LMFDB - Embedded newform 252.6.k.f.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,6,Mod(37,252)] 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("252.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0] 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:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$13.1471 - 22.7714i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.6.k.f.37.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-15.9808 - 27.6796i) q^{5} +(85.7043 - 97.2716i) q^{7} +(130.442 - 225.932i) q^{11} +769.735 q^{13} +(-776.659 + 1345.21i) q^{17} +(-375.024 - 649.561i) q^{19} +(377.427 + 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} -6008.93 q^{29} +(3210.02 - 5559.92i) q^{31} +(-4062.06 - 817.779i) q^{35} +(2387.86 + 4135.90i) q^{37} +5423.27 q^{41} -11896.4 q^{43} +(-8714.00 - 15093.1i) q^{47} +(-2116.54 - 16673.2i) q^{49} +(18825.3 - 32606.4i) q^{53} -8338.26 q^{55} +(11039.0 - 19120.2i) q^{59} +(4086.69 + 7078.36i) q^{61} +(-12301.0 - 21305.9i) q^{65} +(-6500.87 + 11259.8i) q^{67} +12349.6 q^{71} +(-21800.2 + 37759.0i) q^{73} +(-10797.3 - 32051.6i) q^{77} +(-38374.7 - 66467.0i) q^{79} -21893.6 q^{83} +49646.5 q^{85} +(-68483.5 - 118617. i) q^{89} +(65969.7 - 74873.4i) q^{91} +(-11986.4 + 20761.0i) q^{95} -93050.1 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) $ 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−15.9808	−	27.6796i	−0.285873	−	0.495147i	0.686947	−	0.726707i	$-0.258950\pi$
$5$	−15.9808	−	27.6796i	−0.285873	−	0.495147i	−0.972821	+	0.231560i	$0.925617\pi$
$6$		0			0
$7$	85.7043	−	97.2716i	0.661086	−	0.750311i
$7$	85.7043	−	97.2716i	0.661086	−	0.750311i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	130.442	−	225.932i	0.325039	−	0.562984i	−0.656481	−	0.754342i	$-0.727956\pi$
$11$	130.442	−	225.932i	0.325039	−	0.562984i	0.981520	+	0.191359i	$0.0612893\pi$
$12$		0			0
$13$	769.735			1.26323			0.631616	−	0.775281i	$-0.282392\pi$
$13$	769.735			1.26323			0.631616	+	0.775281i	$0.282392\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−776.659	+	1345.21i	−0.651791	+	1.12893i	0.330898	+	0.943667i	$0.392649\pi$
$17$	−776.659	+	1345.21i	−0.651791	+	1.12893i	−0.982688	+	0.185268i	$0.940685\pi$
$18$		0			0
$19$	−375.024	−	649.561i	−0.238328	−	0.412797i	0.721906	−	0.691991i	$-0.243266\pi$
$19$	−375.024	−	649.561i	−0.238328	−	0.412797i	−0.960235	+	0.279194i	$0.909933\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	377.427	+	653.723i	0.148769	+	0.257676i	0.930773	−	0.365598i	$-0.119135\pi$
$23$	377.427	+	653.723i	0.148769	+	0.257676i	−0.782004	+	0.623274i	$0.785802\pi$
$24$		0			0
$25$	1051.73	−	1821.65i	0.336553	−	0.582927i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6008.93			−1.32679			−0.663395	−	0.748269i	$-0.730885\pi$
$29$	−6008.93			−1.32679			−0.663395	+	0.748269i	$0.730885\pi$
$30$		0			0
$31$	3210.02	−	5559.92i	0.599934	−	1.03912i	−0.392896	−	0.919583i	$-0.628527\pi$
$31$	3210.02	−	5559.92i	0.599934	−	1.03912i	0.992830	−	0.119533i	$-0.0381399\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4062.06	−	817.779i	−0.560501	−	0.112841i
$36$		0			0
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	0.973032	−	0.230669i	$-0.0740914\pi$
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	−0.686281	+	0.727336i	$0.740758\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	5423.27			0.503850			0.251925	−	0.967747i	$-0.418936\pi$
$41$	5423.27			0.503850			0.251925	+	0.967747i	$0.418936\pi$
$42$		0			0
$43$	−11896.4			−0.981171			−0.490585	−	0.871393i	$-0.663217\pi$
$43$	−11896.4			−0.981171			−0.490585	+	0.871393i	$0.663217\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	−0.995998	−	0.0893798i	$-0.971512\pi$
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	0.420594	−	0.907249i	$-0.361822\pi$
$48$		0			0
$49$	−2116.54	−	16673.2i	−0.125932	−	0.992039i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	18825.3	−	32606.4i	0.920561	−	1.59446i	0.122012	−	0.992529i	$-0.461065\pi$
$53$	18825.3	−	32606.4i	0.920561	−	1.59446i	0.798549	−	0.601930i	$-0.205601\pi$
$54$		0			0
$55$	−8338.26			−0.371679
$56$		0			0
$57$		0			0
$58$		0			0
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	−0.582342	−	0.812944i	$-0.697864\pi$
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	0.995201	+	0.0978516i	$0.0311971\pi$
$60$		0			0
$61$	4086.69	+	7078.36i	0.140620	+	0.243561i	0.927730	−	0.373251i	$-0.121757\pi$
$61$	4086.69	+	7078.36i	0.140620	+	0.243561i	−0.787110	+	0.616812i	$0.788424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−12301.0	−	21305.9i	−0.361124	−	0.625485i
$66$		0			0
$67$	−6500.87	+	11259.8i	−0.176923	+	0.306440i	−0.940825	−	0.338892i	$-0.889948\pi$
$67$	−6500.87	+	11259.8i	−0.176923	+	0.306440i	0.763902	+	0.645332i	$0.223281\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12349.6			0.290742			0.145371	−	0.989377i	$-0.453562\pi$
$71$	12349.6			0.290742			0.145371	+	0.989377i	$0.453562\pi$
$72$		0			0
$73$	−21800.2	+	37759.0i	−0.478798	+	0.829303i	−0.999704	−	0.0243110i	$-0.992261\pi$
$73$	−21800.2	+	37759.0i	−0.478798	+	0.829303i	0.520906	+	0.853614i	$0.325594\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−10797.3	−	32051.6i	−0.207534	−	0.616060i
$78$		0			0
$79$	−38374.7	−	66467.0i	−0.691796	−	1.19823i	−0.971249	−	0.238066i	$-0.923487\pi$
$79$	−38374.7	−	66467.0i	−0.691796	−	1.19823i	0.279453	−	0.960159i	$-0.409847\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−21893.6			−0.348836			−0.174418	−	0.984672i	$-0.555804\pi$
$83$	−21893.6			−0.348836			−0.174418	+	0.984672i	$0.555804\pi$
$84$		0			0
$85$	49646.5			0.745318
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−68483.5	−	118617.i	−0.916454	−	1.58735i	−0.804758	−	0.593603i	$-0.797705\pi$
$89$	−68483.5	−	118617.i	−0.916454	−	1.58735i	−0.111696	−	0.993742i	$-0.535628\pi$
$90$		0			0
$91$	65969.7	−	74873.4i	0.835104	−	0.947816i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11986.4	+	20761.0i	−0.136263	+	0.236015i
$96$		0			0
$97$	−93050.1			−1.00412			−0.502062	−	0.864832i	$-0.667425\pi$
$97$	−93050.1			−1.00412			−0.502062	+	0.864832i	$0.667425\pi$
$98$		0			0
$99$		0			0

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	252.6.k.f.109.2		8
3.2	odd	2		84.6.i.c.25.3	✓	8
7.2	even	3	inner	252.6.k.f.37.2		8
12.11	even	2		336.6.q.i.193.3		8
21.2	odd	6		84.6.i.c.37.3	yes	8
21.5	even	6		588.6.i.o.373.2		8
21.11	odd	6		588.6.a.n.1.2		4
21.17	even	6		588.6.a.p.1.3		4
21.20	even	2		588.6.i.o.361.2		8
84.23	even	6		336.6.q.i.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.3	✓	8	3.2	odd	2
84.6.i.c.37.3	yes	8	21.2	odd	6
252.6.k.f.37.2		8	7.2	even	3	inner
252.6.k.f.109.2		8	1.1	even	1	trivial
336.6.q.i.193.3		8	12.11	even	2
336.6.q.i.289.3		8	84.23	even	6
588.6.a.n.1.2		4	21.11	odd	6
588.6.a.p.1.3		4	21.17	even	6
588.6.i.o.361.2		8	21.20	even	2
588.6.i.o.373.2		8	21.5	even	6

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-15.9808 - 27.6796i) q^{5} +(85.7043 - 97.2716i) q^{7} +(130.442 - 225.932i) q^{11} +769.735 q^{13} +(-776.659 + 1345.21i) q^{17} +(-375.024 - 649.561i) q^{19} +(377.427 + 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} -6008.93 q^{29} +(3210.02 - 5559.92i) q^{31} +(-4062.06 - 817.779i) q^{35} +(2387.86 + 4135.90i) q^{37} +5423.27 q^{41} -11896.4 q^{43} +(-8714.00 - 15093.1i) q^{47} +(-2116.54 - 16673.2i) q^{49} +(18825.3 - 32606.4i) q^{53} -8338.26 q^{55} +(11039.0 - 19120.2i) q^{59} +(4086.69 + 7078.36i) q^{61} +(-12301.0 - 21305.9i) q^{65} +(-6500.87 + 11259.8i) q^{67} +12349.6 q^{71} +(-21800.2 + 37759.0i) q^{73} +(-10797.3 - 32051.6i) q^{77} +(-38374.7 - 66467.0i) q^{79} -21893.6 q^{83} +49646.5 q^{85} +(-68483.5 - 118617. i) q^{89} +(65969.7 - 74873.4i) q^{91} +(-11986.4 + 20761.0i) q^{95} -93050.1 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 42 q^{7} + 462 q^{11} - 1204 q^{13} - 228 q^{17} + 358 q^{19} + 2148 q^{23} - 5454 q^{25} + 11064 q^{29} + 830 q^{31} - 7692 q^{35} - 3914 q^{37} + 16632 q^{41} - 29036 q^{43} - 41700 q^{47} + 41876 q^{49}+ \cdots - 433356 q^{97}+O(q^{100}) \) 8 * q - 42 * q^7 + 462 * q^11 - 1204 * q^13 - 228 * q^17 + 358 * q^19 + 2148 * q^23 - 5454 * q^25 + 11064 * q^29 + 830 * q^31 - 7692 * q^35 - 3914 * q^37 + 16632 * q^41 - 29036 * q^43 - 41700 * q^47 + 41876 * q^49 - 22164 * q^53 + 7784 * q^55 - 32886 * q^59 + 83732 * q^61 + 93192 * q^65 - 80034 * q^67 - 89544 * q^71 - 22470 * q^73 - 138732 * q^77 - 75286 * q^79 + 34836 * q^83 + 278504 * q^85 - 28944 * q^89 - 12058 * q^91 + 144120 * q^95 - 433356 * q^97

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