LMFDB - Embedded newform 392.2.m.a.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [392,2,Mod(19,392)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("392.19"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(392, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	392.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,-1,0,3,0,0,0,-10,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$3.13013575923$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			19.1
Root			$1.39564 - 0.228425i$ of defining polynomial
Character	$\chi$	$=$	392.19
Dual form			392.2.m.a.227.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.39564 - 0.228425i) q^{2} +(1.89564 + 0.637600i) q^{4} +(-2.50000 - 1.32288i) q^{8} +(-1.50000 + 2.59808i) q^{9} +(2.00000 + 3.46410i) q^{11} +(3.18693 + 2.41733i) q^{16} +(2.68693 - 3.28335i) q^{18} +(-2.00000 - 5.29150i) q^{22} +(-4.58258 - 2.64575i) q^{23} +(2.50000 + 4.33013i) q^{25} +10.5830i q^{29} +(-3.89564 - 4.10170i) q^{32} +(-4.50000 + 3.96863i) q^{36} +(9.16515 + 5.29150i) q^{37} +12.0000 q^{43} +(1.58258 + 7.84190i) q^{44} +(5.79129 + 4.73930i) q^{46} +(-2.50000 - 6.61438i) q^{50} +(-9.16515 + 5.29150i) q^{53} +(2.41742 - 14.7701i) q^{58} +(4.50000 + 6.61438i) q^{64} +(-2.00000 - 3.46410i) q^{67} -5.29150i q^{71} +(7.18693 - 4.51088i) q^{72} +(-11.5826 - 9.47860i) q^{74} +(-13.7477 - 7.93725i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-16.7477 - 2.74110i) q^{86} +(-0.417424 - 11.3060i) q^{88} +(-7.00000 - 7.93725i) q^{92} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 10 q^{8} - 6 q^{9} + 8 q^{11} - q^{16} - 3 q^{18} - 8 q^{22} + 10 q^{25} - 11 q^{32} - 18 q^{36} + 48 q^{43} - 12 q^{44} + 14 q^{46} - 10 q^{50} + 28 q^{58} + 18 q^{64} - 8 q^{67}+ \cdots - 48 q^{99}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 10 * q^8 - 6 * q^9 + 8 * q^11 - q^16 - 3 * q^18 - 8 * q^22 + 10 * q^25 - 11 * q^32 - 18 * q^36 + 48 * q^43 - 12 * q^44 + 14 * q^46 - 10 * q^50 + 28 * q^58 + 18 * q^64 - 8 * q^67 + 15 * q^72 - 28 * q^74 - 18 * q^81 - 12 * q^86 - 20 * q^88 - 28 * q^92 - 48 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−1.39564	−	0.228425i	−0.986869	−	0.161521i
$2$	−1.39564	−	0.228425i	−0.986869	−	0.161521i
$3$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$	1.89564	+	0.637600i	0.947822	+	0.318800i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−2.50000	−	1.32288i	−0.883883	−	0.467707i
$9$	−1.50000	+	2.59808i	−0.500000	+	0.866025i
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	3.18693	+	2.41733i	0.796733	+	0.604332i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$	2.68693	−	3.28335i	0.633316	−	0.773893i
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$	−2.00000	−	5.29150i	−0.426401	−	1.12815i
$23$	−4.58258	−	2.64575i	−0.955533	−	0.551677i	−0.0607377	−	0.998154i	$-0.519345\pi$
$23$	−4.58258	−	2.64575i	−0.955533	−	0.551677i	−0.894795	+	0.446476i	$0.852679\pi$
$24$		0			0
$25$	2.50000	+	4.33013i	0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			10.5830i			1.96521i	0.185695	+	0.982607i	$0.440546\pi$
$29$			10.5830i			1.96521i	−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	−3.89564	−	4.10170i	−0.688659	−	0.725085i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.50000	+	3.96863i	−0.750000	+	0.661438i
$37$	9.16515	+	5.29150i	1.50674	+	0.869918i	0.999969	+	0.00783774i	$0.00249486\pi$
$37$	9.16515	+	5.29150i	1.50674	+	0.869918i	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
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$	1.58258	+	7.84190i	0.238582	+	1.18221i
$45$		0			0
$46$	5.79129	+	4.73930i	0.853879	+	0.698772i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$	−2.50000	−	6.61438i	−0.353553	−	0.935414i
$51$		0			0
$52$		0			0
$53$	−9.16515	+	5.29150i	−1.25893	+	0.726844i	−0.972867	−	0.231367i	$-0.925680\pi$
$53$	−9.16515	+	5.29150i	−1.25893	+	0.726844i	−0.286064	+	0.958211i	$0.592347\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	2.41742	−	14.7701i	0.317423	−	1.93941i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	4.50000	+	6.61438i	0.562500	+	0.826797i
$65$		0			0
$66$		0			0
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	0.717607	−	0.696449i	$-0.245238\pi$
$67$	−2.00000	−	3.46410i	−0.244339	−	0.423207i	−0.961946	+	0.273241i	$0.911904\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	5.29150i		−	0.627986i	−0.949425	−	0.313993i	$-0.898333\pi$
$71$		−	5.29150i		−	0.627986i	0.949425	−	0.313993i	$-0.101667\pi$
$72$	7.18693	−	4.51088i	0.846988	−	0.531612i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	−11.5826	−	9.47860i	−1.34645	−	1.10187i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−13.7477	−	7.93725i	−1.54674	−	0.893011i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−13.7477	−	7.93725i	−1.54674	−	0.893011i	−0.548352	−	0.836247i	$-0.684745\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	−16.7477	−	2.74110i	−1.80595	−	0.295581i
$87$		0			0
$88$	−0.417424	−	11.3060i	−0.0444976	−	1.20522i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		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		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	−12.0000			−1.20605
$100$	1.97822	+	9.80238i	0.197822	+	0.980238i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$	14.0000	−	5.29150i	1.35980	−	0.513956i
$107$	10.0000	−	17.3205i	0.966736	−	1.67444i	0.261861	−	0.965106i	$-0.415664\pi$
$107$	10.0000	−	17.3205i	0.966736	−	1.67444i	0.704875	−	0.709331i	$-0.251003\pi$
$108$		0			0
$109$	9.16515	−	5.29150i	0.877862	−	0.506834i	0.00790932	−	0.999969i	$-0.497482\pi$
$109$	9.16515	−	5.29150i	0.877862	−	0.506834i	0.869953	+	0.493135i	$0.164149\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$	−6.74773	+	20.0616i	−0.626511	+	1.86267i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	15.8745i		−	1.40863i	−0.709885	−	0.704317i	$-0.751253\pi$
$127$		−	15.8745i		−	1.40863i	0.709885	−	0.704317i	$-0.248747\pi$
$128$	−4.76951	−	10.2592i	−0.421569	−	0.906796i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	2.00000	+	5.29150i	0.172774	+	0.457116i
$135$		0			0
$136$		0			0
$137$	−5.00000	−	8.66025i	−0.427179	−	0.739895i	0.569442	−	0.822031i	$-0.307159\pi$
$137$	−5.00000	−	8.66025i	−0.427179	−	0.739895i	−0.996621	+	0.0821359i	$0.973826\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$	−1.20871	+	7.38505i	−0.101433	+	0.619740i
$143$		0			0
$144$	−11.0608	+	4.65390i	−0.921733	+	0.387825i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	14.0000	+	15.8745i	1.15079	+	1.30488i
$149$	9.16515	+	5.29150i	0.750838	+	0.433497i	0.825997	−	0.563675i	$-0.190613\pi$
$149$	9.16515	+	5.29150i	0.750838	+	0.433497i	−0.0751583	+	0.997172i	$0.523946\pi$
$150$		0			0
$151$	4.58258	−	2.64575i	0.372925	−	0.215308i	−0.301811	−	0.953368i	$-0.597591\pi$
$151$	4.58258	−	2.64575i	0.372925	−	0.215308i	0.674735	+	0.738060i	$0.264258\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	17.3739	+	14.2179i	1.38219	+	1.13112i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	4.50000	+	11.9059i	0.353553	+	0.935414i
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	0.146772	+	0.989170i	$0.453112\pi$
$163$	−10.0000	+	17.3205i	−0.783260	+	1.35665i	−0.930033	+	0.367477i	$0.880222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	22.7477	+	7.65120i	1.73450	+	0.583399i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$	−2.00000	+	15.8745i	−0.150756	+	1.19659i
$177$		0			0
$178$		0			0
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	0.781551	−	0.623841i	$-0.214429\pi$
$179$	−2.00000	−	3.46410i	−0.149487	−	0.258919i	−0.931038	+	0.364922i	$0.881096\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	7.95644	+	12.6766i	0.586556	+	0.934528i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.9129	+	13.2288i	1.65792	+	0.957199i	0.973674	+	0.227946i	$0.0732010\pi$
$191$	22.9129	+	13.2288i	1.65792	+	0.957199i	0.684244	+	0.729253i	$0.260132\pi$
$192$		0			0
$193$	9.00000	+	15.5885i	0.647834	+	1.12208i	0.983639	+	0.180150i	$0.0576584\pi$
$193$	9.00000	+	15.5885i	0.647834	+	1.12208i	−0.335805	+	0.941932i	$0.609008\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	10.5830i		−	0.754008i	−0.926212	−	0.377004i	$-0.876954\pi$
$197$		−	10.5830i		−	0.754008i	0.926212	−	0.377004i	$-0.123046\pi$
$198$	16.7477	+	2.74110i	1.19021	+	0.194802i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−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
$207$	13.7477	−	7.93725i	0.955533	−	0.551677i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$	−20.7477	+	4.18710i	−1.42496	+	0.287571i
$213$		0			0
$214$	−17.9129	+	21.8890i	−1.22450	+	1.49630i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−14.0000	+	5.29150i	−0.948200	+	0.358386i
$219$		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
$226$	2.79129	+	0.456850i	0.185674	+	0.0303892i
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	14.0000	−	26.4575i	0.919145	−	1.73702i
$233$	11.0000	−	19.0526i	0.720634	−	1.24817i	−0.240112	−	0.970745i	$-0.577184\pi$
$233$	11.0000	−	19.0526i	0.720634	−	1.24817i	0.960746	−	0.277429i	$-0.0894825\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			26.4575i			1.71139i	0.517477	+	0.855697i	$0.326871\pi$
$239$			26.4575i			1.71139i	−0.517477	+	0.855697i	$0.673129\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	4.47822	−	5.47225i	0.287871	−	0.351770i
$243$		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		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		−	21.1660i		−	1.33070i
$254$	−3.62614	+	22.1552i	−0.227524	+	1.39014i
$255$		0			0
$256$	4.31307	+	15.4077i	0.269567	+	0.962982i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−27.4955	−	15.8745i	−1.70193	−	0.982607i
$262$		0			0
$263$	−4.58258	+	2.64575i	−0.282574	+	0.163144i	−0.634588	−	0.772851i	$-0.718830\pi$
$263$	−4.58258	+	2.64575i	−0.282574	+	0.163144i	0.352014	+	0.935995i	$0.385497\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.58258	−	7.84190i	−0.0966712	−	0.479021i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$	5.00000	+	13.2288i	0.302061	+	0.799178i
$275$	−10.0000	+	17.3205i	−0.603023	+	1.04447i
$276$		0			0
$277$	27.4955	−	15.8745i	1.65204	−	0.953807i	0.675810	−	0.737075i	$-0.263794\pi$
$277$	27.4955	−	15.8745i	1.65204	−	0.953807i	0.976231	−	0.216731i	$-0.0695395\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	3.37386	−	10.0308i	0.200202	−	0.595219i
$285$		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			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	−15.9129	−	25.3531i	−0.924917	−	1.47362i
$297$		0			0
$298$	−11.5826	−	9.47860i	−0.670961	−	0.549081i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−7.00000	+	2.64575i	−0.402805	+	0.152246i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$	−21.0000	−	23.8118i	−1.18134	−	1.33952i
$317$	−9.16515	−	5.29150i	−0.514766	−	0.297200i	0.220024	−	0.975494i	$-0.429386\pi$
$317$	−9.16515	−	5.29150i	−0.514766	−	0.297200i	−0.734791	+	0.678294i	$0.762720\pi$
$318$		0			0
$319$	−36.6606	+	21.1660i	−2.05260	+	1.18507i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−3.56080	−	17.6443i	−0.197822	−	0.980238i
$325$		0			0
$326$	17.9129	−	21.8890i	0.992103	−	1.21232i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	18.0000	−	31.1769i	0.989369	−	1.71364i	0.368744	−	0.929531i	$-0.379788\pi$
$331$	18.0000	−	31.1769i	0.989369	−	1.71364i	0.620625	−	0.784107i	$-0.286879\pi$
$332$		0			0
$333$	−27.4955	+	15.8745i	−1.50674	+	0.869918i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	30.0000			1.63420			0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000			1.63420			0.817102	+	0.576493i	$0.195579\pi$
$338$	18.1434	+	2.96953i	0.986869	+	0.161521i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−30.0000	−	15.8745i	−1.61749	−	0.855896i
$345$		0			0
$346$		0			0
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	0.914702	−	0.404128i	$-0.132425\pi$
$347$	2.00000	+	3.46410i	0.107366	+	0.185963i	−0.807337	+	0.590091i	$0.799092\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	6.41742	−	21.6983i	0.342050	−	1.15652i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	2.00000	+	5.29150i	0.105703	+	0.279665i
$359$	32.0780	+	18.5203i	1.69301	+	0.977462i	0.952063	+	0.305903i	$0.0989582\pi$
$359$	32.0780	+	18.5203i	1.69301	+	0.977462i	0.740951	+	0.671559i	$0.234375\pi$
$360$		0			0
$361$	−9.50000	−	16.4545i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−8.20871	−	19.5094i	−0.427909	−	1.01700i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−27.4955	−	15.8745i	−1.42366	−	0.821951i	−0.427051	−	0.904227i	$-0.640448\pi$
$373$	−27.4955	−	15.8745i	−1.42366	−	0.821951i	−0.996610	+	0.0822766i	$0.973781\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$	−28.9564	−	23.6965i	−1.48154	−	1.21242i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$	−9.00000	−	23.8118i	−0.458088	−	1.21199i
$387$	−18.0000	+	31.1769i	−0.914991	+	1.58481i
$388$		0			0
$389$	−9.16515	+	5.29150i	−0.464692	+	0.268290i	−0.714015	−	0.700130i	$-0.753125\pi$
$389$	−9.16515	+	5.29150i	−0.464692	+	0.268290i	0.249323	+	0.968420i	$0.419792\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−2.41742	+	14.7701i	−0.121788	+	0.744107i
$395$		0			0
$396$	−22.7477	−	7.65120i	−1.14312	−	0.384487i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−2.50000	+	19.8431i	−0.125000	+	0.992157i
$401$	17.0000	−	29.4449i	0.848939	−	1.47041i	−0.0332161	−	0.999448i	$-0.510575\pi$
$401$	17.0000	−	29.4449i	0.848939	−	1.47041i	0.882156	−	0.470958i	$-0.156092\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$			42.3320i			2.09832i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		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		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$			31.7490i			1.54735i	0.633581	+	0.773676i	$0.281584\pi$
$421$			31.7490i			1.54735i	−0.633581	+	0.773676i	$0.718416\pi$
$422$	16.7477	+	2.74110i	0.815267	+	0.133435i
$423$		0			0
$424$	29.9129	−	1.10440i	1.45270	−	0.0536344i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	30.0000	−	26.4575i	1.45010	−	1.27887i
$429$		0			0
$430$		0			0
$431$	22.9129	−	13.2288i	1.10367	−	0.637207i	0.166491	−	0.986043i	$-0.446756\pi$
$431$	22.9129	−	13.2288i	1.10367	−	0.637207i	0.937184	+	0.348836i	$0.113423\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	20.7477	−	4.18710i	0.993636	−	0.200526i
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	−0.524479	−	0.851423i	$-0.675740\pi$
$443$	10.0000	−	17.3205i	0.475114	−	0.822922i	0.999594	+	0.0285009i	$0.00907336\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$	20.9347	+	3.42638i	0.986869	+	0.161521i
$451$		0			0
$452$	−3.79129	−	1.27520i	−0.178327	−	0.0599804i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.00000	+	5.19615i	−0.140334	+	0.243066i	−0.927622	−	0.373519i	$-0.878151\pi$
$457$	−3.00000	+	5.19615i	−0.140334	+	0.243066i	0.787288	+	0.616585i	$0.211484\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$			15.8745i			0.737751i	0.929479	+	0.368875i	$0.120257\pi$
$463$			15.8745i			0.737751i	−0.929479	+	0.368875i	$0.879743\pi$
$464$	−25.5826	+	33.7273i	−1.18764	+	1.56575i
$465$		0			0
$466$	−19.7042	+	24.0779i	−0.912778	+	1.11539i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	24.0000	+	41.5692i	1.10352	+	1.91135i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	31.7490i		−	1.45369i
$478$	6.04356	−	36.9253i	0.276426	−	1.68892i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−7.50000	+	6.61438i	−0.340909	+	0.300654i
$485$		0			0
$486$		0			0
$487$	−32.0780	+	18.5203i	−1.45359	+	0.839233i	−0.998683	−	0.0513038i	$-0.983662\pi$
$487$	−32.0780	+	18.5203i	−1.45359	+	0.839233i	−0.454911	+	0.890537i	$0.650329\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	44.0000			1.98569			0.992846	−	0.119401i	$-0.0380974\pi$
$491$	44.0000			1.98569			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
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	0.109965	+	0.993935i	$0.464926\pi$
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	−0.915756	+	0.401735i	$0.868407\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	−4.83485	+	29.5402i	−0.214935	+	1.31322i
$507$		0			0
$508$	10.1216	−	30.0924i	0.449073	−	1.33513i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	−2.50000	−	22.4889i	−0.110485	−	0.993878i
$513$		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	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	34.7477	+	28.4358i	1.52087	+	1.24460i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	7.00000	−	2.64575i	0.305215	−	0.115360i
$527$		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
$536$	0.417424	+	11.3060i	0.0180300	+	0.488345i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	27.4955	+	15.8745i	1.18212	+	0.682498i	0.956504	−	0.291718i	$-0.0942267\pi$
$541$	27.4955	+	15.8745i	1.18212	+	0.682498i	0.225617	+	0.974216i	$0.427560\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−44.0000			−1.88130			−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000			−1.88130			−0.940652	+	0.339372i	$0.889785\pi$
$548$	−3.95644	−	19.6048i	−0.169011	−	0.837474i
$549$		0			0
$550$	17.9129	−	21.8890i	0.763808	−	0.933351i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−42.0000	+	15.8745i	−1.78441	+	0.674443i
$555$		0			0
$556$		0			0
$557$	9.16515	−	5.29150i	0.388340	−	0.224208i	−0.293101	−	0.956082i	$-0.594687\pi$
$557$	9.16515	−	5.29150i	0.388340	−	0.224208i	0.681441	+	0.731873i	$0.261354\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−36.2867	−	5.93905i	−1.53066	−	0.250524i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−7.00000	+	13.2288i	−0.293713	+	0.555066i
$569$	−11.0000	+	19.0526i	−0.461144	+	0.798725i	−0.999018	−	0.0443003i	$-0.985894\pi$
$569$	−11.0000	+	19.0526i	−0.461144	+	0.798725i	0.537874	+	0.843025i	$0.319228\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	26.4575i		−	1.10335i
$576$	−23.9347	+	1.76978i	−0.997277	+	0.0737406i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	15.2259	−	18.6057i	0.633316	−	0.773893i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−36.6606	−	21.1660i	−1.51833	−	0.876607i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	16.4174	+	39.0188i	0.674752	+	1.60366i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$	14.0000	+	15.8745i	0.573462	+	0.650245i
$597$		0			0
$598$		0			0
$599$	32.0780	−	18.5203i	1.31067	−	0.756717i	0.328465	−	0.944516i	$-0.393469\pi$
$599$	32.0780	−	18.5203i	1.31067	−	0.756717i	0.982208	+	0.187799i	$0.0601353\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	12.0000			0.488678
$604$	10.3739	−	2.09355i	0.422107	−	0.0851854i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	27.4955	−	15.8745i	1.11053	−	0.641165i	0.171564	−	0.985173i	$-0.445118\pi$
$613$	27.4955	−	15.8745i	1.11053	−	0.641165i	0.938967	+	0.344008i	$0.111785\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−26.0000			−1.04672			−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000			−1.04672			−0.523360	+	0.852111i	$0.675322\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		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
$631$		−	47.6235i		−	1.89586i	−0.318475	−	0.947931i	$-0.603171\pi$
$631$		−	47.6235i		−	1.89586i	0.318475	−	0.947931i	$-0.396829\pi$
$632$	23.8693	+	38.0297i	0.949470	+	1.51274i
$633$		0			0
$634$	11.5826	+	9.47860i	0.460003	+	0.376443i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	56.0000	−	21.1660i	2.21706	−	0.837970i
$639$	13.7477	+	7.93725i	0.543852	+	0.313993i
$640$		0			0
$641$	−23.0000	−	39.8372i	−0.908445	−	1.57347i	−0.816224	−	0.577735i	$-0.803937\pi$
$641$	−23.0000	−	39.8372i	−0.908445	−	1.57347i	−0.0922210	−	0.995739i	$-0.529397\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	0.939205	+	25.4385i	0.0368954	+	0.999319i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−30.0000	+	26.4575i	−1.17489	+	1.03616i
$653$	−9.16515	−	5.29150i	−0.358660	−	0.207072i	0.309833	−	0.950791i	$-0.399727\pi$
$653$	−9.16515	−	5.29150i	−0.358660	−	0.207072i	−0.668493	+	0.743719i	$0.733060\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−44.0000			−1.71400			−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000			−1.71400			−0.856998	+	0.515319i	$0.827673\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	−32.2432	+	39.4002i	−1.25317	+	1.53133i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	42.0000	−	15.8745i	1.62747	−	0.615125i
$667$	28.0000	−	48.4974i	1.08416	−	1.87783i
$668$		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	$0.696248\pi$
$674$	−41.8693	−	6.85275i	−1.61275	−	0.263958i
$675$		0			0
$676$	−24.6434	−	8.28880i	−0.947822	−	0.318800i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	26.0000	+	45.0333i	0.994862	+	1.72315i	0.585105	+	0.810958i	$0.301053\pi$
$683$	26.0000	+	45.0333i	0.994862	+	1.72315i	0.409757	+	0.912194i	$0.365613\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	38.2432	+	29.0079i	1.45801	+	1.10592i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−2.00000	−	5.29150i	−0.0759190	−	0.200863i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			52.9150i			1.99857i	0.0377695	+	0.999286i	$0.487975\pi$
$701$			52.9150i			1.99857i	−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$		0			0
$704$	−13.9129	+	28.8172i	−0.524361	+	1.08609i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	45.8258	+	26.4575i	1.72102	+	0.993633i	0.916845	+	0.399244i	$0.130727\pi$
$709$	45.8258	+	26.4575i	1.72102	+	0.993633i	0.804178	+	0.594389i	$0.202606\pi$
$710$		0			0
$711$	41.2432	−	23.8118i	1.54674	−	0.893011i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.58258	−	7.84190i	−0.0591436	−	0.293066i
$717$		0			0
$718$	−40.5390	−	33.1751i	−1.51290	−	1.23808i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	9.50000	+	25.1346i	0.353553	+	0.935414i
$723$		0			0
$724$		0			0
$725$	−45.8258	+	26.4575i	−1.70193	+	0.982607i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$733$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$	7.00000	+	29.1033i	0.258023	+	1.07276i
$737$	8.00000	−	13.8564i	0.294684	−	0.510407i
$738$		0			0
$739$	−26.0000	−	45.0333i	−0.956425	−	1.65658i	−0.731072	−	0.682300i	$-0.760980\pi$
$739$	−26.0000	−	45.0333i	−0.956425	−	1.65658i	−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
$746$	34.7477	+	28.4358i	1.27220	+	1.04111i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−22.9129	−	13.2288i	−0.836103	−	0.482724i	0.0198348	−	0.999803i	$-0.493686\pi$
$751$	−22.9129	−	13.2288i	−0.836103	−	0.482724i	−0.855938	+	0.517079i	$0.827019\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	10.5830i		−	0.384646i	−0.981332	−	0.192323i	$-0.938398\pi$
$757$		−	10.5830i		−	0.384646i	0.981332	−	0.192323i	$-0.0616021\pi$
$758$	−16.7477	−	2.74110i	−0.608305	−	0.0995613i
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$	35.0000	+	39.6863i	1.26626	+	1.43580i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	7.12159	+	35.2886i	0.256312	+	1.27006i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	32.2432	−	39.4002i	1.15896	−	1.41621i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	14.0000	−	5.29150i	0.501924	−	0.189710i
$779$		0			0
$780$		0			0
$781$	18.3303	−	10.5830i	0.655910	−	0.378690i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	6.74773	−	20.0616i	0.240378	−	0.714665i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	30.0000	+	15.8745i	1.06600	+	0.564076i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	8.02178	−	27.1229i	0.283613	−	0.958939i
$801$		0			0
$802$	−30.4519	+	37.2113i	−1.07529	+	1.31398i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	−0.978461	−	0.206430i	$-0.933815\pi$
$809$	−19.0000	−	32.9090i	−0.668004	−	1.15702i	0.310457	−	0.950587i	$-0.399518\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	9.66970	−	59.0804i	0.338923	−	2.07077i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	45.8258	+	26.4575i	1.59933	+	0.923374i	0.991616	+	0.129217i	$0.0412465\pi$
$821$	45.8258	+	26.4575i	1.59933	+	0.923374i	0.607714	+	0.794156i	$0.292087\pi$
$822$		0			0
$823$	−41.2432	+	23.8118i	−1.43765	+	0.830026i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−41.2432	+	23.8118i	−1.43765	+	0.830026i	−0.439961	+	0.898017i	$0.645008\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	44.0000			1.53003			0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000			1.53003			0.765015	+	0.644013i	$0.222732\pi$
$828$	31.1216	−	6.28065i	1.08155	−	0.218268i
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		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			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−83.0000			−2.86207
$842$	7.25227	−	44.3103i	0.249930	−	1.52703i
$843$		0			0
$844$	−22.7477	−	7.65120i	−0.783009	−	0.263365i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−42.0000	−	5.29150i	−1.44229	−	0.181711i
$849$		0			0
$850$		0			0
$851$	−28.0000	−	48.4974i	−0.959828	−	1.66247i
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$	−47.9129	+	30.0725i	−1.63763	+	1.02786i
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$	−35.0000	+	13.2288i	−1.19210	+	0.450573i
$863$	−50.4083	−	29.1033i	−1.71592	−	0.990687i	−0.926041	−	0.377424i	$-0.876810\pi$
$863$	−50.4083	−	29.1033i	−1.71592	−	0.990687i	−0.789879	−	0.613263i	$-0.789857\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	63.4980i		−	2.15402i
$870$		0			0
$871$		0			0
$872$	−29.9129	+	1.10440i	−1.01298	+	0.0373997i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	27.4955	+	15.8745i	0.928456	+	0.536044i	0.886323	−	0.463068i	$-0.153251\pi$
$877$	27.4955	+	15.8745i	0.928456	+	0.536044i	0.0421327	+	0.999112i	$0.486585\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−12.0000			−0.403832			−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000			−0.403832			−0.201916	+	0.979403i	$0.564717\pi$
$884$		0			0
$885$		0			0
$886$	−17.9129	+	21.8890i	−0.601795	+	0.735376i
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	18.0000	−	31.1769i	0.603023	−	1.04447i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−2.79129	−	0.456850i	−0.0931465	−	0.0152453i
$899$		0			0
$900$	−28.4347	−	9.56400i	−0.947822	−	0.318800i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	5.00000	+	2.64575i	0.166298	+	0.0879964i
$905$		0			0
$906$		0			0
$907$	−30.0000	−	51.9615i	−0.996134	−	1.72535i	−0.574148	−	0.818752i	$-0.694667\pi$
$907$	−30.0000	−	51.9615i	−0.996134	−	1.72535i	−0.421986	−	0.906602i	$-0.638667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	58.2065i		−	1.92847i	−0.265052	−	0.964234i	$-0.585389\pi$
$911$		−	58.2065i		−	1.92847i	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
$919$	−32.0780	−	18.5203i	−1.05816	−	0.610927i	−0.133235	−	0.991084i	$-0.542536\pi$
$919$	−32.0780	−	18.5203i	−1.05816	−	0.610927i	−0.924922	+	0.380158i	$0.875870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$			52.9150i			1.73984i
$926$	3.62614	−	22.1552i	0.119162	−	0.728064i
$927$		0			0
$928$	43.4083	−	41.2276i	1.42495	−	1.35336i
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$	33.0000	−	29.1033i	1.08095	−	0.953309i
$933$		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		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−24.0000	−	63.4980i	−0.780307	−	2.06450i
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	−0.981504	−	0.191444i	$-0.938683\pi$
$947$	−10.0000	+	17.3205i	−0.324956	+	0.562841i	0.656547	+	0.754285i	$0.272016\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	58.0000			1.87880			0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000			1.87880			0.939402	+	0.342817i	$0.111381\pi$
$954$	−7.25227	+	44.3103i	−0.234801	+	1.43460i
$955$		0			0
$956$	−16.8693	+	50.1540i	−0.545593	+	1.62210i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$	30.0000	+	51.9615i	0.966736	+	1.67444i
$964$		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	$0.722376\pi$
$968$	11.9782	−	7.51813i	0.384995	−	0.241642i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$	49.0000	−	18.5203i	1.57006	−	0.593427i
$975$		0			0
$976$		0			0
$977$	23.0000	+	39.8372i	0.735835	+	1.27450i	0.954356	+	0.298672i	$0.0965435\pi$
$977$	23.0000	+	39.8372i	0.735835	+	1.27450i	−0.218521	+	0.975832i	$0.570123\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			31.7490i			1.01367i
$982$	−61.4083	−	10.0507i	−1.95962	−	0.320731i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−54.9909	−	31.7490i	−1.74861	−	1.00956i
$990$		0			0
$991$	50.4083	−	29.1033i	1.60127	−	0.924496i	0.610040	−	0.792370i	$-0.291153\pi$
$991$	50.4083	−	29.1033i	1.60127	−	0.924496i	0.991233	−	0.132125i	$-0.0421802\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$	32.2432	−	39.4002i	1.02064	−	1.24719i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.2.m.a.19.1		4
4.3	odd	2		1568.2.q.a.1391.2		4
7.2	even	3		56.2.e.a.27.2	yes	2
7.3	odd	6	inner	392.2.m.a.227.2		4
7.4	even	3	inner	392.2.m.a.227.2		4
7.5	odd	6		56.2.e.a.27.2	yes	2
7.6	odd	2	CM	392.2.m.a.19.1		4
8.3	odd	2	inner	392.2.m.a.19.2		4
8.5	even	2		1568.2.q.a.1391.1		4
21.2	odd	6		504.2.p.a.307.1		2
21.5	even	6		504.2.p.a.307.1		2
28.3	even	6		1568.2.q.a.815.1		4
28.11	odd	6		1568.2.q.a.815.1		4
28.19	even	6		224.2.e.a.111.2		2
28.23	odd	6		224.2.e.a.111.2		2
28.27	even	2		1568.2.q.a.1391.2		4
56.3	even	6	inner	392.2.m.a.227.1		4
56.5	odd	6		224.2.e.a.111.1		2
56.11	odd	6	inner	392.2.m.a.227.1		4
56.13	odd	2		1568.2.q.a.1391.1		4
56.19	even	6		56.2.e.a.27.1	✓	2
56.27	even	2	inner	392.2.m.a.19.2		4
56.37	even	6		224.2.e.a.111.1		2
56.45	odd	6		1568.2.q.a.815.2		4
56.51	odd	6		56.2.e.a.27.1	✓	2
56.53	even	6		1568.2.q.a.815.2		4
84.23	even	6		2016.2.p.a.559.2		2
84.47	odd	6		2016.2.p.a.559.2		2
112.5	odd	12		1792.2.f.d.1791.3		4
112.19	even	12		1792.2.f.d.1791.1		4
112.37	even	12		1792.2.f.d.1791.3		4
112.51	odd	12		1792.2.f.d.1791.1		4
112.61	odd	12		1792.2.f.d.1791.4		4
112.75	even	12		1792.2.f.d.1791.2		4
112.93	even	12		1792.2.f.d.1791.4		4
112.107	odd	12		1792.2.f.d.1791.2		4
168.5	even	6		2016.2.p.a.559.1		2
168.107	even	6		504.2.p.a.307.2		2
168.131	odd	6		504.2.p.a.307.2		2
168.149	odd	6		2016.2.p.a.559.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.2.e.a.27.1	✓	2	56.19	even	6
56.2.e.a.27.1	✓	2	56.51	odd	6
56.2.e.a.27.2	yes	2	7.2	even	3
56.2.e.a.27.2	yes	2	7.5	odd	6
224.2.e.a.111.1		2	56.5	odd	6
224.2.e.a.111.1		2	56.37	even	6
224.2.e.a.111.2		2	28.19	even	6
224.2.e.a.111.2		2	28.23	odd	6
392.2.m.a.19.1		4	1.1	even	1	trivial
392.2.m.a.19.1		4	7.6	odd	2	CM
392.2.m.a.19.2		4	8.3	odd	2	inner
392.2.m.a.19.2		4	56.27	even	2	inner
392.2.m.a.227.1		4	56.3	even	6	inner
392.2.m.a.227.1		4	56.11	odd	6	inner
392.2.m.a.227.2		4	7.3	odd	6	inner
392.2.m.a.227.2		4	7.4	even	3	inner
504.2.p.a.307.1		2	21.2	odd	6
504.2.p.a.307.1		2	21.5	even	6
504.2.p.a.307.2		2	168.107	even	6
504.2.p.a.307.2		2	168.131	odd	6
1568.2.q.a.815.1		4	28.3	even	6
1568.2.q.a.815.1		4	28.11	odd	6
1568.2.q.a.815.2		4	56.45	odd	6
1568.2.q.a.815.2		4	56.53	even	6
1568.2.q.a.1391.1		4	8.5	even	2
1568.2.q.a.1391.1		4	56.13	odd	2
1568.2.q.a.1391.2		4	4.3	odd	2
1568.2.q.a.1391.2		4	28.27	even	2
1792.2.f.d.1791.1		4	112.19	even	12
1792.2.f.d.1791.1		4	112.51	odd	12
1792.2.f.d.1791.2		4	112.75	even	12
1792.2.f.d.1791.2		4	112.107	odd	12
1792.2.f.d.1791.3		4	112.5	odd	12
1792.2.f.d.1791.3		4	112.37	even	12
1792.2.f.d.1791.4		4	112.61	odd	12
1792.2.f.d.1791.4		4	112.93	even	12
2016.2.p.a.559.1		2	168.5	even	6
2016.2.p.a.559.1		2	168.149	odd	6
2016.2.p.a.559.2		2	84.23	even	6
2016.2.p.a.559.2		2	84.47	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	392.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.39564 - 0.228425i) q^{2} +(1.89564 + 0.637600i) q^{4} +(-2.50000 - 1.32288i) q^{8} +(-1.50000 + 2.59808i) q^{9} +(2.00000 + 3.46410i) q^{11} +(3.18693 + 2.41733i) q^{16} +(2.68693 - 3.28335i) q^{18} +(-2.00000 - 5.29150i) q^{22} +(-4.58258 - 2.64575i) q^{23} +(2.50000 + 4.33013i) q^{25} +10.5830i q^{29} +(-3.89564 - 4.10170i) q^{32} +(-4.50000 + 3.96863i) q^{36} +(9.16515 + 5.29150i) q^{37} +12.0000 q^{43} +(1.58258 + 7.84190i) q^{44} +(5.79129 + 4.73930i) q^{46} +(-2.50000 - 6.61438i) q^{50} +(-9.16515 + 5.29150i) q^{53} +(2.41742 - 14.7701i) q^{58} +(4.50000 + 6.61438i) q^{64} +(-2.00000 - 3.46410i) q^{67} -5.29150i q^{71} +(7.18693 - 4.51088i) q^{72} +(-11.5826 - 9.47860i) q^{74} +(-13.7477 - 7.93725i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-16.7477 - 2.74110i) q^{86} +(-0.417424 - 11.3060i) q^{88} +(-7.00000 - 7.93725i) q^{92} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 10 q^{8} - 6 q^{9} + 8 q^{11} - q^{16} - 3 q^{18} - 8 q^{22} + 10 q^{25} - 11 q^{32} - 18 q^{36} + 48 q^{43} - 12 q^{44} + 14 q^{46} - 10 q^{50} + 28 q^{58} + 18 q^{64} - 8 q^{67}+ \cdots - 48 q^{99}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 10 * q^8 - 6 * q^9 + 8 * q^11 - q^16 - 3 * q^18 - 8 * q^22 + 10 * q^25 - 11 * q^32 - 18 * q^36 + 48 * q^43 - 12 * q^44 + 14 * q^46 - 10 * q^50 + 28 * q^58 + 18 * q^64 - 8 * q^67 + 15 * q^72 - 28 * q^74 - 18 * q^81 - 12 * q^86 - 20 * q^88 - 28 * q^92 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more