Embedded newform 43.9.d.a.7.11

• Label: 43.9.d.a.7.11
• Level: $43$
• Weight: $9$
• Character: 43.7
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			7.11
Character	\(\chi\)	\(=\)	43.7
Dual form			43.9.d.a.37.18

$q$-expansion

\(f(q)\)	\(=\)	\(q-8.98674i q^{2} +(117.071 - 67.5912i) q^{3} +175.239 q^{4} +(-869.912 + 502.244i) q^{5} +(-607.424 - 1052.09i) q^{6} +(3700.56 + 2136.52i) q^{7} -3875.43i q^{8} +(5856.64 - 10144.0i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 84 q^{3} - 6474 q^{4} - 3 q^{5} + 1791 q^{6} - 5031 q^{7} + 57858 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	8.98674i		−	0.561671i	−0.959756	−	0.280835i	\(-0.909388\pi\)
							0.959756	−	0.280835i	\(-0.0906115\pi\)
\(3\)	117.071	−	67.5912i	1.44533	−	0.834459i	0.447128	−	0.894470i	\(-0.352447\pi\)
							0.998198	+	0.0600111i	\(0.0191136\pi\)
\(5\)	−869.912	+	502.244i	−1.39186	+	0.803591i	−0.993521	−	0.113648i	\(-0.963747\pi\)
							−0.398339	+	0.917238i	\(0.630413\pi\)
\(7\)	3700.56	+	2136.52i	1.54126	+	0.889845i	0.998760	+	0.0497876i	\(0.0158544\pi\)
							0.542497	+	0.840058i	\(0.317479\pi\)
\(11\)	11087.8			0.757310			0.378655	−	0.925538i	\(-0.376387\pi\)
							0.378655	+	0.925538i	\(0.376387\pi\)
\(13\)	15027.3	−	26028.1i	0.526149	−	0.911316i	−0.473387	−	0.880854i	\(-0.656969\pi\)
							0.999536	−	0.0304618i	\(-0.00969778\pi\)
\(17\)	−15009.9	+	25997.9i	−0.179714	+	0.311274i	−0.941783	−	0.336222i	\(-0.890851\pi\)
							0.762068	+	0.647496i	\(0.224184\pi\)
\(19\)	−63263.8	+	36525.4i	−0.485446	+	0.280272i	−0.722683	−	0.691179i	\(-0.757091\pi\)
							0.237237	+	0.971452i	\(0.423758\pi\)
\(23\)	−134932.	−	233710.i	−0.482175	−	0.835152i	0.517616	−	0.855613i	\(-0.326820\pi\)
							−0.999791	+	0.0204616i	\(0.993486\pi\)
\(29\)	−64879.2	−	37458.0i	−0.0917304	−	0.0529606i	0.453433	−	0.891290i	\(-0.350199\pi\)
							−0.545164	+	0.838330i	\(0.683532\pi\)
\(31\)	375368.	+	650156.i	0.406453	+	0.703997i	0.994489	−	0.104838i	\(-0.0334323\pi\)
							−0.588037	+	0.808834i	\(0.700099\pi\)
\(37\)	−3.05446e6	+	1.76349e6i	−1.62977	+	0.940951i	−0.645616	+	0.763662i	\(0.723399\pi\)
							−0.984159	+	0.177289i	\(0.943267\pi\)
\(41\)	1.05584e6			0.373649			0.186825	−	0.982393i	\(-0.440180\pi\)
							0.186825	+	0.982393i	\(0.440180\pi\)
\(43\)	3.18491e6	−	1.24279e6i	0.931587	−	0.363517i
							\(47\)	−5.14379e6			−1.05412			−0.527062	−	0.849827i	\(-0.676706\pi\)
−0.527062	+	0.849827i	\(0.676706\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.d.a.7.11	✓	56
43.37	odd	6	inner	43.9.d.a.37.18	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.d.a.7.11	✓	56	1.1	even	1	trivial
43.9.d.a.37.18	yes	56	43.37	odd	6	inner