Embedded newform 75.6.i.a.4.7

• Label: 75.6.i.a.4.7
• 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			4.7
Character	\(\chi\)	\(=\)	75.4
Dual form			75.6.i.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.73235 - 1.53763i) q^{2} +(-5.29007 + 7.28115i) q^{3} +(-5.85773 - 4.25589i) q^{4} +(-49.1135 - 26.6995i) q^{5} +(36.2302 - 26.3228i) q^{6} -228.046i q^{7} +(114.769 + 157.966i) q^{8} +(-25.0304 - 77.0356i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 352 q^{4} - 120 q^{5} + 72 q^{6} - 1230 q^{8} + 1944 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.73235	−	1.53763i	−0.836569	−	0.271818i	−0.140759	−	0.990044i	\(-0.544954\pi\)
							−0.695810	+	0.718226i	\(0.744954\pi\)
\(3\)	−5.29007	+	7.28115i	−0.339358	+	0.467086i
							\(5\)	−49.1135	−	26.6995i	−0.878569	−	0.477615i
\(7\)		−	228.046i		−	1.75905i								−0.475855	−	0.879524i	\(-0.657861\pi\)
							0.475855	−	0.879524i	\(-0.342139\pi\)
\(11\)	63.9875	−	196.933i	0.159446	−	0.490724i	−0.839138	−	0.543918i	\(-0.816940\pi\)
							0.998584	+	0.0531943i	\(0.0169403\pi\)
\(13\)	−622.052	+	202.117i	−1.02087	+	0.331699i	−0.771173	−	0.636626i	\(-0.780330\pi\)
							−0.249693	+	0.968325i	\(0.580330\pi\)
\(17\)	−61.9509	−	85.2681i	−0.0519907	−	0.0715590i	0.782230	−	0.622990i	\(-0.214082\pi\)
							−0.834221	+	0.551431i	\(0.814082\pi\)
\(19\)	−1629.48	+	1183.88i	−1.03553	+	0.752359i	−0.969409	−	0.245452i	\(-0.921064\pi\)
							−0.0661250	+	0.997811i	\(0.521064\pi\)
\(23\)	1586.12	+	515.361i	0.625195	+	0.203138i	0.604446	−	0.796646i	\(-0.293395\pi\)
							0.0207498	+	0.999785i	\(0.493395\pi\)
\(29\)	4297.68	+	3122.45i	0.948941	+	0.689446i	0.950556	−	0.310553i	\(-0.100514\pi\)
							−0.00161519	+	0.999999i	\(0.500514\pi\)
\(31\)	−5770.51	+	4192.52i	−1.07847	+	0.783558i	−0.977416	−	0.211323i	\(-0.932223\pi\)
							−0.101058	+	0.994881i	\(0.532223\pi\)
\(37\)	9797.24	−	3183.32i	1.17652	−	0.382275i	0.345447	−	0.938438i	\(-0.387727\pi\)
							0.831073	+	0.556163i	\(0.187727\pi\)
\(41\)	2011.33	+	6190.24i	0.186863	+	0.575106i	0.999975	−	0.00700121i	\(-0.00222857\pi\)
							−0.813112	+	0.582107i	\(0.802229\pi\)
\(43\)		−	18835.1i		−	1.55345i	−0.629838	−	0.776726i	\(-0.716879\pi\)
							0.629838	−	0.776726i	\(-0.283121\pi\)
\(47\)	−3963.19	+	5454.86i	−0.261698	+	0.360196i	−0.919565	−	0.392938i	\(-0.871459\pi\)
							0.657867	+	0.753134i	\(0.271459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.i.a.4.7	✓	96
25.19	even	10	inner	75.6.i.a.19.7	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.6.i.a.4.7	✓	96	1.1	even	1	trivial
75.6.i.a.19.7	yes	96	25.19	even	10	inner