LMFDB - Embedded newform 592.2.w.d.545.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [592,2,Mod(529,592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("592.529"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	592.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$4.72714379966$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 74)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			545.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	592.545
Dual form			592.2.w.d.529.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.36603 + 2.36603i) q^{3} +(-1.50000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-2.23205 + 3.86603i) q^{9} -1.26795 q^{11} +(5.19615 - 3.00000i) q^{13} +(-4.09808 - 2.36603i) q^{15} +(-1.50000 - 0.866025i) q^{17} +(-4.09808 + 2.36603i) q^{19} +(-5.46410 + 9.46410i) q^{21} -4.73205i q^{23} +(-1.00000 + 1.73205i) q^{25} -4.00000 q^{27} -7.73205i q^{29} +4.73205i q^{31} +(-1.73205 - 3.00000i) q^{33} +(-6.00000 - 3.46410i) q^{35} +(-0.500000 + 6.06218i) q^{37} +(14.1962 + 8.19615i) q^{39} +(3.23205 + 5.59808i) q^{41} -2.53590i q^{43} -7.73205i q^{45} +5.66025 q^{47} +(-4.50000 + 7.79423i) q^{49} -4.73205i q^{51} +(4.73205 - 8.19615i) q^{53} +(1.90192 - 1.09808i) q^{55} +(-11.1962 - 6.46410i) q^{57} +(-8.19615 - 4.73205i) q^{59} +(2.30385 - 1.33013i) q^{61} -17.8564 q^{63} +(-5.19615 + 9.00000i) q^{65} +(-2.09808 - 3.63397i) q^{67} +(11.1962 - 6.46410i) q^{69} +(4.73205 + 8.19615i) q^{71} +8.39230 q^{73} -5.46410 q^{75} +(-2.53590 - 4.39230i) q^{77} +(1.90192 - 1.09808i) q^{79} +(1.23205 + 2.13397i) q^{81} +(-2.83013 + 4.90192i) q^{83} +3.00000 q^{85} +(18.2942 - 10.5622i) q^{87} +(4.50000 + 2.59808i) q^{89} +(20.7846 + 12.0000i) q^{91} +(-11.1962 + 6.46410i) q^{93} +(4.09808 - 7.09808i) q^{95} -5.19615i q^{97} +(2.83013 - 4.90192i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 6 q^{5} + 8 q^{7} - 2 q^{9} - 12 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 8 q^{21} - 4 q^{25} - 16 q^{27} - 24 q^{35} - 2 q^{37} + 36 q^{39} + 6 q^{41} - 12 q^{47} - 18 q^{49} + 12 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 6 * q^5 + 8 * q^7 - 2 * q^9 - 12 * q^11 - 6 * q^15 - 6 * q^17 - 6 * q^19 - 8 * q^21 - 4 * q^25 - 16 * q^27 - 24 * q^35 - 2 * q^37 + 36 * q^39 + 6 * q^41 - 12 * q^47 - 18 * q^49 + 12 * q^53 + 18 * q^55 - 24 * q^57 - 12 * q^59 + 30 * q^61 - 16 * q^63 + 2 * q^67 + 24 * q^69 + 12 * q^71 - 8 * q^73 - 8 * q^75 - 24 * q^77 + 18 * q^79 - 2 * q^81 + 6 * q^83 + 12 * q^85 + 42 * q^87 + 18 * q^89 - 24 * q^93 + 6 * q^95 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/592\mathbb{Z}\right)^\times$.

$n$	$113$	$149$	$223$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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
$2$		0			0
$3$	1.36603	+	2.36603i	0.788675	+	1.36603i	0.926779	+	0.375608i	$0.122566\pi$
$3$	1.36603	+	2.36603i	0.788675	+	1.36603i	−0.138104	+	0.990418i	$0.544101\pi$
$4$		0			0
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−1.50000	+	0.866025i	−0.670820	+	0.387298i	0.125567	+	0.992085i	$0.459925\pi$
$6$		0			0
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	0.944911	+	0.327327i	$0.106148\pi$
$7$	2.00000	+	3.46410i	0.755929	+	1.30931i	−0.188982	+	0.981981i	$0.560519\pi$
$8$		0			0
$9$	−2.23205	+	3.86603i	−0.744017	+	1.28868i
$10$		0			0
$11$	−1.26795			−0.382301			−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795			−0.382301			−0.191151	+	0.981561i	$0.561222\pi$
$12$		0			0
$13$	5.19615	−	3.00000i	1.44115	−	0.832050i	0.443227	−	0.896410i	$-0.353834\pi$
$13$	5.19615	−	3.00000i	1.44115	−	0.832050i	0.997927	+	0.0643593i	$0.0205004\pi$
$14$		0			0
$15$	−4.09808	−	2.36603i	−1.05812	−	0.610905i
$16$		0			0
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	0.306944	−	0.951727i	$-0.400693\pi$
$17$	−1.50000	−	0.866025i	−0.363803	−	0.210042i	−0.670748	+	0.741685i	$0.734027\pi$
$18$		0			0
$19$	−4.09808	+	2.36603i	−0.940163	+	0.542803i	−0.890011	−	0.455938i	$-0.849304\pi$
$19$	−4.09808	+	2.36603i	−0.940163	+	0.542803i	−0.0501517	+	0.998742i	$0.515970\pi$
$20$		0			0
$21$	−5.46410	+	9.46410i	−1.19236	+	2.06524i
$22$		0			0
$23$		−	4.73205i		−	0.986701i	−0.869831	−	0.493350i	$-0.835772\pi$
$23$		−	4.73205i		−	0.986701i	0.869831	−	0.493350i	$-0.164228\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$		−	7.73205i		−	1.43581i	−0.696143	−	0.717903i	$-0.745102\pi$
$29$		−	7.73205i		−	1.43581i	0.696143	−	0.717903i	$-0.254898\pi$
$30$		0			0
$31$			4.73205i			0.849901i	0.905216	+	0.424951i	$0.139709\pi$
$31$			4.73205i			0.849901i	−0.905216	+	0.424951i	$0.860291\pi$
$32$		0			0
$33$	−1.73205	−	3.00000i	−0.301511	−	0.522233i
$34$		0			0
$35$	−6.00000	−	3.46410i	−1.01419	−	0.585540i
$36$		0			0
$37$	−0.500000	+	6.06218i	−0.0821995	+	0.996616i
$37$	−0.500000	+	6.06218i	−0.0821995	+	0.996616i
$38$		0			0
$39$	14.1962	+	8.19615i	2.27320	+	1.31243i
$40$		0			0
$41$	3.23205	+	5.59808i	0.504762	+	0.874273i	0.999985	+	0.00550690i	$0.00175291\pi$
$41$	3.23205	+	5.59808i	0.504762	+	0.874273i	−0.495223	+	0.868766i	$0.664914\pi$
$42$		0			0
$43$		−	2.53590i		−	0.386721i	−0.981128	−	0.193360i	$-0.938061\pi$
$43$		−	2.53590i		−	0.386721i	0.981128	−	0.193360i	$-0.0619387\pi$
$44$		0			0
$45$		−	7.73205i		−	1.15263i
$46$		0			0
$47$	5.66025			0.825633			0.412816	−	0.910814i	$-0.364545\pi$
$47$	5.66025			0.825633			0.412816	+	0.910814i	$0.364545\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$		−	4.73205i		−	0.662620i
$52$		0			0
$53$	4.73205	−	8.19615i	0.649997	−	1.12583i	−0.333126	−	0.942882i	$-0.608103\pi$
$53$	4.73205	−	8.19615i	0.649997	−	1.12583i	0.983123	−	0.182946i	$-0.0585633\pi$
$54$		0			0
$55$	1.90192	−	1.09808i	0.256455	−	0.148065i
$56$		0			0
$57$	−11.1962	−	6.46410i	−1.48297	−	0.856191i
$58$		0			0
$59$	−8.19615	−	4.73205i	−1.06705	−	0.616061i	−0.139675	−	0.990197i	$-0.544606\pi$
$59$	−8.19615	−	4.73205i	−1.06705	−	0.616061i	−0.927373	+	0.374137i	$0.877939\pi$
$60$		0			0
$61$	2.30385	−	1.33013i	0.294977	−	0.170305i	−0.345207	−	0.938527i	$-0.612191\pi$
$61$	2.30385	−	1.33013i	0.294977	−	0.170305i	0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$	−17.8564			−2.24970
$64$		0			0
$65$	−5.19615	+	9.00000i	−0.644503	+	1.11631i
$66$		0			0
$67$	−2.09808	−	3.63397i	−0.256321	−	0.443961i	0.708933	−	0.705276i	$-0.249177\pi$
$67$	−2.09808	−	3.63397i	−0.256321	−	0.443961i	−0.965253	+	0.261316i	$0.915844\pi$
$68$		0			0
$69$	11.1962	−	6.46410i	1.34786	−	0.778186i
$70$		0			0
$71$	4.73205	+	8.19615i	0.561591	+	0.972704i	0.997358	+	0.0726447i	$0.0231439\pi$
$71$	4.73205	+	8.19615i	0.561591	+	0.972704i	−0.435767	+	0.900060i	$0.643523\pi$
$72$		0			0
$73$	8.39230			0.982245			0.491122	−	0.871091i	$-0.336587\pi$
$73$	8.39230			0.982245			0.491122	+	0.871091i	$0.336587\pi$
$74$		0			0
$75$	−5.46410			−0.630940
$76$		0			0
$77$	−2.53590	−	4.39230i	−0.288992	−	0.500550i
$78$		0			0
$79$	1.90192	−	1.09808i	0.213983	−	0.123543i	−0.389178	−	0.921163i	$-0.627241\pi$
$79$	1.90192	−	1.09808i	0.213983	−	0.123543i	0.603161	+	0.797619i	$0.293908\pi$
$80$		0			0
$81$	1.23205	+	2.13397i	0.136895	+	0.237108i
$82$		0			0
$83$	−2.83013	+	4.90192i	−0.310647	+	0.538056i	−0.978503	−	0.206235i	$-0.933879\pi$
$83$	−2.83013	+	4.90192i	−0.310647	+	0.538056i	0.667856	+	0.744291i	$0.267212\pi$
$84$		0			0
$85$	3.00000			0.325396
$86$		0			0
$87$	18.2942	−	10.5622i	1.96135	−	1.13238i
$88$		0			0
$89$	4.50000	+	2.59808i	0.476999	+	0.275396i	0.719165	−	0.694839i	$-0.244525\pi$
$89$	4.50000	+	2.59808i	0.476999	+	0.275396i	−0.242166	+	0.970235i	$0.577858\pi$
$90$		0			0
$91$	20.7846	+	12.0000i	2.17882	+	1.25794i
$92$		0			0
$93$	−11.1962	+	6.46410i	−1.16099	+	0.670296i
$94$		0			0
$95$	4.09808	−	7.09808i	0.420454	−	0.728247i
$96$		0			0
$97$		−	5.19615i		−	0.527589i	−0.964579	−	0.263795i	$-0.915026\pi$
$97$		−	5.19615i		−	0.527589i	0.964579	−	0.263795i	$-0.0849741\pi$
$98$		0			0
$99$	2.83013	−	4.90192i	0.284438	−	0.492662i

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.2.w.d.545.2		4
4.3	odd	2		74.2.e.a.27.1	yes	4
12.11	even	2		666.2.s.e.397.2		4
37.11	even	6	inner	592.2.w.d.529.2		4
148.11	odd	6		74.2.e.a.11.1	✓	4
148.23	even	12		2738.2.a.j.1.1		2
148.51	even	12		2738.2.a.f.1.1		2
444.11	even	6		666.2.s.e.307.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.e.a.11.1	✓	4	148.11	odd	6
74.2.e.a.27.1	yes	4	4.3	odd	2
592.2.w.d.529.2		4	37.11	even	6	inner
592.2.w.d.545.2		4	1.1	even	1	trivial
666.2.s.e.307.2		4	444.11	even	6
666.2.s.e.397.2		4	12.11	even	2
2738.2.a.f.1.1		2	148.51	even	12
2738.2.a.j.1.1		2	148.23	even	12

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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.36603 + 2.36603i) q^{3} +(-1.50000 + 0.866025i) q^{5} +(2.00000 + 3.46410i) q^{7} +(-2.23205 + 3.86603i) q^{9} -1.26795 q^{11} +(5.19615 - 3.00000i) q^{13} +(-4.09808 - 2.36603i) q^{15} +(-1.50000 - 0.866025i) q^{17} +(-4.09808 + 2.36603i) q^{19} +(-5.46410 + 9.46410i) q^{21} -4.73205i q^{23} +(-1.00000 + 1.73205i) q^{25} -4.00000 q^{27} -7.73205i q^{29} +4.73205i q^{31} +(-1.73205 - 3.00000i) q^{33} +(-6.00000 - 3.46410i) q^{35} +(-0.500000 + 6.06218i) q^{37} +(14.1962 + 8.19615i) q^{39} +(3.23205 + 5.59808i) q^{41} -2.53590i q^{43} -7.73205i q^{45} +5.66025 q^{47} +(-4.50000 + 7.79423i) q^{49} -4.73205i q^{51} +(4.73205 - 8.19615i) q^{53} +(1.90192 - 1.09808i) q^{55} +(-11.1962 - 6.46410i) q^{57} +(-8.19615 - 4.73205i) q^{59} +(2.30385 - 1.33013i) q^{61} -17.8564 q^{63} +(-5.19615 + 9.00000i) q^{65} +(-2.09808 - 3.63397i) q^{67} +(11.1962 - 6.46410i) q^{69} +(4.73205 + 8.19615i) q^{71} +8.39230 q^{73} -5.46410 q^{75} +(-2.53590 - 4.39230i) q^{77} +(1.90192 - 1.09808i) q^{79} +(1.23205 + 2.13397i) q^{81} +(-2.83013 + 4.90192i) q^{83} +3.00000 q^{85} +(18.2942 - 10.5622i) q^{87} +(4.50000 + 2.59808i) q^{89} +(20.7846 + 12.0000i) q^{91} +(-11.1962 + 6.46410i) q^{93} +(4.09808 - 7.09808i) q^{95} -5.19615i q^{97} +(2.83013 - 4.90192i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 6 q^{5} + 8 q^{7} - 2 q^{9} - 12 q^{11} - 6 q^{15} - 6 q^{17} - 6 q^{19} - 8 q^{21} - 4 q^{25} - 16 q^{27} - 24 q^{35} - 2 q^{37} + 36 q^{39} + 6 q^{41} - 12 q^{47} - 18 q^{49} + 12 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 6 * q^5 + 8 * q^7 - 2 * q^9 - 12 * q^11 - 6 * q^15 - 6 * q^17 - 6 * q^19 - 8 * q^21 - 4 * q^25 - 16 * q^27 - 24 * q^35 - 2 * q^37 + 36 * q^39 + 6 * q^41 - 12 * q^47 - 18 * q^49 + 12 * q^53 + 18 * q^55 - 24 * q^57 - 12 * q^59 + 30 * q^61 - 16 * q^63 + 2 * q^67 + 24 * q^69 + 12 * q^71 - 8 * q^73 - 8 * q^75 - 24 * q^77 + 18 * q^79 - 2 * q^81 + 6 * q^83 + 12 * q^85 + 42 * q^87 + 18 * q^89 - 24 * q^93 + 6 * q^95 - 6 * q^99

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