Embedded newform 75.6.i.a.4.4

• Label: 75.6.i.a.4.4
• 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			4.4
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-8.39483 - 2.72764i) q^{2} +(-5.29007 + 7.28115i) q^{3} +(37.1445 + 26.9871i) q^{4} +(47.9744 + 28.6959i) q^{5} +(64.2696 - 46.6946i) q^{6} +34.5534i q^{7} +(-72.1854 - 99.3547i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−8.39483	−	2.72764i	−1.48401	−	0.482184i	−0.548702	−	0.836018i	\(-0.684878\pi\)
							−0.935308	+	0.353834i	\(0.884878\pi\)
\(3\)	−5.29007	+	7.28115i	−0.339358	+	0.467086i
							\(5\)	47.9744	+	28.6959i	0.858193	+	0.513327i
\(7\)			34.5534i			0.266530i								0.991080	+	0.133265i	\(0.0425461\pi\)
							−0.991080	+	0.133265i	\(0.957454\pi\)
\(11\)	134.452	−	413.800i	0.335031	−	1.03112i	−0.631676	−	0.775233i	\(-0.717633\pi\)
							0.966707	−	0.255887i	\(-0.0823675\pi\)
\(13\)	−263.349	+	85.5673i	−0.432189	+	0.140427i	−0.517029	−	0.855968i	\(-0.672962\pi\)
							0.0848402	+	0.996395i	\(0.472962\pi\)
\(17\)	706.329	+	972.178i	0.592768	+	0.815875i	0.995022	−	0.0996531i	\(-0.0317733\pi\)
							−0.402255	+	0.915528i	\(0.631773\pi\)
\(19\)	−484.513	+	352.019i	−0.307908	+	0.223708i	−0.730998	−	0.682379i	\(-0.760945\pi\)
							0.423090	+	0.906088i	\(0.360945\pi\)
\(23\)	3211.62	+	1043.52i	1.26592	+	0.411321i	0.863599	−	0.504180i	\(-0.168205\pi\)
							0.402316	+	0.915501i	\(0.368205\pi\)
\(29\)	2172.17	+	1578.17i	0.479621	+	0.348465i	0.801179	−	0.598425i	\(-0.204206\pi\)
							−0.321558	+	0.946890i	\(0.604206\pi\)
\(31\)	−3220.63	+	2339.93i	−0.601918	+	0.437319i	−0.846559	−	0.532295i	\(-0.821330\pi\)
							0.244641	+	0.969614i	\(0.421330\pi\)
\(37\)	−15440.4	+	5016.88i	−1.85419	+	0.602462i	−0.858162	+	0.513379i	\(0.828393\pi\)
							−0.996024	+	0.0890825i	\(0.971607\pi\)
\(41\)	−364.662	−	1122.31i	−0.0338790	−	0.104269i	0.932687	−	0.360687i	\(-0.117458\pi\)
							−0.966566	+	0.256418i	\(0.917458\pi\)
\(43\)			4267.34i			0.351954i	0.984394	+	0.175977i	\(0.0563084\pi\)
							−0.984394	+	0.175977i	\(0.943692\pi\)
\(47\)	−5085.86	+	7000.09i	−0.335830	+	0.462231i	−0.943218	−	0.332175i	\(-0.892218\pi\)
							0.607387	+	0.794406i	\(0.292218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.4	✓	96
25.19	even	10	inner	75.6.i.a.19.4	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.4	✓	96	1.1	even	1	trivial
75.6.i.a.19.4	yes	96	25.19	even	10	inner