Embedded newform 731.2.bh.a.13.4

• Label: 731.2.bh.a.13.4
• Level: $731$
• Weight: $2$
• Character: 731.13
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bh (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

Embedding invariants

Embedding label			13.4
Character	\(\chi\)	\(=\)	731.13
Dual form			731.2.bh.a.225.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.49240 - 0.568874i) q^{2} +(0.335866 + 0.0125672i) q^{3} +(4.08650 + 1.96796i) q^{4} +(-1.01898 - 1.38067i) q^{5} +(-0.829963 - 0.222388i) q^{6} +(1.02117 + 3.81105i) q^{7} +(-5.06819 - 4.04174i) q^{8} +(-2.87896 - 0.215748i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{16}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.49240	−	0.568874i	−1.76239	−	0.402255i	−0.785995	−	0.618233i	\(-0.787849\pi\)
							−0.976398	+	0.215978i	\(0.930706\pi\)
\(3\)	0.335866	+	0.0125672i	0.193912	+	0.00725568i	0.134171	−	0.990958i	\(-0.457163\pi\)
							0.0597413	+	0.998214i	\(0.480972\pi\)
\(5\)	−1.01898	−	1.38067i	−0.455703	−	0.617456i	0.515159	−	0.857095i	\(-0.327733\pi\)
							−0.970862	+	0.239639i	\(0.922971\pi\)
\(7\)	1.02117	+	3.81105i	0.385965	+	1.44044i	0.836640	+	0.547753i	\(0.184517\pi\)
							−0.450675	+	0.892688i	\(0.648817\pi\)
\(11\)	1.70364	−	4.86872i	0.513667	−	1.46797i	−0.336613	−	0.941643i	\(-0.609281\pi\)
							0.850279	−	0.526332i	\(-0.176433\pi\)
\(13\)	1.39906	+	3.56475i	0.388030	+	0.988684i	0.982781	+	0.184774i	\(0.0591553\pi\)
							−0.594751	+	0.803910i	\(0.702749\pi\)
\(17\)	−1.09658	+	3.97461i	−0.265960	+	0.963984i
							\(19\)	−0.579657	+	0.0434393i	−0.132983	+	0.00996566i	−0.141055	−	0.990002i	\(-0.545049\pi\)
0.00807253	+	0.999967i	\(0.497430\pi\)
\(23\)	−6.07814	+	1.15005i	−1.26738	+	0.239801i	−0.775771	−	0.631014i	\(-0.782639\pi\)
							−0.491609	+	0.870816i	\(0.663591\pi\)
\(29\)	0.246765	+	6.59493i	0.0458231	+	1.22465i	0.812634	+	0.582775i	\(0.198033\pi\)
							−0.766811	+	0.641873i	\(0.778157\pi\)
\(31\)	−3.74944	−	1.98163i	−0.673419	−	0.355912i	0.0953458	−	0.995444i	\(-0.469604\pi\)
							−0.768764	+	0.639532i	\(0.779128\pi\)
\(37\)	3.08192	−	11.5019i	0.506665	−	1.89090i	0.0555027	−	0.998459i	\(-0.482324\pi\)
							0.451163	−	0.892442i	\(-0.351009\pi\)
\(41\)	−1.19450	−	1.90103i	−0.186549	−	0.296891i	0.740336	−	0.672237i	\(-0.234667\pi\)
							−0.926885	+	0.375347i	\(0.877524\pi\)
\(43\)	0.667076	+	6.52342i	0.101728	+	0.994812i
							\(47\)	−3.06199	−	1.47458i	−0.446637	−	0.215089i	0.197026	−	0.980398i	\(-0.436872\pi\)
−0.643663	+	0.765309i	\(0.722586\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bh.a.13.4	✓	1536
17.4	even	4	inner	731.2.bh.a.701.61	yes	1536
43.10	even	21	inner	731.2.bh.a.268.61	yes	1536
731.225	even	84	inner	731.2.bh.a.225.4	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bh.a.13.4	✓	1536	1.1	even	1	trivial
731.2.bh.a.225.4	yes	1536	731.225	even	84	inner
731.2.bh.a.268.61	yes	1536	43.10	even	21	inner
731.2.bh.a.701.61	yes	1536	17.4	even	4	inner