LMFDB - Embedded newform 21.2.g.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,2,Mod(5,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	21.g (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:	$0.167685844245$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.17
Dual form			21.2.g.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(0.500000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +(3.00000 - 1.73205i) q^{12} +1.73205i q^{13} +(-2.00000 - 3.46410i) q^{16} +(-4.50000 + 2.59808i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 - 4.33013i) q^{25} -5.19615i q^{27} +(4.00000 + 3.46410i) q^{28} +(7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(-0.500000 - 0.866025i) q^{37} +(1.50000 - 2.59808i) q^{39} -5.00000 q^{43} +6.92820i q^{48} +(-6.50000 - 2.59808i) q^{49} +(-3.00000 - 1.73205i) q^{52} +9.00000 q^{57} +(6.00000 - 3.46410i) q^{61} +(7.50000 - 2.59808i) q^{63} +8.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} +(-13.5000 - 7.79423i) q^{73} +(-7.50000 + 4.33013i) q^{75} -10.3923i q^{76} +(6.50000 + 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(4.50000 + 0.866025i) q^{91} +(-7.50000 - 12.9904i) q^{93} +13.8564i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 4 q^{16} - 9 q^{19} - 6 q^{21} + 5 q^{25} + 8 q^{28} + 15 q^{31} - 12 q^{36} - q^{37} + 3 q^{39} - 10 q^{43} - 13 q^{49} - 6 q^{52} + 18 q^{57} + 12 q^{61}+ \cdots - 15 q^{93}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^4 + q^7 + 3 * q^9 + 6 * q^12 - 4 * q^16 - 9 * q^19 - 6 * q^21 + 5 * q^25 + 8 * q^28 + 15 * q^31 - 12 * q^36 - q^37 + 3 * q^39 - 10 * q^43 - 13 * q^49 - 6 * q^52 + 18 * q^57 + 12 * q^61 + 15 * q^63 + 16 * q^64 - 11 * q^67 - 27 * q^73 - 15 * q^75 + 13 * q^79 - 9 * q^81 - 6 * q^84 + 9 * q^91 - 15 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$3$	−1.50000	−	0.866025i	−0.866025	−	0.500000i
$4$	−1.00000	+	1.73205i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$7$	0.500000	−	2.59808i	0.188982	−	0.981981i
$8$		0			0
$9$	1.50000	+	2.59808i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	3.00000	−	1.73205i	0.866025	−	0.500000i
$13$			1.73205i			0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$13$			1.73205i			0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$		0			0
$16$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−4.50000	+	2.59808i	−1.03237	+	0.596040i	−0.114708	+	0.993399i	$0.536593\pi$
$20$		0			0
$21$	−3.00000	+	3.46410i	−0.654654	+	0.755929i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		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		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.987829	−	0.155543i	$-0.0497126\pi$
$31$	7.50000	+	4.33013i	1.34704	+	0.777714i	0.359211	+	0.933257i	$0.383046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.00000			−1.00000
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	0.821995	−	0.569495i	$-0.192861\pi$
$37$	−0.500000	−	0.866025i	−0.0821995	−	0.142374i	−0.904194	+	0.427121i	$0.859528\pi$
$38$		0			0
$39$	1.50000	−	2.59808i	0.240192	−	0.416025i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−5.00000			−0.762493			−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000			−0.762493			−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$			6.92820i			1.00000i
$49$	−6.50000	−	2.59808i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$	−3.00000	−	1.73205i	−0.416025	−	0.240192i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	9.00000			1.19208
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	6.00000	−	3.46410i	0.768221	−	0.443533i	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
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	0.305424	+	0.952217i	$0.401202\pi$
$67$	−5.50000	+	9.52628i	−0.671932	+	1.16382i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−13.5000	−	7.79423i	−1.58006	−	0.912245i	−0.994850	−	0.101361i	$-0.967680\pi$
$73$	−13.5000	−	7.79423i	−1.58006	−	0.912245i	−0.585206	−	0.810885i	$-0.698986\pi$
$74$		0			0
$75$	−7.50000	+	4.33013i	−0.866025	+	0.500000i
$76$		−	10.3923i		−	1.19208i
$77$		0			0
$78$		0			0
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	0.956325	+	0.292306i	$0.0944227\pi$
$79$	6.50000	+	11.2583i	0.731307	+	1.26666i	−0.225018	+	0.974355i	$0.572244\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.00000	−	8.66025i	−0.327327	−	0.944911i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	4.50000	+	0.866025i	0.471728	+	0.0907841i
$92$		0			0
$93$	−7.50000	−	12.9904i	−0.777714	−	1.34704i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$	5.00000	+	8.66025i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	16.5000	−	9.52628i	1.62579	−	0.938652i	0.640464	−	0.767988i	$-0.278742\pi$
$103$	16.5000	−	9.52628i	1.62579	−	0.938652i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	9.00000	+	5.19615i	0.866025	+	0.500000i
$109$	8.50000	−	14.7224i	0.814152	−	1.41015i	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	8.50000	−	14.7224i	0.814152	−	1.41015i	0.909935	−	0.414751i	$-0.136131\pi$
$110$		0			0
$111$			1.73205i			0.164399i
$112$	−10.0000	+	3.46410i	−0.944911	+	0.327327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−4.50000	+	2.59808i	−0.416025	+	0.240192i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−15.0000	+	8.66025i	−1.34704	+	0.777714i
$125$		0			0
$126$		0			0
$127$	−19.0000			−1.68598			−0.842989	−	0.537931i	$-0.819206\pi$
$127$	−19.0000			−1.68598			−0.842989	+	0.537931i	$0.819206\pi$
$128$		0			0
$129$	7.50000	+	4.33013i	0.660338	+	0.381246i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	4.50000	+	12.9904i	0.390199	+	1.12641i
$134$		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
$139$		−	22.5167i		−	1.90984i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$		−	22.5167i		−	1.90984i	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
$148$	2.00000			0.164399
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	−0.935857	−	0.352381i	$-0.885372\pi$
$151$	−2.00000	+	3.46410i	−0.162758	+	0.281905i	0.773099	+	0.634285i	$0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	3.00000	+	5.19615i	0.240192	+	0.416025i
$157$	18.0000	+	10.3923i	1.43656	+	0.829396i	0.997609	−	0.0691164i	$-0.0220180\pi$
$157$	18.0000	+	10.3923i	1.43656	+	0.829396i	0.438948	+	0.898513i	$0.355351\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	0.665771	−	0.746156i	$-0.268103\pi$
$163$	−4.00000	−	6.92820i	−0.313304	−	0.542659i	−0.979076	+	0.203497i	$0.934769\pi$
$164$		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
$169$	10.0000			0.769231
$170$		0			0
$171$	−13.5000	−	7.79423i	−1.03237	−	0.596040i
$172$	5.00000	−	8.66025i	0.381246	−	0.660338i
$173$		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		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$			25.9808i			1.93113i	0.260153	+	0.965567i	$0.416227\pi$
$181$			25.9808i			1.93113i	−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−13.5000	−	2.59808i	−0.981981	−	0.188982i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−12.0000	−	6.92820i	−0.866025	−	0.500000i
$193$	−12.5000	+	21.6506i	−0.899770	+	1.55845i	−0.0719816	+	0.997406i	$0.522932\pi$
$193$	−12.5000	+	21.6506i	−0.899770	+	1.55845i	−0.827788	+	0.561041i	$0.810401\pi$
$194$		0			0
$195$		0			0
$196$	11.0000	−	8.66025i	0.785714	−	0.618590i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−3.00000	−	1.73205i	−0.212664	−	0.122782i	0.389885	−	0.920864i	$-0.372515\pi$
$199$	−3.00000	−	1.73205i	−0.212664	−	0.122782i	−0.602549	+	0.798082i	$0.705848\pi$
$200$		0			0
$201$	16.5000	−	9.52628i	1.16382	−	0.671932i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	6.00000	−	3.46410i	0.416025	−	0.240192i
$209$		0			0
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	15.0000	−	17.3205i	1.01827	−	1.17579i
$218$		0			0
$219$	13.5000	+	23.3827i	0.912245	+	1.58006i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$	−9.00000	+	15.5885i	−0.596040	+	1.03237i
$229$	−25.5000	+	14.7224i	−1.68509	+	0.972886i	−0.726900	+	0.686743i	$0.759040\pi$
$229$	−25.5000	+	14.7224i	−1.68509	+	0.972886i	−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
$237$		−	22.5167i		−	1.46261i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−24.0000	−	13.8564i	−1.54598	−	0.892570i	−0.998443	−	0.0557856i	$-0.982234\pi$
$241$	−24.0000	−	13.8564i	−1.54598	−	0.892570i	−0.547533	−	0.836784i	$-0.684433\pi$
$242$		0			0
$243$	13.5000	−	7.79423i	0.866025	−	0.500000i
$244$			13.8564i			0.887066i
$245$		0			0
$246$		0			0
$247$	−4.50000	−	7.79423i	−0.286328	−	0.495935i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−3.00000	+	15.5885i	−0.188982	+	0.981981i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−2.50000	+	0.866025i	−0.155342	+	0.0538122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−11.0000	−	19.0526i	−0.671932	−	1.16382i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−15.0000	+	8.66025i	−0.911185	+	0.526073i	−0.880812	−	0.473466i	$-0.843003\pi$
$271$	−15.0000	+	8.66025i	−0.911185	+	0.526073i	−0.0303728	+	0.999539i	$0.509669\pi$
$272$		0			0
$273$	−6.00000	−	5.19615i	−0.363137	−	0.314485i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	15.5000	−	26.8468i	0.931305	−	1.61307i	0.150210	−	0.988654i	$-0.452005\pi$
$277$	15.5000	−	26.8468i	0.931305	−	1.61307i	0.781094	−	0.624413i	$-0.214662\pi$
$278$		0			0
$279$			25.9808i			1.55543i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	28.5000	+	16.4545i	1.69415	+	0.978117i	0.951101	+	0.308879i	$0.0999539\pi$
$283$	28.5000	+	16.4545i	1.69415	+	0.978117i	0.743048	+	0.669238i	$0.233379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$	12.0000	−	20.7846i	0.703452	−	1.21842i
$292$	27.0000	−	15.5885i	1.58006	−	0.912245i
$293$		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
$300$		−	17.3205i		−	1.00000i
$301$	−2.50000	+	12.9904i	−0.144098	+	0.748753i
$302$		0			0
$303$		0			0
$304$	18.0000	+	10.3923i	1.03237	+	0.596040i
$305$		0			0
$306$		0			0
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	−33.0000			−1.87730
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−4.50000	+	2.59808i	−0.254355	+	0.146852i	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−4.50000	+	2.59808i	−0.254355	+	0.146852i	0.367402	+	0.930062i	$0.380247\pi$
$314$		0			0
$315$		0			0
$316$	−26.0000			−1.46261
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		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
$325$	7.50000	+	4.33013i	0.416025	+	0.240192i
$326$		0			0
$327$	−25.5000	+	14.7224i	−1.41015	+	0.814152i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	0.851957	−	0.523612i	$-0.175416\pi$
$331$	−0.500000	−	0.866025i	−0.0274825	−	0.0476011i	−0.879440	+	0.476011i	$0.842082\pi$
$332$		0			0
$333$	1.50000	−	2.59808i	0.0821995	−	0.142374i
$334$		0			0
$335$		0			0
$336$	18.0000	+	3.46410i	0.981981	+	0.188982i
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		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		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$		−	34.6410i		−	1.85429i	−0.374701	−	0.927146i	$-0.622255\pi$
$349$		−	34.6410i		−	1.85429i	0.374701	−	0.927146i	$-0.377745\pi$
$350$		0			0
$351$	9.00000			0.480384
$352$		0			0
$353$		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
$358$		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
$361$	4.00000	−	6.92820i	0.210526	−	0.364642i
$362$		0			0
$363$			19.0526i			1.00000i
$364$	−6.00000	+	6.92820i	−0.314485	+	0.363137i
$365$		0			0
$366$		0			0
$367$	−13.5000	−	7.79423i	−0.704694	−	0.406855i	0.104399	−	0.994535i	$-0.466708\pi$
$367$	−13.5000	−	7.79423i	−0.704694	−	0.406855i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	30.0000			1.55543
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	6.50000	+	11.2583i	0.336557	+	0.582934i	−0.647225	+	0.762299i	$0.724071\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	37.0000			1.90056			0.950281	−	0.311393i	$-0.100796\pi$
$379$	37.0000			1.90056			0.950281	+	0.311393i	$0.100796\pi$
$380$		0			0
$381$	28.5000	+	16.4545i	1.46010	+	0.842989i
$382$		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
$387$	−7.50000	−	12.9904i	−0.381246	−	0.660338i
$388$	−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
$397$	16.5000	−	9.52628i	0.828111	−	0.478110i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	16.5000	−	9.52628i	0.828111	−	0.478110i	0.853206	+	0.521575i	$0.174655\pi$
$398$		0			0
$399$	4.50000	−	23.3827i	0.225282	−	1.17060i
$400$	−20.0000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−7.50000	+	12.9904i	−0.373602	+	0.647097i
$404$		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
$412$			38.1051i			1.87730i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−19.5000	+	33.7750i	−0.954919	+	1.65397i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.00000	−	17.3205i	−0.290360	−	0.838198i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−18.0000	+	10.3923i	−0.866025	+	0.500000i
$433$		−	22.5167i		−	1.08208i	−0.840996	−	0.541041i	$-0.818030\pi$
$433$		−	22.5167i		−	1.08208i	0.840996	−	0.541041i	$-0.181970\pi$
$434$		0			0
$435$		0			0
$436$	17.0000	+	29.4449i	0.814152	+	1.41015i
$437$		0			0
$438$		0			0
$439$	27.0000	−	15.5885i	1.28864	−	0.743996i	0.310228	−	0.950662i	$-0.399595\pi$
$439$	27.0000	−	15.5885i	1.28864	−	0.743996i	0.978412	+	0.206666i	$0.0662612\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	−3.00000	−	1.73205i	−0.142374	−	0.0821995i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	4.00000	−	20.7846i	0.188982	−	0.981981i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	6.00000	−	3.46410i	0.281905	−	0.162758i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.725059	+	0.688686i	$0.241812\pi$
$457$	20.5000	+	35.5070i	0.958950	+	1.66095i	0.233890	+	0.972263i	$0.424854\pi$
$458$		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
$463$	23.0000			1.06890			0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000			1.06890			0.534450	+	0.845200i	$0.320519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$		−	10.3923i		−	0.480384i
$469$	22.0000	+	19.0526i	1.01587	+	0.879765i
$470$		0			0
$471$	−18.0000	−	31.1769i	−0.829396	−	1.43656i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			25.9808i			1.19208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	1.50000	−	0.866025i	0.0683941	−	0.0394874i
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	0.430486	+	0.902597i	$0.358342\pi$
$487$	−12.5000	+	21.6506i	−0.566429	+	0.981084i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$			13.8564i			0.626608i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	34.6410i		−	1.55543i
$497$		0			0
$498$		0			0
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−21.5000	−	37.2391i	−0.962472	−	1.66705i	−0.246214	−	0.969216i	$-0.579187\pi$
$500$		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
$506$		0			0
$507$	−15.0000	−	8.66025i	−0.666173	−	0.384615i
$508$	19.0000	−	32.9090i	0.842989	−	1.46010i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−27.0000	+	31.1769i	−1.19441	+	1.37919i
$512$		0			0
$513$	13.5000	+	23.3827i	0.596040	+	1.03237i
$514$		0			0
$515$		0			0
$516$	−15.0000	+	8.66025i	−0.660338	+	0.381246i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−25.5000	+	14.7224i	−1.11504	+	0.643767i	−0.940129	−	0.340818i	$-0.889296\pi$
$523$	−25.5000	+	14.7224i	−1.11504	+	0.643767i	−0.174908	+	0.984585i	$0.555963\pi$
$524$		0			0
$525$	7.50000	+	21.6506i	0.327327	+	0.944911i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−11.5000	+	19.9186i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−27.0000	−	5.19615i	−1.17060	−	0.225282i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−14.5000	−	25.1147i	−0.623404	−	1.07977i	0.365444	−	0.930834i	$-0.380917\pi$
$542$		0			0
$543$	22.5000	−	38.9711i	0.965567	−	1.67241i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−40.0000			−1.71028			−0.855138	−	0.518400i	$-0.826528\pi$
$547$	−40.0000			−1.71028			−0.855138	+	0.518400i	$0.826528\pi$
$548$		0			0
$549$	18.0000	+	10.3923i	0.768221	+	0.443533i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	32.5000	−	11.2583i	1.38204	−	0.478753i
$554$		0			0
$555$		0			0
$556$	39.0000	+	22.5167i	1.65397	+	0.954919i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	8.66025i		−	0.366290i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	15.5000	−	26.8468i	0.648655	−	1.12350i	0.983444	−	0.181210i	$-0.0580014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	12.0000	+	20.7846i	0.500000	+	0.866025i
$577$	28.5000	+	16.4545i	1.18647	+	0.685009i	0.957503	−	0.288425i	$-0.0931316\pi$
$577$	28.5000	+	16.4545i	1.18647	+	0.685009i	0.228968	+	0.973434i	$0.426465\pi$
$578$		0			0
$579$	37.5000	−	21.6506i	1.55845	−	0.899770i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−24.0000	+	3.46410i	−0.989743	+	0.142857i
$589$	−45.0000			−1.85419
$590$		0			0
$591$		0			0
$592$	−2.00000	+	3.46410i	−0.0821995	+	0.142374i
$593$		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
$596$		0			0
$597$	3.00000	+	5.19615i	0.122782	+	0.212664i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$			1.73205i			0.0706518i	0.999376	+	0.0353259i	$0.0112469\pi$
$601$			1.73205i			0.0706518i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$	−33.0000			−1.34386
$604$	−4.00000	−	6.92820i	−0.162758	−	0.281905i
$605$		0			0
$606$		0			0
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	−0.588537	−	0.808470i	$-0.700296\pi$
$607$	−4.50000	+	2.59808i	−0.182649	+	0.105453i	0.405887	+	0.913923i	$0.366962\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	5.00000	−	8.66025i	0.201948	−	0.349784i	0.949156	+	0.314806i	$0.101939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	7.50000	+	4.33013i	0.301450	+	0.174042i	0.643094	−	0.765787i	$-0.277650\pi$
$619$	7.50000	+	4.33013i	0.301450	+	0.174042i	−0.341644	+	0.939829i	$0.610984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−12.0000			−0.480384
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−36.0000	+	20.7846i	−1.43656	+	0.829396i
$629$		0			0
$630$		0			0
$631$	44.0000			1.75161			0.875806	−	0.482663i	$-0.160330\pi$
$631$	44.0000			1.75161			0.875806	+	0.482663i	$0.160330\pi$
$632$		0			0
$633$	−24.0000	−	13.8564i	−0.953914	−	0.550743i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.50000	−	11.2583i	0.178296	−	0.446071i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			50.2295i			1.98086i	0.138027	+	0.990429i	$0.455924\pi$
$643$			50.2295i			1.98086i	−0.138027	+	0.990429i	$0.544076\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−37.5000	+	12.9904i	−1.46974	+	0.509133i
$652$	16.0000			0.626608
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	46.7654i		−	1.82449i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		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
$669$	−9.00000	+	15.5885i	−0.347960	+	0.602685i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	37.0000			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$674$		0			0
$675$	−22.5000	−	12.9904i	−0.866025	−	0.500000i
$676$	−10.0000	+	17.3205i	−0.384615	+	0.666173i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	36.0000	+	6.92820i	1.38155	+	0.265880i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	27.0000	−	15.5885i	1.03237	−	0.596040i
$685$		0			0
$686$		0			0
$687$	51.0000			1.94577
$688$	10.0000	+	17.3205i	0.381246	+	0.660338i
$689$		0			0
$690$		0			0
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	16.5000	−	9.52628i	0.627690	−	0.362397i	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
$700$	25.0000	−	8.66025i	0.944911	−	0.327327i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	4.50000	+	2.59808i	0.169721	+	0.0979883i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	0.582115	−	0.813107i	$-0.302225\pi$
$709$	−11.0000	−	19.0526i	−0.413114	−	0.715534i	−0.995228	+	0.0975728i	$0.968892\pi$
$710$		0			0
$711$	−19.5000	+	33.7750i	−0.731307	+	1.26666i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−16.5000	−	47.6314i	−0.614492	−	1.77389i
$722$		0			0
$723$	24.0000	+	41.5692i	0.892570	+	1.54598i
$724$	−45.0000	−	25.9808i	−1.67241	−	0.965567i
$725$		0			0
$726$		0			0
$727$		−	22.5167i		−	0.835097i	−0.908655	−	0.417548i	$-0.862889\pi$
$727$		−	22.5167i		−	0.835097i	0.908655	−	0.417548i	$-0.137111\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$		0			0
$732$	12.0000	−	20.7846i	0.443533	−	0.768221i
$733$	−46.5000	+	26.8468i	−1.71752	+	0.991609i	−0.794121	+	0.607760i	$0.792068\pi$
$733$	−46.5000	+	26.8468i	−1.71752	+	0.991609i	−0.923396	+	0.383849i	$0.874598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−26.5000	+	45.8993i	−0.974818	+	1.68843i	−0.294285	+	0.955718i	$0.595081\pi$
$739$	−26.5000	+	45.8993i	−0.974818	+	1.68843i	−0.680534	+	0.732717i	$0.738252\pi$
$740$		0			0
$741$			15.5885i			0.572656i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	0.948753	+	0.316017i	$0.102346\pi$
$751$	20.5000	+	35.5070i	0.748056	+	1.29567i	−0.200698	+	0.979653i	$0.564321\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	18.0000	−	20.7846i	0.654654	−	0.755929i
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−34.0000	−	29.4449i	−1.23088	−	1.06598i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	24.0000	−	13.8564i	0.866025	−	0.500000i
$769$			25.9808i			0.936890i	0.883493	+	0.468445i	$0.155186\pi$
$769$			25.9808i			0.936890i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	−25.0000	−	43.3013i	−0.899770	−	1.55845i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	37.5000	−	21.6506i	1.34704	−	0.777714i
$776$		0			0
$777$	4.50000	+	0.866025i	0.161437	+	0.0310685i
$778$		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
$787$	−3.00000	−	1.73205i	−0.106938	−	0.0617409i	0.445577	−	0.895244i	$-0.352999\pi$
$787$	−3.00000	−	1.73205i	−0.106938	−	0.0617409i	−0.552515	+	0.833503i	$0.686332\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	+	10.3923i	0.213066	+	0.369042i
$794$		0			0
$795$		0			0
$796$	6.00000	−	3.46410i	0.212664	−	0.122782i
$797$		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
$804$			38.1051i			1.34386i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		−	10.3923i		−	0.364923i	−0.983213	−	0.182462i	$-0.941593\pi$
$811$		−	10.3923i		−	0.364923i	0.983213	−	0.182462i	$-0.0584065\pi$
$812$		0			0
$813$	30.0000			1.05215
$814$		0			0
$815$		0			0
$816$		0			0
$817$	22.5000	−	12.9904i	0.787175	−	0.454476i
$818$		0			0
$819$	4.50000	+	12.9904i	0.157243	+	0.453921i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.0871445	−	0.996196i	$-0.472226\pi$
$823$	26.0000	−	45.0333i	0.906303	−	1.56976i	0.819159	−	0.573567i	$-0.194441\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	49.5000	+	28.5788i	1.71921	+	0.992584i	0.920383	+	0.391018i	$0.127877\pi$
$829$	49.5000	+	28.5788i	1.71921	+	0.992584i	0.798823	+	0.601566i	$0.205456\pi$
$830$		0			0
$831$	−46.5000	+	26.8468i	−1.61307	+	0.931305i
$832$			13.8564i			0.480384i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	22.5000	−	38.9711i	0.777714	−	1.34704i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$		0			0
$844$	−16.0000	+	27.7128i	−0.550743	+	0.953914i
$845$		0			0
$846$		0			0
$847$	−27.5000	+	9.52628i	−0.944911	+	0.327327i
$848$		0			0
$849$	−28.5000	−	49.3634i	−0.978117	−	1.69415i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	46.7654i		−	1.60122i	−0.599189	−	0.800608i	$-0.704510\pi$
$853$		−	46.7654i		−	1.60122i	0.599189	−	0.800608i	$-0.295490\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−15.0000	+	8.66025i	−0.511793	+	0.295484i	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−15.0000	+	8.66025i	−0.511793	+	0.295484i	0.221777	+	0.975097i	$0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	29.4449i		−	1.00000i
$868$	15.0000	+	43.3013i	0.509133	+	1.46974i
$869$		0			0
$870$		0			0
$871$	−16.5000	−	9.52628i	−0.559081	−	0.322786i
$872$		0			0
$873$	−36.0000	+	20.7846i	−1.21842	+	0.703452i
$874$		0			0
$875$		0			0
$876$	−54.0000			−1.82449
$877$	17.0000	+	29.4449i	0.574049	+	0.994282i	0.996144	+	0.0877308i	$0.0279615\pi$
$877$	17.0000	+	29.4449i	0.574049	+	0.994282i	−0.422095	+	0.906552i	$0.638705\pi$
$878$		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
$883$	−47.0000			−1.58168			−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000			−1.58168			−0.790838	+	0.612026i	$0.790355\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	−9.50000	+	49.3634i	−0.318620	+	1.65560i
$890$		0			0
$891$		0			0
$892$	18.0000	+	10.3923i	0.602685	+	0.347960i
$893$		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
$903$	15.0000	−	17.3205i	0.499169	−	0.576390i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	29.5000	−	51.0955i	0.979531	−	1.69660i	0.315442	−	0.948945i	$-0.397847\pi$
$907$	29.5000	−	51.0955i	0.979531	−	1.69660i	0.664089	−	0.747653i	$-0.268820\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−18.0000	−	31.1769i	−0.596040	−	1.03237i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	58.8897i		−	1.94577i
$917$		0			0
$918$		0			0
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	0.857661	−	0.514216i	$-0.171917\pi$
$919$	−0.500000	−	0.866025i	−0.0164935	−	0.0285675i	−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
$925$	−5.00000			−0.164399
$926$		0			0
$927$	49.5000	+	28.5788i	1.62579	+	0.938652i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	36.0000	−	5.19615i	1.17985	−	0.170297i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			50.2295i			1.64093i	0.571700	+	0.820463i	$0.306284\pi$
$937$			50.2295i			1.64093i	−0.571700	+	0.820463i	$0.693716\pi$
$938$		0			0
$939$	9.00000			0.293704
$940$		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	$0.333333\pi$
$948$	39.0000	+	22.5167i	1.26666	+	0.731307i
$949$	13.5000	−	23.3827i	0.438229	−	0.759034i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.0000	+	38.1051i	0.709677	+	1.22920i
$962$		0			0
$963$		0			0
$964$	48.0000	−	27.7128i	1.54598	−	0.892570i
$965$		0			0
$966$		0			0
$967$	−61.0000			−1.96163			−0.980814	−	0.194946i	$-0.937547\pi$
$967$	−61.0000			−1.96163			−0.980814	+	0.194946i	$0.937547\pi$
$968$		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	$0.833333\pi$
$972$			31.1769i			1.00000i
$973$	−58.5000	−	11.2583i	−1.87542	−	0.360925i
$974$		0			0
$975$	−7.50000	−	12.9904i	−0.240192	−	0.416025i
$976$	−24.0000	−	13.8564i	−0.768221	−	0.443533i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	51.0000			1.62830
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	18.0000			0.572656
$989$		0			0
$990$		0			0
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$	8.50000	−	14.7224i	0.270011	−	0.467673i	0.968864	+	0.247592i	$0.0796392\pi$
$992$		0			0
$993$			1.73205i			0.0549650i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−34.5000	−	19.9186i	−1.09263	−	0.630828i	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−34.5000	−	19.9186i	−1.09263	−	0.630828i	−0.934274	+	0.356555i	$0.883951\pi$
$998$		0			0
$999$	−4.50000	+	2.59808i	−0.142374	+	0.0821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.2.g.a.17.1	yes	2
3.2	odd	2	CM	21.2.g.a.17.1	yes	2
4.3	odd	2		336.2.bc.c.17.1		2
5.2	odd	4		525.2.q.d.374.1		4
5.3	odd	4		525.2.q.d.374.2		4
5.4	even	2		525.2.t.c.101.1		2
7.2	even	3		147.2.g.a.68.1		2
7.3	odd	6		147.2.c.a.146.1		2
7.4	even	3		147.2.c.a.146.2		2
7.5	odd	6	inner	21.2.g.a.5.1	✓	2
7.6	odd	2		147.2.g.a.80.1		2
9.2	odd	6		567.2.s.a.458.1		2
9.4	even	3		567.2.i.b.269.1		2
9.5	odd	6		567.2.i.b.269.1		2
9.7	even	3		567.2.s.a.458.1		2
12.11	even	2		336.2.bc.c.17.1		2
15.2	even	4		525.2.q.d.374.1		4
15.8	even	4		525.2.q.d.374.2		4
15.14	odd	2		525.2.t.c.101.1		2
21.2	odd	6		147.2.g.a.68.1		2
21.5	even	6	inner	21.2.g.a.5.1	✓	2
21.11	odd	6		147.2.c.a.146.2		2
21.17	even	6		147.2.c.a.146.1		2
21.20	even	2		147.2.g.a.80.1		2
28.3	even	6		2352.2.k.c.881.2		2
28.11	odd	6		2352.2.k.c.881.1		2
28.19	even	6		336.2.bc.c.257.1		2
35.12	even	12		525.2.q.d.299.2		4
35.19	odd	6		525.2.t.c.26.1		2
35.33	even	12		525.2.q.d.299.1		4
63.5	even	6		567.2.s.a.26.1		2
63.40	odd	6		567.2.s.a.26.1		2
63.47	even	6		567.2.i.b.215.1		2
63.61	odd	6		567.2.i.b.215.1		2
84.11	even	6		2352.2.k.c.881.1		2
84.47	odd	6		336.2.bc.c.257.1		2
84.59	odd	6		2352.2.k.c.881.2		2
105.47	odd	12		525.2.q.d.299.2		4
105.68	odd	12		525.2.q.d.299.1		4
105.89	even	6		525.2.t.c.26.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.g.a.5.1	✓	2	7.5	odd	6	inner
21.2.g.a.5.1	✓	2	21.5	even	6	inner
21.2.g.a.17.1	yes	2	1.1	even	1	trivial
21.2.g.a.17.1	yes	2	3.2	odd	2	CM
147.2.c.a.146.1		2	7.3	odd	6
147.2.c.a.146.1		2	21.17	even	6
147.2.c.a.146.2		2	7.4	even	3
147.2.c.a.146.2		2	21.11	odd	6
147.2.g.a.68.1		2	7.2	even	3
147.2.g.a.68.1		2	21.2	odd	6
147.2.g.a.80.1		2	7.6	odd	2
147.2.g.a.80.1		2	21.20	even	2
336.2.bc.c.17.1		2	4.3	odd	2
336.2.bc.c.17.1		2	12.11	even	2
336.2.bc.c.257.1		2	28.19	even	6
336.2.bc.c.257.1		2	84.47	odd	6
525.2.q.d.299.1		4	35.33	even	12
525.2.q.d.299.1		4	105.68	odd	12
525.2.q.d.299.2		4	35.12	even	12
525.2.q.d.299.2		4	105.47	odd	12
525.2.q.d.374.1		4	5.2	odd	4
525.2.q.d.374.1		4	15.2	even	4
525.2.q.d.374.2		4	5.3	odd	4
525.2.q.d.374.2		4	15.8	even	4
525.2.t.c.26.1		2	35.19	odd	6
525.2.t.c.26.1		2	105.89	even	6
525.2.t.c.101.1		2	5.4	even	2
525.2.t.c.101.1		2	15.14	odd	2
567.2.i.b.215.1		2	63.47	even	6
567.2.i.b.215.1		2	63.61	odd	6
567.2.i.b.269.1		2	9.4	even	3
567.2.i.b.269.1		2	9.5	odd	6
567.2.s.a.26.1		2	63.5	even	6
567.2.s.a.26.1		2	63.40	odd	6
567.2.s.a.458.1		2	9.2	odd	6
567.2.s.a.458.1		2	9.7	even	3
2352.2.k.c.881.1		2	28.11	odd	6
2352.2.k.c.881.1		2	84.11	even	6
2352.2.k.c.881.2		2	28.3	even	6
2352.2.k.c.881.2		2	84.59	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 - 0.866025i) q^{3} +(-1.00000 + 1.73205i) q^{4} +(0.500000 - 2.59808i) q^{7} +(1.50000 + 2.59808i) q^{9} +(3.00000 - 1.73205i) q^{12} +1.73205i q^{13} +(-2.00000 - 3.46410i) q^{16} +(-4.50000 + 2.59808i) q^{19} +(-3.00000 + 3.46410i) q^{21} +(2.50000 - 4.33013i) q^{25} -5.19615i q^{27} +(4.00000 + 3.46410i) q^{28} +(7.50000 + 4.33013i) q^{31} -6.00000 q^{36} +(-0.500000 - 0.866025i) q^{37} +(1.50000 - 2.59808i) q^{39} -5.00000 q^{43} +6.92820i q^{48} +(-6.50000 - 2.59808i) q^{49} +(-3.00000 - 1.73205i) q^{52} +9.00000 q^{57} +(6.00000 - 3.46410i) q^{61} +(7.50000 - 2.59808i) q^{63} +8.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} +(-13.5000 - 7.79423i) q^{73} +(-7.50000 + 4.33013i) q^{75} -10.3923i q^{76} +(6.50000 + 11.2583i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-3.00000 - 8.66025i) q^{84} +(4.50000 + 0.866025i) q^{91} +(-7.50000 - 12.9904i) q^{93} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 4 q^{16} - 9 q^{19} - 6 q^{21} + 5 q^{25} + 8 q^{28} + 15 q^{31} - 12 q^{36} - q^{37} + 3 q^{39} - 10 q^{43} - 13 q^{49} - 6 q^{52} + 18 q^{57} + 12 q^{61}+ \cdots - 15 q^{93}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^4 + q^7 + 3 * q^9 + 6 * q^12 - 4 * q^16 - 9 * q^19 - 6 * q^21 + 5 * q^25 + 8 * q^28 + 15 * q^31 - 12 * q^36 - q^37 + 3 * q^39 - 10 * q^43 - 13 * q^49 - 6 * q^52 + 18 * q^57 + 12 * q^61 + 15 * q^63 + 16 * q^64 - 11 * q^67 - 27 * q^73 - 15 * q^75 + 13 * q^79 - 9 * q^81 - 6 * q^84 + 9 * q^91 - 15 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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