Embedded newform 475.2.bc.a.6.3

• Label: 475.2.bc.a.6.3
• Level: $475$
• Weight: $2$
• Character: 475.6
• 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			6.3
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.19199 - 1.36971i) q^{2} +(-1.37858 + 0.395301i) q^{3} +(2.05198 + 4.20718i) q^{4} +(-2.03188 - 0.933532i) q^{5} +(3.56327 + 1.02175i) q^{6} +(1.36546 - 2.36504i) q^{7} +(0.724330 - 6.89154i) q^{8} +(-0.799932 + 0.499853i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)	\(e\left(\frac{7}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.19199	−	1.36971i	−1.54997	−	0.968530i	−0.990165	−	0.139906i	\(-0.955320\pi\)
							−0.559807	−	0.828623i	\(-0.689125\pi\)
\(3\)	−1.37858	+	0.395301i	−0.795922	+	0.228227i	−0.648867	−	0.760901i	\(-0.724757\pi\)
							−0.147054	+	0.989128i	\(0.546979\pi\)
\(5\)	−2.03188	−	0.933532i	−0.908682	−	0.417488i
							\(7\)	1.36546	−	2.36504i	0.516094	−	0.893900i	−0.483732	−	0.875216i	\(-0.660719\pi\)
0.999825	−	0.0186841i	\(-0.00594769\pi\)
\(11\)	−2.40200	+	2.66769i	−0.724229	+	0.804338i	−0.987034	−	0.160511i	\(-0.948686\pi\)
							0.262805	+	0.964849i	\(0.415352\pi\)
\(13\)	0.0645208	+	1.84763i	0.0178948	+	0.512441i	0.974936	+	0.222488i	\(0.0714177\pi\)
							−0.957041	+	0.289954i	\(0.906360\pi\)
\(17\)	0.321897	−	0.0225092i	0.0780714	−	0.00545928i	−0.0306673	−	0.999530i	\(-0.509763\pi\)
							0.108739	+	0.994070i	\(0.465319\pi\)
\(19\)	−4.02292	+	1.67812i	−0.922922	+	0.384987i
							\(23\)	−2.55662	+	0.359309i	−0.533092	+	0.0749211i	−0.400583	−	0.916261i	\(-0.631192\pi\)
−0.132509	+	0.991182i	\(0.542303\pi\)
\(29\)	3.37880	+	0.236269i	0.627428	+	0.0438740i	0.379931	−	0.925015i	\(-0.375948\pi\)
							0.247496	+	0.968889i	\(0.420392\pi\)
\(31\)	8.51444	−	3.79087i	1.52924	−	0.680861i	0.542040	−	0.840353i	\(-0.317652\pi\)
							0.987199	+	0.159492i	\(0.0509856\pi\)
\(37\)	0.407467	−	1.25405i	0.0669871	−	0.206165i	−0.911960	−	0.410279i	\(-0.865431\pi\)
							0.978947	+	0.204114i	\(0.0654313\pi\)
\(41\)	5.22035	−	6.68175i	0.815282	−	1.04351i	−0.182893	−	0.983133i	\(-0.558546\pi\)
							0.998175	−	0.0603810i	\(-0.0192316\pi\)
\(43\)	0.627112	+	3.55653i	0.0956337	+	0.542365i	0.994551	+	0.104248i	\(0.0332435\pi\)
							−0.898918	+	0.438118i	\(0.855645\pi\)
\(47\)	3.45438	+	0.241554i	0.503873	+	0.0352342i	0.319432	−	0.947609i	\(-0.396508\pi\)
							0.184441	+	0.982844i	\(0.440952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.3	✓	1152
19.16	even	9	inner	475.2.bc.a.206.46	yes	1152
25.21	even	5	inner	475.2.bc.a.196.46	yes	1152
475.396	even	45	inner	475.2.bc.a.396.3	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.3	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.46	yes	1152	25.21	even	5	inner
475.2.bc.a.206.46	yes	1152	19.16	even	9	inner
475.2.bc.a.396.3	yes	1152	475.396	even	45	inner