Embedded newform 731.2.z.a.611.13

• Label: 731.2.z.a.611.13
• Level: $731$
• Weight: $2$
• Character: 731.611
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

Embedding invariants

Embedding label			611.13
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16355 - 1.45905i) q^{2} +(-0.00644316 + 0.0427476i) q^{3} +(-0.329925 + 1.44549i) q^{4} +(0.740612 - 1.08628i) q^{5} +(0.0698677 - 0.0403381i) q^{6} +(2.40880 + 1.39072i) q^{7} +(-0.869832 + 0.418889i) q^{8} +(2.86493 + 0.883715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16355	−	1.45905i	−0.822755	−	1.03170i	−0.998879	−	0.0473338i	\(-0.984928\pi\)
							0.176124	−	0.984368i	\(-0.443644\pi\)
\(3\)	−0.00644316	+	0.0427476i	−0.00371996	+	0.0246803i	−0.990614	−	0.136690i	\(-0.956353\pi\)
							0.986894	+	0.161371i	\(0.0515915\pi\)
\(5\)	0.740612	−	1.08628i	0.331212	−	0.485798i	−0.624122	−	0.781327i	\(-0.714543\pi\)
							0.955334	+	0.295528i	\(0.0954957\pi\)
\(7\)	2.40880	+	1.39072i	0.910442	+	0.525644i	0.880573	−	0.473910i	\(-0.157158\pi\)
							0.0298685	+	0.999554i	\(0.490491\pi\)
\(11\)	−3.09236	+	0.705812i	−0.932383	+	0.212810i	−0.661623	−	0.749837i	\(-0.730132\pi\)
							−0.270760	+	0.962647i	\(0.587275\pi\)
\(13\)	−0.0558263	+	0.744950i	−0.0154834	+	0.206612i	0.984097	+	0.177634i	\(0.0568443\pi\)
							−0.999580	+	0.0289780i	\(0.990775\pi\)
\(17\)	2.09959	−	3.54848i	0.509226	−	0.860633i
							\(19\)	1.37335	−	0.423621i	0.315067	−	0.0971854i	−0.133189	−	0.991091i	\(-0.542522\pi\)
0.448256	+	0.893905i	\(0.352045\pi\)
\(23\)	5.99104	+	6.45680i	1.24922	+	1.34634i	0.914798	+	0.403913i	\(0.132350\pi\)
							0.334420	+	0.942424i	\(0.391460\pi\)
\(29\)	−0.105590	−	0.700546i	−0.0196076	−	0.130088i	0.976885	−	0.213764i	\(-0.0685722\pi\)
							−0.996493	+	0.0836754i	\(0.973334\pi\)
\(31\)	5.44517	+	2.13707i	0.977982	+	0.383829i	0.799865	−	0.600180i	\(-0.204904\pi\)
							0.178117	+	0.984009i	\(0.443000\pi\)
\(37\)	−6.51451	+	3.76116i	−1.07098	+	0.618330i	−0.928449	−	0.371460i	\(-0.878857\pi\)
							−0.142531	+	0.989790i	\(0.545524\pi\)
\(41\)	−3.24224	+	2.58560i	−0.506353	+	0.403803i	−0.843071	−	0.537803i	\(-0.819255\pi\)
							0.336718	+	0.941606i	\(0.390683\pi\)
\(43\)	5.04253	−	4.19201i	0.768978	−	0.639275i
							\(47\)	1.72426	−	7.55449i	0.251510	−	1.10194i	−0.678558	−	0.734547i	\(-0.737395\pi\)
0.930068	−	0.367389i	\(-0.119748\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.611.13	yes	768
17.16	even	2	inner	731.2.z.a.611.14	yes	768
43.24	even	21	inner	731.2.z.a.67.14	yes	768
731.67	even	42	inner	731.2.z.a.67.13	✓	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.13	✓	768	731.67	even	42	inner
731.2.z.a.67.14	yes	768	43.24	even	21	inner
731.2.z.a.611.13	yes	768	1.1	even	1	trivial
731.2.z.a.611.14	yes	768	17.16	even	2	inner