Embedded newform 43.9.f.a.22.19

• Label: 43.9.f.a.22.19
• 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			22.19
Character	\(\chi\)	\(=\)	43.22
Dual form			43.9.f.a.2.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.66094 - 7.60201i) q^{2} +(-55.7168 - 115.697i) q^{3} +(115.225 + 144.488i) q^{4} +(474.386 - 108.275i) q^{5} -1083.51 q^{6} +3609.61i q^{7} +(3626.10 - 827.633i) q^{8} +(-6190.76 + 7762.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.66094	−	7.60201i	0.228808	−	0.475126i	−0.754681	−	0.656092i	\(-0.772208\pi\)
							0.983489	+	0.180966i	\(0.0579226\pi\)
\(3\)	−55.7168	−	115.697i	−0.687862	−	1.42836i	−0.893196	−	0.449668i	\(-0.851542\pi\)
							0.205334	−	0.978692i	\(-0.434172\pi\)
\(5\)	474.386	−	108.275i	0.759017	−	0.173241i	0.174539	−	0.984650i	\(-0.444156\pi\)
							0.584478	+	0.811410i	\(0.301299\pi\)
\(7\)			3609.61i			1.50338i	0.659518	+	0.751689i	\(0.270761\pi\)
							−0.659518	+	0.751689i	\(0.729239\pi\)
\(11\)	−13594.7	+	17047.2i	−0.928533	+	1.16434i	0.0575923	+	0.998340i	\(0.481658\pi\)
							−0.986125	+	0.166003i	\(0.946914\pi\)
\(13\)	8864.19	+	38836.6i	0.310360	+	1.35978i	0.853919	+	0.520405i	\(0.174219\pi\)
							−0.543559	+	0.839371i	\(0.682924\pi\)
\(17\)	−7750.43	+	33956.9i	−0.0927962	+	0.406567i	−0.999897	−	0.0143465i	\(-0.995433\pi\)
							0.907101	+	0.420913i	\(0.138290\pi\)
\(19\)	179840.	−	143418.i	1.37998	−	1.10050i	0.396787	−	0.917911i	\(-0.370125\pi\)
							0.983190	−	0.182585i	\(-0.0584463\pi\)
\(23\)	52421.7	−	65734.7i	0.187327	−	0.234900i	−0.679296	−	0.733865i	\(-0.737715\pi\)
							0.866623	+	0.498964i	\(0.166286\pi\)
\(29\)	−49700.3	+	103204.i	−0.0702696	+	0.145916i	−0.933145	−	0.359501i	\(-0.882947\pi\)
							0.862875	+	0.505417i	\(0.168661\pi\)
\(31\)	−910457.	−	438453.i	−0.985854	−	0.474762i	−0.129739	−	0.991548i	\(-0.541414\pi\)
							−0.856115	+	0.516786i	\(0.827128\pi\)
\(37\)			394872.i			0.210693i	0.994436	+	0.105346i	\(0.0335951\pi\)
							−0.994436	+	0.105346i	\(0.966405\pi\)
\(41\)	2.30530e6	+	1.11017e6i	0.815816	+	0.392876i	0.794776	−	0.606902i	\(-0.207588\pi\)
							0.0210396	+	0.999779i	\(0.493302\pi\)
\(43\)	−3.20972e6	−	1.17723e6i	−0.938845	−	0.344340i
							\(47\)	2.95171e6	+	3.70133e6i	0.604899	+	0.758519i	0.986133	−	0.165960i	\(-0.0530722\pi\)
−0.381234	+	0.924479i	\(0.624501\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.22.19	yes	174
43.2	odd	14	inner	43.9.f.a.2.19	✓	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.19	✓	174	43.2	odd	14	inner
43.9.f.a.22.19	yes	174	1.1	even	1	trivial