Embedded newform 75.6.i.a.4.9

• Label: 75.6.i.a.4.9
• Level: $75$
• Weight: $6$
• Character: 75.4
• Analytic conductor: $12.029$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.i (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.0287864860\)
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			4.9
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.40554 - 1.10653i) q^{2} +(5.29007 - 7.28115i) q^{3} +(-15.5152 - 11.2725i) q^{4} +(54.3939 - 12.8960i) q^{5} +(-26.0723 + 18.9427i) q^{6} +236.745i q^{7} +(107.716 + 148.259i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\)	\(26\)	\(52\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{10}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.40554	−	1.10653i	−0.602020	−	0.195608i	−0.00787907	−	0.999969i	\(-0.502508\pi\)
							−0.594141	+	0.804361i	\(0.702508\pi\)
\(3\)	5.29007	−	7.28115i	0.339358	−	0.467086i
							\(5\)	54.3939	−	12.8960i	0.973027	−	0.230690i
\(7\)			236.745i			1.82615i								0.407793	+	0.913074i	\(0.366299\pi\)
							−0.407793	+	0.913074i	\(0.633701\pi\)
\(11\)	38.5651	−	118.691i	0.0960975	−	0.295758i	−0.891441	−	0.453137i	\(-0.850305\pi\)
							0.987538	+	0.157380i	\(0.0503047\pi\)
\(13\)	244.815	−	79.5452i	0.401772	−	0.130544i	−0.101159	−	0.994870i	\(-0.532255\pi\)
							0.502931	+	0.864327i	\(0.332255\pi\)
\(17\)	503.811	+	693.436i	0.422810	+	0.581948i	0.966284	−	0.257478i	\(-0.0828915\pi\)
							−0.543474	+	0.839426i	\(0.682891\pi\)
\(19\)	856.708	−	622.435i	0.544439	−	0.395558i	−0.281292	−	0.959622i	\(-0.590763\pi\)
							0.825731	+	0.564064i	\(0.190763\pi\)
\(23\)	2183.89	+	709.590i	0.860819	+	0.279697i	0.705971	−	0.708241i	\(-0.250511\pi\)
							0.154848	+	0.987938i	\(0.450511\pi\)
\(29\)	−247.467	−	179.795i	−0.0546415	−	0.0396993i	0.560129	−	0.828405i	\(-0.310751\pi\)
							−0.614771	+	0.788706i	\(0.710751\pi\)
\(31\)	3527.40	−	2562.81i	0.659251	−	0.478974i	−0.207159	−	0.978307i	\(-0.566422\pi\)
							0.866410	+	0.499334i	\(0.166422\pi\)
\(37\)	1728.74	−	561.702i	0.207599	−	0.0674531i	−0.203371	−	0.979102i	\(-0.565190\pi\)
							0.410971	+	0.911649i	\(0.365190\pi\)
\(41\)	6418.72	+	19754.8i	0.596333	+	1.83532i	0.547975	+	0.836494i	\(0.315399\pi\)
							0.0483577	+	0.998830i	\(0.484601\pi\)
\(43\)		−	17486.4i		−	1.44222i	−0.692823	−	0.721108i	\(-0.743633\pi\)
							0.692823	−	0.721108i	\(-0.256367\pi\)
\(47\)	3889.21	−	5353.04i	0.256813	−	0.353473i	−0.661070	−	0.750325i	\(-0.729897\pi\)
							0.917883	+	0.396852i	\(0.129897\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.9	✓	96
25.19	even	10	inner	75.6.i.a.19.9	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.9	✓	96	1.1	even	1	trivial
75.6.i.a.19.9	yes	96	25.19	even	10	inner