Embedded newform 475.2.bc.a.396.5

• Label: 475.2.bc.a.396.5
• Level: $475$
• Weight: $2$
• Character: 475.396
• 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			396.5
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.04037 + 1.27497i) q^{2} +(2.34026 + 0.671059i) q^{3} +(1.66084 - 3.40523i) q^{4} +(-1.72152 + 1.42702i) q^{5} +(-5.63058 + 1.61454i) q^{6} +(-2.03230 - 3.52005i) q^{7} +(0.449834 + 4.27989i) q^{8} +(2.48235 + 1.55115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.04037	+	1.27497i	−1.44276	+	0.901538i	−0.442774	+	0.896633i	\(0.646006\pi\)
							−0.999988	+	0.00490438i	\(0.998439\pi\)
\(3\)	2.34026	+	0.671059i	1.35115	+	0.387436i	0.871660	−	0.490112i	\(-0.163044\pi\)
							0.479490	+	0.877547i	\(0.340822\pi\)
\(5\)	−1.72152	+	1.42702i	−0.769885	+	0.638182i
							\(7\)	−2.03230	−	3.52005i	−0.768137	−	1.33045i	−0.938572	−	0.345083i	\(-0.887851\pi\)
0.170435	−	0.985369i	\(-0.445483\pi\)
\(11\)	−3.19498	−	3.54838i	−0.963322	−	1.06988i	−0.997514	−	0.0704627i	\(-0.977552\pi\)
							0.0341920	−	0.999415i	\(-0.489114\pi\)
\(13\)	0.0693342	−	1.98547i	0.0192298	−	0.550670i	−0.951100	−	0.308884i	\(-0.900044\pi\)
							0.970330	−	0.241786i	\(-0.0777333\pi\)
\(17\)	3.81487	+	0.266762i	0.925242	+	0.0646992i	0.524423	−	0.851458i	\(-0.324281\pi\)
							0.400819	+	0.916157i	\(0.368726\pi\)
\(19\)	2.70510	−	3.41796i	0.620592	−	0.784133i
							\(23\)	9.21254	+	1.29474i	1.92095	+	0.269972i	0.994356	−	0.106097i	\(-0.0338355\pi\)
0.926592	+	0.376069i	\(0.122724\pi\)
\(29\)	−8.12034	+	0.567830i	−1.50791	+	0.105443i	−0.799632	−	0.600490i	\(-0.794972\pi\)
							−0.708278	+	0.705934i	\(0.750528\pi\)
\(31\)	−6.73380	−	2.99808i	−1.20943	−	0.538472i	−0.299842	−	0.953989i	\(-0.596934\pi\)
							−0.909585	+	0.415517i	\(0.863601\pi\)
\(37\)	1.98344	+	6.10439i	0.326075	+	1.00355i	0.970953	+	0.239270i	\(0.0769080\pi\)
							−0.644878	+	0.764285i	\(0.723092\pi\)
\(41\)	−2.23255	−	2.85753i	−0.348665	−	0.446271i	0.581426	−	0.813599i	\(-0.302495\pi\)
							−0.930091	+	0.367328i	\(0.880273\pi\)
\(43\)	1.76056	−	9.98462i	0.268483	−	1.52264i	−0.490449	−	0.871470i	\(-0.663167\pi\)
							0.758932	−	0.651170i	\(-0.225722\pi\)
\(47\)	3.90042	−	0.272744i	0.568935	−	0.0397838i	0.217607	−	0.976036i	\(-0.430175\pi\)
							0.351328	+	0.936253i	\(0.385730\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.5	yes	1152
19.6	even	9	inner	475.2.bc.a.196.44	yes	1152
25.6	even	5	inner	475.2.bc.a.206.44	yes	1152
475.6	even	45	inner	475.2.bc.a.6.5	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.5	✓	1152	475.6	even	45	inner
475.2.bc.a.196.44	yes	1152	19.6	even	9	inner
475.2.bc.a.206.44	yes	1152	25.6	even	5	inner
475.2.bc.a.396.5	yes	1152	1.1	even	1	trivial