Embedded newform 475.2.bc.a.16.32

• Label: 475.2.bc.a.16.32
• 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			16.32
Character	\(\chi\)	\(=\)	475.16
Dual form			475.2.bc.a.386.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.235316 + 0.943803i) q^{2} +(-0.767384 - 0.982207i) q^{3} +(0.930506 - 0.494759i) q^{4} +(1.56582 - 1.59631i) q^{5} +(0.746431 - 0.955388i) q^{6} +(1.73336 + 3.00227i) q^{7} +(1.98764 + 2.20750i) q^{8} +(0.349914 - 1.40343i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.235316	+	0.943803i	0.166394	+	0.667369i	0.994311	+	0.106518i	\(0.0339703\pi\)
							−0.827917	+	0.560851i	\(0.810474\pi\)
\(3\)	−0.767384	−	0.982207i	−0.443049	−	0.567077i	0.513708	−	0.857965i	\(-0.328272\pi\)
							−0.956757	+	0.290888i	\(0.906049\pi\)
\(5\)	1.56582	−	1.59631i	0.700257	−	0.713891i
							\(7\)	1.73336	+	3.00227i	0.655150	+	1.13475i	0.981856	+	0.189627i	\(0.0607279\pi\)
−0.326706	+	0.945126i	\(0.605939\pi\)
\(11\)	−4.69845	+	2.09189i	−1.41664	+	0.630728i	−0.965185	−	0.261568i	\(-0.915761\pi\)
							−0.451452	+	0.892295i	\(0.649094\pi\)
\(13\)	2.54941	−	0.731032i	0.707080	−	0.202752i	0.0972219	−	0.995263i	\(-0.469004\pi\)
							0.609858	+	0.792511i	\(0.291227\pi\)
\(17\)	0.933471	−	0.583297i	0.226400	−	0.141470i	−0.411977	−	0.911194i	\(-0.635162\pi\)
							0.638377	+	0.769724i	\(0.279606\pi\)
\(19\)	2.56785	−	3.52224i	0.589105	−	0.808057i
							\(23\)	0.515454	−	1.05684i	0.107480	−	0.220366i	−0.838275	−	0.545248i	\(-0.816436\pi\)
0.945755	+	0.324882i	\(0.105324\pi\)
\(29\)	6.59530	+	4.12120i	1.22472	+	0.765288i	0.978962	−	0.204045i	\(-0.0654088\pi\)
							0.245754	+	0.969332i	\(0.420964\pi\)
\(31\)	−7.87467	+	1.67381i	−1.41433	+	0.300626i	−0.850809	−	0.525475i	\(-0.823888\pi\)
							−0.563523	+	0.826100i	\(0.690554\pi\)
\(37\)	4.56145	−	3.31409i	0.749898	−	0.544833i	−0.145897	−	0.989300i	\(-0.546607\pi\)
							0.895795	+	0.444467i	\(0.146607\pi\)
\(41\)	−5.26675	+	7.80827i	−0.822528	+	1.21945i	0.150466	+	0.988615i	\(0.451923\pi\)
							−0.972994	+	0.230832i	\(0.925855\pi\)
\(43\)	1.13747	−	6.45089i	0.173462	−	0.983752i	−0.766442	−	0.642313i	\(-0.777975\pi\)
							0.939904	−	0.341438i	\(-0.110914\pi\)
\(47\)	−5.50976	−	3.44288i	−0.803681	−	0.502195i	0.0648120	−	0.997897i	\(-0.479355\pi\)
							−0.868493	+	0.495702i	\(0.834911\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.16.32	✓	1152
19.6	even	9	inner	475.2.bc.a.291.32	yes	1152
25.11	even	5	inner	475.2.bc.a.111.32	yes	1152
475.386	even	45	inner	475.2.bc.a.386.32	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.16.32	✓	1152	1.1	even	1	trivial
475.2.bc.a.111.32	yes	1152	25.11	even	5	inner
475.2.bc.a.291.32	yes	1152	19.6	even	9	inner
475.2.bc.a.386.32	yes	1152	475.386	even	45	inner