Embedded newform 475.2.bc.a.6.6

• Label: 475.2.bc.a.6.6
• 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			6.6
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91392 - 1.19595i) q^{2} +(-2.99079 + 0.857597i) q^{3} +(1.35606 + 2.78033i) q^{4} +(0.364394 - 2.20618i) q^{5} +(6.74979 + 1.93547i) q^{6} +(-1.35748 + 2.35123i) q^{7} +(0.257940 - 2.45413i) q^{8} +(5.66524 - 3.54003i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91392	−	1.19595i	−1.35335	−	0.845666i	−0.357257	−	0.934006i	\(-0.616288\pi\)
							−0.996090	+	0.0883404i	\(0.971844\pi\)
\(3\)	−2.99079	+	0.857597i	−1.72674	+	0.495134i	−0.984556	−	0.175067i	\(-0.943986\pi\)
							−0.742180	+	0.670201i	\(0.766208\pi\)
\(5\)	0.364394	−	2.20618i	0.162962	−	0.986632i
							\(7\)	−1.35748	+	2.35123i	−0.513080	+	0.888681i	0.486805	+	0.873511i	\(0.338162\pi\)
−0.999885	+	0.0151699i	\(0.995171\pi\)
\(11\)	−0.206157	+	0.228961i	−0.0621588	+	0.0690344i	−0.773422	−	0.633891i	\(-0.781457\pi\)
							0.711263	+	0.702926i	\(0.248123\pi\)
\(13\)	−0.225565	−	6.45934i	−0.0625605	−	1.79150i	−0.474467	−	0.880273i	\(-0.657359\pi\)
							0.411906	−	0.911226i	\(-0.364863\pi\)
\(17\)	1.39891	−	0.0978210i	0.339284	−	0.0237251i	0.100902	−	0.994896i	\(-0.467827\pi\)
							0.238383	+	0.971171i	\(0.423383\pi\)
\(19\)	−3.90664	−	1.93344i	−0.896244	−	0.443562i
							\(23\)	6.04246	−	0.849213i	1.25994	−	0.177073i	0.522591	−	0.852584i	\(-0.324966\pi\)
0.737350	+	0.675511i	\(0.236077\pi\)
\(29\)	6.50372	+	0.454785i	1.20771	+	0.0844514i	0.659380	−	0.751810i	\(-0.270819\pi\)
							0.548331	+	0.836261i	\(0.315263\pi\)
\(31\)	−7.52579	+	3.35070i	−1.35167	+	0.601803i	−0.949502	−	0.313762i	\(-0.898410\pi\)
							−0.402170	+	0.915565i	\(0.631744\pi\)
\(37\)	−2.30510	+	7.09438i	−0.378956	+	1.16631i	0.561814	+	0.827264i	\(0.310104\pi\)
							−0.940770	+	0.339044i	\(0.889896\pi\)
\(41\)	−2.87162	+	3.67551i	−0.448472	+	0.574018i	−0.958121	−	0.286364i	\(-0.907553\pi\)
							0.509648	+	0.860383i	\(0.329775\pi\)
\(43\)	0.355925	+	2.01855i	0.0542781	+	0.307826i	0.999845	−	0.0176012i	\(-0.00560291\pi\)
							−0.945567	+	0.325427i	\(0.894492\pi\)
\(47\)	−2.84068	−	0.198640i	−0.414355	−	0.0289746i	−0.138942	−	0.990300i	\(-0.544370\pi\)
							−0.275413	+	0.961326i	\(0.588815\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.6	✓	1152
19.16	even	9	inner	475.2.bc.a.206.43	yes	1152
25.21	even	5	inner	475.2.bc.a.196.43	yes	1152
475.396	even	45	inner	475.2.bc.a.396.6	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.6	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.43	yes	1152	25.21	even	5	inner
475.2.bc.a.206.43	yes	1152	19.16	even	9	inner
475.2.bc.a.396.6	yes	1152	475.396	even	45	inner