Embedded newform 475.2.n.b.39.11

• Label: 475.2.n.b.39.11
• Level: $475$
• Weight: $2$
• Character: 475.39
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.n (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(24\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			39.11
Character	\(\chi\)	\(=\)	475.39
Dual form			475.2.n.b.134.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222497 - 0.306241i) q^{2} +(-2.07855 - 0.675363i) q^{3} +(0.573755 - 1.76584i) q^{4} +(1.71339 + 1.43676i) q^{5} +(0.255648 + 0.786804i) q^{6} +3.25699i q^{7} +(-1.38845 + 0.451133i) q^{8} +(1.43722 + 1.04420i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 26 q^{4} - 4 q^{5} - 2 q^{6} + 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(401\)
\(\chi(n)\)	\(e\left(\frac{3}{10}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222497	−	0.306241i	−0.157329	−	0.216545i	0.723075	−	0.690770i	\(-0.242728\pi\)
							−0.880404	+	0.474225i	\(0.842728\pi\)
\(3\)	−2.07855	−	0.675363i	−1.20005	−	0.389921i	−0.360273	−	0.932847i	\(-0.617316\pi\)
							−0.839781	+	0.542926i	\(0.817316\pi\)
\(5\)	1.71339	+	1.43676i	0.766252	+	0.642540i
							\(7\)			3.25699i			1.23103i	0.788127	+	0.615513i	\(0.211051\pi\)
−0.788127	+	0.615513i	\(0.788949\pi\)
\(11\)	−2.65054	+	1.92573i	−0.799168	+	0.580629i	−0.910670	−	0.413135i	\(-0.864434\pi\)
							0.111502	+	0.993764i	\(0.464434\pi\)
\(13\)	0.0900153	−	0.123895i	0.0249658	−	0.0343624i	−0.796352	−	0.604834i	\(-0.793239\pi\)
							0.821317	+	0.570471i	\(0.193239\pi\)
\(17\)	−6.79139	+	2.20666i	−1.64715	+	0.535193i	−0.978120	−	0.208039i	\(-0.933292\pi\)
							−0.669034	+	0.743232i	\(0.733292\pi\)
\(19\)	0.309017	+	0.951057i	0.0708934	+	0.218187i
							\(23\)	5.39419	+	7.42446i	1.12477	+	1.54811i	0.797647	+	0.603124i	\(0.206078\pi\)
0.327119	+	0.944983i	\(0.393922\pi\)
\(29\)	1.27308	−	3.91815i	0.236406	−	0.727582i	−0.760526	−	0.649307i	\(-0.775059\pi\)
							0.996932	−	0.0782744i	\(-0.0249410\pi\)
\(31\)	1.65495	+	5.09343i	0.297239	+	0.914806i	0.982460	+	0.186472i	\(0.0597053\pi\)
							−0.685222	+	0.728335i	\(0.740295\pi\)
\(37\)	1.69925	−	2.33882i	0.279355	−	0.384499i	−0.646165	−	0.763198i	\(-0.723628\pi\)
							0.925520	+	0.378699i	\(0.123628\pi\)
\(41\)	−2.88615	−	2.09691i	−0.450740	−	0.327482i	0.339148	−	0.940733i	\(-0.389861\pi\)
							−0.789888	+	0.613251i	\(0.789861\pi\)
\(43\)		−	0.218244i		−	0.0332819i	−0.999862	−	0.0166410i	\(-0.994703\pi\)
							0.999862	−	0.0166410i	\(-0.00529723\pi\)
\(47\)	−6.00201	−	1.95017i	−0.875482	−	0.284461i	−0.163402	−	0.986560i	\(-0.552247\pi\)
							−0.712080	+	0.702098i	\(0.752247\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.n.b.39.11	✓	96
25.9	even	10	inner	475.2.n.b.134.11	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.n.b.39.11	✓	96	1.1	even	1	trivial
475.2.n.b.134.11	yes	96	25.9	even	10	inner