LMFDB - Embedded newform 756.2.j.a.505.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(253,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("756.253");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$756 = 2^{2} \cdot 3^{3} \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	756.j (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$3^{3}$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.3
Root			$0.500000 - 2.05195i$ of defining polynomial
Character	$\chi$	$=$	756.505
Dual form			756.2.j.a.253.3

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.02704 + 1.77889i) q^{5} +(-0.500000 + 0.866025i) q^{7} +(-2.52704 + 4.37697i) q^{11} +(0.500000 + 0.866025i) q^{13} -0.273346 q^{17} -5.38151 q^{19} +(-2.66372 - 4.61369i) q^{23} +(0.390369 - 0.676139i) q^{25} +(-4.16372 + 7.21177i) q^{29} +(5.08113 + 8.80077i) q^{31} -2.05408 q^{35} +8.16225 q^{37} +(-2.52704 - 4.37697i) q^{41} +(-2.30039 + 3.98439i) q^{43} +(0.690757 - 1.19643i) q^{47} +(-0.500000 - 0.866025i) q^{49} -3.43560 q^{53} -10.3815 q^{55} +(0.890369 + 1.54216i) q^{59} +(-0.390369 + 0.676139i) q^{61} +(-1.02704 + 1.77889i) q^{65} +(4.19076 + 7.25860i) q^{67} +7.78074 q^{71} +9.38151 q^{73} +(-2.52704 - 4.37697i) q^{77} +(-6.47150 + 11.2090i) q^{79} +(-2.86333 + 4.95943i) q^{83} +(-0.280738 - 0.486253i) q^{85} +13.8171 q^{89} -1.00000 q^{91} +(-5.52704 - 9.57312i) q^{95} +(1.10963 - 1.92194i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 3 q^{5} - 3 q^{7} - 6 q^{11} + 3 q^{13} + 6 q^{19} - 6 q^{23} - 6 q^{25} - 15 q^{29} + 3 q^{31} + 6 q^{35} - 6 q^{37} - 6 q^{41} - 3 q^{43} - 15 q^{47} - 3 q^{49} + 36 q^{53} - 24 q^{55} - 3 q^{59}+ \cdots + 15 q^{97}+O(q^{100})$ 6 * q - 3 * q^5 - 3 * q^7 - 6 * q^11 + 3 * q^13 + 6 * q^19 - 6 * q^23 - 6 * q^25 - 15 * q^29 + 3 * q^31 + 6 * q^35 - 6 * q^37 - 6 * q^41 - 3 * q^43 - 15 * q^47 - 3 * q^49 + 36 * q^53 - 24 * q^55 - 3 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 + 30 * q^71 + 18 * q^73 - 6 * q^77 - 3 * q^79 - 18 * q^83 + 15 * q^85 - 12 * q^89 - 6 * q^91 - 24 * q^95 + 15 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$ .

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.02704	+	1.77889i	0.459307	+	0.795543i	0.998924	−	0.0463670i	$-0.0147644\pi$
$5$	1.02704	+	1.77889i	0.459307	+	0.795543i	−0.539617	+	0.841910i	$0.681431\pi$
$6$		0			0
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$7$	−0.500000	+	0.866025i	−0.188982	+	0.327327i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.52704	+	4.37697i	−0.761932	+	1.31970i	0.179922	+	0.983681i	$0.442416\pi$
$11$	−2.52704	+	4.37697i	−0.761932	+	1.31970i	−0.941854	+	0.336024i	$0.890918\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	0.926995	−	0.375073i	$-0.122382\pi$
$13$	0.500000	+	0.866025i	0.138675	+	0.240192i	−0.788320	+	0.615265i	$0.789049\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.273346			−0.0662962			−0.0331481	−	0.999450i	$-0.510553\pi$
$17$	−0.273346			−0.0662962			−0.0331481	+	0.999450i	$0.510553\pi$
$18$		0			0
$19$	−5.38151			−1.23460			−0.617302	−	0.786726i	$-0.711774\pi$
$19$	−5.38151			−1.23460			−0.617302	+	0.786726i	$0.711774\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.66372	−	4.61369i	−0.555423	−	0.962021i	−0.997870	−	0.0652265i	$-0.979223\pi$
$23$	−2.66372	−	4.61369i	−0.555423	−	0.962021i	0.442447	−	0.896794i	$-0.354110\pi$
$24$		0			0
$25$	0.390369	−	0.676139i	0.0780738	−	0.135228i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.16372	+	7.21177i	−0.773183	+	1.33919i	0.162628	+	0.986687i	$0.448003\pi$
$29$	−4.16372	+	7.21177i	−0.773183	+	1.33919i	−0.935810	+	0.352504i	$0.885330\pi$
$30$		0			0
$31$	5.08113	+	8.80077i	0.912597	+	1.58066i	0.810382	+	0.585903i	$0.199260\pi$
$31$	5.08113	+	8.80077i	0.912597	+	1.58066i	0.102216	+	0.994762i	$0.467407\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.05408			−0.347204
$36$		0			0
$37$	8.16225			1.34187			0.670933	−	0.741518i	$-0.265894\pi$
$37$	8.16225			1.34187			0.670933	+	0.741518i	$0.265894\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.52704	−	4.37697i	−0.394658	−	0.683567i	0.598400	−	0.801198i	$-0.295803\pi$
$41$	−2.52704	−	4.37697i	−0.394658	−	0.683567i	−0.993057	+	0.117631i	$0.962470\pi$
$42$		0			0
$43$	−2.30039	+	3.98439i	−0.350806	+	0.607614i	−0.986391	−	0.164417i	$-0.947426\pi$
$43$	−2.30039	+	3.98439i	−0.350806	+	0.607614i	0.635585	+	0.772031i	$0.280759\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.690757	−	1.19643i	0.100757	−	0.174517i	−0.811240	−	0.584714i	$-0.801207\pi$
$47$	0.690757	−	1.19643i	0.100757	−	0.174517i	0.911997	+	0.410197i	$0.134540\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.0714286	−	0.123718i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.43560			−0.471916			−0.235958	−	0.971763i	$-0.575823\pi$
$53$	−3.43560			−0.471916			−0.235958	+	0.971763i	$0.575823\pi$
$54$		0			0
$55$	−10.3815			−1.39984
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.890369	+	1.54216i	0.115916	+	0.200773i	0.918146	−	0.396243i	$-0.129686\pi$
$59$	0.890369	+	1.54216i	0.115916	+	0.200773i	−0.802229	+	0.597016i	$0.796353\pi$
$60$		0			0
$61$	−0.390369	+	0.676139i	−0.0499816	+	0.0865707i	−0.889934	−	0.456090i	$-0.849250\pi$
$61$	−0.390369	+	0.676139i	−0.0499816	+	0.0865707i	0.839952	+	0.542660i	$0.182583\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.02704	+	1.77889i	−0.127389	+	0.220644i
$66$		0			0
$67$	4.19076	+	7.25860i	0.511982	+	0.886780i	0.999904	+	0.0138919i	$0.00442207\pi$
$67$	4.19076	+	7.25860i	0.511982	+	0.886780i	−0.487921	+	0.872888i	$0.662245\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.78074			0.923404			0.461702	−	0.887035i	$-0.347239\pi$
$71$	7.78074			0.923404			0.461702	+	0.887035i	$0.347239\pi$
$72$		0			0
$73$	9.38151			1.09802			0.549012	−	0.835815i	$-0.315004\pi$
$73$	9.38151			1.09802			0.549012	+	0.835815i	$0.315004\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.52704	−	4.37697i	−0.287983	−	0.498801i
$78$		0			0
$79$	−6.47150	+	11.2090i	−0.728100	+	1.26111i	0.229585	+	0.973289i	$0.426263\pi$
$79$	−6.47150	+	11.2090i	−0.728100	+	1.26111i	−0.957685	+	0.287818i	$0.907070\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.86333	+	4.95943i	−0.314291	+	0.544368i	−0.979287	−	0.202479i	$-0.935100\pi$
$83$	−2.86333	+	4.95943i	−0.314291	+	0.544368i	0.664996	+	0.746847i	$0.268433\pi$
$84$		0			0
$85$	−0.280738	−	0.486253i	−0.0304503	−	0.0527415i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.8171			1.46461			0.732306	−	0.680976i	$-0.238444\pi$
$89$	13.8171			1.46461			0.732306	+	0.680976i	$0.238444\pi$
$90$		0			0
$91$	−1.00000			−0.104828
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−5.52704	−	9.57312i	−0.567063	−	0.982181i
$96$		0			0
$97$	1.10963	−	1.92194i	0.112666	−	0.195143i	−0.804178	−	0.594388i	$-0.797394\pi$
$97$	1.10963	−	1.92194i	0.112666	−	0.195143i	0.916844	+	0.399245i	$0.130728\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.36333	−	2.36135i	0.135656	−	0.234963i	−0.790192	−	0.612860i	$-0.790019\pi$
$101$	1.36333	−	2.36135i	0.135656	−	0.234963i	0.925848	+	0.377896i	$0.123352\pi$
$102$		0			0
$103$	−8.99115	−	15.5731i	−0.885924	−	1.53447i	−0.844651	−	0.535317i	$-0.820192\pi$
$103$	−8.99115	−	15.5731i	−0.885924	−	1.53447i	−0.0412728	−	0.999148i	$-0.513141\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.10817			−0.107131			−0.0535653	−	0.998564i	$-0.517059\pi$
$107$	−1.10817			−0.107131			−0.0535653	+	0.998564i	$0.517059\pi$
$108$		0			0
$109$	3.38151			0.323890			0.161945	−	0.986800i	$-0.448223\pi$
$109$	3.38151			0.323890			0.161945	+	0.986800i	$0.448223\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.43560	+	16.3429i	0.887626	+	1.53741i	0.842673	+	0.538425i	$0.180980\pi$
$113$	9.43560	+	16.3429i	0.887626	+	1.53741i	0.0449531	+	0.998989i	$0.485686\pi$
$114$		0			0
$115$	5.47150	−	9.47691i	0.510220	−	0.883726i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.136673	−	0.236725i	0.0125288	−	0.0217005i
$120$		0			0
$121$	−7.27188	−	12.5953i	−0.661080	−	1.14502i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.8741			1.06205
$126$		0			0
$127$	17.1623			1.52290			0.761452	−	0.648221i	$-0.224487\pi$
$127$	17.1623			1.52290			0.761452	+	0.648221i	$0.224487\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−8.94445	−	15.4922i	−0.781481	−	1.35356i	−0.931079	−	0.364817i	$-0.881132\pi$
$131$	−8.94445	−	15.4922i	−0.781481	−	1.35356i	0.149599	−	0.988747i	$-0.452202\pi$
$132$		0			0
$133$	2.69076	−	4.66053i	0.233318	−	0.404119i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.24630	+	3.89071i	−0.191915	+	0.332406i	−0.945885	−	0.324503i	$-0.894803\pi$
$137$	−2.24630	+	3.89071i	−0.191915	+	0.332406i	0.753970	+	0.656909i	$0.228136\pi$
$138$		0			0
$139$	−9.07227	−	15.7136i	−0.769500	−	1.33281i	−0.937834	−	0.347083i	$-0.887172\pi$
$139$	−9.07227	−	15.7136i	−0.769500	−	1.33281i	0.168334	−	0.985730i	$-0.446161\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.05408			−0.422644
$144$		0			0
$145$	−17.1052			−1.42051
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2.25370	−	3.90352i	−0.184630	−	0.319788i	0.758822	−	0.651298i	$-0.225775\pi$
$149$	−2.25370	−	3.90352i	−0.184630	−	0.319788i	−0.943452	+	0.331510i	$0.892442\pi$
$150$		0			0
$151$	5.49115	−	9.51094i	0.446863	−	0.773990i	−0.551317	−	0.834296i	$-0.685874\pi$
$151$	5.49115	−	9.51094i	0.446863	−	0.773990i	0.998180	+	0.0603064i	$0.0192078\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.4371	+	18.0775i	−0.838325	+	1.45202i
$156$		0			0
$157$	−2.08998	−	3.61995i	−0.166799	−	0.288904i	0.770494	−	0.637447i	$-0.220010\pi$
$157$	−2.08998	−	3.61995i	−0.166799	−	0.288904i	−0.937293	+	0.348544i	$0.886676\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.32743			0.419860
$162$		0			0
$163$	−5.61849			−0.440074			−0.220037	−	0.975492i	$-0.570618\pi$
$163$	−5.61849			−0.440074			−0.220037	+	0.975492i	$0.570618\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.44592	−	9.43260i	−0.421418	−	0.729917i	0.574661	−	0.818392i	$-0.305134\pi$
$167$	−5.44592	−	9.43260i	−0.421418	−	0.729917i	−0.996078	+	0.0884750i	$0.971801\pi$
$168$		0			0
$169$	6.00000	−	10.3923i	0.461538	−	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.30039	−	12.6446i	0.555038	−	0.961354i	−0.442862	−	0.896590i	$-0.646037\pi$
$173$	7.30039	−	12.6446i	0.555038	−	0.961354i	0.997901	−	0.0647648i	$-0.0206297\pi$
$174$		0			0
$175$	0.390369	+	0.676139i	0.0295091	+	0.0511113i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	23.4897			1.75570			0.877851	−	0.478934i	$-0.158977\pi$
$179$	23.4897			1.75570			0.877851	+	0.478934i	$0.158977\pi$
$180$		0			0
$181$	−1.39922			−0.104003			−0.0520017	−	0.998647i	$-0.516560\pi$
$181$	−1.39922			−0.104003			−0.0520017	+	0.998647i	$0.516560\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.38298	+	14.5197i	0.616329	+	1.06751i
$186$		0			0
$187$	0.690757	−	1.19643i	0.0505132	−	0.0874914i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	11.6819	−	20.2336i	0.845273	−	1.46406i	−0.0401112	−	0.999195i	$-0.512771\pi$
$191$	11.6819	−	20.2336i	0.845273	−	1.46406i	0.885384	−	0.464860i	$-0.153895\pi$
$192$		0			0
$193$	7.27188	+	12.5953i	0.523442	+	0.906628i	0.999628	+	0.0272830i	$0.00868552\pi$
$193$	7.27188	+	12.5953i	0.523442	+	0.906628i	−0.476186	+	0.879345i	$0.657981\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	17.3422			1.23558			0.617791	−	0.786343i	$-0.288028\pi$
$197$	17.3422			1.23558			0.617791	+	0.786343i	$0.288028\pi$
$198$		0			0
$199$	−11.5438			−0.818316			−0.409158	−	0.912464i	$-0.634178\pi$
$199$	−11.5438			−0.818316			−0.409158	+	0.912464i	$0.634178\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.16372	−	7.21177i	−0.292236	−	0.506167i
$204$		0			0
$205$	5.19076	−	8.99066i	0.362538	−	0.627935i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	13.5993	−	23.5547i	0.940684	−	1.62931i
$210$		0			0
$211$	12.2630	+	21.2402i	0.844222	+	1.46223i	0.886295	+	0.463120i	$0.153270\pi$
$211$	12.2630	+	21.2402i	0.844222	+	1.46223i	−0.0420736	+	0.999115i	$0.513396\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.45038			−0.644511
$216$		0			0
$217$	−10.1623			−0.689859
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.136673	−	0.236725i	−0.00919363	−	0.0159238i
$222$		0			0
$223$	−4.28074	+	7.41446i	−0.286659	+	0.496509i	−0.973010	−	0.230762i	$-0.925878\pi$
$223$	−4.28074	+	7.41446i	−0.286659	+	0.496509i	0.686351	+	0.727271i	$0.259211\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.5993	+	18.3586i	−0.703501	+	1.21850i	0.263729	+	0.964597i	$0.415048\pi$
$227$	−10.5993	+	18.3586i	−0.703501	+	1.21850i	−0.967230	+	0.253903i	$0.918286\pi$
$228$		0			0
$229$	2.28074	+	3.95035i	0.150715	+	0.261047i	0.931491	−	0.363765i	$-0.118509\pi$
$229$	2.28074	+	3.95035i	0.150715	+	0.261047i	−0.780775	+	0.624812i	$0.785176\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−13.5074			−0.884899			−0.442449	−	0.896794i	$-0.645890\pi$
$233$	−13.5074			−0.884899			−0.442449	+	0.896794i	$0.645890\pi$
$234$		0			0
$235$	2.83775			0.185114
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.82743	−	11.8255i	−0.441630	−	0.764925i	0.556181	−	0.831061i	$-0.312266\pi$
$239$	−6.82743	−	11.8255i	−0.441630	−	0.764925i	−0.997811	+	0.0661361i	$0.978933\pi$
$240$		0			0
$241$	−1.60963	+	2.78796i	−0.103685	+	0.179588i	−0.913200	−	0.407511i	$-0.866397\pi$
$241$	−1.60963	+	2.78796i	−0.103685	+	0.179588i	0.809515	+	0.587099i	$0.199730\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.02704	−	1.77889i	0.0656153	−	0.113649i
$246$		0			0
$247$	−2.69076	−	4.66053i	−0.171209	−	0.296542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.38151			−0.276559			−0.138279	−	0.990393i	$-0.544157\pi$
$251$	−4.38151			−0.276559			−0.138279	+	0.990393i	$0.544157\pi$
$252$		0			0
$253$	26.9253			1.69278
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.72665	+	9.91886i	0.357219	+	0.618721i	0.987495	−	0.157650i	$-0.0503917\pi$
$257$	5.72665	+	9.91886i	0.357219	+	0.618721i	−0.630276	+	0.776371i	$0.717058\pi$
$258$		0			0
$259$	−4.08113	+	7.06872i	−0.253589	+	0.439229i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.41741	+	5.91913i	−0.210727	+	0.364989i	−0.951942	−	0.306278i	$-0.900916\pi$
$263$	−3.41741	+	5.91913i	−0.210727	+	0.364989i	0.741216	+	0.671267i	$0.234250\pi$
$264$		0			0
$265$	−3.52850	−	6.11155i	−0.216754	−	0.375429i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.67257			−0.589747			−0.294873	−	0.955536i	$-0.595278\pi$
$269$	−9.67257			−0.589747			−0.294873	+	0.955536i	$0.595278\pi$
$270$		0			0
$271$	−12.8377			−0.779838			−0.389919	−	0.920849i	$-0.627497\pi$
$271$	−12.8377			−0.779838			−0.389919	+	0.920849i	$0.627497\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.97296	+	3.41726i	0.118974	+	0.206069i
$276$		0			0
$277$	−5.79153	+	10.0312i	−0.347980	+	0.602718i	−0.985890	−	0.167392i	$-0.946465\pi$
$277$	−5.79153	+	10.0312i	−0.347980	+	0.602718i	0.637911	+	0.770110i	$0.279799\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.46410	+	4.26795i	−0.146996	+	0.254605i	−0.930116	−	0.367266i	$-0.880294\pi$
$281$	−2.46410	+	4.26795i	−0.146996	+	0.254605i	0.783120	+	0.621871i	$0.213627\pi$
$282$		0			0
$283$	9.30039	+	16.1087i	0.552851	+	0.957565i	0.998067	+	0.0621426i	$0.0197934\pi$
$283$	9.30039	+	16.1087i	0.552851	+	0.957565i	−0.445217	+	0.895423i	$0.646873\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.05408			0.298333
$288$		0			0
$289$	−16.9253			−0.995605
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.3801	−	21.4429i	−0.723250	−	1.25271i	−0.959690	−	0.281060i	$-0.909314\pi$
$293$	−12.3801	−	21.4429i	−0.723250	−	1.25271i	0.236440	−	0.971646i	$-0.424019\pi$
$294$		0			0
$295$	−1.82889	+	3.16774i	−0.106482	+	0.184433i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.66372	−	4.61369i	0.154047	−	0.266817i
$300$		0			0
$301$	−2.30039	−	3.98439i	−0.132592	−	0.229656i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.60370			−0.0918277
$306$		0			0
$307$	21.9430			1.25235			0.626176	−	0.779681i	$-0.284619\pi$
$307$	21.9430			1.25235			0.626176	+	0.779681i	$0.284619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	13.5811	+	23.5232i	0.770115	+	1.33388i	0.937499	+	0.347987i	$0.113135\pi$
$311$	13.5811	+	23.5232i	0.770115	+	1.33388i	−0.167384	+	0.985892i	$0.553532\pi$
$312$		0			0
$313$	4.27188	−	7.39912i	0.241461	−	0.418223i	−0.719670	−	0.694317i	$-0.755707\pi$
$313$	4.27188	−	7.39912i	0.241461	−	0.418223i	0.961131	+	0.276094i	$0.0890400\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.199612	−	0.345738i	0.0112113	−	0.0194186i	−0.860365	−	0.509678i	$-0.829765\pi$
$317$	0.199612	−	0.345738i	0.0112113	−	0.0194186i	0.871577	+	0.490259i	$0.163098\pi$
$318$		0			0
$319$	−21.0438	−	36.4489i	−1.17822	−	2.04075i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.47102			0.0818495
$324$		0			0
$325$	0.780738			0.0433076
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.690757	+	1.19643i	0.0380827	+	0.0659611i
$330$		0			0
$331$	2.80924	−	4.86575i	0.154410	−	0.267446i	−0.778434	−	0.627726i	$-0.783986\pi$
$331$	2.80924	−	4.86575i	0.154410	−	0.267446i	0.932844	+	0.360281i	$0.117319\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.60817	+	14.9098i	−0.470314	+	0.814609i
$336$		0			0
$337$	14.4911	+	25.0994i	0.789383	+	1.36725i	0.926345	+	0.376675i	$0.122933\pi$
$337$	14.4911	+	25.0994i	0.789383	+	1.36725i	−0.136962	+	0.990576i	$0.543734\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−51.3609			−2.78135
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.2345	+	29.8511i	0.925198	+	1.60249i	0.791243	+	0.611502i	$0.209434\pi$
$347$	17.2345	+	29.8511i	0.925198	+	1.60249i	0.133955	+	0.990987i	$0.457232\pi$
$348$		0			0
$349$	−8.78074	+	15.2087i	−0.470022	+	0.814102i	−0.999412	−	0.0342762i	$-0.989087\pi$
$349$	−8.78074	+	15.2087i	−0.470022	+	0.814102i	0.529390	+	0.848378i	$0.322421\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	16.4445	−	28.4826i	0.875250	−	1.51598i	0.0187537	−	0.999824i	$-0.494030\pi$
$353$	16.4445	−	28.4826i	0.875250	−	1.51598i	0.856496	−	0.516153i	$-0.172636\pi$
$354$		0			0
$355$	7.99115	+	13.8411i	0.424126	+	0.734608i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.96362			−0.156414			−0.0782071	−	0.996937i	$-0.524920\pi$
$359$	−2.96362			−0.156414			−0.0782071	+	0.996937i	$0.524920\pi$
$360$		0			0
$361$	9.96070			0.524247
$362$		0			0
$363$		0			0
$364$		0			0
$365$	9.63521	+	16.6887i	0.504330	+	0.873525i
$366$		0			0
$367$	−6.68190	+	11.5734i	−0.348792	+	0.604126i	−0.986035	−	0.166537i	$-0.946742\pi$
$367$	−6.68190	+	11.5734i	−0.348792	+	0.604126i	0.637243	+	0.770663i	$0.280075\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.71780	−	2.97532i	0.0891837	−	0.154471i
$372$		0			0
$373$	−2.30039	−	3.98439i	−0.119110	−	0.206304i	0.800305	−	0.599592i	$-0.204671\pi$
$373$	−2.30039	−	3.98439i	−0.119110	−	0.206304i	−0.919415	+	0.393289i	$0.871337\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.32743			−0.428884
$378$		0			0
$379$	−2.21926			−0.113996			−0.0569979	−	0.998374i	$-0.518153\pi$
$379$	−2.21926			−0.113996			−0.0569979	+	0.998374i	$0.518153\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−15.5167	−	26.8758i	−0.792868	−	1.37329i	−0.924184	−	0.381947i	$-0.875254\pi$
$383$	−15.5167	−	26.8758i	−0.792868	−	1.37329i	0.131317	−	0.991340i	$-0.458079\pi$
$384$		0			0
$385$	5.19076	−	8.99066i	0.264545	−	0.458206i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.4174	+	21.5076i	−0.629588	+	1.09048i	0.358047	+	0.933704i	$0.383443\pi$
$389$	−12.4174	+	21.5076i	−0.629588	+	1.09048i	−0.987634	+	0.156774i	$0.949890\pi$
$390$		0			0
$391$	0.728116	+	1.26113i	0.0368224	+	0.0637783i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−26.5860			−1.33769
$396$		0			0
$397$	17.7237			0.889528			0.444764	−	0.895648i	$-0.353287\pi$
$397$	17.7237			0.889528			0.444764	+	0.895648i	$0.353287\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	15.0885	+	26.1341i	0.753485	+	1.30507i	0.946124	+	0.323804i	$0.104962\pi$
$401$	15.0885	+	26.1341i	0.753485	+	1.30507i	−0.192640	+	0.981270i	$0.561705\pi$
$402$		0			0
$403$	−5.08113	+	8.80077i	−0.253109	+	0.438398i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.6264	+	35.7259i	−1.02241	+	1.77087i
$408$		0			0
$409$	−8.38151	−	14.5172i	−0.414439	−	0.717830i	0.580930	−	0.813953i	$-0.302689\pi$
$409$	−8.38151	−	14.5172i	−0.414439	−	0.717830i	−0.995369	+	0.0961236i	$0.969356\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.78074			−0.0876244
$414$		0			0
$415$	−11.7630			−0.577424
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.44445	+	2.50187i	0.0705662	+	0.122224i	0.899150	−	0.437641i	$-0.144186\pi$
$419$	1.44445	+	2.50187i	0.0705662	+	0.122224i	−0.828583	+	0.559866i	$0.810853\pi$
$420$		0			0
$421$	0.0899807	−	0.155851i	0.00438539	−	0.00759572i	−0.863824	−	0.503793i	$-0.831937\pi$
$421$	0.0899807	−	0.155851i	0.00438539	−	0.00759572i	0.868210	+	0.496197i	$0.165271\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.106706	+	0.184820i	−0.00517600	+	0.00896509i
$426$		0			0
$427$	−0.390369	−	0.676139i	−0.0188913	−	0.0327207i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4.76595			0.229568			0.114784	−	0.993390i	$-0.463382\pi$
$431$	4.76595			0.229568			0.114784	+	0.993390i	$0.463382\pi$
$432$		0			0
$433$	−27.7630			−1.33421			−0.667103	−	0.744965i	$-0.732466\pi$
$433$	−27.7630			−1.33421			−0.667103	+	0.744965i	$0.732466\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.3348	+	24.8286i	0.685728	+	1.18771i
$438$		0			0
$439$	2.32889	−	4.03376i	0.111152	−	0.192521i	−0.805083	−	0.593162i	$-0.797879\pi$
$439$	2.32889	−	4.03376i	0.111152	−	0.192521i	0.916235	+	0.400641i	$0.131213\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.38151	+	2.39285i	−0.0656377	+	0.113688i	−0.896977	−	0.442078i	$-0.854242\pi$
$443$	−1.38151	+	2.39285i	−0.0656377	+	0.113688i	0.831339	+	0.555766i	$0.187575\pi$
$444$		0			0
$445$	14.1908	+	24.5791i	0.672706	+	1.16516i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−19.9430			−0.941168			−0.470584	−	0.882355i	$-0.655957\pi$
$449$	−19.9430			−0.941168			−0.470584	+	0.882355i	$0.655957\pi$
$450$		0			0
$451$	25.5438			1.20281
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.02704	−	1.77889i	−0.0481485	−	0.0833956i
$456$		0			0
$457$	−13.6908	+	23.7131i	−0.640427	+	1.10925i	0.344911	+	0.938635i	$0.387909\pi$
$457$	−13.6908	+	23.7131i	−0.640427	+	1.10925i	−0.985338	+	0.170616i	$0.945424\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−3.02558	+	5.24046i	−0.140915	+	0.244072i	−0.927842	−	0.372975i	$-0.878338\pi$
$461$	−3.02558	+	5.24046i	−0.140915	+	0.244072i	0.786926	+	0.617047i	$0.211671\pi$
$462$		0			0
$463$	8.77188	+	15.1933i	0.407664	+	0.706095i	0.994628	−	0.103519i	$-0.0330101\pi$
$463$	8.77188	+	15.1933i	0.407664	+	0.706095i	−0.586964	+	0.809613i	$0.699677\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	23.6156			1.09280			0.546399	−	0.837525i	$-0.315998\pi$
$467$	23.6156			1.09280			0.546399	+	0.837525i	$0.315998\pi$
$468$		0			0
$469$	−8.38151			−0.387022
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.6264	−	20.1374i	−0.534580	−	0.925920i
$474$		0			0
$475$	−2.10078	+	3.63865i	−0.0963902	+	0.166953i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.1264	−	33.1278i	0.873906	−	1.51365i	0.0159814	−	0.999872i	$-0.494913\pi$
$479$	19.1264	−	33.1278i	0.873906	−	1.51365i	0.857924	−	0.513776i	$-0.171754\pi$
$480$		0			0
$481$	4.08113	+	7.06872i	0.186083	+	0.322306i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.55855			0.206993
$486$		0			0
$487$	−6.57918			−0.298131			−0.149066	−	0.988827i	$-0.547627\pi$
$487$	−6.57918			−0.298131			−0.149066	+	0.988827i	$0.547627\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.02704	−	1.77889i	−0.0463498	−	0.0802801i	0.841920	−	0.539603i	$-0.181426\pi$
$491$	−1.02704	−	1.77889i	−0.0463498	−	0.0802801i	−0.888270	+	0.459323i	$0.848092\pi$
$492$		0			0
$493$	1.13814	−	1.97131i	0.0512591	−	0.0887833i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.89037	+	6.73832i	−0.174507	+	0.302255i
$498$		0			0
$499$	8.16225	+	14.1374i	0.365393	+	0.632879i	0.988839	−	0.148987i	$-0.0476014\pi$
$499$	8.16225	+	14.1374i	0.365393	+	0.632879i	−0.623446	+	0.781866i	$0.714268\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5.60078			0.249726			0.124863	−	0.992174i	$-0.460151\pi$
$503$	5.60078			0.249726			0.124863	+	0.992174i	$0.460151\pi$
$504$		0			0
$505$	5.60078			0.249231
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.336285	+	0.582462i	0.0149056	+	0.0258172i	0.873382	−	0.487036i	$-0.161922\pi$
$509$	0.336285	+	0.582462i	0.0149056	+	0.0258172i	−0.858476	+	0.512853i	$0.828589\pi$
$510$		0			0
$511$	−4.69076	+	8.12463i	−0.207507	+	0.359412i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.4686	−	31.9885i	0.813822	−	1.40958i
$516$		0			0
$517$	3.49115	+	6.04684i	0.153540	+	0.265940i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−26.4533			−1.15894			−0.579470	−	0.814993i	$-0.696740\pi$
$521$	−26.4533			−1.15894			−0.579470	+	0.814993i	$0.696740\pi$
$522$		0			0
$523$	27.3068			1.19404			0.597021	−	0.802225i	$-0.296351\pi$
$523$	27.3068			1.19404			0.597021	+	0.802225i	$0.296351\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.38891	−	2.40566i	−0.0605017	−	0.104792i
$528$		0			0
$529$	−2.69076	+	4.66053i	−0.116989	+	0.202632i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	2.52704	−	4.37697i	0.109458	−	0.189587i
$534$		0			0
$535$	−1.13814	−	1.97131i	−0.0492059	−	0.0852271i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.05408			0.217695
$540$		0			0
$541$	−19.3245			−0.830825			−0.415413	−	0.909633i	$-0.636363\pi$
$541$	−19.3245			−0.830825			−0.415413	+	0.909633i	$0.636363\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.47296	+	6.01534i	0.148765	+	0.257669i
$546$		0			0
$547$	9.17111	−	15.8848i	0.392128	−	0.679186i	−0.600602	−	0.799548i	$-0.705072\pi$
$547$	9.17111	−	15.8848i	0.392128	−	0.679186i	0.992730	+	0.120362i	$0.0384056\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	22.4071	−	38.8102i	0.954574	−	1.65337i
$552$		0			0
$553$	−6.47150	−	11.2090i	−0.275196	−	0.476653i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−9.19863			−0.389758			−0.194879	−	0.980827i	$-0.562431\pi$
$557$	−9.19863			−0.389758			−0.194879	+	0.980827i	$0.562431\pi$
$558$		0			0
$559$	−4.60078			−0.194592
$560$		0			0
$561$		0			0
$562$		0			0
$563$	16.5811	+	28.7194i	0.698811	+	1.21038i	0.968879	+	0.247535i	$0.0796206\pi$
$563$	16.5811	+	28.7194i	0.698811	+	1.21038i	−0.270068	+	0.962841i	$0.587046\pi$
$564$		0			0
$565$	−19.3815	+	33.5698i	−0.815386	+	1.41229i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.1008	−	22.6912i	0.549213	−	0.951265i	−0.449116	−	0.893474i	$-0.648261\pi$
$569$	13.1008	−	22.6912i	0.549213	−	0.951265i	0.998329	−	0.0577914i	$-0.0184058\pi$
$570$		0			0
$571$	−4.89037	−	8.47037i	−0.204656	−	0.354474i	0.745367	−	0.666654i	$-0.232274\pi$
$571$	−4.89037	−	8.47037i	−0.204656	−	0.354474i	−0.950023	+	0.312180i	$0.898941\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.15933			−0.173456
$576$		0			0
$577$	36.3068			1.51147			0.755736	−	0.654877i	$-0.227279\pi$
$577$	36.3068			1.51147			0.755736	+	0.654877i	$0.227279\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.86333	−	4.95943i	−0.118791	−	0.205752i
$582$		0			0
$583$	8.68190	−	15.0375i	0.359568	−	0.622789i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.0737	+	20.9123i	−0.498336	+	0.863144i	−0.999998	−	0.00191995i	$-0.999389\pi$
$587$	−12.0737	+	20.9123i	−0.498336	+	0.863144i	0.501662	+	0.865064i	$0.332722\pi$
$588$		0			0
$589$	−27.3442	−	47.3615i	−1.12670	−	1.95150i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−41.4897			−1.70378			−0.851889	−	0.523723i	$-0.824543\pi$
$593$	−41.4897			−1.70378			−0.851889	+	0.523723i	$0.824543\pi$
$594$		0			0
$595$	0.561476			0.0230183
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.3422	−	19.6453i	−0.463430	−	0.802685i	0.535699	−	0.844409i	$-0.320048\pi$
$599$	−11.3422	−	19.6453i	−0.463430	−	0.802685i	−0.999129	+	0.0417243i	$0.986715\pi$
$600$		0			0
$601$	20.1249	−	34.8573i	0.820912	−	1.42186i	−0.0840927	−	0.996458i	$-0.526799\pi$
$601$	20.1249	−	34.8573i	0.820912	−	1.42186i	0.905004	−	0.425403i	$-0.139867\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.9371	−	25.8717i	0.607278	−	1.05184i
$606$		0			0
$607$	−8.66225	−	15.0035i	−0.351590	−	0.608972i	0.634938	−	0.772563i	$-0.281026\pi$
$607$	−8.66225	−	15.0035i	−0.351590	−	0.608972i	−0.986528	+	0.163591i	$0.947692\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.38151			0.0558901
$612$		0			0
$613$	−32.9646			−1.33143			−0.665713	−	0.746207i	$-0.731873\pi$
$613$	−32.9646			−1.33143			−0.665713	+	0.746207i	$0.731873\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.4700	−	23.3308i	−0.542283	−	0.939262i	−0.998772	−	0.0495330i	$-0.984227\pi$
$617$	−13.4700	−	23.3308i	−0.542283	−	0.939262i	0.456489	−	0.889729i	$-0.349107\pi$
$618$		0			0
$619$	0.991146	−	1.71671i	0.0398375	−	0.0690006i	−0.845419	−	0.534103i	$-0.820649\pi$
$619$	0.991146	−	1.71671i	0.0398375	−	0.0690006i	0.885257	+	0.465103i	$0.153983\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.90856	+	11.9660i	−0.276785	+	0.479407i
$624$		0			0
$625$	10.2434	+	17.7421i	0.409735	+	0.709682i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2.23112			−0.0889606
$630$		0			0
$631$	−25.4868			−1.01461			−0.507306	−	0.861766i	$-0.669359\pi$
$631$	−25.4868			−1.01461			−0.507306	+	0.861766i	$0.669359\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.6264	+	30.5297i	0.699481	+	1.21154i
$636$		0			0
$637$	0.500000	−	0.866025i	0.0198107	−	0.0343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20.0423	+	34.7143i	−0.791623	+	1.37113i	0.133338	+	0.991071i	$0.457430\pi$
$641$	−20.0423	+	34.7143i	−0.791623	+	1.37113i	−0.924961	+	0.380061i	$0.875903\pi$
$642$		0			0
$643$	−3.50885	−	6.07751i	−0.138376	−	0.239674i	0.788506	−	0.615027i	$-0.210855\pi$
$643$	−3.50885	−	6.07751i	−0.138376	−	0.239674i	−0.926882	+	0.375353i	$0.877521\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.8142			0.464464			0.232232	−	0.972660i	$-0.425397\pi$
$647$	11.8142			0.464464			0.232232	+	0.972660i	$0.425397\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	0.136673	+	0.236725i	0.00534843	+	0.00926375i	0.868687	−	0.495361i	$-0.164964\pi$
$653$	0.136673	+	0.236725i	0.00534843	+	0.00926375i	−0.863339	+	0.504625i	$0.831631\pi$
$654$		0			0
$655$	18.3727	−	31.8224i	0.717879	−	1.24340i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7.39970	−	12.8167i	0.288251	−	0.499266i	−0.685141	−	0.728410i	$-0.740259\pi$
$659$	7.39970	−	12.8167i	0.288251	−	0.499266i	0.973392	+	0.229144i	$0.0735928\pi$
$660$		0			0
$661$	4.50885	+	7.80956i	0.175374	+	0.303757i	0.940291	−	0.340372i	$-0.110553\pi$
$661$	4.50885	+	7.80956i	0.175374	+	0.303757i	−0.764917	+	0.644129i	$0.777220\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	11.0541			0.428659
$666$		0			0
$667$	44.3638			1.71777
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.97296	−	3.41726i	−0.0761652	−	0.131922i
$672$		0			0
$673$	−11.9803	+	20.7506i	−0.461809	+	0.799876i	−0.999051	−	0.0435519i	$-0.986133\pi$
$673$	−11.9803	+	20.7506i	−0.461809	+	0.799876i	0.537243	+	0.843428i	$0.319466\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.32889	+	5.76581i	−0.127940	+	0.221598i	−0.922878	−	0.385092i	$-0.874170\pi$
$677$	−3.32889	+	5.76581i	−0.127940	+	0.221598i	0.794938	+	0.606690i	$0.207503\pi$
$678$		0			0
$679$	1.10963	+	1.92194i	0.0425837	+	0.0737572i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.0728			1.15070			0.575351	−	0.817907i	$-0.304865\pi$
$683$	30.0728			1.15070			0.575351	+	0.817907i	$0.304865\pi$
$684$		0			0
$685$	−9.22820			−0.352591
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1.71780	−	2.97532i	−0.0654429	−	0.113351i
$690$		0			0
$691$	1.63814	−	2.83733i	0.0623176	−	0.107937i	−0.833183	−	0.552997i	$-0.813484\pi$
$691$	1.63814	−	2.83733i	0.0623176	−	0.107937i	0.895501	+	0.445060i	$0.146817\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.6352	−	32.2771i	0.706874	−	1.22434i
$696$		0			0
$697$	0.690757	+	1.19643i	0.0261643	+	0.0453179i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	32.2891			1.21954			0.609771	−	0.792578i	$-0.291261\pi$
$701$	32.2891			1.21954			0.609771	+	0.792578i	$0.291261\pi$
$702$		0			0
$703$	−43.9253			−1.65667
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.36333	+	2.36135i	0.0512732	+	0.0888078i
$708$		0			0
$709$	24.5438	−	42.5111i	0.921761	−	1.59654i	0.125071	−	0.992148i	$-0.460084\pi$
$709$	24.5438	−	42.5111i	0.921761	−	1.59654i	0.796690	−	0.604388i	$-0.206582\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	27.0693	−	46.8855i	1.01376	−	1.75588i
$714$		0			0
$715$	−5.19076	−	8.99066i	−0.194123	−	0.336231i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	41.9617			1.56491			0.782453	−	0.622710i	$-0.213968\pi$
$719$	41.9617			1.56491			0.782453	+	0.622710i	$0.213968\pi$
$720$		0			0
$721$	17.9823			0.669696
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.25077	+	5.63050i	0.120731	+	0.209112i
$726$		0			0
$727$	14.2434	−	24.6703i	0.528258	−	0.914969i	−0.471200	−	0.882027i	$-0.656179\pi$
$727$	14.2434	−	24.6703i	0.528258	−	0.914969i	0.999457	−	0.0329425i	$-0.0104878\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	0.628802	−	1.08912i	0.0232571	−	0.0402825i
$732$		0			0
$733$	−14.2630	−	24.7043i	−0.526817	−	0.912474i	−0.999512	−	0.0312475i	$-0.990052\pi$
$733$	−14.2630	−	24.7043i	−0.526817	−	0.912474i	0.472695	−	0.881226i	$-0.343281\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.3609			−1.56038
$738$		0			0
$739$	7.85934			0.289110			0.144555	−	0.989497i	$-0.453825\pi$
$739$	7.85934			0.289110			0.144555	+	0.989497i	$0.453825\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.37364	−	5.84332i	−0.123767	−	0.214371i	0.797483	−	0.603341i	$-0.206164\pi$
$743$	−3.37364	−	5.84332i	−0.123767	−	0.214371i	−0.921250	+	0.388970i	$0.872831\pi$
$744$		0			0
$745$	4.62928	−	8.01815i	0.169604	−	0.293762i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	0.554084	−	0.959702i	0.0202458	−	0.0350667i
$750$		0			0
$751$	−11.0900	−	19.2084i	−0.404679	−	0.700925i	0.589605	−	0.807692i	$-0.299283\pi$
$751$	−11.0900	−	19.2084i	−0.404679	−	0.700925i	−0.994284	+	0.106767i	$0.965950\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.5586			0.820990
$756$		0			0
$757$	−20.3815			−0.740779			−0.370389	−	0.928877i	$-0.620776\pi$
$757$	−20.3815			−0.740779			−0.370389	+	0.928877i	$0.620776\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	20.3274	+	35.2081i	0.736869	+	1.27629i	0.953899	+	0.300129i	$0.0970298\pi$
$761$	20.3274	+	35.2081i	0.736869	+	1.27629i	−0.217030	+	0.976165i	$0.569637\pi$
$762$		0			0
$763$	−1.69076	+	2.92848i	−0.0612095	+	0.106018i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−0.890369	+	1.54216i	−0.0321494	+	0.0556843i
$768$		0			0
$769$	16.9518	+	29.3615i	0.611299	+	1.05880i	0.991022	+	0.133701i	$0.0426861\pi$
$769$	16.9518	+	29.3615i	0.611299	+	1.05880i	−0.379723	+	0.925100i	$0.623981\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.74825			0.314653			0.157326	−	0.987547i	$-0.449713\pi$
$773$	8.74825			0.314653			0.157326	+	0.987547i	$0.449713\pi$
$774$		0			0
$775$	7.93406			0.285000
$776$		0			0
$777$		0			0
$778$		0			0
$779$	13.5993	+	23.5547i	0.487246	+	0.843935i
$780$		0			0
$781$	−19.6623	+	34.0560i	−0.703571	+	1.21862i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.29300	−	7.43569i	0.153224	−	0.265391i
$786$		0			0
$787$	−4.64260	−	8.04122i	−0.165491	−	0.286639i	0.771339	−	0.636425i	$-0.219588\pi$
$787$	−4.64260	−	8.04122i	−0.165491	−	0.286639i	−0.936830	+	0.349786i	$0.886254\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−18.8712			−0.670983
$792$		0			0
$793$	−0.780738			−0.0277248
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11.2271	+	19.4460i	0.397685	+	0.688811i	0.993440	−	0.114355i	$-0.0364802\pi$
$797$	11.2271	+	19.4460i	0.397685	+	0.688811i	−0.595754	+	0.803167i	$0.703147\pi$
$798$		0			0
$799$	−0.188816	+	0.327039i	−0.00667983	+	0.0115698i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−23.7075	+	41.0626i	−0.836619	+	1.44907i
$804$		0			0
$805$	5.47150	+	9.47691i	0.192845	+	0.334017i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	5.40215			0.189929			0.0949647	−	0.995481i	$-0.469726\pi$
$809$	5.40215			0.189929			0.0949647	+	0.995481i	$0.469726\pi$
$810$		0			0
$811$	−0.0177088			−0.000621841			−0.000310920	−	1.00000i	$-0.500099\pi$
$811$	−0.0177088			−0.000621841			−0.000310920		1.00000i	$0.500099\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.77042	−	9.99466i	−0.202129	−	0.350098i
$816$		0			0
$817$	12.3796	−	21.4420i	0.433106	−	0.750162i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.528505	+	0.915397i	−0.0184449	+	0.0319476i	−0.875101	−	0.483941i	$-0.839205\pi$
$821$	−0.528505	+	0.915397i	−0.0184449	+	0.0319476i	0.856656	+	0.515889i	$0.172538\pi$
$822$		0			0
$823$	−6.76303	−	11.7139i	−0.235744	−	0.408321i	0.723744	−	0.690068i	$-0.242419\pi$
$823$	−6.76303	−	11.7139i	−0.235744	−	0.408321i	−0.959489	+	0.281747i	$0.909086\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	49.3068			1.71457			0.857283	−	0.514846i	$-0.172151\pi$
$827$	49.3068			1.71457			0.857283	+	0.514846i	$0.172151\pi$
$828$		0			0
$829$	−53.7060			−1.86529			−0.932644	−	0.360799i	$-0.882504\pi$
$829$	−53.7060			−1.86529			−0.932644	+	0.360799i	$0.882504\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.136673	+	0.236725i	0.00473544	+	0.00820203i
$834$		0			0
$835$	11.1864	−	19.3754i	0.387120	−	0.670512i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−18.7163	+	32.4176i	−0.646160	+	1.11918i	0.337873	+	0.941192i	$0.390293\pi$
$839$	−18.7163	+	32.4176i	−0.646160	+	1.11918i	−0.984032	+	0.177990i	$0.943041\pi$
$840$		0			0
$841$	−20.1730	−	34.9407i	−0.695622	−	1.20485i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	24.6490			0.847952
$846$		0			0
$847$	14.5438			0.499730
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−21.7419	−	37.6581i	−0.745303	−	1.29090i
$852$		0			0
$853$	10.3092	−	17.8561i	0.352982	−	0.611382i	−0.633789	−	0.773506i	$-0.718501\pi$
$853$	10.3092	−	17.8561i	0.352982	−	0.611382i	0.986770	+	0.162124i	$0.0518344\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10.4445	−	18.0903i	0.356776	−	0.617954i	−0.630644	−	0.776072i	$-0.717209\pi$
$857$	10.4445	−	18.0903i	0.356776	−	0.617954i	0.987420	+	0.158118i	$0.0505427\pi$
$858$		0			0
$859$	−13.4430	−	23.2839i	−0.458669	−	0.794438i	0.540222	−	0.841523i	$-0.318340\pi$
$859$	−13.4430	−	23.2839i	−0.458669	−	0.794438i	−0.998891	+	0.0470847i	$0.985007\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26.8142			−0.912766			−0.456383	−	0.889784i	$-0.650855\pi$
$863$	−26.8142			−0.912766			−0.456383	+	0.889784i	$0.650855\pi$
$864$		0			0
$865$	29.9912			1.01973
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−32.7075	−	56.6510i	−1.10953	−	1.92175i
$870$		0			0
$871$	−4.19076	+	7.25860i	−0.141998	+	0.245948i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.93706	+	10.2833i	−0.200709	+	0.347639i
$876$		0			0
$877$	18.9538	+	32.8289i	0.640024	+	1.10855i	0.985427	+	0.170099i	$0.0544089\pi$
$877$	18.9538	+	32.8289i	0.640024	+	1.10855i	−0.345403	+	0.938454i	$0.612258\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−7.47782			−0.251934			−0.125967	−	0.992034i	$-0.540203\pi$
$881$	−7.47782			−0.251934			−0.125967	+	0.992034i	$0.540203\pi$
$882$		0			0
$883$	5.07472			0.170778			0.0853889	−	0.996348i	$-0.472787\pi$
$883$	5.07472			0.170778			0.0853889	+	0.996348i	$0.472787\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.26157	+	16.0415i	0.310973	+	0.538621i	0.978573	−	0.205899i	$-0.0660118\pi$
$887$	9.26157	+	16.0415i	0.310973	+	0.538621i	−0.667600	+	0.744520i	$0.732678\pi$
$888$		0			0
$889$	−8.58113	+	14.8629i	−0.287802	+	0.498487i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.71732	+	6.43859i	−0.124395	+	0.215459i
$894$		0			0
$895$	24.1249	+	41.7855i	0.806406	+	1.39674i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−84.6255			−2.82242
$900$		0			0
$901$	0.939108			0.0312862
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.43706	−	2.48906i	−0.0477695	−	0.0827393i
$906$		0			0
$907$	24.8245	−	42.9973i	0.824284	−	1.42770i	−0.0781810	−	0.996939i	$-0.524911\pi$
$907$	24.8245	−	42.9973i	0.824284	−	1.42770i	0.902465	−	0.430763i	$-0.141755\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	11.9808	−	20.7514i	0.396943	−	0.687525i	−0.596404	−	0.802684i	$-0.703405\pi$
$911$	11.9808	−	20.7514i	0.396943	−	0.687525i	0.993347	+	0.115159i	$0.0367379\pi$
$912$		0			0
$913$	−14.4715	−	25.0654i	−0.478937	−	0.829543i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	17.8889			0.590744
$918$		0			0
$919$	−18.8377			−0.621400			−0.310700	−	0.950508i	$-0.600563\pi$
$919$	−18.8377			−0.621400			−0.310700	+	0.950508i	$0.600563\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	3.89037	+	6.73832i	0.128053	+	0.221794i
$924$		0			0
$925$	3.18629	−	5.51882i	0.104765	−	0.181458i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	7.45185	−	12.9070i	0.244487	−	0.423464i	−0.717500	−	0.696558i	$-0.754714\pi$
$929$	7.45185	−	12.9070i	0.244487	−	0.423464i	0.961987	+	0.273094i	$0.0880471\pi$
$930$		0			0
$931$	2.69076	+	4.66053i	0.0881860	+	0.152743i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.83775			0.0928043
$936$		0			0
$937$	3.94299			0.128812			0.0644059	−	0.997924i	$-0.479485\pi$
$937$	3.94299			0.128812			0.0644059	+	0.997924i	$0.479485\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.4056	−	37.0756i	−0.697804	−	1.20863i	−0.969226	−	0.246171i	$-0.920827\pi$
$941$	−21.4056	−	37.0756i	−0.697804	−	1.20863i	0.271423	−	0.962460i	$-0.412506\pi$
$942$		0			0
$943$	−13.4626	+	23.3180i	−0.438404	+	0.759338i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.5919	+	23.5419i	−0.441678	+	0.765009i	−0.997814	−	0.0660823i	$-0.978950\pi$
$947$	−13.5919	+	23.5419i	−0.441678	+	0.765009i	0.556136	+	0.831091i	$0.312283\pi$
$948$		0			0
$949$	4.69076	+	8.12463i	0.152268	+	0.263737i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.12295			−0.0363760			−0.0181880	−	0.999835i	$-0.505790\pi$
$953$	−1.12295			−0.0363760			−0.0181880	+	0.999835i	$0.505790\pi$
$954$		0			0
$955$	47.9912			1.55296
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.24630	−	3.89071i	−0.0725369	−	0.125638i
$960$		0			0
$961$	−36.1357	+	62.5889i	−1.16567	+	2.01900i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.9371	+	25.8717i	−0.480841	+	0.832841i
$966$		0			0
$967$	4.75223	+	8.23111i	0.152822	+	0.264695i	0.932264	−	0.361780i	$-0.117831\pi$
$967$	4.75223	+	8.23111i	0.152822	+	0.264695i	−0.779442	+	0.626474i	$0.784497\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.5979			0.789383			0.394691	−	0.918814i	$-0.370852\pi$
$971$	24.5979			0.789383			0.394691	+	0.918814i	$0.370852\pi$
$972$		0			0
$973$	18.1445			0.581687
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.7016	−	21.9997i	−0.406359	−	0.703834i	0.588120	−	0.808774i	$-0.299868\pi$
$977$	−12.7016	−	21.9997i	−0.406359	−	0.703834i	−0.994479	+	0.104940i	$0.966535\pi$
$978$		0			0
$979$	−34.9164	+	60.4770i	−1.11593	+	1.93285i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.4267	−	19.7917i	0.364457	−	0.631257i	−0.624232	−	0.781239i	$-0.714588\pi$
$983$	11.4267	−	19.7917i	0.364457	−	0.631257i	0.988689	+	0.149982i	$0.0479214\pi$
$984$		0			0
$985$	17.8112	+	30.8499i	0.567512	+	0.982959i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	24.5103			0.779383
$990$		0			0
$991$	24.4690			0.777285			0.388642	−	0.921389i	$-0.372944\pi$
$991$	24.4690			0.777285			0.388642	+	0.921389i	$0.372944\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−11.8559	−	20.5351i	−0.375858	−	0.651006i
$996$		0			0
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	−0.995077	−	0.0991016i	$-0.968403\pi$
$997$	−13.0000	+	22.5167i	−0.411714	+	0.713110i	0.583363	+	0.812211i	$0.301736\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.j.a.505.3		6
3.2	odd	2		252.2.j.b.169.2	yes	6
4.3	odd	2		3024.2.r.i.2017.3		6
7.2	even	3		5292.2.i.d.2125.3		6
7.3	odd	6		5292.2.l.d.3313.3		6
7.4	even	3		5292.2.l.g.3313.1		6
7.5	odd	6		5292.2.i.g.2125.1		6
7.6	odd	2		5292.2.j.e.3529.1		6
9.2	odd	6		2268.2.a.g.1.3		3
9.4	even	3	inner	756.2.j.a.253.3		6
9.5	odd	6		252.2.j.b.85.2	✓	6
9.7	even	3		2268.2.a.j.1.1		3
12.11	even	2		1008.2.r.g.673.2		6
21.2	odd	6		1764.2.i.f.1537.2		6
21.5	even	6		1764.2.i.e.1537.2		6
21.11	odd	6		1764.2.l.d.961.1		6
21.17	even	6		1764.2.l.g.961.3		6
21.20	even	2		1764.2.j.d.1177.2		6
36.7	odd	6		9072.2.a.bz.1.1		3
36.11	even	6		9072.2.a.bt.1.3		3
36.23	even	6		1008.2.r.g.337.2		6
36.31	odd	6		3024.2.r.i.1009.3		6
63.4	even	3		5292.2.i.d.1549.3		6
63.5	even	6		1764.2.l.g.949.3		6
63.13	odd	6		5292.2.j.e.1765.1		6
63.23	odd	6		1764.2.l.d.949.1		6
63.31	odd	6		5292.2.i.g.1549.1		6
63.32	odd	6		1764.2.i.f.373.2		6
63.40	odd	6		5292.2.l.d.361.3		6
63.41	even	6		1764.2.j.d.589.2		6
63.58	even	3		5292.2.l.g.361.1		6
63.59	even	6		1764.2.i.e.373.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.j.b.85.2	✓	6	9.5	odd	6
252.2.j.b.169.2	yes	6	3.2	odd	2
756.2.j.a.253.3		6	9.4	even	3	inner
756.2.j.a.505.3		6	1.1	even	1	trivial
1008.2.r.g.337.2		6	36.23	even	6
1008.2.r.g.673.2		6	12.11	even	2
1764.2.i.e.373.2		6	63.59	even	6
1764.2.i.e.1537.2		6	21.5	even	6
1764.2.i.f.373.2		6	63.32	odd	6
1764.2.i.f.1537.2		6	21.2	odd	6
1764.2.j.d.589.2		6	63.41	even	6
1764.2.j.d.1177.2		6	21.20	even	2
1764.2.l.d.949.1		6	63.23	odd	6
1764.2.l.d.961.1		6	21.11	odd	6
1764.2.l.g.949.3		6	63.5	even	6
1764.2.l.g.961.3		6	21.17	even	6
2268.2.a.g.1.3		3	9.2	odd	6
2268.2.a.j.1.1		3	9.7	even	3
3024.2.r.i.1009.3		6	36.31	odd	6
3024.2.r.i.2017.3		6	4.3	odd	2
5292.2.i.d.1549.3		6	63.4	even	3
5292.2.i.d.2125.3		6	7.2	even	3
5292.2.i.g.1549.1		6	63.31	odd	6
5292.2.i.g.2125.1		6	7.5	odd	6
5292.2.j.e.1765.1		6	63.13	odd	6
5292.2.j.e.3529.1		6	7.6	odd	2
5292.2.l.d.361.3		6	63.40	odd	6
5292.2.l.d.3313.3		6	7.3	odd	6
5292.2.l.g.361.1		6	63.58	even	3
5292.2.l.g.3313.1		6	7.4	even	3
9072.2.a.bt.1.3		3	36.11	even	6
9072.2.a.bz.1.1		3	36.7	odd	6

Overview	Random
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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

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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	756.2.j.a.505.3

Level	$756$
Weight	$2$
Character	756.505
Analytic conductor	$6.037$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$