Embedded newform 475.2.bc.a.396.15

• Label: 475.2.bc.a.396.15
• Level: $475$
• Weight: $2$
• Character: 475.396
• 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			396.15
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.978277 + 0.611295i) q^{2} +(-2.30987 - 0.662346i) q^{3} +(-0.293398 + 0.601556i) q^{4} +(2.15159 + 0.608806i) q^{5} +(2.66459 - 0.764058i) q^{6} +(0.629622 + 1.09054i) q^{7} +(-0.321864 - 3.06233i) q^{8} +(2.35267 + 1.47011i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.978277	+	0.611295i	−0.691746	+	0.432251i	−0.829716	−	0.558185i	\(-0.811498\pi\)
							0.137970	+	0.990436i	\(0.455942\pi\)
\(3\)	−2.30987	−	0.662346i	−1.33361	−	0.382406i	−0.468265	−	0.883588i	\(-0.655121\pi\)
							−0.865342	+	0.501182i	\(0.832899\pi\)
\(5\)	2.15159	+	0.608806i	0.962222	+	0.272266i
							\(7\)	0.629622	+	1.09054i	0.237975	+	0.412184i	0.960133	−	0.279544i	\(-0.0901831\pi\)
−0.722158	+	0.691728i	\(0.756850\pi\)
\(11\)	0.499803	+	0.555088i	0.150696	+	0.167365i	0.813766	−	0.581192i	\(-0.197414\pi\)
							−0.663070	+	0.748557i	\(0.730747\pi\)
\(13\)	0.209026	−	5.98572i	0.0579734	−	1.66014i	−0.527579	−	0.849506i	\(-0.676900\pi\)
							0.585552	−	0.810635i	\(-0.300878\pi\)
\(17\)	−1.59316	−	0.111404i	−0.386397	−	0.0270195i	−0.124763	−	0.992187i	\(-0.539817\pi\)
							−0.261634	+	0.965167i	\(0.584261\pi\)
\(19\)	−1.77415	+	3.98150i	−0.407019	+	0.913420i
							\(23\)	7.02875	+	0.987827i	1.46560	+	0.205976i	0.826407	−	0.563073i	\(-0.190381\pi\)
0.639189	+	0.769049i	\(0.279270\pi\)
\(29\)	−0.0907475	+	0.00634568i	−0.0168514	+	0.00117836i	−0.0781797	−	0.996939i	\(-0.524911\pi\)
							0.0613283	+	0.998118i	\(0.480466\pi\)
\(31\)	5.43647	+	2.42047i	0.976418	+	0.434729i	0.831993	−	0.554786i	\(-0.187200\pi\)
							0.144425	+	0.989516i	\(0.453867\pi\)
\(37\)	−0.862743	−	2.65525i	−0.141834	−	0.436520i	0.854756	−	0.519030i	\(-0.173707\pi\)
							−0.996590	+	0.0825094i	\(0.973707\pi\)
\(41\)	4.92546	+	6.30431i	0.769228	+	0.984567i	0.999933	+	0.0115437i	\(0.00367455\pi\)
							−0.230705	+	0.973024i	\(0.574103\pi\)
\(43\)	−1.24043	+	7.03482i	−0.189164	+	1.07280i	0.731325	+	0.682030i	\(0.238902\pi\)
							−0.920488	+	0.390770i	\(0.872209\pi\)
\(47\)	0.369623	−	0.0258465i	0.0539150	−	0.00377010i	−0.0427735	−	0.999085i	\(-0.513619\pi\)
							0.0966885	+	0.995315i	\(0.469175\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.396.15	yes	1152
19.6	even	9	inner	475.2.bc.a.196.34	yes	1152
25.6	even	5	inner	475.2.bc.a.206.34	yes	1152
475.6	even	45	inner	475.2.bc.a.6.15	✓	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.15	✓	1152	475.6	even	45	inner
475.2.bc.a.196.34	yes	1152	19.6	even	9	inner
475.2.bc.a.206.34	yes	1152	25.6	even	5	inner
475.2.bc.a.396.15	yes	1152	1.1	even	1	trivial