Embedded newform 475.2.bc.a.6.9

• Label: 475.2.bc.a.6.9
• 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			6.9
Character	\(\chi\)	\(=\)	475.6
Dual form			475.2.bc.a.396.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.76581 - 1.10340i) q^{2} +(-0.597589 + 0.171356i) q^{3} +(1.02386 + 2.09922i) q^{4} +(1.42897 - 1.71990i) q^{5} +(1.24431 + 0.356799i) q^{6} +(1.89729 - 3.28621i) q^{7} +(0.0730432 - 0.694959i) q^{8} +(-2.21639 + 1.38496i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 18 q^{2} - 18 q^{3} - 18 q^{4} - 24 q^{5} - 18 q^{6} - 30 q^{7} - 9 q^{8} - 18 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.76581	−	1.10340i	−1.24862	−	0.780224i	−0.265663	−	0.964066i	\(-0.585591\pi\)
							−0.982956	+	0.183842i	\(0.941147\pi\)
\(3\)	−0.597589	+	0.171356i	−0.345018	+	0.0989324i	−0.443672	−	0.896189i	\(-0.646324\pi\)
							0.0986532	+	0.995122i	\(0.468547\pi\)
\(5\)	1.42897	−	1.71990i	0.639056	−	0.769161i
							\(7\)	1.89729	−	3.28621i	0.717109	−	1.24207i	−0.245031	−	0.969515i	\(-0.578798\pi\)
0.962140	−	0.272554i	\(-0.0878685\pi\)
\(11\)	0.789636	−	0.876979i	0.238084	−	0.264419i	−0.612248	−	0.790666i	\(-0.709735\pi\)
							0.850332	+	0.526247i	\(0.176401\pi\)
\(13\)	−0.0671754	−	1.92365i	−0.0186311	−	0.533525i	−0.972446	−	0.233129i	\(-0.925103\pi\)
							0.953815	−	0.300396i	\(-0.0971188\pi\)
\(17\)	5.10163	−	0.356741i	1.23733	−	0.0865224i	0.563943	−	0.825814i	\(-0.309284\pi\)
							0.673386	+	0.739292i	\(0.264839\pi\)
\(19\)	3.47549	+	2.63077i	0.797333	+	0.603540i
							\(23\)	0.229863	−	0.0323051i	0.0479297	−	0.00673608i	−0.115167	−	0.993346i	\(-0.536740\pi\)
0.163097	+	0.986610i	\(0.447852\pi\)
\(29\)	−5.81965	−	0.406950i	−1.08068	−	0.0755686i	−0.481704	−	0.876334i	\(-0.659982\pi\)
							−0.598978	+	0.800765i	\(0.704426\pi\)
\(31\)	−3.69231	+	1.64392i	−0.663159	+	0.295257i	−0.710579	−	0.703618i	\(-0.751567\pi\)
							0.0474198	+	0.998875i	\(0.484900\pi\)
\(37\)	3.56716	−	10.9786i	0.586438	−	1.80487i	−0.00697934	−	0.999976i	\(-0.502222\pi\)
							0.593417	−	0.804895i	\(-0.297778\pi\)
\(41\)	−3.26718	+	4.18180i	−0.510247	+	0.653087i	−0.972292	−	0.233770i	\(-0.924894\pi\)
							0.462045	+	0.886857i	\(0.347116\pi\)
\(43\)	−0.657073	−	3.72645i	−0.100203	−	0.568278i	−0.993028	−	0.117875i	\(-0.962392\pi\)
							0.892826	−	0.450402i	\(-0.148719\pi\)
\(47\)	−10.4724	−	0.732303i	−1.52756	−	0.106817i	−0.718908	−	0.695105i	\(-0.755358\pi\)
							−0.808652	+	0.588288i	\(0.799802\pi\)