Embedded newform 464.2.bi.a.45.11

• Label: 464.2.bi.a.45.11
• Level: $464$
• Weight: $2$
• Character: 464.45
• Analytic conductor: $3.705$
• Analytic rank: $0$
• Dimension: $696$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	464.bi (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.70505865379\)
Analytic rank:	\(0\)
Dimension:	\(696\)
Relative dimension:	\(58\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			45.11
Character	\(\chi\)	\(=\)	464.45
Dual form			464.2.bi.a.165.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19521 + 0.755956i) q^{2} +(-0.0156654 + 0.00176507i) q^{3} +(0.857060 - 1.80706i) q^{4} +(0.0497111 - 0.0173947i) q^{5} +(0.0173892 - 0.0139520i) q^{6} +(-0.883781 - 0.704792i) q^{7} +(0.341687 + 2.80771i) q^{8} +(-2.92454 + 0.667508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 696 q - 10 q^{2} - 10 q^{3} - 14 q^{4} - 10 q^{5} - 22 q^{6} - 10 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/464\mathbb{Z}\right)^\times\).

\(n\)	\(117\)	\(175\)	\(321\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{1}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19521	+	0.755956i	−0.845142	+	0.534542i
							\(3\)	−0.0156654	+	0.00176507i	−0.00904444	+	0.00101906i	−0.116486	−	0.993192i	\(-0.537163\pi\)
0.107441	+	0.994211i	\(0.465734\pi\)
\(5\)	0.0497111	−	0.0173947i	0.0222315	−	0.00777914i	−0.319140	−	0.947707i	\(-0.603394\pi\)
							0.341372	+	0.939928i	\(0.389108\pi\)
\(7\)	−0.883781	−	0.704792i	−0.334038	−	0.266386i	0.442078	−	0.896977i	\(-0.354241\pi\)
							−0.776116	+	0.630590i	\(0.782813\pi\)
\(11\)	−0.557050	−	0.886540i	−0.167957	−	0.267302i	0.752131	−	0.659013i	\(-0.229026\pi\)
							−0.920088	+	0.391712i	\(0.871883\pi\)
\(13\)	−1.96626	+	1.23548i	−0.545342	+	0.342661i	−0.776361	−	0.630288i	\(-0.782937\pi\)
							0.231020	+	0.972949i	\(0.425794\pi\)
\(17\)	−1.14690			−0.278164			−0.139082	−	0.990281i	\(-0.544415\pi\)
							−0.139082	+	0.990281i	\(0.544415\pi\)
\(19\)	−0.435755	+	3.86743i	−0.0999690	+	0.887249i	0.839207	+	0.543812i	\(0.183020\pi\)
							−0.939176	+	0.343437i	\(0.888409\pi\)
\(23\)	3.13228	+	6.50425i	0.653126	+	1.35623i	0.919779	+	0.392437i	\(0.128368\pi\)
							−0.266653	+	0.963792i	\(0.585918\pi\)
\(29\)	−3.10966	+	4.39659i	−0.577449	+	0.816427i
							\(31\)	−3.83532	−	1.84699i	−0.688844	−	0.331730i	0.0565066	−	0.998402i	\(-0.482004\pi\)
−0.745351	+	0.666672i	\(0.767718\pi\)
\(37\)	−7.34920	−	4.61781i	−1.20820	−	0.759163i	−0.231305	−	0.972881i	\(-0.574300\pi\)
							−0.976895	+	0.213719i	\(0.931442\pi\)
\(41\)		−	4.58224i		−	0.715625i	−0.933793	−	0.357813i	\(-0.883523\pi\)
							0.933793	−	0.357813i	\(-0.116477\pi\)
\(43\)	1.13155	−	3.23379i	0.172560	−	0.493148i	−0.824828	−	0.565384i	\(-0.808728\pi\)
							0.997388	+	0.0722364i	\(0.0230136\pi\)
\(47\)	−1.81448	+	7.94977i	−0.264669	+	1.15959i	0.651451	+	0.758690i	\(0.274160\pi\)
							−0.916121	+	0.400902i	\(0.868697\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.2.bi.a.45.11	✓	696
16.5	even	4	inner	464.2.bi.a.277.30	yes	696
29.20	even	7	inner	464.2.bi.a.397.30	yes	696
464.165	even	28	inner	464.2.bi.a.165.11	yes	696

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.2.bi.a.45.11	✓	696	1.1	even	1	trivial
464.2.bi.a.165.11	yes	696	464.165	even	28	inner
464.2.bi.a.277.30	yes	696	16.5	even	4	inner
464.2.bi.a.397.30	yes	696	29.20	even	7	inner