Embedded newform 475.2.bc.a.396.3

• Label: 475.2.bc.a.396.3
• Level: $475$
• Weight: $2$
• Character: 475.396
• 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			396.3
Character	\(\chi\)	\(=\)	475.396
Dual form			475.2.bc.a.6.3

$q$-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 \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{5}\right)\)	\(e\left(\frac{2}{9}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.3	✓	1152	475.6	even	45	inner
475.2.bc.a.196.46	yes	1152	19.6	even	9	inner
475.2.bc.a.206.46	yes	1152	25.6	even	5	inner
475.2.bc.a.396.3	yes	1152	1.1	even	1	trivial