Embedded newform 43.5.h.a.3.9

• Label: 43.5.h.a.3.9
• Level: $43$
• Weight: $5$
• Character: 43.3
• Analytic conductor: $4.445$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.44490841261\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(14\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	43.3
Dual form			43.5.h.a.29.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.597906 - 1.24156i) q^{2} +(8.51855 + 0.638377i) q^{3} +(8.79185 + 11.0246i) q^{4} +(-9.70706 + 31.4695i) q^{5} +(5.88588 - 10.1946i) q^{6} +(-17.8087 + 10.2819i) q^{7} +(40.4402 - 9.23021i) q^{8} +(-7.93707 - 1.19632i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q - 14 q^{2} - 20 q^{3} + 220 q^{4} - 11 q^{5} - 22 q^{6} - 150 q^{7} + 322 q^{8} - 660 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{1}{42}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.597906	−	1.24156i	0.149476	−	0.310391i	−0.812763	−	0.582594i	\(-0.802038\pi\)
							0.962240	+	0.272203i	\(0.0877522\pi\)
\(3\)	8.51855	+	0.638377i	0.946506	+	0.0709308i	0.539031	−	0.842286i	\(-0.318790\pi\)
							0.407475	+	0.913217i	\(0.366409\pi\)
\(5\)	−9.70706	+	31.4695i	−0.388282	+	1.25878i	0.524877	+	0.851178i	\(0.324111\pi\)
							−0.913159	+	0.407603i	\(0.866365\pi\)
\(7\)	−17.8087	+	10.2819i	−0.363443	+	0.209834i	−0.670590	−	0.741828i	\(-0.733959\pi\)
							0.307147	+	0.951662i	\(0.400626\pi\)
\(11\)	73.8724	−	92.6330i	0.610515	−	0.765562i	−0.376460	−	0.926433i	\(-0.622859\pi\)
							0.986975	+	0.160871i	\(0.0514302\pi\)
\(13\)	106.233	−	98.5700i	0.628599	−	0.583254i	−0.300183	−	0.953882i	\(-0.597048\pi\)
							0.928782	+	0.370627i	\(0.120857\pi\)
\(17\)	188.559	−	58.1629i	0.652455	−	0.201256i	0.0491811	−	0.998790i	\(-0.484339\pi\)
							0.603274	+	0.797534i	\(0.293863\pi\)
\(19\)	−94.2528	−	625.326i	−0.261088	−	1.73221i	−0.607673	−	0.794187i	\(-0.707897\pi\)
							0.346585	−	0.938019i	\(-0.387341\pi\)
\(23\)	4.42618	+	11.2777i	0.00836707	+	0.0213189i	0.934996	−	0.354658i	\(-0.115403\pi\)
							−0.926629	+	0.375977i	\(0.877307\pi\)
\(29\)	226.023	−	16.9381i	0.268755	−	0.0201404i	0.0603291	−	0.998179i	\(-0.480785\pi\)
							0.208426	+	0.978038i	\(0.433166\pi\)
\(31\)	−1073.50	+	731.903i	−1.11707	+	0.761606i	−0.973648	−	0.228056i	\(-0.926763\pi\)
							−0.143422	+	0.989662i	\(0.545811\pi\)
\(37\)	802.059	+	463.069i	0.585872	+	0.338254i	0.763464	−	0.645851i	\(-0.223497\pi\)
							−0.177591	+	0.984104i	\(0.556831\pi\)
\(41\)	1125.50	+	542.010i	0.669539	+	0.322433i	0.737590	−	0.675249i	\(-0.235964\pi\)
							−0.0680507	+	0.997682i	\(0.521678\pi\)
\(43\)	−1398.43	−	1209.63i	−0.756315	−	0.654208i
							\(47\)	1278.89	+	1603.68i	0.578945	+	0.725974i	0.981933	−	0.189231i	\(-0.0605996\pi\)
−0.402987	+	0.915206i	\(0.632028\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.5.h.a.3.9	✓	168
43.29	odd	42	inner	43.5.h.a.29.9	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.h.a.3.9	✓	168	1.1	even	1	trivial
43.5.h.a.29.9	yes	168	43.29	odd	42	inner