Embedded newform 43.7.d.a.37.5

• Label: 43.7.d.a.37.5
• Level: $43$
• Weight: $7$
• Character: 43.37
• 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			37.5
Character	\(\chi\)	\(=\)	43.37
Dual form			43.7.d.a.7.17

$q$-expansion

\(f(q)\)	\(=\)	\(q-9.90547i q^{2} +(12.0709 + 6.96915i) q^{3} -34.1183 q^{4} +(-41.4735 - 23.9447i) q^{5} +(69.0327 - 119.568i) q^{6} +(-415.762 + 240.040i) q^{7} -295.992i q^{8} +(-267.362 - 463.084i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 3 q^{3} - 1194 q^{4} - 3 q^{5} + 111 q^{6} - 147 q^{7} + 3852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	9.90547i		−	1.23818i	−0.785319	−	0.619092i	\(-0.787501\pi\)
							0.785319	−	0.619092i	\(-0.212499\pi\)
\(3\)	12.0709	+	6.96915i	0.447071	+	0.258117i	0.706593	−	0.707621i	\(-0.250231\pi\)
							−0.259521	+	0.965737i	\(0.583565\pi\)
\(5\)	−41.4735	−	23.9447i	−0.331788	−	0.191558i	0.324847	−	0.945767i	\(-0.394687\pi\)
							−0.656635	+	0.754209i	\(0.728021\pi\)
\(7\)	−415.762	+	240.040i	−1.21213	+	0.699826i	−0.963224	−	0.268701i	\(-0.913406\pi\)
							−0.248910	+	0.968527i	\(0.580072\pi\)
\(11\)	−1050.18			−0.789015			−0.394508	−	0.918893i	\(-0.629085\pi\)
							−0.394508	+	0.918893i	\(0.629085\pi\)
\(13\)	−1567.47	−	2714.94i	−0.713459	−	1.23575i	−0.963551	−	0.267525i	\(-0.913794\pi\)
							0.250092	−	0.968222i	\(-0.419539\pi\)
\(17\)	1146.21	+	1985.29i	0.233301	+	0.404090i	0.958778	−	0.284157i	\(-0.0917138\pi\)
							−0.725476	+	0.688247i	\(0.758380\pi\)
\(19\)	10015.3	+	5782.31i	1.46016	+	0.843025i	0.999018	−	0.0442998i	\(-0.0141057\pi\)
							0.461144	+	0.887325i	\(0.347439\pi\)
\(23\)	−8694.71	+	15059.7i	−0.714614	+	1.23775i	0.248494	+	0.968634i	\(0.420065\pi\)
							−0.963108	+	0.269115i	\(0.913269\pi\)
\(29\)	23889.4	−	13792.6i	0.979517	−	0.565524i	0.0773924	−	0.997001i	\(-0.475341\pi\)
							0.902124	+	0.431477i	\(0.142007\pi\)
\(31\)	23677.4	−	41010.4i	0.794783	−	1.37660i	−0.128194	−	0.991749i	\(-0.540918\pi\)
							0.922977	−	0.384855i	\(-0.125749\pi\)
\(37\)	−62039.1	−	35818.3i	−1.22479	−	0.707131i	−0.258852	−	0.965917i	\(-0.583344\pi\)
							−0.965935	+	0.258786i	\(0.916678\pi\)
\(41\)	−23687.0			−0.343683			−0.171842	−	0.985125i	\(-0.554972\pi\)
							−0.171842	+	0.985125i	\(0.554972\pi\)
\(43\)	−32889.9	+	72385.2i	−0.413673	+	0.910426i
							\(47\)	154554.			1.48863			0.744313	−	0.667831i	\(-0.232777\pi\)
0.744313	+	0.667831i	\(0.232777\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.d.a.37.5	yes	42
43.7	odd	6	inner	43.7.d.a.7.17	✓	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.d.a.7.17	✓	42	43.7	odd	6	inner
43.7.d.a.37.5	yes	42	1.1	even	1	trivial