Embedded newform 731.2.z.a.67.11

• Label: 731.2.z.a.67.11
• 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			67.11
Character	\(\chi\)	\(=\)	731.67
Dual form			731.2.z.a.611.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29543 + 1.62442i) q^{2} +(-0.00546009 - 0.0362253i) q^{3} +(-0.515560 - 2.25882i) q^{4} +(-2.06352 - 3.02662i) q^{5} +(0.0659185 + 0.0380581i) q^{6} +(0.0270939 - 0.0156427i) q^{7} +(0.593233 + 0.285686i) q^{8} +(2.86544 - 0.883870i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29543	+	1.62442i	−0.916011	+	1.14864i	0.0724814	+	0.997370i	\(0.476908\pi\)
							−0.988492	+	0.151271i	\(0.951663\pi\)
\(3\)	−0.00546009	−	0.0362253i	−0.00315239	−	0.0209147i	0.987199	−	0.159491i	\(-0.0509854\pi\)
							−0.990352	+	0.138577i	\(0.955747\pi\)
\(5\)	−2.06352	−	3.02662i	−0.922832	−	1.35355i	−0.935387	−	0.353625i	\(-0.884949\pi\)
							0.0125551	−	0.999921i	\(-0.496003\pi\)
\(7\)	0.0270939	−	0.0156427i	0.0102405	−	0.00591237i	−0.494871	−	0.868966i	\(-0.664785\pi\)
							0.505112	+	0.863054i	\(0.331451\pi\)
\(11\)	−4.28552	−	0.978143i	−1.29213	−	0.294921i	−0.479421	−	0.877585i	\(-0.659153\pi\)
							−0.812713	+	0.582664i	\(0.802010\pi\)
\(13\)	0.221445	+	2.95498i	0.0614178	+	0.819563i	0.939826	+	0.341653i	\(0.110987\pi\)
							−0.878409	+	0.477910i	\(0.841394\pi\)
\(17\)	−4.04668	−	0.790184i	−0.981464	−	0.191648i
							\(19\)	5.97788	+	1.84393i	1.37142	+	0.423027i	0.890996	−	0.454010i	\(-0.150007\pi\)
0.480423	+	0.877037i	\(0.340483\pi\)
\(23\)	−0.363581	+	0.391848i	−0.0758120	+	0.0817059i	−0.769819	−	0.638263i	\(-0.779653\pi\)
							0.694007	+	0.719969i	\(0.255844\pi\)
\(29\)	−1.13199	+	7.51026i	−0.210205	+	1.39462i	0.596000	+	0.802984i	\(0.296756\pi\)
							−0.806205	+	0.591636i	\(0.798482\pi\)
\(31\)	−5.67094	+	2.22568i	−1.01853	+	0.399744i	−0.815085	−	0.579342i	\(-0.803310\pi\)
							−0.203447	+	0.979086i	\(0.565214\pi\)
\(37\)	−7.89017	−	4.55539i	−1.29714	−	0.748902i	−0.317227	−	0.948350i	\(-0.602752\pi\)
							−0.979908	+	0.199448i	\(0.936085\pi\)
\(41\)	7.05382	+	5.62523i	1.10162	+	0.878514i	0.993296	−	0.115601i	\(-0.0368795\pi\)
							0.108326	+	0.994115i	\(0.465451\pi\)
\(43\)	−5.80735	+	3.04543i	−0.885613	+	0.464423i
							\(47\)	−0.136544	−	0.598237i	−0.0199169	−	0.0872618i	0.963993	−	0.265929i	\(-0.0856786\pi\)
−0.983910	+	0.178667i	\(0.942822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.67.11	✓	768
17.16	even	2	inner	731.2.z.a.67.12	yes	768
43.9	even	21	inner	731.2.z.a.611.12	yes	768
731.611	even	42	inner	731.2.z.a.611.11	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.11	✓	768	1.1	even	1	trivial
731.2.z.a.67.12	yes	768	17.16	even	2	inner
731.2.z.a.611.11	yes	768	731.611	even	42	inner
731.2.z.a.611.12	yes	768	43.9	even	21	inner