Embedded newform 475.2.bc.a.6.13

• Label: 475.2.bc.a.6.13
• 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			6.13
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.30322 - 0.814342i) q^{2} +(1.85231 - 0.531140i) q^{3} +(0.158485 + 0.324943i) q^{4} +(-0.982153 - 2.00882i) q^{5} +(-2.84649 - 0.816218i) q^{6} +(0.367774 - 0.637004i) q^{7} +(-0.263190 + 2.50409i) q^{8} +(0.604785 - 0.377912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.30322	−	0.814342i	−0.921515	−	0.575827i	−0.0158751	−	0.999874i	\(-0.505053\pi\)
							−0.905640	+	0.424047i	\(0.860609\pi\)
\(3\)	1.85231	−	0.531140i	1.06943	−	0.306654i	0.305659	−	0.952141i	\(-0.401123\pi\)
							0.763771	+	0.645487i	\(0.223346\pi\)
\(5\)	−0.982153	−	2.00882i	−0.439232	−	0.898374i
							\(7\)	0.367774	−	0.637004i	0.139006	−	0.240765i	−0.788115	−	0.615528i	\(-0.788943\pi\)
0.927120	+	0.374763i	\(0.122276\pi\)
\(11\)	1.34614	−	1.49504i	0.405877	−	0.450772i	−0.505204	−	0.863000i	\(-0.668583\pi\)
							0.911081	+	0.412228i	\(0.135249\pi\)
\(13\)	−0.168125	−	4.81446i	−0.0466294	−	1.33529i	−0.766975	−	0.641677i	\(-0.778239\pi\)
							0.720346	−	0.693615i	\(-0.243983\pi\)
\(17\)	−1.55778	+	0.108930i	−0.377816	+	0.0264195i	−0.257402	−	0.966304i	\(-0.582867\pi\)
							−0.120414	+	0.992724i	\(0.538422\pi\)
\(19\)	−4.25336	−	0.953362i	−0.975789	−	0.218716i
							\(23\)	0.267146	−	0.0375450i	0.0557039	−	0.00782867i	−0.111266	−	0.993791i	\(-0.535491\pi\)
0.166970	+	0.985962i	\(0.446602\pi\)
\(29\)	−2.78965	−	0.195072i	−0.518026	−	0.0362239i	−0.191649	−	0.981464i	\(-0.561383\pi\)
							−0.326377	+	0.945240i	\(0.605828\pi\)
\(31\)	8.77582	−	3.90724i	1.57618	−	0.701762i	0.582380	−	0.812917i	\(-0.302122\pi\)
							0.993803	+	0.111155i	\(0.0354550\pi\)
\(37\)	0.0553274	−	0.170280i	0.00909576	−	0.0279939i	−0.946406	−	0.322980i	\(-0.895315\pi\)
							0.955501	+	0.294987i	\(0.0953152\pi\)
\(41\)	−1.24163	+	1.58921i	−0.193910	+	0.248193i	−0.875133	−	0.483883i	\(-0.839226\pi\)
							0.681223	+	0.732076i	\(0.261448\pi\)
\(43\)	−1.18807	−	6.73786i	−0.181178	−	1.02751i	−0.930768	−	0.365611i	\(-0.880860\pi\)
							0.749589	−	0.661903i	\(-0.230251\pi\)
\(47\)	−7.58492	−	0.530389i	−1.10637	−	0.0773652i	−0.495138	−	0.868814i	\(-0.664883\pi\)
							−0.611235	+	0.791449i	\(0.709327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.13	✓	1152
19.16	even	9	inner	475.2.bc.a.206.36	yes	1152
25.21	even	5	inner	475.2.bc.a.196.36	yes	1152
475.396	even	45	inner	475.2.bc.a.396.13	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.13	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.36	yes	1152	25.21	even	5	inner
475.2.bc.a.206.36	yes	1152	19.16	even	9	inner
475.2.bc.a.396.13	yes	1152	475.396	even	45	inner