LMFDB - Embedded newform 2700.2.i.a.901.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(901,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("2700.901");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.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:	$21.5596085457$
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 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			901.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2700.901
Dual form			2700.2.i.a.1801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{7} +(-2.00000 + 3.46410i) q^{13} -6.00000 q^{17} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{23} +(1.50000 + 2.59808i) q^{29} +(5.00000 - 8.66025i) q^{31} +10.0000 q^{37} +(4.50000 - 7.79423i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-4.50000 - 7.79423i) q^{47} +(3.00000 - 5.19615i) q^{49} -6.00000 q^{53} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(5.50000 - 9.52628i) q^{67} -12.0000 q^{71} +4.00000 q^{73} +(5.00000 + 8.66025i) q^{79} +(4.50000 + 7.79423i) q^{83} -9.00000 q^{89} +4.00000 q^{91} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{7} - 4 q^{13} - 12 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{29} + 10 q^{31} + 20 q^{37} + 9 q^{41} - 4 q^{43} - 9 q^{47} + 6 q^{49} - 12 q^{53} - 6 q^{59} + q^{61} + 11 q^{67} - 24 q^{71} + 8 q^{73}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q - q^7 - 4 * q^13 - 12 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^29 + 10 * q^31 + 20 * q^37 + 9 * q^41 - 4 * q^43 - 9 * q^47 + 6 * q^49 - 12 * q^53 - 6 * q^59 + q^61 + 11 * q^67 - 24 * q^71 + 8 * q^73 + 10 * q^79 + 9 * q^83 - 18 * q^89 + 8 * q^91 - 10 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	0.755929	−	0.654654i	$-0.227186\pi$
$7$	−0.500000	−	0.866025i	−0.188982	−	0.327327i	−0.944911	+	0.327327i	$0.893852\pi$
$8$		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
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	0.443227	+	0.896410i	$0.353834\pi$
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.00000			−1.45521			−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000			−1.45521			−0.727607	+	0.685994i	$0.759367\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	$-0.732076\pi$
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	0.978961	+	0.204046i	$0.0654092\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	5.00000	−	8.66025i	0.898027	−	1.55543i	0.0680129	−	0.997684i	$-0.478334\pi$
$31$	5.00000	−	8.66025i	0.898027	−	1.55543i	0.830014	−	0.557743i	$-0.188333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	−0.264704	−	0.964330i	$-0.585274\pi$
$41$	4.50000	−	7.79423i	0.702782	−	1.21725i	0.967486	−	0.252924i	$-0.0813924\pi$
$42$		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
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	−0.981543	−	0.191243i	$-0.938748\pi$
$47$	−4.50000	−	7.79423i	−0.656392	−	1.13691i	0.325150	−	0.945662i	$-0.394585\pi$
$48$		0			0
$49$	3.00000	−	5.19615i	0.428571	−	0.742307i
$50$		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
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	−0.992524	−	0.122047i	$-0.961054\pi$
$59$	−3.00000	+	5.19615i	−0.390567	+	0.676481i	0.601958	+	0.798528i	$0.294388\pi$
$60$		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
$65$		0			0
$66$		0			0
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	5.50000	−	9.52628i	0.671932	−	1.16382i	0.977356	−	0.211604i	$-0.0678686\pi$
$68$		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
$73$	4.00000			0.468165			0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000			0.468165			0.234082	+	0.972217i	$0.424791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	0.997274	+	0.0737937i	$0.0235106\pi$
$79$	5.00000	+	8.66025i	0.562544	+	0.974355i	−0.434730	+	0.900561i	$0.643156\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	$-0.00222321\pi$
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	−0.506036	+	0.862512i	$0.668890\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.00000			−0.953998			−0.476999	−	0.878904i	$-0.658275\pi$
$89$	−9.00000			−0.953998			−0.476999	+	0.878904i	$0.658275\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	−0.999961	−	0.00888289i	$-0.997172\pi$
$97$	−5.00000	−	8.66025i	−0.507673	−	0.879316i	0.492287	−	0.870433i	$-0.336161\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	−0.598858	−	0.800855i	$-0.704379\pi$
$103$	4.00000	−	6.92820i	0.394132	−	0.682656i	0.992990	+	0.118199i	$0.0377120\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.00000			0.870063			0.435031	−	0.900415i	$-0.356737\pi$
$107$	9.00000			0.870063			0.435031	+	0.900415i	$0.356737\pi$
$108$		0			0
$109$	−19.0000			−1.81987			−0.909935	−	0.414751i	$-0.863869\pi$
$109$	−19.0000			−1.81987			−0.909935	+	0.414751i	$0.863869\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	−0.432670	−	0.901553i	$-0.642428\pi$
$113$	6.00000	−	10.3923i	0.564433	−	0.977626i	0.997102	−	0.0760733i	$-0.0242383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.00000	+	5.19615i	0.275010	+	0.476331i
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	7.00000			0.621150			0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000			0.621150			0.310575	+	0.950549i	$0.399478\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	−0.704692	−	0.709514i	$-0.748915\pi$
$131$	3.00000	−	5.19615i	0.262111	−	0.453990i	0.966803	+	0.255524i	$0.0822479\pi$
$132$		0			0
$133$	−1.00000	−	1.73205i	−0.0867110	−	0.150188i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	−0.999893	−	0.0146279i	$-0.995344\pi$
$137$	−6.00000	−	10.3923i	−0.512615	−	0.887875i	0.487278	−	0.873247i	$-0.337990\pi$
$138$		0			0
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	0.0346338	+	0.999400i	$0.488974\pi$
$139$	−10.0000	+	17.3205i	−0.848189	+	1.46911i	−0.882823	+	0.469706i	$0.844360\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		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	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	−0.823359	−	0.567521i	$-0.807902\pi$
$157$	1.00000	−	1.73205i	0.0798087	−	0.138233i	0.903167	+	0.429289i	$0.141236\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.00000			−0.236433
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.50000	−	12.9904i	0.580367	−	1.00523i	−0.415068	−	0.909790i	$-0.636242\pi$
$167$	7.50000	−	12.9904i	0.580367	−	1.00523i	0.995436	−	0.0954356i	$-0.0304244\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.0000			1.79384			0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000			1.79384			0.896922	+	0.442189i	$0.145798\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	0.953912	−	0.300088i	$-0.0970159\pi$
$191$	3.00000	+	5.19615i	0.217072	+	0.375980i	−0.736839	+	0.676068i	$0.763683\pi$
$192$		0			0
$193$	−2.00000	+	3.46410i	−0.143963	+	0.249351i	−0.928986	−	0.370116i	$-0.879318\pi$
$193$	−2.00000	+	3.46410i	−0.143963	+	0.249351i	0.785022	+	0.619467i	$0.212651\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	2.00000			0.141776			0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000			0.141776			0.0708881	+	0.997484i	$0.477417\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.50000	−	2.59808i	0.105279	−	0.182349i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	−0.788935	−	0.614477i	$-0.789367\pi$
$211$	2.00000	−	3.46410i	0.137686	−	0.238479i	0.926620	+	0.375999i	$0.122700\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−10.0000			−0.678844
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	5.50000	+	9.52628i	0.368307	+	0.637927i	0.989301	−	0.145889i	$-0.0466041\pi$
$223$	5.50000	+	9.52628i	0.368307	+	0.637927i	−0.620994	+	0.783815i	$0.713271\pi$
$224$		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
$229$	0.500000	−	0.866025i	0.0330409	−	0.0572286i	−0.849032	−	0.528341i	$-0.822814\pi$
$229$	0.500000	−	0.866025i	0.0330409	−	0.0572286i	0.882073	+	0.471113i	$0.156147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−30.0000			−1.96537			−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000			−1.96537			−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$239$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$240$		0			0
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	0.956456	−	0.291877i	$-0.0942799\pi$
$241$	3.50000	+	6.06218i	0.225455	+	0.390499i	−0.731001	+	0.682376i	$0.760947\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.00000	+	6.92820i	−0.254514	+	0.440831i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	−0.944294	−	0.329104i	$-0.893253\pi$
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	0.757159	+	0.653231i	$0.226587\pi$
$258$		0			0
$259$	−5.00000	−	8.66025i	−0.310685	−	0.538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	0.952517	+	0.304487i	$0.0984850\pi$
$263$	12.0000	+	20.7846i	0.739952	+	1.28163i	−0.212565	+	0.977147i	$0.568182\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.00000			−0.548740			−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000			−0.548740			−0.274370	+	0.961624i	$0.588469\pi$
$270$		0			0
$271$	14.0000			0.850439			0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000			0.850439			0.425220	+	0.905090i	$0.360197\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	0.894503	−	0.447062i	$-0.147530\pi$
$277$	1.00000	+	1.73205i	0.0600842	+	0.104069i	−0.834419	+	0.551131i	$0.814196\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.5000	+	28.5788i	0.984307	+	1.70487i	0.644974	+	0.764204i	$0.276868\pi$
$281$	16.5000	+	28.5788i	0.984307	+	1.70487i	0.339333	+	0.940666i	$0.389799\pi$
$282$		0			0
$283$	−0.500000	+	0.866025i	−0.0297219	+	0.0514799i	−0.880504	−	0.474039i	$-0.842796\pi$
$283$	−0.500000	+	0.866025i	−0.0297219	+	0.0514799i	0.850782	+	0.525519i	$0.176129\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.00000			−0.531253
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	−0.764990	−	0.644042i	$-0.777256\pi$
$293$	3.00000	−	5.19615i	0.175262	−	0.303562i	0.940252	+	0.340480i	$0.110589\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$	−2.00000	+	3.46410i	−0.115278	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−29.0000			−1.65512			−0.827559	−	0.561379i	$-0.810271\pi$
$307$	−29.0000			−1.65512			−0.827559	+	0.561379i	$0.810271\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	−0.489585	−	0.871956i	$-0.662852\pi$
$311$	9.00000	−	15.5885i	0.510343	−	0.883940i	0.999928	−	0.0119847i	$-0.00381495\pi$
$312$		0			0
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	0.546335	−	0.837567i	$-0.316023\pi$
$313$	−8.00000	−	13.8564i	−0.452187	−	0.783210i	−0.998522	+	0.0543564i	$0.982689\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	0.983866	−	0.178908i	$-0.0572566\pi$
$317$	6.00000	+	10.3923i	0.336994	+	0.583690i	−0.646872	+	0.762598i	$0.723923\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−12.0000			−0.667698
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.50000	+	7.79423i	−0.248093	+	0.429710i
$330$		0			0
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	−0.963135	−	0.269019i	$-0.913301\pi$
$331$	−13.0000	−	22.5167i	−0.714545	−	1.23763i	0.248590	−	0.968609i	$-0.420033\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	−0.736270	−	0.676688i	$-0.763415\pi$
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	0.954164	+	0.299285i	$0.0967480\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−13.0000			−0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	0.340293	+	0.940319i	$0.389474\pi$
$347$	−12.0000	+	20.7846i	−0.644194	+	1.11578i	−0.984487	+	0.175457i	$0.943860\pi$
$348$		0			0
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	−0.990282	−	0.139072i	$-0.955588\pi$
$349$	−11.5000	−	19.9186i	−0.615581	−	1.06622i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	0.661004	−	0.750382i	$-0.270130\pi$
$353$	−6.00000	−	10.3923i	−0.319348	−	0.553127i	−0.980352	+	0.197256i	$0.936797\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	0.893873	+	0.448320i	$0.147978\pi$
$367$	16.0000	+	27.7128i	0.835193	+	1.44660i	−0.0586798	+	0.998277i	$0.518689\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.00000	+	5.19615i	0.155752	+	0.269771i
$372$		0			0
$373$	7.00000	−	12.1244i	0.362446	−	0.627775i	−0.625917	−	0.779890i	$-0.715275\pi$
$373$	7.00000	−	12.1244i	0.362446	−	0.627775i	0.988363	+	0.152115i	$0.0486083\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.0000			−0.618031
$378$		0			0
$379$	14.0000			0.719132			0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000			0.719132			0.359566	+	0.933120i	$0.382925\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.50000	−	12.9904i	−0.380265	−	0.658638i	0.610835	−	0.791758i	$-0.290834\pi$
$389$	−7.50000	−	12.9904i	−0.380265	−	0.658638i	−0.991100	+	0.133120i	$0.957501\pi$
$390$		0			0
$391$	−9.00000	+	15.5885i	−0.455150	+	0.788342i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	22.0000			1.10415			0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000			1.10415			0.552074	+	0.833795i	$0.313837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$	20.0000	+	34.6410i	0.996271	+	1.72559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	−0.715523	−	0.698589i	$-0.753812\pi$
$409$	5.00000	−	8.66025i	0.247234	−	0.428222i	0.962757	+	0.270367i	$0.0871450\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6.00000			0.295241
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	−0.681426	−	0.731887i	$-0.738640\pi$
$419$	6.00000	−	10.3923i	0.293119	−	0.507697i	0.974546	+	0.224189i	$0.0719734\pi$
$420$		0			0
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	11.0000	+	19.0526i	0.536107	+	0.928565i	−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.500000	−	0.866025i	0.0241967	−	0.0419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.00000			−0.289010			−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000			−0.289010			−0.144505	+	0.989504i	$0.546159\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.00000	−	5.19615i	0.143509	−	0.248566i
$438$		0			0
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	0.978412	+	0.206666i	$0.0662612\pi$
$439$	14.0000	+	24.2487i	0.668184	+	1.15733i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	13.5000	+	23.3827i	0.641404	+	1.11094i	0.985119	+	0.171871i	$0.0549812\pi$
$443$	13.5000	+	23.3827i	0.641404	+	1.11094i	−0.343715	+	0.939074i	$0.611685\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	0.725059	−	0.688686i	$-0.241812\pi$
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	0.938386	+	0.345589i	$0.112321\pi$
$461$	16.5000	+	28.5788i	0.768482	+	1.33105i	−0.169904	+	0.985461i	$0.554346\pi$
$462$		0			0
$463$	−2.00000	+	3.46410i	−0.0929479	+	0.160990i	−0.908750	−	0.417340i	$-0.862962\pi$
$463$	−2.00000	+	3.46410i	−0.0929479	+	0.160990i	0.815802	+	0.578331i	$0.196296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−15.0000	−	25.9808i	−0.685367	−	1.18709i	−0.973321	−	0.229447i	$-0.926308\pi$
$479$	−15.0000	−	25.9808i	−0.685367	−	1.18709i	0.287954	−	0.957644i	$-0.407025\pi$
$480$		0			0
$481$	−20.0000	+	34.6410i	−0.911922	+	1.57949i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	16.0000			0.725029			0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000			0.725029			0.362515	+	0.931978i	$0.381918\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	0.457263	+	0.889332i	$0.348830\pi$
$491$	−12.0000	+	20.7846i	−0.541552	+	0.937996i	−0.998815	+	0.0486647i	$0.984503\pi$
$492$		0			0
$493$	−9.00000	−	15.5885i	−0.405340	−	0.702069i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.00000	+	10.3923i	0.269137	+	0.466159i
$498$		0			0
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	−0.817781	−	0.575529i	$-0.804796\pi$
$499$	2.00000	−	3.46410i	0.0895323	−	0.155074i	0.907314	+	0.420455i	$0.138129\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.00000			0.133763			0.0668817	−	0.997761i	$-0.478695\pi$
$503$	3.00000			0.133763			0.0668817	+	0.997761i	$0.478695\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	−0.948353	−	0.317217i	$-0.897252\pi$
$509$	−4.50000	+	7.79423i	−0.199459	+	0.345473i	0.748894	+	0.662690i	$0.230585\pi$
$510$		0			0
$511$	−2.00000	−	3.46410i	−0.0884748	−	0.153243i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−27.0000			−1.18289			−0.591446	−	0.806345i	$-0.701443\pi$
$521$	−27.0000			−1.18289			−0.591446	+	0.806345i	$0.701443\pi$
$522$		0			0
$523$	7.00000			0.306089			0.153044	−	0.988219i	$-0.451092\pi$
$523$	7.00000			0.306089			0.153044	+	0.988219i	$0.451092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−30.0000	+	51.9615i	−1.30682	+	2.26348i
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	18.0000	+	31.1769i	0.779667	+	1.35042i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.00000			−0.300954			−0.150477	−	0.988614i	$-0.548081\pi$
$541$	−7.00000			−0.300954			−0.150477	+	0.988614i	$0.548081\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	0.781449	−	0.623970i	$-0.214481\pi$
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	−0.931098	+	0.364770i	$0.881148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.00000	+	5.19615i	0.127804	+	0.221364i
$552$		0			0
$553$	5.00000	−	8.66025i	0.212622	−	0.368271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.00000			−0.254228			−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000			−0.254228			−0.127114	+	0.991888i	$0.540571\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	−0.0824933	−	0.996592i	$-0.526288\pi$
$563$	19.5000	−	33.7750i	0.821827	−	1.42345i	0.904320	−	0.426855i	$-0.140378\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	0.922032	−	0.387113i	$-0.126528\pi$
$569$	3.00000	+	5.19615i	0.125767	+	0.217834i	−0.796266	+	0.604947i	$0.793194\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−38.0000			−1.58196			−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000			−1.58196			−0.790980	+	0.611842i	$0.790429\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.50000	−	7.79423i	0.186691	−	0.323359i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	13.5000	+	23.3827i	0.557205	+	0.965107i	0.997728	+	0.0673658i	$0.0214594\pi$
$587$	13.5000	+	23.3827i	0.557205	+	0.965107i	−0.440524	+	0.897741i	$0.645207\pi$
$588$		0			0
$589$	10.0000	−	17.3205i	0.412043	−	0.713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000			−0.246390			−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000			−0.246390			−0.123195	+	0.992382i	$0.539314\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	3.00000	−	5.19615i	0.122577	−	0.212309i	−0.798206	−	0.602384i	$-0.794218\pi$
$599$	3.00000	−	5.19615i	0.122577	−	0.212309i	0.920783	+	0.390075i	$0.127551\pi$
$600$		0			0
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	0.998302	−	0.0582563i	$-0.0185541\pi$
$601$	11.0000	+	19.0526i	0.448699	+	0.777170i	−0.549602	+	0.835426i	$0.685221\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−3.50000	+	6.06218i	−0.142061	+	0.246056i	−0.928272	−	0.371901i	$-0.878706\pi$
$607$	−3.50000	+	6.06218i	−0.142061	+	0.246056i	0.786212	+	0.617957i	$0.212039\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	36.0000			1.45640
$612$		0			0
$613$	22.0000			0.888572			0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000			0.888572			0.444286	+	0.895885i	$0.353457\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	−0.719605	−	0.694383i	$-0.755677\pi$
$617$	6.00000	−	10.3923i	0.241551	−	0.418378i	0.961156	+	0.276005i	$0.0890106\pi$
$618$		0			0
$619$	−13.0000	−	22.5167i	−0.522514	−	0.905021i	−0.999657	−	0.0261952i	$-0.991661\pi$
$619$	−13.0000	−	22.5167i	−0.522514	−	0.905021i	0.477143	−	0.878826i	$-0.341672\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.50000	+	7.79423i	0.180289	+	0.312269i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−60.0000			−2.39236
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	12.0000	+	20.7846i	0.475457	+	0.823516i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	0.995400	−	0.0958109i	$-0.0305444\pi$
$641$	10.5000	+	18.1865i	0.414725	+	0.718325i	−0.580674	+	0.814136i	$0.697211\pi$
$642$		0			0
$643$	−15.5000	+	26.8468i	−0.611260	+	1.05873i	0.379768	+	0.925082i	$0.376004\pi$
$643$	−15.5000	+	26.8468i	−0.611260	+	1.05873i	−0.991028	+	0.133652i	$0.957330\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.00000			0.353827			0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000			0.353827			0.176913	+	0.984226i	$0.443389\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	−0.918736	−	0.394872i	$-0.870789\pi$
$653$	−3.00000	+	5.19615i	−0.117399	+	0.203341i	0.801337	+	0.598213i	$0.204122\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	0.968052	+	0.250748i	$0.0806766\pi$
$659$	18.0000	+	31.1769i	0.701180	+	1.21448i	−0.266872	+	0.963732i	$0.585990\pi$
$660$		0			0
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	−0.752252	−	0.658876i	$-0.771032\pi$
$661$	5.00000	−	8.66025i	0.194477	−	0.336845i	0.946729	+	0.322031i	$0.104366\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9.00000			0.348481
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.0000	−	24.2487i	−0.539660	−	0.934719i	−0.998922	−	0.0464181i	$-0.985219\pi$
$673$	−14.0000	−	24.2487i	−0.539660	−	0.934719i	0.459262	−	0.888301i	$-0.348114\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−5.00000	+	8.66025i	−0.191882	+	0.332350i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	12.0000	−	20.7846i	0.457164	−	0.791831i
$690$		0			0
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	0.945319	−	0.326146i	$-0.105750\pi$
$691$	5.00000	+	8.66025i	0.190209	+	0.329452i	−0.755110	+	0.655598i	$0.772417\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−27.0000	+	46.7654i	−1.02270	+	1.77136i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21.0000			0.793159			0.396580	−	0.918000i	$-0.370197\pi$
$701$	21.0000			0.793159			0.396580	+	0.918000i	$0.370197\pi$
$702$		0			0
$703$	20.0000			0.754314
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−9.00000	+	15.5885i	−0.338480	+	0.586264i
$708$		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
$713$	−15.0000	−	25.9808i	−0.561754	−	0.972987i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	42.0000			1.56634			0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000			1.56634			0.783168	+	0.621810i	$0.213603\pi$
$720$		0			0
$721$	−8.00000			−0.297936
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−9.50000	−	16.4545i	−0.352335	−	0.610263i	0.634323	−	0.773068i	$-0.281279\pi$
$727$	−9.50000	−	16.4545i	−0.352335	−	0.610263i	−0.986658	+	0.162805i	$0.947946\pi$
$728$		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	−0.707303	−	0.706910i	$-0.750088\pi$
$733$	7.00000	−	12.1244i	0.258551	−	0.447823i	0.965854	+	0.259087i	$0.0834217\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	2.00000			0.0735712			0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000			0.0735712			0.0367856	+	0.999323i	$0.488288\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−16.5000	+	28.5788i	−0.605326	+	1.04846i	0.386674	+	0.922217i	$0.373624\pi$
$743$	−16.5000	+	28.5788i	−0.605326	+	1.04846i	−0.992000	+	0.126239i	$0.959709\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.50000	−	7.79423i	−0.164426	−	0.284795i
$750$		0			0
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	−0.682341	−	0.731034i	$-0.739038\pi$
$751$	8.00000	−	13.8564i	0.291924	−	0.505627i	0.974265	+	0.225407i	$0.0723712\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−20.0000			−0.726912			−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000			−0.726912			−0.363456	+	0.931611i	$0.618403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.5000	+	38.9711i	−0.815624	+	1.41270i	0.0932544	+	0.995642i	$0.470273\pi$
$761$	−22.5000	+	38.9711i	−0.815624	+	1.41270i	−0.908879	+	0.417061i	$0.863060\pi$
$762$		0			0
$763$	9.50000	+	16.4545i	0.343923	+	0.595692i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.0000	−	20.7846i	−0.433295	−	0.750489i
$768$		0			0
$769$	15.5000	−	26.8468i	0.558944	−	0.968120i	−0.438641	−	0.898663i	$-0.644540\pi$
$769$	15.5000	−	26.8468i	0.558944	−	0.968120i	0.997585	−	0.0694574i	$-0.0221268\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−36.0000			−1.29483			−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000			−1.29483			−0.647415	+	0.762138i	$0.724150\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9.00000	−	15.5885i	0.322458	−	0.558514i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	0.250832	+	0.968031i	$0.419296\pi$
$787$	−20.0000	+	34.6410i	−0.712923	+	1.23482i	−0.963755	+	0.266788i	$0.914038\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	−4.00000			−0.142044
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12.0000	−	20.7846i	0.425062	−	0.736229i	−0.571364	−	0.820696i	$-0.693586\pi$
$797$	12.0000	−	20.7846i	0.425062	−	0.736229i	0.996426	+	0.0844678i	$0.0269190\pi$
$798$		0			0
$799$	27.0000	+	46.7654i	0.955191	+	1.65444i
$800$		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
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		0			0
$811$	−22.0000			−0.772524			−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000			−0.772524			−0.386262	+	0.922389i	$0.626234\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.00000	−	6.92820i	−0.139942	−	0.242387i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.50000	+	2.59808i	0.0523504	+	0.0906735i	0.891013	−	0.453978i	$-0.149995\pi$
$821$	1.50000	+	2.59808i	0.0523504	+	0.0906735i	−0.838663	+	0.544651i	$0.816662\pi$
$822$		0			0
$823$	−24.5000	+	42.4352i	−0.854016	+	1.47920i	0.0235383	+	0.999723i	$0.492507\pi$
$823$	−24.5000	+	42.4352i	−0.854016	+	1.47920i	−0.877555	+	0.479477i	$0.840826\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	27.0000			0.938882			0.469441	−	0.882964i	$-0.344455\pi$
$827$	27.0000			0.938882			0.469441	+	0.882964i	$0.344455\pi$
$828$		0			0
$829$	−7.00000			−0.243120			−0.121560	−	0.992584i	$-0.538790\pi$
$829$	−7.00000			−0.243120			−0.121560	+	0.992584i	$0.538790\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.0000	+	31.1769i	−0.623663	+	1.08022i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.0000	−	25.9808i	−0.517858	−	0.896956i	−0.999785	−	0.0207443i	$-0.993396\pi$
$839$	−15.0000	−	25.9808i	−0.517858	−	0.896956i	0.481927	−	0.876211i	$-0.339937\pi$
$840$		0			0
$841$	10.0000	−	17.3205i	0.344828	−	0.597259i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−11.0000			−0.377964
$848$		0			0
$849$		0			0
$850$		0			0
$851$	15.0000	−	25.9808i	0.514193	−	0.890609i
$852$		0			0
$853$	−2.00000	−	3.46410i	−0.0684787	−	0.118609i	0.829753	−	0.558131i	$-0.188481\pi$
$853$	−2.00000	−	3.46410i	−0.0684787	−	0.118609i	−0.898232	+	0.439522i	$0.855148\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.0000	+	41.5692i	0.819824	+	1.41998i	0.905811	+	0.423681i	$0.139262\pi$
$857$	24.0000	+	41.5692i	0.819824	+	1.41998i	−0.0859870	+	0.996296i	$0.527404\pi$
$858$		0			0
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	−0.984655	−	0.174512i	$-0.944165\pi$
$859$	−10.0000	+	17.3205i	−0.341196	+	0.590968i	0.643459	+	0.765480i	$0.277499\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.0000			0.510606			0.255303	−	0.966861i	$-0.417825\pi$
$863$	15.0000			0.510606			0.255303	+	0.966861i	$0.417825\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	22.0000	+	38.1051i	0.745442	+	1.29114i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−2.00000	+	3.46410i	−0.0675352	+	0.116974i	−0.897816	−	0.440371i	$-0.854847\pi$
$877$	−2.00000	+	3.46410i	−0.0675352	+	0.116974i	0.830281	+	0.557346i	$0.188180\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$	−29.0000			−0.975928			−0.487964	−	0.872864i	$-0.662260\pi$
$883$	−29.0000			−0.975928			−0.487964	+	0.872864i	$0.662260\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	−0.109881	−	0.993945i	$-0.535047\pi$
$887$	24.0000	−	41.5692i	0.805841	−	1.39576i	0.915722	−	0.401813i	$-0.131620\pi$
$888$		0			0
$889$	−3.50000	−	6.06218i	−0.117386	−	0.203319i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−9.00000	−	15.5885i	−0.301174	−	0.521648i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.0000			1.00056
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.500000	−	0.866025i	−0.0166022	−	0.0287559i	0.857605	−	0.514309i	$-0.171952\pi$
$907$	−0.500000	−	0.866025i	−0.0166022	−	0.0287559i	−0.874207	+	0.485553i	$0.838618\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.00000	−	5.19615i	−0.0993944	−	0.172156i	0.812040	−	0.583602i	$-0.198357\pi$
$911$	−3.00000	−	5.19615i	−0.0993944	−	0.172156i	−0.911434	+	0.411446i	$0.865024\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.00000			−0.198137
$918$		0			0
$919$	−34.0000			−1.12156			−0.560778	−	0.827966i	$-0.689498\pi$
$919$	−34.0000			−1.12156			−0.560778	+	0.827966i	$0.689498\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.0000	−	41.5692i	0.789970	−	1.36827i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	3.00000	+	5.19615i	0.0984268	+	0.170480i	0.911034	−	0.412332i	$-0.135286\pi$
$929$	3.00000	+	5.19615i	0.0984268	+	0.170480i	−0.812607	+	0.582812i	$0.801952\pi$
$930$		0			0
$931$	6.00000	−	10.3923i	0.196642	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	16.0000			0.522697			0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000			0.522697			0.261349	+	0.965244i	$0.415833\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	22.5000	−	38.9711i	0.733479	−	1.27042i	−0.221908	−	0.975068i	$-0.571229\pi$
$941$	22.5000	−	38.9711i	0.733479	−	1.27042i	0.955387	−	0.295355i	$-0.0954381\pi$
$942$		0			0
$943$	−13.5000	−	23.3827i	−0.439620	−	0.761445i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.50000	−	12.9904i	−0.243717	−	0.422131i	0.718053	−	0.695988i	$-0.245034\pi$
$947$	−7.50000	−	12.9904i	−0.243717	−	0.422131i	−0.961770	+	0.273858i	$0.911700\pi$
$948$		0			0
$949$	−8.00000	+	13.8564i	−0.259691	+	0.449798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	24.0000			0.777436			0.388718	−	0.921357i	$-0.372918\pi$
$953$	24.0000			0.777436			0.388718	+	0.921357i	$0.372918\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.00000	+	10.3923i	−0.193750	+	0.335585i
$960$		0			0
$961$	−34.5000	−	59.7558i	−1.11290	−	1.92760i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.500000	+	0.866025i	−0.0160789	+	0.0278495i	−0.873953	−	0.486011i	$-0.838452\pi$
$967$	−0.500000	+	0.866025i	−0.0160789	+	0.0278495i	0.857874	+	0.513860i	$0.171785\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−42.0000			−1.34784			−0.673922	−	0.738802i	$-0.735392\pi$
$971$	−42.0000			−1.34784			−0.673922	+	0.738802i	$0.735392\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.0000	+	20.7846i	−0.383914	+	0.664959i	−0.991618	−	0.129205i	$-0.958757\pi$
$977$	−12.0000	+	20.7846i	−0.383914	+	0.664959i	0.607704	+	0.794164i	$0.292091\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.50000	−	7.79423i	−0.143528	−	0.248597i	0.785295	−	0.619122i	$-0.212511\pi$
$983$	−4.50000	−	7.79423i	−0.143528	−	0.248597i	−0.928823	+	0.370525i	$0.879178\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12.0000			−0.381578
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	0.775923	−	0.630828i	$-0.217285\pi$
$997$	−5.00000	−	8.66025i	−0.158352	−	0.274273i	−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.i.a.901.1		2
3.2	odd	2		900.2.i.a.301.1		2
5.2	odd	4		2700.2.s.a.1549.2		4
5.3	odd	4		2700.2.s.a.1549.1		4
5.4	even	2		540.2.i.a.361.1		2
9.2	odd	6		900.2.i.a.601.1		2
9.4	even	3		8100.2.a.h.1.1		1
9.5	odd	6		8100.2.a.i.1.1		1
9.7	even	3	inner	2700.2.i.a.1801.1		2
15.2	even	4		900.2.s.a.49.1		4
15.8	even	4		900.2.s.a.49.2		4
15.14	odd	2		180.2.i.a.121.1	yes	2
20.19	odd	2		2160.2.q.e.1441.1		2
45.2	even	12		900.2.s.a.349.2		4
45.4	even	6		1620.2.a.b.1.1		1
45.7	odd	12		2700.2.s.a.2449.1		4
45.13	odd	12		8100.2.d.f.649.1		2
45.14	odd	6		1620.2.a.e.1.1		1
45.22	odd	12		8100.2.d.f.649.2		2
45.23	even	12		8100.2.d.e.649.1		2
45.29	odd	6		180.2.i.a.61.1	✓	2
45.32	even	12		8100.2.d.e.649.2		2
45.34	even	6		540.2.i.a.181.1		2
45.38	even	12		900.2.s.a.349.1		4
45.43	odd	12		2700.2.s.a.2449.2		4
60.59	even	2		720.2.q.a.481.1		2
180.59	even	6		6480.2.a.t.1.1		1
180.79	odd	6		2160.2.q.e.721.1		2
180.119	even	6		720.2.q.a.241.1		2
180.139	odd	6		6480.2.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.i.a.61.1	✓	2	45.29	odd	6
180.2.i.a.121.1	yes	2	15.14	odd	2
540.2.i.a.181.1		2	45.34	even	6
540.2.i.a.361.1		2	5.4	even	2
720.2.q.a.241.1		2	180.119	even	6
720.2.q.a.481.1		2	60.59	even	2
900.2.i.a.301.1		2	3.2	odd	2
900.2.i.a.601.1		2	9.2	odd	6
900.2.s.a.49.1		4	15.2	even	4
900.2.s.a.49.2		4	15.8	even	4
900.2.s.a.349.1		4	45.38	even	12
900.2.s.a.349.2		4	45.2	even	12
1620.2.a.b.1.1		1	45.4	even	6
1620.2.a.e.1.1		1	45.14	odd	6
2160.2.q.e.721.1		2	180.79	odd	6
2160.2.q.e.1441.1		2	20.19	odd	2
2700.2.i.a.901.1		2	1.1	even	1	trivial
2700.2.i.a.1801.1		2	9.7	even	3	inner
2700.2.s.a.1549.1		4	5.3	odd	4
2700.2.s.a.1549.2		4	5.2	odd	4
2700.2.s.a.2449.1		4	45.7	odd	12
2700.2.s.a.2449.2		4	45.43	odd	12
6480.2.a.h.1.1		1	180.139	odd	6
6480.2.a.t.1.1		1	180.59	even	6
8100.2.a.h.1.1		1	9.4	even	3
8100.2.a.i.1.1		1	9.5	odd	6
8100.2.d.e.649.1		2	45.23	even	12
8100.2.d.e.649.2		2	45.32	even	12
8100.2.d.f.649.1		2	45.13	odd	12
8100.2.d.f.649.2		2	45.22	odd	12

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{7} +(-2.00000 + 3.46410i) q^{13} -6.00000 q^{17} +2.00000 q^{19} +(1.50000 - 2.59808i) q^{23} +(1.50000 + 2.59808i) q^{29} +(5.00000 - 8.66025i) q^{31} +10.0000 q^{37} +(4.50000 - 7.79423i) q^{41} +(-2.00000 - 3.46410i) q^{43} +(-4.50000 - 7.79423i) q^{47} +(3.00000 - 5.19615i) q^{49} -6.00000 q^{53} +(-3.00000 + 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +(5.50000 - 9.52628i) q^{67} -12.0000 q^{71} +4.00000 q^{73} +(5.00000 + 8.66025i) q^{79} +(4.50000 + 7.79423i) q^{83} -9.00000 q^{89} +4.00000 q^{91} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{7} - 4 q^{13} - 12 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{29} + 10 q^{31} + 20 q^{37} + 9 q^{41} - 4 q^{43} - 9 q^{47} + 6 q^{49} - 12 q^{53} - 6 q^{59} + q^{61} + 11 q^{67} - 24 q^{71} + 8 q^{73}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q - q^7 - 4 * q^13 - 12 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^29 + 10 * q^31 + 20 * q^37 + 9 * q^41 - 4 * q^43 - 9 * q^47 + 6 * q^49 - 12 * q^53 - 6 * q^59 + q^61 + 11 * q^67 - 24 * q^71 + 8 * q^73 + 10 * q^79 + 9 * q^83 - 18 * q^89 + 8 * q^91 - 10 * q^97

Introduction

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