Embedded newform 475.2.bc.a.291.32

• Label: 475.2.bc.a.291.32
• Level: $475$
• Weight: $2$
• Character: 475.291
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.bc (of order \(45\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(48\) over \(\Q(\zeta_{45})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{45}]$

Embedding invariants

Embedding label			291.32
Character	\(\chi\)	\(=\)	475.291
Dual form			475.2.bc.a.111.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.970326 - 0.0678518i) q^{2} +(0.0435001 - 1.24568i) q^{3} +(-1.04361 + 0.146669i) q^{4} +(-0.925421 + 2.03558i) q^{5} +(-0.0423123 - 1.21167i) q^{6} +(1.73336 - 3.00227i) q^{7} +(-2.90557 + 0.617597i) q^{8} +(1.44287 + 0.100895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.970326	−	0.0678518i	0.686124	−	0.0479785i	0.277560	−	0.960708i	\(-0.410474\pi\)
							0.408564	+	0.912730i	\(0.366030\pi\)
\(3\)	0.0435001	−	1.24568i	0.0251148	−	0.719193i	−0.919898	−	0.392159i	\(-0.871728\pi\)
							0.945012	−	0.327034i	\(-0.106049\pi\)
\(5\)	−0.925421	+	2.03558i	−0.413861	+	0.910340i
							\(7\)	1.73336	−	3.00227i	0.655150	−	1.13475i	−0.326706	−	0.945126i	\(-0.605939\pi\)
0.981856	−	0.189627i	\(-0.0607279\pi\)
\(11\)	0.537600	−	5.11492i	0.162093	−	1.54221i	−0.547024	−	0.837117i	\(-0.684239\pi\)
							0.709117	−	0.705091i	\(-0.249094\pi\)
\(13\)	1.48306	−	2.19873i	0.411328	−	0.609819i	−0.565266	−	0.824909i	\(-0.691227\pi\)
							0.976594	+	0.215090i	\(0.0690044\pi\)
\(17\)	−0.412340	−	1.02058i	−0.100007	−	0.247526i	0.868883	−	0.495017i	\(-0.164838\pi\)
							−0.968890	+	0.247491i	\(0.920394\pi\)
\(19\)	−0.00711364	−	4.35889i	−0.00163198	−	0.999999i
							\(23\)	−0.845828	+	0.816807i	−0.176367	+	0.170316i	−0.777603	−	0.628756i	\(-0.783564\pi\)
0.601235	+	0.799072i	\(0.294676\pi\)
\(29\)	−2.91333	+	7.21074i	−0.540991	+	1.33900i	0.370940	+	0.928657i	\(0.379036\pi\)
							−0.911931	+	0.410344i	\(0.865409\pi\)
\(31\)	5.38690	+	5.98276i	0.967516	+	1.07453i	0.997185	+	0.0749752i	\(0.0238878\pi\)
							−0.0296698	+	0.999560i	\(0.509446\pi\)
\(37\)	4.56145	−	3.31409i	0.749898	−	0.544833i	−0.145897	−	0.989300i	\(-0.546607\pi\)
							0.895795	+	0.444467i	\(0.146607\pi\)
\(41\)	−9.05362	+	2.59609i	−1.41394	+	0.405440i	−0.893732	−	0.448600i	\(-0.851923\pi\)
							−0.520206	+	0.854041i	\(0.674145\pi\)
\(43\)	1.13747	+	6.45089i	0.173462	+	0.983752i	0.939904	+	0.341438i	\(0.110914\pi\)
							−0.766442	+	0.642313i	\(0.777975\pi\)
\(47\)	2.43381	−	6.02390i	0.355008	−	0.878676i	−0.638983	−	0.769221i	\(-0.720645\pi\)
							0.993992	−	0.109456i	\(-0.0349108\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.291.32	yes	1152
19.16	even	9	inner	475.2.bc.a.16.32	✓	1152
25.11	even	5	inner	475.2.bc.a.386.32	yes	1152
475.111	even	45	inner	475.2.bc.a.111.32	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.32	✓	1152	19.16	even	9	inner
475.2.bc.a.111.32	yes	1152	475.111	even	45	inner
475.2.bc.a.291.32	yes	1152	1.1	even	1	trivial
475.2.bc.a.386.32	yes	1152	25.11	even	5	inner