Embedded newform 731.2.z.a.67.14

• Label: 731.2.z.a.67.14
• 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			67.14
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.14

$q$-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	0.990614	−	0.136690i	\(-0.0436466\pi\)
							−0.986894	+	0.161371i	\(0.948408\pi\)
\(5\)	−0.740612	−	1.08628i	−0.331212	−	0.485798i	0.624122	−	0.781327i	\(-0.285457\pi\)
							−0.955334	+	0.295528i	\(0.904504\pi\)
\(7\)	−2.40880	+	1.39072i	−0.910442	+	0.525644i	−0.880573	−	0.473910i	\(-0.842842\pi\)
							−0.0298685	+	0.999554i	\(0.509509\pi\)
\(11\)	3.09236	+	0.705812i	0.932383	+	0.212810i	0.661623	−	0.749837i	\(-0.269868\pi\)
							0.270760	+	0.962647i	\(0.412725\pi\)
\(13\)	−0.0558263	−	0.744950i	−0.0154834	−	0.206612i	−0.999580	−	0.0289780i	\(-0.990775\pi\)
							0.984097	−	0.177634i	\(-0.0568443\pi\)
\(17\)	4.07025	+	0.658050i	0.987182	+	0.159601i
							\(19\)	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	\(0.608540\pi\)
							−0.914798	+	0.403913i	\(0.867650\pi\)
\(29\)	0.105590	−	0.700546i	0.0196076	−	0.130088i	−0.976885	−	0.213764i	\(-0.931428\pi\)
							0.996493	+	0.0836754i	\(0.0266659\pi\)
\(31\)	−5.44517	+	2.13707i	−0.977982	+	0.383829i	−0.799865	−	0.600180i	\(-0.795096\pi\)
							−0.178117	+	0.984009i	\(0.557000\pi\)
\(37\)	6.51451	+	3.76116i	1.07098	+	0.618330i	0.928449	−	0.371460i	\(-0.121143\pi\)
							0.142531	+	0.989790i	\(0.454476\pi\)
\(41\)	3.24224	+	2.58560i	0.506353	+	0.403803i	0.843071	−	0.537803i	\(-0.180745\pi\)
							−0.336718	+	0.941606i	\(0.609317\pi\)
\(43\)	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.67.14	yes	768
17.16	even	2	inner	731.2.z.a.67.13	✓	768
43.9	even	21	inner	731.2.z.a.611.13	yes	768
731.611	even	42	inner	731.2.z.a.611.14	yes	768

    

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