Embedded newform 43.7.f.a.22.4

• Label: 43.7.f.a.22.4
• Level: $43$
• Weight: $7$
• Character: 43.22
• Analytic conductor: $9.892$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.89232559565\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(21\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			22.4
Character	\(\chi\)	\(=\)	43.22
Dual form			43.7.f.a.2.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.02991 + 10.4447i) q^{2} +(20.6257 + 42.8298i) q^{3} +(-43.8887 - 55.0347i) q^{4} +(127.013 - 28.9898i) q^{5} -551.091 q^{6} +619.099i q^{7} +(72.2435 - 16.4891i) q^{8} +(-954.446 + 1196.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.02991	+	10.4447i	−0.628739	+	1.30559i	0.306601	+	0.951838i	\(0.400808\pi\)
							−0.935340	+	0.353751i	\(0.884906\pi\)
\(3\)	20.6257	+	42.8298i	0.763916	+	1.58629i	0.809354	+	0.587322i	\(0.199818\pi\)
							−0.0454371	+	0.998967i	\(0.514468\pi\)
\(5\)	127.013	−	28.9898i	1.01610	−	0.231919i	0.318132	−	0.948046i	\(-0.396944\pi\)
							0.697969	+	0.716128i	\(0.254087\pi\)
\(7\)			619.099i			1.80495i	0.430739	+	0.902477i	\(0.358253\pi\)
							−0.430739	+	0.902477i	\(0.641747\pi\)
\(11\)	1279.58	−	1604.54i	0.961368	−	1.20552i	−0.0172542	−	0.999851i	\(-0.505492\pi\)
							0.978622	−	0.205666i	\(-0.0659361\pi\)
\(13\)	58.3552	+	255.671i	0.0265613	+	0.116373i	0.986471	−	0.163938i	\(-0.0524197\pi\)
							−0.959909	+	0.280311i	\(0.909563\pi\)
\(17\)	−499.657	+	2189.14i	−0.101701	+	0.445582i	0.898280	+	0.439424i	\(0.144817\pi\)
							−0.999981	+	0.00615793i	\(0.998040\pi\)
\(19\)	9757.92	−	7781.68i	1.42265	−	1.13452i	0.452567	−	0.891731i	\(-0.350508\pi\)
							0.970079	−	0.242791i	\(-0.0780630\pi\)
\(23\)	5083.42	−	6374.40i	0.417803	−	0.523909i	−0.527739	−	0.849406i	\(-0.676960\pi\)
							0.945543	+	0.325497i	\(0.105532\pi\)
\(29\)	8210.48	−	17049.2i	0.336647	−	0.699055i	−0.662083	−	0.749430i	\(-0.730328\pi\)
							0.998730	+	0.0503756i	\(0.0160418\pi\)
\(31\)	−5612.02	−	2702.61i	−0.188380	−	0.0907189i	0.337316	−	0.941391i	\(-0.390481\pi\)
							−0.525696	+	0.850672i	\(0.676195\pi\)
\(37\)		−	7288.20i		−	0.143885i	−0.997409	−	0.0719424i	\(-0.977080\pi\)
							0.997409	−	0.0719424i	\(-0.0229198\pi\)
\(41\)	21045.2	+	10134.9i	0.305353	+	0.147050i	0.580285	−	0.814413i	\(-0.302941\pi\)
							−0.274932	+	0.961464i	\(0.588655\pi\)
\(43\)	−79504.8	+	585.981i	−0.999973	+	0.00737018i
							\(47\)	5976.39	+	7494.15i	0.0575632	+	0.0721820i	0.809780	−	0.586734i	\(-0.199587\pi\)
−0.752216	+	0.658916i	\(0.771015\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.7.f.a.22.4	yes	126
43.2	odd	14	inner	43.7.f.a.2.4	✓	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.7.f.a.2.4	✓	126	43.2	odd	14	inner
43.7.f.a.22.4	yes	126	1.1	even	1	trivial