Embedded newform 475.2.bc.a.111.4

• Label: 475.2.bc.a.111.4
• 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			111.4
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.46168 - 0.172137i) q^{2} +(-0.0802746 - 2.29876i) q^{3} +(4.04969 + 0.569146i) q^{4} +(1.58431 + 1.57796i) q^{5} +(-0.198093 + 5.67263i) q^{6} +(-0.754362 - 1.30659i) q^{7} +(-5.04352 - 1.07203i) q^{8} +(-2.28518 + 0.159795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.46168	−	0.172137i	−1.74067	−	0.121719i	−0.836245	−	0.548356i	\(-0.815254\pi\)
							−0.904423	+	0.426636i	\(0.859698\pi\)
\(3\)	−0.0802746	−	2.29876i	−0.0463466	−	1.32719i	−0.770450	−	0.637500i	\(-0.779968\pi\)
							0.724103	−	0.689691i	\(-0.242254\pi\)
\(5\)	1.58431	+	1.57796i	0.708524	+	0.705687i
							\(7\)	−0.754362	−	1.30659i	−0.285122	−	0.493846i	0.687517	−	0.726169i	\(-0.258701\pi\)
−0.972639	+	0.232323i	\(0.925367\pi\)
\(11\)	0.143096	+	1.36147i	0.0431452	+	0.410499i	0.994684	+	0.102971i	\(0.0328350\pi\)
							−0.951539	+	0.307528i	\(0.900498\pi\)
\(13\)	−0.976423	−	1.44761i	−0.270811	−	0.401494i	0.668960	−	0.743298i	\(-0.266740\pi\)
							−0.939771	+	0.341805i	\(0.888962\pi\)
\(17\)	1.57491	−	3.89805i	0.381972	−	0.945415i	−0.606646	−	0.794972i	\(-0.707486\pi\)
							0.988619	−	0.150443i	\(-0.0480700\pi\)
\(19\)	3.54066	−	2.54238i	0.812284	−	0.583262i
							\(23\)	4.81938	+	4.65402i	1.00491	+	0.970431i	0.999563	−	0.0295546i	\(-0.00940888\pi\)
0.00534769	+	0.999986i	\(0.498298\pi\)
\(29\)	−1.01677	−	2.51660i	−0.188810	−	0.467322i	0.802853	−	0.596177i	\(-0.203314\pi\)
							−0.991663	+	0.128855i	\(0.958870\pi\)
\(31\)	4.17154	−	4.63297i	0.749231	−	0.832106i	−0.241147	−	0.970489i	\(-0.577524\pi\)
							0.990379	+	0.138383i	\(0.0441904\pi\)
\(37\)	−7.68150	−	5.58093i	−1.26283	−	0.917500i	−0.263937	−	0.964540i	\(-0.585021\pi\)
							−0.998893	+	0.0470400i	\(0.985021\pi\)
\(41\)	−3.04928	−	0.874367i	−0.476217	−	0.136553i	0.0289386	−	0.999581i	\(-0.490787\pi\)
							−0.505156	+	0.863028i	\(0.668565\pi\)
\(43\)	0.767542	−	4.35295i	0.117049	−	0.663818i	−0.868666	−	0.495398i	\(-0.835022\pi\)
							0.985715	−	0.168420i	\(-0.0538666\pi\)
\(47\)	−0.790098	−	1.95556i	−0.115248	−	0.285248i	0.858525	−	0.512772i	\(-0.171381\pi\)
							−0.973772	+	0.227524i	\(0.926937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.4	yes	1152
19.6	even	9	inner	475.2.bc.a.386.4	yes	1152
25.16	even	5	inner	475.2.bc.a.16.4	✓	1152
475.291	even	45	inner	475.2.bc.a.291.4	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.4	✓	1152	25.16	even	5	inner
475.2.bc.a.111.4	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.4	yes	1152	475.291	even	45	inner
475.2.bc.a.386.4	yes	1152	19.6	even	9	inner