LMFDB - Embedded newform 1728.2.r.f.1441.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(289,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1441.5
Root			$0.583700 - 2.17840i$ of defining polynomial
Character	$\chi$	$=$	1728.1441
Dual form			1728.2.r.f.289.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 + 0.866025i) q^{5} +(1.80664 + 3.12920i) q^{7} +O(q^{10})$	$q+(1.50000 + 0.866025i) q^{5} +(1.80664 + 3.12920i) q^{7} +(0.635828 - 0.367095i) q^{11} +(0.527909 + 0.304788i) q^{13} +5.52420 q^{17} -2.00000i q^{19} +(2.36788 - 4.10129i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(6.78630 - 3.91807i) q^{29} +(-4.70951 + 8.15710i) q^{31} +6.25839i q^{35} +2.34163i q^{37} +(-4.26210 + 7.38217i) q^{41} +(-8.88403 + 5.12920i) q^{43} +(5.88032 + 10.1850i) q^{47} +(-3.02791 + 5.24449i) q^{49} -13.0323i q^{53} +1.27166 q^{55} +(1.04788 + 0.604996i) q^{59} +(9.78630 - 5.65012i) q^{61} +(0.527909 + 0.914365i) q^{65} +(-5.46826 - 3.15710i) q^{67} +2.63999 q^{71} -2.05582 q^{73} +(2.29743 + 1.32642i) q^{77} +(1.24540 + 2.15710i) q^{79} +(-10.6161 + 6.12920i) q^{83} +(8.28630 + 4.78410i) q^{85} +1.94418 q^{89} +2.20257i q^{91} +(1.73205 - 3.00000i) q^{95} +(7.78630 + 13.4863i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 18 q^{5}+O(q^{10}) $ 12 * q + 18 * q^5	$ 12 q + 18 q^{5} - 6 q^{13} - 12 q^{25} - 18 q^{29} - 18 q^{41} - 24 q^{49} + 18 q^{61} - 6 q^{65} + 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) $ 12 * q + 18 * q^5 - 6 * q^13 - 12 * q^25 - 18 * q^29 - 18 * q^41 - 24 * q^49 + 18 * q^61 - 6 * q^65 + 90 * q^77 + 48 * q^89 - 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.50000	+	0.866025i	0.670820	+	0.387298i	0.796387	−	0.604787i	$-0.206742\pi$
$5$	1.50000	+	0.866025i	0.670820	+	0.387298i	−0.125567	+	0.992085i	$0.540075\pi$
$6$		0			0
$7$	1.80664	+	3.12920i	0.682846	+	1.18272i	0.974108	+	0.226081i	$0.0725915\pi$
$7$	1.80664	+	3.12920i	0.682846	+	1.18272i	−0.291262	+	0.956643i	$0.594075\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.635828	−	0.367095i	0.191709	−	0.110683i	−0.401073	−	0.916046i	$-0.631363\pi$
$11$	0.635828	−	0.367095i	0.191709	−	0.110683i	0.592783	+	0.805363i	$0.298029\pi$
$12$		0			0
$13$	0.527909	+	0.304788i	0.146416	+	0.0845331i	0.571418	−	0.820659i	$-0.306393\pi$
$13$	0.527909	+	0.304788i	0.146416	+	0.0845331i	−0.425003	+	0.905192i	$0.639727\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.52420			1.33982			0.669908	−	0.742444i	$-0.266334\pi$
$17$	5.52420			1.33982			0.669908	+	0.742444i	$0.266334\pi$
$18$		0			0
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.36788	−	4.10129i	0.493737	−	0.855177i	−0.506237	−	0.862394i	$-0.668964\pi$
$23$	2.36788	−	4.10129i	0.493737	−	0.855177i	0.999974	+	0.00721700i	$0.00229726\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.78630	−	3.91807i	1.26018	−	0.727568i	0.287074	−	0.957908i	$-0.407317\pi$
$29$	6.78630	−	3.91807i	1.26018	−	0.727568i	0.973110	+	0.230341i	$0.0739841\pi$
$30$		0			0
$31$	−4.70951	+	8.15710i	−0.845852	+	1.46506i	0.0390267	+	0.999238i	$0.487574\pi$
$31$	−4.70951	+	8.15710i	−0.845852	+	1.46506i	−0.884879	+	0.465821i	$0.845759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			6.25839i			1.05786i
$36$		0			0
$37$			2.34163i			0.384961i	0.981301	+	0.192481i	$0.0616532\pi$
$37$			2.34163i			0.384961i	−0.981301	+	0.192481i	$0.938347\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.26210	+	7.38217i	−0.665628	+	1.15290i	0.313486	+	0.949593i	$0.398503\pi$
$41$	−4.26210	+	7.38217i	−0.665628	+	1.15290i	−0.979115	+	0.203309i	$0.934830\pi$
$42$		0			0
$43$	−8.88403	+	5.12920i	−1.35480	+	0.782195i	−0.988918	−	0.148466i	$-0.952567\pi$
$43$	−8.88403	+	5.12920i	−1.35480	+	0.782195i	−0.365884	+	0.930661i	$0.619233\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.88032	+	10.1850i	0.857733	+	1.48564i	0.874086	+	0.485771i	$0.161461\pi$
$47$	5.88032	+	10.1850i	0.857733	+	1.48564i	−0.0163535	+	0.999866i	$0.505206\pi$
$48$		0			0
$49$	−3.02791	+	5.24449i	−0.432558	+	0.749213i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	13.0323i		−	1.79012i	−0.445942	−	0.895062i	$-0.647131\pi$
$53$		−	13.0323i		−	1.79012i	0.445942	−	0.895062i	$-0.352869\pi$
$54$		0			0
$55$	1.27166			0.171470
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.04788	+	0.604996i	0.136423	+	0.0787637i	0.566658	−	0.823953i	$-0.308236\pi$
$59$	1.04788	+	0.604996i	0.136423	+	0.0787637i	−0.430235	+	0.902717i	$0.641569\pi$
$60$		0			0
$61$	9.78630	−	5.65012i	1.25301	−	0.723424i	0.281302	−	0.959619i	$-0.409234\pi$
$61$	9.78630	−	5.65012i	1.25301	−	0.723424i	0.971706	+	0.236195i	$0.0759005\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.527909	+	0.914365i	0.0654790	+	0.113413i
$66$		0			0
$67$	−5.46826	−	3.15710i	−0.668055	−	0.385702i	0.127284	−	0.991866i	$-0.459374\pi$
$67$	−5.46826	−	3.15710i	−0.668055	−	0.385702i	−0.795339	+	0.606165i	$0.792707\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.63999			0.313309			0.156655	−	0.987653i	$-0.449929\pi$
$71$	2.63999			0.313309			0.156655	+	0.987653i	$0.449929\pi$
$72$		0			0
$73$	−2.05582			−0.240615			−0.120308	−	0.992737i	$-0.538388\pi$
$73$	−2.05582			−0.240615			−0.120308	+	0.992737i	$0.538388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	2.29743	+	1.32642i	0.261816	+	0.151160i
$78$		0			0
$79$	1.24540	+	2.15710i	0.140119	+	0.242693i	0.927541	−	0.373721i	$-0.121918\pi$
$79$	1.24540	+	2.15710i	0.140119	+	0.242693i	−0.787422	+	0.616414i	$0.788585\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.6161	+	6.12920i	−1.16527	+	0.672767i	−0.952560	−	0.304350i	$-0.901561\pi$
$83$	−10.6161	+	6.12920i	−1.16527	+	0.672767i	−0.212706	+	0.977116i	$0.568228\pi$
$84$		0			0
$85$	8.28630	+	4.78410i	0.898775	+	0.518908i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.94418			0.206083			0.103041	−	0.994677i	$-0.467143\pi$
$89$	1.94418			0.206083			0.103041	+	0.994677i	$0.467143\pi$
$90$		0			0
$91$			2.20257i			0.230892i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.73205	−	3.00000i	0.177705	−	0.307794i
$96$		0			0
$97$	7.78630	+	13.4863i	0.790579	+	1.36932i	0.925609	+	0.378481i	$0.123554\pi$
$97$	7.78630	+	13.4863i	0.790579	+	1.36932i	−0.135030	+	0.990842i	$0.543113\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.70257	+	5.60178i	−0.965442	+	0.557398i	−0.897844	−	0.440314i	$-0.854867\pi$
$101$	−9.70257	+	5.60178i	−0.965442	+	0.557398i	−0.0675984	+	0.997713i	$0.521534\pi$
$102$		0			0
$103$	−0.684168	+	1.18501i	−0.0674130	+	0.116763i	−0.897762	−	0.440481i	$-0.854808\pi$
$103$	−0.684168	+	1.18501i	−0.0674130	+	0.116763i	0.830349	+	0.557244i	$0.188141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	13.4684i		−	1.30204i	−0.759062	−	0.651019i	$-0.774342\pi$
$107$		−	13.4684i		−	1.30204i	0.759062	−	0.651019i	$-0.225658\pi$
$108$		0			0
$109$		−	1.42084i		−	0.136092i	−0.997682	−	0.0680458i	$-0.978324\pi$
$109$		−	1.42084i		−	0.136092i	0.997682	−	0.0680458i	$-0.0216764\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.75839	−	8.24177i	0.447632	−	0.775321i	−0.550600	−	0.834769i	$-0.685601\pi$
$113$	4.75839	−	8.24177i	0.447632	−	0.775321i	0.998231	+	0.0594485i	$0.0189342\pi$
$114$		0			0
$115$	7.10364	−	4.10129i	0.662418	−	0.382447i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	9.98025	+	17.2863i	0.914888	+	1.58463i
$120$		0			0
$121$	−5.23048	+	9.05946i	−0.475498	+	0.823587i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$	2.24495			0.199207			0.0996035	−	0.995027i	$-0.468243\pi$
$127$	2.24495			0.199207			0.0996035	+	0.995027i	$0.468243\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.223773	+	0.129195i	0.0195511	+	0.0112878i	0.509744	−	0.860326i	$-0.329740\pi$
$131$	0.223773	+	0.129195i	0.0195511	+	0.0112878i	−0.490193	+	0.871614i	$0.663074\pi$
$132$		0			0
$133$	6.25839	−	3.61328i	0.542671	−	0.313311i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00371	−	1.73848i	−0.0857527	−	0.148528i	0.819959	−	0.572422i	$-0.193996\pi$
$137$	−1.00371	−	1.73848i	−0.0857527	−	0.148528i	−0.905712	+	0.423894i	$0.860663\pi$
$138$		0			0
$139$	1.95582	+	1.12920i	0.165891	+	0.0957771i	0.580647	−	0.814156i	$-0.302800\pi$
$139$	1.95582	+	1.12920i	0.165891	+	0.0957771i	−0.414756	+	0.909933i	$0.636133\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.447546			0.0374256
$144$		0			0
$145$	13.5726			1.12714
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.7863	+	7.38217i	1.04749	+	0.604771i	0.921947	−	0.387316i	$-0.126598\pi$
$149$	12.7863	+	7.38217i	1.04749	+	0.604771i	0.125548	+	0.992088i	$0.459931\pi$
$150$		0			0
$151$	2.82827	+	4.89871i	0.230162	+	0.398652i	0.957856	−	0.287250i	$-0.0927412\pi$
$151$	2.82827	+	4.89871i	0.230162	+	0.398652i	−0.727694	+	0.685902i	$0.759408\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−14.1285	+	8.15710i	−1.13483	+	0.655194i
$156$		0			0
$157$	13.7584	+	7.94341i	1.09804	+	0.633953i	0.935705	−	0.352783i	$-0.114765\pi$
$157$	13.7584	+	7.94341i	1.09804	+	0.633953i	0.162334	+	0.986736i	$0.448098\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	17.1116			1.34859
$162$		0			0
$163$		−	8.51678i		−	0.667086i	−0.942735	−	0.333543i	$-0.891756\pi$
$163$		−	8.51678i		−	0.667086i	0.942735	−	0.333543i	$-0.108244\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.95582	+	3.38759i	−0.151346	+	0.262139i	−0.931723	−	0.363171i	$-0.881694\pi$
$167$	−1.95582	+	3.38759i	−0.151346	+	0.262139i	0.780376	+	0.625310i	$0.215027\pi$
$168$		0			0
$169$	−6.31421	−	10.9365i	−0.485708	−	0.841271i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−10.5000	+	6.06218i	−0.798300	+	0.460899i	−0.842876	−	0.538107i	$-0.819140\pi$
$173$	−10.5000	+	6.06218i	−0.798300	+	0.460899i	0.0445762	+	0.999006i	$0.485806\pi$
$174$		0			0
$175$	3.61328	−	6.25839i	0.273139	−	0.473090i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$		0			0
$181$		−	2.34163i		−	0.174052i	−0.996206	−	0.0870259i	$-0.972264\pi$
$181$		−	2.34163i		−	0.174052i	0.996206	−	0.0870259i	$-0.0277363\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.02791	+	3.51244i	−0.149095	+	0.258240i
$186$		0			0
$187$	3.51244	−	2.02791i	0.256855	−	0.148295i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.684168	−	1.18501i	−0.0495046	−	0.0857445i	0.840211	−	0.542259i	$-0.182431\pi$
$191$	−0.684168	−	1.18501i	−0.0495046	−	0.0857445i	−0.889716	+	0.456515i	$0.849098\pi$
$192$		0			0
$193$	7.73048	−	13.3896i	0.556452	−	0.963804i	−0.441337	−	0.897342i	$-0.645496\pi$
$193$	7.73048	−	13.3896i	0.556452	−	0.963804i	0.997789	−	0.0664620i	$-0.0211711\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	12.9356i		−	0.921625i	−0.887498	−	0.460812i	$-0.847558\pi$
$197$		−	12.9356i		−	0.921625i	0.887498	−	0.460812i	$-0.152442\pi$
$198$		0			0
$199$	−18.7413			−1.32854			−0.664269	−	0.747493i	$-0.731257\pi$
$199$	−18.7413			−1.32854			−0.664269	+	0.747493i	$0.731257\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	24.5208	+	14.1571i	1.72102	+	0.993634i
$204$		0			0
$205$	−12.7863	+	7.38217i	−0.893034	+	0.515593i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.734191	−	1.27166i	−0.0507851	−	0.0879623i
$210$		0			0
$211$	7.20032	+	4.15710i	0.495690	+	0.286187i	0.726932	−	0.686709i	$-0.240946\pi$
$211$	7.20032	+	4.15710i	0.495690	+	0.286187i	−0.231242	+	0.972896i	$0.574279\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−17.7681			−1.21177
$216$		0			0
$217$	−34.0336			−2.31035
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.91627	+	1.68371i	0.196170	+	0.113259i
$222$		0			0
$223$	−3.99909	−	6.92662i	−0.267799	−	0.463841i	0.700494	−	0.713658i	$-0.252963\pi$
$223$	−3.99909	−	6.92662i	−0.267799	−	0.463841i	−0.968293	+	0.249817i	$0.919630\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.56032	−	2.63290i	0.302679	−	0.174752i	−0.340967	−	0.940075i	$-0.610754\pi$
$227$	4.56032	−	2.63290i	0.302679	−	0.174752i	0.643646	+	0.765323i	$0.277421\pi$
$228$		0			0
$229$	−14.0168	−	8.09259i	−0.926255	−	0.534774i	−0.0406298	−	0.999174i	$-0.512936\pi$
$229$	−14.0168	−	8.09259i	−0.926255	−	0.534774i	−0.885625	+	0.464401i	$0.846270\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−28.0968			−1.84068			−0.920341	−	0.391116i	$-0.872089\pi$
$233$	−28.0968			−1.84068			−0.920341	+	0.391116i	$0.872089\pi$
$234$		0			0
$235$			20.3700i			1.32879i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.29608	−	16.1013i	0.601314	−	1.04151i	−0.391309	−	0.920259i	$-0.627978\pi$
$239$	9.29608	−	16.1013i	0.601314	−	1.04151i	0.992622	−	0.121246i	$-0.0386891\pi$
$240$		0			0
$241$	4.55582	+	7.89091i	0.293466	+	0.508298i	0.974627	−	0.223836i	$-0.0718579\pi$
$241$	4.55582	+	7.89091i	0.293466	+	0.508298i	−0.681161	+	0.732134i	$0.738525\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.08373	+	5.24449i	−0.580338	+	0.335058i
$246$		0			0
$247$	0.609577	−	1.05582i	0.0387864	−	0.0671801i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	21.5800i		−	1.36212i	−0.732228	−	0.681059i	$-0.761520\pi$
$251$		−	21.5800i		−	1.36212i	0.732228	−	0.681059i	$-0.238480\pi$
$252$		0			0
$253$		−	3.47695i		−	0.218594i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.70999	+	4.69384i	−0.169045	+	0.292794i	−0.938084	−	0.346407i	$-0.887402\pi$
$257$	−2.70999	+	4.69384i	−0.169045	+	0.292794i	0.769040	+	0.639201i	$0.220735\pi$
$258$		0			0
$259$	−7.32741	+	4.23048i	−0.455303	+	0.262869i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−5.00787	−	8.67389i	−0.308798	−	0.534855i	0.669301	−	0.742991i	$-0.266594\pi$
$263$	−5.00787	−	8.67389i	−0.308798	−	0.534855i	−0.978100	+	0.208136i	$0.933260\pi$
$264$		0			0
$265$	11.2863	−	19.5484i	0.693312	−	1.20085i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			12.1115i			0.738452i	0.929340	+	0.369226i	$0.120377\pi$
$269$			12.1115i			0.738452i	−0.929340	+	0.369226i	$0.879623\pi$
$270$		0			0
$271$	20.0572			1.21839			0.609193	−	0.793022i	$-0.291493\pi$
$271$	20.0572			1.21839			0.609193	+	0.793022i	$0.291493\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.27166	−	0.734191i	−0.0766837	−	0.0442734i
$276$		0			0
$277$	−4.75839	+	2.74726i	−0.285904	+	0.165067i	−0.636093	−	0.771612i	$-0.719451\pi$
$277$	−4.75839	+	2.74726i	−0.285904	+	0.165067i	0.350189	+	0.936679i	$0.386117\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.58373	−	7.93925i	−0.273442	−	0.473616i	0.696299	−	0.717752i	$-0.254829\pi$
$281$	−4.58373	−	7.93925i	−0.273442	−	0.473616i	−0.969741	+	0.244136i	$0.921496\pi$
$282$		0			0
$283$	−19.7239	−	11.3876i	−1.17246	−	0.676922i	−0.218204	−	0.975903i	$-0.570020\pi$
$283$	−19.7239	−	11.3876i	−1.17246	−	0.676922i	−0.954259	+	0.298981i	$0.903353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−30.8003			−1.81809
$288$		0			0
$289$	13.5168			0.795105
$290$		0			0
$291$		0			0
$292$		0			0
$293$	18.7026	+	10.7979i	1.09262	+	0.630822i	0.934272	−	0.356562i	$-0.116051\pi$
$293$	18.7026	+	10.7979i	1.09262	+	0.630822i	0.158344	+	0.987384i	$0.449384\pi$
$294$		0			0
$295$	1.04788	+	1.81499i	0.0610101	+	0.105673i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.50005	−	1.44340i	0.144582	−	0.0834742i
$300$		0			0
$301$	−32.1005	−	18.5332i	−1.85024	−	1.06824i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	19.5726			1.12072
$306$		0			0
$307$			8.51678i			0.486078i	0.970016	+	0.243039i	$0.0781444\pi$
$307$			8.51678i			0.486078i	−0.970016	+	0.243039i	$0.921856\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.3965	−	21.4713i	0.702939	−	1.21753i	−0.264491	−	0.964388i	$-0.585204\pi$
$311$	12.3965	−	21.4713i	0.702939	−	1.21753i	0.967430	−	0.253138i	$-0.0814628\pi$
$312$		0			0
$313$	−7.55582	−	13.0871i	−0.427080	−	0.739724i	0.569532	−	0.821969i	$-0.307124\pi$
$313$	−7.55582	−	13.0871i	−0.427080	−	0.739724i	−0.996612	+	0.0822447i	$0.973791\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5837	+	6.11052i	−0.594441	+	0.343201i	−0.766852	−	0.641824i	$-0.778178\pi$
$317$	−10.5837	+	6.11052i	−0.594441	+	0.343201i	0.172410	+	0.985025i	$0.444844\pi$
$318$		0			0
$319$	2.87661	−	4.98244i	0.161059	−	0.278963i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	11.0484i		−	0.614749i
$324$		0			0
$325$		−	1.21915i		−	0.0676264i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−21.2473	+	36.8013i	−1.17140	+	2.02892i
$330$		0			0
$331$	21.4076	−	12.3597i	1.17667	−	0.679349i	0.221426	−	0.975177i	$-0.428929\pi$
$331$	21.4076	−	12.3597i	1.17667	−	0.679349i	0.955241	+	0.295828i	$0.0955955\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5.46826	−	9.47131i	−0.298763	−	0.517473i
$336$		0			0
$337$	−6.75839	+	11.7059i	−0.368153	+	0.637660i	−0.989277	−	0.146053i	$-0.953343\pi$
$337$	−6.75839	+	11.7059i	−0.368153	+	0.637660i	0.621124	+	0.783712i	$0.286676\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.91535i			0.374487i
$342$		0			0
$343$	3.41160			0.184209
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−13.3044	−	7.68130i	−0.714218	−	0.412354i	0.0984028	−	0.995147i	$-0.468627\pi$
$347$	−13.3044	−	7.68130i	−0.714218	−	0.412354i	−0.812621	+	0.582793i	$0.801960\pi$
$348$		0			0
$349$	−27.3310	+	15.7796i	−1.46299	+	0.844660i	−0.999149	−	0.0412558i	$-0.986864\pi$
$349$	−27.3310	+	15.7796i	−1.46299	+	0.844660i	−0.463846	+	0.885916i	$0.653531\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−13.9963	−	24.2423i	−0.744947	−	1.29029i	−0.950219	−	0.311582i	$-0.899141\pi$
$353$	−13.9963	−	24.2423i	−0.744947	−	1.29029i	0.205272	−	0.978705i	$-0.434192\pi$
$354$		0			0
$355$	3.95999	+	2.28630i	0.210174	+	0.121344i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−13.1290			−0.692921			−0.346460	−	0.938065i	$-0.612616\pi$
$359$	−13.1290			−0.692921			−0.346460	+	0.938065i	$0.612616\pi$
$360$		0			0
$361$	15.0000			0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.08373	−	1.78039i	−0.161410	−	0.0931899i
$366$		0			0
$367$	−6.73992	−	11.6739i	−0.351821	−	0.609372i	0.634748	−	0.772720i	$-0.281104\pi$
$367$	−6.73992	−	11.6739i	−0.351821	−	0.609372i	−0.986569	+	0.163348i	$0.947771\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	40.7806	−	23.5447i	2.11722	−	1.22238i
$372$		0			0
$373$	−4.84212	−	2.79560i	−0.250715	−	0.144751i	0.369376	−	0.929280i	$-0.379571\pi$
$373$	−4.84212	−	2.79560i	−0.250715	−	0.144751i	−0.620092	+	0.784529i	$0.712905\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4.77673			0.246014
$378$		0			0
$379$		−	33.0894i		−	1.69969i	−0.527035	−	0.849844i	$-0.676696\pi$
$379$		−	33.0894i		−	1.69969i	0.527035	−	0.849844i	$-0.323304\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.61237	−	13.1850i	0.388974	−	0.673723i	−0.603338	−	0.797486i	$-0.706163\pi$
$383$	7.61237	−	13.1850i	0.388974	−	0.673723i	0.992312	+	0.123763i	$0.0394963\pi$
$384$		0			0
$385$	2.29743	+	3.97926i	0.117088	+	0.202802i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.29743	+	3.05847i	−0.268590	+	0.155071i	−0.628247	−	0.778014i	$-0.716227\pi$
$389$	−5.29743	+	3.05847i	−0.268590	+	0.155071i	0.359657	+	0.933085i	$0.382894\pi$
$390$		0			0
$391$	13.0806	−	22.6563i	0.661516	−	1.14578i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			4.31421i			0.217071i
$396$		0			0
$397$			22.2054i			1.11446i	0.830358	+	0.557230i	$0.188136\pi$
$397$			22.2054i			1.11446i	−0.830358	+	0.557230i	$0.811864\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0521	−	26.0710i	0.751666	−	1.30192i	−0.195348	−	0.980734i	$-0.562584\pi$
$401$	15.0521	−	26.0710i	0.751666	−	1.30192i	0.947015	−	0.321190i	$-0.104083\pi$
$402$		0			0
$403$	−4.97238	+	2.87080i	−0.247692	+	0.143005i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.859601	+	1.48887i	0.0426088	+	0.0738007i
$408$		0			0
$409$	4.55582	−	7.89091i	0.225271	−	0.390180i	−0.731130	−	0.682238i	$-0.761007\pi$
$409$	4.55582	−	7.89091i	0.225271	−	0.390180i	0.956401	+	0.292058i	$0.0943400\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			4.37204i			0.215134i
$414$		0			0
$415$	−21.2322			−1.04225
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−19.7722	−	11.4155i	−0.965936	−	0.557683i	−0.0679411	−	0.997689i	$-0.521643\pi$
$419$	−19.7722	−	11.4155i	−0.965936	−	0.557683i	−0.897995	+	0.440006i	$0.854976\pi$
$420$		0			0
$421$	−21.2473	+	12.2671i	−1.03553	+	0.597863i	−0.918563	−	0.395274i	$-0.870650\pi$
$421$	−21.2473	+	12.2671i	−1.03553	+	0.597863i	−0.116965	+	0.993136i	$0.537316\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.52420	−	9.56819i	−0.267963	−	0.464126i
$426$		0			0
$427$	35.3607	+	20.4155i	1.71122	+	0.987975i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.920789			0.0443529			0.0221764	−	0.999754i	$-0.492940\pi$
$431$	0.920789			0.0443529			0.0221764	+	0.999754i	$0.492940\pi$
$432$		0			0
$433$	2.05582			0.0987963			0.0493981	−	0.998779i	$-0.484270\pi$
$433$	2.05582			0.0987963			0.0493981	+	0.998779i	$0.484270\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.20257	−	4.73576i	−0.392382	−	0.226542i
$438$		0			0
$439$	−11.0281	−	19.1013i	−0.526344	−	0.911655i	−0.999529	−	0.0306915i	$-0.990229\pi$
$439$	−11.0281	−	19.1013i	−0.526344	−	0.911655i	0.473185	−	0.880963i	$-0.343104\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−23.5517	+	13.5976i	−1.11897	+	0.646040i	−0.941139	−	0.338020i	$-0.890243\pi$
$443$	−23.5517	+	13.5976i	−1.11897	+	0.646040i	−0.177836	+	0.984060i	$0.556910\pi$
$444$		0			0
$445$	2.91627	+	1.68371i	0.138245	+	0.0798156i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9.10422			0.429655			0.214827	−	0.976652i	$-0.431081\pi$
$449$	9.10422			0.429655			0.214827	+	0.976652i	$0.431081\pi$
$450$		0			0
$451$			6.25839i			0.294696i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.90748	+	3.30386i	−0.0894242	+	0.154887i
$456$		0			0
$457$	−14.8421	−	25.7073i	−0.694285	−	1.20254i	−0.970421	−	0.241418i	$-0.922387\pi$
$457$	−14.8421	−	25.7073i	−0.694285	−	1.20254i	0.276136	−	0.961119i	$-0.410946\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.3589	+	11.7542i	−0.948208	+	0.547448i	−0.892524	−	0.451000i	$-0.851067\pi$
$461$	−20.3589	+	11.7542i	−0.948208	+	0.547448i	−0.0556845	+	0.998448i	$0.517734\pi$
$462$		0			0
$463$	−3.95075	+	6.84290i	−0.183607	+	0.318016i	−0.943106	−	0.332492i	$-0.892111\pi$
$463$	−3.95075	+	6.84290i	−0.183607	+	0.318016i	0.759499	+	0.650508i	$0.225444\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			3.06324i			0.141750i	0.997485	+	0.0708748i	$0.0225791\pi$
$467$			3.06324i			0.141750i	−0.997485	+	0.0708748i	$0.977421\pi$
$468$		0			0
$469$		−	22.8150i		−	1.05350i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−3.76581	+	6.52257i	−0.173152	+	0.299908i
$474$		0			0
$475$	−3.46410	+	2.00000i	−0.158944	+	0.0917663i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.10364	+	12.3039i	0.324573	+	0.562178i	0.981426	−	0.191841i	$-0.0614459\pi$
$479$	7.10364	+	12.3039i	0.324573	+	0.562178i	−0.656852	+	0.754019i	$0.728113\pi$
$480$		0			0
$481$	−0.713701	+	1.23617i	−0.0325419	+	0.0563643i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			26.9725i			1.22476i
$486$		0			0
$487$	22.1088			1.00184			0.500922	−	0.865492i	$-0.332994\pi$
$487$	22.1088			1.00184			0.500922	+	0.865492i	$0.332994\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	34.2340	+	19.7650i	1.54496	+	0.891983i	0.998514	+	0.0544890i	$0.0173530\pi$
$491$	34.2340	+	19.7650i	1.54496	+	0.891983i	0.546446	+	0.837494i	$0.315980\pi$
$492$		0			0
$493$	37.4889	−	21.6442i	1.68841	−	0.974806i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.76952	+	8.26105i	0.213942	+	0.370559i
$498$		0			0
$499$	10.2652	+	5.92662i	0.459534	+	0.265312i	0.711848	−	0.702333i	$-0.247858\pi$
$499$	10.2652	+	5.92662i	0.459534	+	0.265312i	−0.252314	+	0.967645i	$0.581192\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−23.4246			−1.04445			−0.522226	−	0.852807i	$-0.674898\pi$
$503$	−23.4246			−1.04445			−0.522226	+	0.852807i	$0.674898\pi$
$504$		0			0
$505$	−19.4051			−0.863518
$506$		0			0
$507$		0			0
$508$		0			0
$509$	6.78630	+	3.91807i	0.300797	+	0.173665i	0.642801	−	0.766033i	$-0.277772\pi$
$509$	6.78630	+	3.91807i	0.300797	+	0.173665i	−0.342004	+	0.939699i	$0.611106\pi$
$510$		0			0
$511$	−3.71413	−	6.43305i	−0.164303	−	0.284582i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.05250	+	1.18501i	−0.0904441	+	0.0522179i
$516$		0			0
$517$	7.47774	+	4.31728i	0.328871	+	0.189874i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.4535			−1.20276			−0.601381	−	0.798963i	$-0.705383\pi$
$521$	−27.4535			−1.20276			−0.601381	+	0.798963i	$0.705383\pi$
$522$		0			0
$523$		−	3.83255i		−	0.167586i	−0.996483	−	0.0837928i	$-0.973297\pi$
$523$		−	3.83255i		−	0.167586i	0.996483	−	0.0837928i	$-0.0267034\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−26.0163	+	45.0615i	−1.13329	+	1.96291i
$528$		0			0
$529$	0.286299	+	0.495885i	0.0124478	+	0.0215602i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.50000	+	2.59808i	−0.194917	+	0.112535i
$534$		0			0
$535$	11.6640	−	20.2026i	0.504277	−	0.873433i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			4.44613i			0.191508i
$540$		0			0
$541$		−	15.9707i		−	0.686632i	−0.939220	−	0.343316i	$-0.888450\pi$
$541$		−	15.9707i		−	0.686632i	0.939220	−	0.343316i	$-0.111550\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.23048	−	2.13126i	0.0527080	−	0.0912930i
$546$		0			0
$547$	9.02905	−	5.21292i	0.386054	−	0.222888i	−0.294395	−	0.955684i	$-0.595118\pi$
$547$	9.02905	−	5.21292i	0.386054	−	0.222888i	0.680449	+	0.732795i	$0.261785\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−7.83614	−	13.5726i	−0.333831	−	0.578212i
$552$		0			0
$553$	−4.50000	+	7.79423i	−0.191359	+	0.331444i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	25.0471i		−	1.06128i	−0.847597	−	0.530640i	$-0.821951\pi$
$557$		−	25.0471i		−	1.06128i	0.847597	−	0.530640i	$-0.178049\pi$
$558$		0			0
$559$	−6.25327			−0.264485
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.28405	−	1.31870i	−0.0962612	−	0.0555764i	0.451097	−	0.892475i	$-0.351033\pi$
$563$	−2.28405	−	1.31870i	−0.0962612	−	0.0555764i	−0.547358	+	0.836899i	$0.684366\pi$
$564$		0			0
$565$	14.2752	−	8.24177i	0.600561	−	0.346734i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−10.0242	−	17.3624i	−0.420236	−	0.727871i	0.575726	−	0.817643i	$-0.304719\pi$
$569$	−10.0242	−	17.3624i	−0.420236	−	0.727871i	−0.995962	+	0.0897720i	$0.971386\pi$
$570$		0			0
$571$	−2.00416	−	1.15710i	−0.0838716	−	0.0484233i	0.457478	−	0.889221i	$-0.348753\pi$
$571$	−2.00416	−	1.15710i	−0.0838716	−	0.0484233i	−0.541349	+	0.840798i	$0.682086\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−9.47152			−0.394989
$576$		0			0
$577$	24.4610			1.01832			0.509162	−	0.860671i	$-0.329956\pi$
$577$	24.4610			1.01832			0.509162	+	0.860671i	$0.329956\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−38.3589	−	22.1465i	−1.59140	−	0.918792i
$582$		0			0
$583$	−4.78410	−	8.28630i	−0.198137	−	0.343183i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−17.0839	+	9.86339i	−0.705127	+	0.407106i	−0.809254	−	0.587459i	$-0.800129\pi$
$587$	−17.0839	+	9.86339i	−0.705127	+	0.407106i	0.104127	+	0.994564i	$0.466795\pi$
$588$		0			0
$589$	16.3142	+	9.41901i	0.672215	+	0.388104i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8.46096			−0.347450			−0.173725	−	0.984794i	$-0.555580\pi$
$593$	−8.46096			−0.347450			−0.173725	+	0.984794i	$0.555580\pi$
$594$		0			0
$595$			34.5726i			1.41734i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.22748	−	5.59016i	0.131871	−	0.228408i	−0.792527	−	0.609837i	$-0.791235\pi$
$599$	3.22748	−	5.59016i	0.131871	−	0.228408i	0.924398	+	0.381430i	$0.124568\pi$
$600$		0			0
$601$	2.52791	+	4.37847i	0.103116	+	0.178601i	0.912967	−	0.408034i	$-0.133786\pi$
$601$	2.52791	+	4.37847i	0.103116	+	0.178601i	−0.809851	+	0.586635i	$0.800452\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−15.6914	+	9.05946i	−0.637948	+	0.368319i
$606$		0			0
$607$	−0.0220880	+	0.0382575i	−0.000896524	+	0.00155282i	−0.866473	−	0.499223i	$-0.833619\pi$
$607$	−0.0220880	+	0.0382575i	−0.000896524	+	0.00155282i	0.865577	+	0.500776i	$0.166952\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			7.16901i			0.290027i
$612$		0			0
$613$			26.8887i			1.08602i	0.839725	+	0.543012i	$0.182716\pi$
$613$			26.8887i			1.08602i	−0.839725	+	0.543012i	$0.817284\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	9.20628	−	15.9457i	0.370631	−	0.641952i	−0.619032	−	0.785366i	$-0.712475\pi$
$617$	9.20628	−	15.9457i	0.370631	−	0.641952i	0.989663	+	0.143414i	$0.0458081\pi$
$618$		0			0
$619$	−1.65330	+	0.954531i	−0.0664516	+	0.0383658i	−0.532858	−	0.846205i	$-0.678882\pi$
$619$	−1.65330	+	0.954531i	−0.0664516	+	0.0383658i	0.466406	+	0.884571i	$0.345549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	3.51244	+	6.08373i	0.140723	+	0.243739i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			12.9356i			0.515777i
$630$		0			0
$631$	1.51752			0.0604114			0.0302057	−	0.999544i	$-0.490384\pi$
$631$	1.51752			0.0604114			0.0302057	+	0.999544i	$0.490384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.36742	+	1.94418i	0.133632	+	0.0771525i
$636$		0			0
$637$	−3.19692	+	1.84574i	−0.126667	+	0.0731310i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.49258	+	14.7096i	0.335437	+	0.580994i	0.983569	−	0.180535i	$-0.0577828\pi$
$641$	8.49258	+	14.7096i	0.335437	+	0.580994i	−0.648132	+	0.761528i	$0.724449\pi$
$642$		0			0
$643$	−9.02905	−	5.21292i	−0.356071	−	0.205578i	0.311285	−	0.950317i	$-0.399241\pi$
$643$	−9.02905	−	5.21292i	−0.356071	−	0.205578i	−0.667356	+	0.744739i	$0.732574\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	44.2802			1.74083			0.870417	−	0.492315i	$-0.163849\pi$
$647$	44.2802			1.74083			0.870417	+	0.492315i	$0.163849\pi$
$648$		0			0
$649$	0.888365			0.0348714
$650$		0			0
$651$		0			0
$652$		0			0
$653$	28.6452	+	16.5383i	1.12097	+	0.647194i	0.941649	−	0.336596i	$-0.109276\pi$
$653$	28.6452	+	16.5383i	1.12097	+	0.647194i	0.179324	+	0.983790i	$0.442609\pi$
$654$		0			0
$655$	0.223773	+	0.387586i	0.00874353	+	0.0151442i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	27.8882	−	16.1013i	1.08637	−	0.627217i	0.153764	−	0.988108i	$-0.450861\pi$
$659$	27.8882	−	16.1013i	1.08637	−	0.627217i	0.932608	+	0.360891i	$0.117527\pi$
$660$		0			0
$661$	−11.2752	−	6.50972i	−0.438553	−	0.253199i	0.264430	−	0.964405i	$-0.414816\pi$
$661$	−11.2752	−	6.50972i	−0.438553	−	0.253199i	−0.702984	+	0.711206i	$0.748149\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.5168			0.485380
$666$		0			0
$667$		−	37.1101i		−	1.43691i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.14827	−	7.18501i	0.160142	−	0.277374i
$672$		0			0
$673$	3.73048	+	6.46138i	0.143800	+	0.249068i	0.928924	−	0.370269i	$-0.120735\pi$
$673$	3.73048	+	6.46138i	0.143800	+	0.249068i	−0.785125	+	0.619337i	$0.787401\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	30.8700	−	17.8228i	1.18643	−	0.684987i	0.228938	−	0.973441i	$-0.426475\pi$
$677$	30.8700	−	17.8228i	1.18643	−	0.684987i	0.957494	+	0.288455i	$0.0931414\pi$
$678$		0			0
$679$	−28.1341	+	48.7297i	−1.07969	+	1.87007i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27.4535i		−	1.05048i	−0.850954	−	0.525240i	$-0.823975\pi$
$683$		−	27.4535i		−	1.05048i	0.850954	−	0.525240i	$-0.176025\pi$
$684$		0			0
$685$		−	3.47695i		−	0.132847i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.97209	−	6.87986i	0.151325	−	0.262102i
$690$		0			0
$691$	−19.3730	+	11.1850i	−0.736984	+	0.425498i	−0.820972	−	0.570969i	$-0.806568\pi$
$691$	−19.3730	+	11.1850i	−0.736984	+	0.425498i	0.0839877	+	0.996467i	$0.473234\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.95582	+	3.38759i	0.0741886	+	0.128498i
$696$		0			0
$697$	−23.5447	+	40.7806i	−0.891819	+	1.54468i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	10.4633i		−	0.395193i	−0.980283	−	0.197596i	$-0.936686\pi$
$701$		−	10.4633i		−	0.395193i	0.980283	−	0.197596i	$-0.0633136\pi$
$702$		0			0
$703$	4.68325			0.176632
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−35.0581	−	20.2408i	−1.31850	−	0.761235i
$708$		0			0
$709$	20.3884	−	11.7712i	0.765701	−	0.442078i	−0.0656378	−	0.997844i	$-0.520908\pi$
$709$	20.3884	−	11.7712i	0.765701	−	0.442078i	0.831339	+	0.555766i	$0.187575\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	22.3031	+	38.6301i	0.835257	+	1.44671i
$714$		0			0
$715$	0.671318	+	0.387586i	0.0251059	+	0.0144949i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	45.0076			1.67850			0.839251	−	0.543745i	$-0.182994\pi$
$719$	45.0076			1.67850			0.839251	+	0.543745i	$0.182994\pi$
$720$		0			0
$721$	−4.94418			−0.184131
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13.5726	−	7.83614i	−0.504074	−	0.291027i
$726$		0			0
$727$	8.37529	+	14.5064i	0.310622	+	0.538014i	0.978497	−	0.206260i	$-0.0661292\pi$
$727$	8.37529	+	14.5064i	0.310622	+	0.538014i	−0.667875	+	0.744274i	$0.732796\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−49.0771	+	28.3347i	−1.81518	+	1.04800i
$732$		0			0
$733$	−21.0798	−	12.1704i	−0.778601	−	0.449525i	0.0573335	−	0.998355i	$-0.481740\pi$
$733$	−21.0798	−	12.1704i	−0.778601	−	0.449525i	−0.835934	+	0.548830i	$0.815073\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.63583			−0.170763
$738$		0			0
$739$		−	32.0558i		−	1.17919i	−0.807698	−	0.589596i	$-0.799287\pi$
$739$		−	32.0558i		−	1.17919i	0.807698	−	0.589596i	$-0.200713\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−8.98071	+	15.5550i	−0.329470	+	0.570659i	−0.982407	−	0.186753i	$-0.940203\pi$
$743$	−8.98071	+	15.5550i	−0.329470	+	0.570659i	0.652937	+	0.757413i	$0.273537\pi$
$744$		0			0
$745$	12.7863	+	22.1465i	0.468454	+	0.811386i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	42.1452	−	24.3325i	1.53995	−	0.889092i
$750$		0			0
$751$	10.9273	−	18.9266i	0.398742	−	0.690642i	−0.594829	−	0.803853i	$-0.702780\pi$
$751$	10.9273	−	18.9266i	0.398742	−	0.690642i	0.993571	+	0.113210i	$0.0361134\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			9.79743i			0.356565i
$756$		0			0
$757$			52.1292i			1.89467i	0.320248	+	0.947334i	$0.396234\pi$
$757$			52.1292i			1.89467i	−0.320248	+	0.947334i	$0.603766\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	16.0447	−	27.7902i	0.581620	−	1.00739i	−0.413668	−	0.910428i	$-0.635753\pi$
$761$	16.0447	−	27.7902i	0.581620	−	1.00739i	0.995288	−	0.0969668i	$-0.0309141\pi$
$762$		0			0
$763$	4.44608	−	2.56695i	0.160959	−	0.0929297i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.368791	+	0.638765i	0.0133163	+	0.0230645i
$768$		0			0
$769$	−1.61164	+	2.79143i	−0.0581171	+	0.100662i	−0.893620	−	0.448824i	$-0.851843\pi$
$769$	−1.61164	+	2.79143i	−0.0581171	+	0.100662i	0.835503	+	0.549486i	$0.185176\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	19.2331i		−	0.691765i	−0.938278	−	0.345883i	$-0.887580\pi$
$773$		−	19.2331i		−	0.691765i	0.938278	−	0.345883i	$-0.112420\pi$
$774$		0			0
$775$	18.8380			0.676682
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.7643	+	8.52420i	0.528988	+	0.305411i
$780$		0			0
$781$	1.67858	−	0.969129i	0.0600643	−	0.0346782i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13.7584	+	23.8302i	0.491058	+	0.850537i
$786$		0			0
$787$	−31.4974	−	18.1850i	−1.12276	−	0.648226i	−0.180656	−	0.983546i	$-0.557822\pi$
$787$	−31.4974	−	18.1850i	−1.12276	−	0.648226i	−0.942104	+	0.335321i	$0.891155\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	34.3868			1.22265
$792$		0			0
$793$	6.88836			0.244613
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.9538	+	7.47885i	0.458845	+	0.264915i	0.711559	−	0.702627i	$-0.247990\pi$
$797$	12.9538	+	7.47885i	0.458845	+	0.264915i	−0.252713	+	0.967541i	$0.581323\pi$
$798$		0			0
$799$	32.4841	+	56.2640i	1.14920	+	1.99048i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.30715	+	0.754681i	−0.0461282	+	0.0266321i
$804$		0			0
$805$	25.6675	+	14.8191i	0.904659	+	0.522305i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.0409808			0.00144081			0.000720404	−	1.00000i	$-0.499771\pi$
$809$	0.0409808			0.00144081			0.000720404		1.00000i	$0.499771\pi$
$810$		0			0
$811$			43.2568i			1.51895i	0.650535	+	0.759476i	$0.274545\pi$
$811$			43.2568i			1.51895i	−0.650535	+	0.759476i	$0.725455\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.37575	−	12.7752i	0.258361	−	0.447495i
$816$		0			0
$817$	10.2584	+	17.7681i	0.358896	+	0.621626i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	24.7863	−	14.3104i	0.865048	−	0.499436i	−0.000651604	−	1.00000i	$-0.500207\pi$
$821$	24.7863	−	14.3104i	0.865048	−	0.499436i	0.865699	+	0.500564i	$0.166874\pi$
$822$		0			0
$823$	−21.2059	+	36.7297i	−0.739191	+	1.28032i	0.213669	+	0.976906i	$0.431459\pi$
$823$	−21.2059	+	36.7297i	−0.739191	+	1.28032i	−0.952860	+	0.303411i	$0.901875\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	14.1116i		−	0.490710i	−0.969433	−	0.245355i	$-0.921096\pi$
$827$		−	14.1116i		−	0.490710i	0.969433	−	0.245355i	$-0.0789045\pi$
$828$		0			0
$829$		−	22.6088i		−	0.785237i	−0.919701	−	0.392618i	$-0.871569\pi$
$829$		−	22.6088i		−	0.785237i	0.919701	−	0.392618i	$-0.128431\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.7268	+	28.9716i	−0.579548	+	1.00381i
$834$		0			0
$835$	−5.86747	+	3.38759i	−0.203052	+	0.117232i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−7.05530	−	12.2201i	−0.243576	−	0.421886i	0.718154	−	0.695884i	$-0.244987\pi$
$839$	−7.05530	−	12.2201i	−0.243576	−	0.421886i	−0.961730	+	0.273998i	$0.911654\pi$
$840$		0			0
$841$	16.2026	−	28.0637i	0.558709	−	0.967713i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	21.8731i		−	0.752456i
$846$		0			0
$847$	−37.7984			−1.29877
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.60368	+	5.54469i	0.329210	+	0.190070i
$852$		0			0
$853$	−36.7863	+	21.2386i	−1.25954	+	0.727195i	−0.972985	−	0.230869i	$-0.925843\pi$
$853$	−36.7863	+	21.2386i	−1.25954	+	0.727195i	−0.286554	+	0.958064i	$0.592510\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.8421	+	29.1714i	0.575316	+	0.996476i	0.996007	+	0.0892724i	$0.0284542\pi$
$857$	16.8421	+	29.1714i	0.575316	+	0.996476i	−0.420691	+	0.907204i	$0.638213\pi$
$858$		0			0
$859$	−39.8165	−	22.9881i	−1.35852	−	0.784344i	−0.369098	−	0.929390i	$-0.620333\pi$
$859$	−39.8165	−	22.9881i	−1.35852	−	0.784344i	−0.989425	+	0.145047i	$0.953667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	31.8786			1.08516			0.542581	−	0.840004i	$-0.317447\pi$
$863$	31.8786			1.08516			0.542581	+	0.840004i	$0.317447\pi$
$864$		0			0
$865$	−21.0000			−0.714021
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.58373	+	0.914365i	0.0537242	+	0.0310177i
$870$		0			0
$871$	−1.92450	−	3.33333i	−0.0652091	−	0.112945i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	37.9395	−	21.9044i	1.28259	−	0.740503i
$876$		0			0
$877$	45.3925	+	26.2073i	1.53279	+	0.884959i	0.999231	+	0.0391990i	$0.0124806\pi$
$877$	45.3925	+	26.2073i	1.53279	+	0.884959i	0.533563	+	0.845760i	$0.320853\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−34.0558			−1.14737			−0.573685	−	0.819076i	$-0.694487\pi$
$881$	−34.0558			−1.14737			−0.573685	+	0.819076i	$0.694487\pi$
$882$		0			0
$883$			10.0000i			0.336527i	0.985742	+	0.168263i	$0.0538159\pi$
$883$			10.0000i			0.336527i	−0.985742	+	0.168263i	$0.946184\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−18.0530	+	31.2687i	−0.606161	+	1.04990i	0.385706	+	0.922622i	$0.373958\pi$
$887$	−18.0530	+	31.2687i	−0.606161	+	1.04990i	−0.991867	+	0.127280i	$0.959375\pi$
$888$		0			0
$889$	4.05582	+	7.02488i	0.136028	+	0.235607i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.3700	−	11.7606i	0.681657	−	0.393555i
$894$		0			0
$895$	−5.19615	+	9.00000i	−0.173688	+	0.300837i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			73.8087i			2.46166i
$900$		0			0
$901$		−	71.9930i		−	2.39843i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.02791	−	3.51244i	0.0674100	−	0.116757i
$906$		0			0
$907$	−36.6935	+	21.1850i	−1.21839	+	0.703437i	−0.964573	−	0.263816i	$-0.915019\pi$
$907$	−36.6935	+	21.1850i	−1.21839	+	0.703437i	−0.253815	+	0.967253i	$0.581686\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.3321	−	43.8765i	−0.839289	−	1.45369i	−0.890490	−	0.455004i	$-0.849638\pi$
$911$	−25.3321	−	43.8765i	−0.839289	−	1.45369i	0.0512002	−	0.998688i	$-0.483695\pi$
$912$		0			0
$913$	−4.50000	+	7.79423i	−0.148928	+	0.257951i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.933638i			0.0308315i
$918$		0			0
$919$	−25.3372			−0.835796			−0.417898	−	0.908494i	$-0.637233\pi$
$919$	−25.3372			−0.835796			−0.417898	+	0.908494i	$0.637233\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.39367	+	0.804638i	0.0458734	+	0.0264850i
$924$		0			0
$925$	4.05582	−	2.34163i	0.133354	−	0.0769922i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−7.67466	−	13.2929i	−0.251798	−	0.436126i	0.712223	−	0.701953i	$-0.247688\pi$
$929$	−7.67466	−	13.2929i	−0.251798	−	0.436126i	−0.964021	+	0.265827i	$0.914355\pi$
$930$		0			0
$931$	10.4890	+	6.05582i	0.343763	+	0.198471i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.02488			0.229738
$936$		0			0
$937$	−11.5390			−0.376964			−0.188482	−	0.982077i	$-0.560357\pi$
$937$	−11.5390			−0.376964			−0.188482	+	0.982077i	$0.560357\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−28.6675	−	16.5512i	−0.934532	−	0.539552i	−0.0462901	−	0.998928i	$-0.514740\pi$
$941$	−28.6675	−	16.5512i	−0.934532	−	0.539552i	−0.888242	+	0.459376i	$0.848073\pi$
$942$		0			0
$943$	20.1843	+	34.9602i	0.657290	+	1.13846i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.9526	−	8.63290i	0.485895	−	0.280532i	−0.236975	−	0.971516i	$-0.576156\pi$
$947$	14.9526	−	8.63290i	0.485895	−	0.280532i	0.722870	+	0.690984i	$0.242823\pi$
$948$		0			0
$949$	−1.08528	−	0.626589i	−0.0352298	−	0.0203399i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	29.2159			0.946394			0.473197	−	0.880957i	$-0.343100\pi$
$953$	29.2159			0.946394			0.473197	+	0.880957i	$0.343100\pi$
$954$		0			0
$955$		−	2.37003i		−	0.0766922i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.62669	−	6.28160i	0.117112	−	0.202844i
$960$		0			0
$961$	−28.8589	−	49.9851i	−0.930932	−	1.61242i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	23.1914	−	13.3896i	0.746559	−	0.431026i
$966$		0			0
$967$	−2.00416	+	3.47131i	−0.0644495	+	0.111630i	−0.896450	−	0.443146i	$-0.853862\pi$
$967$	−2.00416	+	3.47131i	−0.0644495	+	0.111630i	0.832000	+	0.554775i	$0.187196\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			20.2791i			0.650787i	0.945579	+	0.325393i	$0.105497\pi$
$971$			20.2791i			0.650787i	−0.945579	+	0.325393i	$0.894503\pi$
$972$		0			0
$973$			8.16021i			0.261604i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.65417	−	2.86511i	0.0529217	−	0.0916631i	−0.838351	−	0.545131i	$-0.816480\pi$
$977$	1.65417	−	2.86511i	0.0529217	−	0.0916631i	0.891273	+	0.453468i	$0.149813\pi$
$978$		0			0
$979$	1.23617	−	0.713701i	0.0395080	−	0.0228100i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.5963	+	35.6739i	0.656921	+	1.13782i	0.981409	+	0.191930i	$0.0614748\pi$
$983$	20.5963	+	35.6739i	0.656921	+	1.13782i	−0.324488	+	0.945890i	$0.605192\pi$
$984$		0			0
$985$	11.2026	−	19.4034i	0.356944	−	0.618245i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			48.5813i			1.54479i
$990$		0			0
$991$	44.2802			1.40661			0.703303	−	0.710890i	$-0.251708\pi$
$991$	44.2802			1.40661			0.703303	+	0.710890i	$0.251708\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.1120	−	16.2305i	−0.891211	−	0.514541i
$996$		0			0
$997$	30.8478	−	17.8100i	0.976959	−	0.564047i	0.0756081	−	0.997138i	$-0.475910\pi$
$997$	30.8478	−	17.8100i	0.976959	−	0.564047i	0.901351	+	0.433090i	$0.142577\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.r.f.1441.5		12
3.2	odd	2		576.2.r.e.481.1	yes	12
4.3	odd	2	inner	1728.2.r.f.1441.2		12
8.3	odd	2		1728.2.r.e.1441.2		12
8.5	even	2		1728.2.r.e.1441.5		12
9.2	odd	6		576.2.r.f.97.6	yes	12
9.4	even	3		5184.2.d.r.2593.2		12
9.5	odd	6		5184.2.d.q.2593.8		12
9.7	even	3		1728.2.r.e.289.5		12
12.11	even	2		576.2.r.e.481.6	yes	12
24.5	odd	2		576.2.r.f.481.6	yes	12
24.11	even	2		576.2.r.f.481.1	yes	12
36.7	odd	6		1728.2.r.e.289.2		12
36.11	even	6		576.2.r.f.97.1	yes	12
36.23	even	6		5184.2.d.q.2593.11		12
36.31	odd	6		5184.2.d.r.2593.5		12
72.5	odd	6		5184.2.d.q.2593.2		12
72.11	even	6		576.2.r.e.97.6	yes	12
72.13	even	6		5184.2.d.r.2593.8		12
72.29	odd	6		576.2.r.e.97.1	✓	12
72.43	odd	6	inner	1728.2.r.f.289.2		12
72.59	even	6		5184.2.d.q.2593.5		12
72.61	even	6	inner	1728.2.r.f.289.5		12
72.67	odd	6		5184.2.d.r.2593.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.e.97.1	✓	12	72.29	odd	6
576.2.r.e.97.6	yes	12	72.11	even	6
576.2.r.e.481.1	yes	12	3.2	odd	2
576.2.r.e.481.6	yes	12	12.11	even	2
576.2.r.f.97.1	yes	12	36.11	even	6
576.2.r.f.97.6	yes	12	9.2	odd	6
576.2.r.f.481.1	yes	12	24.11	even	2
576.2.r.f.481.6	yes	12	24.5	odd	2
1728.2.r.e.289.2		12	36.7	odd	6
1728.2.r.e.289.5		12	9.7	even	3
1728.2.r.e.1441.2		12	8.3	odd	2
1728.2.r.e.1441.5		12	8.5	even	2
1728.2.r.f.289.2		12	72.43	odd	6	inner
1728.2.r.f.289.5		12	72.61	even	6	inner
1728.2.r.f.1441.2		12	4.3	odd	2	inner
1728.2.r.f.1441.5		12	1.1	even	1	trivial
5184.2.d.q.2593.2		12	72.5	odd	6
5184.2.d.q.2593.5		12	72.59	even	6
5184.2.d.q.2593.8		12	9.5	odd	6
5184.2.d.q.2593.11		12	36.23	even	6
5184.2.d.r.2593.2		12	9.4	even	3
5184.2.d.r.2593.5		12	36.31	odd	6
5184.2.d.r.2593.8		12	72.13	even	6
5184.2.d.r.2593.11		12	72.67	odd	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 + 0.866025i) q^{5} +(1.80664 + 3.12920i) q^{7} +O(q^{10})\)	\(q+(1.50000 + 0.866025i) q^{5} +(1.80664 + 3.12920i) q^{7} +(0.635828 - 0.367095i) q^{11} +(0.527909 + 0.304788i) q^{13} +5.52420 q^{17} -2.00000i q^{19} +(2.36788 - 4.10129i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(6.78630 - 3.91807i) q^{29} +(-4.70951 + 8.15710i) q^{31} +6.25839i q^{35} +2.34163i q^{37} +(-4.26210 + 7.38217i) q^{41} +(-8.88403 + 5.12920i) q^{43} +(5.88032 + 10.1850i) q^{47} +(-3.02791 + 5.24449i) q^{49} -13.0323i q^{53} +1.27166 q^{55} +(1.04788 + 0.604996i) q^{59} +(9.78630 - 5.65012i) q^{61} +(0.527909 + 0.914365i) q^{65} +(-5.46826 - 3.15710i) q^{67} +2.63999 q^{71} -2.05582 q^{73} +(2.29743 + 1.32642i) q^{77} +(1.24540 + 2.15710i) q^{79} +(-10.6161 + 6.12920i) q^{83} +(8.28630 + 4.78410i) q^{85} +1.94418 q^{89} +2.20257i q^{91} +(1.73205 - 3.00000i) q^{95} +(7.78630 + 13.4863i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 18 q^{5}+O(q^{10}) \) 12 * q + 18 * q^5	\( 12 q + 18 q^{5} - 6 q^{13} - 12 q^{25} - 18 q^{29} - 18 q^{41} - 24 q^{49} + 18 q^{61} - 6 q^{65} + 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) 12 * q + 18 * q^5 - 6 * q^13 - 12 * q^25 - 18 * q^29 - 18 * q^41 - 24 * q^49 + 18 * q^61 - 6 * q^65 + 90 * q^77 + 48 * q^89 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more