Embedded newform 475.2.e.f.26.6

• Label: 475.2.e.f.26.6
• Level: $475$
• Weight: $2$
• Character: 475.26
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(6\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 3 x^{11} + 17 x^{10} - 18 x^{9} + 109 x^{8} - 93 x^{7} + 484 x^{6} - 147 x^{5} + 1009 x^{4} - 552 x^{3} + 1107 x^{2} + 33 x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			26.6
Root			\(0.590804 - 1.02330i\) of defining polynomial
Character	\(\chi\)	\(=\)	475.26
Dual form			475.2.e.f.201.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.740597 + 1.28275i) q^{2} +(0.0908038 + 0.157277i) q^{3} +(-0.0969683 + 0.167954i) q^{4} +(-0.134498 + 0.232958i) q^{6} -1.30422 q^{7} +2.67513 q^{8} +(1.48351 - 2.56951i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{2} - 3 q^{3} - 2 q^{4} + q^{6} + 4 q^{7} + 12 q^{8} - 7 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.740597	+	1.28275i	0.523681	+	0.907043i	0.999620	+	0.0275641i	\(0.00877505\pi\)
							−0.475939	+	0.879478i	\(0.657892\pi\)
\(3\)	0.0908038	+	0.157277i	0.0524256	+	0.0908038i	0.891047	−	0.453910i	\(-0.149971\pi\)
							−0.838622	+	0.544714i	\(0.816638\pi\)
\(5\)		0			0
							\(7\)	−1.30422			−0.492948			−0.246474	−	0.969149i	\(-0.579272\pi\)
−0.246474	+	0.969149i	\(0.579272\pi\)
\(11\)	4.98247			1.50227			0.751136	−	0.660147i	\(-0.229506\pi\)
							0.751136	+	0.660147i	\(0.229506\pi\)
\(13\)	0.203067	−	0.351723i	0.0563207	−	0.0975504i	−0.836490	−	0.547982i	\(-0.815396\pi\)
							0.892811	+	0.450431i	\(0.148730\pi\)
\(17\)	1.37510	+	2.38173i	0.333510	+	0.577656i	0.983197	−	0.182546i	\(-0.0584337\pi\)
							−0.649688	+	0.760201i	\(0.725100\pi\)
\(19\)	−4.35785	+	0.0955054i	−0.999760	+	0.0219104i
							\(23\)	−3.47860	+	6.02510i	−0.725337	+	1.25632i	0.233498	+	0.972357i	\(0.424983\pi\)
−0.958835	+	0.283964i	\(0.908350\pi\)
\(29\)	2.00728	−	3.47672i	0.372743	−	0.645610i	−0.617243	−	0.786772i	\(-0.711751\pi\)
							0.989986	+	0.141162i	\(0.0450839\pi\)
\(31\)	−2.57321			−0.462163			−0.231081	−	0.972934i	\(-0.574226\pi\)
							−0.231081	+	0.972934i	\(0.574226\pi\)
\(37\)	3.71348			0.610492			0.305246	−	0.952274i	\(-0.401261\pi\)
							0.305246	+	0.952274i	\(0.401261\pi\)
\(41\)	0.607965	+	1.05303i	0.0949482	+	0.164455i	0.909587	−	0.415514i	\(-0.136398\pi\)
							−0.814639	+	0.579969i	\(0.803065\pi\)
\(43\)	−1.56130	−	2.70426i	−0.238097	−	0.412396i	0.722071	−	0.691819i	\(-0.243190\pi\)
							−0.960168	+	0.279423i	\(0.909857\pi\)
\(47\)	3.25405	−	5.63617i	0.474651	−	0.822120i	−0.524927	−	0.851147i	\(-0.675907\pi\)
							0.999579	+	0.0290269i	\(0.00924083\pi\)