Embedded newform 475.2.bc.a.6.5

• Label: 475.2.bc.a.6.5
• Level: $475$
• Weight: $2$
• Character: 475.6
• 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			6.5
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.5

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.5	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.44	yes	1152	25.21	even	5	inner
475.2.bc.a.206.44	yes	1152	19.16	even	9	inner
475.2.bc.a.396.5	yes	1152	475.396	even	45	inner