	+	0.667667i	−0.950447	−	0.310887i	$-0.899374\pi$
$853$	−33.7750	+	19.5000i	−1.15643	+	0.667667i	−0.205987	+	0.978555i	$0.566040\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.9904	−	7.50000i	0.443743	−	0.256195i	−0.261441	−	0.965219i	$-0.584198\pi$
$857$	12.9904	−	7.50000i	0.443743	−	0.256195i	0.705184	+	0.709024i	$0.250864\pi$
$858$		0			0
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	6.50000	−	11.2583i	0.221777	−	0.384129i	0.955348	+	0.295484i	$0.0954809\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			36.0000i			1.22545i	0.790295	+	0.612727i	$0.209928\pi$
$863$			36.0000i			1.22545i	−0.790295	+	0.612727i	$0.790072\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			8.00000i			0.271694i
$868$		0			0
$869$	26.0000	+	45.0333i	0.881990	+	1.52765i
$870$		0			0
$871$	−1.50000	−	2.59808i	−0.0508256	−	0.0880325i
$872$		0			0
$873$		−	10.0000i		−	0.338449i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−4.33013	+	2.50000i	−0.146218	+	0.0844190i	−0.571324	−	0.820724i	$-0.693570\pi$
$877$	−4.33013	+	2.50000i	−0.146218	+	0.0844190i	0.425106	+	0.905143i	$0.360237\pi$
$878$		0			0
$879$	15.0000	+	25.9808i	0.505937	+	0.876309i
$880$		0			0
$881$	6.00000			0.202145			0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000			0.202145			0.101073	+	0.994879i	$0.467773\pi$
$882$		0			0
$883$	21.6506	+	12.5000i	0.728602	+	0.420658i	0.817910	−	0.575346i	$-0.195132\pi$
$883$	21.6506	+	12.5000i	0.728602	+	0.420658i	−0.0893086	+	0.996004i	$0.528466\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.6506	−	12.5000i	−0.726957	−	0.419709i	0.0903508	−	0.995910i	$-0.471201\pi$
$887$	−21.6506	−	12.5000i	−0.726957	−	0.419709i	−0.817308	+	0.576201i	$0.804535\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.00000	+	3.46410i	0.0670025	+	0.116052i
$892$		0			0
$893$	18.1865	+	24.5000i	0.608589	+	0.819861i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−4.33013	−	2.50000i	−0.144579	−	0.0834726i
$898$		0			0
$899$	14.0000	−	24.2487i	0.466926	−	0.808740i
$900$		0			0
$901$	33.0000			1.09939
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	30.3109	−	17.5000i	1.00646	−	0.581078i	0.0963043	−	0.995352i	$-0.469298\pi$
$907$	30.3109	−	17.5000i	1.00646	−	0.581078i	0.910152	+	0.414274i	$0.135964\pi$
$908$		0			0
$909$	1.00000	+	1.73205i	0.0331679	+	0.0574485i
$910$		0			0
$911$	−24.0000			−0.795155			−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000			−0.795155			−0.397578	+	0.917568i	$0.630149\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$	13.5000	−	23.3827i	0.444840	−	0.770486i
$922$		0			0
$923$			11.0000i			0.362069i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	11.5000	+	19.9186i	0.377303	+	0.653508i	0.990669	−	0.136291i	$-0.0435183\pi$
$929$	11.5000	+	19.9186i	0.377303	+	0.653508i	−0.613366	+	0.789799i	$0.710185\pi$
$930$		0			0
$931$	28.0000	+	12.1244i	0.917663	+	0.397360i
$932$		0			0
$933$	−17.3205	+	10.0000i	−0.567048	+	0.327385i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−19.9186	−	11.5000i	−0.650712	−	0.375689i	0.138017	−	0.990430i	$-0.455927\pi$
$937$	−19.9186	−	11.5000i	−0.650712	−	0.375689i	−0.788729	+	0.614741i	$0.789260\pi$
$938$		0			0
$939$	11.0000			0.358971
$940$		0			0
$941$	−9.50000	+	16.4545i	−0.309691	+	0.536401i	−0.978295	−	0.207218i	$-0.933559\pi$
$941$	−9.50000	+	16.4545i	−0.309691	+	0.536401i	0.668604	+	0.743619i	$0.266892\pi$
$942$		0			0
$943$			25.0000i			0.814112i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.3827	−	13.5000i	0.759835	−	0.438691i	−0.0694014	−	0.997589i	$-0.522109\pi$
$947$	23.3827	−	13.5000i	0.759835	−	0.438691i	0.829237	+	0.558898i	$0.188776\pi$
$948$		0			0
$949$	15.0000			0.486921
$950$		0			0
$951$	−15.0000			−0.486408
$952$		0			0
$953$	21.6506	−	12.5000i	0.701333	−	0.404915i	−0.106511	−	0.994312i	$-0.533968\pi$
$953$	21.6506	−	12.5000i	0.701333	−	0.404915i	0.807844	+	0.589397i	$0.200635\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$			28.0000i			0.905111i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.0000			−0.483871
$962$		0			0
$963$	−34.6410	−	20.0000i	−1.11629	−	0.644491i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−38.9711	+	22.5000i	−1.25323	+	0.723551i	−0.971749	−	0.236016i	$-0.924158\pi$
$967$	−38.9711	+	22.5000i	−1.25323	+	0.723551i	−0.281478	+	0.959568i	$0.590825\pi$
$968$		0			0
$969$	10.5000	−	7.79423i	0.337309	−	0.250387i
$970$		0			0
$971$	−17.5000	−	30.3109i	−0.561602	−	0.972723i	−0.997357	−	0.0726575i	$-0.976852\pi$
$971$	−17.5000	−	30.3109i	−0.561602	−	0.972723i	0.435755	−	0.900065i	$-0.356481\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	10.0000i		−	0.319928i	−0.987123	−	0.159964i	$-0.948862\pi$
$977$		−	10.0000i		−	0.319928i	0.987123	−	0.159964i	$-0.0511379\pi$
$978$		0			0
$979$	−6.00000	+	10.3923i	−0.191761	+	0.332140i
$980$		0			0
$981$	−6.00000			−0.191565
$982$		0			0
$983$	38.9711	−	22.5000i	1.24299	−	0.717639i	0.273285	−	0.961933i	$-0.411890\pi$
$983$	38.9711	−	22.5000i	1.24299	−	0.717639i	0.969701	+	0.244294i	$0.0785563\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	25.0000			0.794954
$990$		0			0
$991$	2.50000	+	4.33013i	0.0794151	+	0.137551i	0.902998	−	0.429645i	$-0.141361\pi$
$991$	2.50000	+	4.33013i	0.0794151	+	0.137551i	−0.823583	+	0.567196i	$0.808028\pi$
$992$		0			0
$993$	17.3205	−	10.0000i	0.549650	−	0.317340i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	45.8993	+	26.5000i	1.45365	+	0.839263i	0.998686	−	0.0512480i	$-0.0163199\pi$
$997$	45.8993	+	26.5000i	1.45365	+	0.839263i	0.454961	+	0.890511i	$0.349653\pi$
$998$		0			0
$999$	−50.0000			−1.58193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.s.a.349.2		4
5.2	odd	4		1900.2.i.a.501.1		2
5.3	odd	4		76.2.e.a.45.1	✓	2
5.4	even	2	inner	1900.2.s.a.349.1		4
15.8	even	4		684.2.k.b.577.1		2
19.11	even	3	inner	1900.2.s.a.49.1		4
20.3	even	4		304.2.i.a.273.1		2
40.3	even	4		1216.2.i.g.577.1		2
40.13	odd	4		1216.2.i.c.577.1		2
60.23	odd	4		2736.2.s.g.577.1		2
95.8	even	12		1444.2.e.b.429.1		2
95.18	even	4		1444.2.e.b.653.1		2
95.49	even	6	inner	1900.2.s.a.49.2		4
95.68	odd	12		76.2.e.a.49.1	yes	2
95.83	odd	12		1444.2.a.b.1.1		1
95.87	odd	12		1900.2.i.a.201.1		2
95.88	even	12		1444.2.a.c.1.1		1
285.68	even	12		684.2.k.b.505.1		2
380.83	even	12		5776.2.a.k.1.1		1
380.163	even	12		304.2.i.a.49.1		2
380.183	odd	12		5776.2.a.f.1.1		1
760.163	even	12		1216.2.i.g.961.1		2
760.733	odd	12		1216.2.i.c.961.1		2
1140.923	odd	12		2736.2.s.g.1873.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.2.e.a.45.1	✓	2	5.3	odd	4
76.2.e.a.49.1	yes	2	95.68	odd	12
304.2.i.a.49.1		2	380.163	even	12
304.2.i.a.273.1		2	20.3	even	4
684.2.k.b.505.1		2	285.68	even	12
684.2.k.b.577.1		2	15.8	even	4
1216.2.i.c.577.1		2	40.13	odd	4
1216.2.i.c.961.1		2	760.733	odd	12
1216.2.i.g.577.1		2	40.3	even	4
1216.2.i.g.961.1		2	760.163	even	12
1444.2.a.b.1.1		1	95.83	odd	12
1444.2.a.c.1.1		1	95.88	even	12
1444.2.e.b.429.1		2	95.8	even	12
1444.2.e.b.653.1		2	95.18	even	4
1900.2.i.a.201.1		2	95.87	odd	12
1900.2.i.a.501.1		2	5.2	odd	4
1900.2.s.a.49.1		4	19.11	even	3	inner
1900.2.s.a.49.2		4	95.49	even	6	inner
1900.2.s.a.349.1		4	5.4	even	2	inner
1900.2.s.a.349.2		4	1.1	even	1	trivial
2736.2.s.g.577.1		2	60.23	odd	4
2736.2.s.g.1873.1		2	1140.923	odd	12
5776.2.a.f.1.1		1	380.183	odd	12
5776.2.a.k.1.1		1	380.83	even	12

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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{3} +(-1.00000 + 1.73205i) q^{9} -4.00000 q^{11} +(-0.866025 - 0.500000i) q^{13} +(2.59808 - 1.50000i) q^{17} +(4.00000 + 1.73205i) q^{19} +(4.33013 + 2.50000i) q^{23} +5.00000i q^{27} +(3.50000 - 6.06218i) q^{29} +4.00000 q^{31} +(-3.46410 + 2.00000i) q^{33} +10.0000i q^{37} -1.00000 q^{39} +(2.50000 + 4.33013i) q^{41} +(4.33013 - 2.50000i) q^{43} +(6.06218 + 3.50000i) q^{47} +7.00000 q^{49} +(1.50000 - 2.59808i) q^{51} +(9.52628 + 5.50000i) q^{53} +(4.33013 - 0.500000i) q^{57} +(1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(2.59808 + 1.50000i) q^{67} +5.00000 q^{69} +(-5.50000 - 9.52628i) q^{71} +(-12.9904 + 7.50000i) q^{73} +(-6.50000 - 11.2583i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.00000i q^{87} +(1.50000 - 2.59808i) q^{89} +(3.46410 - 2.00000i) q^{93} +(-4.33013 + 2.50000i) q^{97} +(4.00000 - 6.92820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 16 q^{11} + 16 q^{19} + 14 q^{29} + 16 q^{31} - 4 q^{39} + 10 q^{41} + 28 q^{49} + 6 q^{51} + 6 q^{59} - 22 q^{61} + 20 q^{69} - 22 q^{71} - 26 q^{79} - 2 q^{81} + 6 q^{89} + 16 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 16 * q^11 + 16 * q^19 + 14 * q^29 + 16 * q^31 - 4 * q^39 + 10 * q^41 + 28 * q^49 + 6 * q^51 + 6 * q^59 - 22 * q^61 + 20 * q^69 - 22 * q^71 - 26 * q^79 - 2 * q^81 + 6 * q^89 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more