LMFDB - Embedded newform 300.2.e.a.251.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(251,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("300.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$300 = 2^{2} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	300.e (of order $2$ , degree $1$ , 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:	$2.39551206064$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + x^{2} + 4$ x^4 + x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			251.3
Root			$0.866025 - 1.11803i$ of defining polynomial
Character	$\chi$	$=$	300.251
Dual form			300.2.e.a.251.4

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 1.11803i) q^{2} +1.73205 q^{3} +(-0.500000 - 1.93649i) q^{4} +(1.50000 - 1.93649i) q^{6} +(-2.59808 - 1.11803i) q^{8} +3.00000 q^{9} +(-0.866025 - 3.35410i) q^{12} +(-3.50000 + 1.93649i) q^{16} -4.47214i q^{17} +(2.59808 - 3.35410i) q^{18} +7.74597i q^{19} -3.46410 q^{23} +(-4.50000 - 1.93649i) q^{24} +5.19615 q^{27} +7.74597i q^{31} +(-0.866025 + 5.59017i) q^{32} +(-5.00000 - 3.87298i) q^{34} +(-1.50000 - 5.80948i) q^{36} +(8.66025 + 6.70820i) q^{38} +(-3.00000 + 3.87298i) q^{46} -10.3923 q^{47} +(-6.06218 + 3.35410i) q^{48} +7.00000 q^{49} -7.74597i q^{51} -4.47214i q^{53} +(4.50000 - 5.80948i) q^{54} +13.4164i q^{57} -2.00000 q^{61} +(8.66025 + 6.70820i) q^{62} +(5.50000 + 5.80948i) q^{64} +(-8.66025 + 2.23607i) q^{68} -6.00000 q^{69} +(-7.79423 - 3.35410i) q^{72} +(15.0000 - 3.87298i) q^{76} -7.74597i q^{79} +9.00000 q^{81} +3.46410 q^{83} +(1.73205 + 6.70820i) q^{92} +13.4164i q^{93} +(-9.00000 + 11.6190i) q^{94} +(-1.50000 + 9.68246i) q^{96} +(6.06218 - 7.82624i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{4} + 6 q^{6} + 12 q^{9} - 14 q^{16} - 18 q^{24} - 20 q^{34} - 6 q^{36} - 12 q^{46} + 28 q^{49} + 18 q^{54} - 8 q^{61} + 22 q^{64} - 24 q^{69} + 60 q^{76} + 36 q^{81} - 36 q^{94} - 6 q^{96}+O(q^{100})$ 4 * q - 2 * q^4 + 6 * q^6 + 12 * q^9 - 14 * q^16 - 18 * q^24 - 20 * q^34 - 6 * q^36 - 12 * q^46 + 28 * q^49 + 18 * q^54 - 8 * q^61 + 22 * q^64 - 24 * q^69 + 60 * q^76 + 36 * q^81 - 36 * q^94 - 6 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$ .

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0.866025	−	1.11803i	0.612372	−	0.790569i
$2$	0.866025	−	1.11803i	0.612372	−	0.790569i
$3$	1.73205			1.00000
$3$	1.73205			1.00000
$4$	−0.500000	−	1.93649i	−0.250000	−	0.968246i
$5$		0			0
$5$		0			0
$6$	1.50000	−	1.93649i	0.612372	−	0.790569i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	−2.59808	−	1.11803i	−0.918559	−	0.395285i
$9$	3.00000			1.00000
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.866025	−	3.35410i	−0.250000	−	0.968246i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	−3.50000	+	1.93649i	−0.875000	+	0.484123i
$17$		−	4.47214i		−	1.08465i	−0.840168	−	0.542326i	$-0.817544\pi$
$17$		−	4.47214i		−	1.08465i	0.840168	−	0.542326i	$-0.182456\pi$
$18$	2.59808	−	3.35410i	0.612372	−	0.790569i
$19$			7.74597i			1.77705i	0.458831	+	0.888523i	$0.348268\pi$
$19$			7.74597i			1.77705i	−0.458831	+	0.888523i	$0.651732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.46410			−0.722315			−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410			−0.722315			−0.361158	+	0.932505i	$0.617618\pi$
$24$	−4.50000	−	1.93649i	−0.918559	−	0.395285i
$25$		0			0
$26$		0			0
$27$	5.19615			1.00000
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			7.74597i			1.39122i	0.718421	+	0.695608i	$0.244865\pi$
$31$			7.74597i			1.39122i	−0.718421	+	0.695608i	$0.755135\pi$
$32$	−0.866025	+	5.59017i	−0.153093	+	0.988212i
$33$		0			0
$34$	−5.00000	−	3.87298i	−0.857493	−	0.664211i
$35$		0			0
$36$	−1.50000	−	5.80948i	−0.250000	−	0.968246i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	8.66025	+	6.70820i	1.40488	+	1.08821i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−3.00000	+	3.87298i	−0.442326	+	0.571040i
$47$	−10.3923			−1.51587			−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923			−1.51587			−0.757937	+	0.652328i	$0.773792\pi$
$48$	−6.06218	+	3.35410i	−0.875000	+	0.484123i
$49$	7.00000			1.00000
$50$		0			0
$51$		−	7.74597i		−	1.08465i
$52$		0			0
$53$		−	4.47214i		−	0.614295i	−0.951662	−	0.307148i	$-0.900625\pi$
$53$		−	4.47214i		−	0.614295i	0.951662	−	0.307148i	$-0.0993745\pi$
$54$	4.50000	−	5.80948i	0.612372	−	0.790569i
$55$		0			0
$56$		0			0
$57$			13.4164i			1.77705i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.66025	+	6.70820i	1.09985	+	0.851943i
$63$		0			0
$64$	5.50000	+	5.80948i	0.687500	+	0.726184i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$	−8.66025	+	2.23607i	−1.05021	+	0.271163i
$69$	−6.00000			−0.722315
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	−7.79423	−	3.35410i	−0.918559	−	0.395285i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$		0			0
$76$	15.0000	−	3.87298i	1.72062	−	0.444262i
$77$		0			0
$78$		0			0
$79$		−	7.74597i		−	0.871489i	−0.900070	−	0.435745i	$-0.856485\pi$
$79$		−	7.74597i		−	0.871489i	0.900070	−	0.435745i	$-0.143515\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$	3.46410			0.380235			0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410			0.380235			0.190117	+	0.981761i	$0.439113\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1.73205	+	6.70820i	0.180579	+	0.699379i
$93$			13.4164i			1.39122i
$94$	−9.00000	+	11.6190i	−0.928279	+	1.19840i
$95$		0			0
$96$	−1.50000	+	9.68246i	−0.153093	+	0.988212i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	6.06218	−	7.82624i	0.612372	−	0.790569i
$99$		0			0
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$	−8.66025	−	6.70820i	−0.857493	−	0.664211i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−5.00000	−	3.87298i	−0.485643	−	0.376177i
$107$	10.3923			1.00466			0.502331	−	0.864675i	$-0.332476\pi$
$107$	10.3923			1.00466			0.502331	+	0.864675i	$0.332476\pi$
$108$	−2.59808	−	10.0623i	−0.250000	−	0.968246i
$109$	−14.0000			−1.34096			−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000			−1.34096			−0.670478	+	0.741929i	$0.733911\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.47214i			0.420703i	0.977626	+	0.210352i	$0.0674609\pi$
$113$			4.47214i			0.420703i	−0.977626	+	0.210352i	$0.932539\pi$
$114$	15.0000	+	11.6190i	1.40488	+	1.08821i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$	−1.73205	+	2.23607i	−0.156813	+	0.202444i
$123$		0			0
$124$	15.0000	−	3.87298i	1.34704	−	0.347804i
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	11.2583	−	1.11803i	0.995105	−	0.0988212i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−5.00000	+	11.6190i	−0.428746	+	0.996317i
$137$		−	22.3607i		−	1.91040i	−0.295958	−	0.955201i	$-0.595639\pi$
$137$		−	22.3607i		−	1.91040i	0.295958	−	0.955201i	$-0.404361\pi$
$138$	−5.19615	+	6.70820i	−0.442326	+	0.571040i
$139$		−	23.2379i		−	1.97101i	−0.169638	−	0.985506i	$-0.554260\pi$
$139$		−	23.2379i		−	1.97101i	0.169638	−	0.985506i	$-0.445740\pi$
$140$		0			0
$141$	−18.0000			−1.51587
$142$		0			0
$143$		0			0
$144$	−10.5000	+	5.80948i	−0.875000	+	0.484123i
$145$		0			0
$146$		0			0
$147$	12.1244			1.00000
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		−	23.2379i		−	1.89107i	−0.325515	−	0.945537i	$-0.605538\pi$
$151$		−	23.2379i		−	1.89107i	0.325515	−	0.945537i	$-0.394462\pi$
$152$	8.66025	−	20.1246i	0.702439	−	1.63232i
$153$		−	13.4164i		−	1.08465i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−8.66025	−	6.70820i	−0.688973	−	0.533676i
$159$		−	7.74597i		−	0.614295i
$160$		0			0
$161$		0			0
$162$	7.79423	−	10.0623i	0.612372	−	0.790569i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	3.00000	−	3.87298i	0.232845	−	0.300602i
$167$	−24.2487			−1.87642			−0.938211	−	0.346064i	$-0.887518\pi$
$167$	−24.2487			−1.87642			−0.938211	+	0.346064i	$0.887518\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$			23.2379i			1.77705i
$172$		0			0
$173$		−	22.3607i		−	1.70005i	−0.526742	−	0.850026i	$-0.676586\pi$
$173$		−	22.3607i		−	1.70005i	0.526742	−	0.850026i	$-0.323414\pi$
$174$		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	$-0.195291\pi$
$181$	22.0000			1.63525			0.817624	+	0.575753i	$0.195291\pi$
$182$		0			0
$183$	−3.46410			−0.256074
$184$	9.00000	+	3.87298i	0.663489	+	0.285520i
$185$		0			0
$186$	15.0000	+	11.6190i	1.09985	+	0.851943i
$187$		0			0
$188$	5.19615	+	20.1246i	0.378968	+	1.46774i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	9.52628	+	10.0623i	0.687500	+	0.726184i
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	−3.50000	−	13.5554i	−0.250000	−	0.968246i
$197$			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
$199$			23.2379i			1.64729i	0.567105	+	0.823646i	$0.308063\pi$
$199$			23.2379i			1.64729i	−0.567105	+	0.823646i	$0.691937\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−15.0000	+	3.87298i	−1.05021	+	0.271163i
$205$		0			0
$206$		0			0
$207$	−10.3923			−0.722315
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	7.74597i		−	0.533254i	−0.963800	−	0.266627i	$-0.914091\pi$
$211$		−	7.74597i		−	0.533254i	0.963800	−	0.266627i	$-0.0859092\pi$
$212$	−8.66025	+	2.23607i	−0.594789	+	0.153574i
$213$		0			0
$214$	9.00000	−	11.6190i	0.615227	−	0.794255i
$215$		0			0
$216$	−13.5000	−	5.80948i	−0.918559	−	0.395285i
$217$		0			0
$218$	−12.1244	+	15.6525i	−0.821165	+	1.06012i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$		0			0
$226$	5.00000	+	3.87298i	0.332595	+	0.257627i
$227$	24.2487			1.60944			0.804722	−	0.593652i	$-0.202314\pi$
$227$	24.2487			1.60944			0.804722	+	0.593652i	$0.202314\pi$
$228$	25.9808	−	6.70820i	1.72062	−	0.444262i
$229$	26.0000			1.71813			0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000			1.71813			0.859064	+	0.511868i	$0.171046\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.3607i			1.46490i	0.680823	+	0.732448i	$0.261622\pi$
$233$			22.3607i			1.46490i	−0.680823	+	0.732448i	$0.738378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	13.4164i		−	0.871489i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$	−9.52628	+	12.2984i	−0.612372	+	0.790569i
$243$	15.5885			1.00000
$244$	1.00000	+	3.87298i	0.0640184	+	0.247942i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	8.66025	−	20.1246i	0.549927	−	1.27791i
$249$	6.00000			0.380235
$250$		0			0
$251$		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
$256$	8.50000	−	13.5554i	0.531250	−	0.847215i
$257$			31.3050i			1.95275i	0.216085	+	0.976375i	$0.430671\pi$
$257$			31.3050i			1.95275i	−0.216085	+	0.976375i	$0.569329\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−31.1769			−1.92245			−0.961225	−	0.275764i	$-0.911069\pi$
$263$	−31.1769			−1.92245			−0.961225	+	0.275764i	$0.911069\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$			7.74597i			0.470534i	0.971931	+	0.235267i	$0.0755965\pi$
$271$			7.74597i			0.470534i	−0.971931	+	0.235267i	$0.924404\pi$
$272$	8.66025	+	15.6525i	0.525105	+	0.949071i
$273$		0			0
$274$	−25.0000	−	19.3649i	−1.51031	−	1.16988i
$275$		0			0
$276$	3.00000	+	11.6190i	0.180579	+	0.699379i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−25.9808	−	20.1246i	−1.55822	−	1.20699i
$279$			23.2379i			1.39122i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−15.5885	+	20.1246i	−0.928279	+	1.19840i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2.59808	+	16.7705i	−0.153093	+	0.988212i
$289$	−3.00000			−0.176471
$290$		0			0
$291$		0			0
$292$		0			0
$293$			31.3050i			1.82885i	0.404750	+	0.914427i	$0.367359\pi$
$293$			31.3050i			1.82885i	−0.404750	+	0.914427i	$0.632641\pi$
$294$	10.5000	−	13.5554i	0.612372	−	0.790569i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−25.9808	−	20.1246i	−1.49502	−	1.15804i
$303$		0			0
$304$	−15.0000	−	27.1109i	−0.860309	−	1.55492i
$305$		0			0
$306$	−15.0000	−	11.6190i	−0.857493	−	0.664211i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$		0			0
$316$	−15.0000	+	3.87298i	−0.843816	+	0.217872i
$317$			22.3607i			1.25590i	0.778253	+	0.627950i	$0.216106\pi$
$317$			22.3607i			1.25590i	−0.778253	+	0.627950i	$0.783894\pi$
$318$	−8.66025	−	6.70820i	−0.485643	−	0.376177i
$319$		0			0
$320$		0			0
$321$	18.0000			1.00466
$322$		0			0
$323$	34.6410			1.92748
$324$	−4.50000	−	17.4284i	−0.250000	−	0.968246i
$325$		0			0
$326$		0			0
$327$	−24.2487			−1.34096
$328$		0			0
$329$		0			0
$330$		0			0
$331$			23.2379i			1.27727i	0.769510	+	0.638635i	$0.220501\pi$
$331$			23.2379i			1.27727i	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−1.73205	−	6.70820i	−0.0950586	−	0.368161i
$333$		0			0
$334$	−21.0000	+	27.1109i	−1.14907	+	1.48344i
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−11.2583	+	14.5344i	−0.612372	+	0.790569i
$339$			7.74597i			0.420703i
$340$		0			0
$341$		0			0
$342$	25.9808	+	20.1246i	1.40488	+	1.08821i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−25.0000	−	19.3649i	−1.34401	−	1.04106i
$347$	10.3923			0.557888			0.278944	−	0.960307i	$-0.410016\pi$
$347$	10.3923			0.557888			0.278944	+	0.960307i	$0.410016\pi$
$348$		0			0
$349$	34.0000			1.81998			0.909989	−	0.414632i	$-0.136090\pi$
$349$	34.0000			1.81998			0.909989	+	0.414632i	$0.136090\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	31.3050i		−	1.66619i	−0.553127	−	0.833097i	$-0.686565\pi$
$353$		−	31.3050i		−	1.66619i	0.553127	−	0.833097i	$-0.313435\pi$
$354$		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
$361$	−41.0000			−2.15789
$362$	19.0526	−	24.5967i	1.00138	−	1.29278i
$363$	−19.0526			−1.00000
$364$		0			0
$365$		0			0
$366$	−3.00000	+	3.87298i	−0.156813	+	0.202444i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	12.1244	−	6.70820i	0.632026	−	0.349689i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	25.9808	−	6.70820i	1.34704	−	0.347804i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	27.0000	+	11.6190i	1.39242	+	0.599202i
$377$		0			0
$378$		0			0
$379$			38.7298i			1.98942i	0.102733	+	0.994709i	$0.467241\pi$
$379$			38.7298i			1.98942i	−0.102733	+	0.994709i	$0.532759\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	38.1051			1.94708			0.973540	−	0.228515i	$-0.0733872\pi$
$383$	38.1051			1.94708			0.973540	+	0.228515i	$0.0733872\pi$
$384$	19.5000	−	1.93649i	0.995105	−	0.0988212i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$			15.4919i			0.783461i
$392$	−18.1865	−	7.82624i	−0.918559	−	0.395285i
$393$		0			0
$394$	5.00000	+	3.87298i	0.251896	+	0.195118i
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$	25.9808	+	20.1246i	1.30230	+	1.00876i
$399$		0			0
$400$		0			0
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−8.66025	+	20.1246i	−0.428746	+	0.996317i
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$		−	38.7298i		−	1.91040i
$412$		0			0
$413$		0			0
$414$	−9.00000	+	11.6190i	−0.442326	+	0.571040i
$415$		0			0
$416$		0			0
$417$		−	40.2492i		−	1.97101i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$	−8.66025	−	6.70820i	−0.421575	−	0.326550i
$423$	−31.1769			−1.51587
$424$	−5.00000	+	11.6190i	−0.242821	+	0.564266i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−5.19615	−	20.1246i	−0.251166	−	0.972760i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−18.1865	+	10.0623i	−0.875000	+	0.484123i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	7.00000	+	27.1109i	0.335239	+	1.29838i
$437$		−	26.8328i		−	1.28359i
$438$		0			0
$439$		−	38.7298i		−	1.84847i	−0.381819	−	0.924237i	$-0.624702\pi$
$439$		−	38.7298i		−	1.84847i	0.381819	−	0.924237i	$-0.375298\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$		0			0
$443$	−38.1051			−1.81043			−0.905214	−	0.424955i	$-0.860290\pi$
$443$	−38.1051			−1.81043			−0.905214	+	0.424955i	$0.860290\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	8.66025	−	2.23607i	0.407344	−	0.105176i
$453$		−	40.2492i		−	1.89107i
$454$	21.0000	−	27.1109i	0.985579	−	1.27238i
$455$		0			0
$456$	15.0000	−	34.8569i	0.702439	−	1.63232i
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$	22.5167	−	29.0689i	1.05213	−	1.35830i
$459$		−	23.2379i		−	1.08465i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	25.0000	+	19.3649i	1.15810	+	0.897062i
$467$	24.2487			1.12210			0.561048	−	0.827783i	$-0.310398\pi$
$467$	24.2487			1.12210			0.561048	+	0.827783i	$0.310398\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−15.0000	−	11.6190i	−0.688973	−	0.533676i
$475$		0			0
$476$		0			0
$477$		−	13.4164i		−	0.614295i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	1.73205	−	2.23607i	0.0788928	−	0.101850i
$483$		0			0
$484$	5.50000	+	21.3014i	0.250000	+	0.968246i
$485$		0			0
$486$	13.5000	−	17.4284i	0.612372	−	0.790569i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	5.19615	+	2.23607i	0.235219	+	0.101222i
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−15.0000	−	27.1109i	−0.673520	−	1.21731i
$497$		0			0
$498$	5.19615	−	6.70820i	0.232845	−	0.300602i
$499$			7.74597i			0.346757i	0.984855	+	0.173379i	$0.0554684\pi$
$499$			7.74597i			0.346757i	−0.984855	+	0.173379i	$0.944532\pi$
$500$		0			0
$501$	−42.0000			−1.87642
$502$		0			0
$503$	−3.46410			−0.154457			−0.0772283	−	0.997013i	$-0.524607\pi$
$503$	−3.46410			−0.154457			−0.0772283	+	0.997013i	$0.524607\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−22.5167			−1.00000
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−7.79423	−	21.2426i	−0.344459	−	0.938801i
$513$			40.2492i			1.77705i
$514$	35.0000	+	27.1109i	1.54378	+	1.19581i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	38.7298i		−	1.70005i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$	−27.0000	+	34.8569i	−1.17726	+	1.51983i
$527$	34.6410			1.50899
$528$		0			0
$529$	−11.0000			−0.478261
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.66025	+	6.70820i	0.371990	+	0.288142i
$543$	38.1051			1.63525
$544$	25.0000	+	3.87298i	1.07187	+	0.166053i
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−43.3013	+	11.1803i	−1.84974	+	0.477600i
$549$	−6.00000			−0.256074
$550$		0			0
$551$		0			0
$552$	15.5885	+	6.70820i	0.663489	+	0.285520i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−45.0000	+	11.6190i	−1.90843	+	0.492753i
$557$			22.3607i			0.947452i	0.880672	+	0.473726i	$0.157091\pi$
$557$			22.3607i			0.947452i	−0.880672	+	0.473726i	$0.842909\pi$
$558$	25.9808	+	20.1246i	1.09985	+	0.851943i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	31.1769			1.31395			0.656975	−	0.753912i	$-0.271836\pi$
$563$	31.1769			1.31395			0.656975	+	0.753912i	$0.271836\pi$
$564$	9.00000	+	34.8569i	0.378968	+	1.46774i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		−	38.7298i		−	1.62079i	−0.585882	−	0.810397i	$-0.699252\pi$
$571$		−	38.7298i		−	1.62079i	0.585882	−	0.810397i	$-0.300748\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	16.5000	+	17.4284i	0.687500	+	0.726184i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−2.59808	+	3.35410i	−0.108066	+	0.139512i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	35.0000	+	27.1109i	1.44584	+	1.11994i
$587$	−45.0333			−1.85872			−0.929362	−	0.369170i	$-0.879642\pi$
$587$	−45.0333			−1.85872			−0.929362	+	0.369170i	$0.879642\pi$
$588$	−6.06218	−	23.4787i	−0.250000	−	0.968246i
$589$	−60.0000			−2.47226
$590$		0			0
$591$			7.74597i			0.318626i
$592$		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
$597$			40.2492i			1.64729i
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$		0			0
$604$	−45.0000	+	11.6190i	−1.83102	+	0.472768i
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−43.3013	−	6.70820i	−1.75610	−	0.272054i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−25.9808	+	6.70820i	−1.05021	+	0.271163i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			49.1935i			1.98046i	0.139459	+	0.990228i	$0.455464\pi$
$617$			49.1935i			1.98046i	−0.139459	+	0.990228i	$0.544536\pi$
$618$		0			0
$619$		−	23.2379i		−	0.934010i	−0.884255	−	0.467005i	$-0.845333\pi$
$619$		−	23.2379i		−	0.934010i	0.884255	−	0.467005i	$-0.154667\pi$
$620$		0			0
$621$	−18.0000			−0.722315
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			38.7298i			1.54181i	0.636950	+	0.770905i	$0.280196\pi$
$631$			38.7298i			1.54181i	−0.636950	+	0.770905i	$0.719804\pi$
$632$	−8.66025	+	20.1246i	−0.344486	+	0.800514i
$633$		−	13.4164i		−	0.533254i
$634$	25.0000	+	19.3649i	0.992877	+	0.769079i
$635$		0			0
$636$	−15.0000	+	3.87298i	−0.594789	+	0.153574i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	15.5885	−	20.1246i	0.615227	−	0.794255i
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$	30.0000	−	38.7298i	1.18033	−	1.52380i
$647$	−24.2487			−0.953315			−0.476658	−	0.879089i	$-0.658152\pi$
$647$	−24.2487			−0.953315			−0.476658	+	0.879089i	$0.658152\pi$
$648$	−23.3827	−	10.0623i	−0.918559	−	0.395285i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			49.1935i			1.92509i	0.271122	+	0.962545i	$0.412605\pi$
$653$			49.1935i			1.92509i	−0.271122	+	0.962545i	$0.587395\pi$
$654$	−21.0000	+	27.1109i	−0.821165	+	1.06012i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	22.0000			0.855701			0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000			0.855701			0.427850	+	0.903850i	$0.359271\pi$
$662$	25.9808	+	20.1246i	1.00977	+	0.782165i
$663$		0			0
$664$	−9.00000	−	3.87298i	−0.349268	−	0.150301i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	12.1244	+	46.9574i	0.469105	+	1.81684i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	6.50000	+	25.1744i	0.250000	+	0.968246i
$677$		−	31.3050i		−	1.20315i	−0.798817	−	0.601574i	$-0.794541\pi$
$677$		−	31.3050i		−	1.20315i	0.798817	−	0.601574i	$-0.205459\pi$
$678$	8.66025	+	6.70820i	0.332595	+	0.257627i
$679$		0			0
$680$		0			0
$681$	42.0000			1.60944
$682$		0			0
$683$	−38.1051			−1.45805			−0.729026	−	0.684486i	$-0.760027\pi$
$683$	−38.1051			−1.45805			−0.729026	+	0.684486i	$0.760027\pi$
$684$	45.0000	−	11.6190i	1.72062	−	0.444262i
$685$		0			0
$686$		0			0
$687$	45.0333			1.71813
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	7.74597i		−	0.294670i	−0.989087	−	0.147335i	$-0.952930\pi$
$691$		−	7.74597i		−	0.294670i	0.989087	−	0.147335i	$-0.0470696\pi$
$692$	−43.3013	+	11.1803i	−1.64607	+	0.425013i
$693$		0			0
$694$	9.00000	−	11.6190i	0.341635	−	0.441049i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	29.4449	−	38.0132i	1.11450	−	1.43882i
$699$			38.7298i			1.46490i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−35.0000	−	27.1109i	−1.31724	−	1.02033i
$707$		0			0
$708$		0			0
$709$	26.0000			0.976450			0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000			0.976450			0.488225	+	0.872718i	$0.337644\pi$
$710$		0			0
$711$		−	23.2379i		−	0.871489i
$712$		0			0
$713$		−	26.8328i		−	1.00490i
$714$		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
$722$	−35.5070	+	45.8394i	−1.32144	+	1.70597i
$723$	3.46410			0.128831
$724$	−11.0000	−	42.6028i	−0.408812	−	1.58332i
$725$		0			0
$726$	−16.5000	+	21.3014i	−0.612372	+	0.790569i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$	1.73205	+	6.70820i	0.0640184	+	0.247942i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	3.00000	−	19.3649i	0.110581	−	0.713800i
$737$		0			0
$738$		0			0
$739$		−	54.2218i		−	1.99458i	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$		−	54.2218i		−	1.99458i	0.0735712	−	0.997290i	$-0.476560\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−31.1769			−1.14377			−0.571885	−	0.820334i	$-0.693788\pi$
$743$	−31.1769			−1.14377			−0.571885	+	0.820334i	$0.693788\pi$
$744$	15.0000	−	34.8569i	0.549927	−	1.27791i
$745$		0			0
$746$		0			0
$747$	10.3923			0.380235
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	54.2218i		−	1.97858i	−0.145962	−	0.989290i	$-0.546628\pi$
$751$		−	54.2218i		−	1.97858i	0.145962	−	0.989290i	$-0.453372\pi$
$752$	36.3731	−	20.1246i	1.32639	−	0.733869i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$	43.3013	+	33.5410i	1.57277	+	1.21826i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	33.0000	−	42.6028i	1.19234	−	1.53930i
$767$		0			0
$768$	14.7224	−	23.4787i	0.531250	−	0.847215i
$769$	46.0000			1.65880			0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000			1.65880			0.829401	+	0.558653i	$0.188682\pi$
$770$		0			0
$771$			54.2218i			1.95275i
$772$		0			0
$773$		−	4.47214i		−	0.160852i	−0.996761	−	0.0804258i	$-0.974372\pi$
$773$		−	4.47214i		−	0.160852i	0.996761	−	0.0804258i	$-0.0256280\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	17.3205	+	13.4164i	0.619380	+	0.479770i
$783$		0			0
$784$	−24.5000	+	13.5554i	−0.875000	+	0.484123i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	8.66025	−	2.23607i	0.308509	−	0.0796566i
$789$	−54.0000			−1.92245
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	45.0000	−	11.6190i	1.59498	−	0.411823i
$797$		−	49.1935i		−	1.74252i	−0.490819	−	0.871262i	$-0.663302\pi$
$797$		−	49.1935i		−	1.74252i	0.490819	−	0.871262i	$-0.336698\pi$
$798$		0			0
$799$			46.4758i			1.64420i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$			23.2379i			0.815993i	0.912983	+	0.407997i	$0.133772\pi$
$811$			23.2379i			0.815993i	−0.912983	+	0.407997i	$0.866228\pi$
$812$		0			0
$813$			13.4164i			0.470534i
$814$		0			0
$815$		0			0
$816$	15.0000	+	27.1109i	0.525105	+	0.949071i
$817$		0			0
$818$	−22.5167	+	29.0689i	−0.787277	+	1.01637i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−43.3013	−	33.5410i	−1.51031	−	1.16988i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−45.0333			−1.56596			−0.782981	−	0.622046i	$-0.786302\pi$
$827$	−45.0333			−1.56596			−0.782981	+	0.622046i	$0.786302\pi$
$828$	5.19615	+	20.1246i	0.180579	+	0.699379i
$829$	34.0000			1.18087			0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000			1.18087			0.590434	+	0.807086i	$0.298956\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	31.3050i		−	1.08465i
$834$	−45.0000	−	34.8569i	−1.55822	−	1.20699i
$835$		0			0
$836$		0			0
$837$			40.2492i			1.39122i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$	32.9090	−	42.4853i	1.13412	−	1.46414i
$843$		0			0
$844$	−15.0000	+	3.87298i	−0.516321	+	0.133314i
$845$		0			0
$846$	−27.0000	+	34.8569i	−0.928279	+	1.19840i
$847$		0			0
$848$	8.66025	+	15.6525i	0.297394	+	0.537508i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−27.0000	−	11.6190i	−0.922841	−	0.397128i
$857$		−	58.1378i		−	1.98595i	−0.118331	−	0.992974i	$-0.537755\pi$
$857$		−	58.1378i		−	1.98595i	0.118331	−	0.992974i	$-0.462245\pi$
$858$		0			0
$859$			38.7298i			1.32144i	0.750630	+	0.660722i	$0.229750\pi$
$859$			38.7298i			1.32144i	−0.750630	+	0.660722i	$0.770250\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38.1051			1.29711			0.648557	−	0.761166i	$-0.275373\pi$
$863$	38.1051			1.29711			0.648557	+	0.761166i	$0.275373\pi$
$864$	−4.50000	+	29.0474i	−0.153093	+	0.988212i
$865$		0			0
$866$		0			0
$867$	−5.19615			−0.176471
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	36.3731	+	15.6525i	1.23175	+	0.530060i
$873$		0			0
$874$	−30.0000	−	23.2379i	−1.01477	−	0.786034i
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$	−43.3013	−	33.5410i	−1.46135	−	1.13195i
$879$			54.2218i			1.82885i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	18.1865	−	23.4787i	0.612372	−	0.790569i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−33.0000	+	42.6028i	−1.10866	+	1.43127i
$887$	58.8897			1.97732			0.988662	−	0.150160i	$-0.0479788\pi$
$887$	58.8897			1.97732			0.988662	+	0.150160i	$0.0479788\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	80.4984i		−	2.69378i
$894$		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	−	11.6190i	0.166298	−	0.386441i
$905$		0			0
$906$	−45.0000	−	34.8569i	−1.49502	−	1.15804i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	−12.1244	−	46.9574i	−0.402361	−	1.55834i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−25.9808	−	46.9574i	−0.860309	−	1.55492i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−13.0000	−	50.3488i	−0.429532	−	1.66357i
$917$		0			0
$918$	−25.9808	−	20.1246i	−0.857493	−	0.664211i
$919$			23.2379i			0.766548i	0.923635	+	0.383274i	$0.125203\pi$
$919$			23.2379i			0.766548i	−0.923635	+	0.383274i	$0.874797\pi$
$920$		0			0
$921$		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		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$			54.2218i			1.77705i
$932$	43.3013	−	11.1803i	1.41838	−	0.366224i
$933$		0			0
$934$	21.0000	−	27.1109i	0.687141	−	0.887095i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−58.8897			−1.91366			−0.956830	−	0.290650i	$-0.906129\pi$
$947$	−58.8897			−1.91366			−0.956830	+	0.290650i	$0.906129\pi$
$948$	−25.9808	+	6.70820i	−0.843816	+	0.217872i
$949$		0			0
$950$		0			0
$951$			38.7298i			1.25590i
$952$		0			0
$953$			58.1378i			1.88327i	0.336640	+	0.941634i	$0.390710\pi$
$953$			58.1378i			1.88327i	−0.336640	+	0.941634i	$0.609290\pi$
$954$	−15.0000	−	11.6190i	−0.485643	−	0.376177i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−29.0000			−0.935484
$962$		0			0
$963$	31.1769			1.00466
$964$	−1.00000	−	3.87298i	−0.0322078	−	0.124740i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	28.5788	+	12.2984i	0.918559	+	0.395285i
$969$	60.0000			1.92748
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−7.79423	−	30.1869i	−0.250000	−	0.968246i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	7.00000	−	3.87298i	0.224065	−	0.123971i
$977$		−	4.47214i		−	0.143076i	−0.997438	−	0.0715382i	$-0.977209\pi$
$977$		−	4.47214i		−	0.143076i	0.997438	−	0.0715382i	$-0.0227908\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−42.0000			−1.34096
$982$		0			0
$983$	−3.46410			−0.110488			−0.0552438	−	0.998473i	$-0.517594\pi$
$983$	−3.46410			−0.110488			−0.0552438	+	0.998473i	$0.517594\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	54.2218i		−	1.72241i	−0.508257	−	0.861206i	$-0.669710\pi$
$991$		−	54.2218i		−	1.72241i	0.508257	−	0.861206i	$-0.330290\pi$
$992$	−43.3013	−	6.70820i	−1.37482	−	0.212986i
$993$			40.2492i			1.27727i
$994$		0			0
$995$		0			0
$996$	−3.00000	−	11.6190i	−0.0950586	−	0.368161i
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$	8.66025	+	6.70820i	0.274136	+	0.212344i
$999$		0			0

Display

a_p

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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

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a_p

) (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.2.e.a.251.3		4
3.2	odd	2	inner	300.2.e.a.251.2		4
4.3	odd	2	inner	300.2.e.a.251.1		4
5.2	odd	4		60.2.h.b.59.4	yes	4
5.3	odd	4		60.2.h.b.59.1	✓	4
5.4	even	2	inner	300.2.e.a.251.2		4
12.11	even	2	inner	300.2.e.a.251.4		4
15.2	even	4		60.2.h.b.59.1	✓	4
15.8	even	4		60.2.h.b.59.4	yes	4
15.14	odd	2	CM	300.2.e.a.251.3		4
20.3	even	4		60.2.h.b.59.2	yes	4
20.7	even	4		60.2.h.b.59.3	yes	4
20.19	odd	2	inner	300.2.e.a.251.4		4
40.3	even	4		960.2.o.a.959.3		4
40.13	odd	4		960.2.o.a.959.1		4
40.27	even	4		960.2.o.a.959.2		4
40.37	odd	4		960.2.o.a.959.4		4
60.23	odd	4		60.2.h.b.59.3	yes	4
60.47	odd	4		60.2.h.b.59.2	yes	4
60.59	even	2	inner	300.2.e.a.251.1		4
120.53	even	4		960.2.o.a.959.4		4
120.77	even	4		960.2.o.a.959.1		4
120.83	odd	4		960.2.o.a.959.2		4
120.107	odd	4		960.2.o.a.959.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.2.h.b.59.1	✓	4	5.3	odd	4
60.2.h.b.59.1	✓	4	15.2	even	4
60.2.h.b.59.2	yes	4	20.3	even	4
60.2.h.b.59.2	yes	4	60.47	odd	4
60.2.h.b.59.3	yes	4	20.7	even	4
60.2.h.b.59.3	yes	4	60.23	odd	4
60.2.h.b.59.4	yes	4	5.2	odd	4
60.2.h.b.59.4	yes	4	15.8	even	4
300.2.e.a.251.1		4	4.3	odd	2	inner
300.2.e.a.251.1		4	60.59	even	2	inner
300.2.e.a.251.2		4	3.2	odd	2	inner
300.2.e.a.251.2		4	5.4	even	2	inner
300.2.e.a.251.3		4	1.1	even	1	trivial
300.2.e.a.251.3		4	15.14	odd	2	CM
300.2.e.a.251.4		4	12.11	even	2	inner
300.2.e.a.251.4		4	20.19	odd	2	inner
960.2.o.a.959.1		4	40.13	odd	4
960.2.o.a.959.1		4	120.77	even	4
960.2.o.a.959.2		4	40.27	even	4
960.2.o.a.959.2		4	120.83	odd	4
960.2.o.a.959.3		4	40.3	even	4
960.2.o.a.959.3		4	120.107	odd	4
960.2.o.a.959.4		4	40.37	odd	4
960.2.o.a.959.4		4	120.53	even	4

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	300.2.e.a.251.3

Level	$300$
Weight	$2$
Character	300.251
Analytic conductor	$2.396$
Analytic rank	$0$
Dimension	$4$
CM discriminant	-15
Inner twists	$8$