Embedded newform 75.6.i.a.4.1

• Label: 75.6.i.a.4.1
• 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			4.1
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-10.3671 - 3.36849i) q^{2} +(-5.29007 + 7.28115i) q^{3} +(70.2423 + 51.0340i) q^{4} +(-12.9290 - 54.3860i) q^{5} +(79.3693 - 57.6652i) q^{6} +32.5585i q^{7} +(-351.272 - 483.484i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−10.3671	−	3.36849i	−1.83267	−	0.595470i	−0.999071	−	0.0430969i	\(-0.986278\pi\)
							−0.833597	−	0.552373i	\(-0.813722\pi\)
\(3\)	−5.29007	+	7.28115i	−0.339358	+	0.467086i
							\(5\)	−12.9290	−	54.3860i	−0.231281	−	0.972887i
\(7\)			32.5585i			0.251142i								0.992085	+	0.125571i	\(0.0400763\pi\)
							−0.992085	+	0.125571i	\(0.959924\pi\)
\(11\)	−135.953	+	418.419i	−0.338771	+	1.04263i	0.626064	+	0.779772i	\(0.284665\pi\)
							−0.964835	+	0.262857i	\(0.915335\pi\)
\(13\)	−476.063	+	154.682i	−0.781278	+	0.253853i	−0.672386	−	0.740201i	\(-0.734730\pi\)
							−0.108892	+	0.994054i	\(0.534730\pi\)
\(17\)	−968.699	−	1333.30i	−0.812955	−	1.11894i	−0.990861	−	0.134889i	\(-0.956932\pi\)
							0.177905	−	0.984048i	\(-0.443068\pi\)
\(19\)	605.637	−	440.021i	0.384883	−	0.279634i	−0.378472	−	0.925613i	\(-0.623551\pi\)
							0.763355	+	0.645979i	\(0.223551\pi\)
\(23\)	2524.19	+	820.160i	0.994954	+	0.323280i	0.760847	−	0.648931i	\(-0.224784\pi\)
							0.234107	+	0.972211i	\(0.424784\pi\)
\(29\)	2414.92	+	1754.54i	0.533221	+	0.387408i	0.821561	−	0.570120i	\(-0.193103\pi\)
							−0.288340	+	0.957528i	\(0.593103\pi\)
\(31\)	6450.77	−	4686.76i	1.20561	−	0.875928i	0.210787	−	0.977532i	\(-0.432397\pi\)
							0.994825	+	0.101604i	\(0.0323974\pi\)
\(37\)	4805.77	−	1561.49i	0.577110	−	0.187515i	−0.00589529	−	0.999983i	\(-0.501877\pi\)
							0.583006	+	0.812468i	\(0.301877\pi\)
\(41\)	6290.78	+	19361.0i	0.584447	+	1.79874i	0.601480	+	0.798888i	\(0.294578\pi\)
							−0.0170332	+	0.999855i	\(0.505422\pi\)
\(43\)			14294.8i			1.17898i	0.807776	+	0.589490i	\(0.200671\pi\)
							−0.807776	+	0.589490i	\(0.799329\pi\)
\(47\)	11084.7	−	15256.8i	0.731949	−	1.00744i	−0.267093	−	0.963671i	\(-0.586063\pi\)
							0.999042	−	0.0437705i	\(-0.0139370\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.1	✓	96
25.19	even	10	inner	75.6.i.a.19.1	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.1	✓	96	1.1	even	1	trivial
75.6.i.a.19.1	yes	96	25.19	even	10	inner