LMFDB - Embedded newform 57.2.j.a.41.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [57,2,Mod(2,57)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(57, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 1]))

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

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

chi := DirichletCharacter("57.2");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 57 = 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	57.j (of order $18$, degree $6$, 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:	$0.455147291521$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label			41.1
Root			$-0.173648 - 0.984808i$ of defining polynomial
Character	$\chi$	$=$	57.41
Dual form			57.2.j.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.592396 - 1.62760i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.47178 + 4.28125i) q^{7} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})$	$q+(-0.592396 - 1.62760i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.47178 + 4.28125i) q^{7} +(-2.29813 + 1.92836i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(1.08512 - 2.98135i) q^{13} +(-3.75877 + 1.36808i) q^{16} +(0.500000 + 4.33013i) q^{19} +(5.50387 - 6.55926i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.57398 - 6.35532i) q^{28} +(-4.66772 + 2.69491i) q^{31} +(4.59627 + 3.85673i) q^{36} -11.7352i q^{37} -5.49525 q^{39} +(-0.0957998 + 0.543308i) q^{43} +(4.45336 + 5.30731i) q^{48} +(-8.71941 + 15.1025i) q^{49} +(-6.24897 - 1.10186i) q^{52} +(6.75150 - 3.37895i) q^{57} +(0.762641 + 4.32515i) q^{61} +(-13.9363 - 5.07239i) q^{63} +(4.00000 + 6.92820i) q^{64} +(-1.42514 - 1.69842i) q^{67} +(14.4547 - 5.26108i) q^{73} +8.66025i q^{75} +(8.35504 - 2.48865i) q^{76} +(-5.09879 - 14.0088i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-14.8307 - 8.56250i) q^{84} +(15.4461 - 2.72356i) q^{91} +(7.15136 + 6.00070i) q^{93} +(-3.34002 + 3.98048i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) $ 6 * q - 18 * q^12 - 15 * q^13 + 3 * q^19 + 9 * q^21 + 27 * q^27 + 30 * q^28 - 39 * q^43 - 21 * q^49 - 12 * q^52 - 42 * q^61 - 36 * q^63 + 24 * q^64 + 33 * q^67 + 51 * q^73 + 12 * q^79 + 48 * q^91 + 54 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/57\mathbb{Z}\right)^\times$.

$n$	$20$	$40$
$\chi(n)$	$-1$	$e\left(\frac{13}{18}\right)$

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		0.642788	−	0.766044i	$-0.277778\pi$
$2$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$3$	−0.592396	−	1.62760i	−0.342020	−	0.939693i
$3$	−0.592396	−	1.62760i	−0.342020	−	0.939693i
$4$	−0.347296	−	1.96962i	−0.173648	−	0.984808i
$5$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$5$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$6$		0			0
$7$	2.47178	+	4.28125i	0.934246	+	1.61816i	0.775974	+	0.630765i	$0.217259\pi$
$7$	2.47178	+	4.28125i	0.934246	+	1.61816i	0.158272	+	0.987396i	$0.449408\pi$
$8$		0			0
$9$	−2.29813	+	1.92836i	−0.766044	+	0.642788i
$10$		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
$13$	1.08512	−	2.98135i	0.300959	−	0.826877i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.08512	−	2.98135i	0.300959	−	0.826877i	0.994334	−	0.106301i	$-0.0339006\pi$
$14$		0			0
$15$		0			0
$16$	−3.75877	+	1.36808i	−0.939693	+	0.342020i
$17$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$17$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$18$		0			0
$19$	0.500000	+	4.33013i	0.114708	+	0.993399i
$19$	0.500000	+	4.33013i	0.114708	+	0.993399i
$20$		0			0
$21$	5.50387	−	6.55926i	1.20104	−	1.43135i
$22$		0			0
$23$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$23$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$24$		0			0
$25$	−4.69846	−	1.71010i	−0.939693	−	0.342020i
$26$		0			0
$27$	4.50000	+	2.59808i	0.866025	+	0.500000i
$28$	7.57398	−	6.35532i	1.43135	−	1.20104i
$29$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$29$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$30$		0			0
$31$	−4.66772	+	2.69491i	−0.838347	+	0.484020i	−0.856702	−	0.515812i	$-0.827490\pi$
$31$	−4.66772	+	2.69491i	−0.838347	+	0.484020i	0.0183550	+	0.999832i	$0.494157\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	4.59627	+	3.85673i	0.766044	+	0.642788i
$37$		−	11.7352i		−	1.92925i	−0.263620	−	0.964626i	$-0.584917\pi$
$37$		−	11.7352i		−	1.92925i	0.263620	−	0.964626i	$-0.415083\pi$
$38$		0			0
$39$	−5.49525			−0.879945
$40$		0			0
$41$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$41$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$42$		0			0
$43$	−0.0957998	+	0.543308i	−0.0146093	+	0.0828537i	−0.991241	−	0.132068i	$-0.957838\pi$
$43$	−0.0957998	+	0.543308i	−0.0146093	+	0.0828537i	0.976631	+	0.214921i	$0.0689495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$47$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$48$	4.45336	+	5.30731i	0.642788	+	0.766044i
$49$	−8.71941	+	15.1025i	−1.24563	+	2.15749i
$50$		0			0
$51$		0			0
$52$	−6.24897	−	1.10186i	−0.866576	−	0.152801i
$53$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$53$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	6.75150	−	3.37895i	0.894258	−	0.447553i
$58$		0			0
$59$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$59$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$60$		0			0
$61$	0.762641	+	4.32515i	0.0976462	+	0.553779i	0.993904	+	0.110246i	$0.0351639\pi$
$61$	0.762641	+	4.32515i	0.0976462	+	0.553779i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−13.9363	−	5.07239i	−1.75581	−	0.639062i
$64$	4.00000	+	6.92820i	0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	−1.42514	−	1.69842i	−0.174109	−	0.207495i	0.671932	−	0.740613i	$-0.265465\pi$
$67$	−1.42514	−	1.69842i	−0.174109	−	0.207495i	−0.846041	+	0.533118i	$0.821020\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$71$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$72$		0			0
$73$	14.4547	−	5.26108i	1.69180	−	0.615763i	0.696946	−	0.717124i	$-0.254542\pi$
$73$	14.4547	−	5.26108i	1.69180	−	0.615763i	0.994850	+	0.101361i	$0.0323196\pi$
$74$		0			0
$75$			8.66025i			1.00000i
$76$	8.35504	−	2.48865i	0.958388	−	0.285467i
$77$		0			0
$78$		0			0
$79$	−5.09879	−	14.0088i	−0.573659	−	1.57612i	−0.798677	−	0.601760i	$-0.794466\pi$
$79$	−5.09879	−	14.0088i	−0.573659	−	1.57612i	0.225018	−	0.974355i	$-0.427756\pi$
$80$		0			0
$81$	1.56283	−	8.86327i	0.173648	−	0.984808i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	−14.8307	−	8.56250i	−1.61816	−	0.934246i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$89$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$90$		0			0
$91$	15.4461	−	2.72356i	1.61919	−	0.285507i
$92$		0			0
$93$	7.15136	+	6.00070i	0.741561	+	0.622244i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.34002	+	3.98048i	−0.339128	+	0.404157i	−0.908474	−	0.417941i	$-0.862752\pi$
$97$	−3.34002	+	3.98048i	−0.339128	+	0.404157i	0.569346	+	0.822098i	$0.307196\pi$
$98$		0			0
$99$		0			0
$100$	−1.73648	+	9.84808i	−0.173648	+	0.984808i
$101$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$101$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$102$		0			0
$103$	17.5783	+	10.1488i	1.73204	+	0.999995i	0.867536	+	0.497374i	$0.165702\pi$
$103$	17.5783	+	10.1488i	1.73204	+	0.999995i	0.864507	+	0.502621i	$0.167631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$107$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$108$	3.55438	−	9.76557i	0.342020	−	0.939693i
$109$	−8.52869	−	1.50384i	−0.816900	−	0.144041i	−0.250441	−	0.968132i	$-0.580576\pi$
$109$	−8.52869	−	1.50384i	−0.816900	−	0.144041i	−0.566458	+	0.824090i	$0.691687\pi$
$110$		0			0
$111$	−19.1001	+	6.95188i	−1.81290	+	0.659843i
$112$	−15.1480	−	12.7106i	−1.43135	−	1.20104i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.25537	+	8.94405i	0.300959	+	0.826877i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.50000	−	9.52628i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	6.92902	+	8.25768i	0.622244	+	0.741561i
$125$		0			0
$126$		0			0
$127$	−4.14677	+	11.3932i	−0.367967	+	1.01098i	0.608167	+	0.793809i	$0.291905\pi$
$127$	−4.14677	+	11.3932i	−0.367967	+	1.01098i	−0.976134	+	0.217171i	$0.930317\pi$
$128$		0			0
$129$	0.941037	−	0.165930i	0.0828537	−	0.0146093i
$130$		0			0
$131$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$131$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$132$		0			0
$133$	−17.3025	+	12.8438i	−1.50031	+	1.11369i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$137$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	13.4179	+	4.88371i	1.13809	+	0.414230i	0.841223	−	0.540689i	$-0.181836\pi$
$139$	13.4179	+	4.88371i	1.13809	+	0.414230i	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
$147$	29.7460	+	5.24503i	2.45341	+	0.432603i
$148$	−23.1138	+	4.07559i	−1.89994	+	0.335011i
$149$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$149$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$150$		0			0
$151$		−	15.5885i		−	1.26857i	−0.773099	−	0.634285i	$-0.781294\pi$
$151$		−	15.5885i		−	1.26857i	0.773099	−	0.634285i	$-0.218706\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.90848	+	10.8235i	0.152801	+	0.866576i
$157$	1.05004	−	5.95507i	0.0838023	−	0.475267i	−0.913806	−	0.406150i	$-0.866871\pi$
$157$	1.05004	−	5.95507i	0.0838023	−	0.475267i	0.997609	−	0.0691164i	$-0.0220180\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	12.6348	−	21.8840i	0.989630	−	1.71409i	0.370420	−	0.928864i	$-0.379214\pi$
$163$	12.6348	−	21.8840i	0.989630	−	1.71409i	0.619210	−	0.785225i	$-0.287453\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$167$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$168$		0			0
$169$	2.24763	+	1.88598i	0.172894	+	0.145076i
$170$		0			0
$171$	−9.49912	−	8.98703i	−0.726416	−	0.687255i
$172$	1.10338			0.0841318
$173$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$173$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$174$		0			0
$175$	−4.29220	−	24.3423i	−0.324460	−	1.84010i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−12.2467	−	14.5951i	−0.910294	−	1.08485i	−0.996073	−	0.0885316i	$-0.971783\pi$
$181$	−12.2467	−	14.5951i	−0.910294	−	1.08485i	0.0857797	−	0.996314i	$-0.472662\pi$
$182$		0			0
$183$	6.58781	−	3.80347i	0.486985	−	0.281161i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$			25.6875i			1.86849i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	8.90673	−	10.6146i	0.642788	−	0.766044i
$193$	7.70052	+	21.1570i	0.554296	+	1.52292i	0.827788	+	0.561041i	$0.189599\pi$
$193$	7.70052	+	21.1570i	0.554296	+	1.52292i	−0.273492	+	0.961874i	$0.588179\pi$
$194$		0			0
$195$		0			0
$196$	32.7743	+	11.9289i	2.34102	+	0.852061i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−14.7253	+	12.3560i	−1.04385	+	0.875895i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−14.7253	+	12.3560i	−1.04385	+	0.875895i	−0.0514178	+	0.998677i	$0.516374\pi$
$200$		0			0
$201$	−1.92009	+	3.32570i	−0.135433	+	0.234577i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			12.6907i			0.879945i
$209$		0			0
$210$		0			0
$211$	−18.5510	+	22.1082i	−1.27710	+	1.52199i	−0.550743	+	0.834675i	$0.685655\pi$
$211$	−18.5510	+	22.1082i	−1.27710	+	1.52199i	−0.726359	+	0.687315i	$0.758789\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−23.0752	−	13.3224i	−1.56644	−	0.904387i
$218$		0			0
$219$	−17.1258	−	20.4098i	−1.15726	−	1.37916i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.0483	−	3.35873i	−1.27557	−	0.224917i	−0.505471	−	0.862844i	$-0.668681\pi$
$223$	−19.0483	−	3.35873i	−1.27557	−	0.224917i	−0.770097	+	0.637927i	$0.779792\pi$
$224$		0			0
$225$	14.0954	−	5.13030i	0.939693	−	0.342020i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−9.00000	−	12.1244i	−0.596040	−	0.802955i
$229$	30.2131			1.99654			0.998268	−	0.0588329i	$-0.0187379\pi$
$229$	30.2131			1.99654			0.998268	+	0.0588329i	$0.0187379\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$233$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.7802	+	16.5975i	−1.28486	+	1.07813i
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−7.26904	+	19.9715i	−0.468240	+	1.28648i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−7.26904	+	19.9715i	−0.468240	+	1.28648i	−0.919150	+	0.393909i	$0.871123\pi$
$242$		0			0
$243$	−15.3516	+	2.70691i	−0.984808	+	0.173648i
$244$	8.25402	−	3.00422i	0.528410	−	0.192325i
$245$		0			0
$246$		0			0
$247$	13.4522	+	3.20804i	0.855942	+	0.204123i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$251$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$252$	−5.15064	+	29.2108i	−0.324460	+	1.84010i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	12.2567	−	10.2846i	0.766044	−	0.642788i
$257$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$257$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$258$		0			0
$259$	50.2413	−	29.0068i	3.12184	−	1.80240i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$263$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.85029	+	3.39684i	−0.174109	+	0.207495i
$269$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$269$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$270$		0			0
$271$	−0.173648	+	0.984808i	−0.0105484	+	0.0598228i	−0.989628	−	0.143657i	$-0.954114\pi$
$271$	−0.173648	+	0.984808i	−0.0105484	+	0.0598228i	0.979079	+	0.203479i	$0.0652250\pi$
$272$		0			0
$273$	−13.5831	−	23.5266i	−0.822084	−	1.42389i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	−0.781094	−	0.624413i	$-0.785338\pi$
$277$	2.50000	−	4.33013i	0.150210	−	0.260172i	0.931305	+	0.364241i	$0.118672\pi$
$278$		0			0
$279$	5.53028	−	15.1943i	0.331089	−	0.909660i
$280$		0			0
$281$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$281$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$282$		0			0
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	0.469344	−	0.883016i	$-0.344491\pi$
$283$	−5.36231	−	4.49951i	−0.318756	−	0.267468i	−0.788100	+	0.615547i	$0.788935\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.95202	+	16.7417i	0.173648	+	0.984808i
$290$		0			0
$291$	8.45723	+	3.07818i	0.495772	+	0.180446i
$292$	−15.3824	−	26.6431i	−0.900186	−	1.55917i
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	17.0574	−	3.00767i	0.984808	−	0.173648i
$301$	−2.56283	+	0.932795i	−0.147719	+	0.0537654i
$302$		0			0
$303$		0			0
$304$	−7.80335	−	15.5919i	−0.447553	−	0.894258i
$305$		0			0
$306$		0			0
$307$	10.0707	+	27.6691i	0.574767	+	1.57916i	0.796879	+	0.604139i	$0.206483\pi$
$307$	10.0707	+	27.6691i	0.574767	+	1.57916i	−0.222112	+	0.975021i	$0.571295\pi$
$308$		0			0
$309$	6.10488	−	34.6225i	0.347295	−	1.96961i
$310$		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
$313$	9.95858	−	8.35624i	0.562892	−	0.472323i	−0.316387	−	0.948630i	$-0.602470\pi$
$313$	9.95858	−	8.35624i	0.562892	−	0.472323i	0.879279	+	0.476308i	$0.158025\pi$
$314$		0			0
$315$		0			0
$316$	−25.8212	+	14.9079i	−1.45256	+	0.838633i
$317$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$317$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−18.0000			−1.00000
$325$	−10.1968	+	12.1521i	−0.565617	+	0.674077i
$326$		0			0
$327$	2.60472	+	14.7721i	0.144041	+	0.816900i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.32279	+	3.65046i	0.347532	+	0.200648i	0.663598	−	0.748090i	$-0.269029\pi$
$331$	6.32279	+	3.65046i	0.347532	+	0.200648i	−0.316066	+	0.948737i	$0.602362\pi$
$332$		0			0
$333$	22.6297	+	26.9690i	1.24010	+	1.47789i
$334$		0			0
$335$		0			0
$336$	−11.7142	+	32.1845i	−0.639062	+	1.75581i
$337$	30.6052	+	5.39652i	1.66717	+	0.293967i	0.926049	−	0.377403i	$-0.123183\pi$
$337$	30.6052	+	5.39652i	1.66717	+	0.293967i	0.741122	+	0.671370i	$0.234294\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−51.6049			−2.78640
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$347$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$348$		0			0
$349$	5.77110	+	9.99583i	0.308920	+	0.535065i	0.978126	−	0.208012i	$-0.0666992\pi$
$349$	5.77110	+	9.99583i	0.308920	+	0.535065i	−0.669207	+	0.743076i	$0.733366\pi$
$350$		0			0
$351$	12.6288	−	10.5968i	0.674077	−	0.565617i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$359$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$360$		0			0
$361$	−18.5000	+	4.33013i	−0.973684	+	0.227901i
$362$		0			0
$363$	−12.2467	+	14.5951i	−0.642788	+	0.766044i
$364$	−10.7287	−	29.4770i	−0.562339	−	1.54501i
$365$		0			0
$366$		0			0
$367$	−20.1370	−	7.32926i	−1.05114	−	0.382584i	−0.242048	−	0.970264i	$-0.577819\pi$
$367$	−20.1370	−	7.32926i	−1.05114	−	0.382584i	−0.809093	+	0.587680i	$0.800041\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	9.33544	−	16.1695i	0.484020	−	0.838347i
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−6.00000	+	3.46410i	−0.310668	+	0.179364i	0.336557	+	0.941663i	$0.390737\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			1.26158i			0.0648029i	0.999475	+	0.0324014i	$0.0103155\pi$
$379$			1.26158i			0.0648029i	−0.999475	+	0.0324014i	$0.989684\pi$
$380$		0			0
$381$	21.0000			1.07586
$382$		0			0
$383$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$383$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.827534	−	1.43333i	−0.0420659	−	0.0728603i
$388$	9.00000	+	5.19615i	0.456906	+	0.263795i
$389$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$389$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−20.2028	−	16.9522i	−1.01395	−	0.850805i	−0.0250943	−	0.999685i	$-0.507989\pi$
$397$	−20.2028	−	16.9522i	−1.01395	−	0.850805i	−0.988855	+	0.148880i	$0.952433\pi$
$398$		0			0
$399$	31.1544	+	20.5528i	1.55967	+	1.02893i
$400$	20.0000			1.00000
$401$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$401$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$402$		0			0
$403$	2.96942	+	16.8404i	0.147917	+	0.838880i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	8.90673	+	10.6146i	0.440409	+	0.524859i	0.939895	−	0.341463i	$-0.110922\pi$
$409$	8.90673	+	10.6146i	0.440409	+	0.524859i	−0.499486	+	0.866322i	$0.666478\pi$
$410$		0			0
$411$		0			0
$412$	13.8844	−	38.1472i	0.684037	−	1.87938i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	24.7320i		−	1.21113i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−12.4403	−	34.1795i	−0.606304	−	1.66581i	−0.738231	−	0.674548i	$-0.764339\pi$
$421$	−12.4403	−	34.1795i	−0.606304	−	1.66581i	0.131927	−	0.991259i	$-0.457883\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−16.6320	+	13.9559i	−0.804878	+	0.675373i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$431$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$432$	−20.4688	−	3.60921i	−0.984808	−	0.173648i
$433$	37.7952	−	6.66431i	1.81632	−	0.320266i	0.840996	−	0.541041i	$-0.181970\pi$
$433$	37.7952	−	6.66431i	1.81632	−	0.320266i	0.975325	+	0.220774i	$0.0708584\pi$
$434$		0			0
$435$		0			0
$436$			17.3205i			0.829502i
$437$		0			0
$438$		0			0
$439$	21.2046	−	25.2706i	1.01204	−	1.20610i	0.0336266	−	0.999434i	$-0.489294\pi$
$439$	21.2046	−	25.2706i	1.01204	−	1.20610i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$	−9.08466	−	51.5216i	−0.432603	−	2.45341i
$442$		0			0
$443$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$443$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$444$	20.3259	+	35.2056i	0.964626	+	1.67078i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−19.7743	+	34.2500i	−0.934246	+	1.61816i
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−25.3717	+	9.23454i	−1.19207	+	0.433877i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	23.6144			1.10464			0.552318	−	0.833633i	$-0.313743\pi$
$457$	23.6144			1.10464			0.552318	+	0.833633i	$0.313743\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$461$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$	17.0266	+	29.4910i	0.791294	+	1.37056i	0.925166	+	0.379563i	$0.123926\pi$
$463$	17.0266	+	29.4910i	0.791294	+	1.37056i	−0.133871	+	0.990999i	$0.542741\pi$
$464$		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	$0.666667\pi$
$468$	16.4858	−	9.51806i	0.762054	−	0.439972i
$469$	3.74872	−	10.2995i	0.173100	−	0.475588i
$470$		0			0
$471$	−10.3145	+	1.81872i	−0.475267	+	0.0838023i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	5.05572	−	21.2000i	0.231972	−	0.972722i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$479$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$480$		0			0
$481$	−34.9867	−	12.7341i	−1.59526	−	0.580626i
$482$		0			0
$483$		0			0
$484$	−16.8530	+	14.1413i	−0.766044	+	0.642788i
$485$		0			0
$486$		0			0
$487$	−34.5000	+	19.9186i	−1.56334	+	0.902597i	−0.566429	+	0.824110i	$0.691675\pi$
$487$	−34.5000	+	19.9186i	−1.56334	+	0.902597i	−0.996915	+	0.0784867i	$0.974991\pi$
$488$		0			0
$489$	−43.1031	−	7.60024i	−1.94919	−	0.343695i
$490$		0			0
$491$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$491$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	13.8580	−	16.5154i	0.622244	−	0.741561i
$497$		0			0
$498$		0			0
$499$	−6.29648	+	35.7091i	−0.281869	+	1.59856i	0.434389	+	0.900725i	$0.356964\pi$
$499$	−6.29648	+	35.7091i	−0.281869	+	1.59856i	−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$503$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.73813	−	4.77547i	0.0771931	−	0.212086i
$508$	23.8803	+	4.21074i	1.05952	+	0.186822i
$509$		0			0		0.984808	−	0.173648i	$-0.0555556\pi$
$509$		0			0		−0.984808	+	0.173648i	$0.944444\pi$
$510$		0			0
$511$	58.2529	+	48.8800i	2.57696	+	2.16232i
$512$		0			0
$513$	−9.00000	+	20.7846i	−0.397360	+	0.917663i
$514$		0			0
$515$		0			0
$516$	−0.653637	−	1.79585i	−0.0287748	−	0.0790581i
$517$		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
$523$	−25.4800	−	30.3659i	−1.11416	−	1.32781i	−0.939254	−	0.343222i	$-0.888482\pi$
$523$	−25.4800	−	30.3659i	−1.11416	−	1.32781i	−0.174908	−	0.984585i	$-0.555963\pi$
$524$		0			0
$525$	−37.0767	+	21.4063i	−1.61816	+	0.934246i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−21.6129	+	7.86646i	−0.939693	+	0.342020i
$530$		0			0
$531$		0			0
$532$	31.3063	+	29.6186i	1.35730	+	1.28413i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	34.9281	−	29.3082i	1.50168	−	1.26006i	0.623404	−	0.781900i	$-0.285749\pi$
$541$	34.9281	−	29.3082i	1.50168	−	1.26006i	0.878274	−	0.478157i	$-0.158695\pi$
$542$		0			0
$543$	−16.5000	+	28.5788i	−0.708083	+	1.22644i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−35.9131	+	6.33245i	−1.53553	+	0.270756i	−0.876517	−	0.481371i	$-0.840139\pi$
$547$	−35.9131	+	6.33245i	−1.53553	+	0.270756i	−0.659018	+	0.752128i	$0.729028\pi$
$548$		0			0
$549$	−10.0931	−	8.46913i	−0.430764	−	0.361453i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	47.3722	−	56.4560i	2.01447	−	2.40075i
$554$		0			0
$555$		0			0
$556$	4.95904	−	28.1241i	0.210310	−	1.19273i
$557$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$557$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$558$		0			0
$559$	1.51584	+	0.875168i	0.0641130	+	0.0370157i
$560$		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
$567$	41.8089	−	15.2172i	1.75581	−	0.639062i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$	−41.2036			−1.72432			−0.862158	−	0.506640i	$-0.830887\pi$
$571$	−41.2036			−1.72432			−0.862158	+	0.506640i	$0.830887\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−22.5526	−	8.20848i	−0.939693	−	0.342020i
$577$	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
$579$	29.8733	−	25.0667i	1.24149	−	1.04174i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$587$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$588$		−	60.4098i		−	2.49126i
$589$	−14.0032	−	18.8644i	−0.576990	−	0.777292i
$590$		0			0
$591$		0			0
$592$	16.0547	+	44.1099i	0.659843	+	1.81290i
$593$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$593$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	28.8338	+	16.6472i	1.18009	+	0.681326i
$598$		0			0
$599$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$599$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$600$		0			0
$601$	−15.9232	+	9.19329i	−0.649523	+	0.375002i	−0.788273	−	0.615325i	$-0.789025\pi$
$601$	−15.9232	+	9.19329i	−0.649523	+	0.375002i	0.138751	+	0.990327i	$0.455691\pi$
$602$		0			0
$603$	6.55035	+	1.15500i	0.266751	+	0.0470353i
$604$	−30.7033	+	5.41381i	−1.24930	+	0.220285i
$605$		0			0
$606$		0			0
$607$		−	49.1579i		−	1.99526i	−0.0688294	−	0.997628i	$-0.521926\pi$
$607$		−	49.1579i		−	1.99526i	0.0688294	−	0.997628i	$-0.478074\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6.42498	−	36.4379i	0.259503	−	1.47171i	−0.524742	−	0.851261i	$-0.675838\pi$
$613$	6.42498	−	36.4379i	0.259503	−	1.47171i	0.784245	−	0.620451i	$-0.213051\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$617$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$618$		0			0
$619$	−8.51872	+	14.7549i	−0.342396	+	0.593048i	−0.984877	−	0.173254i	$-0.944572\pi$
$619$	−8.51872	+	14.7549i	−0.342396	+	0.593048i	0.642481	+	0.766302i	$0.277905\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	20.6554	−	7.51795i	0.826877	−	0.300959i
$625$	19.1511	+	16.0697i	0.766044	+	0.642788i
$626$		0			0
$627$		0			0
$628$	−12.0939			−0.482598
$629$		0			0
$630$		0			0
$631$	−3.14636	−	17.8439i	−0.125255	−	0.710355i	−0.981156	−	0.193216i	$-0.938108\pi$
$631$	−3.14636	−	17.8439i	−0.125255	−	0.710355i	0.855901	−	0.517139i	$-0.173003\pi$
$632$		0			0
$633$	46.9727	+	17.0967i	1.86700	+	0.679532i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	35.5641	+	42.3836i	1.40910	+	1.67930i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$641$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$642$		0			0
$643$	−25.3008	+	9.20875i	−0.997767	+	0.363157i	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−25.3008	+	9.20875i	−0.997767	+	0.363157i	−0.209044	+	0.977906i	$0.567035\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−8.01392	+	45.4492i	−0.314090	+	1.78129i
$652$	−47.4911	−	17.2854i	−1.85990	−	0.676947i
$653$		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
$657$	−23.0736	+	39.9646i	−0.900186	+	1.55917i
$658$		0			0
$659$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$659$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$660$		0			0
$661$	34.1147	−	6.01535i	1.32691	−	0.233970i	0.535126	−	0.844772i	$-0.320264\pi$
$661$	34.1147	−	6.01535i	1.32691	−	0.233970i	0.791783	+	0.610802i	$0.209153\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	5.81749	+	32.9926i	0.224917	+	1.27557i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−26.0862	−	15.0609i	−1.00555	−	0.580554i	−0.0956642	−	0.995414i	$-0.530497\pi$
$673$	−26.0862	−	15.0609i	−1.00555	−	0.580554i	−0.909886	+	0.414859i	$0.863831\pi$
$674$		0			0
$675$	−16.7001	−	19.9024i	−0.642788	−	0.766044i
$676$	2.93407	−	5.08195i	0.112849	−	0.195460i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−25.2973	−	4.46059i	−0.970820	−	0.171182i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−14.4020	+	21.8308i	−0.550673	+	0.834721i
$685$		0			0
$686$		0			0
$687$	−17.8981	−	49.1746i	−0.682855	−	1.87613i
$688$	−0.383199	−	2.17323i	−0.0146093	−	0.0828537i
$689$		0			0
$690$		0			0
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−20.5000	−	35.5070i	−0.779857	−	1.35075i	0.152167	−	0.988355i	$-0.451375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−46.4543	+	16.9080i	−1.75581	+	0.639062i
$701$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$701$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$702$		0			0
$703$	50.8149	−	5.86759i	1.91652	−	0.221300i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−41.2904	−	15.0285i	−1.55070	−	0.564407i	−0.582115	−	0.813107i	$-0.697775\pi$
$709$	−41.2904	−	15.0285i	−1.55070	−	0.564407i	−0.968581	+	0.248700i	$0.919997\pi$
$710$		0			0
$711$	38.7318	+	22.3618i	1.45256	+	0.838633i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$719$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$720$		0			0
$721$			100.343i			3.73697i
$722$		0			0
$723$	36.8117			1.36904
$724$	−24.4935	+	29.1902i	−0.910294	+	1.08485i
$725$		0			0
$726$		0			0
$727$	−4.00480	+	22.7124i	−0.148530	+	0.842355i	0.815935	+	0.578144i	$0.196223\pi$
$727$	−4.00480	+	22.7124i	−0.148530	+	0.842355i	−0.964465	+	0.264211i	$0.914888\pi$
$728$		0			0
$729$	13.5000	+	23.3827i	0.500000	+	0.866025i
$730$		0			0
$731$		0			0
$732$	−9.77930	−	11.6545i	−0.361453	−	0.430764i
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	−0.129275	−	0.991609i	$-0.541265\pi$
$733$	21.5000	−	37.2391i	0.794121	−	1.37546i	0.923396	−	0.383849i	$-0.125402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	2.09652	+	1.75919i	0.0771216	+	0.0647127i	0.680534	−	0.732717i	$-0.261748\pi$
$739$	2.09652	+	1.75919i	0.0771216	+	0.0647127i	−0.603412	+	0.797430i	$0.706193\pi$
$740$		0			0
$741$	−2.74763	−	23.7951i	−0.100937	−	0.874136i
$742$		0			0
$743$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$743$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	30.9839	+	36.9251i	1.13062	+	1.34742i	0.929919	+	0.367764i	$0.119877\pi$
$751$	30.9839	+	36.9251i	1.13062	+	1.34742i	0.200698	+	0.979653i	$0.435679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	50.5945	−	8.92118i	1.84010	−	0.324460i
$757$	48.0096	−	17.4740i	1.74494	−	0.635105i	0.745432	−	0.666581i	$-0.232243\pi$
$757$	48.0096	−	17.4740i	1.74494	−	0.635105i	0.999505	+	0.0314762i	$0.0100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$	−14.6427	−	40.2306i	−0.530103	−	1.45645i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−24.0000	−	13.8564i	−0.866025	−	0.500000i
$769$	−42.0795	+	35.3089i	−1.51743	+	1.27327i	−0.669994	+	0.742367i	$0.733703\pi$
$769$	−42.0795	+	35.3089i	−1.51743	+	1.27327i	−0.847432	+	0.530904i	$0.821852\pi$
$770$		0			0
$771$		0			0
$772$	38.9968	−	22.5148i	1.40353	−	0.810326i
$773$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$773$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$774$		0			0
$775$	26.5397	−	4.67966i	0.953333	−	0.168098i
$776$		0			0
$777$	−76.9741	−	64.5890i	−2.76143	−	2.31712i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	12.1129	−	68.6955i	0.432603	−	2.45341i
$785$		0			0
$786$		0			0
$787$	−47.2397	−	27.2738i	−1.68391	−	0.972208i	−0.959017	−	0.283350i	$-0.908554\pi$
$787$	−47.2397	−	27.2738i	−1.68391	−	0.972208i	−0.724897	−	0.688858i	$-0.758113\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	13.7223	+	2.41962i	0.487295	+	0.0859232i
$794$		0			0
$795$		0			0
$796$	29.4507	+	24.7121i	1.04385	+	0.875895i
$797$		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
$804$	7.21719	+	2.62684i	0.254531	+	0.0926415i
$805$		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
$811$	18.3643	−	50.4555i	0.644857	−	1.77173i	0.00895645	−	0.999960i	$-0.497149\pi$
$811$	18.3643	−	50.4555i	0.644857	−	1.77173i	0.635901	−	0.771771i	$-0.280629\pi$
$812$		0			0
$813$	1.70574	−	0.300767i	0.0598228	−	0.0105484i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.40049	−	0.143172i	−0.0839826	−	0.00500894i
$818$		0			0
$819$	−30.2452	+	36.0448i	−1.05685	+	1.25951i
$820$		0			0
$821$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$821$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$822$		0			0
$823$	44.1656	+	16.0749i	1.53951	+	0.560337i	0.965929	−	0.258808i	$-0.0833297\pi$
$823$	44.1656	+	16.0749i	1.53951	+	0.560337i	0.573586	+	0.819146i	$0.305552\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$827$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$828$		0			0
$829$	−14.5657	+	8.40949i	−0.505886	+	0.292074i	−0.731141	−	0.682226i	$-0.761012\pi$
$829$	−14.5657	+	8.40949i	−0.505886	+	0.292074i	0.225255	+	0.974300i	$0.427679\pi$
$830$		0			0
$831$	−8.52869	−	1.50384i	−0.295857	−	0.0521675i
$832$	24.9959	−	4.40745i	0.866576	−	0.152801i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−28.0063			−0.968040
$838$		0			0
$839$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$839$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$840$		0			0
$841$	−5.03580	+	28.5594i	−0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	49.9873	+	28.8602i	1.72063	+	0.993409i
$845$		0			0
$846$		0			0
$847$	27.1896	−	47.0938i	0.934246	−	1.61816i
$848$		0			0
$849$	−4.14677	+	11.3932i	−0.142317	+	0.391012i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−37.4473	−	31.4220i	−1.28217	−	1.07587i	−0.992941	−	0.118609i	$-0.962157\pi$
$853$	−37.4473	−	31.4220i	−1.28217	−	1.07587i	−0.289229	−	0.957260i	$-0.593399\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$857$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$858$		0			0
$859$	8.10917	+	45.9894i	0.276681	+	1.56914i	0.733571	+	0.679613i	$0.237852\pi$
$859$	8.10917	+	45.9894i	0.276681	+	1.56914i	−0.456889	+	0.889524i	$0.651036\pi$
$860$		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
$867$	25.5000	−	14.7224i	0.866025	−	0.500000i
$868$	−18.2262	+	50.0760i	−0.618637	+	1.69969i
$869$		0			0
$870$		0			0
$871$	−6.61004	+	2.40586i	−0.223973	+	0.0815194i
$872$		0			0
$873$		−	15.5885i		−	0.527589i
$874$		0			0
$875$		0			0
$876$	−34.2517	+	40.8195i	−1.15726	+	1.37916i
$877$	11.6095	+	31.8969i	0.392026	+	1.07708i	0.966075	+	0.258261i	$0.0831496\pi$
$877$	11.6095	+	31.8969i	0.392026	+	1.07708i	−0.574049	+	0.818821i	$0.694628\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	−9.67049	+	8.11451i	−0.325438	+	0.273075i	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−9.67049	+	8.11451i	−0.325438	+	0.273075i	0.465400	+	0.885100i	$0.345910\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.342020	−	0.939693i	$-0.388889\pi$
$887$		0			0		−0.342020	+	0.939693i	$0.611111\pi$
$888$		0			0
$889$	−59.0269	+	10.4080i	−1.97970	+	0.349074i
$890$		0			0
$891$		0			0
$892$			38.6843i			1.29525i
$893$		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
$903$	3.03643	+	3.61867i	0.101046	+	0.120422i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	56.2893	+	9.92533i	1.86906	+	0.329565i	0.989304	−	0.145868i	$-0.0465973\pi$
$907$	56.2893	+	9.92533i	1.86906	+	0.329565i	0.879752	+	0.475433i	$0.157708\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−20.7547	+	21.9373i	−0.687255	+	0.726416i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−10.4929	−	59.5081i	−0.346695	−	1.96620i
$917$		0			0
$918$		0			0
$919$	−29.9372	−	51.8528i	−0.987538	−	1.71047i	−0.630065	−	0.776542i	$-0.716972\pi$
$919$	−29.9372	−	51.8528i	−0.987538	−	1.71047i	−0.357472	−	0.933924i	$-0.616361\pi$
$920$		0			0
$921$	39.0683	−	32.7822i	1.28734	−	1.08021i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−20.0684	+	55.1373i	−0.659843	+	1.81290i
$926$		0			0
$927$	−59.9680	+	10.5740i	−1.96961	+	0.347295i
$928$		0			0
$929$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$929$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$930$		0			0
$931$	−69.7553	−	30.2049i	−2.28614	−	0.989926i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	40.7720	+	14.8398i	1.33196	+	0.484795i	0.907273	−	0.420543i	$-0.138160\pi$
$937$	40.7720	+	14.8398i	1.33196	+	0.484795i	0.424691	+	0.905338i	$0.360383\pi$
$938$		0			0
$939$	−19.5000	−	11.2583i	−0.636358	−	0.367402i
$940$		0			0
$941$		0			0		−0.642788	−	0.766044i	$-0.722222\pi$
$941$		0			0		0.642788	+	0.766044i	$0.277778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$947$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$948$	39.5604	+	33.1951i	1.28486	+	1.07813i
$949$		−	48.8034i		−	1.58423i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$953$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.974937	+	1.68864i	−0.0314496	+	0.0544723i
$962$		0			0
$963$		0			0
$964$	41.8607	+	7.38117i	1.34824	+	0.237732i
$965$		0			0
$966$		0			0
$967$	17.2610	+	14.4837i	0.555078	+	0.465766i	0.876656	−	0.481117i	$-0.159769\pi$
$967$	17.2610	+	14.4837i	0.555078	+	0.465766i	−0.321578	+	0.946883i	$0.604213\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.642788	−	0.766044i	$-0.277778\pi$
$971$		0			0		−0.642788	+	0.766044i	$0.722222\pi$
$972$	10.6631	+	29.2967i	0.342020	+	0.939693i
$973$	12.2577	+	69.5167i	0.392963	+	2.22860i
$974$		0			0
$975$	25.8192	+	9.39743i	0.826877	+	0.300959i
$976$	−8.78375	−	15.2139i	−0.281161	−	0.486985i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	22.5000	−	12.9904i	0.718370	−	0.414751i
$982$		0			0
$983$		0			0		−0.984808	−	0.173648i	$-0.944444\pi$
$983$		0			0		0.984808	+	0.173648i	$0.0555556\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.64672	−	27.6098i	0.0523891	−	0.878384i
$989$		0			0
$990$		0			0
$991$	12.1457	+	33.3700i	0.385820	+	1.06003i	0.968864	+	0.247592i	$0.0796392\pi$
$991$	12.1457	+	33.3700i	0.385820	+	1.06003i	−0.583044	+	0.812440i	$0.698139\pi$
$992$		0			0
$993$	2.19588	−	12.4535i	0.0696842	−	0.395199i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.8350	+	20.8391i	−0.786533	+	0.659980i	−0.944885	−	0.327403i	$-0.893827\pi$
$997$	−24.8350	+	20.8391i	−0.786533	+	0.659980i	0.158352	+	0.987383i	$0.449382\pi$
$998$		0			0
$999$	30.4889	−	52.8083i	0.964626	−	1.67078i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	57.2.j.a.41.1	yes	6
3.2	odd	2	CM	57.2.j.a.41.1	yes	6
4.3	odd	2		912.2.cc.a.497.1		6
12.11	even	2		912.2.cc.a.497.1		6
19.5	even	9		1083.2.d.a.1082.4		6
19.13	odd	18	inner	57.2.j.a.32.1	✓	6
19.14	odd	18		1083.2.d.a.1082.1		6
57.5	odd	18		1083.2.d.a.1082.4		6
57.14	even	18		1083.2.d.a.1082.1		6
57.32	even	18	inner	57.2.j.a.32.1	✓	6
76.51	even	18		912.2.cc.a.545.1		6
228.203	odd	18		912.2.cc.a.545.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.j.a.32.1	✓	6	19.13	odd	18	inner
57.2.j.a.32.1	✓	6	57.32	even	18	inner
57.2.j.a.41.1	yes	6	1.1	even	1	trivial
57.2.j.a.41.1	yes	6	3.2	odd	2	CM
912.2.cc.a.497.1		6	4.3	odd	2
912.2.cc.a.497.1		6	12.11	even	2
912.2.cc.a.545.1		6	76.51	even	18
912.2.cc.a.545.1		6	228.203	odd	18
1083.2.d.a.1082.1		6	19.14	odd	18
1083.2.d.a.1082.1		6	57.14	even	18
1083.2.d.a.1082.4		6	19.5	even	9
1083.2.d.a.1082.4		6	57.5	odd	18

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:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.j (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.592396 - 1.62760i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.47178 + 4.28125i) q^{7} +(-2.29813 + 1.92836i) q^{9} +O(q^{10})\)	\(q+(-0.592396 - 1.62760i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(2.47178 + 4.28125i) q^{7} +(-2.29813 + 1.92836i) q^{9} +(-3.00000 + 1.73205i) q^{12} +(1.08512 - 2.98135i) q^{13} +(-3.75877 + 1.36808i) q^{16} +(0.500000 + 4.33013i) q^{19} +(5.50387 - 6.55926i) q^{21} +(-4.69846 - 1.71010i) q^{25} +(4.50000 + 2.59808i) q^{27} +(7.57398 - 6.35532i) q^{28} +(-4.66772 + 2.69491i) q^{31} +(4.59627 + 3.85673i) q^{36} -11.7352i q^{37} -5.49525 q^{39} +(-0.0957998 + 0.543308i) q^{43} +(4.45336 + 5.30731i) q^{48} +(-8.71941 + 15.1025i) q^{49} +(-6.24897 - 1.10186i) q^{52} +(6.75150 - 3.37895i) q^{57} +(0.762641 + 4.32515i) q^{61} +(-13.9363 - 5.07239i) q^{63} +(4.00000 + 6.92820i) q^{64} +(-1.42514 - 1.69842i) q^{67} +(14.4547 - 5.26108i) q^{73} +8.66025i q^{75} +(8.35504 - 2.48865i) q^{76} +(-5.09879 - 14.0088i) q^{79} +(1.56283 - 8.86327i) q^{81} +(-14.8307 - 8.56250i) q^{84} +(15.4461 - 2.72356i) q^{91} +(7.15136 + 6.00070i) q^{93} +(-3.34002 + 3.98048i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q - 18 q^{12} - 15 q^{13} + 3 q^{19} + 9 q^{21} + 27 q^{27} + 30 q^{28} - 39 q^{43} - 21 q^{49} - 12 q^{52} - 42 q^{61} - 36 q^{63} + 24 q^{64} + 33 q^{67} + 51 q^{73} + 12 q^{79} + 48 q^{91} + 54 q^{93}+O(q^{100}) \) 6 * q - 18 * q^12 - 15 * q^13 + 3 * q^19 + 9 * q^21 + 27 * q^27 + 30 * q^28 - 39 * q^43 - 21 * q^49 - 12 * q^52 - 42 * q^61 - 36 * q^63 + 24 * q^64 + 33 * q^67 + 51 * q^73 + 12 * q^79 + 48 * q^91 + 54 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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