Embedded newform 43.9.f.a.2.20

• Label: 43.9.f.a.2.20
• 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			2.20
Character	\(\chi\)	\(=\)	43.2
Dual form			43.9.f.a.22.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.47023 + 9.28252i) q^{2} +(45.7755 - 95.0539i) q^{3} +(93.4311 - 117.159i) q^{4} +(498.592 + 113.800i) q^{5} +1086.97 q^{6} +1744.42i q^{7} +(4076.58 + 930.453i) q^{8} +(-2849.12 - 3572.69i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 174 q - 7 q^{2} - 7 q^{3} + 4021 q^{4} - 7 q^{5} + 1780 q^{6} - 5383 q^{8} + 56490 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.47023	+	9.28252i	0.279389	+	0.580158i	0.992689	−	0.120702i	\(-0.0385144\pi\)
							−0.713300	+	0.700859i	\(0.752800\pi\)
\(3\)	45.7755	−	95.0539i	0.565130	−	1.17350i	−0.401150	−	0.916012i	\(-0.631389\pi\)
							0.966280	−	0.257492i	\(-0.0828962\pi\)
\(5\)	498.592	+	113.800i	0.797747	+	0.182080i	0.601918	−	0.798558i	\(-0.294403\pi\)
							0.195828	+	0.980638i	\(0.437260\pi\)
\(7\)			1744.42i			0.726538i	0.931684	+	0.363269i	\(0.118339\pi\)
							−0.931684	+	0.363269i	\(0.881661\pi\)
\(11\)	10887.9	+	13653.0i	0.743659	+	0.932518i	0.999414	−	0.0342231i	\(-0.0108957\pi\)
							−0.255756	+	0.966741i	\(0.582324\pi\)
\(13\)	6288.43	−	27551.4i	0.220175	−	0.964651i	−0.737171	−	0.675707i	\(-0.763839\pi\)
							0.957346	−	0.288944i	\(-0.0933043\pi\)
\(17\)	−18570.6	−	81363.1i	−0.222347	−	0.974164i	−0.955706	−	0.294323i	\(-0.904906\pi\)
							0.733360	−	0.679841i	\(-0.237951\pi\)
\(19\)	−30312.4	−	24173.3i	−0.232598	−	0.185491i	0.500254	−	0.865879i	\(-0.333240\pi\)
							−0.732852	+	0.680388i	\(0.761811\pi\)
\(23\)	−28506.6	−	35746.1i	−0.101867	−	0.127737i	0.728285	−	0.685275i	\(-0.240318\pi\)
							−0.830152	+	0.557537i	\(0.811746\pi\)
\(29\)	543762.	+	1.12913e6i	0.768806	+	1.59644i	0.802241	+	0.597001i	\(0.203641\pi\)
							−0.0334349	+	0.999441i	\(0.510645\pi\)
\(31\)	−1.02751e6	+	494824.i	−1.11260	+	0.535802i	−0.897600	−	0.440811i	\(-0.854691\pi\)
							−0.215004	+	0.976613i	\(0.568977\pi\)
\(37\)		−	3.13557e6i		−	1.67305i	−0.547927	−	0.836526i	\(-0.684583\pi\)
							0.547927	−	0.836526i	\(-0.315417\pi\)
\(41\)	−2.18126e6	+	1.05044e6i	−0.771918	+	0.371736i	−0.778015	−	0.628245i	\(-0.783774\pi\)
							0.00609710	+	0.999981i	\(0.498059\pi\)
\(43\)	−3.40038e6	+	354389.i	−0.994613	+	0.103659i
							\(47\)	207835.	−	260617.i	0.0425919	−	0.0534086i	−0.760079	−	0.649830i	\(-0.774840\pi\)
0.802671	+	0.596422i	\(0.203411\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.9.f.a.2.20	✓	174
43.22	odd	14	inner	43.9.f.a.22.20	yes	174

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.9.f.a.2.20	✓	174	1.1	even	1	trivial
43.9.f.a.22.20	yes	174	43.22	odd	14	inner