Embedded newform 43.7.h.a.19.8

• Label: 43.7.h.a.19.8
• Level: $43$
• Weight: $7$
• Character: 43.19
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $252$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(252\)
Relative dimension:	\(21\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			19.8
Character	\(\chi\)	\(=\)	43.19
Dual form			43.7.h.a.34.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.38443 - 2.69899i) q^{2} +(4.57117 + 30.3277i) q^{3} +(-10.0715 - 44.1263i) q^{4} +(1.30105 + 1.90829i) q^{5} +(66.3835 - 114.980i) q^{6} +(-187.206 + 108.083i) q^{7} +(-205.216 + 426.135i) q^{8} +(-202.263 + 62.3898i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/43\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{19}{42}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.38443	−	2.69899i	−0.423054	−	0.337374i	0.388710	−	0.921360i	\(-0.372921\pi\)
							−0.811764	+	0.583986i	\(0.801492\pi\)
\(3\)	4.57117	+	30.3277i	0.169303	+	1.12325i	0.898509	+	0.438955i	\(0.144651\pi\)
							−0.729206	+	0.684294i	\(0.760111\pi\)
\(5\)	1.30105	+	1.90829i	0.0104084	+	0.0152663i	0.831408	−	0.555663i	\(-0.187535\pi\)
							−0.820999	+	0.570929i	\(0.806583\pi\)
\(7\)	−187.206	+	108.083i	−0.545790	+	0.315112i	−0.747422	−	0.664349i	\(-0.768709\pi\)
							0.201632	+	0.979461i	\(0.435375\pi\)
\(11\)	409.987	−	1796.27i	0.308029	−	1.34956i	−0.549657	−	0.835391i	\(-0.685241\pi\)
							0.857686	−	0.514174i	\(-0.171901\pi\)
\(13\)	−136.180	−	1817.20i	−0.0619846	−	0.827127i	−0.938396	−	0.345563i	\(-0.887688\pi\)
							0.876411	−	0.481564i	\(-0.159931\pi\)
\(17\)	−6610.56	−	4507.00i	−1.34552	−	0.917363i	−0.345744	−	0.938329i	\(-0.612373\pi\)
							−0.999780	+	0.0209664i	\(0.993326\pi\)
\(19\)	1503.43	−	4874.00i	0.219191	−	0.710599i	−0.777468	−	0.628923i	\(-0.783496\pi\)
							0.996659	−	0.0816769i	\(-0.0260275\pi\)
\(23\)	−5848.05	−	5426.20i	−0.480648	−	0.445977i	0.402257	−	0.915527i	\(-0.368226\pi\)
							−0.882906	+	0.469550i	\(0.844416\pi\)
\(29\)	−1467.17	+	9734.05i	−0.0601571	+	0.399116i	0.938406	+	0.345534i	\(0.112302\pi\)
							−0.998563	+	0.0535826i	\(0.982936\pi\)
\(31\)	−8522.42	−	21714.8i	−0.286074	−	0.728904i	−0.999600	−	0.0282755i	\(-0.990998\pi\)
							0.713527	−	0.700628i	\(-0.247097\pi\)
\(37\)	18502.8	+	10682.6i	0.365286	+	0.210898i	0.671397	−	0.741098i	\(-0.265695\pi\)
							−0.306111	+	0.951996i	\(0.599028\pi\)
\(41\)	−30301.8	+	37997.3i	−0.439660	+	0.551316i	−0.951454	−	0.307792i	\(-0.900410\pi\)
							0.511794	+	0.859108i	\(0.328981\pi\)
\(43\)	−79370.5	−	4657.00i	−0.998283	−	0.0585735i
							\(47\)	28951.2	+	126843.i	0.278851	+	1.22173i	0.899248	+	0.437440i	\(0.144115\pi\)
−0.620396	+	0.784288i	\(0.713028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.19.8	✓	252
43.34	odd	42	inner	43.7.h.a.34.8	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.19.8	✓	252	1.1	even	1	trivial
43.7.h.a.34.8	yes	252	43.34	odd	42	inner