LMFDB - Embedded newform 378.2.k.c.269.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.k (of order $6$, degree $2$, 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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.2.k.c.215.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +(1.50000 + 0.866025i) q^{10} +(2.59808 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.73205 + 3.00000i) q^{17} +(6.00000 - 3.46410i) q^{19} +1.73205 q^{20} +(-5.19615 + 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} +(2.50000 - 0.866025i) q^{28} -9.00000i q^{29} +(3.00000 + 1.73205i) q^{31} +(-0.866025 - 0.500000i) q^{32} +3.46410i q^{34} +(-0.866025 + 4.50000i) q^{35} +(-2.00000 - 3.46410i) q^{37} +(3.46410 - 6.00000i) q^{38} +(1.50000 - 0.866025i) q^{40} -3.46410 q^{41} -8.00000 q^{43} +(-3.00000 + 5.19615i) q^{46} +(-1.73205 - 3.00000i) q^{47} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} +(-2.59808 - 1.50000i) q^{53} +(1.73205 - 2.00000i) q^{56} +(-4.50000 - 7.79423i) q^{58} +(-6.06218 + 10.5000i) q^{59} +(3.00000 - 1.73205i) q^{61} +3.46410 q^{62} -1.00000 q^{64} +(-7.00000 + 12.1244i) q^{67} +(1.73205 + 3.00000i) q^{68} +(1.50000 + 4.33013i) q^{70} +6.00000i q^{71} +(-10.5000 - 6.06218i) q^{73} +(-3.46410 - 2.00000i) q^{74} -6.92820i q^{76} +(-5.50000 - 9.52628i) q^{79} +(0.866025 - 1.50000i) q^{80} +(-3.00000 + 1.73205i) q^{82} +17.3205 q^{83} -6.00000 q^{85} +(-6.92820 + 4.00000i) q^{86} +(-5.19615 - 9.00000i) q^{89} +6.00000i q^{92} +(-3.00000 - 1.73205i) q^{94} +(10.3923 + 6.00000i) q^{95} +6.92820i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 + 8 * q^7	$ 4 q + 2 q^{4} + 8 q^{7} + 6 q^{10} - 2 q^{16} + 24 q^{19} + 4 q^{25} + 10 q^{28} + 12 q^{31} - 8 q^{37} + 6 q^{40} - 32 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{58} + 12 q^{61} - 4 q^{64} - 28 q^{67} + 6 q^{70} - 42 q^{73} - 22 q^{79} - 12 q^{82} - 24 q^{85} - 12 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^7 + 6 * q^10 - 2 * q^16 + 24 * q^19 + 4 * q^25 + 10 * q^28 + 12 * q^31 - 8 * q^37 + 6 * q^40 - 32 * q^43 - 12 * q^46 + 4 * q^49 - 18 * q^58 + 12 * q^61 - 4 * q^64 - 28 * q^67 + 6 * q^70 - 42 * q^73 - 22 * q^79 - 12 * q^82 - 24 * q^85 - 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	0.992085	−	0.125567i	$-0.0400750\pi$
$5$	0.866025	+	1.50000i	0.387298	+	0.670820i	−0.604787	+	0.796387i	$0.706742\pi$
$6$		0			0
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	1.50000	+	0.866025i	0.474342	+	0.273861i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	2.59808	+	0.500000i	0.694365	+	0.133631i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−1.73205	+	3.00000i	−0.420084	+	0.727607i	−0.995947	−	0.0899392i	$-0.971333\pi$
$17$	−1.73205	+	3.00000i	−0.420084	+	0.727607i	0.575863	+	0.817546i	$0.304666\pi$
$18$		0			0
$19$	6.00000	−	3.46410i	1.37649	−	0.794719i	0.384759	−	0.923017i	$-0.374285\pi$
$19$	6.00000	−	3.46410i	1.37649	−	0.794719i	0.991736	+	0.128298i	$0.0409513\pi$
$20$	1.73205			0.387298
$21$		0			0
$22$		0			0
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.931831	−	0.362892i	$-0.881789\pi$
$23$	−5.19615	+	3.00000i	−1.08347	+	0.625543i	−0.151642	+	0.988436i	$0.548456\pi$
$24$		0			0
$25$	1.00000	−	1.73205i	0.200000	−	0.346410i
$26$		0			0
$27$		0			0
$28$	2.50000	−	0.866025i	0.472456	−	0.163663i
$29$		−	9.00000i		−	1.67126i	−0.549294	−	0.835629i	$-0.685103\pi$
$29$		−	9.00000i		−	1.67126i	0.549294	−	0.835629i	$-0.314897\pi$
$30$		0			0
$31$	3.00000	+	1.73205i	0.538816	+	0.311086i	0.744599	−	0.667512i	$-0.232641\pi$
$31$	3.00000	+	1.73205i	0.538816	+	0.311086i	−0.205783	+	0.978598i	$0.565974\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$			3.46410i			0.594089i
$35$	−0.866025	+	4.50000i	−0.146385	+	0.760639i
$36$		0			0
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	0.653476	−	0.756948i	$-0.273310\pi$
$37$	−2.00000	−	3.46410i	−0.328798	−	0.569495i	−0.982274	+	0.187453i	$0.939977\pi$
$38$	3.46410	−	6.00000i	0.561951	−	0.973329i
$39$		0			0
$40$	1.50000	−	0.866025i	0.237171	−	0.136931i
$41$	−3.46410			−0.541002			−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410			−0.541002			−0.270501	+	0.962720i	$0.587189\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000	+	5.19615i	−0.442326	+	0.766131i
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	0.711608	−	0.702577i	$-0.247967\pi$
$47$	−1.73205	−	3.00000i	−0.252646	−	0.437595i	−0.964253	+	0.264982i	$0.914634\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		−	2.00000i		−	0.282843i
$51$		0			0
$52$		0			0
$53$	−2.59808	−	1.50000i	−0.356873	−	0.206041i	0.310835	−	0.950464i	$-0.399391\pi$
$53$	−2.59808	−	1.50000i	−0.356873	−	0.206041i	−0.667708	+	0.744423i	$0.732725\pi$
$54$		0			0
$55$		0			0
$56$	1.73205	−	2.00000i	0.231455	−	0.267261i
$57$		0			0
$58$	−4.50000	−	7.79423i	−0.590879	−	1.02343i
$59$	−6.06218	+	10.5000i	−0.789228	+	1.36698i	0.137212	+	0.990542i	$0.456186\pi$
$59$	−6.06218	+	10.5000i	−0.789228	+	1.36698i	−0.926440	+	0.376442i	$0.877147\pi$
$60$		0			0
$61$	3.00000	−	1.73205i	0.384111	−	0.221766i	−0.295495	−	0.955344i	$-0.595484\pi$
$61$	3.00000	−	1.73205i	0.384111	−	0.221766i	0.679605	+	0.733578i	$0.262151\pi$
$62$	3.46410			0.439941
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	0.0212861	+	0.999773i	$0.493224\pi$
$67$	−7.00000	+	12.1244i	−0.855186	+	1.48123i	−0.876472	+	0.481452i	$0.840109\pi$
$68$	1.73205	+	3.00000i	0.210042	+	0.363803i
$69$		0			0
$70$	1.50000	+	4.33013i	0.179284	+	0.517549i
$71$			6.00000i			0.712069i	0.934473	+	0.356034i	$0.115871\pi$
$71$			6.00000i			0.712069i	−0.934473	+	0.356034i	$0.884129\pi$
$72$		0			0
$73$	−10.5000	−	6.06218i	−1.22893	−	0.709524i	−0.262126	−	0.965034i	$-0.584423\pi$
$73$	−10.5000	−	6.06218i	−1.22893	−	0.709524i	−0.966807	+	0.255510i	$0.917757\pi$
$74$	−3.46410	−	2.00000i	−0.402694	−	0.232495i
$75$		0			0
$76$		−	6.92820i		−	0.794719i
$77$		0			0
$78$		0			0
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	−0.989705	−	0.143120i	$-0.954286\pi$
$79$	−5.50000	−	9.52628i	−0.618798	−	1.07179i	0.370907	−	0.928670i	$-0.379047\pi$
$80$	0.866025	−	1.50000i	0.0968246	−	0.167705i
$81$		0			0
$82$	−3.00000	+	1.73205i	−0.331295	+	0.191273i
$83$	17.3205			1.90117			0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205			1.90117			0.950586	+	0.310460i	$0.100483\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$	−6.92820	+	4.00000i	−0.747087	+	0.431331i
$87$		0			0
$88$		0			0
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	−0.998218	−	0.0596775i	$-0.980993\pi$
$89$	−5.19615	−	9.00000i	−0.550791	−	0.953998i	0.447427	−	0.894321i	$-0.352341\pi$
$90$		0			0
$91$		0			0
$92$			6.00000i			0.625543i
$93$		0			0
$94$	−3.00000	−	1.73205i	−0.309426	−	0.178647i
$95$	10.3923	+	6.00000i	1.06623	+	0.615587i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$	4.33013	+	5.50000i	0.437409	+	0.555584i
$99$		0			0
$100$	−1.00000	−	1.73205i	−0.100000	−	0.173205i
$101$	2.59808	−	4.50000i	0.258518	−	0.447767i	−0.707327	−	0.706887i	$-0.750099\pi$
$101$	2.59808	−	4.50000i	0.258518	−	0.447767i	0.965845	+	0.259120i	$0.0834325\pi$
$102$		0			0
$103$	9.00000	−	5.19615i	0.886796	−	0.511992i	0.0139031	−	0.999903i	$-0.495574\pi$
$103$	9.00000	−	5.19615i	0.886796	−	0.511992i	0.872893	+	0.487911i	$0.162241\pi$
$104$		0			0
$105$		0			0
$106$	−3.00000			−0.291386
$107$	2.59808	−	1.50000i	0.251166	−	0.145010i	−0.369132	−	0.929377i	$-0.620345\pi$
$107$	2.59808	−	1.50000i	0.251166	−	0.145010i	0.620298	+	0.784366i	$0.287012\pi$
$108$		0			0
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	0.173316	+	0.984866i	$0.444552\pi$
$109$	−8.00000	+	13.8564i	−0.766261	+	1.32720i	−0.939577	+	0.342337i	$0.888782\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	−	2.59808i	0.0472456	−	0.245495i
$113$		−	18.0000i		−	1.69330i	−0.532152	−	0.846649i	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		0			0
$115$	−9.00000	−	5.19615i	−0.839254	−	0.484544i
$116$	−7.79423	−	4.50000i	−0.723676	−	0.417815i
$117$		0			0
$118$			12.1244i			1.11614i
$119$	−8.66025	+	3.00000i	−0.793884	+	0.275010i
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$	1.73205	−	3.00000i	0.156813	−	0.271607i
$123$		0			0
$124$	3.00000	−	1.73205i	0.269408	−	0.155543i
$125$	12.1244			1.08444
$126$		0			0
$127$	−7.00000			−0.621150			−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000			−0.621150			−0.310575	+	0.950549i	$0.600522\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$		0			0
$131$	5.19615	+	9.00000i	0.453990	+	0.786334i	0.998630	−	0.0523366i	$-0.0166669\pi$
$131$	5.19615	+	9.00000i	0.453990	+	0.786334i	−0.544640	+	0.838670i	$0.683334\pi$
$132$		0			0
$133$	18.0000	+	3.46410i	1.56080	+	0.300376i
$134$			14.0000i			1.20942i
$135$		0			0
$136$	3.00000	+	1.73205i	0.257248	+	0.148522i
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$			10.3923i			0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$			10.3923i			0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	3.46410	+	3.00000i	0.292770	+	0.253546i
$141$		0			0
$142$	3.00000	+	5.19615i	0.251754	+	0.436051i
$143$		0			0
$144$		0			0
$145$	13.5000	−	7.79423i	1.12111	−	0.647275i
$146$	−12.1244			−1.00342
$147$		0			0
$148$	−4.00000			−0.328798
$149$	12.9904	−	7.50000i	1.06421	−	0.614424i	0.137619	−	0.990485i	$-0.456055\pi$
$149$	12.9904	−	7.50000i	1.06421	−	0.614424i	0.926595	+	0.376061i	$0.122722\pi$
$150$		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	$0.0611289\pi$
$152$	−3.46410	−	6.00000i	−0.280976	−	0.486664i
$153$		0			0
$154$		0			0
$155$			6.00000i			0.481932i
$156$		0			0
$157$	9.00000	+	5.19615i	0.718278	+	0.414698i	0.814119	−	0.580699i	$-0.197221\pi$
$157$	9.00000	+	5.19615i	0.718278	+	0.414698i	−0.0958404	+	0.995397i	$0.530554\pi$
$158$	−9.52628	−	5.50000i	−0.757870	−	0.437557i
$159$		0			0
$160$		−	1.73205i		−	0.136931i
$161$	−15.5885	−	3.00000i	−1.22854	−	0.236433i
$162$		0			0
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	0.824202	−	0.566296i	$-0.191624\pi$
$163$	−1.00000	−	1.73205i	−0.0783260	−	0.135665i	−0.902528	+	0.430632i	$0.858291\pi$
$164$	−1.73205	+	3.00000i	−0.135250	+	0.234261i
$165$		0			0
$166$	15.0000	−	8.66025i	1.16423	−	0.672166i
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$	−5.19615	+	3.00000i	−0.398527	+	0.230089i
$171$		0			0
$172$	−4.00000	+	6.92820i	−0.304997	+	0.528271i
$173$	−2.59808	−	4.50000i	−0.197528	−	0.342129i	0.750198	−	0.661213i	$-0.229958\pi$
$173$	−2.59808	−	4.50000i	−0.197528	−	0.342129i	−0.947726	+	0.319084i	$0.896625\pi$
$174$		0			0
$175$	5.00000	−	1.73205i	0.377964	−	0.130931i
$176$		0			0
$177$		0			0
$178$	−9.00000	−	5.19615i	−0.674579	−	0.389468i
$179$	12.9904	+	7.50000i	0.970947	+	0.560576i	0.899525	−	0.436870i	$-0.143913\pi$
$179$	12.9904	+	7.50000i	0.970947	+	0.560576i	0.0714220	+	0.997446i	$0.477246\pi$
$180$		0			0
$181$		−	17.3205i		−	1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$		−	17.3205i		−	1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$		0			0
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$	3.46410	−	6.00000i	0.254686	−	0.441129i
$186$		0			0
$187$		0			0
$188$	−3.46410			−0.252646
$189$		0			0
$190$	12.0000			0.870572
$191$	−5.19615	+	3.00000i	−0.375980	+	0.217072i	−0.676068	−	0.736839i	$-0.736317\pi$
$191$	−5.19615	+	3.00000i	−0.375980	+	0.217072i	0.300088	+	0.953912i	$0.402984\pi$
$192$		0			0
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	−0.987945	−	0.154805i	$-0.950525\pi$
$193$	−5.00000	+	8.66025i	−0.359908	+	0.623379i	0.628037	+	0.778183i	$0.283859\pi$
$194$	3.46410	+	6.00000i	0.248708	+	0.430775i
$195$		0			0
$196$	6.50000	+	2.59808i	0.464286	+	0.185577i
$197$		−	3.00000i		−	0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$		−	3.00000i		−	0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$		0			0
$199$	7.50000	+	4.33013i	0.531661	+	0.306955i	0.741693	−	0.670740i	$-0.234023\pi$
$199$	7.50000	+	4.33013i	0.531661	+	0.306955i	−0.210032	+	0.977695i	$0.567357\pi$
$200$	−1.73205	−	1.00000i	−0.122474	−	0.0707107i
$201$		0			0
$202$		−	5.19615i		−	0.365600i
$203$	15.5885	−	18.0000i	1.09410	−	1.26335i
$204$		0			0
$205$	−3.00000	−	5.19615i	−0.209529	−	0.362915i
$206$	5.19615	−	9.00000i	0.362033	−	0.627060i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$	−2.59808	+	1.50000i	−0.178437	+	0.103020i
$213$		0			0
$214$	1.50000	−	2.59808i	0.102538	−	0.177601i
$215$	−6.92820	−	12.0000i	−0.472500	−	0.818393i
$216$		0			0
$217$	3.00000	+	8.66025i	0.203653	+	0.587896i
$218$			16.0000i			1.08366i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			15.5885i			1.04388i	0.852982	+	0.521940i	$0.174792\pi$
$223$			15.5885i			1.04388i	−0.852982	+	0.521940i	$0.825208\pi$
$224$	−0.866025	−	2.50000i	−0.0578638	−	0.167038i
$225$		0			0
$226$	−9.00000	−	15.5885i	−0.598671	−	1.03693i
$227$	−12.9904	+	22.5000i	−0.862202	+	1.49338i	0.00759708	+	0.999971i	$0.497582\pi$
$227$	−12.9904	+	22.5000i	−0.862202	+	1.49338i	−0.869799	+	0.493406i	$0.835752\pi$
$228$		0			0
$229$	−3.00000	+	1.73205i	−0.198246	+	0.114457i	−0.595837	−	0.803105i	$-0.703180\pi$
$229$	−3.00000	+	1.73205i	−0.198246	+	0.114457i	0.397591	+	0.917563i	$0.369846\pi$
$230$	−10.3923			−0.685248
$231$		0			0
$232$	−9.00000			−0.590879
$233$	25.9808	−	15.0000i	1.70206	−	0.982683i	0.758380	−	0.651813i	$-0.225991\pi$
$233$	25.9808	−	15.0000i	1.70206	−	0.982683i	0.943676	−	0.330870i	$-0.107342\pi$
$234$		0			0
$235$	3.00000	−	5.19615i	0.195698	−	0.338960i
$236$	6.06218	+	10.5000i	0.394614	+	0.683492i
$237$		0			0
$238$	−6.00000	+	6.92820i	−0.388922	+	0.449089i
$239$		−	24.0000i		−	1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$		−	24.0000i		−	1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$		0			0
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	0.547533	−	0.836784i	$-0.315567\pi$
$241$	1.50000	+	0.866025i	0.0966235	+	0.0557856i	−0.450910	+	0.892570i	$0.648900\pi$
$242$	−9.52628	−	5.50000i	−0.612372	−	0.353553i
$243$		0			0
$244$		−	3.46410i		−	0.221766i
$245$	−9.52628	+	7.50000i	−0.608612	+	0.479157i
$246$		0			0
$247$		0			0
$248$	1.73205	−	3.00000i	0.109985	−	0.190500i
$249$		0			0
$250$	10.5000	−	6.06218i	0.664078	−	0.383406i
$251$	8.66025			0.546630			0.273315	−	0.961925i	$-0.411880\pi$
$251$	8.66025			0.546630			0.273315	+	0.961925i	$0.411880\pi$
$252$		0			0
$253$		0			0
$254$	−6.06218	+	3.50000i	−0.380375	+	0.219610i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	10.3923	+	18.0000i	0.648254	+	1.12281i	0.983540	+	0.180693i	$0.0578339\pi$
$257$	10.3923	+	18.0000i	0.648254	+	1.12281i	−0.335285	+	0.942117i	$0.608833\pi$
$258$		0			0
$259$	2.00000	−	10.3923i	0.124274	−	0.645746i
$260$		0			0
$261$		0			0
$262$	9.00000	+	5.19615i	0.556022	+	0.321019i
$263$	15.5885	+	9.00000i	0.961225	+	0.554964i	0.896550	−	0.442943i	$-0.146065\pi$
$263$	15.5885	+	9.00000i	0.961225	+	0.554964i	0.0646755	+	0.997906i	$0.479399\pi$
$264$		0			0
$265$		−	5.19615i		−	0.319197i
$266$	17.3205	−	6.00000i	1.06199	−	0.367884i
$267$		0			0
$268$	7.00000	+	12.1244i	0.427593	+	0.740613i
$269$	−0.866025	+	1.50000i	−0.0528025	+	0.0914566i	−0.891219	−	0.453574i	$-0.850149\pi$
$269$	−0.866025	+	1.50000i	−0.0528025	+	0.0914566i	0.838416	+	0.545031i	$0.183482\pi$
$270$		0			0
$271$	10.5000	−	6.06218i	0.637830	−	0.368251i	−0.145948	−	0.989292i	$-0.546623\pi$
$271$	10.5000	−	6.06218i	0.637830	−	0.368251i	0.783778	+	0.621041i	$0.213290\pi$
$272$	3.46410			0.210042
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	−0.960810	−	0.277207i	$-0.910591\pi$
$277$	−4.00000	+	6.92820i	−0.240337	+	0.416275i	0.720473	+	0.693482i	$0.243925\pi$
$278$	5.19615	+	9.00000i	0.311645	+	0.539784i
$279$		0			0
$280$	4.50000	+	0.866025i	0.268926	+	0.0517549i
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		0			0
$283$	6.00000	+	3.46410i	0.356663	+	0.205919i	0.667616	−	0.744506i	$-0.267315\pi$
$283$	6.00000	+	3.46410i	0.356663	+	0.205919i	−0.310953	+	0.950425i	$0.600648\pi$
$284$	5.19615	+	3.00000i	0.308335	+	0.178017i
$285$		0			0
$286$		0			0
$287$	−6.92820	−	6.00000i	−0.408959	−	0.354169i
$288$		0			0
$289$	2.50000	+	4.33013i	0.147059	+	0.254713i
$290$	7.79423	−	13.5000i	0.457693	−	0.792747i
$291$		0			0
$292$	−10.5000	+	6.06218i	−0.614466	+	0.354762i
$293$	−29.4449			−1.72019			−0.860094	−	0.510136i	$-0.829595\pi$
$293$	−29.4449			−1.72019			−0.860094	+	0.510136i	$0.829595\pi$
$294$		0			0
$295$	−21.0000			−1.22267
$296$	−3.46410	+	2.00000i	−0.201347	+	0.116248i
$297$		0			0
$298$	7.50000	−	12.9904i	0.434463	−	0.752513i
$299$		0			0
$300$		0			0
$301$	−16.0000	−	13.8564i	−0.922225	−	0.798670i
$302$		−	8.00000i		−	0.460348i
$303$		0			0
$304$	−6.00000	−	3.46410i	−0.344124	−	0.198680i
$305$	5.19615	+	3.00000i	0.297531	+	0.171780i
$306$		0			0
$307$			6.92820i			0.395413i	0.980261	+	0.197707i	$0.0633494\pi$
$307$			6.92820i			0.395413i	−0.980261	+	0.197707i	$0.936651\pi$
$308$		0			0
$309$		0			0
$310$	3.00000	+	5.19615i	0.170389	+	0.295122i
$311$	−6.92820	+	12.0000i	−0.392862	+	0.680458i	−0.992826	−	0.119570i	$-0.961848\pi$
$311$	−6.92820	+	12.0000i	−0.392862	+	0.680458i	0.599963	+	0.800027i	$0.295182\pi$
$312$		0			0
$313$	−19.5000	+	11.2583i	−1.10221	+	0.636358i	−0.936799	−	0.349867i	$-0.886227\pi$
$313$	−19.5000	+	11.2583i	−1.10221	+	0.636358i	−0.165406	+	0.986226i	$0.552893\pi$
$314$	10.3923			0.586472
$315$		0			0
$316$	−11.0000			−0.618798
$317$	5.19615	−	3.00000i	0.291845	−	0.168497i	−0.346929	−	0.937892i	$-0.612775\pi$
$317$	5.19615	−	3.00000i	0.291845	−	0.168497i	0.638774	+	0.769395i	$0.279442\pi$
$318$		0			0
$319$		0			0
$320$	−0.866025	−	1.50000i	−0.0484123	−	0.0838525i
$321$		0			0
$322$	−15.0000	+	5.19615i	−0.835917	+	0.289570i
$323$			24.0000i			1.33540i
$324$		0			0
$325$		0			0
$326$	−1.73205	−	1.00000i	−0.0959294	−	0.0553849i
$327$		0			0
$328$			3.46410i			0.191273i
$329$	1.73205	−	9.00000i	0.0954911	−	0.496186i
$330$		0			0
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	0.695266	−	0.718752i	$-0.255287\pi$
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	−0.970091	+	0.242742i	$0.921953\pi$
$332$	8.66025	−	15.0000i	0.475293	−	0.823232i
$333$		0			0
$334$	−15.0000	+	8.66025i	−0.820763	+	0.473868i
$335$	−24.2487			−1.32485
$336$		0			0
$337$	−23.0000			−1.25289			−0.626445	−	0.779466i	$-0.715491\pi$
$337$	−23.0000			−1.25289			−0.626445	+	0.779466i	$0.715491\pi$
$338$	11.2583	−	6.50000i	0.612372	−	0.353553i
$339$		0			0
$340$	−3.00000	+	5.19615i	−0.162698	+	0.281801i
$341$		0			0
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$			8.00000i			0.431331i
$345$		0			0
$346$	−4.50000	−	2.59808i	−0.241921	−	0.139673i
$347$	28.5788	+	16.5000i	1.53419	+	0.885766i	0.999162	+	0.0409337i	$0.0130332\pi$
$347$	28.5788	+	16.5000i	1.53419	+	0.885766i	0.535031	+	0.844833i	$0.320300\pi$
$348$		0			0
$349$			3.46410i			0.185429i	0.995693	+	0.0927146i	$0.0295544\pi$
$349$			3.46410i			0.185429i	−0.995693	+	0.0927146i	$0.970446\pi$
$350$	3.46410	−	4.00000i	0.185164	−	0.213809i
$351$		0			0
$352$		0			0
$353$	−1.73205	+	3.00000i	−0.0921878	+	0.159674i	−0.908431	−	0.418034i	$-0.862719\pi$
$353$	−1.73205	+	3.00000i	−0.0921878	+	0.159674i	0.816244	+	0.577708i	$0.196053\pi$
$354$		0			0
$355$	−9.00000	+	5.19615i	−0.477670	+	0.275783i
$356$	−10.3923			−0.550791
$357$		0			0
$358$	15.0000			0.792775
$359$	−20.7846	+	12.0000i	−1.09697	+	0.633336i	−0.935423	−	0.353529i	$-0.884981\pi$
$359$	−20.7846	+	12.0000i	−1.09697	+	0.633336i	−0.161546	+	0.986865i	$0.551648\pi$
$360$		0			0
$361$	14.5000	−	25.1147i	0.763158	−	1.32183i
$362$	−8.66025	−	15.0000i	−0.455173	−	0.788382i
$363$		0			0
$364$		0			0
$365$		−	21.0000i		−	1.09919i
$366$		0			0
$367$	−7.50000	−	4.33013i	−0.391497	−	0.226031i	0.291312	−	0.956628i	$-0.405908\pi$
$367$	−7.50000	−	4.33013i	−0.391497	−	0.226031i	−0.682808	+	0.730597i	$0.739242\pi$
$368$	5.19615	+	3.00000i	0.270868	+	0.156386i
$369$		0			0
$370$		−	6.92820i		−	0.360180i
$371$	−2.59808	−	7.50000i	−0.134885	−	0.389381i
$372$		0			0
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	0.625917	−	0.779890i	$-0.284725\pi$
$373$	−7.00000	−	12.1244i	−0.362446	−	0.627775i	−0.988363	+	0.152115i	$0.951392\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	+	1.73205i	−0.154713	+	0.0893237i
$377$		0			0
$378$		0			0
$379$	−14.0000			−0.719132			−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000			−0.719132			−0.359566	+	0.933120i	$0.617075\pi$
$380$	10.3923	−	6.00000i	0.533114	−	0.307794i
$381$		0			0
$382$	−3.00000	+	5.19615i	−0.153493	+	0.265858i
$383$	−8.66025	−	15.0000i	−0.442518	−	0.766464i	0.555357	−	0.831612i	$-0.312581\pi$
$383$	−8.66025	−	15.0000i	−0.442518	−	0.766464i	−0.997876	+	0.0651476i	$0.979248\pi$
$384$		0			0
$385$		0			0
$386$			10.0000i			0.508987i
$387$		0			0
$388$	6.00000	+	3.46410i	0.304604	+	0.175863i
$389$	−18.1865	−	10.5000i	−0.922094	−	0.532371i	−0.0377914	−	0.999286i	$-0.512032\pi$
$389$	−18.1865	−	10.5000i	−0.922094	−	0.532371i	−0.884302	+	0.466915i	$0.845366\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$		0			0
$394$	−1.50000	−	2.59808i	−0.0755689	−	0.130889i
$395$	9.52628	−	16.5000i	0.479319	−	0.830205i
$396$		0			0
$397$	−24.0000	+	13.8564i	−1.20453	+	0.695433i	−0.961558	−	0.274601i	$-0.911454\pi$
$397$	−24.0000	+	13.8564i	−1.20453	+	0.695433i	−0.242967	+	0.970034i	$0.578121\pi$
$398$	8.66025			0.434099
$399$		0			0
$400$	−2.00000			−0.100000
$401$	−5.19615	+	3.00000i	−0.259483	+	0.149813i	−0.624099	−	0.781345i	$-0.714534\pi$
$401$	−5.19615	+	3.00000i	−0.259483	+	0.149813i	0.364615	+	0.931158i	$0.381200\pi$
$402$		0			0
$403$		0			0
$404$	−2.59808	−	4.50000i	−0.129259	−	0.223883i
$405$		0			0
$406$	4.50000	−	23.3827i	0.223331	−	1.16046i
$407$		0			0
$408$		0			0
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	0.127609	−	0.991825i	$-0.459270\pi$
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	−0.795141	+	0.606425i	$0.792603\pi$
$410$	−5.19615	−	3.00000i	−0.256620	−	0.148159i
$411$		0			0
$412$		−	10.3923i		−	0.511992i
$413$	−30.3109	+	10.5000i	−1.49150	+	0.516671i
$414$		0			0
$415$	15.0000	+	25.9808i	0.736321	+	1.27535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3.46410			0.169232			0.0846162	−	0.996414i	$-0.473034\pi$
$419$	3.46410			0.169232			0.0846162	+	0.996414i	$0.473034\pi$
$420$		0			0
$421$	32.0000			1.55958			0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000			1.55958			0.779792	+	0.626038i	$0.215325\pi$
$422$	13.8564	−	8.00000i	0.674519	−	0.389434i
$423$		0			0
$424$	−1.50000	+	2.59808i	−0.0728464	+	0.126174i
$425$	3.46410	+	6.00000i	0.168034	+	0.291043i
$426$		0			0
$427$	9.00000	+	1.73205i	0.435541	+	0.0838198i
$428$		−	3.00000i		−	0.145010i
$429$		0			0
$430$	−12.0000	−	6.92820i	−0.578691	−	0.334108i
$431$	−31.1769	−	18.0000i	−1.50174	−	0.867029i	−0.999998	−	0.00201168i	$-0.999360\pi$
$431$	−31.1769	−	18.0000i	−1.50174	−	0.867029i	−0.501741	−	0.865018i	$-0.667307\pi$
$432$		0			0
$433$		−	1.73205i		−	0.0832370i	−0.999134	−	0.0416185i	$-0.986749\pi$
$433$		−	1.73205i		−	0.0832370i	0.999134	−	0.0416185i	$-0.0132514\pi$
$434$	6.92820	+	6.00000i	0.332564	+	0.288009i
$435$		0			0
$436$	8.00000	+	13.8564i	0.383131	+	0.663602i
$437$	−20.7846	+	36.0000i	−0.994263	+	1.72211i
$438$		0			0
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−27.0000	+	15.5885i	−1.28864	+	0.743996i	−0.310228	+	0.950662i	$0.600405\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	7.79423	−	4.50000i	0.370315	−	0.213801i	−0.303281	−	0.952901i	$-0.598082\pi$
$443$	7.79423	−	4.50000i	0.370315	−	0.213801i	0.673596	+	0.739100i	$0.264749\pi$
$444$		0			0
$445$	9.00000	−	15.5885i	0.426641	−	0.738964i
$446$	7.79423	+	13.5000i	0.369067	+	0.639244i
$447$		0			0
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$			6.00000i			0.283158i	0.989927	+	0.141579i	$0.0452178\pi$
$449$			6.00000i			0.283158i	−0.989927	+	0.141579i	$0.954782\pi$
$450$		0			0
$451$		0			0
$452$	−15.5885	−	9.00000i	−0.733219	−	0.423324i
$453$		0			0
$454$			25.9808i			1.21934i
$455$		0			0
$456$		0			0
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.725059	+	0.688686i	$0.241812\pi$
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.233890	+	0.972263i	$0.424854\pi$
$458$	−1.73205	+	3.00000i	−0.0809334	+	0.140181i
$459$		0			0
$460$	−9.00000	+	5.19615i	−0.419627	+	0.242272i
$461$	22.5167			1.04871			0.524353	−	0.851501i	$-0.324307\pi$
$461$	22.5167			1.04871			0.524353	+	0.851501i	$0.324307\pi$
$462$		0			0
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$	−7.79423	+	4.50000i	−0.361838	+	0.208907i
$465$		0			0
$466$	15.0000	−	25.9808i	0.694862	−	1.20354i
$467$	−2.59808	−	4.50000i	−0.120225	−	0.208235i	0.799632	−	0.600491i	$-0.205028\pi$
$467$	−2.59808	−	4.50000i	−0.120225	−	0.208235i	−0.919856	+	0.392256i	$0.871695\pi$
$468$		0			0
$469$	−35.0000	+	12.1244i	−1.61615	+	0.559851i
$470$		−	6.00000i		−	0.276759i
$471$		0			0
$472$	10.5000	+	6.06218i	0.483302	+	0.279034i
$473$		0			0
$474$		0			0
$475$		−	13.8564i		−	0.635776i
$476$	−1.73205	+	9.00000i	−0.0793884	+	0.412514i
$477$		0			0
$478$	−12.0000	−	20.7846i	−0.548867	−	0.950666i
$479$	6.92820	−	12.0000i	0.316558	−	0.548294i	−0.663210	−	0.748434i	$-0.730806\pi$
$479$	6.92820	−	12.0000i	0.316558	−	0.548294i	0.979767	+	0.200140i	$0.0641396\pi$
$480$		0			0
$481$		0			0
$482$	1.73205			0.0788928
$483$		0			0
$484$	−11.0000			−0.500000
$485$	−10.3923	+	6.00000i	−0.471890	+	0.272446i
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	−0.877132	−	0.480250i	$-0.840546\pi$
$487$	−0.500000	+	0.866025i	−0.0226572	+	0.0392434i	0.854475	+	0.519493i	$0.173879\pi$
$488$	−1.73205	−	3.00000i	−0.0784063	−	0.135804i
$489$		0			0
$490$	−4.50000	+	11.2583i	−0.203289	+	0.508600i
$491$			3.00000i			0.135388i	0.997706	+	0.0676941i	$0.0215642\pi$
$491$			3.00000i			0.135388i	−0.997706	+	0.0676941i	$0.978436\pi$
$492$		0			0
$493$	27.0000	+	15.5885i	1.21602	+	0.702069i
$494$		0			0
$495$		0			0
$496$		−	3.46410i		−	0.155543i
$497$	−10.3923	+	12.0000i	−0.466159	+	0.538274i
$498$		0			0
$499$	7.00000	+	12.1244i	0.313363	+	0.542761i	0.979088	−	0.203436i	$-0.0652110\pi$
$499$	7.00000	+	12.1244i	0.313363	+	0.542761i	−0.665725	+	0.746197i	$0.731878\pi$
$500$	6.06218	−	10.5000i	0.271109	−	0.469574i
$501$		0			0
$502$	7.50000	−	4.33013i	0.334741	−	0.193263i
$503$	27.7128			1.23565			0.617827	−	0.786314i	$-0.288013\pi$
$503$	27.7128			1.23565			0.617827	+	0.786314i	$0.288013\pi$
$504$		0			0
$505$	9.00000			0.400495
$506$		0			0
$507$		0			0
$508$	−3.50000	+	6.06218i	−0.155287	+	0.268966i
$509$	17.3205	+	30.0000i	0.767718	+	1.32973i	0.938798	+	0.344469i	$0.111941\pi$
$509$	17.3205	+	30.0000i	0.767718	+	1.32973i	−0.171080	+	0.985257i	$0.554726\pi$
$510$		0			0
$511$	−10.5000	−	30.3109i	−0.464493	−	1.34087i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	18.0000	+	10.3923i	0.793946	+	0.458385i
$515$	15.5885	+	9.00000i	0.686909	+	0.396587i
$516$		0			0
$517$		0			0
$518$	−3.46410	−	10.0000i	−0.152204	−	0.439375i
$519$		0			0
$520$		0			0
$521$	1.73205	−	3.00000i	0.0758825	−	0.131432i	−0.825587	−	0.564275i	$-0.809156\pi$
$521$	1.73205	−	3.00000i	0.0758825	−	0.131432i	0.901470	+	0.432842i	$0.142489\pi$
$522$		0			0
$523$	18.0000	−	10.3923i	0.787085	−	0.454424i	−0.0518503	−	0.998655i	$-0.516512\pi$
$523$	18.0000	−	10.3923i	0.787085	−	0.454424i	0.838935	+	0.544231i	$0.183179\pi$
$524$	10.3923			0.453990
$525$		0			0
$526$	18.0000			0.784837
$527$	−10.3923	+	6.00000i	−0.452696	+	0.261364i
$528$		0			0
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$	−2.59808	−	4.50000i	−0.112853	−	0.195468i
$531$		0			0
$532$	12.0000	−	13.8564i	0.520266	−	0.600751i
$533$		0			0
$534$		0			0
$535$	4.50000	+	2.59808i	0.194552	+	0.112325i
$536$	12.1244	+	7.00000i	0.523692	+	0.302354i
$537$		0			0
$538$			1.73205i			0.0746740i
$539$		0			0
$540$		0			0
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	0.956504	+	0.291718i	$0.0942267\pi$
$541$	17.0000	+	29.4449i	0.730887	+	1.26593i	−0.225617	+	0.974216i	$0.572440\pi$
$542$	6.06218	−	10.5000i	0.260393	−	0.451014i
$543$		0			0
$544$	3.00000	−	1.73205i	0.128624	−	0.0742611i
$545$	−27.7128			−1.18709
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−31.1769	−	54.0000i	−1.32818	−	2.30048i
$552$		0			0
$553$	5.50000	−	28.5788i	0.233884	−	1.21530i
$554$			8.00000i			0.339887i
$555$		0			0
$556$	9.00000	+	5.19615i	0.381685	+	0.220366i
$557$	15.5885	+	9.00000i	0.660504	+	0.381342i	0.792469	−	0.609912i	$-0.208795\pi$
$557$	15.5885	+	9.00000i	0.660504	+	0.381342i	−0.131965	+	0.991254i	$0.542129\pi$
$558$		0			0
$559$		0			0
$560$	4.33013	−	1.50000i	0.182981	−	0.0633866i
$561$		0			0
$562$	9.00000	+	15.5885i	0.379642	+	0.657559i
$563$	2.59808	−	4.50000i	0.109496	−	0.189652i	−0.806070	−	0.591820i	$-0.798410\pi$
$563$	2.59808	−	4.50000i	0.109496	−	0.189652i	0.915566	+	0.402167i	$0.131743\pi$
$564$		0			0
$565$	27.0000	−	15.5885i	1.13590	−	0.655811i
$566$	6.92820			0.291214
$567$		0			0
$568$	6.00000			0.251754
$569$	36.3731	−	21.0000i	1.52484	−	0.880366i	0.525271	−	0.850935i	$-0.323964\pi$
$569$	36.3731	−	21.0000i	1.52484	−	0.880366i	0.999567	−	0.0294311i	$-0.00936956\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$		0			0
$573$		0			0
$574$	−9.00000	−	1.73205i	−0.375653	−	0.0722944i
$575$			12.0000i			0.500435i
$576$		0			0
$577$	−13.5000	−	7.79423i	−0.562012	−	0.324478i	0.191940	−	0.981407i	$-0.438522\pi$
$577$	−13.5000	−	7.79423i	−0.562012	−	0.324478i	−0.753953	+	0.656929i	$0.771855\pi$
$578$	4.33013	+	2.50000i	0.180110	+	0.103986i
$579$		0			0
$580$		−	15.5885i		−	0.647275i
$581$	34.6410	+	30.0000i	1.43715	+	1.24461i
$582$		0			0
$583$		0			0
$584$	−6.06218	+	10.5000i	−0.250855	+	0.434493i
$585$		0			0
$586$	−25.5000	+	14.7224i	−1.05340	+	0.608178i
$587$	36.3731			1.50128			0.750639	−	0.660713i	$-0.229746\pi$
$587$	36.3731			1.50128			0.750639	+	0.660713i	$0.229746\pi$
$588$		0			0
$589$	24.0000			0.988903
$590$	−18.1865	+	10.5000i	−0.748728	+	0.432278i
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$	12.1244	+	21.0000i	0.497888	+	0.862367i	0.999997	−	0.00243746i	$-0.000775869\pi$
$593$	12.1244	+	21.0000i	0.497888	+	0.862367i	−0.502109	+	0.864804i	$0.667443\pi$
$594$		0			0
$595$	−12.0000	−	10.3923i	−0.491952	−	0.426043i
$596$		−	15.0000i		−	0.614424i
$597$		0			0
$598$		0			0
$599$	−20.7846	−	12.0000i	−0.849236	−	0.490307i	0.0111569	−	0.999938i	$-0.496449\pi$
$599$	−20.7846	−	12.0000i	−0.849236	−	0.490307i	−0.860393	+	0.509631i	$0.829782\pi$
$600$		0			0
$601$			1.73205i			0.0706518i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$			1.73205i			0.0706518i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$	−20.7846	−	4.00000i	−0.847117	−	0.163028i
$603$		0			0
$604$	−4.00000	−	6.92820i	−0.162758	−	0.281905i
$605$	9.52628	−	16.5000i	0.387298	−	0.670820i
$606$		0			0
$607$	−28.5000	+	16.4545i	−1.15678	+	0.667867i	−0.950530	−	0.310633i	$-0.899459\pi$
$607$	−28.5000	+	16.4545i	−1.15678	+	0.667867i	−0.206249	+	0.978499i	$0.566126\pi$
$608$	−6.92820			−0.280976
$609$		0			0
$610$	6.00000			0.242933
$611$		0			0
$612$		0			0
$613$	−4.00000	+	6.92820i	−0.161558	+	0.279827i	−0.935428	−	0.353518i	$-0.884985\pi$
$613$	−4.00000	+	6.92820i	−0.161558	+	0.279827i	0.773869	+	0.633345i	$0.218319\pi$
$614$	3.46410	+	6.00000i	0.139800	+	0.242140i
$615$		0			0
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$	18.0000	+	10.3923i	0.723481	+	0.417702i	0.816033	−	0.578006i	$-0.196169\pi$
$619$	18.0000	+	10.3923i	0.723481	+	0.417702i	−0.0925515	+	0.995708i	$0.529502\pi$
$620$	5.19615	+	3.00000i	0.208683	+	0.120483i
$621$		0			0
$622$			13.8564i			0.555591i
$623$	5.19615	−	27.0000i	0.208179	−	1.08173i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$	−11.2583	+	19.5000i	−0.449973	+	0.779377i
$627$		0			0
$628$	9.00000	−	5.19615i	0.359139	−	0.207349i
$629$	13.8564			0.552491
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$	−9.52628	+	5.50000i	−0.378935	+	0.218778i
$633$		0			0
$634$	3.00000	−	5.19615i	0.119145	−	0.206366i
$635$	−6.06218	−	10.5000i	−0.240570	−	0.416680i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−1.50000	−	0.866025i	−0.0592927	−	0.0342327i
$641$	−10.3923	−	6.00000i	−0.410471	−	0.236986i	0.280521	−	0.959848i	$-0.409493\pi$
$641$	−10.3923	−	6.00000i	−0.410471	−	0.236986i	−0.690992	+	0.722862i	$0.742826\pi$
$642$		0			0
$643$		−	27.7128i		−	1.09289i	−0.837496	−	0.546443i	$-0.815981\pi$
$643$		−	27.7128i		−	1.09289i	0.837496	−	0.546443i	$-0.184019\pi$
$644$	−10.3923	+	12.0000i	−0.409514	+	0.472866i
$645$		0			0
$646$	12.0000	+	20.7846i	0.472134	+	0.817760i
$647$	−5.19615	+	9.00000i	−0.204282	+	0.353827i	−0.949904	−	0.312543i	$-0.898819\pi$
$647$	−5.19615	+	9.00000i	−0.204282	+	0.353827i	0.745622	+	0.666369i	$0.232153\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.00000			−0.0783260
$653$	−12.9904	+	7.50000i	−0.508353	+	0.293498i	−0.732156	−	0.681137i	$-0.761486\pi$
$653$	−12.9904	+	7.50000i	−0.508353	+	0.293498i	0.223803	+	0.974634i	$0.428153\pi$
$654$		0			0
$655$	−9.00000	+	15.5885i	−0.351659	+	0.609091i
$656$	1.73205	+	3.00000i	0.0676252	+	0.117130i
$657$		0			0
$658$	−3.00000	−	8.66025i	−0.116952	−	0.337612i
$659$		−	27.0000i		−	1.05177i	−0.850555	−	0.525885i	$-0.823734\pi$
$659$		−	27.0000i		−	1.05177i	0.850555	−	0.525885i	$-0.176266\pi$
$660$		0			0
$661$	−12.0000	−	6.92820i	−0.466746	−	0.269476i	0.248131	−	0.968727i	$-0.420184\pi$
$661$	−12.0000	−	6.92820i	−0.466746	−	0.269476i	−0.714877	+	0.699251i	$0.753517\pi$
$662$	−8.66025	−	5.00000i	−0.336590	−	0.194331i
$663$		0			0
$664$		−	17.3205i		−	0.672166i
$665$	10.3923	+	30.0000i	0.402996	+	1.16335i
$666$		0			0
$667$	27.0000	+	46.7654i	1.04544	+	1.81076i
$668$	−8.66025	+	15.0000i	−0.335075	+	0.580367i
$669$		0			0
$670$	−21.0000	+	12.1244i	−0.811301	+	0.468405i
$671$		0			0
$672$		0			0
$673$	−5.00000			−0.192736			−0.0963679	−	0.995346i	$-0.530723\pi$
$673$	−5.00000			−0.192736			−0.0963679	+	0.995346i	$0.530723\pi$
$674$	−19.9186	+	11.5000i	−0.767235	+	0.442963i
$675$		0			0
$676$	6.50000	−	11.2583i	0.250000	−	0.433013i
$677$	12.9904	+	22.5000i	0.499261	+	0.864745i	1.00000		0.000853228i	$-0.000271591\pi$
$677$	12.9904	+	22.5000i	0.499261	+	0.864745i	−0.500739	+	0.865598i	$0.666938\pi$
$678$		0			0
$679$	−12.0000	+	13.8564i	−0.460518	+	0.531760i
$680$			6.00000i			0.230089i
$681$		0			0
$682$		0			0
$683$	7.79423	+	4.50000i	0.298238	+	0.172188i	0.641651	−	0.766997i	$-0.278250\pi$
$683$	7.79423	+	4.50000i	0.298238	+	0.172188i	−0.343413	+	0.939184i	$0.611583\pi$
$684$		0			0
$685$		0			0
$686$	−0.866025	+	18.5000i	−0.0330650	+	0.706333i
$687$		0			0
$688$	4.00000	+	6.92820i	0.152499	+	0.264135i
$689$		0			0
$690$		0			0
$691$	−24.0000	+	13.8564i	−0.913003	+	0.527123i	−0.881396	−	0.472378i	$-0.843396\pi$
$691$	−24.0000	+	13.8564i	−0.913003	+	0.527123i	−0.0316069	+	0.999500i	$0.510062\pi$
$692$	−5.19615			−0.197528
$693$		0			0
$694$	33.0000			1.25266
$695$	−15.5885	+	9.00000i	−0.591304	+	0.341389i
$696$		0			0
$697$	6.00000	−	10.3923i	0.227266	−	0.393637i
$698$	1.73205	+	3.00000i	0.0655591	+	0.113552i
$699$		0			0
$700$	1.00000	−	5.19615i	0.0377964	−	0.196396i
$701$		−	27.0000i		−	1.01978i	−0.860241	−	0.509888i	$-0.829687\pi$
$701$		−	27.0000i		−	1.01978i	0.860241	−	0.509888i	$-0.170313\pi$
$702$		0			0
$703$	−24.0000	−	13.8564i	−0.905177	−	0.522604i
$704$		0			0
$705$		0			0
$706$			3.46410i			0.130373i
$707$	12.9904	−	4.50000i	0.488554	−	0.169240i
$708$		0			0
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	−0.999549	−	0.0300298i	$-0.990440\pi$
$709$	−14.0000	−	24.2487i	−0.525781	−	0.910679i	0.473768	−	0.880650i	$-0.342894\pi$
$710$	−5.19615	+	9.00000i	−0.195008	+	0.337764i
$711$		0			0
$712$	−9.00000	+	5.19615i	−0.337289	+	0.194734i
$713$	−20.7846			−0.778390
$714$		0			0
$715$		0			0
$716$	12.9904	−	7.50000i	0.485473	−	0.280288i
$717$		0			0
$718$	−12.0000	+	20.7846i	−0.447836	+	0.775675i
$719$	−24.2487	−	42.0000i	−0.904324	−	1.56634i	−0.821822	−	0.569745i	$-0.807042\pi$
$719$	−24.2487	−	42.0000i	−0.904324	−	1.56634i	−0.0825027	−	0.996591i	$-0.526291\pi$
$720$		0			0
$721$	27.0000	+	5.19615i	1.00553	+	0.193515i
$722$		−	29.0000i		−	1.07927i
$723$		0			0
$724$	−15.0000	−	8.66025i	−0.557471	−	0.321856i
$725$	−15.5885	−	9.00000i	−0.578941	−	0.334252i
$726$		0			0
$727$		−	12.1244i		−	0.449667i	−0.974397	−	0.224834i	$-0.927816\pi$
$727$		−	12.1244i		−	0.449667i	0.974397	−	0.224834i	$-0.0721839\pi$
$728$		0			0
$729$		0			0
$730$	−10.5000	−	18.1865i	−0.388622	−	0.673114i
$731$	13.8564	−	24.0000i	0.512498	−	0.887672i
$732$		0			0
$733$	−6.00000	+	3.46410i	−0.221615	+	0.127950i	−0.606698	−	0.794933i	$-0.707506\pi$
$733$	−6.00000	+	3.46410i	−0.221615	+	0.127950i	0.385083	+	0.922882i	$0.374173\pi$
$734$	−8.66025			−0.319656
$735$		0			0
$736$	6.00000			0.221163
$737$		0			0
$738$		0			0
$739$	5.00000	−	8.66025i	0.183928	−	0.318573i	−0.759287	−	0.650756i	$-0.774452\pi$
$739$	5.00000	−	8.66025i	0.183928	−	0.318573i	0.943215	+	0.332184i	$0.107785\pi$
$740$	−3.46410	−	6.00000i	−0.127343	−	0.220564i
$741$		0			0
$742$	−6.00000	−	5.19615i	−0.220267	−	0.190757i
$743$		−	36.0000i		−	1.32071i	−0.750953	−	0.660356i	$-0.770405\pi$
$743$		−	36.0000i		−	1.32071i	0.750953	−	0.660356i	$-0.229595\pi$
$744$		0			0
$745$	22.5000	+	12.9904i	0.824336	+	0.475931i
$746$	−12.1244	−	7.00000i	−0.443904	−	0.256288i
$747$		0			0
$748$		0			0
$749$	7.79423	+	1.50000i	0.284795	+	0.0548088i
$750$		0			0
$751$	−3.50000	−	6.06218i	−0.127717	−	0.221212i	0.795075	−	0.606511i	$-0.207432\pi$
$751$	−3.50000	−	6.06218i	−0.127717	−	0.221212i	−0.922792	+	0.385299i	$0.874098\pi$
$752$	−1.73205	+	3.00000i	−0.0631614	+	0.109399i
$753$		0			0
$754$		0			0
$755$	13.8564			0.504286
$756$		0			0
$757$	−38.0000			−1.38113			−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000			−1.38113			−0.690567	+	0.723269i	$0.742639\pi$
$758$	−12.1244	+	7.00000i	−0.440376	+	0.254251i
$759$		0			0
$760$	6.00000	−	10.3923i	0.217643	−	0.376969i
$761$	17.3205	+	30.0000i	0.627868	+	1.08750i	0.987979	+	0.154590i	$0.0494055\pi$
$761$	17.3205	+	30.0000i	0.627868	+	1.08750i	−0.360111	+	0.932910i	$0.617261\pi$
$762$		0			0
$763$	−40.0000	+	13.8564i	−1.44810	+	0.501636i
$764$			6.00000i			0.217072i
$765$		0			0
$766$	−15.0000	−	8.66025i	−0.541972	−	0.312908i
$767$		0			0
$768$		0			0
$769$			48.4974i			1.74886i	0.485150	+	0.874431i	$0.338765\pi$
$769$			48.4974i			1.74886i	−0.485150	+	0.874431i	$0.661235\pi$
$770$		0			0
$771$		0			0
$772$	5.00000	+	8.66025i	0.179954	+	0.311689i
$773$	−6.92820	+	12.0000i	−0.249190	+	0.431610i	−0.963301	−	0.268422i	$-0.913498\pi$
$773$	−6.92820	+	12.0000i	−0.249190	+	0.431610i	0.714111	+	0.700032i	$0.246831\pi$
$774$		0			0
$775$	6.00000	−	3.46410i	0.215526	−	0.124434i
$776$	6.92820			0.248708
$777$		0			0
$778$	−21.0000			−0.752886
$779$	−20.7846	+	12.0000i	−0.744686	+	0.429945i
$780$		0			0
$781$		0			0
$782$	−10.3923	−	18.0000i	−0.371628	−	0.643679i
$783$		0			0
$784$	5.50000	−	4.33013i	0.196429	−	0.154647i
$785$			18.0000i			0.642448i
$786$		0			0
$787$	3.00000	+	1.73205i	0.106938	+	0.0617409i	0.552515	−	0.833503i	$-0.313668\pi$
$787$	3.00000	+	1.73205i	0.106938	+	0.0617409i	−0.445577	+	0.895244i	$0.647001\pi$

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.k (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(0.866025 + 1.50000i) q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} +(1.50000 + 0.866025i) q^{10} +(2.59808 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-1.73205 + 3.00000i) q^{17} +(6.00000 - 3.46410i) q^{19} +1.73205 q^{20} +(-5.19615 + 3.00000i) q^{23} +(1.00000 - 1.73205i) q^{25} +(2.50000 - 0.866025i) q^{28} -9.00000i q^{29} +(3.00000 + 1.73205i) q^{31} +(-0.866025 - 0.500000i) q^{32} +3.46410i q^{34} +(-0.866025 + 4.50000i) q^{35} +(-2.00000 - 3.46410i) q^{37} +(3.46410 - 6.00000i) q^{38} +(1.50000 - 0.866025i) q^{40} -3.46410 q^{41} -8.00000 q^{43} +(-3.00000 + 5.19615i) q^{46} +(-1.73205 - 3.00000i) q^{47} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} +(-2.59808 - 1.50000i) q^{53} +(1.73205 - 2.00000i) q^{56} +(-4.50000 - 7.79423i) q^{58} +(-6.06218 + 10.5000i) q^{59} +(3.00000 - 1.73205i) q^{61} +3.46410 q^{62} -1.00000 q^{64} +(-7.00000 + 12.1244i) q^{67} +(1.73205 + 3.00000i) q^{68} +(1.50000 + 4.33013i) q^{70} +6.00000i q^{71} +(-10.5000 - 6.06218i) q^{73} +(-3.46410 - 2.00000i) q^{74} -6.92820i q^{76} +(-5.50000 - 9.52628i) q^{79} +(0.866025 - 1.50000i) q^{80} +(-3.00000 + 1.73205i) q^{82} +17.3205 q^{83} -6.00000 q^{85} +(-6.92820 + 4.00000i) q^{86} +(-5.19615 - 9.00000i) q^{89} +6.00000i q^{92} +(-3.00000 - 1.73205i) q^{94} +(10.3923 + 6.00000i) q^{95} +6.92820i q^{97} +(4.33013 + 5.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 + 8 * q^7	\( 4 q + 2 q^{4} + 8 q^{7} + 6 q^{10} - 2 q^{16} + 24 q^{19} + 4 q^{25} + 10 q^{28} + 12 q^{31} - 8 q^{37} + 6 q^{40} - 32 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{58} + 12 q^{61} - 4 q^{64} - 28 q^{67} + 6 q^{70} - 42 q^{73} - 22 q^{79} - 12 q^{82} - 24 q^{85} - 12 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^7 + 6 * q^10 - 2 * q^16 + 24 * q^19 + 4 * q^25 + 10 * q^28 + 12 * q^31 - 8 * q^37 + 6 * q^40 - 32 * q^43 - 12 * q^46 + 4 * q^49 - 18 * q^58 + 12 * q^61 - 4 * q^64 - 28 * q^67 + 6 * q^70 - 42 * q^73 - 22 * q^79 - 12 * q^82 - 24 * q^85 - 12 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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