LMFDB - Embedded newform 784.2.x.i.557.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,2,Mod(165,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 8]))

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

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

chi := DirichletCharacter("784.165");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.x (of order $12$, degree $4$, 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:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label			557.1
Root			$-1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	784.557
Dual form			784.2.x.i.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} +(1.32288 - 2.50000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})$	$q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} +(1.32288 - 2.50000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +(0.236361 + 0.882113i) q^{11} +(-3.68693 + 1.55130i) q^{16} +(4.18693 + 0.685275i) q^{18} +(0.531373 - 1.17712i) q^{22} +(-4.58258 + 2.64575i) q^{23} +(-4.33013 - 2.50000i) q^{25} +(4.29150 + 4.29150i) q^{29} +(5.42458 + 1.60436i) q^{32} +(-3.96863 - 4.50000i) q^{36} +(-11.3264 - 3.03490i) q^{37} +(-8.64575 + 8.64575i) q^{43} +(-1.63585 + 0.812383i) q^{44} +(7.38505 + 1.20871i) q^{46} +(2.50000 + 6.61438i) q^{50} +(0.106697 + 0.398200i) q^{53} +(-0.853179 - 8.54050i) q^{58} +(-4.50000 - 6.61438i) q^{64} +(-13.5745 + 3.63729i) q^{67} +16.0000i q^{71} +(0.313068 + 8.47950i) q^{72} +(9.67800 + 13.4660i) q^{74} +(7.93725 + 13.7477i) q^{79} +(4.50000 - 7.79423i) q^{81} +(17.2059 - 1.71883i) q^{86} +(2.51796 + 0.576022i) q^{88} +(-7.00000 - 7.93725i) q^{92} +(-1.93725 - 1.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{4}+O(q^{10}) $ 8 * q - 6 * q^4	$ 8 q - 6 q^{4} + 8 q^{11} - 2 q^{16} + 6 q^{18} + 36 q^{22} - 8 q^{29} - 12 q^{37} - 48 q^{43} + 26 q^{44} + 20 q^{50} + 20 q^{53} - 26 q^{58} - 36 q^{64} - 8 q^{67} + 30 q^{72} - 22 q^{74} + 36 q^{81} + 2 q^{86} - 34 q^{88} - 56 q^{92} + 48 q^{99}+O(q^{100}) $ 8 * q - 6 * q^4 + 8 * q^11 - 2 * q^16 + 6 * q^18 + 36 * q^22 - 8 * q^29 - 12 * q^37 - 48 * q^43 + 26 * q^44 + 20 * q^50 + 20 * q^53 - 26 * q^58 - 36 * q^64 - 8 * q^67 + 30 * q^72 - 22 * q^74 + 36 * q^81 + 2 * q^86 - 34 * q^88 - 56 * q^92 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	−1.09445	−	0.895644i	−0.773893	−	0.633316i
$2$	−1.09445	−	0.895644i	−0.773893	−	0.633316i
$3$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$3$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$4$	0.395644	+	1.96048i	0.197822	+	0.980238i
$5$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$5$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	1.32288	−	2.50000i	0.467707	−	0.883883i
$9$	−2.59808	+	1.50000i	−0.866025	+	0.500000i
$10$		0			0
$11$	0.236361	+	0.882113i	0.0712656	+	0.265967i	0.992361	−	0.123371i	$-0.0393705\pi$
$11$	0.236361	+	0.882113i	0.0712656	+	0.265967i	−0.921095	+	0.389338i	$0.872704\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$	−3.68693	+	1.55130i	−0.921733	+	0.387825i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	4.18693	+	0.685275i	0.986869	+	0.161521i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$		0			0
$21$		0			0
$22$	0.531373	−	1.17712i	0.113289	−	0.250964i
$23$	−4.58258	+	2.64575i	−0.955533	+	0.551677i	−0.894795	−	0.446476i	$-0.852679\pi$
$23$	−4.58258	+	2.64575i	−0.955533	+	0.551677i	−0.0607377	+	0.998154i	$0.519345\pi$
$24$		0			0
$25$	−4.33013	−	2.50000i	−0.866025	−	0.500000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.29150	+	4.29150i	0.796912	+	0.796912i	0.982607	−	0.185695i	$-0.0594537\pi$
$29$	4.29150	+	4.29150i	0.796912	+	0.796912i	−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	5.42458	+	1.60436i	0.958939	+	0.283613i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.96863	−	4.50000i	−0.661438	−	0.750000i
$37$	−11.3264	−	3.03490i	−1.86205	−	0.498935i	−0.862080	−	0.506772i	$-0.830838\pi$
$37$	−11.3264	−	3.03490i	−1.86205	−	0.498935i	−0.999969	+	0.00783774i	$0.997505\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−8.64575	+	8.64575i	−1.31846	+	1.31846i	−0.403473	+	0.914991i	$0.632197\pi$
$43$	−8.64575	+	8.64575i	−1.31846	+	1.31846i	−0.914991	+	0.403473i	$0.867803\pi$
$44$	−1.63585	+	0.812383i	−0.246613	+	0.122471i
$45$		0			0
$46$	7.38505	+	1.20871i	1.08887	+	0.178215i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		0			0
$49$		0			0
$50$	2.50000	+	6.61438i	0.353553	+	0.935414i
$51$		0			0
$52$		0			0
$53$	0.106697	+	0.398200i	0.0146560	+	0.0546970i	0.972867	−	0.231367i	$-0.0743197\pi$
$53$	0.106697	+	0.398200i	0.0146560	+	0.0546970i	−0.958211	+	0.286064i	$0.907653\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.853179	−	8.54050i	−0.112028	−	1.12142i
$59$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$59$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$61$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$62$		0			0
$63$		0			0
$64$	−4.50000	−	6.61438i	−0.562500	−	0.826797i
$65$		0			0
$66$		0			0
$67$	−13.5745	+	3.63729i	−1.65839	+	0.444365i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−13.5745	+	3.63729i	−1.65839	+	0.444365i	−0.696449	+	0.717607i	$0.745238\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			16.0000i			1.89885i	0.313993	+	0.949425i	$0.398333\pi$
$71$			16.0000i			1.89885i	−0.313993	+	0.949425i	$0.601667\pi$
$72$	0.313068	+	8.47950i	0.0368954	+	0.999319i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	9.67800	+	13.4660i	1.12504	+	1.56539i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.93725	+	13.7477i	0.893011	+	1.54674i	0.836247	+	0.548352i	$0.184745\pi$
$79$	7.93725	+	13.7477i	0.893011	+	1.54674i	0.0567635	+	0.998388i	$0.481922\pi$
$80$		0			0
$81$	4.50000	−	7.79423i	0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$	17.2059	−	1.71883i	1.85536	−	0.185346i
$87$		0			0
$88$	2.51796	+	0.576022i	0.268415	+	0.0614041i
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$	−7.00000	−	7.93725i	−0.729800	−	0.827516i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	−1.93725	−	1.93725i	−0.194701	−	0.194701i
$100$	3.18800	−	9.47822i	0.318800	−	0.947822i
$101$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$101$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$	0.239870	−	0.531373i	0.0232983	−	0.0516115i
$107$	10.0461	+	2.69184i	0.971192	+	0.260230i	0.709331	−	0.704875i	$-0.248997\pi$
$107$	10.0461	+	2.69184i	0.971192	+	0.260230i	0.261861	+	0.965106i	$0.415664\pi$
$108$		0			0
$109$	−5.06590	+	1.35740i	−0.485225	+	0.130016i	−0.493135	−	0.869953i	$-0.664149\pi$
$109$	−5.06590	+	1.35740i	−0.485225	+	0.130016i	0.00790932	+	0.999969i	$0.497482\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	21.1660			1.99113			0.995565	−	0.0940721i	$-0.0299884\pi$
$113$	21.1660			1.99113			0.995565	+	0.0940721i	$0.0299884\pi$
$114$		0			0
$115$		0			0
$116$	−6.71548	+	10.1113i	−0.623517	+	0.938810i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	8.80402	−	5.08301i	0.800366	−	0.462091i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−0.999100	+	11.2695i	−0.0883088	+	0.996093i
$129$		0			0
$130$		0			0
$131$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$131$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$132$		0			0
$133$		0			0
$134$	18.1144	+	8.17712i	1.56484	+	0.706396i
$135$		0			0
$136$		0			0
$137$	−18.3303	−	10.5830i	−1.56606	−	0.904167i	−0.996621	−	0.0821359i	$-0.973826\pi$
$137$	−18.3303	−	10.5830i	−1.56606	−	0.904167i	−0.569442	−	0.822031i	$-0.692841\pi$
$138$		0			0
$139$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$139$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$140$		0			0
$141$		0			0
$142$	14.3303	−	17.5112i	1.20257	−	1.46951i
$143$		0			0
$144$	7.25198	−	9.56080i	0.604332	−	0.796733i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.46863	−	23.4059i	0.120720	−	1.92395i
$149$	−22.2546	−	5.96310i	−1.82317	−	0.488516i	−0.825997	−	0.563675i	$-0.809387\pi$
$149$	−22.2546	−	5.96310i	−1.82317	−	0.488516i	−0.997172	+	0.0751583i	$0.976054\pi$
$150$		0			0
$151$	4.58258	+	2.64575i	0.372925	+	0.215308i	0.674735	−	0.738060i	$-0.264258\pi$
$151$	4.58258	+	2.64575i	0.372925	+	0.215308i	−0.301811	+	0.953368i	$0.597591\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$157$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$158$	3.62614	−	22.1552i	0.288480	−	1.76257i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−11.9059	+	4.50000i	−0.935414	+	0.353553i
$163$	6.56549	−	24.5027i	0.514249	−	1.91920i	0.146772	−	0.989170i	$-0.453112\pi$
$163$	6.56549	−	24.5027i	0.514249	−	1.91920i	0.367477	−	0.930033i	$-0.380222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$		0			0
$172$	−20.3704	−	13.5291i	−1.55323	−	1.03159i
$173$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$173$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$174$		0			0
$175$		0			0
$176$	−2.23987	−	2.88562i	−0.168837	−	0.217512i
$177$		0			0
$178$		0			0
$179$	−15.3388	+	4.11001i	−1.14647	+	0.307197i	−0.781551	−	0.623841i	$-0.785571\pi$
$179$	−15.3388	+	4.11001i	−1.14647	+	0.307197i	−0.364922	+	0.931038i	$0.618904\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	0.552200	+	14.9564i	0.0407088	+	1.10260i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.2288	+	22.9129i	0.957199	+	1.65792i	0.729253	+	0.684244i	$0.239868\pi$
$191$	13.2288	+	22.9129i	0.957199	+	1.65792i	0.227946	+	0.973674i	$0.426799\pi$
$192$		0			0
$193$	10.5830	−	18.3303i	0.761781	−	1.31944i	−0.180150	−	0.983639i	$-0.557658\pi$
$193$	10.5830	−	18.3303i	0.761781	−	1.31944i	0.941932	−	0.335805i	$-0.109008\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	7.70850	−	7.70850i	0.549208	−	0.549208i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$	7.70850	−	7.70850i	0.549208	−	0.549208i	0.926212	+	0.377004i	$0.123046\pi$
$198$	0.385139	+	3.85532i	0.0273706	+	0.273986i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−11.9782	+	7.51813i	−0.846988	+	0.531612i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.93725	−	13.7477i	0.551677	−	0.955533i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−19.2288	−	19.2288i	−1.32376	−	1.32376i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	−19.2288	−	19.2288i	−1.32376	−	1.32376i	−0.413057	−	0.910705i	$-0.635539\pi$
$212$	−0.738447	+	0.366723i	−0.0507168	+	0.0251866i
$213$		0			0
$214$	−8.58402	−	11.9438i	−0.586791	−	0.816462i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	6.76013	+	3.05163i	0.457854	+	0.206683i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$	−23.1652	−	18.9572i	−1.54092	−	1.26101i
$227$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$227$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$228$		0			0
$229$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$229$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$230$		0			0
$231$		0			0
$232$	16.4059	−	5.05163i	1.07710	−	0.331656i
$233$	18.3303	−	10.5830i	1.20086	−	0.693316i	0.240112	−	0.970745i	$-0.422816\pi$
$233$	18.3303	−	10.5830i	1.20086	−	0.693316i	0.960746	+	0.277429i	$0.0894825\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	−14.1881	−	2.32217i	−0.912048	−	0.149275i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$251$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$	−3.41699	−	3.41699i	−0.214825	−	0.214825i
$254$	17.5112	+	14.3303i	1.09875	+	0.899163i
$255$		0			0
$256$	11.1869	−	11.4391i	0.699183	−	0.714943i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−17.5869	−	4.71240i	−1.08860	−	0.291690i
$262$		0			0
$263$	27.7128	+	16.0000i	1.70885	+	0.986602i	0.935995	+	0.352014i	$0.114503\pi$
$263$	27.7128	+	16.0000i	1.70885	+	0.986602i	0.772851	+	0.634588i	$0.218830\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−12.5015	−	25.1735i	−0.763651	−	1.53772i
$269$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$269$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$	10.5830	+	28.0000i	0.639343	+	1.69154i
$275$	1.18181	−	4.41056i	0.0712656	−	0.265967i
$276$		0			0
$277$	7.64060	+	28.5151i	0.459079	+	1.71331i	0.675810	+	0.737075i	$0.263794\pi$
$277$	7.64060	+	28.5151i	0.459079	+	1.71331i	−0.216731	+	0.976231i	$0.569540\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	21.1660i		−	1.26266i	−0.775515	−	0.631329i	$-0.782510\pi$
$281$		−	21.1660i		−	1.26266i	0.775515	−	0.631329i	$-0.217490\pi$
$282$		0			0
$283$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$283$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$284$	−31.3676	+	6.33030i	−1.86133	+	0.375634i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.5000	+	3.96863i	−0.972272	+	0.233854i
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$293$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$294$		0			0
$295$		0			0
$296$	−22.5707	+	24.3012i	−1.31189	+	1.41248i
$297$		0			0
$298$	19.0157	+	26.4585i	1.10155	+	1.53270i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−2.64575	−	7.00000i	−0.152246	−	0.402805i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$307$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		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
$316$	−23.8118	+	21.0000i	−1.33952	+	1.18134i
$317$	4.28561	−	15.9941i	0.240704	−	0.898318i	−0.734791	−	0.678294i	$-0.762720\pi$
$317$	4.28561	−	15.9941i	0.240704	−	0.898318i	0.975494	−	0.220024i	$-0.0706137\pi$
$318$		0			0
$319$	−2.77124	+	4.79993i	−0.155160	+	0.268745i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	17.0608	+	5.73840i	0.947822	+	0.318800i
$325$		0			0
$326$	−29.1313	+	20.9367i	−1.61344	+	1.15958i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	20.9743	+	5.62005i	1.15285	+	0.308906i	0.784107	−	0.620625i	$-0.213121\pi$
$331$	20.9743	+	5.62005i	1.15285	+	0.308906i	0.368744	+	0.929531i	$0.379788\pi$
$332$		0			0
$333$	33.9792	−	9.10470i	1.86205	−	0.498935i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−21.1660			−1.15299			−0.576493	−	0.817102i	$-0.695579\pi$
$337$	−21.1660			−1.15299			−0.576493	+	0.817102i	$0.695579\pi$
$338$	−11.6434	+	14.2279i	−0.633316	+	0.773893i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	10.1771	+	33.0516i	0.548714	+	1.78202i
$345$		0			0
$346$		0			0
$347$	7.51094	+	28.0312i	0.403208	+	1.50479i	0.807337	+	0.590091i	$0.200908\pi$
$347$	7.51094	+	28.0312i	0.403208	+	1.50479i	−0.404128	+	0.914702i	$0.632425\pi$
$348$		0			0
$349$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$349$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$350$		0			0
$351$		0			0
$352$	−0.133062	+	5.16430i	−0.00709224	+	0.275258i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	20.4686	+	9.23987i	1.08180	+	0.488342i
$359$	−32.0780	+	18.5203i	−1.69301	+	0.977462i	−0.740951	+	0.671559i	$0.765625\pi$
$359$	−32.0780	+	18.5203i	−1.69301	+	0.977462i	−0.952063	+	0.305903i	$0.901042\pi$
$360$		0			0
$361$	16.4545	+	9.50000i	0.866025	+	0.500000i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$	12.7913	−	16.8637i	0.666792	−	0.879079i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−6.65870	−	1.78419i	−0.344774	−	0.0923820i	0.0822766	−	0.996610i	$-0.473781\pi$
$373$	−6.65870	−	1.78419i	−0.344774	−	0.0923820i	−0.427051	+	0.904227i	$0.640448\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12.5203	+	12.5203i	−0.643123	+	0.643123i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$	−12.5203	+	12.5203i	−0.643123	+	0.643123i	0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$	6.04356	−	36.9253i	0.309215	−	1.88926i
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$	−28.0000	+	10.5830i	−1.42516	+	0.538661i
$387$	9.49369	−	35.4309i	0.482592	−	1.80106i
$388$		0			0
$389$	8.89131	+	33.1828i	0.450807	+	1.68244i	0.700130	+	0.714015i	$0.253125\pi$
$389$	8.89131	+	33.1828i	0.450807	+	1.68244i	−0.249323	+	0.968420i	$0.580208\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−15.3406	+	1.53250i	−0.772850	+	0.0772062i
$395$		0			0
$396$	3.03148	−	4.56440i	0.152337	−	0.229370i
$397$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$397$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$398$		0			0
$399$		0			0
$400$	19.8431	+	2.50000i	0.992157	+	0.125000i
$401$	10.5830	+	18.3303i	0.528490	+	0.915372i	0.999448	+	0.0332161i	$0.0105750\pi$
$401$	10.5830	+	18.3303i	0.528490	+	0.915372i	−0.470958	+	0.882156i	$0.656092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	10.7085i		−	0.530801i
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−21.0000	+	7.93725i	−1.03209	+	0.390095i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	−28.8745	+	28.8745i	−1.40726	+	1.40726i	−0.633581	+	0.773676i	$0.718416\pi$
$421$	−28.8745	+	28.8745i	−1.40726	+	1.40726i	−0.773676	+	0.633581i	$0.781584\pi$
$422$	3.82280	+	38.2670i	0.186091	+	1.86281i
$423$		0			0
$424$	1.13665	+	0.260026i	0.0552005	+	0.0126280i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.30262	+	20.7601i	−0.0629643	+	1.00348i
$429$		0			0
$430$		0			0
$431$	16.0000	−	27.7128i	0.770693	−	1.33488i	−0.166491	−	0.986043i	$-0.553244\pi$
$431$	16.0000	−	27.7128i	0.770693	−	1.33488i	0.937184	−	0.348836i	$-0.113423\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$	−4.66545	−	9.39453i	−0.223435	−	0.449916i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−38.9594	−	10.4391i	−1.85102	−	0.495978i	−0.851423	−	0.524479i	$-0.824260\pi$
$443$	−38.9594	−	10.4391i	−1.85102	−	0.495978i	−0.999594	+	0.0285009i	$0.990927\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$	−16.4168	−	13.4347i	−0.773893	−	0.633316i
$451$		0			0
$452$	8.37420	+	41.4955i	0.393889	+	1.95178i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.19615	−	3.00000i	0.243066	−	0.140334i	−0.373519	−	0.927622i	$-0.621849\pi$
$457$	5.19615	−	3.00000i	0.243066	−	0.140334i	0.616585	+	0.787288i	$0.288516\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$461$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$	15.8745			0.737751			0.368875	−	0.929479i	$-0.379743\pi$
$463$	15.8745			0.737751			0.368875	+	0.929479i	$0.379743\pi$
$464$	−22.4799	−	9.16507i	−1.04360	−	0.425478i
$465$		0			0
$466$	−29.5402	−	4.83485i	−1.36842	−	0.223970i
$467$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$467$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−9.67005	−	5.58301i	−0.444629	−	0.256707i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.874508	−	0.874508i	−0.0400410	−	0.0400410i
$478$	−17.5112	−	14.3303i	−0.800944	−	0.655453i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	13.4484	+	15.2490i	0.611289	+	0.693137i
$485$		0			0
$486$		0			0
$487$	32.0780	+	18.5203i	1.45359	+	0.839233i	0.998683	−	0.0513038i	$-0.0163377\pi$
$487$	32.0780	+	18.5203i	1.45359	+	0.839233i	0.454911	+	0.890537i	$0.349671\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.6458	−	24.6458i	1.11225	−	1.11225i	0.119401	−	0.992846i	$-0.461903\pi$
$491$	24.6458	−	24.6458i	1.11225	−	1.11225i	0.992846	−	0.119401i	$-0.0380974\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.74640	+	6.51764i	−0.0781794	+	0.291770i	−0.993935	−	0.109965i	$-0.964926\pi$
$499$	−1.74640	+	6.51764i	−0.0781794	+	0.291770i	0.915756	+	0.401735i	$0.131593\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$	0.679321	+	6.80014i	0.0301995	+	0.302303i
$507$		0			0
$508$	−6.33030	−	31.3676i	−0.280862	−	1.39171i
$509$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$509$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$510$		0			0
$511$		0			0
$512$	−22.4889	+	2.50000i	−0.993878	+	0.110485i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$	15.0274	+	20.9091i	0.657730	+	0.915166i
$523$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$523$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$524$		0			0
$525$		0			0
$526$	−16.0000	−	42.3320i	−0.697633	−	1.84576i
$527$		0			0
$528$		0			0
$529$	2.50000	−	4.33013i	0.108696	−	0.188266i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−8.86421	+	38.7480i	−0.382876	+	1.67366i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−12.0329	+	44.9074i	−0.517335	+	1.93072i	−0.225617	+	0.974216i	$0.572440\pi$
$541$	−12.0329	+	44.9074i	−0.517335	+	1.93072i	−0.291718	+	0.956504i	$0.594227\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.0627	−	14.0627i	−0.601280	−	0.601280i	0.339372	−	0.940652i	$-0.389785\pi$
$547$	−14.0627	−	14.0627i	−0.601280	−	0.601280i	−0.940652	+	0.339372i	$0.889785\pi$
$548$	13.4955	−	40.1232i	0.576497	−	1.71398i
$549$		0			0
$550$	−5.24372	+	3.76867i	−0.223593	+	0.160696i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	17.1771	−	38.0516i	0.729786	−	1.61666i
$555$		0			0
$556$		0			0
$557$	−38.6469	+	10.3554i	−1.63752	+	0.438773i	−0.956082	−	0.293101i	$-0.905313\pi$
$557$	−38.6469	+	10.3554i	−1.63752	+	0.438773i	−0.681441	+	0.731873i	$0.738646\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−18.9572	+	23.1652i	−0.799662	+	0.977163i
$563$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$563$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	40.0000	+	21.1660i	1.67836	+	0.888106i
$569$	19.0526	−	11.0000i	0.798725	−	0.461144i	−0.0443003	−	0.999018i	$-0.514106\pi$
$569$	19.0526	−	11.0000i	0.798725	−	0.461144i	0.843025	+	0.537874i	$0.180772\pi$
$570$		0			0
$571$	−7.98366	−	29.7954i	−0.334106	−	1.24690i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−7.98366	−	29.7954i	−0.334106	−	1.24690i	0.570730	−	0.821138i	$-0.306660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26.4575			1.10335
$576$	21.6129	+	10.4347i	0.900538	+	0.434777i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	3.88323	−	23.7259i	0.161521	−	0.986869i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.326038	+	0.188238i	−0.0135031	+	0.00779603i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	46.4677	−	6.38119i	1.90981	−	0.262265i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$	2.88562	−	45.9889i	0.118200	−	1.88378i
$597$		0			0
$598$		0			0
$599$	−27.7128	−	16.0000i	−1.13231	−	0.653742i	−0.187799	−	0.982208i	$-0.560135\pi$
$599$	−27.7128	−	16.0000i	−1.13231	−	0.653742i	−0.944516	+	0.328465i	$0.893469\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	29.8118	−	29.8118i	1.21403	−	1.21403i
$604$	−3.37386	+	10.0308i	−0.137281	+	0.408148i
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.14401	−	4.26950i	−0.0462061	−	0.172444i	0.938967	−	0.344008i	$-0.111785\pi$
$613$	−1.14401	−	4.26950i	−0.0462061	−	0.172444i	−0.985173	+	0.171564i	$0.945118\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			26.0000i			1.04672i	0.852111	+	0.523360i	$0.175322\pi$
$617$			26.0000i			1.04672i	−0.852111	+	0.523360i	$0.824678\pi$
$618$		0			0
$619$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$619$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$620$		0			0
$621$		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
$631$		−	16.0000i		−	0.636950i	−0.947931	−	0.318475i	$-0.896829\pi$
$631$		−	16.0000i		−	0.636950i	0.947931	−	0.318475i	$-0.103171\pi$
$632$	44.8693	−	1.65660i	1.78481	−	0.0658961i
$633$		0			0
$634$	−19.0154	+	13.6664i	−0.755198	+	0.542761i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	7.33202	−	2.77124i	0.290277	−	0.109715i
$639$	−24.0000	−	41.5692i	−0.949425	−	1.64445i
$640$		0			0
$641$	−10.5830	+	18.3303i	−0.418004	+	0.724003i	−0.995739	−	0.0922210i	$-0.970603\pi$
$641$	−10.5830	+	18.3303i	−0.418004	+	0.724003i	0.577735	+	0.816224i	$0.303937\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	−13.5326	−	21.5608i	−0.531612	−	0.846988i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	50.6346	+	3.17712i	1.98301	+	0.124426i
$653$	7.21381	−	26.9223i	0.282298	−	1.05355i	−0.668493	−	0.743719i	$-0.733060\pi$
$653$	7.21381	−	26.9223i	0.282298	−	1.05355i	0.950791	−	0.309833i	$-0.100273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	35.2288	+	35.2288i	1.37232	+	1.37232i	0.856998	+	0.515319i	$0.172327\pi$
$659$	35.2288	+	35.2288i	1.37232	+	1.37232i	0.515319	+	0.856998i	$0.327673\pi$
$660$		0			0
$661$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$661$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$662$	−17.9218	−	24.9364i	−0.696549	−	0.969179i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−45.3431	−	20.4686i	−1.75701	−	0.793143i
$667$	−31.0204	−	8.31189i	−1.20111	−	0.321838i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−30.0000			−1.15642			−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000			−1.15642			−0.578208	+	0.815890i	$0.696248\pi$
$674$	23.1652	+	18.9572i	0.892288	+	0.730204i
$675$		0			0
$676$	25.4862	−	5.14337i	0.980238	−	0.197822i
$677$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$677$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.54825	−	31.9025i	−0.327090	−	1.22072i	−0.912194	−	0.409757i	$-0.865613\pi$
$683$	−8.54825	−	31.9025i	−0.327090	−	1.22072i	0.585105	−	0.810958i	$-0.301053\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	18.4641	−	45.2885i	0.703939	−	1.72661i
$689$		0			0
$690$		0			0
$691$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$691$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$692$		0			0
$693$		0			0
$694$	16.8856	−	37.4059i	0.640969	−	1.41991i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−25.4575	−	25.4575i	−0.961517	−	0.961517i	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	−25.4575	−	25.4575i	−0.961517	−	0.961517i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$		0			0
$703$		0			0
$704$	4.77100	−	5.53289i	0.179814	−	0.208529i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.2397	+	10.7822i	1.51123	+	0.404934i	0.916845	−	0.399244i	$-0.130727\pi$
$709$	40.2397	+	10.7822i	1.51123	+	0.404934i	0.594389	+	0.804178i	$0.297394\pi$
$710$		0			0
$711$	−41.2432	−	23.8118i	−1.54674	−	0.893011i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−14.1263	−	28.4452i	−0.527923	−	1.06305i
$717$		0			0
$718$	51.6954	+	8.46099i	1.92925	+	0.315761i
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$		0			0
$722$	−9.50000	−	25.1346i	−0.353553	−	0.935414i
$723$		0			0
$724$		0			0
$725$	−7.85399	−	29.3115i	−0.291690	−	1.08860i
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$733$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$734$		0			0
$735$		0			0
$736$	−29.1033	+	7.00000i	−1.07276	+	0.258023i
$737$	−6.41699	−	11.1146i	−0.236373	−	0.409410i
$738$		0			0
$739$	−46.3592	+	12.4219i	−1.70535	+	0.456947i	−0.974277	−	0.225354i	$-0.927646\pi$
$739$	−46.3592	+	12.4219i	−1.70535	+	0.456947i	−0.731072	+	0.682300i	$0.760980\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			37.0405i			1.35888i	0.733729	+	0.679442i	$0.237778\pi$
$743$			37.0405i			1.35888i	−0.733729	+	0.679442i	$0.762222\pi$
$744$		0			0
$745$		0			0
$746$	5.68962	+	7.91654i	0.208312	+	0.289845i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	24.0000	+	41.5692i	0.875772	+	1.51688i	0.855938	+	0.517079i	$0.172981\pi$
$751$	24.0000	+	41.5692i	0.875772	+	1.51688i	0.0198348	+	0.999803i	$0.493686\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−32.2915	+	32.2915i	−1.17365	+	1.17365i	−0.192323	+	0.981332i	$0.561602\pi$
$757$	−32.2915	+	32.2915i	−1.17365	+	1.17365i	−0.981332	+	0.192323i	$0.938398\pi$
$758$	24.9165	−	2.48911i	0.905008	−	0.0904086i
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	−39.6863	+	35.0000i	−1.43580	+	1.26626i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	40.1232	+	13.4955i	1.44407	+	0.485712i
$773$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$773$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$774$	−42.1239	+	30.2745i	−1.51411	+	1.08819i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	19.9889	−	44.2804i	0.716636	−	1.58753i
$779$		0			0
$780$		0			0
$781$	−14.1138	+	3.78178i	−0.505032	+	0.135323i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$787$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$788$	18.1621	+	12.0625i	0.647000	+	0.429709i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−7.40588	+	2.28039i	−0.263157	+	0.0810301i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$797$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	−19.4782	−	20.5085i	−0.688659	−	0.725085i
$801$		0			0
$802$	4.83485	−	29.5402i	0.170724	−	1.04310i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32.9090	−	19.0000i	−1.15702	−	0.668004i	−0.206430	−	0.978461i	$-0.566185\pi$
$809$	−32.9090	−	19.0000i	−1.15702	−	0.668004i	−0.950587	+	0.310457i	$0.899518\pi$
$810$		0			0
$811$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$811$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$	−9.59100	+	11.7199i	−0.336164	+	0.410783i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	51.1679	+	13.7104i	1.78577	+	0.478496i	0.991616	−	0.129217i	$-0.0412465\pi$
$821$	51.1679	+	13.7104i	1.78577	+	0.478496i	0.794156	+	0.607714i	$0.207913\pi$
$822$		0			0
$823$	−27.7128	−	16.0000i	−0.966008	−	0.557725i	−0.0679910	−	0.997686i	$-0.521659\pi$
$823$	−27.7128	−	16.0000i	−0.966008	−	0.557725i	−0.898017	+	0.439961i	$0.854992\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.47974	+	3.47974i	−0.121002	+	0.121002i	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−3.47974	+	3.47974i	−0.121002	+	0.121002i	0.644013	+	0.765015i	$0.277268\pi$
$828$	30.0924	+	10.1216i	1.04578	+	0.351750i
$829$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$829$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$			7.83399i			0.270138i
$842$	57.4630	−	5.74044i	1.98031	−	0.197829i
$843$		0			0
$844$	30.0898	−	45.3053i	1.03573	−	1.55947i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.01111	−	1.30262i	−0.0347218	−	0.0447320i
$849$		0			0
$850$		0			0
$851$	59.9337	−	16.0592i	2.05450	−	0.550502i
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$	20.0193	−	21.5543i	0.684247	−	0.736709i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$859$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$860$		0			0
$861$		0			0
$862$	−42.3320	+	16.0000i	−1.44183	+	0.544962i
$863$	−29.1033	−	50.4083i	−0.990687	−	1.71592i	−0.613263	−	0.789879i	$-0.710143\pi$
$863$	−29.1033	−	50.4083i	−0.990687	−	1.71592i	−0.377424	−	0.926041i	$-0.623190\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−10.2510	+	10.2510i	−0.347741	+	0.347741i
$870$		0			0
$871$		0			0
$872$	−3.30805	+	14.4604i	−0.112025	+	0.489692i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−14.9611	+	55.8356i	−0.505201	+	1.88543i	−0.0421327	+	0.999112i	$0.513415\pi$
$877$	−14.9611	+	55.8356i	−0.505201	+	1.88543i	−0.463068	+	0.886323i	$0.653251\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	23.1033	+	23.1033i	0.777487	+	0.777487i	0.979403	−	0.201916i	$-0.0647168\pi$
$883$	23.1033	+	23.1033i	0.777487	+	0.777487i	−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$	33.2894	+	46.3189i	1.11838	+	1.55611i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	7.93901	+	2.12725i	0.265967	+	0.0712656i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	2.18890	+	1.79129i	0.0730446	+	0.0597760i
$899$		0			0
$900$	5.93466	+	29.4071i	0.197822	+	0.980238i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	28.0000	−	52.9150i	0.931266	−	1.75993i
$905$		0			0
$906$		0			0
$907$	−11.9492	−	44.5949i	−0.396766	−	1.48075i	−0.818752	−	0.574148i	$-0.805333\pi$
$907$	−11.9492	−	44.5949i	−0.396766	−	1.48075i	0.421986	−	0.906602i	$-0.361333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.0000			0.530104			0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000			0.530104			0.265052	+	0.964234i	$0.414611\pi$
$912$		0			0
$913$		0			0
$914$	−8.37386	−	1.37055i	−0.276983	−	0.0453338i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	41.5692	−	24.0000i	1.37124	−	0.791687i	0.380158	−	0.924922i	$-0.375870\pi$
$919$	41.5692	−	24.0000i	1.37124	−	0.791687i	0.991084	+	0.133235i	$0.0425364\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	41.4575	+	41.4575i	1.36311	+	1.36311i
$926$	−17.3739	−	14.2179i	−0.570941	−	0.467229i
$927$		0			0
$928$	16.3945	+	30.1647i	0.538176	+	0.990204i
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$		0			0
$932$	28.0000	+	31.7490i	0.917170	+	1.03997i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$941$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	5.58301	+	14.7712i	0.181519	+	0.480255i
$947$	−14.3128	+	53.4161i	−0.465103	+	1.73579i	0.191444	+	0.981504i	$0.438683\pi$
$947$	−14.3128	+	53.4161i	−0.465103	+	1.73579i	−0.656547	+	0.754285i	$0.727984\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.1660i			0.685634i	0.939402	+	0.342817i	$0.111381\pi$
$953$			21.1660i			0.685634i	−0.939402	+	0.342817i	$0.888619\pi$
$954$	0.173858	+	1.74035i	0.00562886	+	0.0563460i
$955$		0			0
$956$	6.33030	+	31.3676i	0.204737	+	1.01450i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	−30.1383	+	8.07553i	−0.971192	+	0.260230i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	47.6235i		−	1.53147i	−0.643157	−	0.765735i	$-0.722376\pi$
$967$		−	47.6235i		−	1.53147i	0.643157	−	0.765735i	$-0.277624\pi$
$968$	−1.06088	−	28.7342i	−0.0340981	−	0.923554i
$969$		0			0
$970$		0			0
$971$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$971$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$972$		0			0
$973$		0			0
$974$	−18.5203	−	49.0000i	−0.593427	−	1.57006i
$975$		0			0
$976$		0			0
$977$	−23.0000	+	39.8372i	−0.735835	+	1.27450i	0.218521	+	0.975832i	$0.429877\pi$
$977$	−23.0000	+	39.8372i	−0.735835	+	1.27450i	−0.954356	+	0.298672i	$0.903456\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	11.1255	−	11.1255i	0.355210	−	0.355210i
$982$	−49.0474	+	4.89974i	−1.56516	+	0.156357i
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	16.7453	−	62.4943i	0.532470	−	1.98720i
$990$		0			0
$991$	−29.1033	+	50.4083i	−0.924496	+	1.60127i	−0.132125	+	0.991233i	$0.542180\pi$
$991$	−29.1033	+	50.4083i	−0.924496	+	1.60127i	−0.792370	+	0.610040i	$0.791153\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$997$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$998$	7.74883	−	5.56909i	0.245285	−	0.176286i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.i.557.1		8
7.2	even	3	inner	784.2.x.i.765.1		8
7.3	odd	6		784.2.m.f.589.2	yes	4
7.4	even	3		784.2.m.f.589.2	yes	4
7.5	odd	6	inner	784.2.x.i.765.1		8
7.6	odd	2	CM	784.2.x.i.557.1		8
16.5	even	4	inner	784.2.x.i.165.1		8
112.5	odd	12	inner	784.2.x.i.373.1		8
112.37	even	12	inner	784.2.x.i.373.1		8
112.53	even	12		784.2.m.f.197.2	✓	4
112.69	odd	4	inner	784.2.x.i.165.1		8
112.101	odd	12		784.2.m.f.197.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.2.m.f.197.2	✓	4	112.53	even	12
784.2.m.f.197.2	✓	4	112.101	odd	12
784.2.m.f.589.2	yes	4	7.3	odd	6
784.2.m.f.589.2	yes	4	7.4	even	3
784.2.x.i.165.1		8	16.5	even	4	inner
784.2.x.i.165.1		8	112.69	odd	4	inner
784.2.x.i.373.1		8	112.5	odd	12	inner
784.2.x.i.373.1		8	112.37	even	12	inner
784.2.x.i.557.1		8	1.1	even	1	trivial
784.2.x.i.557.1		8	7.6	odd	2	CM
784.2.x.i.765.1		8	7.2	even	3	inner
784.2.x.i.765.1		8	7.5	odd	6	inner

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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} +(1.32288 - 2.50000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +O(q^{10})\)	\(q+(-1.09445 - 0.895644i) q^{2} +(0.395644 + 1.96048i) q^{4} +(1.32288 - 2.50000i) q^{8} +(-2.59808 + 1.50000i) q^{9} +(0.236361 + 0.882113i) q^{11} +(-3.68693 + 1.55130i) q^{16} +(4.18693 + 0.685275i) q^{18} +(0.531373 - 1.17712i) q^{22} +(-4.58258 + 2.64575i) q^{23} +(-4.33013 - 2.50000i) q^{25} +(4.29150 + 4.29150i) q^{29} +(5.42458 + 1.60436i) q^{32} +(-3.96863 - 4.50000i) q^{36} +(-11.3264 - 3.03490i) q^{37} +(-8.64575 + 8.64575i) q^{43} +(-1.63585 + 0.812383i) q^{44} +(7.38505 + 1.20871i) q^{46} +(2.50000 + 6.61438i) q^{50} +(0.106697 + 0.398200i) q^{53} +(-0.853179 - 8.54050i) q^{58} +(-4.50000 - 6.61438i) q^{64} +(-13.5745 + 3.63729i) q^{67} +16.0000i q^{71} +(0.313068 + 8.47950i) q^{72} +(9.67800 + 13.4660i) q^{74} +(7.93725 + 13.7477i) q^{79} +(4.50000 - 7.79423i) q^{81} +(17.2059 - 1.71883i) q^{86} +(2.51796 + 0.576022i) q^{88} +(-7.00000 - 7.93725i) q^{92} +(-1.93725 - 1.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4}+O(q^{10}) \) 8 * q - 6 * q^4	\( 8 q - 6 q^{4} + 8 q^{11} - 2 q^{16} + 6 q^{18} + 36 q^{22} - 8 q^{29} - 12 q^{37} - 48 q^{43} + 26 q^{44} + 20 q^{50} + 20 q^{53} - 26 q^{58} - 36 q^{64} - 8 q^{67} + 30 q^{72} - 22 q^{74} + 36 q^{81} + 2 q^{86} - 34 q^{88} - 56 q^{92} + 48 q^{99}+O(q^{100}) \) 8 * q - 6 * q^4 + 8 * q^11 - 2 * q^16 + 6 * q^18 + 36 * q^22 - 8 * q^29 - 12 * q^37 - 48 * q^43 + 26 * q^44 + 20 * q^50 + 20 * q^53 - 26 * q^58 - 36 * q^64 - 8 * q^67 + 30 * q^72 - 22 * q^74 + 36 * q^81 + 2 * q^86 - 34 * q^88 - 56 * q^92 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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