Embedded newform 95.11.m.a.7.10

• Label: 95.11.m.a.7.10
• 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			7.10
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.5993 + 50.7533i) q^{2} +(410.062 + 109.876i) q^{3} +(-1504.15 - 868.420i) q^{4} +(1812.79 - 2545.47i) q^{5} +(-11153.1 + 19317.8i) q^{6} +(-7833.14 - 7833.14i) q^{7} +(26484.9 - 26484.9i) q^{8} +(104940. + 60587.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.5993	+	50.7533i	−0.424978	+	1.58604i	0.338992	+	0.940789i	\(0.389914\pi\)
							−0.763970	+	0.645252i	\(0.776753\pi\)
\(3\)	410.062	+	109.876i	1.68750	+	0.452164i	0.969742	−	0.244132i	\(-0.0785030\pi\)
							0.717755	+	0.696296i	\(0.245170\pi\)
\(5\)	1812.79	−	2545.47i	0.580093	−	0.814550i
							\(7\)	−7833.14	−	7833.14i	−0.466064	−	0.466064i	0.434573	−	0.900637i	\(-0.356899\pi\)
−0.900637	+	0.434573i	\(0.856899\pi\)
\(11\)	37153.7			0.230695			0.115348	−	0.993325i	\(-0.463202\pi\)
							0.115348	+	0.993325i	\(0.463202\pi\)
\(13\)	−43015.9	−	160538.i	−0.115854	−	0.432374i	0.883495	−	0.468441i	\(-0.155184\pi\)
							−0.999349	+	0.0360661i	\(0.988517\pi\)
\(17\)	165762.	−	618631.i	0.116745	−	0.435700i	−0.882666	−	0.470000i	\(-0.844254\pi\)
							0.999412	+	0.0343008i	\(0.0109204\pi\)
\(19\)	−16475.5	−	2.47604e6i	−0.00665380	−	0.999978i
							\(23\)	−537002.	−	2.00412e6i	−0.0834328	−	0.311375i	0.911580	−	0.411123i	\(-0.134863\pi\)
−0.995013	+	0.0997473i	\(0.968197\pi\)
\(29\)	4.51346e6	+	2.60585e6i	0.220049	+	0.127045i	0.605973	−	0.795485i	\(-0.292784\pi\)
							−0.385924	+	0.922531i	\(0.626117\pi\)
\(31\)	1.06781e7			0.372980			0.186490	−	0.982457i	\(-0.440289\pi\)
							0.186490	+	0.982457i	\(0.440289\pi\)
\(37\)	−4.02366e7	−	4.02366e7i	−0.580247	−	0.580247i	0.354724	−	0.934971i	\(-0.384575\pi\)
							−0.934971	+	0.354724i	\(0.884575\pi\)
\(41\)	−8.25314e7	−	1.42949e8i	−0.712360	−	1.23384i	−0.963969	−	0.266015i	\(-0.914293\pi\)
							0.251609	−	0.967829i	\(-0.419040\pi\)
\(43\)	2.73724e8	+	7.33442e7i	1.86196	+	0.498911i	0.999968	−	0.00800706i	\(-0.00254875\pi\)
							0.861994	+	0.506918i	\(0.169215\pi\)
\(47\)	2.02001e8	−	5.41260e7i	0.880774	−	0.236003i	0.210033	−	0.977694i	\(-0.432643\pi\)
							0.670741	+	0.741692i	\(0.265976\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.10	✓	392
5.3	odd	4	inner	95.11.m.a.83.89	yes	392
19.11	even	3	inner	95.11.m.a.87.89	yes	392
95.68	odd	12	inner	95.11.m.a.68.10	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.10	✓	392	1.1	even	1	trivial
95.11.m.a.68.10	yes	392	95.68	odd	12	inner
95.11.m.a.83.89	yes	392	5.3	odd	4	inner
95.11.m.a.87.89	yes	392	19.11	even	3	inner