Embedded newform 475.2.bc.a.16.2

• Label: 475.2.bc.a.16.2
• 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			16.2
Character	\(\chi\)	\(=\)	475.16
Dual form			475.2.bc.a.386.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.639582 - 2.56522i) q^{2} +(-1.46382 - 1.87360i) q^{3} +(-4.40541 + 2.34240i) q^{4} +(-1.20542 - 1.88334i) q^{5} +(-3.86997 + 4.95334i) q^{6} +(2.24445 + 3.88750i) q^{7} +(5.28837 + 5.87333i) q^{8} +(-0.641851 + 2.57432i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.639582	−	2.56522i	−0.452253	−	1.81389i	−0.570984	−	0.820961i	\(-0.693438\pi\)
							0.118732	−	0.992926i	\(-0.462117\pi\)
\(3\)	−1.46382	−	1.87360i	−0.845135	−	1.08172i	−0.995655	−	0.0931162i	\(-0.970317\pi\)
							0.150521	−	0.988607i	\(-0.451905\pi\)
\(5\)	−1.20542	−	1.88334i	−0.539080	−	0.842255i
							\(7\)	2.24445	+	3.88750i	0.848322	+	1.46934i	0.882705	+	0.469928i	\(0.155720\pi\)
−0.0343828	+	0.999409i	\(0.510947\pi\)
\(11\)	−2.01767	+	0.898325i	−0.608351	+	0.270855i	−0.687704	−	0.725991i	\(-0.741381\pi\)
							0.0793532	+	0.996847i	\(0.474715\pi\)
\(13\)	4.67767	−	1.34130i	1.29735	−	0.372010i	0.445260	−	0.895401i	\(-0.353111\pi\)
							0.852092	+	0.523391i	\(0.175333\pi\)
\(17\)	−4.89710	+	3.06005i	−1.18772	+	0.742170i	−0.972160	−	0.234320i	\(-0.924714\pi\)
							−0.215561	+	0.976490i	\(0.569158\pi\)
\(19\)	−3.30881	+	2.83756i	−0.759094	+	0.650981i
							\(23\)	−1.04693	+	2.14652i	−0.218300	+	0.447581i	−0.979637	−	0.200775i	\(-0.935654\pi\)
0.761338	+	0.648356i	\(0.224543\pi\)
\(29\)	0.262539	+	0.164053i	0.0487524	+	0.0304638i	0.554076	−	0.832466i	\(-0.313072\pi\)
							−0.505324	+	0.862930i	\(0.668627\pi\)
\(31\)	−3.56254	+	0.757242i	−0.639851	+	0.136005i	−0.516402	−	0.856346i	\(-0.672729\pi\)
							−0.123449	+	0.992351i	\(0.539396\pi\)
\(37\)	−5.50350	+	3.99852i	−0.904769	+	0.657353i	−0.939687	−	0.342037i	\(-0.888883\pi\)
							0.0349173	+	0.999390i	\(0.488883\pi\)
\(41\)	3.03726	−	4.50292i	0.474340	−	0.703237i	−0.513608	−	0.858025i	\(-0.671691\pi\)
							0.987948	+	0.154787i	\(0.0494693\pi\)
\(43\)	−0.237442	+	1.34660i	−0.0362095	+	0.205355i	−0.997545	−	0.0700245i	\(-0.977692\pi\)
							0.961336	+	0.275379i	\(0.0888034\pi\)
\(47\)	3.23541	+	2.02171i	0.471933	+	0.294897i	0.744959	−	0.667110i	\(-0.232469\pi\)
							−0.273026	+	0.962007i	\(0.588024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.16.2	✓	1152
19.6	even	9	inner	475.2.bc.a.291.2	yes	1152
25.11	even	5	inner	475.2.bc.a.111.2	yes	1152
475.386	even	45	inner	475.2.bc.a.386.2	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.2	✓	1152	1.1	even	1	trivial
475.2.bc.a.111.2	yes	1152	25.11	even	5	inner
475.2.bc.a.291.2	yes	1152	19.6	even	9	inner
475.2.bc.a.386.2	yes	1152	475.386	even	45	inner