Embedded newform 475.2.bc.a.111.19

• Label: 475.2.bc.a.111.19
• 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			111.19
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.920737 - 0.0643842i) q^{2} +(-0.115834 - 3.31704i) q^{3} +(-1.13693 - 0.159784i) q^{4} +(-0.216571 + 2.22556i) q^{5} +(-0.106913 + 3.06158i) q^{6} +(0.0485700 + 0.0841256i) q^{7} +(2.84215 + 0.604118i) q^{8} +(-7.99664 + 0.559180i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.920737	−	0.0643842i	−0.651059	−	0.0455265i	−0.259592	−	0.965718i	\(-0.583588\pi\)
							−0.391467	+	0.920192i	\(0.628032\pi\)
\(3\)	−0.115834	−	3.31704i	−0.0668766	−	1.91509i	−0.319595	−	0.947554i	\(-0.603547\pi\)
							0.252719	−	0.967540i	\(-0.418675\pi\)
\(5\)	−0.216571	+	2.22556i	−0.0968534	+	0.995299i
							\(7\)	0.0485700	+	0.0841256i	0.0183577	+	0.0317965i	0.875058	−	0.484017i	\(-0.160823\pi\)
−0.856701	+	0.515814i	\(0.827490\pi\)
\(11\)	0.398495	+	3.79143i	0.120151	+	1.14316i	0.873937	+	0.486039i	\(0.161559\pi\)
							−0.753786	+	0.657120i	\(0.771774\pi\)
\(13\)	3.12203	+	4.62860i	0.865896	+	1.28374i	0.957642	+	0.287961i	\(0.0929774\pi\)
							−0.0917460	+	0.995782i	\(0.529245\pi\)
\(17\)	1.08644	−	2.68903i	0.263500	−	0.652186i	−0.736198	−	0.676767i	\(-0.763381\pi\)
							0.999698	+	0.0245805i	\(0.00782501\pi\)
\(19\)	−0.378048	+	4.34247i	−0.0867301	+	0.996232i
							\(23\)	−2.43673	−	2.35312i	−0.508093	−	0.490660i	0.395848	−	0.918316i	\(-0.370451\pi\)
−0.903942	+	0.427656i	\(0.859339\pi\)
\(29\)	−1.96649	−	4.86723i	−0.365168	−	0.903822i	−0.992180	−	0.124813i	\(-0.960167\pi\)
							0.627012	−	0.779009i	\(-0.284278\pi\)
\(31\)	−0.160213	+	0.177934i	−0.0287751	+	0.0319580i	−0.757361	−	0.652997i	\(-0.773511\pi\)
							0.728585	+	0.684955i	\(0.240178\pi\)
\(37\)	2.57050	+	1.86758i	0.422587	+	0.307027i	0.778678	−	0.627424i	\(-0.215891\pi\)
							−0.356091	+	0.934451i	\(0.615891\pi\)
\(41\)	6.12785	+	1.75713i	0.957009	+	0.274418i	0.717565	−	0.696492i	\(-0.245257\pi\)
							0.239445	+	0.970910i	\(0.423035\pi\)
\(43\)	−1.77626	+	10.0737i	−0.270877	+	1.53622i	0.480884	+	0.876784i	\(0.340316\pi\)
							−0.751761	+	0.659436i	\(0.770795\pi\)
\(47\)	2.99463	+	7.41197i	0.436812	+	1.08115i	0.971699	+	0.236224i	\(0.0759098\pi\)
							−0.534887	+	0.844924i	\(0.679646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.19	yes	1152
19.6	even	9	inner	475.2.bc.a.386.19	yes	1152
25.16	even	5	inner	475.2.bc.a.16.19	✓	1152
475.291	even	45	inner	475.2.bc.a.291.19	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.19	✓	1152	25.16	even	5	inner
475.2.bc.a.111.19	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.19	yes	1152	475.291	even	45	inner
475.2.bc.a.386.19	yes	1152	19.6	even	9	inner