Embedded newform 75.6.i.a.4.13

• Label: 75.6.i.a.4.13
• 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			4.13
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.222946 - 0.0724396i) q^{2} +(-5.29007 + 7.28115i) q^{3} +(-25.8441 - 18.7768i) q^{4} +(-31.5757 - 46.1300i) q^{5} +(1.70684 - 1.24009i) q^{6} +161.430i q^{7} +(8.81087 + 12.1271i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.222946	−	0.0724396i	−0.0394117	−	0.0128056i	0.289245	−	0.957255i	\(-0.406596\pi\)
							−0.328656	+	0.944450i	\(0.606596\pi\)
\(3\)	−5.29007	+	7.28115i	−0.339358	+	0.467086i
							\(5\)	−31.5757	−	46.1300i	−0.564844	−	0.825198i
\(7\)			161.430i			1.24520i								0.782540	+	0.622600i	\(0.213924\pi\)
							−0.782540	+	0.622600i	\(0.786076\pi\)
\(11\)	38.3620	−	118.066i	0.0955916	−	0.294201i	−0.891816	−	0.452399i	\(-0.850568\pi\)
							0.987407	+	0.158198i	\(0.0505684\pi\)
\(13\)	859.616	−	279.306i	1.41074	−	0.458376i	0.498090	−	0.867125i	\(-0.334035\pi\)
							0.912647	+	0.408749i	\(0.134035\pi\)
\(17\)	529.936	+	729.394i	0.444735	+	0.612125i	0.971256	−	0.238037i	\(-0.0765040\pi\)
							−0.526521	+	0.850162i	\(0.676504\pi\)
\(19\)	−597.358	+	434.006i	−0.379621	+	0.275811i	−0.761189	−	0.648530i	\(-0.775384\pi\)
							0.381568	+	0.924341i	\(0.375384\pi\)
\(23\)	2851.89	+	926.636i	1.12412	+	0.365250i	0.811340	−	0.584575i	\(-0.198739\pi\)
							0.312783	+	0.949825i	\(0.398739\pi\)
\(29\)	2074.33	+	1507.09i	0.458019	+	0.332770i	0.792754	−	0.609542i	\(-0.208647\pi\)
							−0.334735	+	0.942312i	\(0.608647\pi\)
\(31\)	5208.20	−	3783.98i	0.973382	−	0.707204i	0.0171625	−	0.999853i	\(-0.494537\pi\)
							0.956220	+	0.292649i	\(0.0945367\pi\)
\(37\)	−9453.31	+	3071.57i	−1.13522	+	0.368855i	−0.815557	−	0.578676i	\(-0.803569\pi\)
							−0.319662	+	0.947532i	\(0.603569\pi\)
\(41\)	−3145.63	−	9681.24i	−0.292245	−	0.899439i	−0.984133	−	0.177433i	\(-0.943221\pi\)
							0.691888	−	0.722005i	\(-0.256779\pi\)
\(43\)			14650.8i			1.20834i	0.796854	+	0.604171i	\(0.206496\pi\)
							−0.796854	+	0.604171i	\(0.793504\pi\)
\(47\)	9750.01	−	13419.7i	0.643814	−	0.886134i	−0.354998	−	0.934867i	\(-0.615518\pi\)
							0.998812	+	0.0487331i	\(0.0155184\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.13	✓	96
25.19	even	10	inner	75.6.i.a.19.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.13	✓	96	1.1	even	1	trivial
75.6.i.a.19.13	yes	96	25.19	even	10	inner