LMFDB - Embedded newform 784.2.x.i.165.1

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			165.1
Root			$-0.228425 - 1.39564i$ of defining polynomial
Character	$\chi$	$=$	784.165
Dual form			784.2.x.i.765.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.228425 - 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(1.32288 + 2.50000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})$	$q+(-0.228425 - 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(1.32288 + 2.50000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(-0.882113 + 0.236361i) q^{11} +(3.18693 - 2.41733i) q^{16} +(-2.68693 - 3.28335i) q^{18} +(0.531373 + 1.17712i) q^{22} +(4.58258 - 2.64575i) q^{23} +(4.33013 + 2.50000i) q^{25} +(4.29150 - 4.29150i) q^{29} +(-4.10170 - 3.89564i) q^{32} +(-3.96863 + 4.50000i) q^{36} +(3.03490 - 11.3264i) q^{37} +(-8.64575 - 8.64575i) q^{43} +(1.52147 - 1.01049i) q^{44} +(-4.73930 - 5.79129i) q^{46} +(2.50000 - 6.61438i) q^{50} +(-0.398200 + 0.106697i) q^{53} +(-6.96970 - 5.00912i) q^{58} +(-4.50000 + 6.61438i) q^{64} +(3.63729 + 13.5745i) q^{67} -16.0000i q^{71} +(7.18693 + 4.51088i) q^{72} +(-16.5009 - 1.64841i) q^{74} +(7.93725 + 13.7477i) q^{79} +(4.50000 - 7.79423i) q^{81} +(-10.0915 + 14.0413i) q^{86} +(-1.75783 - 1.89261i) q^{88} +(-7.00000 + 7.93725i) q^{92} +(-1.93725 + 1.93725i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.228425	−	1.39564i	−0.161521	−	0.986869i
$2$	−0.228425	−	1.39564i	−0.161521	−	0.986869i
$3$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$4$	−1.89564	+	0.637600i	−0.947822	+	0.318800i
$5$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$5$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$6$		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
$11$	−0.882113	+	0.236361i	−0.265967	+	0.0712656i	−0.389338	−	0.921095i	$-0.627296\pi$
$11$	−0.882113	+	0.236361i	−0.265967	+	0.0712656i	0.123371	+	0.992361i	$0.460630\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$	3.18693	−	2.41733i	0.796733	−	0.604332i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$	−2.68693	−	3.28335i	−0.633316	−	0.773893i
$19$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$19$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$20$		0			0
$21$		0			0
$22$	0.531373	+	1.17712i	0.113289	+	0.250964i
$23$	4.58258	−	2.64575i	0.955533	−	0.551677i	0.0607377	−	0.998154i	$-0.480655\pi$
$23$	4.58258	−	2.64575i	0.955533	−	0.551677i	0.894795	+	0.446476i	$0.147321\pi$
$24$		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	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	4.29150	−	4.29150i	0.796912	−	0.796912i	0.982607	+	0.185695i	$0.0594537\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	−4.10170	−	3.89564i	−0.725085	−	0.688659i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.96863	+	4.50000i	−0.661438	+	0.750000i
$37$	3.03490	−	11.3264i	0.498935	−	1.86205i	−0.00783774	−	0.999969i	$-0.502495\pi$
$37$	3.03490	−	11.3264i	0.498935	−	1.86205i	0.506772	−	0.862080i	$-0.330838\pi$
$38$		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	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−8.64575	−	8.64575i	−1.31846	−	1.31846i	−0.403473	−	0.914991i	$-0.632197\pi$
$44$	1.52147	−	1.01049i	0.229370	−	0.152337i
$45$		0			0
$46$	−4.73930	−	5.79129i	−0.698772	−	0.853879i
$47$		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
$53$	−0.398200	+	0.106697i	−0.0546970	+	0.0146560i	−0.286064	−	0.958211i	$-0.592347\pi$
$53$	−0.398200	+	0.106697i	−0.0546970	+	0.0146560i	0.231367	+	0.972867i	$0.425680\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−6.96970	−	5.00912i	−0.915166	−	0.657730i
$59$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$59$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$60$		0			0
$61$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$61$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$62$		0			0
$63$		0			0
$64$	−4.50000	+	6.61438i	−0.562500	+	0.826797i
$65$		0			0
$66$		0			0
$67$	3.63729	+	13.5745i	0.444365	+	1.65839i	0.717607	+	0.696449i	$0.245238\pi$
$67$	3.63729	+	13.5745i	0.444365	+	1.65839i	−0.273241	+	0.961946i	$0.588096\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	16.0000i		−	1.89885i	−0.313993	−	0.949425i	$-0.601667\pi$
$71$		−	16.0000i		−	1.89885i	0.313993	−	0.949425i	$-0.398333\pi$
$72$	7.18693	+	4.51088i	0.846988	+	0.531612i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	−16.5009	−	1.64841i	−1.91819	−	0.191623i
$75$		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	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$		0			0
$86$	−10.0915	+	14.0413i	−1.08819	+	1.51411i
$87$		0			0
$88$	−1.75783	−	1.89261i	−0.187385	−	0.201752i
$89$		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
$100$	−9.80238	−	1.97822i	−0.980238	−	0.197822i
$101$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$101$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$102$		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
$107$	−2.69184	+	10.0461i	−0.260230	+	0.971192i	0.704875	+	0.709331i	$0.251003\pi$
$107$	−2.69184	+	10.0461i	−0.260230	+	0.971192i	−0.965106	+	0.261861i	$0.915664\pi$
$108$		0			0
$109$	1.35740	+	5.06590i	0.130016	+	0.485225i	0.999969	−	0.00790932i	$-0.00251764\pi$
$109$	1.35740	+	5.06590i	0.130016	+	0.485225i	−0.869953	+	0.493135i	$0.835851\pi$
$110$		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
$116$	−5.39890	+	10.8714i	−0.501275	+	1.00939i
$117$		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	$0.751253\pi$
$128$	10.2592	+	4.76951i	0.906796	+	0.421569i
$129$		0			0
$130$		0			0
$131$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$131$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$132$		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	$0.0261741\pi$
$137$	18.3303	+	10.5830i	1.56606	+	0.904167i	0.569442	+	0.822031i	$0.307159\pi$
$138$		0			0
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$		0			0
$142$	−22.3303	+	3.65480i	−1.87392	+	0.306704i
$143$		0			0
$144$	4.65390	−	11.0608i	0.387825	−	0.921733i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.46863	+	23.4059i	0.120720	+	1.92395i
$149$	5.96310	−	22.2546i	0.488516	−	1.82317i	−0.0751583	−	0.997172i	$-0.523946\pi$
$149$	5.96310	−	22.2546i	0.488516	−	1.82317i	0.563675	−	0.825997i	$-0.309387\pi$
$150$		0			0
$151$	−4.58258	−	2.64575i	−0.372925	−	0.215308i	0.301811	−	0.953368i	$-0.402409\pi$
$151$	−4.58258	−	2.64575i	−0.372925	−	0.215308i	−0.674735	+	0.738060i	$0.735742\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$157$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$158$	17.3739	−	14.2179i	1.38219	−	1.13112i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−11.9059	−	4.50000i	−0.935414	−	0.353553i
$163$	−24.5027	−	6.56549i	−1.91920	−	0.514249i	−0.989170	−	0.146772i	$-0.953112\pi$
$163$	−24.5027	−	6.56549i	−1.91920	−	0.514249i	−0.930033	−	0.367477i	$-0.880222\pi$
$164$		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
$172$	21.9018	+	10.8767i	1.67000	+	0.829343i
$173$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$173$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$174$		0			0
$175$		0			0
$176$	−2.23987	+	2.88562i	−0.168837	+	0.217512i
$177$		0			0
$178$		0			0
$179$	4.11001	+	15.3388i	0.307197	+	1.14647i	0.931038	+	0.364922i	$0.118904\pi$
$179$	4.11001	+	15.3388i	0.307197	+	1.14647i	−0.623841	+	0.781551i	$0.714429\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$	12.6766	+	7.95644i	0.934528	+	0.586556i
$185$		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	0.926212	−	0.377004i	$-0.123046\pi$
$197$	7.70850	+	7.70850i	0.549208	+	0.549208i	−0.377004	+	0.926212i	$0.623046\pi$
$198$	3.14623	+	2.26120i	0.223593	+	0.160696i
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−0.521780	+	14.1325i	−0.0368954	+	0.999319i
$201$		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	−0.413057	+	0.910705i	$0.635539\pi$
$211$	−19.2288	+	19.2288i	−1.32376	+	1.32376i	−0.910705	+	0.413057i	$0.864461\pi$
$212$	0.686815	−	0.456153i	0.0471707	−	0.0313287i
$213$		0			0
$214$	14.6356	+	1.46207i	1.00047	+	0.0999453i
$215$		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
$226$	−4.83485	−	29.5402i	−0.321609	−	1.96499i
$227$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$227$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$228$		0			0
$229$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$229$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$230$		0			0
$231$		0			0
$232$	16.4059	+	5.05163i	1.07710	+	0.331656i
$233$	−18.3303	+	10.5830i	−1.20086	+	0.693316i	−0.960746	−	0.277429i	$-0.910518\pi$
$233$	−18.3303	+	10.5830i	−1.20086	+	0.693316i	−0.240112	+	0.970745i	$0.577184\pi$
$234$		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	$0.166667\pi$
$242$	9.10513	+	11.1262i	0.585300	+	0.715219i
$243$		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	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$	−3.41699	+	3.41699i	−0.214825	+	0.214825i
$254$	3.65480	+	22.3303i	0.229323	+	1.40113i
$255$		0			0
$256$	4.31307	−	15.4077i	0.269567	−	0.962982i
$257$		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
$261$	4.71240	−	17.5869i	0.291690	−	1.08860i
$262$		0			0
$263$	−27.7128	−	16.0000i	−1.70885	−	0.986602i	−0.935995	−	0.352014i	$-0.885497\pi$
$263$	−27.7128	−	16.0000i	−1.70885	−	0.986602i	−0.772851	−	0.634588i	$-0.781170\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−15.5501	−	23.4134i	−0.949876	−	1.43020i
$269$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$269$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$270$		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
$275$	−4.41056	−	1.18181i	−0.265967	−	0.0712656i
$276$		0			0
$277$	−28.5151	+	7.64060i	−1.71331	+	0.459079i	−0.976231	−	0.216731i	$-0.930460\pi$
$277$	−28.5151	+	7.64060i	−1.71331	+	0.459079i	−0.737075	+	0.675810i	$0.763794\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			21.1660i			1.26266i	0.775515	+	0.631329i	$0.217490\pi$
$281$			21.1660i			1.26266i	−0.775515	+	0.631329i	$0.782510\pi$
$282$		0			0
$283$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$283$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$284$	10.2016	+	30.3303i	0.605354	+	1.79977i
$285$		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	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$	32.3308	−	7.39617i	1.87919	−	0.429894i
$297$		0			0
$298$	−32.4216	−	3.23886i	−1.87813	−	0.187622i
$299$		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	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		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
$317$	−15.9941	−	4.28561i	−0.898318	−	0.240704i	−0.220024	−	0.975494i	$-0.570614\pi$
$317$	−15.9941	−	4.28561i	−0.898318	−	0.240704i	−0.678294	+	0.734791i	$0.737280\pi$
$318$		0			0
$319$	−2.77124	+	4.79993i	−0.155160	+	0.268745i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−3.56080	+	17.6443i	−0.197822	+	0.980238i
$325$		0			0
$326$	−3.56604	+	35.6968i	−0.197505	+	1.97706i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−5.62005	+	20.9743i	−0.308906	+	1.15285i	0.620625	+	0.784107i	$0.286879\pi$
$331$	−5.62005	+	20.9743i	−0.308906	+	1.15285i	−0.929531	+	0.368744i	$0.879788\pi$
$332$		0			0
$333$	−9.10470	−	33.9792i	−0.498935	−	1.86205i
$334$		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	$0.695579\pi$
$338$	18.1434	−	2.96953i	0.986869	−	0.161521i
$339$		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
$347$	−28.0312	+	7.51094i	−1.50479	+	0.403208i	−0.914702	−	0.404128i	$-0.867575\pi$
$347$	−28.0312	+	7.51094i	−1.50479	+	0.403208i	−0.590091	+	0.807337i	$0.700908\pi$
$348$		0			0
$349$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$349$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$	4.53894	+	2.46691i	0.241926	+	0.131487i
$353$		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	$-0.234375\pi$
$359$	32.0780	−	18.5203i	1.69301	−	0.977462i	0.952063	−	0.305903i	$-0.0989582\pi$
$360$		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	$0.166667\pi$
$368$	8.20871	−	19.5094i	0.427909	−	1.01700i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.78419	−	6.65870i	0.0923820	−	0.344774i	−0.904227	−	0.427051i	$-0.859552\pi$
$373$	1.78419	−	6.65870i	0.0923820	−	0.344774i	0.996610	+	0.0822766i	$0.0262191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−12.5203	−	12.5203i	−0.643123	−	0.643123i	0.308199	−	0.951322i	$-0.400274\pi$
$379$	−12.5203	−	12.5203i	−0.643123	−	0.643123i	−0.951322	+	0.308199i	$0.900274\pi$
$380$		0			0
$381$		0			0
$382$	28.9564	−	23.6965i	1.48154	−	1.21242i
$383$		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
$387$	−35.4309	−	9.49369i	−1.80106	−	0.482592i
$388$		0			0
$389$	−33.1828	+	8.89131i	−1.68244	+	0.450807i	−0.968420	−	0.249323i	$-0.919792\pi$
$389$	−33.1828	+	8.89131i	−1.68244	+	0.450807i	−0.714015	+	0.700130i	$0.753125\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	8.99750	−	12.5191i	0.453288	−	0.630705i
$395$		0			0
$396$	2.43715	−	4.90754i	0.122471	−	0.246613i
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$		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	$-0.750000\pi$
$419$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$	−28.8745	−	28.8745i	−1.40726	−	1.40726i	−0.773676	−	0.633581i	$-0.781584\pi$
$421$	−28.8745	−	28.8745i	−1.40726	−	1.40726i	−0.633581	−	0.773676i	$-0.718416\pi$
$422$	31.2288	+	22.4442i	1.52020	+	1.09256i
$423$		0			0
$424$	−0.793512	−	0.854353i	−0.0385364	−	0.0414910i
$425$		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
$436$	−5.80318	−	8.73766i	−0.277922	−	0.418458i
$437$		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
$443$	10.4391	−	38.9594i	0.495978	−	1.85102i	−0.0285009	−	0.999594i	$-0.509073\pi$
$443$	10.4391	−	38.9594i	0.495978	−	1.85102i	0.524479	−	0.851423i	$-0.324260\pi$
$444$		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	$0.515028\pi$
$450$	−3.42638	−	20.9347i	−0.161521	−	0.986869i
$451$		0			0
$452$	−40.1232	+	13.4955i	−1.88724	+	0.634773i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.19615	+	3.00000i	−0.243066	+	0.140334i	−0.616585	−	0.787288i	$-0.711484\pi$
$457$	−5.19615	+	3.00000i	−0.243066	+	0.140334i	0.373519	+	0.927622i	$0.378151\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$461$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$	15.8745			0.737751			0.368875	−	0.929479i	$-0.379743\pi$
$463$	15.8745			0.737751			0.368875	+	0.929479i	$0.379743\pi$
$464$	3.30276	−	24.0507i	0.153327	−	1.11653i
$465$		0			0
$466$	18.9572	+	23.1652i	0.878176	+	1.07310i
$467$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$467$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$468$		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
$478$	−3.65480	−	22.3303i	−0.167167	−	1.02136i
$479$		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	−0.454911	−	0.890537i	$-0.650329\pi$
$487$	−32.0780	−	18.5203i	−1.45359	−	0.839233i	−0.998683	+	0.0513038i	$0.983662\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.6458	+	24.6458i	1.11225	+	1.11225i	0.992846	+	0.119401i	$0.0380974\pi$
$491$	24.6458	+	24.6458i	1.11225	+	1.11225i	0.119401	+	0.992846i	$0.461903\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	6.51764	+	1.74640i	0.291770	+	0.0781794i	0.401735	−	0.915756i	$-0.368407\pi$
$499$	6.51764	+	1.74640i	0.291770	+	0.0781794i	−0.109965	+	0.993935i	$0.535074\pi$
$500$		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
$506$	5.54944	+	3.98838i	0.246702	+	0.177305i
$507$		0			0
$508$	30.3303	−	10.2016i	1.34569	−	0.452623i
$509$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$509$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$510$		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	$0.666667\pi$
$522$	−25.6215	−	2.55954i	−1.12142	−	0.112028i
$523$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$523$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$524$		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
$536$	−29.1247	+	27.0506i	−1.25799	+	1.16841i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	44.9074	+	12.0329i	1.93072	+	0.517335i	0.974216	+	0.225617i	$0.0724399\pi$
$541$	44.9074	+	12.0329i	1.93072	+	0.517335i	0.956504	+	0.291718i	$0.0942267\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−14.0627	+	14.0627i	−0.601280	+	0.601280i	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−14.0627	+	14.0627i	−0.601280	+	0.601280i	0.339372	+	0.940652i	$0.389785\pi$
$548$	−41.4955	−	8.37420i	−1.77260	−	0.357728i
$549$		0			0
$550$	−0.641898	+	6.42553i	−0.0273706	+	0.273986i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	17.1771	+	38.0516i	0.729786	+	1.61666i
$555$		0			0
$556$		0			0
$557$	10.3554	+	38.6469i	0.438773	+	1.63752i	0.731873	+	0.681441i	$0.238646\pi$
$557$	10.3554	+	38.6469i	0.438773	+	1.63752i	−0.293101	+	0.956082i	$0.594687\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	29.5402	−	4.83485i	1.24608	−	0.203946i
$563$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$563$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$564$		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	−0.843025	−	0.537874i	$-0.819228\pi$
$569$	−19.0526	+	11.0000i	−0.798725	+	0.461144i	0.0443003	+	0.999018i	$0.485894\pi$
$570$		0			0
$571$	29.7954	−	7.98366i	1.24690	−	0.334106i	0.425762	−	0.904835i	$-0.360006\pi$
$571$	29.7954	−	7.98366i	1.24690	−	0.334106i	0.821138	+	0.570730i	$0.193340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	26.4575			1.10335
$576$	−1.76978	+	23.9347i	−0.0737406	+	0.997277i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	18.6057	−	15.2259i	0.773893	−	0.633316i
$579$		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	$-0.250000\pi$
$587$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−17.7076	−	43.4328i	−0.727777	−	1.78508i
$593$		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	0.944516	−	0.328465i	$-0.106531\pi$
$599$	27.7128	+	16.0000i	1.13231	+	0.653742i	0.187799	+	0.982208i	$0.439865\pi$
$600$		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
$604$	10.3739	+	2.09355i	0.422107	+	0.0851854i
$605$		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
$613$	4.26950	−	1.14401i	0.172444	−	0.0462061i	−0.171564	−	0.985173i	$-0.554882\pi$
$613$	4.26950	−	1.14401i	0.172444	−	0.0462061i	0.344008	+	0.938967i	$0.388215\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	26.0000i		−	1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$		−	26.0000i		−	1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$		0			0
$619$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$619$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$620$		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	$0.103171\pi$
$631$			16.0000i			0.636950i	−0.947931	+	0.318475i	$0.896829\pi$
$632$	−23.8693	+	38.0297i	−0.949470	+	1.51274i
$633$		0			0
$634$	−2.32773	+	23.3010i	−0.0924458	+	0.925402i
$635$		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	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		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	$0.666667\pi$
$648$	25.4385	+	0.939205i	0.999319	+	0.0368954i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	50.6346	−	3.17712i	1.98301	−	0.124426i
$653$	−26.9223	−	7.21381i	−1.05355	−	0.282298i	−0.309833	−	0.950791i	$-0.600273\pi$
$653$	−26.9223	−	7.21381i	−1.05355	−	0.282298i	−0.743719	+	0.668493i	$0.766940\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	35.2288	−	35.2288i	1.37232	−	1.37232i	0.515319	−	0.856998i	$-0.327673\pi$
$659$	35.2288	−	35.2288i	1.37232	−	1.37232i	0.856998	−	0.515319i	$-0.172327\pi$
$660$		0			0
$661$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$661$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$662$	30.5564	+	3.05253i	1.18761	+	0.118640i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−45.3431	+	20.4686i	−1.75701	+	0.793143i
$667$	8.31189	−	31.0204i	0.321838	−	1.20111i
$668$		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	$0.696248\pi$
$674$	4.83485	+	29.5402i	0.186231	+	1.13785i
$675$		0			0
$676$	−8.28880	−	24.6434i	−0.318800	−	0.947822i
$677$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$677$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	31.9025	−	8.54825i	1.22072	−	0.327090i	0.409757	−	0.912194i	$-0.365613\pi$
$683$	31.9025	−	8.54825i	1.22072	−	0.327090i	0.810958	+	0.585105i	$0.198947\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−48.4530	−	6.65382i	−1.84725	−	0.253674i
$689$		0			0
$690$		0			0
$691$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$691$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$692$		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	−0.999286	−	0.0377695i	$-0.987975\pi$
$701$	−25.4575	+	25.4575i	−0.961517	+	0.961517i	0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$		0			0
$704$	2.40612	−	6.89825i	0.0906842	−	0.259988i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10.7822	+	40.2397i	−0.404934	+	1.51123i	0.399244	+	0.916845i	$0.369273\pi$
$709$	−10.7822	+	40.2397i	−0.404934	+	1.51123i	−0.804178	+	0.594389i	$0.797394\pi$
$710$		0			0
$711$	41.2432	+	23.8118i	1.54674	+	0.893011i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−17.5711	−	26.4563i	−0.656663	−	0.988718i
$717$		0			0
$718$	−33.1751	−	40.5390i	−1.23808	−	1.51290i
$719$		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
$725$	29.3115	−	7.85399i	1.08860	−	0.291690i
$726$		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		0.258819	−	0.965926i	$-0.416667\pi$
$733$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$734$		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
$739$	12.4219	+	46.3592i	0.456947	+	1.70535i	0.682300	+	0.731072i	$0.260980\pi$
$739$	12.4219	+	46.3592i	0.456947	+	1.70535i	−0.225354	+	0.974277i	$0.572354\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	37.0405i		−	1.35888i	−0.733729	−	0.679442i	$-0.762222\pi$
$743$		−	37.0405i		−	1.35888i	0.733729	−	0.679442i	$-0.237778\pi$
$744$		0			0
$745$		0			0
$746$	−9.70073	−	0.969084i	−0.355169	−	0.0354807i
$747$		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	−0.981332	−	0.192323i	$-0.938398\pi$
$757$	−32.2915	−	32.2915i	−1.17365	−	1.17365i	−0.192323	−	0.981332i	$-0.561602\pi$
$758$	−14.6139	+	20.3338i	−0.530800	+	0.738556i
$759$		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
$772$	−8.37420	+	41.4955i	−0.301394	+	1.49345i
$773$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$773$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$774$	−5.15650	+	51.6176i	−0.185346	+	1.85536i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	19.9889	+	44.2804i	0.716636	+	1.58753i
$779$		0			0
$780$		0			0
$781$	3.78178	+	14.1138i	0.135323	+	0.505032i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$787$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$788$	−19.5275	−	9.69763i	−0.695639	−	0.345464i
$789$		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	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	−8.02178	−	27.1229i	−0.283613	−	0.958939i
$801$		0			0
$802$	23.1652	−	18.9572i	0.817990	−	0.669402i
$803$		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	0.950587	−	0.310457i	$-0.100482\pi$
$809$	32.9090	+	19.0000i	1.15702	+	0.668004i	0.206430	+	0.978461i	$0.433815\pi$
$810$		0			0
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$	14.9452	−	2.44609i	0.523831	−	0.0857354i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13.7104	+	51.1679i	−0.478496	+	1.78577i	0.129217	+	0.991616i	$0.458754\pi$
$821$	−13.7104	+	51.1679i	−0.478496	+	1.78577i	−0.607714	+	0.794156i	$0.707913\pi$
$822$		0			0
$823$	27.7128	+	16.0000i	0.966008	+	0.557725i	0.898017	−	0.439961i	$-0.145008\pi$
$823$	27.7128	+	16.0000i	0.966008	+	0.557725i	0.0679910	+	0.997686i	$0.478341\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.47974	−	3.47974i	−0.121002	−	0.121002i	0.644013	−	0.765015i	$-0.277268\pi$
$827$	−3.47974	−	3.47974i	−0.121002	−	0.121002i	−0.765015	+	0.644013i	$0.777268\pi$
$828$	−6.28065	+	31.1216i	−0.218268	+	1.08155i
$829$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$829$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$830$		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
$842$	−33.7029	+	46.8942i	−1.16148	+	1.61608i
$843$		0			0
$844$	24.1906	−	48.7111i	0.832675	−	1.67671i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−1.01111	+	1.30262i	−0.0347218	+	0.0447320i
$849$		0			0
$850$		0			0
$851$	−16.0592	−	59.9337i	−0.550502	−	2.05450i
$852$		0			0
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$	−28.6762	+	6.56012i	−0.980132	+	0.224220i
$857$		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		−0.965926	−	0.258819i	$-0.916667\pi$
$859$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$860$		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
$872$	−10.8691	+	10.0951i	−0.368073	+	0.341862i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	55.8356	+	14.9611i	1.88543	+	0.505201i	0.999112	+	0.0421327i	$0.0134152\pi$
$877$	55.8356	+	14.9611i	1.88543	+	0.505201i	0.886323	+	0.463068i	$0.153251\pi$
$878$		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	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	23.1033	−	23.1033i	0.777487	−	0.777487i	0.979403	+	0.201916i	$0.0647168\pi$
$884$		0			0
$885$		0			0
$886$	−56.7580	−	5.67002i	−1.90682	−	0.190488i
$887$		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
$891$	−2.12725	+	7.93901i	−0.0712656	+	0.265967i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.456850	+	2.79129i	0.0152453	+	0.0931465i
$899$		0			0
$900$	−28.4347	+	9.56400i	−0.947822	+	0.318800i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	28.0000	+	52.9150i	0.931266	+	1.75993i
$905$		0			0
$906$		0			0
$907$	44.5949	−	11.9492i	1.48075	−	0.396766i	0.574148	−	0.818752i	$-0.305333\pi$
$907$	44.5949	−	11.9492i	1.48075	−	0.396766i	0.906602	+	0.421986i	$0.138667\pi$
$908$		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
$914$	5.37386	+	6.56670i	0.177752	+	0.217207i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−41.5692	+	24.0000i	−1.37124	+	0.791687i	−0.991084	−	0.133235i	$-0.957464\pi$
$919$	−41.5692	+	24.0000i	−1.37124	+	0.791687i	−0.380158	+	0.924922i	$0.624130\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	41.4575	−	41.4575i	1.36311	−	1.36311i
$926$	−3.62614	−	22.1552i	−0.119162	−	0.728064i
$927$		0			0
$928$	−34.3206	+	0.884298i	−1.12663	+	0.0290285i
$929$		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		0.965926	−	0.258819i	$-0.0833333\pi$
$941$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	5.58301	−	14.7712i	0.181519	−	0.480255i
$947$	53.4161	+	14.3128i	1.73579	+	0.465103i	0.981504	−	0.191444i	$-0.0613171\pi$
$947$	53.4161	+	14.3128i	1.73579	+	0.465103i	0.754285	+	0.656547i	$0.227984\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	21.1660i		−	0.685634i	−0.939402	−	0.342817i	$-0.888619\pi$
$953$		−	21.1660i		−	0.685634i	0.939402	−	0.342817i	$-0.111381\pi$
$954$	1.42026	+	1.02074i	0.0459827	+	0.0330477i
$955$		0			0
$956$	−30.3303	+	10.2016i	−0.980952	+	0.329943i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$		0			0
$963$	8.07553	+	30.1383i	0.260230	+	0.971192i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			47.6235i			1.53147i	0.643157	+	0.765735i	$0.277624\pi$
$967$			47.6235i			1.53147i	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−24.3541	−	15.2859i	−0.782772	−	0.491307i
$969$		0			0
$970$		0			0
$971$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$971$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$972$		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
$982$	28.7670	−	40.0264i	0.917991	−	1.27729i
$983$		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
$989$	−62.4943	−	16.7453i	−1.98720	−	0.532470i
$990$		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		−0.258819	−	0.965926i	$-0.583333\pi$
$997$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$998$	0.948555	−	9.49523i	0.0300260	−	0.300566i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.i.165.1		8
7.2	even	3	inner	784.2.x.i.373.1		8
7.3	odd	6		784.2.m.f.197.2	✓	4
7.4	even	3		784.2.m.f.197.2	✓	4
7.5	odd	6	inner	784.2.x.i.373.1		8
7.6	odd	2	CM	784.2.x.i.165.1		8
16.13	even	4	inner	784.2.x.i.557.1		8
112.13	odd	4	inner	784.2.x.i.557.1		8
112.45	odd	12		784.2.m.f.589.2	yes	4
112.61	odd	12	inner	784.2.x.i.765.1		8
112.93	even	12	inner	784.2.x.i.765.1		8
112.109	even	12		784.2.m.f.589.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.2.m.f.197.2	✓	4	7.3	odd	6
784.2.m.f.197.2	✓	4	7.4	even	3
784.2.m.f.589.2	yes	4	112.45	odd	12
784.2.m.f.589.2	yes	4	112.109	even	12
784.2.x.i.165.1		8	1.1	even	1	trivial
784.2.x.i.165.1		8	7.6	odd	2	CM
784.2.x.i.373.1		8	7.2	even	3	inner
784.2.x.i.373.1		8	7.5	odd	6	inner
784.2.x.i.557.1		8	16.13	even	4	inner
784.2.x.i.557.1		8	112.13	odd	4	inner
784.2.x.i.765.1		8	112.61	odd	12	inner
784.2.x.i.765.1		8	112.93	even	12	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)

\(f(q)\)	\(=\)	\(q+(-0.228425 - 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(1.32288 + 2.50000i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)	\(q+(-0.228425 - 1.39564i) q^{2} +(-1.89564 + 0.637600i) q^{4} +(1.32288 + 2.50000i) q^{8} +(2.59808 - 1.50000i) q^{9} +(-0.882113 + 0.236361i) q^{11} +(3.18693 - 2.41733i) q^{16} +(-2.68693 - 3.28335i) q^{18} +(0.531373 + 1.17712i) q^{22} +(4.58258 - 2.64575i) q^{23} +(4.33013 + 2.50000i) q^{25} +(4.29150 - 4.29150i) q^{29} +(-4.10170 - 3.89564i) q^{32} +(-3.96863 + 4.50000i) q^{36} +(3.03490 - 11.3264i) q^{37} +(-8.64575 - 8.64575i) q^{43} +(1.52147 - 1.01049i) q^{44} +(-4.73930 - 5.79129i) q^{46} +(2.50000 - 6.61438i) q^{50} +(-0.398200 + 0.106697i) q^{53} +(-6.96970 - 5.00912i) q^{58} +(-4.50000 + 6.61438i) q^{64} +(3.63729 + 13.5745i) q^{67} -16.0000i q^{71} +(7.18693 + 4.51088i) q^{72} +(-16.5009 - 1.64841i) q^{74} +(7.93725 + 13.7477i) q^{79} +(4.50000 - 7.79423i) q^{81} +(-10.0915 + 14.0413i) q^{86} +(-1.75783 - 1.89261i) q^{88} +(-7.00000 + 7.93725i) q^{92} +(-1.93725 + 1.93725i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{4}+O(q^{10}) \) 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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