LMFDB - Embedded newform 405.2.j.c.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(109,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("405.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.j (of order $6$, 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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 25 $ x^4 - 5*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			109.1
Root			$-1.93649 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	405.109
Dual form			405.2.j.c.379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 + 1.11803i) q^{2} +(1.50000 - 2.59808i) q^{4} +(-1.93649 - 1.11803i) q^{5} +2.23607i q^{8} +O(q^{10})$	$q+(-1.93649 + 1.11803i) q^{2} +(1.50000 - 2.59808i) q^{4} +(-1.93649 - 1.11803i) q^{5} +2.23607i q^{8} +5.00000 q^{10} +(0.500000 + 0.866025i) q^{16} +4.47214i q^{17} -4.00000 q^{19} +(-5.80948 + 3.35410i) q^{20} +(7.74597 + 4.47214i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-4.00000 + 6.92820i) q^{31} +(-5.80948 - 3.35410i) q^{32} +(-5.00000 - 8.66025i) q^{34} +(7.74597 - 4.47214i) q^{38} +(2.50000 - 4.33013i) q^{40} -20.0000 q^{46} +(-7.74597 + 4.47214i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(-9.68246 - 5.59017i) q^{50} +4.47214i q^{53} +(-1.00000 - 1.73205i) q^{61} -17.8885i q^{62} +13.0000 q^{64} +(11.6190 + 6.70820i) q^{68} +(-6.00000 + 10.3923i) q^{76} +(8.00000 + 13.8564i) q^{79} -2.23607i q^{80} +(15.4919 - 8.94427i) q^{83} +(5.00000 - 8.66025i) q^{85} +(23.2379 - 13.4164i) q^{92} +(10.0000 - 17.3205i) q^{94} +(7.74597 + 4.47214i) q^{95} -15.6525i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4}+O(q^{10}) $ 4 * q + 6 * q^4	$ 4 q + 6 q^{4} + 20 q^{10} + 2 q^{16} - 16 q^{19} + 10 q^{25} - 16 q^{31} - 20 q^{34} + 10 q^{40} - 80 q^{46} - 14 q^{49} - 4 q^{61} + 52 q^{64} - 24 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100}) $ 4 * q + 6 * q^4 + 20 * q^10 + 2 * q^16 - 16 * q^19 + 10 * q^25 - 16 * q^31 - 20 * q^34 + 10 * q^40 - 80 * q^46 - 14 * q^49 - 4 * q^61 + 52 * q^64 - 24 * q^76 + 32 * q^79 + 20 * q^85 + 40 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$	−1.93649	+	1.11803i	−1.36931	+	0.790569i	−0.990839	−	0.135045i	$-0.956882\pi$
$2$	−1.93649	+	1.11803i	−1.36931	+	0.790569i	−0.378467	+	0.925615i	$0.623549\pi$
$3$		0			0
$3$		0			0
$4$	1.50000	−	2.59808i	0.750000	−	1.29904i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$5$	−1.93649	−	1.11803i	−0.866025	−	0.500000i
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$			2.23607i			0.790569i
$9$		0			0
$10$	5.00000			1.58114
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	−5.80948	+	3.35410i	−1.29904	+	0.750000i
$21$		0			0
$22$		0			0
$23$	7.74597	+	4.47214i	1.61515	+	0.932505i	0.988152	+	0.153481i	$0.0490483\pi$
$23$	7.74597	+	4.47214i	1.61515	+	0.932505i	0.626994	+	0.779024i	$0.284285\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$	−5.80948	−	3.35410i	−1.02698	−	0.592927i
$33$		0			0
$34$	−5.00000	−	8.66025i	−0.857493	−	1.48522i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	7.74597	−	4.47214i	1.25656	−	0.725476i
$39$		0			0
$40$	2.50000	−	4.33013i	0.395285	−	0.684653i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$		0			0
$46$	−20.0000			−2.94884
$47$	−7.74597	+	4.47214i	−1.12987	+	0.652328i	−0.943901	−	0.330228i	$-0.892874\pi$
$47$	−7.74597	+	4.47214i	−1.12987	+	0.652328i	−0.185964	+	0.982556i	$0.559541\pi$
$48$		0			0
$49$	−3.50000	+	6.06218i	−0.500000	+	0.866025i
$50$	−9.68246	−	5.59017i	−1.36931	−	0.790569i
$51$		0			0
$52$		0			0
$53$			4.47214i			0.614295i	0.951662	+	0.307148i	$0.0993745\pi$
$53$			4.47214i			0.614295i	−0.951662	+	0.307148i	$0.900625\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	0.794879	−	0.606768i	$-0.207534\pi$
$61$	−1.00000	−	1.73205i	−0.128037	−	0.221766i	−0.922916	+	0.385002i	$0.874201\pi$
$62$		−	17.8885i		−	2.27185i
$63$		0			0
$64$	13.0000			1.62500
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	11.6190	+	6.70820i	1.40900	+	0.813489i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$	−6.00000	+	10.3923i	−0.688247	+	1.19208i
$77$		0			0
$78$		0			0
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.827401	+	0.561611i	$0.189818\pi$
$79$	8.00000	+	13.8564i	0.900070	+	1.55897i	0.0726692	+	0.997356i	$0.476848\pi$
$80$		−	2.23607i		−	0.250000i
$81$		0			0
$82$		0			0
$83$	15.4919	−	8.94427i	1.70046	−	0.981761i	0.755172	−	0.655527i	$-0.227553\pi$
$83$	15.4919	−	8.94427i	1.70046	−	0.981761i	0.945289	−	0.326234i	$-0.105780\pi$
$84$		0			0
$85$	5.00000	−	8.66025i	0.542326	−	0.939336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	23.2379	−	13.4164i	2.42272	−	1.39876i
$93$		0			0
$94$	10.0000	−	17.3205i	1.03142	−	1.78647i
$95$	7.74597	+	4.47214i	0.794719	+	0.458831i
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		−	15.6525i		−	1.58114i
$99$		0			0
$100$	15.0000			1.50000
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−5.00000	−	8.66025i	−0.485643	−	0.841158i
$107$			17.8885i			1.72935i	0.502331	+	0.864675i	$0.332476\pi$
$107$			17.8885i			1.72935i	−0.502331	+	0.864675i	$0.667524\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.87298	−	2.23607i	−0.364340	−	0.210352i	0.306643	−	0.951825i	$-0.400794\pi$
$113$	−3.87298	−	2.23607i	−0.364340	−	0.210352i	−0.670983	+	0.741473i	$0.734128\pi$
$114$		0			0
$115$	−10.0000	−	17.3205i	−0.932505	−	1.61515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$	3.87298	+	2.23607i	0.350643	+	0.202444i
$123$		0			0
$124$	12.0000	+	20.7846i	1.07763	+	1.86651i
$125$		−	11.1803i		−	1.00000i
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−13.5554	+	7.82624i	−1.19814	+	0.691748i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−10.0000			−0.857493
$137$	−19.3649	+	11.1803i	−1.65446	+	0.955201i	−0.679249	+	0.733908i	$0.737694\pi$
$137$	−19.3649	+	11.1803i	−1.65446	+	0.955201i	−0.975207	+	0.221293i	$0.928972\pi$
$138$		0			0
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	−0.768655	−	0.639664i	$-0.779074\pi$
$139$	2.00000	−	3.46410i	0.169638	−	0.293821i	0.938293	+	0.345843i	$0.112407\pi$
$140$		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
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		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	$0.938871\pi$
$152$		−	8.94427i		−	0.725476i
$153$		0			0
$154$		0			0
$155$	15.4919	−	8.94427i	1.24434	−	0.718421i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−30.9839	−	17.8885i	−2.46494	−	1.42314i
$159$		0			0
$160$	7.50000	+	12.9904i	0.592927	+	1.02698i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−20.0000	+	34.6410i	−1.55230	+	2.68866i
$167$	7.74597	+	4.47214i	0.599401	+	0.346064i	0.768806	−	0.639482i	$-0.220851\pi$
$167$	7.74597	+	4.47214i	0.599401	+	0.346064i	−0.169405	+	0.985547i	$0.554185\pi$
$168$		0			0
$169$	−6.50000	−	11.2583i	−0.500000	−	0.866025i
$170$			22.3607i			1.71499i
$171$		0			0
$172$		0			0
$173$	−19.3649	+	11.1803i	−1.47229	+	0.850026i	−0.999514	−	0.0311588i	$-0.990080\pi$
$173$	−19.3649	+	11.1803i	−1.47229	+	0.850026i	−0.472773	+	0.881184i	$0.656747\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		0			0
$184$	−10.0000	+	17.3205i	−0.737210	+	1.27688i
$185$		0			0
$186$		0			0
$187$		0			0
$188$			26.8328i			1.95698i
$189$		0			0
$190$	−20.0000			−1.45095
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	10.5000	+	18.1865i	0.750000	+	1.29904i
$197$			4.47214i			0.318626i	0.987228	+	0.159313i	$0.0509280\pi$
$197$			4.47214i			0.318626i	−0.987228	+	0.159313i	$0.949072\pi$
$198$		0			0
$199$	−16.0000			−1.13421			−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000			−1.13421			−0.567105	+	0.823646i	$0.691937\pi$
$200$	−9.68246	+	5.59017i	−0.684653	+	0.395285i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	14.0000	−	24.2487i	0.963800	−	1.66935i	0.250994	−	0.967989i	$-0.419243\pi$
$211$	14.0000	−	24.2487i	0.963800	−	1.66935i	0.712806	−	0.701361i	$-0.247424\pi$
$212$	11.6190	+	6.70820i	0.797993	+	0.460721i
$213$		0			0
$214$	−20.0000	−	34.6410i	−1.36717	−	2.36801i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−27.1109	+	15.6525i	−1.83618	+	1.06012i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$		0			0
$226$	10.0000			0.665190
$227$	15.4919	−	8.94427i	1.02824	−	0.593652i	0.111757	−	0.993736i	$-0.464352\pi$
$227$	15.4919	−	8.94427i	1.02824	−	0.593652i	0.916479	+	0.400083i	$0.131019\pi$
$228$		0			0
$229$	−13.0000	+	22.5167i	−0.859064	+	1.48794i	0.0137585	+	0.999905i	$0.495620\pi$
$229$	−13.0000	+	22.5167i	−0.859064	+	1.48794i	−0.872823	+	0.488037i	$0.837713\pi$
$230$	38.7298	+	22.3607i	2.55377	+	1.47442i
$231$		0			0
$232$		0			0
$233$		−	22.3607i		−	1.46490i	−0.680823	−	0.732448i	$-0.738378\pi$
$233$		−	22.3607i		−	1.46490i	0.680823	−	0.732448i	$-0.261622\pi$
$234$		0			0
$235$	20.0000			1.30466
$236$		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
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	0.832019	−	0.554747i	$-0.187185\pi$
$241$	−1.00000	−	1.73205i	−0.0644157	−	0.111571i	−0.896435	+	0.443176i	$0.853852\pi$
$242$			24.5967i			1.58114i
$243$		0			0
$244$	−6.00000			−0.384111
$245$	13.5554	−	7.82624i	0.866025	−	0.500000i
$246$		0			0
$247$		0			0
$248$	−15.4919	−	8.94427i	−0.983739	−	0.567962i
$249$		0			0
$250$	12.5000	+	21.6506i	0.790569	+	1.36931i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	4.50000	−	7.79423i	0.281250	−	0.487139i
$257$	−27.1109	−	15.6525i	−1.69113	−	0.976375i	−0.953608	−	0.301052i	$-0.902662\pi$
$257$	−27.1109	−	15.6525i	−1.69113	−	0.976375i	−0.737523	−	0.675322i	$-0.764005\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.74597	+	4.47214i	−0.477637	+	0.275764i	−0.719431	−	0.694564i	$-0.755597\pi$
$263$	−7.74597	+	4.47214i	−0.477637	+	0.275764i	0.241794	+	0.970328i	$0.422264\pi$
$264$		0			0
$265$	5.00000	−	8.66025i	0.307148	−	0.531995i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	32.0000			1.94386			0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000			1.94386			0.971931	+	0.235267i	$0.0755965\pi$
$272$	−3.87298	+	2.23607i	−0.234834	+	0.135582i
$273$		0			0
$274$	25.0000	−	43.3013i	1.51031	−	2.61593i
$275$		0			0
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$			8.94427i			0.536442i
$279$		0			0
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−27.1109	−	15.6525i	−1.58383	−	0.914427i	−0.994292	−	0.106690i	$-0.965975\pi$
$293$	−27.1109	−	15.6525i	−1.58383	−	0.914427i	−0.589542	−	0.807737i	$-0.700692\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	15.4919	+	8.94427i	0.891461	+	0.514685i
$303$		0			0
$304$	−2.00000	−	3.46410i	−0.114708	−	0.198680i
$305$			4.47214i			0.256074i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	−20.0000	+	34.6410i	−1.13592	+	1.96748i
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	48.0000			2.70021
$317$	−19.3649	+	11.1803i	−1.08764	+	0.627950i	−0.932948	−	0.360012i	$-0.882773\pi$
$317$	−19.3649	+	11.1803i	−1.08764	+	0.627950i	−0.154694	+	0.987962i	$0.549439\pi$
$318$		0			0
$319$		0			0
$320$	−25.1744	−	14.5344i	−1.40729	−	0.812500i
$321$		0			0
$322$		0			0
$323$		−	17.8885i		−	0.995345i
$324$		0			0
$325$		0			0
$326$		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	$0.553834\pi$
$332$		−	53.6656i		−	2.94528i
$333$		0			0
$334$	−20.0000			−1.09435
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	25.1744	+	14.5344i	1.36931	+	0.790569i
$339$		0			0
$340$	−15.0000	−	25.9808i	−0.813489	−	1.40900i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	25.0000	−	43.3013i	1.34401	−	2.32789i
$347$	30.9839	+	17.8885i	1.66330	+	0.960307i	0.971123	+	0.238581i	$0.0766823\pi$
$347$	30.9839	+	17.8885i	1.66330	+	0.960307i	0.692179	+	0.721726i	$0.256651\pi$
$348$		0			0
$349$	17.0000	+	29.4449i	0.909989	+	1.57615i	0.814076	+	0.580758i	$0.197244\pi$
$349$	17.0000	+	29.4449i	0.909989	+	1.57615i	0.0959126	+	0.995390i	$0.469423\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	27.1109	−	15.6525i	1.44297	−	0.833097i	0.444920	−	0.895570i	$-0.353232\pi$
$353$	27.1109	−	15.6525i	1.44297	−	0.833097i	0.998047	+	0.0624731i	$0.0198987\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	42.6028	−	24.5967i	2.23915	−	1.29278i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$			8.94427i			0.466252i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$	−10.0000	−	17.3205i	−0.515711	−	0.893237i
$377$		0			0
$378$		0			0
$379$	−4.00000			−0.205466			−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000			−0.205466			−0.102733	+	0.994709i	$0.532759\pi$
$380$	23.2379	−	13.4164i	1.19208	−	0.688247i
$381$		0			0
$382$		0			0
$383$	7.74597	+	4.47214i	0.395800	+	0.228515i	0.684670	−	0.728853i	$-0.259946\pi$
$383$	7.74597	+	4.47214i	0.395800	+	0.228515i	−0.288870	+	0.957368i	$0.593280\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$	−20.0000	+	34.6410i	−1.01144	+	1.75187i
$392$	−13.5554	−	7.82624i	−0.684653	−	0.395285i
$393$		0			0
$394$	−5.00000	−	8.66025i	−0.251896	−	0.436297i
$395$		−	35.7771i		−	1.80014i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	30.9839	−	17.8885i	1.55308	−	0.896672i
$399$		0			0
$400$	−2.50000	+	4.33013i	−0.125000	+	0.216506i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	0.341994	+	0.939702i	$0.388898\pi$
$409$	−13.0000	+	22.5167i	−0.642809	+	1.11338i	−0.984803	+	0.173675i	$0.944436\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−40.0000			−1.96352
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$419$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−19.0000	−	32.9090i	−0.926003	−	1.60388i	−0.789940	−	0.613185i	$-0.789888\pi$
$421$	−19.0000	−	32.9090i	−0.926003	−	1.60388i	−0.136064	−	0.990700i	$-0.543445\pi$
$422$			62.6099i			3.04780i
$423$		0			0
$424$	−10.0000			−0.485643
$425$	−19.3649	+	11.1803i	−0.939336	+	0.542326i
$426$		0			0
$427$		0			0
$428$	46.4758	+	26.8328i	2.24649	+	1.29701i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	21.0000	−	36.3731i	1.00572	−	1.74195i
$437$	−30.9839	−	17.8885i	−1.48216	−	0.855725i
$438$		0			0
$439$	8.00000	+	13.8564i	0.381819	+	0.661330i	0.991322	−	0.131453i	$-0.0419644\pi$
$439$	8.00000	+	13.8564i	0.381819	+	0.661330i	−0.609503	+	0.792784i	$0.708631\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.4919	−	8.94427i	0.736044	−	0.424955i	−0.0845852	−	0.996416i	$-0.526957\pi$
$443$	15.4919	−	8.94427i	0.736044	−	0.424955i	0.820629	+	0.571461i	$0.193623\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−11.6190	+	6.70820i	−0.546509	+	0.315527i
$453$		0			0
$454$	−20.0000	+	34.6410i	−0.938647	+	1.62578i
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		−	58.1378i		−	2.71660i
$459$		0			0
$460$	−60.0000			−2.79751
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$		0			0
$465$		0			0
$466$	25.0000	+	43.3013i	1.15810	+	2.00589i
$467$		−	35.7771i		−	1.65557i	−0.561048	−	0.827783i	$-0.689602\pi$
$467$		−	35.7771i		−	1.65557i	0.561048	−	0.827783i	$-0.310398\pi$
$468$		0			0
$469$		0			0
$470$	−38.7298	+	22.3607i	−1.78647	+	1.03142i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−10.0000	−	17.3205i	−0.458831	−	0.794719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$	3.87298	+	2.23607i	0.176410	+	0.101850i
$483$		0			0
$484$	−16.5000	−	28.5788i	−0.750000	−	1.29904i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	3.87298	−	2.23607i	0.175322	−	0.101222i
$489$		0			0
$490$	−17.5000	+	30.3109i	−0.790569	+	1.36931i
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−8.00000			−0.359211
$497$		0			0
$498$		0			0
$499$	−22.0000	+	38.1051i	−0.984855	+	1.70582i	−0.342277	+	0.939599i	$0.611198\pi$
$499$	−22.0000	+	38.1051i	−0.984855	+	1.70582i	−0.642578	+	0.766220i	$0.722135\pi$
$500$	−29.0474	−	16.7705i	−1.29904	−	0.750000i
$501$		0			0
$502$		0			0
$503$			44.7214i			1.99403i	0.0772283	+	0.997013i	$0.475393\pi$
$503$			44.7214i			1.99403i	−0.0772283	+	0.997013i	$0.524607\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		−	11.1803i		−	0.494106i
$513$		0			0
$514$	70.0000			3.08757
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	10.0000	−	17.3205i	0.436021	−	0.755210i
$527$	−30.9839	−	17.8885i	−1.34968	−	0.779237i
$528$		0			0
$529$	28.5000	+	49.3634i	1.23913	+	2.14624i
$530$			22.3607i			0.971286i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	20.0000	−	34.6410i	0.864675	−	1.49766i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$	−61.9677	+	35.7771i	−2.66174	+	1.53676i
$543$		0			0
$544$	15.0000	−	25.9808i	0.643120	−	1.11392i
$545$	−27.1109	−	15.6525i	−1.16130	−	0.670478i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$			67.0820i			2.86560i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−6.00000	−	10.3923i	−0.254457	−	0.440732i
$557$		−	22.3607i		−	0.947452i	−0.880672	−	0.473726i	$-0.842909\pi$
$557$		−	22.3607i		−	0.947452i	0.880672	−	0.473726i	$-0.157091\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	30.9839	+	17.8885i	1.30581	+	0.753912i	0.981395	−	0.192001i	$-0.0614977\pi$
$563$	30.9839	+	17.8885i	1.30581	+	0.753912i	0.324420	+	0.945913i	$0.394831\pi$
$564$		0			0
$565$	5.00000	+	8.66025i	0.210352	+	0.364340i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	14.0000	−	24.2487i	0.585882	−	1.01478i	−0.408883	−	0.912587i	$-0.634082\pi$
$571$	14.0000	−	24.2487i	0.585882	−	1.01478i	0.994765	−	0.102190i	$-0.0325850\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			44.7214i			1.86501i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	5.80948	−	3.35410i	0.241642	−	0.139512i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	70.0000			2.89167
$587$	15.4919	−	8.94427i	0.639421	−	0.369170i	−0.144971	−	0.989436i	$-0.546309\pi$
$587$	15.4919	−	8.94427i	0.639421	−	0.369170i	0.784391	+	0.620266i	$0.212975\pi$
$588$		0			0
$589$	16.0000	−	27.7128i	0.659269	−	1.14189i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			4.47214i			0.183649i	0.995775	+	0.0918243i	$0.0292698\pi$
$593$			4.47214i			0.183649i	−0.995775	+	0.0918243i	$0.970730\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$	−19.0000	−	32.9090i	−0.775026	−	1.34238i	−0.934780	−	0.355228i	$-0.884403\pi$
$601$	−19.0000	−	32.9090i	−0.775026	−	1.34238i	0.159754	−	0.987157i	$-0.448930\pi$
$602$		0			0
$603$		0			0
$604$	−24.0000			−0.976546
$605$	−21.3014	+	12.2984i	−0.866025	+	0.500000i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$	23.2379	+	13.4164i	0.942421	+	0.544107i
$609$		0			0
$610$	−5.00000	−	8.66025i	−0.202444	−	0.350643i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.6028	+	24.5967i	1.71512	+	0.990228i	0.927292	+	0.374338i	$0.122130\pi$
$617$	42.6028	+	24.5967i	1.71512	+	0.990228i	0.787833	+	0.615889i	$0.211203\pi$
$618$		0			0
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.0376891	−	0.999290i	$-0.512000\pi$
$620$		−	53.6656i		−	2.15526i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$	−30.9839	+	17.8885i	−1.23247	+	0.711568i
$633$		0			0
$634$	25.0000	−	43.3013i	0.992877	−	1.71971i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	35.0000			1.38350
$641$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$641$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$642$		0			0
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$	20.0000	+	34.6410i	0.786889	+	1.36293i
$647$			44.7214i			1.75818i	0.476658	+	0.879089i	$0.341848\pi$
$647$			44.7214i			1.75818i	−0.476658	+	0.879089i	$0.658152\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	42.6028	+	24.5967i	1.66718	+	0.962545i	0.969149	+	0.246474i	$0.0792721\pi$
$653$	42.6028	+	24.5967i	1.66718	+	0.962545i	0.698028	+	0.716071i	$0.254061\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	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	$0.0259377\pi$
$662$	−54.2218	−	31.3050i	−2.10739	−	1.21670i
$663$		0			0
$664$	20.0000	+	34.6410i	0.776151	+	1.34433i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	23.2379	−	13.4164i	0.899101	−	0.519096i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$	−39.0000			−1.50000
$677$	27.1109	−	15.6525i	1.04196	−	0.601574i	0.121569	−	0.992583i	$-0.461207\pi$
$677$	27.1109	−	15.6525i	1.04196	−	0.601574i	0.920387	+	0.391009i	$0.127874\pi$
$678$		0			0
$679$		0			0
$680$	19.3649	+	11.1803i	0.742611	+	0.428746i
$681$		0			0
$682$		0			0
$683$		−	35.7771i		−	1.36897i	−0.729026	−	0.684486i	$-0.760027\pi$
$683$		−	35.7771i		−	1.36897i	0.729026	−	0.684486i	$-0.239973\pi$
$684$		0			0
$685$	50.0000			1.91040
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	26.0000	+	45.0333i	0.989087	+	1.71315i	0.622139	+	0.782907i	$0.286264\pi$
$691$	26.0000	+	45.0333i	0.989087	+	1.71315i	0.366947	+	0.930242i	$0.380403\pi$
$692$			67.0820i			2.55008i
$693$		0			0
$694$	−80.0000			−3.03676
$695$	−7.74597	+	4.47214i	−0.293821	+	0.169638i
$696$		0			0
$697$		0			0
$698$	−65.8407	−	38.0132i	−2.49211	−	1.43882i
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−35.0000	+	60.6218i	−1.31724	+	2.28153i
$707$		0			0
$708$		0			0
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	0.511683	−	0.859174i	$-0.329022\pi$
$709$	−13.0000	−	22.5167i	−0.488225	−	0.845631i	−0.999908	+	0.0135434i	$0.995689\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−61.9677	+	35.7771i	−2.32071	+	1.33986i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	5.80948	−	3.35410i	0.216206	−	0.124827i
$723$		0			0
$724$	−33.0000	+	57.1577i	−1.22644	+	2.12425i
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$	−30.0000	−	51.9615i	−1.10581	−	1.91533i
$737$		0			0
$738$		0			0
$739$	−4.00000			−0.147142			−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000			−0.147142			−0.0735712	+	0.997290i	$0.523440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−38.7298	−	22.3607i	−1.42086	−	0.820334i	−0.424487	−	0.905434i	$-0.639546\pi$
$743$	−38.7298	−	22.3607i	−1.42086	−	0.820334i	−0.996372	+	0.0851001i	$0.972879\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	−0.929731	−	0.368238i	$-0.879961\pi$
$751$	−4.00000	+	6.92820i	−0.145962	+	0.252814i	0.783769	+	0.621052i	$0.213294\pi$
$752$	−7.74597	−	4.47214i	−0.282466	−	0.163082i
$753$		0			0
$754$		0			0
$755$			17.8885i			0.651031i
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	7.74597	−	4.47214i	0.281346	−	0.162435i
$759$		0			0
$760$	−10.0000	+	17.3205i	−0.362738	+	0.628281i
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−20.0000			−0.722629
$767$		0			0
$768$		0			0
$769$	23.0000	−	39.8372i	0.829401	−	1.43657i	−0.0691074	−	0.997609i	$-0.522015\pi$
$769$	23.0000	−	39.8372i	0.829401	−	1.43657i	0.898509	−	0.438956i	$-0.144652\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			4.47214i			0.160852i	0.996761	+	0.0804258i	$0.0256280\pi$
$773$			4.47214i			0.160852i	−0.996761	+	0.0804258i	$0.974372\pi$
$774$		0			0
$775$	−40.0000			−1.43684
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		−	89.4427i		−	3.19847i
$783$		0			0
$784$	−7.00000			−0.250000
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	11.6190	+	6.70820i	0.413908	+	0.238970i
$789$		0			0
$790$	40.0000	+	69.2820i	1.42314	+	2.46494i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−24.0000	+	41.5692i	−0.850657	+	1.47338i
$797$	42.6028	+	24.5967i	1.50907	+	0.871262i	0.999944	+	0.0105689i	$0.00336426\pi$
$797$	42.6028	+	24.5967i	1.50907	+	0.871262i	0.509125	+	0.860693i	$0.329969\pi$
$798$		0			0
$799$	−20.0000	−	34.6410i	−0.707549	−	1.22551i
$800$		−	33.5410i		−	1.18585i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	−52.0000			−1.82597			−0.912983	−	0.407997i	$-0.866228\pi$
$811$	−52.0000			−1.82597			−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	58.1378i		−	2.03274i
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	35.7771i		−	1.24409i	−0.782981	−	0.622046i	$-0.786302\pi$
$827$		−	35.7771i		−	1.24409i	0.782981	−	0.622046i	$-0.213698\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$	77.4597	−	44.7214i	2.68866	−	1.55230i
$831$		0			0
$832$		0			0
$833$	−27.1109	−	15.6525i	−0.939336	−	0.542326i
$834$		0			0
$835$	−10.0000	−	17.3205i	−0.346064	−	0.599401i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$	73.5867	+	42.4853i	2.53596	+	1.46414i
$843$		0			0
$844$	−42.0000	−	72.7461i	−1.44570	−	2.50403i
$845$			29.0689i			1.00000i
$846$		0			0
$847$		0			0
$848$	−3.87298	+	2.23607i	−0.132999	+	0.0767869i
$849$		0			0
$850$	25.0000	−	43.3013i	0.857493	−	1.48522i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$	−40.0000			−1.36717
$857$	50.3488	−	29.0689i	1.71988	−	0.992974i	0.800775	−	0.598965i	$-0.204421\pi$
$857$	50.3488	−	29.0689i	1.71988	−	0.992974i	0.919107	−	0.394009i	$-0.128912\pi$
$858$		0			0
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	0.196887	+	0.980426i	$0.436917\pi$
$859$	−22.0000	+	38.1051i	−0.750630	+	1.30013i	−0.947518	+	0.319704i	$0.896417\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44.7214i			1.52233i	0.648557	+	0.761166i	$0.275373\pi$
$863$			44.7214i			1.52233i	−0.648557	+	0.761166i	$0.724627\pi$
$864$		0			0
$865$	50.0000			1.70005
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$			31.3050i			1.06012i
$873$		0			0
$874$	80.0000			2.70604
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	−30.9839	−	17.8885i	−1.04565	−	0.603709i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−20.0000	+	34.6410i	−0.671913	+	1.16379i
$887$	7.74597	+	4.47214i	0.260084	+	0.150160i	0.624373	−	0.781126i	$-0.285355\pi$
$887$	7.74597	+	4.47214i	0.260084	+	0.150160i	−0.364289	+	0.931286i	$0.618688\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	30.9839	−	17.8885i	1.03684	−	0.598617i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−20.0000			−0.666297
$902$		0			0
$903$		0			0
$904$	5.00000	−	8.66025i	0.166298	−	0.288036i
$905$	42.6028	+	24.5967i	1.41617	+	0.817624i
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		−	53.6656i		−	1.78096i
$909$		0			0
$910$		0			0
$911$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$911$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	39.0000	+	67.5500i	1.28860	+	2.23191i
$917$		0			0
$918$		0			0
$919$	56.0000			1.84727			0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000			1.84727			0.923635	+	0.383274i	$0.125203\pi$
$920$	38.7298	−	22.3607i	1.27688	−	0.737210i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	14.0000	−	24.2487i	0.458831	−	0.794719i
$932$	−58.0948	−	33.5410i	−1.90296	−	1.09867i
$933$		0			0
$934$	40.0000	+	69.2820i	1.30884	+	2.26698i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	30.0000	−	51.9615i	0.978492	−	1.69480i
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.4919	−	8.94427i	0.503420	−	0.290650i	−0.226705	−	0.973964i	$-0.572795\pi$
$947$	15.4919	−	8.94427i	0.503420	−	0.290650i	0.730125	+	0.683314i	$0.239462\pi$
$948$		0			0
$949$		0			0
$950$	38.7298	+	22.3607i	1.25656	+	0.725476i
$951$		0			0
$952$		0			0
$953$			58.1378i			1.88327i	0.336640	+	0.941634i	$0.390710\pi$
$953$			58.1378i			1.88327i	−0.336640	+	0.941634i	$0.609290\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$		0			0
$963$		0			0
$964$	−6.00000			−0.193247
$965$		0			0
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$	21.3014	+	12.2984i	0.684653	+	0.395285i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.00000	−	1.73205i	0.0320092	−	0.0554416i
$977$	−3.87298	−	2.23607i	−0.123908	−	0.0715382i	0.436765	−	0.899576i	$-0.356124\pi$
$977$	−3.87298	−	2.23607i	−0.123908	−	0.0715382i	−0.560673	+	0.828037i	$0.689457\pi$
$978$		0			0
$979$		0			0
$980$		−	46.9574i		−	1.50000i
$981$		0			0
$982$		0			0
$983$	−54.2218	+	31.3050i	−1.72941	+	0.998473i	−0.837081	+	0.547079i	$0.815740\pi$
$983$	−54.2218	+	31.3050i	−1.72941	+	0.998473i	−0.892325	+	0.451394i	$0.850927\pi$
$984$		0			0
$985$	5.00000	−	8.66025i	0.159313	−	0.275939i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$	46.4758	−	26.8328i	1.47561	−	0.851943i
$993$		0			0
$994$		0			0
$995$	30.9839	+	17.8885i	0.982255	+	0.567105i
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		−	98.3870i		−	3.11439i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.j.c.109.1		4
3.2	odd	2	inner	405.2.j.c.109.2		4
5.4	even	2	inner	405.2.j.c.109.2		4
9.2	odd	6	inner	405.2.j.c.379.1		4
9.4	even	3		45.2.b.a.19.1	✓	2
9.5	odd	6		45.2.b.a.19.2	yes	2
9.7	even	3	inner	405.2.j.c.379.2		4
15.14	odd	2	CM	405.2.j.c.109.1		4
36.23	even	6		720.2.f.d.289.1		2
36.31	odd	6		720.2.f.d.289.2		2
45.4	even	6		45.2.b.a.19.2	yes	2
45.13	odd	12		225.2.a.f.1.1		2
45.14	odd	6		45.2.b.a.19.1	✓	2
45.22	odd	12		225.2.a.f.1.2		2
45.23	even	12		225.2.a.f.1.2		2
45.29	odd	6	inner	405.2.j.c.379.2		4
45.32	even	12		225.2.a.f.1.1		2
45.34	even	6	inner	405.2.j.c.379.1		4
63.13	odd	6		2205.2.d.a.1324.1		2
63.41	even	6		2205.2.d.a.1324.2		2
72.5	odd	6		2880.2.f.k.1729.2		2
72.13	even	6		2880.2.f.k.1729.1		2
72.59	even	6		2880.2.f.j.1729.2		2
72.67	odd	6		2880.2.f.j.1729.1		2
180.23	odd	12		3600.2.a.bs.1.1		2
180.59	even	6		720.2.f.d.289.2		2
180.67	even	12		3600.2.a.bs.1.1		2
180.103	even	12		3600.2.a.bs.1.2		2
180.139	odd	6		720.2.f.d.289.1		2
180.167	odd	12		3600.2.a.bs.1.2		2
315.104	even	6		2205.2.d.a.1324.1		2
315.139	odd	6		2205.2.d.a.1324.2		2
360.59	even	6		2880.2.f.j.1729.1		2
360.139	odd	6		2880.2.f.j.1729.2		2
360.149	odd	6		2880.2.f.k.1729.1		2
360.229	even	6		2880.2.f.k.1729.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.b.a.19.1	✓	2	9.4	even	3
45.2.b.a.19.1	✓	2	45.14	odd	6
45.2.b.a.19.2	yes	2	9.5	odd	6
45.2.b.a.19.2	yes	2	45.4	even	6
225.2.a.f.1.1		2	45.13	odd	12
225.2.a.f.1.1		2	45.32	even	12
225.2.a.f.1.2		2	45.22	odd	12
225.2.a.f.1.2		2	45.23	even	12
405.2.j.c.109.1		4	1.1	even	1	trivial
405.2.j.c.109.1		4	15.14	odd	2	CM
405.2.j.c.109.2		4	3.2	odd	2	inner
405.2.j.c.109.2		4	5.4	even	2	inner
405.2.j.c.379.1		4	9.2	odd	6	inner
405.2.j.c.379.1		4	45.34	even	6	inner
405.2.j.c.379.2		4	9.7	even	3	inner
405.2.j.c.379.2		4	45.29	odd	6	inner
720.2.f.d.289.1		2	36.23	even	6
720.2.f.d.289.1		2	180.139	odd	6
720.2.f.d.289.2		2	36.31	odd	6
720.2.f.d.289.2		2	180.59	even	6
2205.2.d.a.1324.1		2	63.13	odd	6
2205.2.d.a.1324.1		2	315.104	even	6
2205.2.d.a.1324.2		2	63.41	even	6
2205.2.d.a.1324.2		2	315.139	odd	6
2880.2.f.j.1729.1		2	72.67	odd	6
2880.2.f.j.1729.1		2	360.59	even	6
2880.2.f.j.1729.2		2	72.59	even	6
2880.2.f.j.1729.2		2	360.139	odd	6
2880.2.f.k.1729.1		2	72.13	even	6
2880.2.f.k.1729.1		2	360.149	odd	6
2880.2.f.k.1729.2		2	72.5	odd	6
2880.2.f.k.1729.2		2	360.229	even	6
3600.2.a.bs.1.1		2	180.23	odd	12
3600.2.a.bs.1.1		2	180.67	even	12
3600.2.a.bs.1.2		2	180.103	even	12
3600.2.a.bs.1.2		2	180.167	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.j (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 + 1.11803i) q^{2} +(1.50000 - 2.59808i) q^{4} +(-1.93649 - 1.11803i) q^{5} +2.23607i q^{8} +O(q^{10})\)	\(q+(-1.93649 + 1.11803i) q^{2} +(1.50000 - 2.59808i) q^{4} +(-1.93649 - 1.11803i) q^{5} +2.23607i q^{8} +5.00000 q^{10} +(0.500000 + 0.866025i) q^{16} +4.47214i q^{17} -4.00000 q^{19} +(-5.80948 + 3.35410i) q^{20} +(7.74597 + 4.47214i) q^{23} +(2.50000 + 4.33013i) q^{25} +(-4.00000 + 6.92820i) q^{31} +(-5.80948 - 3.35410i) q^{32} +(-5.00000 - 8.66025i) q^{34} +(7.74597 - 4.47214i) q^{38} +(2.50000 - 4.33013i) q^{40} -20.0000 q^{46} +(-7.74597 + 4.47214i) q^{47} +(-3.50000 + 6.06218i) q^{49} +(-9.68246 - 5.59017i) q^{50} +4.47214i q^{53} +(-1.00000 - 1.73205i) q^{61} -17.8885i q^{62} +13.0000 q^{64} +(11.6190 + 6.70820i) q^{68} +(-6.00000 + 10.3923i) q^{76} +(8.00000 + 13.8564i) q^{79} -2.23607i q^{80} +(15.4919 - 8.94427i) q^{83} +(5.00000 - 8.66025i) q^{85} +(23.2379 - 13.4164i) q^{92} +(10.0000 - 17.3205i) q^{94} +(7.74597 + 4.47214i) q^{95} -15.6525i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4}+O(q^{10}) \) 4 * q + 6 * q^4	\( 4 q + 6 q^{4} + 20 q^{10} + 2 q^{16} - 16 q^{19} + 10 q^{25} - 16 q^{31} - 20 q^{34} + 10 q^{40} - 80 q^{46} - 14 q^{49} - 4 q^{61} + 52 q^{64} - 24 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100}) \) 4 * q + 6 * q^4 + 20 * q^10 + 2 * q^16 - 16 * q^19 + 10 * q^25 - 16 * q^31 - 20 * q^34 + 10 * q^40 - 80 * q^46 - 14 * q^49 - 4 * q^61 + 52 * q^64 - 24 * q^76 + 32 * q^79 + 20 * q^85 + 40 * q^94

Introduction

L-functions

Modular forms

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Database

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