LMFDB - Embedded newform 1296.2.i.n.433.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(433,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1296.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1296.433
Dual form			1296.2.i.n.865.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +O(q^{10})$	$q+(1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(0.500000 - 0.866025i) q^{13} -3.00000 q^{17} +4.00000 q^{19} +(-2.00000 - 3.46410i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(-2.00000 + 3.46410i) q^{31} -12.0000 q^{35} -1.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-6.00000 - 10.3923i) q^{47} +(-4.50000 + 7.79423i) q^{49} -6.00000 q^{53} +(0.500000 + 0.866025i) q^{61} +(-1.50000 - 2.59808i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +11.0000 q^{73} +(-8.00000 - 13.8564i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} -3.00000 q^{89} -4.00000 q^{91} +(6.00000 - 10.3923i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 3 * q^5 - 4 * q^7	$ 2 q + 3 q^{5} - 4 q^{7} + q^{13} - 6 q^{17} + 8 q^{19} - 4 q^{25} - 9 q^{29} - 4 q^{31} - 24 q^{35} - 2 q^{37} - 6 q^{41} + 8 q^{43} - 12 q^{47} - 9 q^{49} - 12 q^{53} + q^{61} - 3 q^{65} - 4 q^{67} + 24 q^{71} + 22 q^{73} - 16 q^{79} - 12 q^{83} - 9 q^{85} - 6 q^{89} - 8 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 4 * q^7 + q^13 - 6 * q^17 + 8 * q^19 - 4 * q^25 - 9 * q^29 - 4 * q^31 - 24 * q^35 - 2 * q^37 - 6 * q^41 + 8 * q^43 - 12 * q^47 - 9 * q^49 - 12 * q^53 + q^61 - 3 * q^65 - 4 * q^67 + 24 * q^71 + 22 * q^73 - 16 * q^79 - 12 * q^83 - 9 * q^85 - 6 * q^89 - 8 * q^91 + 12 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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	−	2.59808i	0.670820	−	1.16190i	−0.306851	−	0.951757i	$-0.599275\pi$
$5$	1.50000	−	2.59808i	0.670820	−	1.16190i	0.977672	−	0.210138i	$-0.0673912\pi$
$6$		0			0
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	0.188982	−	0.981981i	$-0.439481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	0.500000	−	0.866025i	0.138675	−	0.240192i	0.926995	+	0.375073i	$0.122382\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$	−4.50000	−	7.79423i	−0.835629	−	1.44735i	0.0578882	−	0.998323i	$-0.481563\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−12.0000			−2.02837
$36$		0			0
$37$	−1.00000			−0.164399			−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000			−0.164399			−0.0821995	+	0.996616i	$0.526194\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	−0.999353	−	0.0359748i	$-0.988546\pi$
$41$	−3.00000	+	5.19615i	−0.468521	+	0.811503i	0.530831	+	0.847477i	$0.321880\pi$
$42$		0			0
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.50000	−	2.59808i	−0.186052	−	0.322252i
$66$		0			0
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	0.717607	+	0.696449i	$0.245238\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	11.0000			1.28745			0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000			1.28745			0.643726	+	0.765256i	$0.277388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.00000	−	13.8564i	−0.900070	−	1.55897i	−0.827401	−	0.561611i	$-0.810182\pi$
$79$	−8.00000	−	13.8564i	−0.900070	−	1.55897i	−0.0726692	−	0.997356i	$-0.523152\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	−0.980982	−	0.194099i	$-0.937822\pi$
$83$	−6.00000	−	10.3923i	−0.658586	−	1.14070i	0.322396	−	0.946605i	$-0.395512\pi$
$84$		0			0
$85$	−4.50000	+	7.79423i	−0.488094	+	0.845403i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.00000			−0.317999			−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000			−0.317999			−0.159000	+	0.987279i	$0.550827\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.00000	−	10.3923i	0.615587	−	1.06623i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	0.975796	−	0.218685i	$-0.0701767\pi$
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	−0.677284	+	0.735721i	$0.736843\pi$
$102$		0			0
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	−0.947576	−	0.319531i	$-0.896475\pi$
$103$	−2.00000	+	3.46410i	−0.197066	+	0.341328i	0.750510	+	0.660859i	$0.229808\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	−0.260955	−	0.965351i	$-0.584038\pi$
$113$	7.50000	−	12.9904i	0.705541	−	1.22203i	0.966496	−	0.256681i	$-0.0826291\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	+	10.3923i	0.550019	+	0.952661i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.00000			0.268328
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	0.475380	+	0.879781i	$0.342311\pi$
$131$	−6.00000	+	10.3923i	−0.524222	+	0.907980i	−0.999602	+	0.0281993i	$0.991023\pi$
$132$		0			0
$133$	−8.00000	−	13.8564i	−0.693688	−	1.20150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	0.607233	−	0.794524i	$-0.292279\pi$
$137$	−4.50000	−	7.79423i	−0.384461	−	0.665906i	−0.991694	+	0.128618i	$0.958946\pi$
$138$		0			0
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	−0.0346338	−	0.999400i	$-0.511026\pi$
$139$	10.0000	−	17.3205i	0.848189	−	1.46911i	0.882823	−	0.469706i	$-0.155640\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−27.0000			−2.24223
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	−0.989355	−	0.145519i	$-0.953515\pi$
$149$	−4.50000	+	7.79423i	−0.368654	+	0.638528i	0.620701	+	0.784047i	$0.286848\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	6.00000	+	10.3923i	0.481932	+	0.834730i
$156$		0			0
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	−0.481006	−	0.876717i	$-0.659728\pi$
$157$	6.50000	−	11.2583i	0.518756	−	0.898513i	0.999762	−	0.0217953i	$-0.00693820\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.00000			−0.626608			−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000			−0.626608			−0.313304	+	0.949653i	$0.601436\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	−0.534875	−	0.844931i	$-0.679641\pi$
$167$	6.00000	−	10.3923i	0.464294	−	0.804181i	0.999169	+	0.0407502i	$0.0129748\pi$
$168$		0			0
$169$	6.00000	+	10.3923i	0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.50000	+	2.59808i	0.114043	+	0.197528i	0.917397	−	0.397974i	$-0.130287\pi$
$173$	1.50000	+	2.59808i	0.114043	+	0.197528i	−0.803354	+	0.595502i	$0.796953\pi$
$174$		0			0
$175$	−8.00000	+	13.8564i	−0.604743	+	1.04745i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.50000	+	2.59808i	−0.110282	+	0.191014i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	0.997225	−	0.0744412i	$-0.0237173\pi$
$191$	6.00000	+	10.3923i	0.434145	+	0.751961i	−0.563081	+	0.826402i	$0.690384\pi$
$192$		0			0
$193$	6.50000	−	11.2583i	0.467880	−	0.810392i	−0.531446	−	0.847092i	$-0.678351\pi$
$193$	6.50000	−	11.2583i	0.467880	−	0.810392i	0.999326	+	0.0366998i	$0.0116845\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.00000			−0.213741			−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000			−0.213741			−0.106871	+	0.994273i	$0.534083\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−18.0000	+	31.1769i	−1.26335	+	2.18819i
$204$		0			0
$205$	9.00000	+	15.5885i	0.628587	+	1.08875i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	−0.694857	−	0.719148i	$-0.744533\pi$
$211$	4.00000	−	6.92820i	0.275371	−	0.476957i	0.970229	+	0.242190i	$0.0778659\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.0000			1.63679
$216$		0			0
$217$	16.0000			1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.50000	+	2.59808i	−0.100901	+	0.174766i
$222$		0			0
$223$	4.00000	+	6.92820i	0.267860	+	0.463947i	0.968309	−	0.249756i	$-0.0803503\pi$
$223$	4.00000	+	6.92820i	0.267860	+	0.463947i	−0.700449	+	0.713702i	$0.747017\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	0.595274	−	0.803523i	$-0.297043\pi$
$227$	−6.00000	−	10.3923i	−0.398234	−	0.689761i	−0.993508	+	0.113761i	$0.963710\pi$
$228$		0			0
$229$	−11.5000	+	19.9186i	−0.759941	+	1.31626i	0.182939	+	0.983124i	$0.441439\pi$
$229$	−11.5000	+	19.9186i	−0.759941	+	1.31626i	−0.942880	+	0.333133i	$0.891894\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	21.0000			1.37576			0.687878	−	0.725826i	$-0.258542\pi$
$233$	21.0000			1.37576			0.687878	+	0.725826i	$0.258542\pi$
$234$		0			0
$235$	−36.0000			−2.34838
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	−0.604087	−	0.796918i	$-0.706462\pi$
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	0.992195	+	0.124696i	$0.0397955\pi$
$240$		0			0
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	0.995809	−	0.0914555i	$-0.0291519\pi$
$241$	6.50000	+	11.2583i	0.418702	+	0.725213i	−0.577107	+	0.816668i	$0.695819\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	13.5000	+	23.3827i	0.862483	+	1.49387i
$246$		0			0
$247$	2.00000	−	3.46410i	0.127257	−	0.220416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	−0.531487	−	0.847066i	$-0.678367\pi$
$257$	7.50000	−	12.9904i	0.467837	−	0.810318i	0.999325	+	0.0367485i	$0.0117000\pi$
$258$		0			0
$259$	2.00000	+	3.46410i	0.124274	+	0.215249i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$	−9.00000	+	15.5885i	−0.552866	+	0.957591i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	21.0000			1.28039			0.640196	−	0.768211i	$-0.278853\pi$
$269$	21.0000			1.28039			0.640196	+	0.768211i	$0.278853\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	0.976231	−	0.216731i	$-0.0695395\pi$
$277$	5.00000	+	8.66025i	0.300421	+	0.520344i	−0.675810	+	0.737075i	$0.736206\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.5000	+	23.3827i	0.805342	+	1.39489i	0.916060	+	0.401042i	$0.131352\pi$
$281$	13.5000	+	23.3827i	0.805342	+	1.39489i	−0.110717	+	0.993852i	$0.535315\pi$
$282$		0			0
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	−0.919327	−	0.393494i	$-0.871266\pi$
$283$	−2.00000	+	3.46410i	−0.118888	+	0.205919i	0.800439	+	0.599414i	$0.204600\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	24.0000			1.41668
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	−0.967009	−	0.254741i	$-0.918010\pi$
$293$	−4.50000	+	7.79423i	−0.262893	+	0.455344i	0.704117	+	0.710084i	$0.251343\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	16.0000	−	27.7128i	0.922225	−	1.59734i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	3.00000			0.171780
$306$		0			0
$307$	−20.0000			−1.14146			−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000			−1.14146			−0.570730	+	0.821138i	$0.693340\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	−0.294384	−	0.955687i	$-0.595114\pi$
$311$	12.0000	−	20.7846i	0.680458	−	1.17859i	0.974841	−	0.222900i	$-0.0715523\pi$
$312$		0			0
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	−0.983118	−	0.182973i	$-0.941428\pi$
$313$	−11.5000	−	19.9186i	−0.650018	−	1.12586i	0.333099	−	0.942892i	$-0.391906\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5000	−	18.1865i	−0.589739	−	1.02146i	−0.994266	−	0.106932i	$-0.965897\pi$
$317$	−10.5000	−	18.1865i	−0.589739	−	1.02146i	0.404528	−	0.914526i	$-0.367436\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−24.0000	+	41.5692i	−1.32316	+	2.29179i
$330$		0			0
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	0.998298	+	0.0583130i	$0.0185721\pi$
$331$	10.0000	+	17.3205i	0.549650	+	0.952021i	−0.448649	+	0.893708i	$0.648095\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.00000	+	10.3923i	0.327815	+	0.567792i
$336$		0			0
$337$	−1.00000	+	1.73205i	−0.0544735	+	0.0943508i	−0.891976	−	0.452082i	$-0.850681\pi$
$337$	−1.00000	+	1.73205i	−0.0544735	+	0.0943508i	0.837503	+	0.546433i	$0.184015\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	8.00000			0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	0.615581	−	0.788074i	$-0.288921\pi$
$349$	−7.00000	−	12.1244i	−0.374701	−	0.649002i	−0.990282	+	0.139072i	$0.955588\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	0.999711	−	0.0240566i	$-0.00765819\pi$
$353$	9.00000	+	15.5885i	0.479022	+	0.829690i	−0.520689	+	0.853746i	$0.674325\pi$
$354$		0			0
$355$	18.0000	−	31.1769i	0.955341	−	1.65470i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	16.5000	−	28.5788i	0.863649	−	1.49588i
$366$		0			0
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	0.951336	−	0.308155i	$-0.0997115\pi$
$367$	4.00000	+	6.92820i	0.208798	+	0.361649i	−0.742538	+	0.669804i	$0.766378\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9.00000			−0.463524
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6.00000	+	10.3923i	−0.306586	+	0.531022i	−0.977613	−	0.210411i	$-0.932520\pi$
$383$	−6.00000	+	10.3923i	−0.306586	+	0.531022i	0.671027	+	0.741433i	$0.265853\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−48.0000			−2.41514
$396$		0			0
$397$	−25.0000			−1.25471			−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000			−1.25471			−0.627357	+	0.778732i	$0.715863\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	−0.826139	−	0.563466i	$-0.809468\pi$
$401$	1.50000	−	2.59808i	0.0749064	−	0.129742i	0.901046	+	0.433724i	$0.142801\pi$
$402$		0			0
$403$	2.00000	+	3.46410i	0.0996271	+	0.172559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	−0.371750	−	0.928333i	$-0.621242\pi$
$409$	12.5000	−	21.6506i	0.618085	−	1.07056i	0.989835	−	0.142222i	$-0.0454247\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−36.0000			−1.76717
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.0000	+	20.7846i	−0.586238	+	1.01539i	0.408481	+	0.912767i	$0.366058\pi$
$419$	−12.0000	+	20.7846i	−0.586238	+	1.01539i	−0.994720	+	0.102628i	$0.967275\pi$
$420$		0			0
$421$	6.50000	+	11.2583i	0.316791	+	0.548697i	0.979817	−	0.199899i	$-0.0640614\pi$
$421$	6.50000	+	11.2583i	0.316791	+	0.548697i	−0.663026	+	0.748596i	$0.730728\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	6.00000	+	10.3923i	0.291043	+	0.504101i
$426$		0			0
$427$	2.00000	−	3.46410i	0.0967868	−	0.167640i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$	11.0000			0.528626			0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000			0.528626			0.264313	+	0.964437i	$0.414855\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$	−4.50000	+	7.79423i	−0.213320	+	0.369482i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−18.0000			−0.849473			−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000			−0.849473			−0.424736	+	0.905317i	$0.639633\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000	+	10.3923i	−0.281284	+	0.487199i
$456$		0			0
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	0.877483	−	0.479608i	$-0.159221\pi$
$457$	0.500000	+	0.866025i	0.0233890	+	0.0405110i	−0.854094	+	0.520119i	$0.825888\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	0.576685	−	0.816967i	$-0.304346\pi$
$461$	−9.00000	−	15.5885i	−0.419172	−	0.726027i	−0.995856	+	0.0909401i	$0.971013\pi$
$462$		0			0
$463$	4.00000	−	6.92820i	0.185896	−	0.321981i	−0.757982	−	0.652275i	$-0.773815\pi$
$463$	4.00000	−	6.92820i	0.185896	−	0.321981i	0.943878	+	0.330294i	$0.107148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	24.0000			1.11059			0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000			1.11059			0.555294	+	0.831654i	$0.312606\pi$
$468$		0			0
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−8.00000	−	13.8564i	−0.367065	−	0.635776i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	0.695773	−	0.718262i	$-0.255062\pi$
$479$	−6.00000	−	10.3923i	−0.274147	−	0.474837i	−0.969920	+	0.243426i	$0.921729\pi$
$480$		0			0
$481$	−0.500000	+	0.866025i	−0.0227980	+	0.0394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.00000			−0.272446
$486$		0			0
$487$	4.00000			0.181257			0.0906287	−	0.995885i	$-0.471112\pi$
$487$	4.00000			0.181257			0.0906287	+	0.995885i	$0.471112\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$	13.5000	+	23.3827i	0.608009	+	1.05310i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−24.0000	−	41.5692i	−1.07655	−	1.86463i
$498$		0			0
$499$	−20.0000	+	34.6410i	−0.895323	+	1.55074i	−0.0619186	+	0.998081i	$0.519722\pi$
$499$	−20.0000	+	34.6410i	−0.895323	+	1.55074i	−0.833404	+	0.552664i	$0.813611\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	−0.314459	−	0.949271i	$-0.601823\pi$
$509$	15.0000	−	25.9808i	0.664863	−	1.15158i	0.979322	−	0.202306i	$-0.0648436\pi$
$510$		0			0
$511$	−22.0000	−	38.1051i	−0.973223	−	1.68567i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.00000	+	10.3923i	0.264392	+	0.457940i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.00000	−	10.3923i	0.261364	−	0.452696i
$528$		0			0
$529$	11.5000	+	19.9186i	0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.00000	+	5.19615i	0.129944	+	0.225070i
$534$		0			0
$535$	−18.0000	+	31.1769i	−0.778208	+	1.34790i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.00000			−0.0429934			−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000			−0.0429934			−0.0214967	+	0.999769i	$0.506843\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	16.5000	−	28.5788i	0.706782	−	1.22418i
$546$		0			0
$547$	22.0000	+	38.1051i	0.940652	+	1.62926i	0.764231	+	0.644942i	$0.223119\pi$
$547$	22.0000	+	38.1051i	0.940652	+	1.62926i	0.176421	+	0.984315i	$0.443548\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−18.0000	−	31.1769i	−0.766826	−	1.32818i
$552$		0			0
$553$	−32.0000	+	55.4256i	−1.36078	+	2.35694i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.00000			−0.127114			−0.0635570	−	0.997978i	$-0.520244\pi$
$557$	−3.00000			−0.127114			−0.0635570	+	0.997978i	$0.520244\pi$
$558$		0			0
$559$	8.00000			0.338364
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6.00000	−	10.3923i	0.252870	−	0.437983i	−0.711445	−	0.702742i	$-0.751959\pi$
$563$	6.00000	−	10.3923i	0.252870	−	0.437983i	0.964315	+	0.264758i	$0.0852922\pi$
$564$		0			0
$565$	−22.5000	−	38.9711i	−0.946582	−	1.63953i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7.50000	+	12.9904i	0.314416	+	0.544585i	0.979313	−	0.202350i	$-0.0648579\pi$
$569$	7.50000	+	12.9904i	0.314416	+	0.544585i	−0.664897	+	0.746935i	$0.731525\pi$
$570$		0			0
$571$	−8.00000	+	13.8564i	−0.334790	+	0.579873i	−0.983444	−	0.181210i	$-0.941999\pi$
$571$	−8.00000	+	13.8564i	−0.334790	+	0.579873i	0.648655	+	0.761083i	$0.275332\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−25.0000			−1.04076			−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000			−1.04076			−0.520382	+	0.853934i	$0.674210\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−24.0000	+	41.5692i	−0.995688	+	1.72458i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0000	−	20.7846i	−0.495293	−	0.857873i	0.504692	−	0.863299i	$-0.331606\pi$
$587$	−12.0000	−	20.7846i	−0.495293	−	0.857873i	−0.999985	+	0.00542667i	$0.998273\pi$
$588$		0			0
$589$	−8.00000	+	13.8564i	−0.329634	+	0.570943i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	33.0000			1.35515			0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000			1.35515			0.677574	+	0.735455i	$0.263031\pi$
$594$		0			0
$595$	36.0000			1.47586
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	0.219061	+	0.975711i	$0.429701\pi$
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	−0.954521	+	0.298143i	$0.903633\pi$
$600$		0			0
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	−0.963405	−	0.268049i	$-0.913621\pi$
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	0.249565	−	0.968358i	$-0.419712\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.5000	−	28.5788i	−0.670820	−	1.16190i
$606$		0			0
$607$	10.0000	−	17.3205i	0.405887	−	0.703018i	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	10.0000	−	17.3205i	0.405887	−	0.703018i	0.994424	+	0.105453i	$0.0336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$	38.0000			1.53481			0.767403	−	0.641165i	$-0.221549\pi$
$613$	38.0000			1.53481			0.767403	+	0.641165i	$0.221549\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	−0.834251	−	0.551385i	$-0.814100\pi$
$617$	1.50000	−	2.59808i	0.0603877	−	0.104595i	0.894639	+	0.446790i	$0.147433\pi$
$618$		0			0
$619$	4.00000	+	6.92820i	0.160774	+	0.278468i	0.935146	−	0.354262i	$-0.115268\pi$
$619$	4.00000	+	6.92820i	0.160774	+	0.278468i	−0.774373	+	0.632730i	$0.781934\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	+	10.3923i	0.240385	+	0.416359i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.00000			0.119618
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	24.0000	−	41.5692i	0.952411	−	1.64962i
$636$		0			0
$637$	4.50000	+	7.79423i	0.178296	+	0.308819i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.5000	−	38.9711i	−0.888697	−	1.53927i	−0.841417	−	0.540386i	$-0.818278\pi$
$641$	−22.5000	−	38.9711i	−0.888697	−	1.53927i	−0.0472793	−	0.998882i	$-0.515055\pi$
$642$		0			0
$643$	−2.00000	+	3.46410i	−0.0788723	+	0.136611i	−0.902764	−	0.430137i	$-0.858465\pi$
$643$	−2.00000	+	3.46410i	−0.0788723	+	0.136611i	0.823891	+	0.566748i	$0.191799\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−36.0000			−1.41531			−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000			−1.41531			−0.707653	+	0.706560i	$0.750246\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	−0.986634	−	0.162951i	$-0.947899\pi$
$653$	−9.00000	+	15.5885i	−0.352197	+	0.610023i	0.634437	+	0.772975i	$0.281232\pi$
$654$		0			0
$655$	18.0000	+	31.1769i	0.703318	+	1.21818i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	0.958902	−	0.283738i	$-0.0915745\pi$
$659$	6.00000	+	10.3923i	0.233727	+	0.404827i	−0.725175	+	0.688565i	$0.758241\pi$
$660$		0			0
$661$	−11.5000	+	19.9186i	−0.447298	+	0.774743i	−0.998209	−	0.0598209i	$-0.980947\pi$
$661$	−11.5000	+	19.9186i	−0.447298	+	0.774743i	0.550911	+	0.834564i	$0.314280\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−48.0000			−1.86136
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	0.740334	−	0.672239i	$-0.234667\pi$
$673$	−5.50000	−	9.52628i	−0.212009	−	0.367211i	−0.952343	+	0.305028i	$0.901334\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.00000	+	5.19615i	0.115299	+	0.199704i	0.917899	−	0.396813i	$-0.129884\pi$
$677$	3.00000	+	5.19615i	0.115299	+	0.199704i	−0.802600	+	0.596518i	$0.796551\pi$
$678$		0			0
$679$	−4.00000	+	6.92820i	−0.153506	+	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−36.0000			−1.37750			−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000			−1.37750			−0.688751	+	0.724998i	$0.741841\pi$
$684$		0			0
$685$	−27.0000			−1.03162
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.00000	+	5.19615i	−0.114291	+	0.197958i
$690$		0			0
$691$	−14.0000	−	24.2487i	−0.532585	−	0.922464i	−0.999276	−	0.0380440i	$-0.987887\pi$
$691$	−14.0000	−	24.2487i	−0.532585	−	0.922464i	0.466691	−	0.884420i	$-0.345446\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−30.0000	−	51.9615i	−1.13796	−	1.97101i
$696$		0			0
$697$	9.00000	−	15.5885i	0.340899	−	0.590455i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−51.0000			−1.92624			−0.963122	−	0.269066i	$-0.913285\pi$
$701$	−51.0000			−1.92624			−0.963122	+	0.269066i	$0.913285\pi$
$702$		0			0
$703$	−4.00000			−0.150863
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.0000	−	20.7846i	0.451306	−	0.781686i
$708$		0			0
$709$	−23.5000	−	40.7032i	−0.882561	−	1.52864i	−0.848484	−	0.529221i	$-0.822484\pi$
$709$	−23.5000	−	40.7032i	−0.882561	−	1.52864i	−0.0340772	−	0.999419i	$-0.510849\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	16.0000			0.595871
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−18.0000	+	31.1769i	−0.668503	+	1.15788i
$726$		0			0
$727$	−14.0000	−	24.2487i	−0.519231	−	0.899335i	−0.999750	−	0.0223506i	$-0.992885\pi$
$727$	−14.0000	−	24.2487i	−0.519231	−	0.899335i	0.480519	−	0.876984i	$-0.340448\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	−	20.7846i	−0.443836	−	0.768747i
$732$		0			0
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	−0.965854	−	0.259087i	$-0.916578\pi$
$733$	−7.00000	+	12.1244i	−0.258551	+	0.447823i	0.707303	+	0.706910i	$0.249912\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.00000	−	10.3923i	0.220119	−	0.381257i	−0.734725	−	0.678365i	$-0.762689\pi$
$743$	6.00000	−	10.3923i	0.220119	−	0.381257i	0.954844	+	0.297108i	$0.0960222\pi$
$744$		0			0
$745$	13.5000	+	23.3827i	0.494602	+	0.856675i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000	+	41.5692i	0.876941	+	1.51891i
$750$		0			0
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	−0.623856	−	0.781540i	$-0.714435\pi$
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	0.988761	+	0.149505i	$0.0477681\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	24.0000			0.873449
$756$		0			0
$757$	38.0000			1.38113			0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000			1.38113			0.690567	+	0.723269i	$0.257361\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	−0.697467	−	0.716617i	$-0.745690\pi$
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	0.969342	+	0.245716i	$0.0790230\pi$
$762$		0			0
$763$	−22.0000	−	38.1051i	−0.796453	−	1.37950i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	−0.311577	−	0.950221i	$-0.600857\pi$
$769$	18.5000	−	32.0429i	0.667127	−	1.15550i	0.978704	−	0.205277i	$-0.0658095\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.0000			−0.971123			−0.485561	−	0.874203i	$-0.661385\pi$
$773$	−27.0000			−0.971123			−0.485561	+	0.874203i	$0.661385\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−12.0000	+	20.7846i	−0.429945	+	0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−19.5000	−	33.7750i	−0.695985	−	1.20548i
$786$		0			0
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	−0.426193	−	0.904632i	$-0.640145\pi$
$787$	16.0000	−	27.7128i	0.570338	−	0.987855i	0.996531	−	0.0832226i	$-0.0265213\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−60.0000			−2.13335
$792$		0			0
$793$	1.00000			0.0355110
$794$		0			0
$795$		0			0
$796$		0			0
$797$	25.5000	−	44.1673i	0.903256	−	1.56449i	0.0800155	−	0.996794i	$-0.474503\pi$
$797$	25.5000	−	44.1673i	0.903256	−	1.56449i	0.823241	−	0.567692i	$-0.192164\pi$
$798$		0			0
$799$	18.0000	+	31.1769i	0.636794	+	1.10296i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.0000			−1.37117			−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000			−1.37117			−0.685583	+	0.727994i	$0.740453\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.0000	+	20.7846i	−0.420342	+	0.728053i
$816$		0			0
$817$	16.0000	+	27.7128i	0.559769	+	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	0.966708	−	0.255884i	$-0.0823665\pi$
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	−0.704956	+	0.709251i	$0.749033\pi$
$822$		0			0
$823$	22.0000	−	38.1051i	0.766872	−	1.32826i	−0.172379	−	0.985031i	$-0.555146\pi$
$823$	22.0000	−	38.1051i	0.766872	−	1.32826i	0.939251	−	0.343230i	$-0.111521\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	48.0000			1.66912			0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000			1.66912			0.834562	+	0.550914i	$0.185721\pi$
$828$		0			0
$829$	14.0000			0.486240			0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000			0.486240			0.243120	+	0.969996i	$0.421829\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	13.5000	−	23.3827i	0.467747	−	0.810162i
$834$		0			0
$835$	−18.0000	−	31.1769i	−0.622916	−	1.07892i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	0.995353	−	0.0962912i	$-0.0306980\pi$
$839$	12.0000	+	20.7846i	0.414286	+	0.717564i	−0.581067	+	0.813856i	$0.697365\pi$
$840$		0			0
$841$	−26.0000	+	45.0333i	−0.896552	+	1.55287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	36.0000			1.23844
$846$		0			0
$847$	−44.0000			−1.51186
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.00000	+	8.66025i	0.171197	+	0.296521i	0.938839	−	0.344358i	$-0.111903\pi$
$853$	5.00000	+	8.66025i	0.171197	+	0.296521i	−0.767642	+	0.640879i	$0.778570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	0.978984	+	0.203938i	$0.0653741\pi$
$857$	19.5000	+	33.7750i	0.666107	+	1.15373i	−0.312877	+	0.949794i	$0.601293\pi$
$858$		0			0
$859$	−26.0000	+	45.0333i	−0.887109	+	1.53652i	−0.0438309	+	0.999039i	$0.513956\pi$
$859$	−26.0000	+	45.0333i	−0.887109	+	1.53652i	−0.843278	+	0.537478i	$0.819377\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	2.00000	+	3.46410i	0.0677674	+	0.117377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−6.00000	−	10.3923i	−0.202837	−	0.351324i
$876$		0			0
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	12.5000	−	21.6506i	0.422095	−	0.731090i	0.996144	+	0.0877308i	$0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	−0.591156	−	0.806557i	$-0.701328\pi$
$887$	12.0000	−	20.7846i	0.402921	−	0.697879i	0.994077	+	0.108678i	$0.0346618\pi$
$888$		0			0
$889$	−32.0000	−	55.4256i	−1.07325	−	1.85892i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.0000	−	41.5692i	−0.803129	−	1.39106i
$894$		0			0
$895$	−18.0000	+	31.1769i	−0.601674	+	1.04213i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36.0000			1.20067
$900$		0			0
$901$	18.0000			0.599667
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.0000	+	25.9808i	−0.498617	+	0.863630i
$906$		0			0
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	0.702094	−	0.712084i	$-0.252248\pi$
$907$	−8.00000	−	13.8564i	−0.265636	−	0.460094i	−0.967730	+	0.251990i	$0.918915\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	12.0000	+	20.7846i	0.397578	+	0.688625i	0.993426	−	0.114472i	$-0.0365176\pi$
$911$	12.0000	+	20.7846i	0.397578	+	0.688625i	−0.595849	+	0.803097i	$0.703184\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	48.0000			1.58510
$918$		0			0
$919$	16.0000			0.527791			0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000			0.527791			0.263896	+	0.964551i	$0.414993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.00000	−	10.3923i	0.197492	−	0.342067i
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.50000	+	2.59808i	0.0492134	+	0.0852401i	0.889583	−	0.456774i	$-0.150995\pi$
$929$	1.50000	+	2.59808i	0.0492134	+	0.0852401i	−0.840369	+	0.542014i	$0.817662\pi$
$930$		0			0
$931$	−18.0000	+	31.1769i	−0.589926	+	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−37.0000			−1.20874			−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000			−1.20874			−0.604369	+	0.796705i	$0.706575\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	13.5000	−	23.3827i	0.440087	−	0.762254i	−0.557608	−	0.830104i	$-0.688281\pi$
$941$	13.5000	−	23.3827i	0.440087	−	0.762254i	0.997695	+	0.0678506i	$0.0216141\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	0.751918	−	0.659256i	$-0.229129\pi$
$947$	−6.00000	−	10.3923i	−0.194974	−	0.337705i	−0.946892	+	0.321552i	$0.895796\pi$
$948$		0			0
$949$	5.50000	−	9.52628i	0.178538	−	0.309236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$		0			0
$955$	36.0000			1.16493
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−18.0000	+	31.1769i	−0.581250	+	1.00676i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−19.5000	−	33.7750i	−0.627727	−	1.08726i
$966$		0			0
$967$	−2.00000	+	3.46410i	−0.0643157	+	0.111398i	−0.896390	−	0.443266i	$-0.853820\pi$
$967$	−2.00000	+	3.46410i	−0.0643157	+	0.111398i	0.832075	+	0.554664i	$0.187153\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$	−80.0000			−2.56468
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	−0.685381	−	0.728184i	$-0.740364\pi$
$977$	9.00000	−	15.5885i	0.287936	−	0.498719i	0.973317	+	0.229465i	$0.0736978\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	0.939992	+	0.341197i	$0.110832\pi$
$983$	24.0000	+	41.5692i	0.765481	+	1.32585i	−0.174511	+	0.984655i	$0.555834\pi$
$984$		0			0
$985$	−4.50000	+	7.79423i	−0.143382	+	0.248345i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	52.0000			1.65183			0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000			1.65183			0.825917	+	0.563791i	$0.190658\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.00000	−	10.3923i	0.190213	−	0.329458i
$996$		0			0
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	0.994762	+	0.102214i	$0.0325925\pi$
$997$	18.5000	+	32.0429i	0.585901	+	1.01481i	−0.408862	+	0.912596i	$0.634074\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.i.n.433.1		2
3.2	odd	2		1296.2.i.b.433.1		2
4.3	odd	2		162.2.c.d.109.1		2
9.2	odd	6		1296.2.i.b.865.1		2
9.4	even	3		1296.2.a.c.1.1		1
9.5	odd	6		1296.2.a.l.1.1		1
9.7	even	3	inner	1296.2.i.n.865.1		2
12.11	even	2		162.2.c.a.109.1		2
36.7	odd	6		162.2.c.d.55.1		2
36.11	even	6		162.2.c.a.55.1		2
36.23	even	6		162.2.a.d.1.1	yes	1
36.31	odd	6		162.2.a.a.1.1	✓	1
72.5	odd	6		5184.2.a.h.1.1		1
72.13	even	6		5184.2.a.bd.1.1		1
72.59	even	6		5184.2.a.c.1.1		1
72.67	odd	6		5184.2.a.y.1.1		1
180.23	odd	12		4050.2.c.n.649.1		2
180.59	even	6		4050.2.a.r.1.1		1
180.67	even	12		4050.2.c.g.649.1		2
180.103	even	12		4050.2.c.g.649.2		2
180.139	odd	6		4050.2.a.bh.1.1		1
180.167	odd	12		4050.2.c.n.649.2		2
252.139	even	6		7938.2.a.n.1.1		1
252.167	odd	6		7938.2.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.2.a.a.1.1	✓	1	36.31	odd	6
162.2.a.d.1.1	yes	1	36.23	even	6
162.2.c.a.55.1		2	36.11	even	6
162.2.c.a.109.1		2	12.11	even	2
162.2.c.d.55.1		2	36.7	odd	6
162.2.c.d.109.1		2	4.3	odd	2
1296.2.a.c.1.1		1	9.4	even	3
1296.2.a.l.1.1		1	9.5	odd	6
1296.2.i.b.433.1		2	3.2	odd	2
1296.2.i.b.865.1		2	9.2	odd	6
1296.2.i.n.433.1		2	1.1	even	1	trivial
1296.2.i.n.865.1		2	9.7	even	3	inner
4050.2.a.r.1.1		1	180.59	even	6
4050.2.a.bh.1.1		1	180.139	odd	6
4050.2.c.g.649.1		2	180.67	even	12
4050.2.c.g.649.2		2	180.103	even	12
4050.2.c.n.649.1		2	180.23	odd	12
4050.2.c.n.649.2		2	180.167	odd	12
5184.2.a.c.1.1		1	72.59	even	6
5184.2.a.h.1.1		1	72.5	odd	6
5184.2.a.y.1.1		1	72.67	odd	6
5184.2.a.bd.1.1		1	72.13	even	6
7938.2.a.n.1.1		1	252.139	even	6
7938.2.a.s.1.1		1	252.167	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +O(q^{10})\)	\(q+(1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(0.500000 - 0.866025i) q^{13} -3.00000 q^{17} +4.00000 q^{19} +(-2.00000 - 3.46410i) q^{25} +(-4.50000 - 7.79423i) q^{29} +(-2.00000 + 3.46410i) q^{31} -12.0000 q^{35} -1.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(4.00000 + 6.92820i) q^{43} +(-6.00000 - 10.3923i) q^{47} +(-4.50000 + 7.79423i) q^{49} -6.00000 q^{53} +(0.500000 + 0.866025i) q^{61} +(-1.50000 - 2.59808i) q^{65} +(-2.00000 + 3.46410i) q^{67} +12.0000 q^{71} +11.0000 q^{73} +(-8.00000 - 13.8564i) q^{79} +(-6.00000 - 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} -3.00000 q^{89} -4.00000 q^{91} +(6.00000 - 10.3923i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 3 * q^5 - 4 * q^7	\( 2 q + 3 q^{5} - 4 q^{7} + q^{13} - 6 q^{17} + 8 q^{19} - 4 q^{25} - 9 q^{29} - 4 q^{31} - 24 q^{35} - 2 q^{37} - 6 q^{41} + 8 q^{43} - 12 q^{47} - 9 q^{49} - 12 q^{53} + q^{61} - 3 q^{65} - 4 q^{67} + 24 q^{71} + 22 q^{73} - 16 q^{79} - 12 q^{83} - 9 q^{85} - 6 q^{89} - 8 q^{91} + 12 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 4 * q^7 + q^13 - 6 * q^17 + 8 * q^19 - 4 * q^25 - 9 * q^29 - 4 * q^31 - 24 * q^35 - 2 * q^37 - 6 * q^41 + 8 * q^43 - 12 * q^47 - 9 * q^49 - 12 * q^53 + q^61 - 3 * q^65 - 4 * q^67 + 24 * q^71 + 22 * q^73 - 16 * q^79 - 12 * q^83 - 9 * q^85 - 6 * q^89 - 8 * q^91 + 12 * q^95 - 2 * q^97

Introduction

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Database

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