LMFDB - Embedded newform 592.2.i.a.417.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [592,2,Mod(417,592)] 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("592.417"); 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, 4])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	592.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

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

Embedding invariants

Embedding label			417.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	592.417
Dual form			592.2.i.a.433.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 - 1.73205i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +6.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-3.00000 + 5.19615i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(1.00000 + 1.73205i) q^{19} +(-4.00000 + 6.92820i) q^{21} -6.00000 q^{23} +(-2.00000 + 3.46410i) q^{25} -4.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} +(-6.00000 - 10.3923i) q^{33} +(-6.00000 + 10.3923i) q^{35} +(5.50000 + 2.59808i) q^{37} +(-2.00000 + 3.46410i) q^{39} +(-1.50000 - 2.59808i) q^{41} +4.00000 q^{43} +3.00000 q^{45} +6.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} +6.00000 q^{51} +(3.00000 - 5.19615i) q^{53} +(-9.00000 - 15.5885i) q^{55} +(2.00000 - 3.46410i) q^{57} +(0.500000 + 0.866025i) q^{61} +4.00000 q^{63} +(-3.00000 + 5.19615i) q^{65} +(1.00000 + 1.73205i) q^{67} +(6.00000 + 10.3923i) q^{69} +(-6.00000 - 10.3923i) q^{71} -10.0000 q^{73} +8.00000 q^{75} +(-12.0000 - 20.7846i) q^{77} +(7.00000 + 12.1244i) q^{79} +(5.50000 + 9.52628i) q^{81} +(3.00000 - 5.19615i) q^{83} +9.00000 q^{85} +(-3.00000 - 5.19615i) q^{87} +(-1.50000 + 2.59808i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(2.00000 + 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{95} -13.0000 q^{97} +(-3.00000 + 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 3 q^{5} - 4 q^{7} - q^{9} + 12 q^{11} - 2 q^{13} - 6 q^{15} - 3 q^{17} + 2 q^{19} - 8 q^{21} - 12 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} - 4 q^{31} - 12 q^{33} - 12 q^{35} + 11 q^{37}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 3 * q^5 - 4 * q^7 - q^9 + 12 * q^11 - 2 * q^13 - 6 * q^15 - 3 * q^17 + 2 * q^19 - 8 * q^21 - 12 * q^23 - 4 * q^25 - 8 * q^27 + 6 * q^29 - 4 * q^31 - 12 * q^33 - 12 * q^35 + 11 * q^37 - 4 * q^39 - 3 * q^41 + 8 * q^43 + 6 * q^45 + 12 * q^47 - 9 * q^49 + 12 * q^51 + 6 * q^53 - 18 * q^55 + 4 * q^57 + q^61 + 8 * q^63 - 6 * q^65 + 2 * q^67 + 12 * q^69 - 12 * q^71 - 20 * q^73 + 16 * q^75 - 24 * q^77 + 14 * q^79 + 11 * q^81 + 6 * q^83 + 18 * q^85 - 6 * q^87 - 3 * q^89 - 8 * q^91 + 4 * q^93 + 6 * q^95 - 26 * q^97 - 6 * q^99

Character values

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

$n$	$113$	$149$	$223$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	−0.995782	−	0.0917517i	$-0.970753\pi$
$3$	−1.00000	−	1.73205i	−0.577350	−	1.00000i	0.418432	−	0.908248i	$-0.362580\pi$
$4$		0			0
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−1.50000	−	2.59808i	−0.670820	−	1.16190i	0.306851	−	0.951757i	$-0.400725\pi$
$6$		0			0
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	−0.944911	−	0.327327i	$-0.893852\pi$
$7$	−2.00000	−	3.46410i	−0.755929	−	1.30931i	0.188982	−	0.981981i	$-0.439481\pi$
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	6.00000			1.80907			0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000			1.80907			0.904534	+	0.426401i	$0.140219\pi$
$12$		0			0
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	−1.00000	−	1.73205i	−0.277350	−	0.480384i	−0.970725	+	0.240192i	$0.922790\pi$
$14$		0			0
$15$	−3.00000	+	5.19615i	−0.774597	+	1.34164i
$16$		0			0
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	−0.988583	−	0.150675i	$-0.951855\pi$
$17$	−1.50000	+	2.59808i	−0.363803	+	0.630126i	0.624780	+	0.780801i	$0.285189\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	0.957635	−	0.287984i	$-0.0929851\pi$
$19$	1.00000	+	1.73205i	0.229416	+	0.397360i	−0.728219	+	0.685344i	$0.759652\pi$
$20$		0			0
$21$	−4.00000	+	6.92820i	−0.872872	+	1.51186i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	−2.00000	+	3.46410i	−0.400000	+	0.692820i
$26$		0			0
$27$	−4.00000			−0.769800
$28$		0			0
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$	−6.00000	−	10.3923i	−1.04447	−	1.80907i
$34$		0			0
$35$	−6.00000	+	10.3923i	−1.01419	+	1.75662i
$36$		0			0
$37$	5.50000	+	2.59808i	0.904194	+	0.427121i
$37$	5.50000	+	2.59808i	0.904194	+	0.427121i
$38$		0			0
$39$	−2.00000	+	3.46410i	−0.320256	+	0.554700i
$40$		0			0
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	0.724797	−	0.688963i	$-0.241934\pi$
$41$	−1.50000	−	2.59808i	−0.234261	−	0.405751i	−0.959058	+	0.283211i	$0.908600\pi$
$42$		0			0
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$	3.00000			0.447214
$46$		0			0
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	−4.50000	+	7.79423i	−0.642857	+	1.11346i
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$	−9.00000	−	15.5885i	−1.21356	−	2.10195i
$56$		0			0
$57$	2.00000	−	3.46410i	0.264906	−	0.458831i
$58$		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
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$	4.00000			0.503953
$64$		0			0
$65$	−3.00000	+	5.19615i	−0.372104	+	0.644503i
$66$		0			0
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	0.920623	−	0.390453i	$-0.127682\pi$
$67$	1.00000	+	1.73205i	0.122169	+	0.211604i	−0.798454	+	0.602056i	$0.794348\pi$
$68$		0			0
$69$	6.00000	+	10.3923i	0.722315	+	1.25109i
$70$		0			0
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	−0.964079	−	0.265615i	$-0.914425\pi$
$71$	−6.00000	−	10.3923i	−0.712069	−	1.23334i	0.252010	−	0.967725i	$-0.418908\pi$
$72$		0			0
$73$	−10.0000			−1.17041			−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000			−1.17041			−0.585206	+	0.810885i	$0.698986\pi$
$74$		0			0
$75$	8.00000			0.923760
$76$		0			0
$77$	−12.0000	−	20.7846i	−1.36753	−	2.36863i
$78$		0			0
$79$	7.00000	+	12.1244i	0.787562	+	1.36410i	0.927457	+	0.373930i	$0.121990\pi$
$79$	7.00000	+	12.1244i	0.787562	+	1.36410i	−0.139895	+	0.990166i	$0.544677\pi$
$80$		0			0
$81$	5.50000	+	9.52628i	0.611111	+	1.05848i
$82$		0			0
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$	9.00000			0.976187
$86$		0			0
$87$	−3.00000	−	5.19615i	−0.321634	−	0.557086i
$88$		0			0
$89$	−1.50000	+	2.59808i	−0.159000	+	0.275396i	−0.934508	−	0.355942i	$-0.884160\pi$
$89$	−1.50000	+	2.59808i	−0.159000	+	0.275396i	0.775509	+	0.631337i	$0.217494\pi$
$90$		0			0
$91$	−4.00000	+	6.92820i	−0.419314	+	0.726273i
$92$		0			0
$93$	2.00000	+	3.46410i	0.207390	+	0.359211i
$94$		0			0
$95$	3.00000	−	5.19615i	0.307794	−	0.533114i
$96$		0			0
$97$	−13.0000			−1.31995			−0.659975	−	0.751288i	$-0.729433\pi$
$97$	−13.0000			−1.31995			−0.659975	+	0.751288i	$0.729433\pi$
$98$		0			0
$99$	−3.00000	+	5.19615i	−0.301511	+	0.522233i

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.i.a.417.1		2
4.3	odd	2		74.2.c.b.47.1	✓	2
12.11	even	2		666.2.f.d.343.1		2
37.26	even	3	inner	592.2.i.a.433.1		2
148.27	odd	6		2738.2.a.c.1.1		1
148.47	odd	6		2738.2.a.a.1.1		1
148.63	odd	6		74.2.c.b.63.1	yes	2
444.359	even	6		666.2.f.d.433.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.2.c.b.47.1	✓	2	4.3	odd	2
74.2.c.b.63.1	yes	2	148.63	odd	6
592.2.i.a.417.1		2	1.1	even	1	trivial
592.2.i.a.433.1		2	37.26	even	3	inner
666.2.f.d.343.1		2	12.11	even	2
666.2.f.d.433.1		2	444.359	even	6
2738.2.a.a.1.1		1	148.47	odd	6
2738.2.a.c.1.1		1	148.27	odd	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{3} +(-1.50000 - 2.59808i) q^{5} +(-2.00000 - 3.46410i) q^{7} +(-0.500000 + 0.866025i) q^{9} +6.00000 q^{11} +(-1.00000 - 1.73205i) q^{13} +(-3.00000 + 5.19615i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(1.00000 + 1.73205i) q^{19} +(-4.00000 + 6.92820i) q^{21} -6.00000 q^{23} +(-2.00000 + 3.46410i) q^{25} -4.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} +(-6.00000 - 10.3923i) q^{33} +(-6.00000 + 10.3923i) q^{35} +(5.50000 + 2.59808i) q^{37} +(-2.00000 + 3.46410i) q^{39} +(-1.50000 - 2.59808i) q^{41} +4.00000 q^{43} +3.00000 q^{45} +6.00000 q^{47} +(-4.50000 + 7.79423i) q^{49} +6.00000 q^{51} +(3.00000 - 5.19615i) q^{53} +(-9.00000 - 15.5885i) q^{55} +(2.00000 - 3.46410i) q^{57} +(0.500000 + 0.866025i) q^{61} +4.00000 q^{63} +(-3.00000 + 5.19615i) q^{65} +(1.00000 + 1.73205i) q^{67} +(6.00000 + 10.3923i) q^{69} +(-6.00000 - 10.3923i) q^{71} -10.0000 q^{73} +8.00000 q^{75} +(-12.0000 - 20.7846i) q^{77} +(7.00000 + 12.1244i) q^{79} +(5.50000 + 9.52628i) q^{81} +(3.00000 - 5.19615i) q^{83} +9.00000 q^{85} +(-3.00000 - 5.19615i) q^{87} +(-1.50000 + 2.59808i) q^{89} +(-4.00000 + 6.92820i) q^{91} +(2.00000 + 3.46410i) q^{93} +(3.00000 - 5.19615i) q^{95} -13.0000 q^{97} +(-3.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 3 q^{5} - 4 q^{7} - q^{9} + 12 q^{11} - 2 q^{13} - 6 q^{15} - 3 q^{17} + 2 q^{19} - 8 q^{21} - 12 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} - 4 q^{31} - 12 q^{33} - 12 q^{35} + 11 q^{37}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 3 * q^5 - 4 * q^7 - q^9 + 12 * q^11 - 2 * q^13 - 6 * q^15 - 3 * q^17 + 2 * q^19 - 8 * q^21 - 12 * q^23 - 4 * q^25 - 8 * q^27 + 6 * q^29 - 4 * q^31 - 12 * q^33 - 12 * q^35 + 11 * q^37 - 4 * q^39 - 3 * q^41 + 8 * q^43 + 6 * q^45 + 12 * q^47 - 9 * q^49 + 12 * q^51 + 6 * q^53 - 18 * q^55 + 4 * q^57 + q^61 + 8 * q^63 - 6 * q^65 + 2 * q^67 + 12 * q^69 - 12 * q^71 - 20 * q^73 + 16 * q^75 - 24 * q^77 + 14 * q^79 + 11 * q^81 + 6 * q^83 + 18 * q^85 - 6 * q^87 - 3 * q^89 - 8 * q^91 + 4 * q^93 + 6 * q^95 - 26 * q^97 - 6 * q^99

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