Embedded newform 43.7.d.a.7.2

• Label: 43.7.d.a.7.2
• Level: $43$
• Weight: $7$
• Character: 43.7
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.2
Character	\(\chi\)	\(=\)	43.7
Dual form			43.7.d.a.37.20

$q$-expansion

\(f(q)\)	\(=\)	\(q-13.4327i q^{2} +(-34.2137 + 19.7533i) q^{3} -116.436 q^{4} +(-28.7813 + 16.6169i) q^{5} +(265.339 + 459.580i) q^{6} +(-186.356 - 107.593i) q^{7} +704.357i q^{8} +(415.884 - 720.332i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	13.4327i		−	1.67908i	−0.543297	−	0.839541i	\(-0.682824\pi\)
							0.543297	−	0.839541i	\(-0.317176\pi\)
\(3\)	−34.2137	+	19.7533i	−1.26717	+	0.731603i	−0.974452	−	0.224595i	\(-0.927894\pi\)
							−0.292721	+	0.956198i	\(0.594561\pi\)
\(5\)	−28.7813	+	16.6169i	−0.230251	+	0.132935i	−0.610688	−	0.791872i	\(-0.709107\pi\)
							0.380437	+	0.924807i	\(0.375774\pi\)
\(7\)	−186.356	−	107.593i	−0.543311	−	0.313681i	0.203109	−	0.979156i	\(-0.434896\pi\)
							−0.746420	+	0.665475i	\(0.768229\pi\)
\(11\)	2280.97			1.71372			0.856861	−	0.515547i	\(-0.172411\pi\)
							0.856861	+	0.515547i	\(0.172411\pi\)
\(13\)	−344.553	+	596.783i	−0.156829	+	0.271635i	−0.933723	−	0.357995i	\(-0.883460\pi\)
							0.776895	+	0.629631i	\(0.216794\pi\)
\(17\)	−510.165	+	883.632i	−0.103840	+	0.179856i	−0.913264	−	0.407369i	\(-0.866446\pi\)
							0.809424	+	0.587225i	\(0.199780\pi\)
\(19\)	461.539	−	266.470i	0.0672896	−	0.0388497i	−0.465978	−	0.884796i	\(-0.654297\pi\)
							0.533267	+	0.845947i	\(0.320964\pi\)
\(23\)	4197.86	+	7270.91i	0.345020	+	0.597592i	0.985357	−	0.170502i	\(-0.0545388\pi\)
							−0.640337	+	0.768094i	\(0.721205\pi\)
\(29\)	32834.8	+	18957.2i	1.34630	+	0.777284i	0.987723	−	0.156217i	\(-0.0499300\pi\)
							0.358573	+	0.933502i	\(0.383263\pi\)
\(31\)	12094.9	+	20949.0i	0.405992	+	0.703199i	0.994436	−	0.105339i	\(-0.0335927\pi\)
							−0.588444	+	0.808538i	\(0.700259\pi\)
\(37\)	72457.4	−	41833.3i	1.43047	−	0.825880i	0.433309	−	0.901245i	\(-0.357346\pi\)
							0.997156	+	0.0753657i	\(0.0240124\pi\)
\(41\)	60086.6			0.871818			0.435909	−	0.899991i	\(-0.356427\pi\)
							0.435909	+	0.899991i	\(0.356427\pi\)
\(43\)	−41240.2	+	67975.0i	−0.518699	+	0.854957i
							\(47\)	−184275.			−1.77490			−0.887449	−	0.460906i	\(-0.847524\pi\)
−0.887449	+	0.460906i	\(0.847524\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.7.2	✓	42
43.37	odd	6	inner	43.7.d.a.37.20	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.2	✓	42	1.1	even	1	trivial
43.7.d.a.37.20	yes	42	43.37	odd	6	inner