Embedded newform 731.2.k.b.35.16

• Label: 731.2.k.b.35.16
• Level: $731$
• Weight: $2$
• Character: 731.35
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $180$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.k (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(180\)
Relative dimension:	\(30\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

Embedding invariants

Embedding label			35.16
Character	\(\chi\)	\(=\)	731.35
Dual form			731.2.k.b.188.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0248321 - 0.108796i) q^{2} +(0.413289 + 1.81074i) q^{3} +(1.79072 + 0.862364i) q^{4} +(1.10115 - 1.38080i) q^{5} +0.207265 q^{6} -1.59919 q^{7} +(0.277445 - 0.347905i) q^{8} +(-0.405063 + 0.195068i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 180 q + 2 q^{3} - 34 q^{4} + 2 q^{5} + 6 q^{6} - 54 q^{7} + 14 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0248321	−	0.108796i	0.0175589	−	0.0769307i	−0.965389	−	0.260813i	\(-0.916009\pi\)
							0.982948	+	0.183882i	\(0.0588665\pi\)
\(3\)	0.413289	+	1.81074i	0.238613	+	1.04543i	0.942260	+	0.334882i	\(0.108696\pi\)
							−0.703647	+	0.710549i	\(0.748447\pi\)
\(5\)	1.10115	−	1.38080i	0.492450	−	0.617513i	−0.472058	−	0.881568i	\(-0.656489\pi\)
							0.964508	+	0.264055i	\(0.0850600\pi\)
\(7\)	−1.59919			−0.604436			−0.302218	−	0.953239i	\(-0.597727\pi\)
							−0.302218	+	0.953239i	\(0.597727\pi\)
\(11\)	2.94474	−	1.41811i	0.887872	−	0.427577i	0.0663785	−	0.997795i	\(-0.478856\pi\)
							0.821494	+	0.570218i	\(0.193141\pi\)
\(13\)	−1.13368	+	1.42159i	−0.314426	+	0.394277i	−0.913782	−	0.406205i	\(-0.866852\pi\)
							0.599356	+	0.800482i	\(0.295423\pi\)
\(17\)	−0.623490	−	0.781831i	−0.151218	−	0.189622i
							\(19\)	4.07142	+	1.96069i	0.934048	+	0.449814i	0.838066	−	0.545569i	\(-0.183687\pi\)
0.0959821	+	0.995383i	\(0.469401\pi\)
\(23\)	−1.32006	+	0.635708i	−0.275252	+	0.132554i	−0.566418	−	0.824118i	\(-0.691671\pi\)
							0.291166	+	0.956673i	\(0.405957\pi\)
\(29\)	1.06653	−	4.67279i	0.198050	−	0.867716i	−0.774045	−	0.633131i	\(-0.781770\pi\)
							0.972096	−	0.234585i	\(-0.0753732\pi\)
\(31\)	0.804361	−	3.52414i	0.144468	−	0.632954i	−0.849898	−	0.526947i	\(-0.823336\pi\)
							0.994365	−	0.106006i	\(-0.0338064\pi\)
\(37\)	−7.41506			−1.21903			−0.609514	−	0.792775i	\(-0.708635\pi\)
							−0.609514	+	0.792775i	\(0.708635\pi\)
\(41\)	−0.300416	+	1.31621i	−0.0469171	+	0.205557i	−0.992954	−	0.118502i	\(-0.962191\pi\)
							0.946037	+	0.324060i	\(0.105048\pi\)
\(43\)	−4.06821	+	5.14292i	−0.620397	+	0.784288i
							\(47\)	−5.20258	−	2.50543i	−0.758874	−	0.365455i	0.0140928	−	0.999901i	\(-0.495514\pi\)
−0.772967	+	0.634446i	\(0.781228\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.k.b.35.16	✓	180
43.16	even	7	inner	731.2.k.b.188.16	yes	180

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.k.b.35.16	✓	180	1.1	even	1	trivial
731.2.k.b.188.16	yes	180	43.16	even	7	inner