Embedded newform 43.7.h.a.3.6

• Label: 43.7.h.a.3.6
• Level: $43$
• Weight: $7$
• Character: 43.3
• 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			3.6
Character	\(\chi\)	\(=\)	43.3
Dual form			43.7.h.a.29.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.91348 + 6.04990i) q^{2} +(50.5591 + 3.78888i) q^{3} +(11.7904 + 14.7847i) q^{4} +(-30.3679 + 98.4503i) q^{5} +(-170.225 + 294.839i) q^{6} +(-190.382 + 109.917i) q^{7} +(-542.775 + 123.885i) q^{8} +(1821.01 + 274.473i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 252 q - 14 q^{2} - 11 q^{3} + 1180 q^{4} - 11 q^{5} - 118 q^{6} + 126 q^{7} - 2254 q^{8} - 8178 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.91348	+	6.04990i	−0.364185	+	0.756238i	−0.999877	−	0.0157051i	\(-0.995001\pi\)
							0.635692	+	0.771943i	\(0.280715\pi\)
\(3\)	50.5591	+	3.78888i	1.87256	+	0.140329i	0.962000	−	0.273050i	\(-0.0880325\pi\)
							0.910559	+	0.413379i	\(0.135652\pi\)
\(5\)	−30.3679	+	98.4503i	−0.242943	+	0.787603i	0.749183	+	0.662363i	\(0.230446\pi\)
							−0.992126	+	0.125240i	\(0.960030\pi\)
\(7\)	−190.382	+	109.917i	−0.555051	+	0.320459i	−0.751157	−	0.660124i	\(-0.770504\pi\)
							0.196106	+	0.980583i	\(0.437170\pi\)
\(11\)	1130.33	−	1417.39i	0.849235	−	1.06491i	−0.147880	−	0.989005i	\(-0.547245\pi\)
							0.997115	−	0.0759021i	\(-0.0241836\pi\)
\(13\)	−1204.27	+	1117.40i	−0.548143	+	0.508602i	−0.904917	−	0.425588i	\(-0.860067\pi\)
							0.356774	+	0.934191i	\(0.383877\pi\)
\(17\)	−212.548	+	65.5624i	−0.0432624	+	0.0133447i	−0.316311	−	0.948656i	\(-0.602444\pi\)
							0.273048	+	0.962000i	\(0.411968\pi\)
\(19\)	−1013.49	−	6724.09i	−0.147761	−	0.980331i	−0.932673	−	0.360723i	\(-0.882530\pi\)
							0.784912	−	0.619607i	\(-0.212708\pi\)
\(23\)	4626.66	+	11788.5i	0.380263	+	0.968894i	0.985013	+	0.172479i	\(0.0551777\pi\)
							−0.604750	+	0.796415i	\(0.706727\pi\)
\(29\)	−26241.4	+	1966.52i	−1.07595	+	0.0806314i	−0.600898	−	0.799326i	\(-0.705190\pi\)
							−0.475053	+	0.879957i	\(0.657571\pi\)
\(31\)	34265.6	−	23361.9i	1.15020	−	0.784193i	0.170637	−	0.985334i	\(-0.445417\pi\)
							0.979562	+	0.201141i	\(0.0644651\pi\)
\(37\)	37412.2	+	21599.9i	0.738597	+	0.426429i	0.821559	−	0.570123i	\(-0.193105\pi\)
							−0.0829617	+	0.996553i	\(0.526438\pi\)
\(41\)	−17925.4	−	8632.41i	−0.260086	−	0.125251i	0.299297	−	0.954160i	\(-0.403248\pi\)
							−0.559383	+	0.828909i	\(0.688962\pi\)
\(43\)	59781.5	−	52416.9i	0.751903	−	0.659274i
							\(47\)	−44973.8	−	56395.4i	−0.433178	−	0.543188i	0.516553	−	0.856255i	\(-0.327215\pi\)
−0.949731	+	0.313067i	\(0.898643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.h.a.3.6	✓	252
43.29	odd	42	inner	43.7.h.a.29.6	yes	252

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.h.a.3.6	✓	252	1.1	even	1	trivial
43.7.h.a.29.6	yes	252	43.29	odd	42	inner