LMFDB - Embedded newform 648.2.i.f.433.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,2,Mod(217,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("648.217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	648.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:	$5.17430605098$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			433.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	648.433
Dual form			648.2.i.f.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{11} +(2.50000 - 4.33013i) q^{13} -5.00000 q^{17} +8.00000 q^{19} +(2.00000 - 3.46410i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-1.50000 - 2.59808i) q^{29} +(2.00000 - 3.46410i) q^{31} +3.00000 q^{37} +(3.00000 - 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(3.50000 - 6.06218i) q^{49} -10.0000 q^{53} +4.00000 q^{55} +(-4.00000 + 6.92820i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-2.50000 - 4.33013i) q^{65} +(-4.00000 + 6.92820i) q^{67} +16.0000 q^{71} -5.00000 q^{73} +(-2.00000 - 3.46410i) q^{79} +(-2.00000 - 3.46410i) q^{83} +(-2.50000 + 4.33013i) q^{85} +3.00000 q^{89} +(4.00000 - 6.92820i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5}+O(q^{10}) $ 2 * q + q^5	$ 2 q + q^{5} + 4 q^{11} + 5 q^{13} - 10 q^{17} + 16 q^{19} + 4 q^{23} + 4 q^{25} - 3 q^{29} + 4 q^{31} + 6 q^{37} + 6 q^{41} - 4 q^{43} + 12 q^{47} + 7 q^{49} - 20 q^{53} + 8 q^{55} - 8 q^{59} + 5 q^{61} - 5 q^{65} - 8 q^{67} + 32 q^{71} - 10 q^{73} - 4 q^{79} - 4 q^{83} - 5 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) $ 2 * q + q^5 + 4 * q^11 + 5 * q^13 - 10 * q^17 + 16 * q^19 + 4 * q^23 + 4 * q^25 - 3 * q^29 + 4 * q^31 + 6 * q^37 + 6 * q^41 - 4 * q^43 + 12 * q^47 + 7 * q^49 - 20 * q^53 + 8 * q^55 - 8 * q^59 + 5 * q^61 - 5 * q^65 - 8 * q^67 + 32 * q^71 - 10 * q^73 - 4 * q^79 - 4 * q^83 - 5 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	−0.732294	−	0.680989i	$-0.761550\pi$
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i	0.955901	+	0.293691i	$0.0948835\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	2.50000	−	4.33013i	0.693375	−	1.20096i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.00000			−1.21268			−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000			−1.21268			−0.606339	+	0.795206i	$0.707363\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	−0.578610	−	0.815604i	$-0.696405\pi$
$23$	2.00000	−	3.46410i	0.417029	−	0.722315i	0.995639	+	0.0932891i	$0.0297381\pi$
$24$		0			0
$25$	2.00000	+	3.46410i	0.400000	+	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	0.692480	−	0.721437i	$-0.256518\pi$
$29$	−1.50000	−	2.59808i	−0.278543	−	0.482451i	−0.971023	+	0.238987i	$0.923185\pi$
$30$		0			0
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	−0.628619	−	0.777714i	$-0.716379\pi$
$31$	2.00000	−	3.46410i	0.359211	−	0.622171i	0.987829	+	0.155543i	$0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.00000			0.493197			0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000			0.493197			0.246598	+	0.969118i	$0.420687\pi$
$38$		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
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	0.672264	−	0.740312i	$-0.265322\pi$
$43$	−2.00000	−	3.46410i	−0.304997	−	0.528271i	−0.977261	+	0.212041i	$0.931989\pi$
$44$		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
$49$	3.50000	−	6.06218i	0.500000	−	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	0.478953	+	0.877841i	$0.341016\pi$
$59$	−4.00000	+	6.92820i	−0.520756	+	0.901975i	−0.999709	+	0.0241347i	$0.992317\pi$
$60$		0			0
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	0.980507	−	0.196485i	$-0.0629528\pi$
$61$	2.50000	+	4.33013i	0.320092	+	0.554416i	−0.660415	+	0.750901i	$0.729619\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.50000	−	4.33013i	−0.310087	−	0.537086i
$66$		0			0
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	−0.999915	−	0.0130248i	$-0.995854\pi$
$67$	−4.00000	+	6.92820i	−0.488678	+	0.846415i	0.511237	+	0.859440i	$0.329187\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	16.0000			1.89885			0.949425	−	0.313993i	$-0.101667\pi$
$71$	16.0000			1.89885			0.949425	+	0.313993i	$0.101667\pi$
$72$		0			0
$73$	−5.00000			−0.585206			−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000			−0.585206			−0.292603	+	0.956234i	$0.594521\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.00000	−	3.46410i	−0.219529	−	0.380235i	0.735135	−	0.677920i	$-0.237119\pi$
$83$	−2.00000	−	3.46410i	−0.219529	−	0.380235i	−0.954664	+	0.297686i	$0.903785\pi$
$84$		0			0
$85$	−2.50000	+	4.33013i	−0.271163	+	0.469668i
$86$		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
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		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	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	−17.0000			−1.62830			−0.814152	−	0.580651i	$-0.802798\pi$
$109$	−17.0000			−1.62830			−0.814152	+	0.580651i	$0.802798\pi$
$110$		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
$115$	−2.00000	−	3.46410i	−0.186501	−	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$	−20.0000			−1.77471			−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000			−1.77471			−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	0.269863	+	0.962899i	$0.413022\pi$
$131$	−8.00000	+	13.8564i	−0.698963	+	1.21064i	−0.968826	+	0.247741i	$0.920312\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.50000	−	12.9904i	−0.640768	−	1.10984i	−0.985262	−	0.171054i	$-0.945283\pi$
$137$	−7.50000	−	12.9904i	−0.640768	−	1.10984i	0.344493	−	0.938789i	$-0.388051\pi$
$138$		0			0
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	0.491033	+	0.871141i	$0.336619\pi$
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	−0.999947	+	0.0103230i	$0.996714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	20.0000			1.67248
$144$		0			0
$145$	−3.00000			−0.249136
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2.50000	−	4.33013i	0.204808	−	0.354738i	−0.745264	−	0.666770i	$-0.767676\pi$
$149$	2.50000	−	4.33013i	0.204808	−	0.354738i	0.950072	+	0.312032i	$0.101010\pi$
$150$		0			0
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	0.982873	+	0.184284i	$0.0589965\pi$
$151$	8.00000	+	13.8564i	0.651031	+	1.12762i	−0.331842	+	0.943335i	$0.607670\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000	−	3.46410i	−0.160644	−	0.278243i
$156$		0			0
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	−0.971219	−	0.238190i	$-0.923446\pi$
$157$	−3.50000	+	6.06218i	−0.279330	+	0.483814i	0.691888	+	0.722005i	$0.256779\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−6.00000	−	10.3923i	−0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.50000	+	7.79423i	0.342129	+	0.592584i	0.984828	−	0.173534i	$-0.0555188\pi$
$173$	4.50000	+	7.79423i	0.342129	+	0.592584i	−0.642699	+	0.766119i	$0.722185\pi$
$174$		0			0
$175$		0			0
$176$		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
$181$	6.00000			0.445976			0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000			0.445976			0.222988	+	0.974821i	$0.428419\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.50000	−	2.59808i	0.110282	−	0.191014i
$186$		0			0
$187$	−10.0000	−	17.3205i	−0.731272	−	1.26660i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.863743	−	0.503932i	$-0.831886\pi$
$191$	−12.0000	−	20.7846i	−0.868290	−	1.50392i	−0.00454614	−	0.999990i	$-0.501447\pi$
$192$		0			0
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	−0.993215	−	0.116289i	$-0.962900\pi$
$193$	−5.50000	+	9.52628i	−0.395899	+	0.685717i	0.597317	+	0.802005i	$0.296234\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.0000			1.06871			0.534353	−	0.845262i	$-0.320555\pi$
$197$	15.0000			1.06871			0.534353	+	0.845262i	$0.320555\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.0000	+	27.7128i	1.10674	+	1.91694i
$210$		0			0
$211$	−2.00000	+	3.46410i	−0.137686	+	0.238479i	−0.926620	−	0.375999i	$-0.877300\pi$
$211$	−2.00000	+	3.46410i	−0.137686	+	0.238479i	0.788935	+	0.614477i	$0.210633\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000			−0.272798
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.5000	+	21.6506i	−0.840841	+	1.45638i
$222$		0			0
$223$	10.0000	+	17.3205i	0.669650	+	1.15987i	0.978002	+	0.208595i	$0.0668890\pi$
$223$	10.0000	+	17.3205i	0.669650	+	1.15987i	−0.308353	+	0.951272i	$0.599778\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	−0.771523	−	0.636201i	$-0.780505\pi$
$229$	2.50000	−	4.33013i	0.165205	−	0.286143i	0.936728	+	0.350058i	$0.113838\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.0000			0.720634			0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000			0.720634			0.360317	+	0.932830i	$0.382669\pi$
$234$		0			0
$235$	12.0000			0.782794
$236$		0			0
$237$		0			0
$238$		0			0
$239$	10.0000	−	17.3205i	0.646846	−	1.12037i	−0.337026	−	0.941495i	$-0.609421\pi$
$239$	10.0000	−	17.3205i	0.646846	−	1.12037i	0.983872	−	0.178875i	$-0.0572458\pi$
$240$		0			0
$241$	10.5000	+	18.1865i	0.676364	+	1.17150i	0.976068	+	0.217465i	$0.0697789\pi$
$241$	10.5000	+	18.1865i	0.676364	+	1.17150i	−0.299704	+	0.954032i	$0.596888\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.50000	−	6.06218i	−0.223607	−	0.387298i
$246$		0			0
$247$	20.0000	−	34.6410i	1.27257	−	2.20416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−20.0000			−1.26239			−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000			−1.26239			−0.631194	+	0.775625i	$0.717435\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.50000	−	14.7224i	0.530215	−	0.918360i	−0.469163	−	0.883112i	$-0.655444\pi$
$257$	8.50000	−	14.7224i	0.530215	−	0.918360i	0.999379	−	0.0352486i	$-0.0112223\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.00000	+	3.46410i	0.123325	+	0.213606i	0.921077	−	0.389380i	$-0.127311\pi$
$263$	2.00000	+	3.46410i	0.123325	+	0.213606i	−0.797752	+	0.602986i	$0.793977\pi$
$264$		0			0
$265$	−5.00000	+	8.66025i	−0.307148	+	0.531995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−17.0000			−1.03651			−0.518254	−	0.855227i	$-0.673418\pi$
$269$	−17.0000			−1.03651			−0.518254	+	0.855227i	$0.673418\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.00000	+	13.8564i	−0.482418	+	0.835573i
$276$		0			0
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	−0.980373	−	0.197153i	$-0.936830\pi$
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	0.319447	−	0.947604i	$-0.396503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.50000	+	4.33013i	0.149137	+	0.258314i	0.930909	−	0.365251i	$-0.119017\pi$
$281$	2.50000	+	4.33013i	0.149137	+	0.258314i	−0.781771	+	0.623565i	$0.785684\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	−0.377244	−	0.926114i	$-0.623128\pi$
$293$	10.5000	−	18.1865i	0.613417	−	1.06247i	0.990660	−	0.136355i	$-0.0435386\pi$
$294$		0			0
$295$	4.00000	+	6.92820i	0.232889	+	0.403376i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	−	17.3205i	−0.578315	−	1.00167i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.00000			0.286299
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		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
$313$	−7.50000	−	12.9904i	−0.423925	−	0.734260i	0.572394	−	0.819979i	$-0.306015\pi$
$313$	−7.50000	−	12.9904i	−0.423925	−	0.734260i	−0.996319	+	0.0857188i	$0.972681\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.5000	+	28.5788i	0.926732	+	1.60515i	0.788751	+	0.614713i	$0.210728\pi$
$317$	16.5000	+	28.5788i	0.926732	+	1.60515i	0.137981	+	0.990435i	$0.455939\pi$
$318$		0			0
$319$	6.00000	−	10.3923i	0.335936	−	0.581857i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−40.0000			−2.22566
$324$		0			0
$325$	20.0000			1.10940
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.00000	+	13.8564i	0.439720	+	0.761617i	0.997668	−	0.0682590i	$-0.0217444\pi$
$331$	8.00000	+	13.8564i	0.439720	+	0.761617i	−0.557948	+	0.829876i	$0.688411\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.00000	+	6.92820i	0.218543	+	0.378528i
$336$		0			0
$337$	15.0000	−	25.9808i	0.817102	−	1.41526i	−0.0907066	−	0.995878i	$-0.528913\pi$
$337$	15.0000	−	25.9808i	0.817102	−	1.41526i	0.907809	−	0.419385i	$-0.137754\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.0000			0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	9.00000	+	15.5885i	0.481759	+	0.834431i	0.999781	−	0.0209364i	$-0.00666475\pi$
$349$	9.00000	+	15.5885i	0.481759	+	0.834431i	−0.518022	+	0.855367i	$0.673331\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	7.00000	+	12.1244i	0.372572	+	0.645314i	0.989960	−	0.141344i	$-0.0451425\pi$
$353$	7.00000	+	12.1244i	0.372572	+	0.645314i	−0.617388	+	0.786659i	$0.711809\pi$
$354$		0			0
$355$	8.00000	−	13.8564i	0.424596	−	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.50000	+	4.33013i	−0.130856	+	0.226649i
$366$		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
$371$		0			0
$372$		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
$377$	−15.0000			−0.772539
$378$		0			0
$379$	20.0000			1.02733			0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000			1.02733			0.513665	+	0.857991i	$0.328287\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.00000	+	8.66025i	0.253510	+	0.439092i	0.964490	−	0.264120i	$-0.0850816\pi$
$389$	5.00000	+	8.66025i	0.253510	+	0.439092i	−0.710980	+	0.703213i	$0.751748\pi$
$390$		0			0
$391$	−10.0000	+	17.3205i	−0.505722	+	0.875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000			−0.201262
$396$		0			0
$397$	−13.0000			−0.652451			−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000			−0.652451			−0.326226	+	0.945292i	$0.605777\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−13.5000	+	23.3827i	−0.674158	+	1.16768i	0.302556	+	0.953131i	$0.402160\pi$
$401$	−13.5000	+	23.3827i	−0.674158	+	1.16768i	−0.976714	+	0.214544i	$0.931173\pi$
$402$		0			0
$403$	−10.0000	−	17.3205i	−0.498135	−	0.862796i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	+	10.3923i	0.297409	+	0.515127i
$408$		0			0
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	−0.989697	−	0.143180i	$-0.954267\pi$
$409$	−7.50000	+	12.9904i	−0.370851	+	0.642333i	0.618846	+	0.785513i	$0.287601\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.00000			−0.196352
$416$		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
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	0.877952	−	0.478748i	$-0.158909\pi$
$421$	0.500000	+	0.866025i	0.0243685	+	0.0422075i	−0.853584	+	0.520955i	$0.825576\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.0000	−	17.3205i	−0.485071	−	0.840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.00000			0.192673			0.0963366	−	0.995349i	$-0.469287\pi$
$431$	4.00000			0.192673			0.0963366	+	0.995349i	$0.469287\pi$
$432$		0			0
$433$	−29.0000			−1.39365			−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000			−1.39365			−0.696826	+	0.717241i	$0.745405\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.0000	−	27.7128i	0.765384	−	1.32568i
$438$		0			0
$439$	−6.00000	−	10.3923i	−0.286364	−	0.495998i	0.686575	−	0.727059i	$-0.259113\pi$
$439$	−6.00000	−	10.3923i	−0.286364	−	0.495998i	−0.972939	+	0.231062i	$0.925780\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	0.814595	−	0.580030i	$-0.196959\pi$
$443$	−2.00000	−	3.46410i	−0.0950229	−	0.164584i	−0.909618	+	0.415445i	$0.863626\pi$
$444$		0			0
$445$	1.50000	−	2.59808i	0.0711068	−	0.123161i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	24.0000			1.13012
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	−0.999014	−	0.0443868i	$-0.985867\pi$
$457$	−11.5000	−	19.9186i	−0.537947	−	0.931752i	0.461067	−	0.887365i	$-0.347467\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−7.00000	−	12.1244i	−0.326023	−	0.564688i	0.655696	−	0.755025i	$-0.272375\pi$
$461$	−7.00000	−	12.1244i	−0.326023	−	0.564688i	−0.981719	+	0.190337i	$0.939042\pi$
$462$		0			0
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	−0.618050	−	0.786139i	$-0.712077\pi$
$463$	8.00000	−	13.8564i	0.371792	−	0.643962i	0.989841	+	0.142177i	$0.0454103\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$		0			0
$475$	16.0000	+	27.7128i	0.734130	+	1.27155i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	0.942821	−	0.333300i	$-0.108162\pi$
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	−0.760056	+	0.649857i	$0.774829\pi$
$480$		0			0
$481$	7.50000	−	12.9904i	0.341971	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000			−0.0908153
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−18.0000	+	31.1769i	−0.812329	+	1.40699i	0.0989017	+	0.995097i	$0.468467\pi$
$491$	−18.0000	+	31.1769i	−0.812329	+	1.40699i	−0.911230	+	0.411897i	$0.864866\pi$
$492$		0			0
$493$	7.50000	+	12.9904i	0.337783	+	0.585057i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	−0.968500	−	0.249015i	$-0.919893\pi$
$499$	−6.00000	+	10.3923i	−0.268597	+	0.465223i	0.699903	+	0.714238i	$0.253227\pi$
$500$		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
$505$	−6.00000			−0.266996
$506$		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
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.00000	+	3.46410i	0.0881305	+	0.152647i
$516$		0			0
$517$	−24.0000	+	41.5692i	−1.05552	+	1.82821i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$	−20.0000			−0.874539			−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000			−0.874539			−0.437269	+	0.899331i	$0.644054\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.0000	+	17.3205i	−0.435607	+	0.754493i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.0000	−	25.9808i	−0.649722	−	1.12535i
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	28.0000			1.20605
$540$		0			0
$541$	−21.0000			−0.902861			−0.451430	−	0.892306i	$-0.649086\pi$
$541$	−21.0000			−0.902861			−0.451430	+	0.892306i	$0.649086\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.50000	+	14.7224i	−0.364100	+	0.630640i
$546$		0			0
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	0.569087	−	0.822277i	$-0.307297\pi$
$547$	−10.0000	−	17.3205i	−0.427569	−	0.740571i	−0.996657	+	0.0817056i	$0.973963\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−12.0000	−	20.7846i	−0.511217	−	0.885454i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−25.0000			−1.05928			−0.529642	−	0.848221i	$-0.677674\pi$
$557$	−25.0000			−1.05928			−0.529642	+	0.848221i	$0.677674\pi$
$558$		0			0
$559$	−20.0000			−0.845910
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	−	3.46410i	0.0842900	−	0.145994i	−0.820798	−	0.571218i	$-0.806471\pi$
$563$	2.00000	−	3.46410i	0.0842900	−	0.145994i	0.905088	+	0.425223i	$0.139804\pi$
$564$		0			0
$565$	7.50000	+	12.9904i	0.315527	+	0.546509i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.5000	−	19.9186i	−0.482105	−	0.835030i	0.517684	−	0.855572i	$-0.326794\pi$
$569$	−11.5000	−	19.9186i	−0.482105	−	0.835030i	−0.999789	+	0.0205415i	$0.993461\pi$
$570$		0			0
$571$	−12.0000	+	20.7846i	−0.502184	+	0.869809i	0.497812	+	0.867285i	$0.334137\pi$
$571$	−12.0000	+	20.7846i	−0.502184	+	0.869809i	−0.999997	+	0.00252413i	$0.999197\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.0000			0.667246
$576$		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
$581$		0			0
$582$		0			0
$583$	−20.0000	−	34.6410i	−0.828315	−	1.43468i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	14.0000	+	24.2487i	0.577842	+	1.00085i	0.995726	+	0.0923513i	$0.0294383\pi$
$587$	14.0000	+	24.2487i	0.577842	+	1.00085i	−0.417885	+	0.908500i	$0.637228\pi$
$588$		0			0
$589$	16.0000	−	27.7128i	0.659269	−	1.14189i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.00000			−0.0410651			−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000			−0.0410651			−0.0205325	+	0.999789i	$0.506536\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	−0.717021	−	0.697051i	$-0.754495\pi$
$599$	6.00000	−	10.3923i	0.245153	−	0.424618i	0.962175	+	0.272433i	$0.0878284\pi$
$600$		0			0
$601$	14.5000	+	25.1147i	0.591467	+	1.02445i	0.994035	+	0.109061i	$0.0347845\pi$
$601$	14.5000	+	25.1147i	0.591467	+	1.02445i	−0.402568	+	0.915390i	$0.631882\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.50000	+	4.33013i	0.101639	+	0.176045i
$606$		0			0
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	0.333842	+	0.942629i	$0.391655\pi$
$607$	−16.0000	+	27.7128i	−0.649420	+	1.12483i	−0.983262	+	0.182199i	$0.941678\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	60.0000			2.42734
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.5000	−	38.9711i	0.905816	−	1.56892i	0.0859976	−	0.996295i	$-0.472592\pi$
$617$	22.5000	−	38.9711i	0.905816	−	1.56892i	0.819818	−	0.572624i	$-0.194074\pi$
$618$		0			0
$619$	−20.0000	−	34.6410i	−0.803868	−	1.39234i	−0.917053	−	0.398766i	$-0.869439\pi$
$619$	−20.0000	−	34.6410i	−0.803868	−	1.39234i	0.113185	−	0.993574i	$-0.463895\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−5.50000	+	9.52628i	−0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.0000			−0.598089
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.0000	+	17.3205i	−0.396838	+	0.687343i
$636$		0			0
$637$	−17.5000	−	30.3109i	−0.693375	−	1.20096i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.50000	−	9.52628i	−0.217237	−	0.376265i	0.736725	−	0.676192i	$-0.236371\pi$
$641$	−5.50000	−	9.52628i	−0.217237	−	0.376265i	−0.953962	+	0.299927i	$0.903038\pi$
$642$		0			0
$643$	−18.0000	+	31.1769i	−0.709851	+	1.22950i	0.255062	+	0.966925i	$0.417904\pi$
$643$	−18.0000	+	31.1769i	−0.709851	+	1.22950i	−0.964912	+	0.262573i	$0.915429\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	0.407628	+	0.913148i	$0.366356\pi$
$653$	−15.0000	+	25.9808i	−0.586995	+	1.01671i	−0.994623	+	0.103558i	$0.966977\pi$
$654$		0			0
$655$	8.00000	+	13.8564i	0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	−0.952940	−	0.303160i	$-0.901958\pi$
$661$	−5.50000	+	9.52628i	−0.213925	+	0.370529i	0.739014	+	0.673690i	$0.235292\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−12.0000			−0.464642
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−10.0000	+	17.3205i	−0.386046	+	0.668651i
$672$		0			0
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	0.997586	+	0.0694449i	$0.0221228\pi$
$673$	14.5000	+	25.1147i	0.558934	+	0.968102i	−0.438652	+	0.898657i	$0.644544\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−19.0000	−	32.9090i	−0.730229	−	1.26479i	−0.956785	−	0.290796i	$-0.906080\pi$
$677$	−19.0000	−	32.9090i	−0.730229	−	1.26479i	0.226556	−	0.973998i	$-0.427253\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	20.0000			0.765279			0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000			0.765279			0.382639	+	0.923898i	$0.375015\pi$
$684$		0			0
$685$	−15.0000			−0.573121
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−25.0000	+	43.3013i	−0.952424	+	1.64965i
$690$		0			0
$691$	20.0000	+	34.6410i	0.760836	+	1.31781i	0.942420	+	0.334431i	$0.108544\pi$
$691$	20.0000	+	34.6410i	0.760836	+	1.31781i	−0.181584	+	0.983375i	$0.558123\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	+	10.3923i	0.227593	+	0.394203i
$696$		0			0
$697$	−15.0000	+	25.9808i	−0.568166	+	0.984092i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	39.0000			1.47301			0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000			1.47301			0.736505	+	0.676432i	$0.236475\pi$
$702$		0			0
$703$	24.0000			0.905177
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.5000	−	30.3109i	−0.657226	−	1.13835i	−0.981331	−	0.192328i	$-0.938396\pi$
$709$	−17.5000	−	30.3109i	−0.657226	−	1.13835i	0.324104	−	0.946021i	$-0.394937\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−8.00000	−	13.8564i	−0.299602	−	0.518927i
$714$		0			0
$715$	10.0000	−	17.3205i	0.373979	−	0.647750i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−28.0000			−1.04422			−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000			−1.04422			−0.522112	+	0.852877i	$0.674856\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6.00000	−	10.3923i	0.222834	−	0.385961i
$726$		0			0
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10.0000	+	17.3205i	0.369863	+	0.640622i
$732$		0			0
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	−0.360061	−	0.932929i	$-0.617244\pi$
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	0.987971	−	0.154642i	$-0.0494225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−32.0000			−1.17874
$738$		0			0
$739$	−28.0000			−1.03000			−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000			−1.03000			−0.514998	+	0.857191i	$0.672207\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.0000	+	17.3205i	−0.366864	+	0.635428i	−0.989073	−	0.147423i	$-0.952902\pi$
$743$	−10.0000	+	17.3205i	−0.366864	+	0.635428i	0.622209	+	0.782851i	$0.286235\pi$
$744$		0			0
$745$	−2.50000	−	4.33013i	−0.0915929	−	0.158644i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.5000	−	35.5070i	0.743124	−	1.28713i	−0.207942	−	0.978141i	$-0.566676\pi$
$761$	20.5000	−	35.5070i	0.743124	−	1.28713i	0.951066	−	0.308988i	$-0.0999903\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.0000	+	34.6410i	0.722158	+	1.25081i
$768$		0			0
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	−0.724701	−	0.689063i	$-0.758022\pi$
$769$	6.50000	−	11.2583i	0.234396	−	0.405986i	0.959097	+	0.283078i	$0.0913554\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−9.00000			−0.323708			−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000			−0.323708			−0.161854	+	0.986815i	$0.551747\pi$
$774$		0			0
$775$	16.0000			0.574737
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000	−	41.5692i	0.859889	−	1.48937i
$780$		0			0
$781$	32.0000	+	55.4256i	1.14505	+	1.98328i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3.50000	+	6.06218i	0.124920	+	0.216368i
$786$		0			0
$787$	10.0000	−	17.3205i	0.356462	−	0.617409i	−0.630905	−	0.775860i	$-0.717316\pi$
$787$	10.0000	−	17.3205i	0.356462	−	0.617409i	0.987367	+	0.158450i	$0.0506498\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	25.0000			0.887776
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.5000	−	21.6506i	0.442773	−	0.766905i	−0.555121	−	0.831769i	$-0.687328\pi$
$797$	12.5000	−	21.6506i	0.442773	−	0.766905i	0.997894	+	0.0648645i	$0.0206615\pi$
$798$		0			0
$799$	−30.0000	−	51.9615i	−1.06132	−	1.83827i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.0000	−	17.3205i	−0.352892	−	0.611227i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−17.0000			−0.597688			−0.298844	−	0.954302i	$-0.596601\pi$
$809$	−17.0000			−0.597688			−0.298844	+	0.954302i	$0.596601\pi$
$810$		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
$815$		0			0
$816$		0			0
$817$	−16.0000	−	27.7128i	−0.559769	−	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25.5000	−	44.1673i	−0.889956	−	1.54145i	−0.839926	−	0.542702i	$-0.817401\pi$
$821$	−25.5000	−	44.1673i	−0.889956	−	1.54145i	−0.0500305	−	0.998748i	$-0.515932\pi$
$822$		0			0
$823$	−22.0000	+	38.1051i	−0.766872	+	1.32826i	0.172379	+	0.985031i	$0.444854\pi$
$823$	−22.0000	+	38.1051i	−0.766872	+	1.32826i	−0.939251	+	0.343230i	$0.888479\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−20.0000			−0.695468			−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000			−0.695468			−0.347734	+	0.937593i	$0.613049\pi$
$828$		0			0
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−17.5000	+	30.3109i	−0.606339	+	1.05021i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	0.950813	−	0.309764i	$-0.100250\pi$
$839$	6.00000	+	10.3923i	0.207143	+	0.358782i	−0.743670	+	0.668546i	$0.766917\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000			−0.412813
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6.00000	−	10.3923i	0.205677	−	0.356244i
$852$		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
$857$	−7.50000	−	12.9904i	−0.256195	−	0.443743i	0.709024	−	0.705184i	$-0.249136\pi$
$857$	−7.50000	−	12.9904i	−0.256195	−	0.443743i	−0.965219	+	0.261441i	$0.915802\pi$
$858$		0			0
$859$	6.00000	−	10.3923i	0.204717	−	0.354581i	−0.745325	−	0.666701i	$-0.767706\pi$
$859$	6.00000	−	10.3923i	0.204717	−	0.354581i	0.950043	+	0.312120i	$0.101039\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	20.0000			0.680808			0.340404	−	0.940279i	$-0.389436\pi$
$863$	20.0000			0.680808			0.340404	+	0.940279i	$0.389436\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$		0			0
$868$		0			0
$869$	8.00000	−	13.8564i	0.271381	−	0.470046i
$870$		0			0
$871$	20.0000	+	34.6410i	0.677674	+	1.17377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	22.5000	−	38.9711i	0.759771	−	1.31596i	−0.183196	−	0.983076i	$-0.558644\pi$
$877$	22.5000	−	38.9711i	0.759771	−	1.31596i	0.942967	−	0.332886i	$-0.108022\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.0000			−0.471672			−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000			−0.471672			−0.235836	+	0.971793i	$0.575783\pi$
$882$		0			0
$883$	20.0000			0.673054			0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000			0.673054			0.336527	+	0.941674i	$0.390748\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−26.0000	+	45.0333i	−0.872995	+	1.51207i	−0.0141108	+	0.999900i	$0.504492\pi$
$887$	−26.0000	+	45.0333i	−0.872995	+	1.51207i	−0.858884	+	0.512170i	$0.828842\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	48.0000	+	83.1384i	1.60626	+	2.78212i
$894$		0			0
$895$	−6.00000	+	10.3923i	−0.200558	+	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.0000			−0.400222
$900$		0			0
$901$	50.0000			1.66574
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.00000	−	5.19615i	0.0997234	−	0.172726i
$906$		0			0
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	0.999334	+	0.0364935i	$0.0116188\pi$
$907$	16.0000	+	27.7128i	0.531271	+	0.920189i	−0.468063	+	0.883695i	$0.655048\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−22.0000	−	38.1051i	−0.728893	−	1.26248i	−0.957352	−	0.288925i	$-0.906702\pi$
$911$	−22.0000	−	38.1051i	−0.728893	−	1.26248i	0.228459	−	0.973554i	$-0.426631\pi$
$912$		0			0
$913$	8.00000	−	13.8564i	0.264761	−	0.458580i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	52.0000			1.71532			0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000			1.71532			0.857661	+	0.514216i	$0.171917\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	40.0000	−	69.2820i	1.31662	−	2.28045i
$924$		0			0
$925$	6.00000	+	10.3923i	0.197279	+	0.341697i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26.5000	+	45.8993i	0.869437	+	1.50591i	0.862573	+	0.505932i	$0.168851\pi$
$929$	26.5000	+	45.8993i	0.869437	+	1.50591i	0.00686358	+	0.999976i	$0.497815\pi$
$930$		0			0
$931$	28.0000	−	48.4974i	0.917663	−	1.58944i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−20.0000			−0.654070
$936$		0			0
$937$	27.0000			0.882052			0.441026	−	0.897494i	$-0.354615\pi$
$937$	27.0000			0.882052			0.441026	+	0.897494i	$0.354615\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7.50000	+	12.9904i	−0.244493	+	0.423474i	−0.961989	−	0.273088i	$-0.911955\pi$
$941$	−7.50000	+	12.9904i	−0.244493	+	0.423474i	0.717496	+	0.696563i	$0.245288\pi$
$942$		0			0
$943$	−12.0000	−	20.7846i	−0.390774	−	0.676840i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	0.932002	+	0.362454i	$0.118061\pi$
$947$	24.0000	+	41.5692i	0.779895	+	1.35082i	−0.152106	+	0.988364i	$0.548606\pi$
$948$		0			0
$949$	−12.5000	+	21.6506i	−0.405767	+	0.702809i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	15.0000			0.485898			0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000			0.485898			0.242949	+	0.970039i	$0.421885\pi$
$954$		0			0
$955$	−24.0000			−0.776622
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.50000	+	9.52628i	0.177051	+	0.306662i
$966$		0			0
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	−0.341567	−	0.939857i	$-0.610958\pi$
$967$	20.0000	−	34.6410i	0.643157	−	1.11398i	0.984724	−	0.174123i	$-0.0557089\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	−0.519841	−	0.854263i	$-0.674009\pi$
$977$	15.0000	−	25.9808i	0.479893	−	0.831198i	0.999734	+	0.0230645i	$0.00734232\pi$
$978$		0			0
$979$	6.00000	+	10.3923i	0.191761	+	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	7.50000	−	12.9904i	0.238970	−	0.413908i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.0000			−0.508770
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−10.0000	+	17.3205i	−0.317021	+	0.549097i
$996$		0			0
$997$	−27.5000	−	47.6314i	−0.870934	−	1.50850i	−0.861032	−	0.508551i	$-0.830182\pi$
$997$	−27.5000	−	47.6314i	−0.870934	−	1.50850i	−0.00990158	−	0.999951i	$-0.503152\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.i.f.433.1		2
3.2	odd	2		648.2.i.d.433.1		2
4.3	odd	2		1296.2.i.k.433.1		2
9.2	odd	6		648.2.i.d.217.1		2
9.4	even	3		648.2.a.b.1.1	✓	1
9.5	odd	6		648.2.a.d.1.1	yes	1
9.7	even	3	inner	648.2.i.f.217.1		2
12.11	even	2		1296.2.i.f.433.1		2
36.7	odd	6		1296.2.i.k.865.1		2
36.11	even	6		1296.2.i.f.865.1		2
36.23	even	6		1296.2.a.h.1.1		1
36.31	odd	6		1296.2.a.d.1.1		1
72.5	odd	6		5184.2.a.k.1.1		1
72.13	even	6		5184.2.a.v.1.1		1
72.59	even	6		5184.2.a.l.1.1		1
72.67	odd	6		5184.2.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.a.b.1.1	✓	1	9.4	even	3
648.2.a.d.1.1	yes	1	9.5	odd	6
648.2.i.d.217.1		2	9.2	odd	6
648.2.i.d.433.1		2	3.2	odd	2
648.2.i.f.217.1		2	9.7	even	3	inner
648.2.i.f.433.1		2	1.1	even	1	trivial
1296.2.a.d.1.1		1	36.31	odd	6
1296.2.a.h.1.1		1	36.23	even	6
1296.2.i.f.433.1		2	12.11	even	2
1296.2.i.f.865.1		2	36.11	even	6
1296.2.i.k.433.1		2	4.3	odd	2
1296.2.i.k.865.1		2	36.7	odd	6
5184.2.a.k.1.1		1	72.5	odd	6
5184.2.a.l.1.1		1	72.59	even	6
5184.2.a.u.1.1		1	72.67	odd	6
5184.2.a.v.1.1		1	72.13	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{11} +(2.50000 - 4.33013i) q^{13} -5.00000 q^{17} +8.00000 q^{19} +(2.00000 - 3.46410i) q^{23} +(2.00000 + 3.46410i) q^{25} +(-1.50000 - 2.59808i) q^{29} +(2.00000 - 3.46410i) q^{31} +3.00000 q^{37} +(3.00000 - 5.19615i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(3.50000 - 6.06218i) q^{49} -10.0000 q^{53} +4.00000 q^{55} +(-4.00000 + 6.92820i) q^{59} +(2.50000 + 4.33013i) q^{61} +(-2.50000 - 4.33013i) q^{65} +(-4.00000 + 6.92820i) q^{67} +16.0000 q^{71} -5.00000 q^{73} +(-2.00000 - 3.46410i) q^{79} +(-2.00000 - 3.46410i) q^{83} +(-2.50000 + 4.33013i) q^{85} +3.00000 q^{89} +(4.00000 - 6.92820i) q^{95} +(-1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5}+O(q^{10}) \) 2 * q + q^5	\( 2 q + q^{5} + 4 q^{11} + 5 q^{13} - 10 q^{17} + 16 q^{19} + 4 q^{23} + 4 q^{25} - 3 q^{29} + 4 q^{31} + 6 q^{37} + 6 q^{41} - 4 q^{43} + 12 q^{47} + 7 q^{49} - 20 q^{53} + 8 q^{55} - 8 q^{59} + 5 q^{61} - 5 q^{65} - 8 q^{67} + 32 q^{71} - 10 q^{73} - 4 q^{79} - 4 q^{83} - 5 q^{85} + 6 q^{89} + 8 q^{95} - 2 q^{97}+O(q^{100}) \) 2 * q + q^5 + 4 * q^11 + 5 * q^13 - 10 * q^17 + 16 * q^19 + 4 * q^23 + 4 * q^25 - 3 * q^29 + 4 * q^31 + 6 * q^37 + 6 * q^41 - 4 * q^43 + 12 * q^47 + 7 * q^49 - 20 * q^53 + 8 * q^55 - 8 * q^59 + 5 * q^61 - 5 * q^65 - 8 * q^67 + 32 * q^71 - 10 * q^73 - 4 * q^79 - 4 * q^83 - 5 * q^85 + 6 * q^89 + 8 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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