Embedded newform 475.2.bc.a.111.24

• Label: 475.2.bc.a.111.24
• 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			111.24
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0535088 + 0.00374170i) q^{2} +(-0.0980898 - 2.80893i) q^{3} +(-1.97769 - 0.277946i) q^{4} +(-1.07493 - 1.96074i) q^{5} +(0.00526149 - 0.150669i) q^{6} +(0.464449 + 0.804449i) q^{7} +(-0.209718 - 0.0445770i) q^{8} +(-4.88775 + 0.341785i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0535088	+	0.00374170i	0.0378364	+	0.00264578i	0.0886622	−	0.996062i	\(-0.471741\pi\)
							−0.0508257	+	0.998708i	\(0.516185\pi\)
\(3\)	−0.0980898	−	2.80893i	−0.0566322	−	1.62173i	−0.612505	−	0.790467i	\(-0.709838\pi\)
							0.555873	−	0.831267i	\(-0.312384\pi\)
\(5\)	−1.07493	−	1.96074i	−0.480724	−	0.876872i
							\(7\)	0.464449	+	0.804449i	0.175545	+	0.304053i	0.940350	−	0.340209i	\(-0.110498\pi\)
−0.764805	+	0.644262i	\(0.777165\pi\)
\(11\)	0.136515	+	1.29885i	0.0411608	+	0.391619i	0.995635	+	0.0933272i	\(0.0297503\pi\)
							−0.954475	+	0.298292i	\(0.903583\pi\)
\(13\)	−3.08165	−	4.56873i	−0.854696	−	1.26714i	−0.962038	−	0.272916i	\(-0.912012\pi\)
							0.107342	−	0.994222i	\(-0.465766\pi\)
\(17\)	0.452605	−	1.12024i	0.109773	−	0.271698i	−0.862291	−	0.506413i	\(-0.830971\pi\)
							0.972064	+	0.234715i	\(0.0754157\pi\)
\(19\)	3.12466	+	3.03916i	0.716846	+	0.697231i
							\(23\)	−1.05116	−	1.01509i	−0.219182	−	0.211661i	0.576959	−	0.816773i	\(-0.304239\pi\)
−0.796140	+	0.605112i	\(0.793128\pi\)
\(29\)	1.97514	+	4.88865i	0.366775	+	0.907800i	0.991870	+	0.127252i	\(0.0406156\pi\)
							−0.625096	+	0.780548i	\(0.714940\pi\)
\(31\)	−2.86778	+	3.18499i	−0.515068	+	0.572041i	−0.943432	−	0.331565i	\(-0.892423\pi\)
							0.428364	+	0.903606i	\(0.359090\pi\)
\(37\)	−3.55814	−	2.58514i	−0.584954	−	0.424994i	0.255553	−	0.966795i	\(-0.417743\pi\)
							−0.840507	+	0.541801i	\(0.817743\pi\)
\(41\)	−8.43657	−	2.41915i	−1.31757	−	0.377807i	−0.458058	−	0.888922i	\(-0.651455\pi\)
							−0.859512	+	0.511115i	\(0.829233\pi\)
\(43\)	−0.875940	+	4.96770i	−0.133580	+	0.757567i	0.842259	+	0.539073i	\(0.181225\pi\)
							−0.975838	+	0.218494i	\(0.929886\pi\)
\(47\)	−3.24970	−	8.04328i	−0.474017	−	1.17323i	−0.955027	−	0.296518i	\(-0.904174\pi\)
							0.481010	−	0.876715i	\(-0.340270\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.24	yes	1152
19.6	even	9	inner	475.2.bc.a.386.24	yes	1152
25.16	even	5	inner	475.2.bc.a.16.24	✓	1152
475.291	even	45	inner	475.2.bc.a.291.24	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.24	✓	1152	25.16	even	5	inner
475.2.bc.a.111.24	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.24	yes	1152	475.291	even	45	inner
475.2.bc.a.386.24	yes	1152	19.6	even	9	inner