Embedded newform 731.2.z.a.611.18

• Label: 731.2.z.a.611.18
• 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			611.18
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.792393 - 0.993630i) q^{2} +(0.413423 - 2.74288i) q^{3} +(0.0856288 - 0.375164i) q^{4} +(-1.48606 + 2.17965i) q^{5} +(-3.05300 + 1.76265i) q^{6} +(-0.708079 - 0.408809i) q^{7} +(-2.73071 + 1.31504i) q^{8} +(-4.48577 - 1.38368i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.792393	−	0.993630i	−0.560307	−	0.702602i	0.418308	−	0.908305i	\(-0.362623\pi\)
							−0.978614	+	0.205703i	\(0.934052\pi\)
\(3\)	0.413423	−	2.74288i	0.238690	−	1.58360i	−0.472969	−	0.881079i	\(-0.656818\pi\)
							0.711659	−	0.702525i	\(-0.247944\pi\)
\(5\)	−1.48606	+	2.17965i	−0.664587	+	0.974771i	0.334884	+	0.942259i	\(0.391303\pi\)
							−0.999471	+	0.0325114i	\(0.989649\pi\)
\(7\)	−0.708079	−	0.408809i	−0.267629	−	0.154515i	0.360181	−	0.932882i	\(-0.382715\pi\)
							−0.627809	+	0.778367i	\(0.716048\pi\)
\(11\)	−2.11142	+	0.481917i	−0.636616	+	0.145303i	−0.528635	−	0.848850i	\(-0.677296\pi\)
							−0.107981	+	0.994153i	\(0.534439\pi\)
\(13\)	0.498181	−	6.64777i	0.138171	−	1.84376i	−0.311116	−	0.950372i	\(-0.600703\pi\)
							0.449287	−	0.893387i	\(-0.351678\pi\)
\(17\)	1.73708	+	3.73933i	0.421303	+	0.906920i
							\(19\)	−4.49650	+	1.38699i	−1.03157	+	0.318197i	−0.763914	−	0.645318i	\(-0.776725\pi\)
−0.267655	+	0.963515i	\(0.586249\pi\)
\(23\)	−0.451288	−	0.486373i	−0.0941001	−	0.101416i	0.684246	−	0.729251i	\(-0.260132\pi\)
							−0.778346	+	0.627836i	\(0.783941\pi\)
\(29\)	0.901551	+	5.98140i	0.167414	+	1.11072i	0.901853	+	0.432043i	\(0.142207\pi\)
							−0.734439	+	0.678674i	\(0.762555\pi\)
\(31\)	2.14452	+	0.841664i	0.385168	+	0.151167i	0.550021	−	0.835151i	\(-0.314620\pi\)
							−0.164853	+	0.986318i	\(0.552715\pi\)
\(37\)	−4.64999	+	2.68467i	−0.764454	+	0.441358i	−0.830893	−	0.556433i	\(-0.812170\pi\)
							0.0664387	+	0.997791i	\(0.478836\pi\)
\(41\)	−2.17041	+	1.73084i	−0.338961	+	0.270312i	−0.778145	−	0.628085i	\(-0.783839\pi\)
							0.439184	+	0.898397i	\(0.355268\pi\)
\(43\)	2.61334	+	6.01419i	0.398530	+	0.917155i
							\(47\)	1.69810	−	7.43988i	0.247694	−	1.08522i	−0.686128	−	0.727480i	\(-0.740691\pi\)
0.933822	−	0.357737i	\(-0.116452\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.611.18	yes	768
17.16	even	2	inner	731.2.z.a.611.17	yes	768
43.24	even	21	inner	731.2.z.a.67.17	✓	768
731.67	even	42	inner	731.2.z.a.67.18	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.17	✓	768	43.24	even	21	inner
731.2.z.a.67.18	yes	768	731.67	even	42	inner
731.2.z.a.611.17	yes	768	17.16	even	2	inner
731.2.z.a.611.18	yes	768	1.1	even	1	trivial