Embedded newform 43.9.f.a.22.7

• Label: 43.9.f.a.22.7
• Level: $43$
• Weight: $9$
• Character: 43.22
• Analytic conductor: $17.517$
• Analytic rank: $0$
• Dimension: $174$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(17.5172802326\)
Analytic rank:	\(0\)
Dimension:	\(174\)
Relative dimension:	\(29\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.7
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.83493 + 16.2694i) q^{2} +(-45.2846 - 94.0344i) q^{3} +(-43.6938 - 54.7902i) q^{4} +(843.199 - 192.455i) q^{5} +1884.69 q^{6} -324.806i q^{7} +(-3273.12 + 747.068i) q^{8} +(-2701.06 + 3387.03i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{5}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.83493	+	16.2694i	−0.489683	+	1.01684i	0.498970	+	0.866619i	\(0.333712\pi\)
							−0.988653	+	0.150218i	\(0.952002\pi\)
\(3\)	−45.2846	−	94.0344i	−0.559069	−	1.16092i	−0.968600	−	0.248623i	\(-0.920022\pi\)
							0.409531	−	0.912296i	\(-0.365692\pi\)
\(5\)	843.199	−	192.455i	1.34912	−	0.307928i	0.513902	−	0.857849i	\(-0.328199\pi\)
							0.835216	+	0.549921i	\(0.185342\pi\)
\(7\)		−	324.806i		−	0.135280i	−0.997710	−	0.0676398i	\(-0.978453\pi\)
							0.997710	−	0.0676398i	\(-0.0215469\pi\)
\(11\)	−2507.99	+	3144.92i	−0.171299	+	0.214802i	−0.860069	−	0.510178i	\(-0.829580\pi\)
							0.688770	+	0.724980i	\(0.258151\pi\)
\(13\)	−1699.61	−	7446.49i	−0.0595082	−	0.260722i	0.936419	−	0.350884i	\(-0.114119\pi\)
							−0.995927	+	0.0901615i	\(0.971262\pi\)
\(17\)	4257.88	−	18655.0i	0.0509798	−	0.223357i	−0.943020	−	0.332735i	\(-0.892029\pi\)
							0.994000	+	0.109378i	\(0.0348858\pi\)
\(19\)	−17362.7	+	13846.3i	−0.133231	+	0.106248i	−0.687838	−	0.725864i	\(-0.741440\pi\)
							0.554607	+	0.832112i	\(0.312869\pi\)
\(23\)	161849.	−	202953.i	0.578361	−	0.725242i	−0.403471	−	0.914992i	\(-0.632196\pi\)
							0.981832	+	0.189750i	\(0.0607678\pi\)
\(29\)	522548.	−	1.08508e6i	0.738812	−	1.53416i	−0.103175	−	0.994663i	\(-0.532900\pi\)
							0.841988	−	0.539497i	\(-0.181385\pi\)
\(31\)	−252939.	−	121809.i	−0.273885	−	0.131896i	0.291900	−	0.956449i	\(-0.405713\pi\)
							−0.565785	+	0.824553i	\(0.691427\pi\)
\(37\)		−	1.29460e6i		−	0.690761i	−0.938463	−	0.345380i	\(-0.887750\pi\)
							0.938463	−	0.345380i	\(-0.112250\pi\)
\(41\)	−3.68063e6	−	1.77250e6i	−1.30253	−	0.627264i	−0.351446	−	0.936208i	\(-0.614310\pi\)
							−0.951080	+	0.308944i	\(0.900024\pi\)
\(43\)	294832.	−	3.40606e6i	0.0862384	−	0.996275i
							\(47\)	−3.60348e6	−	4.51862e6i	−0.738467	−	0.926008i	0.260757	−	0.965404i	\(-0.416028\pi\)
−0.999224	+	0.0393966i	\(0.987456\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.7	yes	174
43.2	odd	14	inner	43.9.f.a.2.7	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.7	✓	174	43.2	odd	14	inner
43.9.f.a.22.7	yes	174	1.1	even	1	trivial