Embedded newform 95.11.m.a.7.14

• Label: 95.11.m.a.7.14
• Level: $95$
• Weight: $11$
• Character: 95.7
• Analytic conductor: $60.359$
• Analytic rank: $0$
• Dimension: $392$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 95 = 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	95.m (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(60.3589390040\)
Analytic rank:	\(0\)
Dimension:	\(392\)
Relative dimension:	\(98\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.4652 + 46.5207i) q^{2} +(325.085 + 87.1063i) q^{3} +(-1121.98 - 647.777i) q^{4} +(-2409.61 - 1989.82i) q^{5} +(-8104.49 + 14037.4i) q^{6} +(8414.06 + 8414.06i) q^{7} +(9247.91 - 9247.91i) q^{8} +(46954.8 + 27109.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\)	\(21\)	\(77\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{4}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.4652	+	46.5207i	−0.389537	+	1.45377i	0.441353	+	0.897334i	\(0.354499\pi\)
							−0.830890	+	0.556437i	\(0.812168\pi\)
\(3\)	325.085	+	87.1063i	1.33780	+	0.358462i	0.855617	−	0.517609i	\(-0.173178\pi\)
							0.482181	+	0.876071i	\(0.339845\pi\)
\(5\)	−2409.61	−	1989.82i	−0.771076	−	0.636743i
							\(7\)	8414.06	+	8414.06i	0.500628	+	0.500628i	0.911633	−	0.411005i	\(-0.134822\pi\)
−0.411005	+	0.911633i	\(0.634822\pi\)
\(11\)	−73729.8			−0.457804			−0.228902	−	0.973449i	\(-0.573514\pi\)
							−0.228902	+	0.973449i	\(0.573514\pi\)
\(13\)	−2125.50	−	7932.48i	−0.00572459	−	0.0213645i	0.963004	−	0.269485i	\(-0.0868536\pi\)
							−0.968729	+	0.248121i	\(0.920187\pi\)
\(17\)	−647345.	+	2.41592e6i	−0.455923	+	1.70153i	0.229438	+	0.973323i	\(0.426311\pi\)
							−0.685361	+	0.728204i	\(0.740356\pi\)
\(19\)	−103272.	+	2.47394e6i	−0.0417076	+	0.999130i
							\(23\)	−3.11538e6	−	1.16267e7i	−0.484029	−	1.80642i	−0.584394	−	0.811470i	\(-0.698668\pi\)
0.100365	−	0.994951i	\(-0.467999\pi\)
\(29\)	−7.39283e6	−	4.26825e6i	−0.360430	−	0.208094i	0.308840	−	0.951114i	\(-0.400059\pi\)
							−0.669269	+	0.743020i	\(0.733393\pi\)
\(31\)	2.65075e7			0.925891			0.462946	−	0.886387i	\(-0.346792\pi\)
							0.462946	+	0.886387i	\(0.346792\pi\)
\(37\)	8.30672e6	+	8.30672e6i	0.119790	+	0.119790i	0.764461	−	0.644670i	\(-0.223005\pi\)
							−0.644670	+	0.764461i	\(0.723005\pi\)
\(41\)	−9.81102e7	−	1.69932e8i	−0.846827	−	1.46675i	−0.884025	−	0.467439i	\(-0.845177\pi\)
							0.0371982	−	0.999308i	\(-0.488157\pi\)
\(43\)	−3.01418e7	−	8.07647e6i	−0.205035	−	0.0549388i	0.154840	−	0.987940i	\(-0.450514\pi\)
							−0.359874	+	0.933001i	\(0.617180\pi\)
\(47\)	−2.77126e8	+	7.42557e7i	−1.20834	+	0.323773i	−0.806109	−	0.591767i	\(-0.798431\pi\)
							−0.402227	+	0.915540i	\(0.631764\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.14	✓	392
5.3	odd	4	inner	95.11.m.a.83.85	yes	392
19.11	even	3	inner	95.11.m.a.87.85	yes	392
95.68	odd	12	inner	95.11.m.a.68.14	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.14	✓	392	1.1	even	1	trivial
95.11.m.a.68.14	yes	392	95.68	odd	12	inner
95.11.m.a.83.85	yes	392	5.3	odd	4	inner
95.11.m.a.87.85	yes	392	19.11	even	3	inner