Embedded newform 43.9.f.a.2.3

• Label: 43.9.f.a.2.3
• Level: $43$
• Weight: $9$
• Character: 43.2
• 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			2.3
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.3921 - 25.7324i) q^{2} +(27.0409 - 56.1510i) q^{3} +(-348.980 + 437.608i) q^{4} +(1129.38 + 257.774i) q^{5} -1780.00 q^{6} -11.8724i q^{7} +(8457.04 + 1930.26i) q^{8} +(1668.99 + 2092.85i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{9}{14}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.3921	−	25.7324i	−0.774505	−	1.60828i	−0.793586	−	0.608458i	\(-0.791788\pi\)
							0.0190809	−	0.999818i	\(-0.493926\pi\)
\(3\)	27.0409	−	56.1510i	0.333838	−	0.693223i	−0.664709	−	0.747102i	\(-0.731444\pi\)
							0.998547	+	0.0538797i	\(0.0171588\pi\)
\(5\)	1129.38	+	257.774i	1.80701	+	0.412439i	0.987107	−	0.160061i	\(-0.0511691\pi\)
							0.819905	+	0.572500i	\(0.194026\pi\)
\(7\)		−	11.8724i		−	0.00494477i	−0.999997	−	0.00247239i	\(-0.999213\pi\)
							0.999997	−	0.00247239i	\(-0.000786986\pi\)
\(11\)	−12508.3	−	15684.9i	−0.854331	−	1.07130i	−0.996675	−	0.0814797i	\(-0.974035\pi\)
							0.142344	−	0.989817i	\(-0.454536\pi\)
\(13\)	6786.94	−	29735.5i	0.237630	−	1.04112i	−0.705503	−	0.708707i	\(-0.749279\pi\)
							0.943133	−	0.332417i	\(-0.107864\pi\)
\(17\)	−21094.6	−	92421.4i	−0.252566	−	1.10656i	−0.929005	−	0.370066i	\(-0.879335\pi\)
							0.676439	−	0.736498i	\(-0.263522\pi\)
\(19\)	116284.	+	92733.8i	0.892293	+	0.711580i	0.958155	−	0.286249i	\(-0.0924085\pi\)
							−0.0658627	+	0.997829i	\(0.520980\pi\)
\(23\)	−185694.	−	232853.i	−0.663571	−	0.832092i	0.330156	−	0.943926i	\(-0.392899\pi\)
							−0.993727	+	0.111835i	\(0.964327\pi\)
\(29\)	−275561.	−	572208.i	−0.389606	−	0.809025i	−0.999859	−	0.0168183i	\(-0.994646\pi\)
							0.610253	−	0.792207i	\(-0.291068\pi\)
\(31\)	−757303.	+	364698.i	−0.820017	+	0.394899i	−0.796362	−	0.604821i	\(-0.793245\pi\)
							−0.0236553	+	0.999720i	\(0.507530\pi\)
\(37\)		−	1.84091e6i		−	0.982259i	−0.871087	−	0.491130i	\(-0.836584\pi\)
							0.871087	−	0.491130i	\(-0.163416\pi\)
\(41\)	−517234.	+	249087.i	−0.183042	+	0.0881486i	−0.523161	−	0.852234i	\(-0.675247\pi\)
							0.340118	+	0.940383i	\(0.389533\pi\)
\(43\)	2.80764e6	−	1.95074e6i	0.821234	−	0.570591i
							\(47\)	−1.00831e6	+	1.26438e6i	−0.206634	+	0.259111i	−0.874340	−	0.485314i	\(-0.838705\pi\)
0.667705	+	0.744426i	\(0.267277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.3	✓	174
43.22	odd	14	inner	43.9.f.a.22.3	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.3	✓	174	1.1	even	1	trivial
43.9.f.a.22.3	yes	174	43.22	odd	14	inner