LMFDB - Embedded newform 1296.2.i.b.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1296,2,Mod(433,1296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,0,0,0,-3,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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.b.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		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	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		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	$0.148230\pi$
$29$	4.50000	+	7.79423i	0.835629	+	1.44735i	−0.0578882	+	0.998323i	$0.518437\pi$
$30$		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	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	3.00000	−	5.19615i	0.468521	−	0.811503i	0.999353	+	0.0359748i	$0.0114536\pi$
$42$		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	$0.172597\pi$
$47$	6.00000	+	10.3923i	0.875190	+	1.51587i	0.0186297	+	0.999826i	$0.494070\pi$
$48$		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	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		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	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		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	$0.0621783\pi$
$83$	6.00000	+	10.3923i	0.658586	+	1.14070i	−0.322396	+	0.946605i	$0.604488\pi$
$84$		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	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		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	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		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	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		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	$0.415962\pi$
$113$	−7.50000	+	12.9904i	−0.705541	+	1.22203i	−0.966496	+	0.256681i	$0.917371\pi$
$114$		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	$-0.657689\pi$
$131$	6.00000	−	10.3923i	0.524222	−	0.907980i	0.999602	−	0.0281993i	$-0.00897729\pi$
$132$		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	0.991694	−	0.128618i	$-0.0410540\pi$
$137$	4.50000	+	7.79423i	0.384461	+	0.665906i	−0.607233	+	0.794524i	$0.707721\pi$
$138$		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	−0.620701	−	0.784047i	$-0.713152\pi$
$149$	4.50000	−	7.79423i	0.368654	−	0.638528i	0.989355	+	0.145519i	$0.0464853\pi$
$150$		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	−0.999169	−	0.0407502i	$-0.987025\pi$
$167$	−6.00000	+	10.3923i	−0.464294	+	0.804181i	0.534875	+	0.844931i	$0.320359\pi$
$168$		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	0.803354	−	0.595502i	$-0.203047\pi$
$173$	−1.50000	−	2.59808i	−0.114043	−	0.197528i	−0.917397	+	0.397974i	$0.869713\pi$
$174$		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	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		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	0.563081	−	0.826402i	$-0.309616\pi$
$191$	−6.00000	−	10.3923i	−0.434145	−	0.751961i	−0.997225	+	0.0744412i	$0.976283\pi$
$192$		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	$-0.465917\pi$
$197$	3.00000			0.213741			0.106871	+	0.994273i	$0.465917\pi$
$198$		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	0.993508	−	0.113761i	$-0.0362899\pi$
$227$	6.00000	+	10.3923i	0.398234	+	0.689761i	−0.595274	+	0.803523i	$0.702957\pi$
$228$		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	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		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	−0.992195	−	0.124696i	$-0.960204\pi$
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	0.604087	+	0.796918i	$0.293538\pi$
$240$		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	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	−0.999325	−	0.0367485i	$-0.988300\pi$
$257$	−7.50000	+	12.9904i	−0.467837	+	0.810318i	0.531487	+	0.847066i	$0.321633\pi$
$258$		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	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		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	$-0.721147\pi$
$269$	−21.0000			−1.28039			−0.640196	+	0.768211i	$0.721147\pi$
$270$		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	$-0.868648\pi$
$281$	−13.5000	−	23.3827i	−0.805342	−	1.39489i	0.110717	−	0.993852i	$-0.464685\pi$
$282$		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	−0.704117	−	0.710084i	$-0.748657\pi$
$293$	4.50000	−	7.79423i	0.262893	−	0.455344i	0.967009	+	0.254741i	$0.0819901\pi$
$294$		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	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		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	$0.0341026\pi$
$317$	10.5000	+	18.1865i	0.589739	+	1.02146i	−0.404528	+	0.914526i	$0.632564\pi$
$318$		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	−0.980921	−	0.194409i	$-0.937721\pi$
$347$	−6.00000	+	10.3923i	−0.322097	+	0.557888i	0.658824	+	0.752297i	$0.271054\pi$
$348$		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	0.520689	−	0.853746i	$-0.325675\pi$
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	−0.999711	+	0.0240566i	$0.992342\pi$
$354$		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	−0.671027	−	0.741433i	$-0.734147\pi$
$383$	6.00000	−	10.3923i	0.306586	−	0.531022i	0.977613	+	0.210411i	$0.0674801\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	0.779895	−	0.625910i	$-0.215272\pi$
$389$	−3.00000	−	5.19615i	−0.152106	−	0.263455i	−0.932002	+	0.362454i	$0.881939\pi$
$390$		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	−0.901046	−	0.433724i	$-0.857199\pi$
$401$	−1.50000	+	2.59808i	−0.0749064	+	0.129742i	0.826139	+	0.563466i	$0.190532\pi$
$402$		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	$-0.633942\pi$
$419$	12.0000	−	20.7846i	0.586238	−	1.01539i	0.994720	−	0.102628i	$-0.0327251\pi$
$420$		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	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		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	0.972626	−	0.232377i	$-0.0746503\pi$
$443$	6.00000	+	10.3923i	0.285069	+	0.493753i	−0.687557	+	0.726130i	$0.741317\pi$
$444$		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	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		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	0.995856	−	0.0909401i	$-0.0289872\pi$
$461$	9.00000	+	15.5885i	0.419172	+	0.726027i	−0.576685	+	0.816967i	$0.695654\pi$
$462$		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	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		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	0.969920	−	0.243426i	$-0.0782712\pi$
$479$	6.00000	+	10.3923i	0.274147	+	0.474837i	−0.695773	+	0.718262i	$0.744938\pi$
$480$		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	$0.398177\pi$
$509$	−15.0000	+	25.9808i	−0.664863	+	1.15158i	−0.979322	+	0.202306i	$0.935156\pi$
$510$		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	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		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	$-0.479756\pi$
$557$	3.00000			0.127114			0.0635570	+	0.997978i	$0.479756\pi$
$558$		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	−0.964315	−	0.264758i	$-0.914708\pi$
$563$	−6.00000	+	10.3923i	−0.252870	+	0.437983i	0.711445	+	0.702742i	$0.248041\pi$
$564$		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	0.664897	−	0.746935i	$-0.268475\pi$
$569$	−7.50000	−	12.9904i	−0.314416	−	0.544585i	−0.979313	+	0.202350i	$0.935142\pi$
$570$		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	0.999985	−	0.00542667i	$-0.00172737\pi$
$587$	12.0000	+	20.7846i	0.495293	+	0.857873i	−0.504692	+	0.863299i	$0.668394\pi$
$588$		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	$-0.736969\pi$
$593$	−33.0000			−1.35515			−0.677574	+	0.735455i	$0.736969\pi$
$594$		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	$-0.570299\pi$
$599$	18.0000	−	31.1769i	0.735460	−	1.27385i	0.954521	−	0.298143i	$-0.0963673\pi$
$600$		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	−0.894639	−	0.446790i	$-0.852567\pi$
$617$	−1.50000	+	2.59808i	−0.0603877	+	0.104595i	0.834251	+	0.551385i	$0.185900\pi$
$618$		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	$0.181722\pi$
$641$	22.5000	+	38.9711i	0.888697	+	1.53927i	0.0472793	+	0.998882i	$0.484945\pi$
$642$		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	$-0.249754\pi$
$647$	36.0000			1.41531			0.707653	+	0.706560i	$0.249754\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	−0.634437	−	0.772975i	$-0.718768\pi$
$653$	9.00000	−	15.5885i	0.352197	−	0.610023i	0.986634	+	0.162951i	$0.0521013\pi$
$654$		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	0.725175	−	0.688565i	$-0.241759\pi$
$659$	−6.00000	−	10.3923i	−0.233727	−	0.404827i	−0.958902	+	0.283738i	$0.908425\pi$
$660$		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	0.802600	−	0.596518i	$-0.203449\pi$
$677$	−3.00000	−	5.19615i	−0.115299	−	0.199704i	−0.917899	+	0.396813i	$0.870116\pi$
$678$		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	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		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	$-0.0867150\pi$
$701$	51.0000			1.92624			0.963122	+	0.269066i	$0.0867150\pi$
$702$		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	−0.954844	−	0.297108i	$-0.903978\pi$
$743$	−6.00000	+	10.3923i	−0.220119	+	0.381257i	0.734725	+	0.678365i	$0.237311\pi$
$744$		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	−0.969342	−	0.245716i	$-0.920977\pi$
$761$	−7.50000	+	12.9904i	−0.271875	+	0.470901i	0.697467	+	0.716617i	$0.254310\pi$
$762$		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	$-0.338615\pi$
$773$	27.0000			0.971123			0.485561	+	0.874203i	$0.338615\pi$
$774$		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	$0.525497\pi$
$797$	−25.5000	+	44.1673i	−0.903256	+	1.56449i	−0.823241	+	0.567692i	$0.807836\pi$
$798$		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	$-0.259547\pi$
$809$	39.0000			1.37117			0.685583	+	0.727994i	$0.259547\pi$
$810$		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	0.704956	−	0.709251i	$-0.250967\pi$
$821$	−7.50000	−	12.9904i	−0.261752	−	0.453367i	−0.966708	+	0.255884i	$0.917634\pi$
$822$		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	$-0.814279\pi$
$827$	−48.0000			−1.66912			−0.834562	+	0.550914i	$0.814279\pi$
$828$		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	0.581067	−	0.813856i	$-0.302635\pi$
$839$	−12.0000	−	20.7846i	−0.414286	−	0.717564i	−0.995353	+	0.0962912i	$0.969302\pi$
$840$		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	$-0.934626\pi$
$857$	−19.5000	−	33.7750i	−0.666107	−	1.15373i	0.312877	−	0.949794i	$-0.398707\pi$
$858$		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	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		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	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		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	−0.994077	−	0.108678i	$-0.965338\pi$
$887$	−12.0000	+	20.7846i	−0.402921	+	0.697879i	0.591156	+	0.806557i	$0.298672\pi$
$888$		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	0.595849	−	0.803097i	$-0.296816\pi$
$911$	−12.0000	−	20.7846i	−0.397578	−	0.688625i	−0.993426	+	0.114472i	$0.963482\pi$
$912$		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	0.840369	−	0.542014i	$-0.182338\pi$
$929$	−1.50000	−	2.59808i	−0.0492134	−	0.0852401i	−0.889583	+	0.456774i	$0.849005\pi$
$930$		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	−0.997695	−	0.0678506i	$-0.978386\pi$
$941$	−13.5000	+	23.3827i	−0.440087	+	0.762254i	0.557608	+	0.830104i	$0.311719\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	6.00000	+	10.3923i	0.194974	+	0.337705i	0.946892	−	0.321552i	$-0.104204\pi$
$947$	6.00000	+	10.3923i	0.194974	+	0.337705i	−0.751918	+	0.659256i	$0.770871\pi$
$948$		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	$-0.546566\pi$
$953$	−9.00000			−0.291539			−0.145769	+	0.989319i	$0.546566\pi$
$954$		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	$-0.696029\pi$
$971$	−36.0000			−1.15529			−0.577647	+	0.816286i	$0.696029\pi$
$972$		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	−0.973317	−	0.229465i	$-0.926302\pi$
$977$	−9.00000	+	15.5885i	−0.287936	+	0.498719i	0.685381	+	0.728184i	$0.259636\pi$
$978$		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	$-0.889168\pi$
$983$	−24.0000	−	41.5692i	−0.765481	−	1.32585i	0.174511	−	0.984655i	$-0.444166\pi$
$984$		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.b.433.1		2
3.2	odd	2		1296.2.i.n.433.1		2
4.3	odd	2		162.2.c.a.109.1		2
9.2	odd	6		1296.2.i.n.865.1		2
9.4	even	3		1296.2.a.l.1.1		1
9.5	odd	6		1296.2.a.c.1.1		1
9.7	even	3	inner	1296.2.i.b.865.1		2
12.11	even	2		162.2.c.d.109.1		2
36.7	odd	6		162.2.c.a.55.1		2
36.11	even	6		162.2.c.d.55.1		2
36.23	even	6		162.2.a.a.1.1	✓	1
36.31	odd	6		162.2.a.d.1.1	yes	1
72.5	odd	6		5184.2.a.bd.1.1		1
72.13	even	6		5184.2.a.h.1.1		1
72.59	even	6		5184.2.a.y.1.1		1
72.67	odd	6		5184.2.a.c.1.1		1
180.23	odd	12		4050.2.c.g.649.2		2
180.59	even	6		4050.2.a.bh.1.1		1
180.67	even	12		4050.2.c.n.649.2		2
180.103	even	12		4050.2.c.n.649.1		2
180.139	odd	6		4050.2.a.r.1.1		1
180.167	odd	12		4050.2.c.g.649.1		2
252.139	even	6		7938.2.a.s.1.1		1
252.167	odd	6		7938.2.a.n.1.1		1

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more