Embedded newform 731.2.z.a.67.13

• Label: 731.2.z.a.67.13
• Level: $731$
• Weight: $2$
• Character: 731.67
• 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			67.13
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.13

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{20}{21}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.13	✓	768	1.1	even	1	trivial
731.2.z.a.67.14	yes	768	17.16	even	2	inner
731.2.z.a.611.13	yes	768	731.611	even	42	inner
731.2.z.a.611.14	yes	768	43.9	even	21	inner