Embedded newform 75.6.i.a.4.16

• Label: 75.6.i.a.4.16
• 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			4.16
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.94639 + 1.28226i) q^{2} +(5.29007 - 7.28115i) q^{3} +(-11.9588 - 8.68855i) q^{4} +(54.1701 + 13.8059i) q^{5} +(30.2130 - 21.9510i) q^{6} -87.4923i q^{7} +(-114.101 - 157.047i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.94639	+	1.28226i	0.697629	+	0.226673i	0.636297	−	0.771444i	\(-0.280465\pi\)
							0.0613320	+	0.998117i	\(0.480465\pi\)
\(3\)	5.29007	−	7.28115i	0.339358	−	0.467086i
							\(5\)	54.1701	+	13.8059i	0.969024	+	0.246967i
\(7\)		−	87.4923i		−	0.674877i								−0.941348	−	0.337438i	\(-0.890440\pi\)
							0.941348	−	0.337438i	\(-0.109560\pi\)
\(11\)	10.4195	−	32.0679i	0.0259636	−	0.0799077i	−0.937235	−	0.348698i	\(-0.886624\pi\)
							0.963199	+	0.268790i	\(0.0866239\pi\)
\(13\)	491.123	−	159.575i	0.805994	−	0.261883i	0.123094	−	0.992395i	\(-0.460718\pi\)
							0.682900	+	0.730512i	\(0.260718\pi\)
\(17\)	−1074.13	−	1478.42i	−0.901439	−	1.24072i	−0.970007	−	0.243078i	\(-0.921843\pi\)
							0.0685680	−	0.997646i	\(-0.478157\pi\)
\(19\)	1019.63	−	740.801i	0.647972	−	0.470780i	−0.214607	−	0.976700i	\(-0.568847\pi\)
							0.862580	+	0.505921i	\(0.168847\pi\)
\(23\)	610.065	+	198.222i	0.240468	+	0.0781326i	0.426771	−	0.904360i	\(-0.359651\pi\)
							−0.186304	+	0.982492i	\(0.559651\pi\)
\(29\)	2226.68	+	1617.78i	0.491659	+	0.357211i	0.805822	−	0.592158i	\(-0.201724\pi\)
							−0.314163	+	0.949369i	\(0.601724\pi\)
\(31\)	413.654	−	300.537i	0.0773096	−	0.0561687i	−0.548459	−	0.836177i	\(-0.684785\pi\)
							0.625769	+	0.780009i	\(0.284785\pi\)
\(37\)	−1433.05	+	465.625i	−0.172090	+	0.0559155i	−0.393795	−	0.919198i	\(-0.628838\pi\)
							0.221705	+	0.975114i	\(0.428838\pi\)
\(41\)	328.517	+	1011.07i	0.0305210	+	0.0939340i	0.965157	−	0.261673i	\(-0.0842742\pi\)
							−0.934636	+	0.355607i	\(0.884274\pi\)
\(43\)			14553.7i			1.20033i	0.799875	+	0.600167i	\(0.204899\pi\)
							−0.799875	+	0.600167i	\(0.795101\pi\)
\(47\)	−16772.5	+	23085.4i	−1.10752	+	1.52438i	−0.282512	+	0.959264i	\(0.591168\pi\)
							−0.825013	+	0.565113i	\(0.808832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.16	✓	96
25.19	even	10	inner	75.6.i.a.19.16	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.16	✓	96	1.1	even	1	trivial
75.6.i.a.19.16	yes	96	25.19	even	10	inner