LMFDB - Embedded newform 1764.2.k.a.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,2,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, 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("1764.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1764.k (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:	$14.0856109166$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.2.k.a.1549.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{11} +6.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(1.00000 + 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +2.00000 q^{29} +(-1.00000 - 1.73205i) q^{37} -4.00000 q^{43} +(-6.00000 - 10.3923i) q^{47} +(-3.00000 + 5.19615i) q^{53} -8.00000 q^{55} +(4.00000 - 6.92820i) q^{59} +(3.00000 + 5.19615i) q^{61} +(-12.0000 - 20.7846i) q^{65} +(4.00000 - 6.92820i) q^{67} -14.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-6.00000 - 10.3923i) q^{79} -4.00000 q^{83} -16.0000 q^{85} +(-8.00000 + 13.8564i) q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} + 2 q^{11} + 12 q^{13} + 4 q^{17} - 4 q^{19} + 2 q^{23} - 11 q^{25} + 4 q^{29} - 2 q^{37} - 8 q^{43} - 12 q^{47} - 6 q^{53} - 16 q^{55} + 8 q^{59} + 6 q^{61} - 24 q^{65} + 8 q^{67} - 28 q^{71}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 + 2 * q^11 + 12 * q^13 + 4 * q^17 - 4 * q^19 + 2 * q^23 - 11 * q^25 + 4 * q^29 - 2 * q^37 - 8 * q^43 - 12 * q^47 - 6 * q^53 - 16 * q^55 + 8 * q^59 + 6 * q^61 - 24 * q^65 + 8 * q^67 - 28 * q^71 - 2 * q^73 - 12 * q^79 - 8 * q^83 - 32 * q^85 - 16 * q^95 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−2.00000	−	3.46410i	−0.894427	−	1.54919i	−0.0599153	−	0.998203i	$-0.519083\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	−0.674967	−	0.737848i	$-0.735842\pi$
$11$	1.00000	−	1.73205i	0.301511	−	0.522233i	0.976478	+	0.215615i	$0.0691756\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	−0.514782	−	0.857321i	$-0.672127\pi$
$17$	2.00000	−	3.46410i	0.485071	−	0.840168i	0.999853	+	0.0171533i	$0.00546033\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	0.951247	−	0.308431i	$-0.0998038\pi$
$23$	1.00000	+	1.73205i	0.208514	+	0.361158i	−0.742732	+	0.669588i	$0.766471\pi$
$24$		0			0
$25$	−5.50000	+	9.52628i	−1.10000	+	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.00000			0.371391			0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000			0.371391			0.185695	+	0.982607i	$0.440546\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	0.772043	−	0.635571i	$-0.219235\pi$
$37$	−1.00000	−	1.73205i	−0.164399	−	0.284747i	−0.936442	+	0.350823i	$0.885902\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.856560	−	0.516047i	$-0.827403\pi$
$47$	−6.00000	−	10.3923i	−0.875190	−	1.51587i	−0.0186297	−	0.999826i	$-0.505930\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	−0.995117	−	0.0987002i	$-0.968532\pi$
$53$	−3.00000	+	5.19615i	−0.412082	+	0.713746i	0.583036	+	0.812447i	$0.301865\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$		0			0
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	−0.478953	−	0.877841i	$-0.658984\pi$
$59$	4.00000	−	6.92820i	0.520756	−	0.901975i	0.999709	−	0.0241347i	$-0.00768307\pi$
$60$		0			0
$61$	3.00000	+	5.19615i	0.384111	+	0.665299i	0.991645	−	0.128994i	$-0.0411748\pi$
$61$	3.00000	+	5.19615i	0.384111	+	0.665299i	−0.607535	+	0.794293i	$0.707841\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12.0000	−	20.7846i	−1.48842	−	2.57801i
$66$		0			0
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	−0.511237	−	0.859440i	$-0.670813\pi$
$67$	4.00000	−	6.92820i	0.488678	−	0.846415i	0.999915	+	0.0130248i	$0.00414604\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−14.0000			−1.66149			−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000			−1.66149			−0.830747	+	0.556650i	$0.812086\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.00000	−	10.3923i	−0.675053	−	1.16923i	−0.976453	−	0.215728i	$-0.930788\pi$
$79$	−6.00000	−	10.3923i	−0.675053	−	1.16923i	0.301401	−	0.953498i	$-0.402546\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$		0			0
$85$	−16.0000			−1.73544
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	+	13.8564i	−0.820783	+	1.42164i
$96$		0			0
$97$	2.00000			0.203069			0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000			0.203069			0.101535	+	0.994832i	$0.467625\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000	+	13.8564i	−0.796030	+	1.37876i	0.126153	+	0.992011i	$0.459737\pi$
$101$	−8.00000	+	13.8564i	−0.796030	+	1.37876i	−0.922183	+	0.386753i	$0.873597\pi$
$102$		0			0
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	−0.927030	−	0.374987i	$-0.877647\pi$
$103$	−8.00000	−	13.8564i	−0.788263	−	1.36531i	0.138767	−	0.990325i	$-0.455686\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.861931	+	0.507026i	$0.169255\pi$
$107$	9.00000	+	15.5885i	0.870063	+	1.50699i	0.00813215	+	0.999967i	$0.497411\pi$
$108$		0			0
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	−0.814152	−	0.580651i	$-0.802798\pi$
$109$	1.00000	−	1.73205i	0.0957826	−	0.165900i	0.909935	+	0.414751i	$0.136131\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−10.0000			−0.940721			−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000			−0.940721			−0.470360	+	0.882474i	$0.655876\pi$
$114$		0			0
$115$	4.00000	−	6.92820i	0.373002	−	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	24.0000			2.14663
$126$		0			0
$127$	12.0000			1.06483			0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000			1.06483			0.532414	+	0.846484i	$0.321285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	10.3923i	0.501745	−	0.869048i
$144$		0			0
$145$	−4.00000	−	6.92820i	−0.332182	−	0.575356i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	0.716578	−	0.697507i	$-0.245707\pi$
$149$	−3.00000	−	5.19615i	−0.245770	−	0.425685i	−0.962348	+	0.271821i	$0.912374\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	7.00000	−	12.1244i	0.558661	−	0.967629i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	0.988227	+	0.152992i	$0.0488907\pi$
$163$	8.00000	+	13.8564i	0.626608	+	1.08532i	−0.361619	+	0.932326i	$0.617776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.00000			0.309529			0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000			0.309529			0.154765	+	0.987951i	$0.450538\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	−0.991532	−	0.129861i	$-0.958547\pi$
$173$	−8.00000	−	13.8564i	−0.608229	−	1.05348i	0.383304	−	0.923622i	$-0.374786\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	0.304446	+	0.952529i	$0.401529\pi$
$179$	−9.00000	+	15.5885i	−0.672692	+	1.16514i	−0.977138	+	0.212607i	$0.931805\pi$
$180$		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
$185$	−4.00000	+	6.92820i	−0.294086	+	0.509372i
$186$		0			0
$187$	−4.00000	−	6.92820i	−0.292509	−	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	−0.982828	−	0.184525i	$-0.940925\pi$
$191$	−9.00000	−	15.5885i	−0.651217	−	1.12794i	0.331611	−	0.943416i	$-0.392408\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.0000			1.56744			0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000			1.56744			0.783718	+	0.621117i	$0.213321\pi$
$198$		0			0
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	0.429745	+	0.902950i	$0.358603\pi$
$199$	−8.00000	+	13.8564i	−0.567105	+	0.982255i	−0.996850	+	0.0793045i	$0.974730\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000	+	13.8564i	0.545595	+	0.944999i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.0000	−	20.7846i	0.807207	−	1.39812i
$222$		0			0
$223$	−8.00000			−0.535720			−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000			−0.535720			−0.267860	+	0.963458i	$0.586316\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	0.982592	−	0.185776i	$-0.0594799\pi$
$229$	5.00000	+	8.66025i	0.330409	+	0.572286i	−0.652183	+	0.758062i	$0.726147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.00000	+	12.1244i	0.458585	+	0.794293i	0.998886	−	0.0471787i	$-0.0150230\pi$
$233$	7.00000	+	12.1244i	0.458585	+	0.794293i	−0.540301	+	0.841472i	$0.681690\pi$
$234$		0			0
$235$	−24.0000	+	41.5692i	−1.56559	+	2.71168i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	0.998443	+	0.0557856i	$0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.0000	−	20.7846i	−0.763542	−	1.32249i
$248$		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
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	0.948523	+	0.316709i	$0.102578\pi$
$257$	12.0000	+	20.7846i	0.748539	+	1.29651i	−0.199983	+	0.979799i	$0.564089\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.0000	−	17.3205i	0.609711	−	1.05605i	−0.381577	−	0.924337i	$-0.624619\pi$
$269$	10.0000	−	17.3205i	0.609711	−	1.05605i	0.991288	−	0.131713i	$-0.0420477\pi$
$270$		0			0
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	0.957328	+	0.289003i	$0.0933238\pi$
$271$	12.0000	+	20.7846i	0.728948	+	1.26258i	−0.228380	+	0.973572i	$0.573343\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	11.0000	+	19.0526i	0.663325	+	1.14891i
$276$		0			0
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	−0.319447	−	0.947604i	$-0.603497\pi$
$277$	11.0000	−	19.0526i	0.660926	−	1.14476i	0.980373	−	0.197153i	$-0.0631696\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.0000			0.596550			0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000			0.596550			0.298275	+	0.954480i	$0.403589\pi$
$282$		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
$289$	0.500000	+	0.866025i	0.0294118	+	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−32.0000			−1.86311
$296$		0			0
$297$		0			0
$298$		0			0
$299$	6.00000	+	10.3923i	0.346989	+	0.601003i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	12.0000	−	20.7846i	0.687118	−	1.19012i
$306$		0			0
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	−0.917143	−	0.398559i	$-0.869511\pi$
$311$	−2.00000	+	3.46410i	−0.113410	+	0.196431i	0.803733	+	0.594990i	$0.202844\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	0.999980	+	0.00635137i	$0.00202172\pi$
$317$	9.00000	+	15.5885i	0.505490	+	0.875535i	−0.494489	+	0.869184i	$0.664645\pi$
$318$		0			0
$319$	2.00000	−	3.46410i	0.111979	−	0.193952i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−16.0000			−0.890264
$324$		0			0
$325$	−33.0000	+	57.1577i	−1.83051	+	3.17054i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	0.937829	+	0.347097i	$0.112833\pi$
$331$	14.0000	+	24.2487i	0.769510	+	1.33283i	−0.168320	+	0.985732i	$0.553834\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−32.0000			−1.74835
$336$		0			0
$337$	−6.00000			−0.326841			−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000			−0.326841			−0.163420	+	0.986557i	$0.552253\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3.00000	−	5.19615i	0.161048	−	0.278944i	−0.774197	−	0.632945i	$-0.781846\pi$
$347$	3.00000	−	5.19615i	0.161048	−	0.278944i	0.935245	+	0.354001i	$0.115179\pi$
$348$		0			0
$349$	−22.0000			−1.17763			−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000			−1.17763			−0.588817	+	0.808267i	$0.700406\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	−0.980352	−	0.197256i	$-0.936797\pi$
$353$	−6.00000	+	10.3923i	−0.319348	+	0.553127i	0.661004	+	0.750382i	$0.270130\pi$
$354$		0			0
$355$	28.0000	+	48.4974i	1.48609	+	2.57398i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000	−	1.73205i	−0.0527780	−	0.0914141i	0.838429	−	0.545010i	$-0.183474\pi$
$359$	−1.00000	−	1.73205i	−0.0527780	−	0.0914141i	−0.891207	+	0.453596i	$0.850141\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.00000			0.418739
$366$		0			0
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	−0.995697	−	0.0926670i	$-0.970461\pi$
$367$	−8.00000	+	13.8564i	−0.417597	+	0.723299i	0.578101	+	0.815966i	$0.303794\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	−0.996610	−	0.0822766i	$-0.973781\pi$
$373$	−11.0000	−	19.0526i	−0.569558	−	0.986504i	0.427051	−	0.904227i	$-0.359552\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	0.745548	−	0.666452i	$-0.232188\pi$
$383$	−4.00000	−	6.92820i	−0.204390	−	0.354015i	−0.949938	+	0.312437i	$0.898855\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11.0000	−	19.0526i	0.557722	−	0.966003i	−0.439964	−	0.898015i	$-0.645009\pi$
$389$	11.0000	−	19.0526i	0.557722	−	0.966003i	0.997686	−	0.0679877i	$-0.0216579\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−24.0000	+	41.5692i	−1.20757	+	2.09157i
$396$		0			0
$397$	−9.00000	−	15.5885i	−0.451697	−	0.782362i	0.546795	−	0.837267i	$-0.315848\pi$
$397$	−9.00000	−	15.5885i	−0.451697	−	0.782362i	−0.998492	+	0.0549046i	$0.982515\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.00000			−0.198273
$408$		0			0
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.173064	−	0.984911i	$-0.444633\pi$
$409$	19.0000	−	32.9090i	0.939490	−	1.62724i	0.766426	−	0.642333i	$-0.222033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.00000	+	13.8564i	0.392705	+	0.680184i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	22.0000	+	38.1051i	1.06716	+	1.84837i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.0000	−	25.9808i	0.722525	−	1.25145i	−0.237460	−	0.971397i	$-0.576315\pi$
$431$	15.0000	−	25.9808i	0.722525	−	1.25145i	0.959985	−	0.280052i	$-0.0903517\pi$
$432$		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
$437$	4.00000	−	6.92820i	0.191346	−	0.331421i
$438$		0			0
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	−0.996284	−	0.0861252i	$-0.972552\pi$
$439$	−12.0000	−	20.7846i	−0.572729	−	0.991995i	0.423556	−	0.905870i	$-0.360782\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.0000	+	32.9090i	0.902717	+	1.56355i	0.823952	+	0.566659i	$0.191764\pi$
$443$	19.0000	+	32.9090i	0.902717	+	1.56355i	0.0787648	+	0.996893i	$0.474902\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	14.0000			0.660701			0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000			0.660701			0.330350	+	0.943858i	$0.392833\pi$
$450$		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
$461$	40.0000			1.86299			0.931493	−	0.363760i	$-0.118507\pi$
$461$	40.0000			1.86299			0.931493	+	0.363760i	$0.118507\pi$
$462$		0			0
$463$	4.00000			0.185896			0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000			0.185896			0.0929479	+	0.995671i	$0.470371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	0.943610	−	0.331061i	$-0.107406\pi$
$467$	4.00000	+	6.92820i	0.185098	+	0.320599i	−0.758512	+	0.651660i	$0.774073\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.00000	+	6.92820i	−0.183920	+	0.318559i
$474$		0			0
$475$	44.0000			2.01886
$476$		0			0
$477$		0			0
$478$		0			0
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	−0.345827	−	0.938298i	$-0.612402\pi$
$479$	14.0000	−	24.2487i	0.639676	−	1.10795i	0.985504	−	0.169654i	$-0.0542649\pi$
$480$		0			0
$481$	−6.00000	−	10.3923i	−0.273576	−	0.473848i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−4.00000	−	6.92820i	−0.181631	−	0.314594i
$486$		0			0
$487$	8.00000	−	13.8564i	0.362515	−	0.627894i	−0.625859	−	0.779936i	$-0.715252\pi$
$487$	8.00000	−	13.8564i	0.362515	−	0.627894i	0.988374	+	0.152042i	$0.0485850\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$		0			0
$493$	4.00000	−	6.92820i	0.180151	−	0.312031i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	0.998233	−	0.0594153i	$-0.0189236\pi$
$499$	10.0000	+	17.3205i	0.447661	+	0.775372i	−0.550572	+	0.834788i	$0.685590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	64.0000			2.84796
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	0.921023	+	0.389509i	$0.127355\pi$
$509$	18.0000	+	31.1769i	0.797836	+	1.38189i	−0.123187	+	0.992384i	$0.539311\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−32.0000	+	55.4256i	−1.41009	+	2.44234i
$516$		0			0
$517$	−24.0000			−1.05552
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000	+	10.3923i	−0.262865	+	0.455295i	−0.967002	−	0.254769i	$-0.918001\pi$
$521$	−6.00000	+	10.3923i	−0.262865	+	0.455295i	0.704137	+	0.710064i	$0.251334\pi$
$522$		0			0
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	−0.990873	−	0.134801i	$-0.956961\pi$
$523$	−14.0000	−	24.2487i	−0.612177	−	1.06032i	0.378695	−	0.925521i	$-0.376373\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	36.0000	−	62.3538i	1.55642	−	2.69579i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	0.984325	+	0.176367i	$0.0564345\pi$
$541$	15.0000	+	25.9808i	0.644900	+	1.11700i	−0.339424	+	0.940633i	$0.610232\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.00000			−0.342682
$546$		0			0
$547$	24.0000			1.02617			0.513083	−	0.858339i	$-0.328503\pi$
$547$	24.0000			1.02617			0.513083	+	0.858339i	$0.328503\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.00000	−	6.92820i	−0.170406	−	0.295151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.0000	+	32.9090i	−0.805056	+	1.39440i	0.111198	+	0.993798i	$0.464531\pi$
$557$	−19.0000	+	32.9090i	−0.805056	+	1.39440i	−0.916253	+	0.400599i	$0.868802\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000	+	6.92820i	−0.168580	+	0.291989i	−0.937921	−	0.346850i	$-0.887251\pi$
$563$	−4.00000	+	6.92820i	−0.168580	+	0.291989i	0.769341	+	0.638838i	$0.220585\pi$
$564$		0			0
$565$	20.0000	+	34.6410i	0.841406	+	1.45736i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	0.987786	+	0.155815i	$0.0498003\pi$
$569$	15.0000	+	25.9808i	0.628833	+	1.08917i	−0.358954	+	0.933355i	$0.616866\pi$
$570$		0			0
$571$	12.0000	−	20.7846i	0.502184	−	0.869809i	−0.497812	−	0.867285i	$-0.665863\pi$
$571$	12.0000	−	20.7846i	0.502184	−	0.869809i	0.999997	−	0.00252413i	$-0.000803457\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.0000			−0.917463
$576$		0			0
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	0.134380	+	0.990930i	$0.457096\pi$
$577$	−19.0000	+	32.9090i	−0.790980	+	1.37002i	−0.925361	+	0.379088i	$0.876238\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	6.00000	+	10.3923i	0.248495	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	32.0000			1.32078			0.660391	−	0.750922i	$-0.270391\pi$
$587$	32.0000			1.32078			0.660391	+	0.750922i	$0.270391\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.00000	−	10.3923i	−0.246390	−	0.426761i	0.716131	−	0.697966i	$-0.245911\pi$
$593$	−6.00000	−	10.3923i	−0.246390	−	0.426761i	−0.962522	+	0.271205i	$0.912578\pi$
$594$		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
$601$	46.0000			1.87638			0.938190	−	0.346122i	$-0.112502\pi$
$601$	46.0000			1.87638			0.938190	+	0.346122i	$0.112502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0000	−	24.2487i	0.569181	−	0.985850i
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−36.0000	−	62.3538i	−1.45640	−	2.52257i
$612$		0			0
$613$	−3.00000	+	5.19615i	−0.121169	+	0.209871i	−0.920229	−	0.391381i	$-0.871998\pi$
$613$	−3.00000	+	5.19615i	−0.121169	+	0.209871i	0.799060	+	0.601251i	$0.205331\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.0000			−1.52982			−0.764911	−	0.644136i	$-0.777217\pi$
$617$	−38.0000			−1.52982			−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	−0.993959	−	0.109749i	$-0.964995\pi$
$619$	−10.0000	+	17.3205i	−0.401934	+	0.696170i	0.592025	+	0.805919i	$0.298329\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−20.5000	−	35.5070i	−0.820000	−	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8.00000			−0.318981
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−24.0000	−	41.5692i	−0.952411	−	1.64962i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.00000	+	8.66025i	−0.197488	+	0.342059i	−0.947713	−	0.319123i	$-0.896612\pi$
$641$	−5.00000	+	8.66025i	−0.197488	+	0.342059i	0.750225	+	0.661182i	$0.229945\pi$
$642$		0			0
$643$	−44.0000			−1.73519			−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000			−1.73519			−0.867595	+	0.497271i	$0.834335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.00000	+	10.3923i	−0.235884	+	0.408564i	−0.959529	−	0.281609i	$-0.909132\pi$
$647$	−6.00000	+	10.3923i	−0.235884	+	0.408564i	0.723645	+	0.690172i	$0.242465\pi$
$648$		0			0
$649$	−8.00000	−	13.8564i	−0.314027	−	0.543912i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	0.918736	−	0.394872i	$-0.129211\pi$
$653$	3.00000	+	5.19615i	0.117399	+	0.203341i	−0.801337	+	0.598213i	$0.795878\pi$
$654$		0			0
$655$	−8.00000	+	13.8564i	−0.312586	+	0.541415i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.00000			−0.0779089			−0.0389545	−	0.999241i	$-0.512403\pi$
$659$	−2.00000			−0.0779089			−0.0389545	+	0.999241i	$0.512403\pi$
$660$		0			0
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	−0.568831	−	0.822454i	$-0.692604\pi$
$661$	11.0000	−	19.0526i	0.427850	−	0.741059i	0.996682	+	0.0813955i	$0.0259377\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.00000	+	3.46410i	0.0774403	+	0.134131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	12.0000			0.463255
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.0000	+	20.7846i	0.461197	+	0.798817i	0.999021	−	0.0442400i	$-0.0140866\pi$
$677$	12.0000	+	20.7846i	0.461197	+	0.798817i	−0.537823	+	0.843057i	$0.680753\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	−0.422195	−	0.906505i	$-0.638740\pi$
$683$	15.0000	−	25.9808i	0.573959	−	0.994126i	0.996154	−	0.0876211i	$-0.0279265\pi$
$684$		0			0
$685$	8.00000			0.305664
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	+	31.1769i	−0.685745	+	1.18775i
$690$		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
$695$	8.00000	+	13.8564i	0.303457	+	0.525603i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	50.0000			1.88847			0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000			1.88847			0.944237	+	0.329267i	$0.106802\pi$
$702$		0			0
$703$	−4.00000	+	6.92820i	−0.150863	+	0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	−0.963512	−	0.267664i	$-0.913748\pi$
$709$	−19.0000	−	32.9090i	−0.713560	−	1.23592i	0.249952	−	0.968258i	$-0.419585\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−48.0000			−1.79510
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11.0000	+	19.0526i	−0.408530	+	0.707594i
$726$		0			0
$727$	−16.0000			−0.593407			−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000			−0.593407			−0.296704	+	0.954970i	$0.595887\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	+	13.8564i	−0.295891	+	0.512498i
$732$		0			0
$733$	−3.00000	−	5.19615i	−0.110808	−	0.191924i	0.805289	−	0.592883i	$-0.202010\pi$
$733$	−3.00000	−	5.19615i	−0.110808	−	0.191924i	−0.916096	+	0.400959i	$0.868677\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000	−	13.8564i	−0.294684	−	0.510407i
$738$		0			0
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−30.0000			−1.10059			−0.550297	−	0.834969i	$-0.685485\pi$
$743$	−30.0000			−1.10059			−0.550297	+	0.834969i	$0.685485\pi$
$744$		0			0
$745$	−12.0000	+	20.7846i	−0.439646	+	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.0000			−1.16460
$756$		0			0
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16.0000	−	27.7128i	−0.580000	−	1.00459i	−0.995479	−	0.0949859i	$-0.969719\pi$
$761$	−16.0000	−	27.7128i	−0.580000	−	1.00459i	0.415479	−	0.909603i	$-0.363614\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	24.0000	−	41.5692i	0.866590	−	1.50098i
$768$		0			0
$769$	38.0000			1.37032			0.685158	−	0.728395i	$-0.259733\pi$
$769$	38.0000			1.37032			0.685158	+	0.728395i	$0.259733\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.00000	−	6.92820i	0.143870	−	0.249190i	−0.785081	−	0.619393i	$-0.787379\pi$
$773$	4.00000	−	6.92820i	0.143870	−	0.249190i	0.928951	+	0.370203i	$0.120712\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−14.0000	+	24.2487i	−0.500959	+	0.867687i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−56.0000			−1.99873
$786$		0			0
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	−0.952925	−	0.303207i	$-0.901942\pi$
$787$	−6.00000	+	10.3923i	−0.213877	+	0.370446i	0.739048	+	0.673653i	$0.235276\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	18.0000	+	31.1769i	0.639199	+	1.10712i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	−48.0000			−1.69812
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.00000	+	3.46410i	0.0705785	+	0.122245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.00000	−	8.66025i	0.175791	−	0.304478i	−0.764644	−	0.644453i	$-0.777085\pi$
$809$	5.00000	−	8.66025i	0.175791	−	0.304478i	0.940435	+	0.339975i	$0.110418\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	32.0000	−	55.4256i	1.12091	−	1.94147i
$816$		0			0
$817$	8.00000	+	13.8564i	0.279885	+	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	+	1.73205i	0.0349002	+	0.0604490i	0.882948	−	0.469471i	$-0.155555\pi$
$821$	1.00000	+	1.73205i	0.0349002	+	0.0604490i	−0.848048	+	0.529920i	$0.822222\pi$
$822$		0			0
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	−0.637419	−	0.770517i	$-0.719998\pi$
$823$	10.0000	−	17.3205i	0.348578	−	0.603755i	0.985997	+	0.166762i	$0.0533313\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−18.0000			−0.625921			−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000			−0.625921			−0.312961	+	0.949766i	$0.601321\pi$
$828$		0			0
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	−0.961601	−	0.274450i	$-0.911504\pi$
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	0.718481	+	0.695546i	$0.244838\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8.00000	−	13.8564i	−0.276851	−	0.479521i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	44.0000			1.51905			0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000			1.51905			0.759524	+	0.650479i	$0.225432\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−46.0000	−	79.6743i	−1.58245	−	2.74088i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.00000	−	3.46410i	0.0685591	−	0.118748i
$852$		0			0
$853$	−14.0000			−0.479351			−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000			−0.479351			−0.239675	+	0.970853i	$0.577041\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	−0.0859870	−	0.996296i	$-0.527404\pi$
$857$	24.0000	−	41.5692i	0.819824	−	1.41998i	0.905811	−	0.423681i	$-0.139262\pi$
$858$		0			0
$859$	22.0000	+	38.1051i	0.750630	+	1.30013i	0.947518	+	0.319704i	$0.103583\pi$
$859$	22.0000	+	38.1051i	0.750630	+	1.30013i	−0.196887	+	0.980426i	$0.563083\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	0.721939	−	0.691957i	$-0.243251\pi$
$863$	−7.00000	−	12.1244i	−0.238283	−	0.412718i	−0.960222	+	0.279239i	$0.909918\pi$
$864$		0			0
$865$	−32.0000	+	55.4256i	−1.08803	+	1.88453i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24.0000			−0.814144
$870$		0			0
$871$	24.0000	−	41.5692i	0.813209	−	1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	−0.933860	−	0.357640i	$-0.883582\pi$
$877$	−23.0000	−	39.8372i	−0.776655	−	1.34521i	0.157205	−	0.987566i	$-0.449752\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	0.948999	−	0.315279i	$-0.102098\pi$
$887$	6.00000	+	10.3923i	0.201460	+	0.348939i	−0.747539	+	0.664218i	$0.768765\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−24.0000	+	41.5692i	−0.803129	+	1.39106i
$894$		0			0
$895$	72.0000			2.40669
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	12.0000	+	20.7846i	0.399778	+	0.692436i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12.0000	−	20.7846i	−0.398893	−	0.690904i
$906$		0			0
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	−0.948278	−	0.317441i	$-0.897176\pi$
$907$	−6.00000	+	10.3923i	−0.199227	+	0.345071i	0.749051	+	0.662512i	$0.230510\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18.0000			0.596367			0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000			0.596367			0.298183	+	0.954509i	$0.403619\pi$
$912$		0			0
$913$	−4.00000	+	6.92820i	−0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	0.703378	−	0.710816i	$-0.251674\pi$
$919$	−8.00000	−	13.8564i	−0.263896	−	0.457081i	−0.967274	+	0.253735i	$0.918341\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−84.0000			−2.76489
$924$		0			0
$925$	22.0000			0.723356
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−24.0000	−	41.5692i	−0.787414	−	1.36384i	−0.927546	−	0.373709i	$-0.878086\pi$
$929$	−24.0000	−	41.5692i	−0.787414	−	1.36384i	0.140132	−	0.990133i	$-0.455247\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16.0000	+	27.7128i	−0.523256	+	0.906306i
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.0000	+	24.2487i	−0.456387	+	0.790485i	−0.998767	−	0.0496480i	$-0.984190\pi$
$941$	−14.0000	+	24.2487i	−0.456387	+	0.790485i	0.542380	+	0.840133i	$0.317523\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5.00000	+	8.66025i	0.162478	+	0.281420i	0.935757	−	0.352646i	$-0.114718\pi$
$947$	5.00000	+	8.66025i	0.162478	+	0.281420i	−0.773279	+	0.634066i	$0.781385\pi$
$948$		0			0
$949$	−6.00000	+	10.3923i	−0.194768	+	0.337348i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	22.0000			0.712650			0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000			0.712650			0.356325	+	0.934362i	$0.384030\pi$
$954$		0			0
$955$	−36.0000	+	62.3538i	−1.16493	+	2.01772i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−40.0000			−1.28765
$966$		0			0
$967$	20.0000			0.643157			0.321578	−	0.946883i	$-0.395787\pi$
$967$	20.0000			0.643157			0.321578	+	0.946883i	$0.395787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	−0.995748	−	0.0921142i	$-0.970638\pi$
$971$	−18.0000	−	31.1769i	−0.577647	−	1.00051i	0.418101	−	0.908401i	$-0.362696\pi$
$972$		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
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	−0.174511	−	0.984655i	$-0.555834\pi$
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	0.939992	−	0.341197i	$-0.110832\pi$
$984$		0			0
$985$	−44.0000	−	76.2102i	−1.40196	−	2.42826i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.00000	−	6.92820i	−0.127193	−	0.220304i
$990$		0			0
$991$	−28.0000	+	48.4974i	−0.889449	+	1.54057i	−0.0489218	+	0.998803i	$0.515578\pi$
$991$	−28.0000	+	48.4974i	−0.889449	+	1.54057i	−0.840528	+	0.541769i	$0.817755\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	64.0000			2.02894
$996$		0			0
$997$	19.0000	−	32.9090i	0.601736	−	1.04224i	−0.390822	−	0.920466i	$-0.627809\pi$
$997$	19.0000	−	32.9090i	0.601736	−	1.04224i	0.992558	−	0.121771i	$-0.0388574\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.k.a.361.1		2
3.2	odd	2		588.2.i.d.361.1		2
7.2	even	3	inner	1764.2.k.a.1549.1		2
7.3	odd	6		252.2.a.a.1.1		1
7.4	even	3		1764.2.a.k.1.1		1
7.5	odd	6		1764.2.k.k.1549.1		2
7.6	odd	2		1764.2.k.k.361.1		2
12.11	even	2		2352.2.q.z.1537.1		2
21.2	odd	6		588.2.i.d.373.1		2
21.5	even	6		588.2.i.e.373.1		2
21.11	odd	6		588.2.a.d.1.1		1
21.17	even	6		84.2.a.a.1.1	✓	1
21.20	even	2		588.2.i.e.361.1		2
28.3	even	6		1008.2.a.a.1.1		1
28.11	odd	6		7056.2.a.cd.1.1		1
35.3	even	12		6300.2.k.g.6049.2		2
35.17	even	12		6300.2.k.g.6049.1		2
35.24	odd	6		6300.2.a.w.1.1		1
56.3	even	6		4032.2.a.bn.1.1		1
56.45	odd	6		4032.2.a.bm.1.1		1
63.31	odd	6		2268.2.j.n.1513.1		2
63.38	even	6		2268.2.j.a.757.1		2
63.52	odd	6		2268.2.j.n.757.1		2
63.59	even	6		2268.2.j.a.1513.1		2
84.11	even	6		2352.2.a.a.1.1		1
84.23	even	6		2352.2.q.z.961.1		2
84.47	odd	6		2352.2.q.b.961.1		2
84.59	odd	6		336.2.a.f.1.1		1
84.83	odd	2		2352.2.q.b.1537.1		2
105.17	odd	12		2100.2.k.i.1849.2		2
105.38	odd	12		2100.2.k.i.1849.1		2
105.59	even	6		2100.2.a.r.1.1		1
168.11	even	6		9408.2.a.df.1.1		1
168.53	odd	6		9408.2.a.bn.1.1		1
168.59	odd	6		1344.2.a.a.1.1		1
168.101	even	6		1344.2.a.k.1.1		1
336.59	odd	12		5376.2.c.p.2689.1		2
336.101	even	12		5376.2.c.q.2689.2		2
336.227	odd	12		5376.2.c.p.2689.2		2
336.269	even	12		5376.2.c.q.2689.1		2
420.59	odd	6		8400.2.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.2.a.a.1.1	✓	1	21.17	even	6
252.2.a.a.1.1		1	7.3	odd	6
336.2.a.f.1.1		1	84.59	odd	6
588.2.a.d.1.1		1	21.11	odd	6
588.2.i.d.361.1		2	3.2	odd	2
588.2.i.d.373.1		2	21.2	odd	6
588.2.i.e.361.1		2	21.20	even	2
588.2.i.e.373.1		2	21.5	even	6
1008.2.a.a.1.1		1	28.3	even	6
1344.2.a.a.1.1		1	168.59	odd	6
1344.2.a.k.1.1		1	168.101	even	6
1764.2.a.k.1.1		1	7.4	even	3
1764.2.k.a.361.1		2	1.1	even	1	trivial
1764.2.k.a.1549.1		2	7.2	even	3	inner
1764.2.k.k.361.1		2	7.6	odd	2
1764.2.k.k.1549.1		2	7.5	odd	6
2100.2.a.r.1.1		1	105.59	even	6
2100.2.k.i.1849.1		2	105.38	odd	12
2100.2.k.i.1849.2		2	105.17	odd	12
2268.2.j.a.757.1		2	63.38	even	6
2268.2.j.a.1513.1		2	63.59	even	6
2268.2.j.n.757.1		2	63.52	odd	6
2268.2.j.n.1513.1		2	63.31	odd	6
2352.2.a.a.1.1		1	84.11	even	6
2352.2.q.b.961.1		2	84.47	odd	6
2352.2.q.b.1537.1		2	84.83	odd	2
2352.2.q.z.961.1		2	84.23	even	6
2352.2.q.z.1537.1		2	12.11	even	2
4032.2.a.bm.1.1		1	56.45	odd	6
4032.2.a.bn.1.1		1	56.3	even	6
5376.2.c.p.2689.1		2	336.59	odd	12
5376.2.c.p.2689.2		2	336.227	odd	12
5376.2.c.q.2689.1		2	336.269	even	12
5376.2.c.q.2689.2		2	336.101	even	12
6300.2.a.w.1.1		1	35.24	odd	6
6300.2.k.g.6049.1		2	35.17	even	12
6300.2.k.g.6049.2		2	35.3	even	12
7056.2.a.cd.1.1		1	28.11	odd	6
8400.2.a.e.1.1		1	420.59	odd	6
9408.2.a.bn.1.1		1	168.53	odd	6
9408.2.a.df.1.1		1	168.11	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{5} +(1.00000 - 1.73205i) q^{11} +6.00000 q^{13} +(2.00000 - 3.46410i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(1.00000 + 1.73205i) q^{23} +(-5.50000 + 9.52628i) q^{25} +2.00000 q^{29} +(-1.00000 - 1.73205i) q^{37} -4.00000 q^{43} +(-6.00000 - 10.3923i) q^{47} +(-3.00000 + 5.19615i) q^{53} -8.00000 q^{55} +(4.00000 - 6.92820i) q^{59} +(3.00000 + 5.19615i) q^{61} +(-12.0000 - 20.7846i) q^{65} +(4.00000 - 6.92820i) q^{67} -14.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-6.00000 - 10.3923i) q^{79} -4.00000 q^{83} -16.0000 q^{85} +(-8.00000 + 13.8564i) q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} + 2 q^{11} + 12 q^{13} + 4 q^{17} - 4 q^{19} + 2 q^{23} - 11 q^{25} + 4 q^{29} - 2 q^{37} - 8 q^{43} - 12 q^{47} - 6 q^{53} - 16 q^{55} + 8 q^{59} + 6 q^{61} - 24 q^{65} + 8 q^{67} - 28 q^{71}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 + 2 * q^11 + 12 * q^13 + 4 * q^17 - 4 * q^19 + 2 * q^23 - 11 * q^25 + 4 * q^29 - 2 * q^37 - 8 * q^43 - 12 * q^47 - 6 * q^53 - 16 * q^55 + 8 * q^59 + 6 * q^61 - 24 * q^65 + 8 * q^67 - 28 * q^71 - 2 * q^73 - 12 * q^79 - 8 * q^83 - 32 * q^85 - 16 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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