Embedded newform 475.2.bc.a.6.16

• Label: 475.2.bc.a.6.16
• 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			6.16
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.954295 - 0.596310i) q^{2} +(-0.197845 + 0.0567310i) q^{3} +(-0.321648 - 0.659477i) q^{4} +(-2.23191 + 0.136321i) q^{5} +(0.222632 + 0.0638386i) q^{6} +(0.295393 - 0.511636i) q^{7} +(-0.321554 + 3.05938i) q^{8} +(-2.50822 + 1.56731i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.954295	−	0.596310i	−0.674789	−	0.421655i	0.148781	−	0.988870i	\(-0.452465\pi\)
							−0.823569	+	0.567215i	\(0.808021\pi\)
\(3\)	−0.197845	+	0.0567310i	−0.114226	+	0.0327537i	−0.332263	−	0.943187i	\(-0.607812\pi\)
							0.218038	+	0.975940i	\(0.430034\pi\)
\(5\)	−2.23191	+	0.136321i	−0.998140	+	0.0609644i
							\(7\)	0.295393	−	0.511636i	0.111648	−	0.193380i	−0.804787	−	0.593564i	\(-0.797720\pi\)
0.916435	+	0.400184i	\(0.131054\pi\)
\(11\)	2.24356	−	2.49173i	0.676459	−	0.751284i	−0.302986	−	0.952995i	\(-0.597984\pi\)
							0.979445	+	0.201711i	\(0.0646503\pi\)
\(13\)	0.122965	+	3.52125i	0.0341043	+	0.976619i	0.889025	+	0.457859i	\(0.151384\pi\)
							−0.854920	+	0.518759i	\(0.826394\pi\)
\(17\)	3.49439	−	0.244351i	0.847514	−	0.0592639i	0.360607	−	0.932718i	\(-0.382570\pi\)
							0.486907	+	0.873454i	\(0.338125\pi\)
\(19\)	−3.40648	+	2.71955i	−0.781499	+	0.623906i
							\(23\)	5.85857	−	0.823369i	1.22160	−	0.171684i	0.501253	−	0.865301i	\(-0.332873\pi\)
0.720344	+	0.693617i	\(0.243984\pi\)
\(29\)	9.33111	+	0.652495i	1.73274	+	0.121165i	0.900951	−	0.433920i	\(-0.142870\pi\)
							0.831793	+	0.555086i	\(0.187315\pi\)
\(31\)	−8.83725	+	3.93460i	−1.58722	+	0.706674i	−0.995072	−	0.0991574i	\(-0.968385\pi\)
							−0.592145	+	0.805832i	\(0.701719\pi\)
\(37\)	0.775880	−	2.38791i	0.127554	−	0.392570i	−0.866804	−	0.498649i	\(-0.833830\pi\)
							0.994358	+	0.106079i	\(0.0338296\pi\)
\(41\)	−4.01116	+	5.13405i	−0.626437	+	0.801803i	−0.991628	−	0.129130i	\(-0.958782\pi\)
							0.365190	+	0.930933i	\(0.381004\pi\)
\(43\)	0.376457	+	2.13499i	0.0574091	+	0.325583i	0.999964	−	0.00846765i	\(-0.00269537\pi\)
							−0.942555	+	0.334051i	\(0.891584\pi\)
\(47\)	4.08596	+	0.285718i	0.595998	+	0.0416762i	0.364570	−	0.931176i	\(-0.381216\pi\)
							0.231428	+	0.972852i	\(0.425660\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.bc.a.6.16	✓	1152
19.16	even	9	inner	475.2.bc.a.206.33	yes	1152
25.21	even	5	inner	475.2.bc.a.196.33	yes	1152
475.396	even	45	inner	475.2.bc.a.396.16	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.bc.a.6.16	✓	1152	1.1	even	1	trivial
475.2.bc.a.196.33	yes	1152	25.21	even	5	inner
475.2.bc.a.206.33	yes	1152	19.16	even	9	inner
475.2.bc.a.396.16	yes	1152	475.396	even	45	inner