Embedded newform 75.6.i.a.4.15

• Label: 75.6.i.a.4.15
• 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			4.15
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.74584 + 0.892176i) q^{2} +(5.29007 - 7.28115i) q^{3} +(-19.1449 - 13.9096i) q^{4} +(-10.5312 + 54.9008i) q^{5} +(21.0217 - 15.2732i) q^{6} +106.453i q^{7} +(-94.4636 - 130.018i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.74584	+	0.892176i	0.485400	+	0.157716i	0.541486	−	0.840710i	\(-0.317862\pi\)
							−0.0560859	+	0.998426i	\(0.517862\pi\)
\(3\)	5.29007	−	7.28115i	0.339358	−	0.467086i
							\(5\)	−10.5312	+	54.9008i	−0.188388	+	0.982095i
\(7\)			106.453i			0.821132i								0.911831	+	0.410566i	\(0.134669\pi\)
							−0.911831	+	0.410566i	\(0.865331\pi\)
\(11\)	−123.256	+	379.344i	−0.307134	+	0.945260i	0.671739	+	0.740788i	\(0.265548\pi\)
							−0.978872	+	0.204472i	\(0.934452\pi\)
\(13\)	−410.419	+	133.353i	−0.673548	+	0.218849i	−0.625769	−	0.780009i	\(-0.715215\pi\)
							−0.0477797	+	0.998858i	\(0.515215\pi\)
\(17\)	925.834	+	1274.30i	0.776981	+	1.06942i	0.995609	+	0.0936146i	\(0.0298422\pi\)
							−0.218627	+	0.975808i	\(0.570158\pi\)
\(19\)	−1294.80	+	940.731i	−0.822849	+	0.597835i	−0.917527	−	0.397673i	\(-0.869818\pi\)
							0.0946781	+	0.995508i	\(0.469818\pi\)
\(23\)	936.080	+	304.151i	0.368972	+	0.119886i	0.487633	−	0.873049i	\(-0.337860\pi\)
							−0.118661	+	0.992935i	\(0.537860\pi\)
\(29\)	−3163.59	−	2298.48i	−0.698529	−	0.507511i	0.180924	−	0.983497i	\(-0.442091\pi\)
							−0.879453	+	0.475986i	\(0.842091\pi\)
\(31\)	−148.610	+	107.972i	−0.0277744	+	0.0201793i	−0.601586	−	0.798808i	\(-0.705464\pi\)
							0.573811	+	0.818987i	\(0.305464\pi\)
\(37\)	1083.45	−	352.035i	0.130108	−	0.0422748i	−0.243239	−	0.969966i	\(-0.578210\pi\)
							0.373347	+	0.927692i	\(0.378210\pi\)
\(41\)	−4068.78	−	12522.4i	−0.378011	−	1.16340i	−0.941424	−	0.337224i	\(-0.890512\pi\)
							0.563413	−	0.826175i	\(-0.309488\pi\)
\(43\)			510.913i			0.0421382i	0.999778	+	0.0210691i	\(0.00670700\pi\)
							−0.999778	+	0.0210691i	\(0.993293\pi\)
\(47\)	−1969.79	+	2711.18i	−0.130069	+	0.179025i	−0.869084	−	0.494664i	\(-0.835291\pi\)
							0.739015	+	0.673689i	\(0.235291\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.15	✓	96
25.19	even	10	inner	75.6.i.a.19.15	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.15	✓	96	1.1	even	1	trivial
75.6.i.a.19.15	yes	96	25.19	even	10	inner