Embedded newform 475.2.bc.a.111.18

• Label: 475.2.bc.a.111.18
• 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			111.18
Character	\(\chi\)	\(=\)	475.111
Dual form			475.2.bc.a.291.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.934433 - 0.0653419i) q^{2} +(-0.0140248 - 0.401617i) q^{3} +(-1.11164 - 0.156231i) q^{4} +(-0.112636 - 2.23323i) q^{5} +(-0.0131372 + 0.376201i) q^{6} +(0.363026 + 0.628780i) q^{7} +(2.86104 + 0.608132i) q^{8} +(2.83159 - 0.198004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.934433	−	0.0653419i	−0.660744	−	0.0462037i	−0.264552	−	0.964371i	\(-0.585224\pi\)
							−0.396192	+	0.918168i	\(0.629668\pi\)
\(3\)	−0.0140248	−	0.401617i	−0.00809721	−	0.231874i	−0.996692	−	0.0812730i	\(-0.974101\pi\)
							0.988595	−	0.150601i	\(-0.0481208\pi\)
\(5\)	−0.112636	−	2.23323i	−0.0503725	−	0.998730i
							\(7\)	0.363026	+	0.628780i	0.137211	+	0.237656i	0.926440	−	0.376443i	\(-0.122853\pi\)
−0.789229	+	0.614099i	\(0.789519\pi\)
\(11\)	0.0645854	+	0.614489i	0.0194732	+	0.185275i	0.999935	−	0.0114363i	\(-0.00364036\pi\)
							−0.980461	+	0.196712i	\(0.936974\pi\)
\(13\)	−0.613436	−	0.909456i	−0.170136	−	0.252238i	0.734320	−	0.678803i	\(-0.237501\pi\)
							−0.904457	+	0.426566i	\(0.859723\pi\)
\(17\)	0.426931	−	1.05669i	0.103546	−	0.256285i	−0.866513	−	0.499154i	\(-0.833644\pi\)
							0.970059	+	0.242869i	\(0.0780885\pi\)
\(19\)	−1.93770	−	3.90452i	−0.444539	−	0.895759i
							\(23\)	−2.62556	−	2.53548i	−0.547467	−	0.528683i	0.368663	−	0.929563i	\(-0.379816\pi\)
−0.916131	+	0.400880i	\(0.868704\pi\)
\(29\)	−3.68329	−	9.11646i	−0.683969	−	1.69288i	−0.720411	−	0.693548i	\(-0.756047\pi\)
							0.0364415	−	0.999336i	\(-0.488398\pi\)
\(31\)	0.460866	−	0.511843i	0.0827740	−	0.0919298i	−0.700335	−	0.713814i	\(-0.746966\pi\)
							0.783109	+	0.621885i	\(0.213633\pi\)
\(37\)	−3.97200	−	2.88582i	−0.652992	−	0.474427i	0.211297	−	0.977422i	\(-0.432231\pi\)
							−0.864289	+	0.502995i	\(0.832231\pi\)
\(41\)	10.3114	+	2.95676i	1.61038	+	0.461768i	0.955749	−	0.294183i	\(-0.0950474\pi\)
							0.654628	+	0.755951i	\(0.272825\pi\)
\(43\)	−1.07905	+	6.11959i	−0.164554	+	0.933229i	0.784970	+	0.619534i	\(0.212678\pi\)
							−0.949524	+	0.313696i	\(0.898433\pi\)
\(47\)	1.17779	+	2.91512i	0.171798	+	0.425214i	0.988378	−	0.152018i	\(-0.0485771\pi\)
							−0.816580	+	0.577232i	\(0.804133\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.111.18	yes	1152
19.6	even	9	inner	475.2.bc.a.386.18	yes	1152
25.16	even	5	inner	475.2.bc.a.16.18	✓	1152
475.291	even	45	inner	475.2.bc.a.291.18	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.18	✓	1152	25.16	even	5	inner
475.2.bc.a.111.18	yes	1152	1.1	even	1	trivial
475.2.bc.a.291.18	yes	1152	475.291	even	45	inner
475.2.bc.a.386.18	yes	1152	19.6	even	9	inner