Embedded newform 731.2.z.a.611.10

• Label: 731.2.z.a.611.10
• 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			611.10
Character	\(\chi\)	\(=\)	731.611
Dual form			731.2.z.a.67.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.29990 - 1.63003i) q^{2} +(0.470647 - 3.12254i) q^{3} +(-0.522200 + 2.28791i) q^{4} +(1.19265 - 1.74929i) q^{5} +(-5.70163 + 3.29184i) q^{6} +(-0.150209 - 0.0867232i) q^{7} +(0.651331 - 0.313664i) q^{8} +(-6.66204 - 2.05497i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.29990	−	1.63003i	−0.919171	−	1.15260i	−0.987919	−	0.154970i	\(-0.950472\pi\)
							0.0687479	−	0.997634i	\(-0.478100\pi\)
\(3\)	0.470647	−	3.12254i	0.271728	−	1.80280i	−0.270663	−	0.962674i	\(-0.587243\pi\)
							0.542391	−	0.840126i	\(-0.317519\pi\)
\(5\)	1.19265	−	1.74929i	0.533369	−	0.782308i	−0.461099	−	0.887349i	\(-0.652545\pi\)
							0.994468	+	0.105040i	\(0.0334972\pi\)
\(7\)	−0.150209	−	0.0867232i	−0.0567737	−	0.0327783i	0.471344	−	0.881949i	\(-0.343769\pi\)
							−0.528118	+	0.849171i	\(0.677102\pi\)
\(11\)	2.27020	−	0.518158i	0.684491	−	0.156231i	0.133886	−	0.990997i	\(-0.457254\pi\)
							0.550605	+	0.834766i	\(0.314397\pi\)
\(13\)	−0.0203619	+	0.271710i	−0.00564737	+	0.0753589i	−0.999315	−	0.0369973i	\(-0.988221\pi\)
							0.993668	+	0.112356i	\(0.0358398\pi\)
\(17\)	−3.32847	−	2.43337i	−0.807272	−	0.590179i
							\(19\)	−4.33460	+	1.33705i	−0.994426	+	0.306740i	−0.748924	−	0.662656i	\(-0.769429\pi\)
−0.245502	+	0.969396i	\(0.578953\pi\)
\(23\)	0.898769	+	0.968643i	0.187406	+	0.201976i	0.819765	−	0.572699i	\(-0.194104\pi\)
							−0.632359	+	0.774675i	\(0.717913\pi\)
\(29\)	0.898998	+	5.96446i	0.166940	+	1.10757i	0.902681	+	0.430310i	\(0.141596\pi\)
							−0.735741	+	0.677263i	\(0.763166\pi\)
\(31\)	−5.99294	−	2.35206i	−1.07636	−	0.422442i	−0.240088	−	0.970751i	\(-0.577176\pi\)
							−0.836275	+	0.548310i	\(0.815272\pi\)
\(37\)	−2.20430	+	1.27265i	−0.362385	+	0.209223i	−0.670126	−	0.742247i	\(-0.733760\pi\)
							0.307742	+	0.951470i	\(0.400427\pi\)
\(41\)	5.66110	−	4.51457i	0.884115	−	0.705058i	−0.0722022	−	0.997390i	\(-0.523003\pi\)
							0.956317	+	0.292332i	\(0.0944313\pi\)
\(43\)	−0.844441	−	6.50284i	−0.128776	−	0.991674i
							\(47\)	1.10569	−	4.84433i	0.161281	−	0.706618i	−0.828016	−	0.560704i	\(-0.810531\pi\)
0.989297	−	0.145914i	\(-0.0466123\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.z.a.611.10	yes	768
17.16	even	2	inner	731.2.z.a.611.9	yes	768
43.24	even	21	inner	731.2.z.a.67.9	✓	768
731.67	even	42	inner	731.2.z.a.67.10	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.z.a.67.9	✓	768	43.24	even	21	inner
731.2.z.a.67.10	yes	768	731.67	even	42	inner
731.2.z.a.611.9	yes	768	17.16	even	2	inner
731.2.z.a.611.10	yes	768	1.1	even	1	trivial