Embedded newform 475.2.bc.a.111.23

• Label: 475.2.bc.a.111.23
• 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			111.23
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.339542 - 0.0237431i) q^{2} +(0.0489590 + 1.40200i) q^{3} +(-1.86581 - 0.262223i) q^{4} +(1.65394 - 1.50482i) q^{5} +(0.0166642 - 0.477201i) q^{6} +(1.82609 + 3.16288i) q^{7} +(1.29316 + 0.274870i) q^{8} +(1.02948 - 0.0719880i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.339542	−	0.0237431i	−0.240092	−	0.0167889i	−0.0507966	−	0.998709i	\(-0.516176\pi\)
							−0.189296	+	0.981920i	\(0.560620\pi\)
\(3\)	0.0489590	+	1.40200i	0.0282665	+	0.809447i	0.927906	+	0.372814i	\(0.121607\pi\)
							−0.899640	+	0.436633i	\(0.856171\pi\)
\(5\)	1.65394	−	1.50482i	0.739665	−	0.672975i
							\(7\)	1.82609	+	3.16288i	0.690196	+	1.19546i	0.971773	+	0.235917i	\(0.0758092\pi\)
−0.281577	+	0.959539i	\(0.590857\pi\)
\(11\)	−0.510033	−	4.85264i	−0.153781	−	1.46312i	−0.750604	−	0.660752i	\(-0.770237\pi\)
							0.596823	−	0.802373i	\(-0.296429\pi\)
\(13\)	−1.77220	−	2.62739i	−0.491519	−	0.728706i	0.498922	−	0.866647i	\(-0.333729\pi\)
							−0.990441	+	0.137941i	\(0.955952\pi\)
\(17\)	−2.69668	+	6.67452i	−0.654041	+	1.61881i	0.125235	+	0.992127i	\(0.460031\pi\)
							−0.779276	+	0.626681i	\(0.784413\pi\)
\(19\)	4.30984	−	0.652147i	0.988745	−	0.149613i
							\(23\)	4.81961	+	4.65424i	1.00496	+	0.970476i	0.999565	−	0.0295090i	\(-0.00939437\pi\)
0.00539326	+	0.999985i	\(0.498283\pi\)
\(29\)	2.27056	+	5.61983i	0.421632	+	1.04358i	0.977241	+	0.212132i	\(0.0680406\pi\)
							−0.555609	+	0.831444i	\(0.687515\pi\)
\(31\)	3.60517	−	4.00395i	0.647508	−	0.719131i	−0.326613	−	0.945158i	\(-0.605907\pi\)
							0.974121	+	0.226028i	\(0.0725739\pi\)
\(37\)	6.37114	+	4.62890i	1.04741	+	0.760987i	0.971718	−	0.236145i	\(-0.0758839\pi\)
							0.0756905	+	0.997131i	\(0.475884\pi\)
\(41\)	−1.43178	−	0.410556i	−0.223607	−	0.0641181i	0.161963	−	0.986797i	\(-0.448217\pi\)
							−0.385570	+	0.922679i	\(0.625995\pi\)
\(43\)	−0.199794	+	1.13309i	−0.0304682	+	0.172794i	−0.996245	−	0.0865817i	\(-0.972406\pi\)
							0.965777	+	0.259376i	\(0.0835168\pi\)
\(47\)	−2.50486	−	6.19975i	−0.365372	−	0.904327i	−0.992141	−	0.125122i	\(-0.960068\pi\)
							0.626770	−	0.779205i	\(-0.284377\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.23	yes	1152
19.6	even	9	inner	475.2.bc.a.386.23	yes	1152
25.16	even	5	inner	475.2.bc.a.16.23	✓	1152
475.291	even	45	inner	475.2.bc.a.291.23	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.23	✓	1152	25.16	even	5	inner
475.2.bc.a.111.23	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.23	yes	1152	475.291	even	45	inner
475.2.bc.a.386.23	yes	1152	19.6	even	9	inner