Embedded newform 475.2.bc.a.111.32

• Label: 475.2.bc.a.111.32
• Level: $475$
• Weight: $2$
• Character: 475.111
• 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			111.32
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.32

$q$-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 \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{4}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.32	✓	1152	25.16	even	5	inner
475.2.bc.a.111.32	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.32	yes	1152	475.291	even	45	inner
475.2.bc.a.386.32	yes	1152	19.6	even	9	inner