LMFDB - Embedded newform 399.2.x.a.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [399,2,Mod(65,399)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(399, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 1]))

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

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

chi := DirichletCharacter("399.65");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 399 = 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	399.x (of order $6$, degree $2$, 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:	$3.18603104065$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			65.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	399.65
Dual form			399.2.x.a.221.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +(-3.00000 - 1.73205i) q^{12} +(4.50000 - 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(-4.00000 + 1.73205i) q^{19} +(-1.50000 + 4.33013i) q^{21} +(-2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(-4.00000 + 3.46410i) q^{28} +(9.00000 - 5.19615i) q^{31} +(-3.00000 + 5.19615i) q^{36} +(-4.50000 - 2.59808i) q^{37} +(-4.50000 - 7.79423i) q^{39} +(6.50000 - 11.2583i) q^{43} +(-6.00000 + 3.46410i) q^{48} +(5.50000 + 4.33013i) q^{49} -10.3923i q^{52} +(3.00000 + 6.92820i) q^{57} +13.0000 q^{61} +(7.50000 + 2.59808i) q^{63} -8.00000 q^{64} +(-3.00000 - 1.73205i) q^{67} +17.0000 q^{73} +(7.50000 + 4.33013i) q^{75} +(-1.00000 + 8.66025i) q^{76} +(-4.50000 + 2.59808i) q^{79} +9.00000 q^{81} +(6.00000 + 6.92820i) q^{84} +(-13.5000 + 2.59808i) q^{91} +(-9.00000 - 15.5885i) q^{93} +(12.0000 + 6.92820i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 5 q^{7} - 6 q^{9} - 6 q^{12} + 9 q^{13} - 4 q^{16} - 8 q^{19} - 3 q^{21} - 5 q^{25} - 8 q^{28} + 18 q^{31} - 6 q^{36} - 9 q^{37} - 9 q^{39} + 13 q^{43} - 12 q^{48} + 11 q^{49} + 6 q^{57}+ \cdots + 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 5 * q^7 - 6 * q^9 - 6 * q^12 + 9 * q^13 - 4 * q^16 - 8 * q^19 - 3 * q^21 - 5 * q^25 - 8 * q^28 + 18 * q^31 - 6 * q^36 - 9 * q^37 - 9 * q^39 + 13 * q^43 - 12 * q^48 + 11 * q^49 + 6 * q^57 + 26 * q^61 + 15 * q^63 - 16 * q^64 - 6 * q^67 + 34 * q^73 + 15 * q^75 - 2 * q^76 - 9 * q^79 + 18 * q^81 + 12 * q^84 - 27 * q^91 - 18 * q^93 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/399\mathbb{Z}\right)^\times$.

$n$	$115$	$134$	$211$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$e\left(\frac{1}{6}\right)$

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 $

$2$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$2$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	1.00000	−	1.73205i	0.500000	−	0.866025i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$7$	−2.50000	−	0.866025i	−0.944911	−	0.327327i
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−3.00000	−	1.73205i	−0.866025	−	0.500000i
$13$	4.50000	−	2.59808i	1.24808	−	0.720577i	0.277350	−	0.960769i	$-0.410544\pi$
$13$	4.50000	−	2.59808i	1.24808	−	0.720577i	0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$	−1.50000	+	4.33013i	−0.327327	+	0.944911i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$			5.19615i			1.00000i
$28$	−4.00000	+	3.46410i	−0.755929	+	0.654654i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$	9.00000	−	5.19615i	1.61645	−	0.933257i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	9.00000	−	5.19615i	1.61645	−	0.933257i	0.987829	−	0.155543i	$-0.0497126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.00000	+	5.19615i	−0.500000	+	0.866025i
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	−4.50000	−	2.59808i	−0.739795	−	0.427121i	−0.821995	+	0.569495i	$0.807139\pi$
$38$		0			0
$39$	−4.50000	−	7.79423i	−0.720577	−	1.24808i
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	6.50000	−	11.2583i	0.991241	−	1.71688i	0.381246	−	0.924473i	$-0.375495\pi$
$43$	6.50000	−	11.2583i	0.991241	−	1.71688i	0.609994	−	0.792406i	$-0.291172\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−6.00000	+	3.46410i	−0.866025	+	0.500000i
$49$	5.50000	+	4.33013i	0.785714	+	0.618590i
$50$		0			0
$51$		0			0
$52$		−	10.3923i		−	1.44115i
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	3.00000	+	6.92820i	0.397360	+	0.917663i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	13.0000			1.66448			0.832240	−	0.554416i	$-0.187058\pi$
$61$	13.0000			1.66448			0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−3.00000	−	1.73205i	−0.366508	−	0.211604i	0.305424	−	0.952217i	$-0.401202\pi$
$67$	−3.00000	−	1.73205i	−0.366508	−	0.211604i	−0.671932	+	0.740613i	$0.734535\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$71$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$72$		0			0
$73$	17.0000			1.98970			0.994850	−	0.101361i	$-0.0323196\pi$
$73$	17.0000			1.98970			0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$	7.50000	+	4.33013i	0.866025	+	0.500000i
$76$	−1.00000	+	8.66025i	−0.114708	+	0.993399i
$77$		0			0
$78$		0			0
$79$	−4.50000	+	2.59808i	−0.506290	+	0.292306i	−0.731307	−	0.682048i	$-0.761089\pi$
$79$	−4.50000	+	2.59808i	−0.506290	+	0.292306i	0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	6.00000	+	6.92820i	0.654654	+	0.755929i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−13.5000	+	2.59808i	−1.41518	+	0.272352i
$92$		0			0
$93$	−9.00000	−	15.5885i	−0.933257	−	1.61645i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	12.0000	+	6.92820i	1.21842	+	0.703452i	0.964579	−	0.263795i	$-0.0849741\pi$
$97$	12.0000	+	6.92820i	1.21842	+	0.703452i	0.253837	+	0.967247i	$0.418307\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	0.344865	−	0.938652i	$-0.387925\pi$
$103$	−3.00000	−	1.73205i	−0.295599	−	0.170664i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$	9.00000	+	5.19615i	0.866025	+	0.500000i
$109$			12.1244i			1.16130i	0.814152	+	0.580651i	$0.197202\pi$
$109$			12.1244i			1.16130i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$	−4.50000	+	7.79423i	−0.427121	+	0.739795i
$112$	2.00000	+	10.3923i	0.188982	+	0.981981i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−13.5000	+	7.79423i	−1.24808	+	0.720577i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	+	9.52628i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		−	20.7846i		−	1.86651i
$125$		0			0
$126$		0			0
$127$	10.5000	−	6.06218i	0.931724	−	0.537931i	0.0443678	−	0.999015i	$-0.485873\pi$
$127$	10.5000	−	6.06218i	0.931724	−	0.537931i	0.887357	+	0.461084i	$0.152539\pi$
$128$		0			0
$129$	−19.5000	−	11.2583i	−1.71688	−	0.991241i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	11.5000	−	0.866025i	0.997176	−	0.0750939i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	−0.975417	−	0.220366i	$-0.929275\pi$
$139$	−8.00000	−	13.8564i	−0.678551	−	1.17529i	0.296866	−	0.954919i	$-0.404058\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	7.50000	−	9.52628i	0.618590	−	0.785714i
$148$	−9.00000	+	5.19615i	−0.739795	+	0.427121i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−21.0000	+	12.1244i	−1.70896	+	0.986666i	−0.773099	+	0.634285i	$0.781294\pi$
$151$	−21.0000	+	12.1244i	−1.70896	+	0.986666i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−18.0000			−1.44115
$157$	25.0000			1.99522			0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	25.0000			1.99522			0.997609	+	0.0691164i	$0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.50000	−	14.7224i	0.665771	−	1.15315i	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	8.50000	−	14.7224i	0.665771	−	1.15315i	0.979076	−	0.203497i	$-0.0652307\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	7.00000	−	12.1244i	0.538462	−	0.932643i
$170$		0			0
$171$	12.0000	−	5.19615i	0.917663	−	0.397360i
$172$	−13.0000	−	22.5167i	−0.991241	−	1.71688i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	10.0000	−	8.66025i	0.755929	−	0.654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$179$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$180$		0			0
$181$	−22.5000	+	12.9904i	−1.67241	+	0.965567i	−0.706129	+	0.708083i	$0.749560\pi$
$181$	−22.5000	+	12.9904i	−1.67241	+	0.965567i	−0.966282	+	0.257485i	$0.917106\pi$
$182$		0			0
$183$		−	22.5167i		−	1.66448i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.50000	−	12.9904i	0.327327	−	0.944911i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$			13.8564i			1.00000i
$193$			27.7128i			1.99481i	0.0719816	+	0.997406i	$0.477068\pi$
$193$			27.7128i			1.99481i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$		0			0
$196$	13.0000	−	5.19615i	0.928571	−	0.371154i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	0.602549	+	0.798082i	$0.294152\pi$
$200$		0			0
$201$	−3.00000	+	5.19615i	−0.211604	+	0.366508i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−18.0000	−	10.3923i	−1.24808	−	0.720577i
$209$		0			0
$210$		0			0
$211$	−22.5000	−	12.9904i	−1.54896	−	0.894295i	−0.998221	−	0.0596196i	$-0.981011\pi$
$211$	−22.5000	−	12.9904i	−1.54896	−	0.894295i	−0.550743	−	0.834675i	$-0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−27.0000	+	5.19615i	−1.83288	+	0.352738i
$218$		0			0
$219$		−	29.4449i		−	1.98970i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	16.5000	+	9.52628i	1.10492	+	0.637927i	0.937509	−	0.347960i	$-0.113126\pi$
$223$	16.5000	+	9.52628i	1.10492	+	0.637927i	0.167412	+	0.985887i	$0.446459\pi$
$224$		0			0
$225$	7.50000	−	12.9904i	0.500000	−	0.866025i
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	15.0000	+	1.73205i	0.993399	+	0.114708i
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	0.231287	+	0.972886i	$0.425707\pi$
$229$	−11.0000	+	19.0526i	−0.726900	+	1.25903i	−0.958187	+	0.286143i	$0.907627\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	4.50000	+	7.79423i	0.292306	+	0.506290i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−22.5000	−	12.9904i	−1.44935	−	0.836784i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−22.5000	−	12.9904i	−1.44935	−	0.836784i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		−	15.5885i		−	1.00000i
$244$	13.0000	−	22.5167i	0.832240	−	1.44148i
$245$		0			0
$246$		0			0
$247$	−13.5000	+	18.1865i	−0.858984	+	1.15718i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$	12.0000	−	10.3923i	0.755929	−	0.654654i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	9.00000	+	10.3923i	0.559233	+	0.645746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−6.00000	+	3.46410i	−0.366508	+	0.211604i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	14.0000	+	24.2487i	0.850439	+	1.47300i	0.880812	+	0.473466i	$0.156997\pi$
$271$	14.0000	+	24.2487i	0.850439	+	1.47300i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$	4.50000	+	23.3827i	0.272352	+	1.41518i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.5000	−	26.8468i	−0.931305	−	1.61307i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	−15.5000	−	26.8468i	−0.931305	−	1.61307i	−0.150210	−	0.988654i	$-0.547995\pi$
$278$		0			0
$279$	−27.0000	+	15.5885i	−1.61645	+	0.933257i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$		0			0
$283$	−7.00000			−0.416107			−0.208053	−	0.978117i	$-0.566713\pi$
$283$	−7.00000			−0.416107			−0.208053	+	0.978117i	$0.566713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	17.0000			1.00000
$290$		0			0
$291$	12.0000	−	20.7846i	0.703452	−	1.21842i
$292$	17.0000	−	29.4449i	0.994850	−	1.72313i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	15.0000	−	8.66025i	0.866025	−	0.500000i
$301$	−26.0000	+	22.5167i	−1.49862	+	1.29784i
$302$		0			0
$303$		0			0
$304$	14.0000	+	10.3923i	0.802955	+	0.596040i
$305$		0			0
$306$		0			0
$307$	−1.50000	−	0.866025i	−0.0856095	−	0.0494267i	0.456584	−	0.889680i	$-0.349073\pi$
$307$	−1.50000	−	0.866025i	−0.0856095	−	0.0494267i	−0.542194	+	0.840254i	$0.682406\pi$
$308$		0			0
$309$	−3.00000	+	5.19615i	−0.170664	+	0.295599i
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−35.0000			−1.97832			−0.989158	−	0.146852i	$-0.953086\pi$
$313$	−35.0000			−1.97832			−0.989158	+	0.146852i	$0.953086\pi$
$314$		0			0
$315$		0			0
$316$			10.3923i			0.584613i
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	9.00000	−	15.5885i	0.500000	−	0.866025i
$325$			25.9808i			1.44115i
$326$		0			0
$327$	21.0000			1.16130
$328$		0			0
$329$		0			0
$330$		0			0
$331$	15.0000	+	8.66025i	0.824475	+	0.476011i	0.851957	−	0.523612i	$-0.175416\pi$
$331$	15.0000	+	8.66025i	0.824475	+	0.476011i	−0.0274825	+	0.999622i	$0.508749\pi$
$332$		0			0
$333$	13.5000	+	7.79423i	0.739795	+	0.427121i
$334$		0			0
$335$		0			0
$336$	18.0000	−	3.46410i	0.981981	−	0.188982i
$337$	−19.5000	−	11.2583i	−1.06223	−	0.613280i	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−19.5000	−	11.2583i	−1.06223	−	0.613280i	−0.926049	+	0.377403i	$0.876817\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	23.0000			1.23116			0.615581	−	0.788074i	$-0.288921\pi$
$349$	23.0000			1.23116			0.615581	+	0.788074i	$0.288921\pi$
$350$		0			0
$351$	13.5000	+	23.3827i	0.720577	+	1.24808i
$352$		0			0
$353$		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
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$	16.5000	+	9.52628i	0.866025	+	0.500000i
$364$	−9.00000	+	25.9808i	−0.471728	+	1.36176i
$365$		0			0
$366$		0			0
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	−0.809093	−	0.587680i	$-0.800041\pi$
$367$	2.00000	−	3.46410i	0.104399	−	0.180825i	0.913493	+	0.406855i	$0.133375\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−36.0000			−1.86651
$373$	31.5000	+	18.1865i	1.63101	+	0.941663i	0.983783	+	0.179364i	$0.0574041\pi$
$373$	31.5000	+	18.1865i	1.63101	+	0.941663i	0.647225	+	0.762299i	$0.275929\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			38.1051i			1.95733i	0.205466	+	0.978664i	$0.434129\pi$
$379$			38.1051i			1.95733i	−0.205466	+	0.978664i	$0.565871\pi$
$380$		0			0
$381$	−10.5000	−	18.1865i	−0.537931	−	0.931724i
$382$		0			0
$383$		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
$386$		0			0
$387$	−19.5000	+	33.7750i	−0.991241	+	1.71688i
$388$	24.0000	−	13.8564i	1.21842	−	0.703452i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	17.0000	+	29.4449i	0.853206	+	1.47780i	0.878300	+	0.478110i	$0.158678\pi$
$397$	17.0000	+	29.4449i	0.853206	+	1.47780i	−0.0250943	+	0.999685i	$0.507989\pi$
$398$		0			0
$399$	−1.50000	−	19.9186i	−0.0750939	−	0.997176i
$400$	20.0000			1.00000
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$		0			0
$403$	27.0000	−	46.7654i	1.34497	−	2.32955i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−34.5000	−	19.9186i	−1.70592	−	0.984911i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−34.5000	−	19.9186i	−1.70592	−	0.984911i	−0.766426	−	0.642333i	$-0.777967\pi$
$410$		0			0
$411$		0			0
$412$	−6.00000	+	3.46410i	−0.295599	+	0.170664i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−24.0000	+	13.8564i	−1.17529	+	0.678551i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	31.5000	+	18.1865i	1.53522	+	0.886357i	0.999109	+	0.0422075i	$0.0134391\pi$
$421$	31.5000	+	18.1865i	1.53522	+	0.886357i	0.536107	+	0.844150i	$0.319894\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−32.5000	−	11.2583i	−1.57279	−	0.544829i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$	18.0000	−	10.3923i	0.866025	−	0.500000i
$433$	16.5000	−	9.52628i	0.792939	−	0.457804i	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	16.5000	−	9.52628i	0.792939	−	0.457804i	0.840996	+	0.541041i	$0.181970\pi$
$434$		0			0
$435$		0			0
$436$	21.0000	+	12.1244i	1.00572	+	0.580651i
$437$		0			0
$438$		0			0
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	7.50000	−	4.33013i	0.357955	−	0.206666i	0.668184	+	0.743996i	$0.267072\pi$
$440$		0			0
$441$	−16.5000	−	12.9904i	−0.785714	−	0.618590i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	9.00000	+	15.5885i	0.427121	+	0.739795i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	20.0000	+	6.92820i	0.944911	+	0.327327i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	21.0000	+	36.3731i	0.986666	+	1.70896i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	5.00000	−	8.66025i	0.233890	−	0.405110i	0.958950	+	0.283577i	$0.0915211\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	20.0000			0.929479			0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000			0.929479			0.464739	+	0.885448i	$0.346148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$			31.1769i			1.44115i
$469$	6.00000	+	6.92820i	0.277054	+	0.319915i
$470$		0			0
$471$		−	43.3013i		−	1.99522i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	2.50000	−	21.6506i	0.114708	−	0.993399i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−27.0000			−1.23109
$482$		0			0
$483$		0			0
$484$	11.0000	+	19.0526i	0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−31.5000	+	18.1865i	−1.42740	+	0.824110i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$	−31.5000	+	18.1865i	−1.42740	+	0.824110i	−0.430486	+	0.902597i	$0.641658\pi$
$488$		0			0
$489$	−25.5000	−	14.7224i	−1.15315	−	0.665771i
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−36.0000	−	20.7846i	−1.61645	−	0.933257i
$497$		0			0
$498$		0			0
$499$	16.0000	+	27.7128i	0.716258	+	1.24060i	0.962472	+	0.271380i	$0.0874801\pi$
$499$	16.0000	+	27.7128i	0.716258	+	1.24060i	−0.246214	+	0.969216i	$0.579187\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−21.0000	−	12.1244i	−0.932643	−	0.538462i
$508$		−	24.2487i		−	1.07586i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−42.5000	−	14.7224i	−1.88009	−	0.651282i
$512$		0			0
$513$	−9.00000	−	20.7846i	−0.397360	−	0.917663i
$514$		0			0
$515$		0			0
$516$	−39.0000	+	22.5167i	−1.71688	+	0.991241i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$		−	15.5885i		−	0.681636i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$		−	15.5885i		−	0.681636i	0.940129	−	0.340818i	$-0.110704\pi$
$524$		0			0
$525$	−15.0000	−	17.3205i	−0.654654	−	0.755929i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$	10.0000	−	20.7846i	0.433555	−	0.901127i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−46.0000			−1.97769			−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−46.0000			−1.97769			−0.988847	+	0.148933i	$0.952416\pi$
$542$		0			0
$543$	22.5000	+	38.9711i	0.965567	+	1.67241i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	19.5000	−	11.2583i	0.833760	−	0.481371i	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	19.5000	−	11.2583i	0.833760	−	0.481371i	0.855138	+	0.518400i	$0.173472\pi$
$548$		0			0
$549$	−39.0000			−1.66448
$550$		0			0
$551$		0			0
$552$		0			0
$553$	13.5000	−	2.59808i	0.574078	−	0.110481i
$554$		0			0
$555$		0			0
$556$	−32.0000			−1.35710
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	67.5500i		−	2.85706i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−22.5000	−	7.79423i	−0.944911	−	0.327327i
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	0.983444	−	0.181210i	$-0.0580014\pi$
$571$	8.00000	+	13.8564i	0.334790	+	0.579873i	−0.648655	+	0.761083i	$0.724668\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	24.0000			1.00000
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	48.0000			1.99481
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−9.00000	−	22.5167i	−0.371154	−	0.928571i
$589$	−27.0000	+	36.3731i	−1.11252	+	1.49873i
$590$		0			0
$591$		0			0
$592$			20.7846i			0.854242i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	16.5000	+	9.52628i	0.675300	+	0.389885i
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$		0			0
$601$		−	41.5692i		−	1.69564i	−0.530281	−	0.847822i	$-0.677914\pi$
$601$		−	41.5692i		−	1.69564i	0.530281	−	0.847822i	$-0.322086\pi$
$602$		0			0
$603$	9.00000	+	5.19615i	0.366508	+	0.211604i
$604$			48.4974i			1.97333i
$605$		0			0
$606$		0			0
$607$	−34.5000	−	19.9186i	−1.40031	−	0.808470i	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−34.5000	−	19.9186i	−1.40031	−	0.808470i	−0.994424	+	0.105453i	$0.966371\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	10.0000			0.403896			0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000			0.403896			0.201948	+	0.979396i	$0.435273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	8.50000	+	14.7224i	0.341644	+	0.591744i	0.984738	−	0.174042i	$-0.0556830\pi$
$619$	8.50000	+	14.7224i	0.341644	+	0.591744i	−0.643094	+	0.765787i	$0.722350\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−18.0000	+	31.1769i	−0.720577	+	1.24808i
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	25.0000	−	43.3013i	0.997609	−	1.72791i
$629$		0			0
$630$		0			0
$631$	−0.500000	−	0.866025i	−0.0199047	−	0.0344759i	0.855901	−	0.517139i	$-0.173003\pi$
$631$	−0.500000	−	0.866025i	−0.0199047	−	0.0344759i	−0.875806	+	0.482663i	$0.839670\pi$
$632$		0			0
$633$	−22.5000	+	38.9711i	−0.894295	+	1.54896i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	36.0000	+	5.19615i	1.42637	+	0.205879i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		0			0
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	20.0000	−	34.6410i	0.788723	−	1.36611i	0.926750	−	0.375680i	$-0.122591\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	9.00000	+	46.7654i	0.352738	+	1.83288i
$652$	−17.0000	−	29.4449i	−0.665771	−	1.15315i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−51.0000			−1.98970
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	0.213925	−	0.976850i	$-0.431375\pi$
$661$	−13.5000	−	7.79423i	−0.525089	−	0.303160i	−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	16.5000	−	28.5788i	0.637927	−	1.10492i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	50.2295i		−	1.93620i	−0.250557	−	0.968102i	$-0.580614\pi$
$673$		−	50.2295i		−	1.93620i	0.250557	−	0.968102i	$-0.419386\pi$
$674$		0			0
$675$	−22.5000	−	12.9904i	−0.866025	−	0.500000i
$676$	−14.0000	−	24.2487i	−0.538462	−	0.932643i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	−24.0000	−	27.7128i	−0.921035	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$	3.00000	−	25.9808i	0.114708	−	0.993399i
$685$		0			0
$686$		0			0
$687$	33.0000	+	19.0526i	1.25903	+	0.726900i
$688$	−52.0000			−1.98248
$689$		0			0
$690$		0			0
$691$	24.5000	−	42.4352i	0.932024	−	1.61431i	0.152167	−	0.988355i	$-0.451375\pi$
$691$	24.5000	−	42.4352i	0.932024	−	1.61431i	0.779857	−	0.625958i	$-0.215292\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5.00000	−	25.9808i	−0.188982	−	0.981981i
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$	22.5000	+	2.59808i	0.848604	+	0.0979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	53.0000			1.99046			0.995228	−	0.0975728i	$-0.0311079\pi$
$709$	53.0000			1.99046			0.995228	+	0.0975728i	$0.0311079\pi$
$710$		0			0
$711$	13.5000	−	7.79423i	0.506290	−	0.292306i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	6.00000	+	6.92820i	0.223452	+	0.258020i
$722$		0			0
$723$	−22.5000	+	38.9711i	−0.836784	+	1.44935i
$724$			51.9615i			1.93113i
$725$		0			0
$726$		0			0
$727$	−2.50000	+	4.33013i	−0.0927199	+	0.160596i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$	−2.50000	+	4.33013i	−0.0927199	+	0.160596i	0.815935	+	0.578144i	$0.196223\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	−39.0000	−	22.5167i	−1.44148	−	0.832240i
$733$	3.50000	−	6.06218i	0.129275	−	0.223912i	−0.794121	−	0.607760i	$-0.792068\pi$
$733$	3.50000	−	6.06218i	0.129275	−	0.223912i	0.923396	+	0.383849i	$0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	37.0000			1.36107			0.680534	−	0.732717i	$-0.261748\pi$
$739$	37.0000			1.36107			0.680534	+	0.732717i	$0.261748\pi$
$740$		0			0
$741$	31.5000	+	23.3827i	1.15718	+	0.858984i
$742$		0			0
$743$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$743$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$			17.3205i			0.632034i	0.948753	+	0.316017i	$0.102346\pi$
$751$			17.3205i			0.632034i	−0.948753	+	0.316017i	$0.897654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−18.0000	−	20.7846i	−0.654654	−	0.755929i
$757$	−27.5000	+	47.6314i	−0.999505	+	1.73119i	−0.472493	+	0.881334i	$0.656646\pi$
$757$	−27.5000	+	47.6314i	−0.999505	+	1.73119i	−0.527011	+	0.849858i	$0.676688\pi$
$758$		0			0
$759$		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
$763$	10.5000	−	30.3109i	0.380126	−	1.09733i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	24.0000	+	13.8564i	0.866025	+	0.500000i
$769$	1.00000	+	1.73205i	0.0360609	+	0.0624593i	0.883493	−	0.468445i	$-0.155186\pi$
$769$	1.00000	+	1.73205i	0.0360609	+	0.0624593i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$	48.0000	+	27.7128i	1.72756	+	0.997406i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$			51.9615i			1.86651i
$776$		0			0
$777$	18.0000	−	15.5885i	0.645746	−	0.559233i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	4.00000	−	27.7128i	0.142857	−	0.989743i
$785$		0			0
$786$		0			0
$787$	−43.5000	+	25.1147i	−1.55061	+	0.895244i	−0.552515	+	0.833503i	$0.686332\pi$
$787$	−43.5000	+	25.1147i	−1.55061	+	0.895244i	−0.998092	+	0.0617409i	$0.980335\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	58.5000	−	33.7750i	2.07740	−	1.19939i
$794$		0			0
$795$		0			0
$796$	11.0000	+	19.0526i	0.389885	+	0.675300i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	6.00000	+	10.3923i	0.211604	+	0.366508i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	9.00000	−	5.19615i	0.316033	−	0.182462i	−0.333590	−	0.942718i	$-0.608260\pi$
$811$	9.00000	−	5.19615i	0.316033	−	0.182462i	0.649623	+	0.760257i	$0.274927\pi$
$812$		0			0
$813$	42.0000	−	24.2487i	1.47300	−	0.850439i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−6.50000	+	56.2917i	−0.227406	+	1.96940i
$818$		0			0
$819$	40.5000	−	7.79423i	1.41518	−	0.272352i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−26.0000	+	45.0333i	−0.906303	+	1.56976i	−0.0871445	+	0.996196i	$0.527774\pi$
$823$	−26.0000	+	45.0333i	−0.906303	+	1.56976i	−0.819159	+	0.573567i	$0.805559\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−19.5000	+	11.2583i	−0.677263	+	0.391018i	0.121560	+	0.992584i	$0.461210\pi$
$830$		0			0
$831$	−46.5000	+	26.8468i	−1.61307	+	0.931305i
$832$	−36.0000	+	20.7846i	−1.24808	+	0.720577i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	27.0000	+	46.7654i	0.933257	+	1.61645i
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	14.5000	−	25.1147i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	−45.0000	+	25.9808i	−1.54896	+	0.894295i
$845$		0			0
$846$		0			0
$847$	22.0000	−	19.0526i	0.755929	−	0.654654i
$848$		0			0
$849$			12.1244i			0.416107i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−11.5000	+	19.9186i	−0.393753	+	0.681999i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−11.5000	+	19.9186i	−0.393753	+	0.681999i	0.599189	+	0.800608i	$0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	43.0000			1.46714			0.733571	−	0.679613i	$-0.237852\pi$
$859$	43.0000			1.46714			0.733571	+	0.679613i	$0.237852\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$863$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$	−18.0000	+	51.9615i	−0.610960	+	1.76369i
$869$		0			0
$870$		0			0
$871$	−18.0000			−0.609907
$872$		0			0
$873$	−36.0000	−	20.7846i	−1.21842	−	0.703452i
$874$		0			0
$875$		0			0
$876$	−51.0000	−	29.4449i	−1.72313	−	0.994850i
$877$	46.5000	−	26.8468i	1.57019	−	0.906552i	0.574049	−	0.818821i	$-0.305372\pi$
$877$	46.5000	−	26.8468i	1.57019	−	0.906552i	0.996144	−	0.0877308i	$-0.0279615\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	4.00000	+	6.92820i	0.134611	+	0.233153i	0.925449	−	0.378873i	$-0.123688\pi$
$883$	4.00000	+	6.92820i	0.134611	+	0.233153i	−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−31.5000	+	6.06218i	−1.05648	+	0.203319i
$890$		0			0
$891$		0			0
$892$	33.0000	−	19.0526i	1.10492	−	0.637927i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−15.0000	−	25.9808i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	39.0000	+	45.0333i	1.29784	+	1.49862i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			12.1244i			0.402583i	0.979531	+	0.201291i	$0.0645138\pi$
$907$			12.1244i			0.402583i	−0.979531	+	0.201291i	$0.935486\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	18.0000	−	24.2487i	0.596040	−	0.802955i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	22.0000	+	38.1051i	0.726900	+	1.25903i
$917$		0			0
$918$		0			0
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	0.0164935	+	0.999864i	$0.494750\pi$
$919$	−26.0000	+	45.0333i	−0.857661	+	1.48551i	−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	−1.50000	+	2.59808i	−0.0494267	+	0.0856095i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	22.5000	−	12.9904i	0.739795	−	0.427121i
$926$		0			0
$927$	9.00000	+	5.19615i	0.295599	+	0.170664i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−29.5000	−	7.79423i	−0.966823	−	0.255446i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.571700	+	0.820463i	$0.306284\pi$
$937$	30.5000	+	52.8275i	0.996392	+	1.72580i	0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$			60.6218i			1.97832i
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	18.0000			0.584613
$949$	76.5000	−	44.1673i	2.48330	−	1.43373i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$953$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	38.5000	−	66.6840i	1.24194	−	2.15110i
$962$		0			0
$963$		0			0
$964$	−45.0000	+	25.9808i	−1.44935	+	0.836784i
$965$		0			0
$966$		0			0
$967$	10.0000	−	17.3205i	0.321578	−	0.556990i	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	10.0000	−	17.3205i	0.321578	−	0.556990i	0.980814	+	0.194946i	$0.0624533\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$	−27.0000	−	15.5885i	−0.866025	−	0.500000i
$973$	8.00000	+	41.5692i	0.256468	+	1.33265i
$974$		0			0
$975$	45.0000			1.44115
$976$	−26.0000	−	45.0333i	−0.832240	−	1.44148i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	36.3731i		−	1.16130i
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	18.0000	+	41.5692i	0.572656	+	1.32249i
$989$		0			0
$990$		0			0
$991$		−	15.5885i		−	0.495184i	−0.968864	−	0.247592i	$-0.920361\pi$
$991$		−	15.5885i		−	0.495184i	0.968864	−	0.247592i	$-0.0796392\pi$
$992$		0			0
$993$	15.0000	−	25.9808i	0.476011	−	0.824475i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	29.5000	−	51.0955i	0.934274	−	1.61821i	0.158352	−	0.987383i	$-0.449382\pi$
$997$	29.5000	−	51.0955i	0.934274	−	1.61821i	0.775923	−	0.630828i	$-0.217285\pi$
$998$		0			0
$999$	13.5000	−	23.3827i	0.427121	−	0.739795i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	399.2.x.a.65.1	✓	2
3.2	odd	2	CM	399.2.x.a.65.1	✓	2
7.4	even	3		399.2.bm.a.179.1	yes	2
19.12	odd	6		399.2.bm.a.107.1	yes	2
21.11	odd	6		399.2.bm.a.179.1	yes	2
57.50	even	6		399.2.bm.a.107.1	yes	2
133.88	odd	6	inner	399.2.x.a.221.1	yes	2
399.221	even	6	inner	399.2.x.a.221.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.x.a.65.1	✓	2	1.1	even	1	trivial
399.2.x.a.65.1	✓	2	3.2	odd	2	CM
399.2.x.a.221.1	yes	2	133.88	odd	6	inner
399.2.x.a.221.1	yes	2	399.221	even	6	inner
399.2.bm.a.107.1	yes	2	19.12	odd	6
399.2.bm.a.107.1	yes	2	57.50	even	6
399.2.bm.a.179.1	yes	2	7.4	even	3
399.2.bm.a.179.1	yes	2	21.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 399 = 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	399.x (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(1.00000 - 1.73205i) q^{4} +(-2.50000 - 0.866025i) q^{7} -3.00000 q^{9} +(-3.00000 - 1.73205i) q^{12} +(4.50000 - 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(-4.00000 + 1.73205i) q^{19} +(-1.50000 + 4.33013i) q^{21} +(-2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(-4.00000 + 3.46410i) q^{28} +(9.00000 - 5.19615i) q^{31} +(-3.00000 + 5.19615i) q^{36} +(-4.50000 - 2.59808i) q^{37} +(-4.50000 - 7.79423i) q^{39} +(6.50000 - 11.2583i) q^{43} +(-6.00000 + 3.46410i) q^{48} +(5.50000 + 4.33013i) q^{49} -10.3923i q^{52} +(3.00000 + 6.92820i) q^{57} +13.0000 q^{61} +(7.50000 + 2.59808i) q^{63} -8.00000 q^{64} +(-3.00000 - 1.73205i) q^{67} +17.0000 q^{73} +(7.50000 + 4.33013i) q^{75} +(-1.00000 + 8.66025i) q^{76} +(-4.50000 + 2.59808i) q^{79} +9.00000 q^{81} +(6.00000 + 6.92820i) q^{84} +(-13.5000 + 2.59808i) q^{91} +(-9.00000 - 15.5885i) q^{93} +(12.0000 + 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 5 q^{7} - 6 q^{9} - 6 q^{12} + 9 q^{13} - 4 q^{16} - 8 q^{19} - 3 q^{21} - 5 q^{25} - 8 q^{28} + 18 q^{31} - 6 q^{36} - 9 q^{37} - 9 q^{39} + 13 q^{43} - 12 q^{48} + 11 q^{49} + 6 q^{57}+ \cdots + 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 5 * q^7 - 6 * q^9 - 6 * q^12 + 9 * q^13 - 4 * q^16 - 8 * q^19 - 3 * q^21 - 5 * q^25 - 8 * q^28 + 18 * q^31 - 6 * q^36 - 9 * q^37 - 9 * q^39 + 13 * q^43 - 12 * q^48 + 11 * q^49 + 6 * q^57 + 26 * q^61 + 15 * q^63 - 16 * q^64 - 6 * q^67 + 34 * q^73 + 15 * q^75 - 2 * q^76 - 9 * q^79 + 18 * q^81 + 12 * q^84 - 27 * q^91 - 18 * q^93 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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