Embedded newform 475.2.bc.a.16.4

• Label: 475.2.bc.a.16.4
• Level: $475$
• Weight: $2$
• Character: 475.16
• 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			16.4
Character	\(\chi\)	\(=\)	475.16
Dual form			475.2.bc.a.386.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.596988 - 2.39439i) q^{2} +(1.41612 + 1.81255i) q^{3} +(-3.61080 + 1.91990i) q^{4} +(-2.02846 + 0.940935i) q^{5} +(3.49455 - 4.47282i) q^{6} +(-0.754362 - 1.30659i) q^{7} +(3.45017 + 3.83180i) q^{8} +(-0.554184 + 2.22271i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{1}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.596988	−	2.39439i	−0.422134	−	1.69309i	−0.685237	−	0.728320i	\(-0.740302\pi\)
							0.263103	−	0.964768i	\(-0.415254\pi\)
\(3\)	1.41612	+	1.81255i	0.817599	+	1.04648i	0.998020	+	0.0628895i	\(0.0200316\pi\)
							−0.180422	+	0.983589i	\(0.557746\pi\)
\(5\)	−2.02846	+	0.940935i	−0.907154	+	0.420799i
							\(7\)	−0.754362	−	1.30659i	−0.285122	−	0.493846i	0.687517	−	0.726169i	\(-0.258701\pi\)
−0.972639	+	0.232323i	\(0.925367\pi\)
\(11\)	−1.25062	+	0.556811i	−0.377075	+	0.167885i	−0.586518	−	0.809936i	\(-0.699502\pi\)
							0.209443	+	0.977821i	\(0.432835\pi\)
\(13\)	−1.67849	+	0.481298i	−0.465528	+	0.133488i	−0.500199	−	0.865911i	\(-0.666740\pi\)
							0.0346703	+	0.999399i	\(0.488962\pi\)
\(17\)	−3.56535	+	2.22788i	−0.864723	+	0.540339i	−0.888238	−	0.459384i	\(-0.848070\pi\)
							0.0235143	+	0.999724i	\(0.492514\pi\)
\(19\)	−4.35883	−	0.0243220i	−0.999984	−	0.00557985i
							\(23\)	−2.93697	+	6.02168i	−0.612400	+	1.25561i	0.336990	+	0.941508i	\(0.390591\pi\)
−0.949390	+	0.314099i	\(0.898298\pi\)
\(29\)	2.30181	+	1.43833i	0.427436	+	0.267091i	0.726534	−	0.687131i	\(-0.241130\pi\)
							−0.299098	+	0.954222i	\(0.596686\pi\)
\(31\)	−6.09804	+	1.29618i	−1.09524	+	0.232801i	−0.719894	−	0.694084i	\(-0.755810\pi\)
							−0.375347	+	0.926885i	\(0.622476\pi\)
\(37\)	−7.68150	+	5.58093i	−1.26283	+	0.917500i	−0.998893	−	0.0470400i	\(-0.985021\pi\)
							−0.263937	+	0.964540i	\(0.585021\pi\)
\(41\)	−1.77385	+	2.62984i	−0.277029	+	0.410712i	−0.941716	−	0.336410i	\(-0.890787\pi\)
							0.664687	+	0.747122i	\(0.268565\pi\)
\(43\)	0.767542	−	4.35295i	0.117049	−	0.663818i	−0.868666	−	0.495398i	\(-0.835022\pi\)
							0.985715	−	0.168420i	\(-0.0538666\pi\)
\(47\)	1.78865	+	1.11767i	0.260902	+	0.163029i	0.654063	−	0.756440i	\(-0.273063\pi\)
							−0.393161	+	0.919469i	\(0.628619\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.16.4	✓	1152
19.6	even	9	inner	475.2.bc.a.291.4	yes	1152
25.11	even	5	inner	475.2.bc.a.111.4	yes	1152
475.386	even	45	inner	475.2.bc.a.386.4	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.4	✓	1152	1.1	even	1	trivial
475.2.bc.a.111.4	yes	1152	25.11	even	5	inner
475.2.bc.a.291.4	yes	1152	19.6	even	9	inner
475.2.bc.a.386.4	yes	1152	475.386	even	45	inner