Embedded newform 95.11.m.a.7.4

• Label: 95.11.m.a.7.4
• 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			7.4
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.6303 + 58.3330i) q^{2} +(213.494 + 57.2056i) q^{3} +(-2271.63 - 1311.52i) q^{4} +(2345.18 + 2065.38i) q^{5} +(-6673.95 + 11559.6i) q^{6} +(-12299.4 - 12299.4i) q^{7} +(68283.7 - 68283.7i) q^{8} +(-8830.70 - 5098.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.6303	+	58.3330i	−0.488447	+	1.82291i	0.0755658	+	0.997141i	\(0.475924\pi\)
							−0.564012	+	0.825766i	\(0.690743\pi\)
\(3\)	213.494	+	57.2056i	0.878576	+	0.235414i	0.669793	−	0.742548i	\(-0.266383\pi\)
							0.208784	+	0.977962i	\(0.433050\pi\)
\(5\)	2345.18	+	2065.38i	0.750456	+	0.660920i
							\(7\)	−12299.4	−	12299.4i	−0.731801	−	0.731801i	0.239175	−	0.970976i	\(-0.423123\pi\)
−0.970976	+	0.239175i	\(0.923123\pi\)
\(11\)	−55180.6			−0.342628			−0.171314	−	0.985216i	\(-0.554801\pi\)
							−0.171314	+	0.985216i	\(0.554801\pi\)
\(13\)	−35387.8	−	132069.i	−0.0953097	−	0.355701i	0.901757	−	0.432244i	\(-0.142278\pi\)
							−0.997066	+	0.0765436i	\(0.975612\pi\)
\(17\)	102339.	−	381935.i	0.0720771	−	0.268995i	−0.920478	−	0.390795i	\(-0.872200\pi\)
							0.992555	+	0.121800i	\(0.0388666\pi\)
\(19\)	−1.21175e6	+	2.15934e6i	−0.489378	+	0.872072i
							\(23\)	−3.12296e6	−	1.16550e7i	−0.485207	−	1.81082i	−0.579129	−	0.815236i	\(-0.696607\pi\)
0.0939219	−	0.995580i	\(-0.470060\pi\)
\(29\)	2.25586e7	+	1.30242e7i	1.09982	+	0.634984i	0.936175	−	0.351535i	\(-0.114340\pi\)
							0.163649	+	0.986519i	\(0.447674\pi\)
\(31\)	1.84801e7			0.645500			0.322750	−	0.946484i	\(-0.395393\pi\)
							0.322750	+	0.946484i	\(0.395393\pi\)
\(37\)	7.95644e7	+	7.95644e7i	1.14739	+	1.14739i	0.987065	+	0.160323i	\(0.0512535\pi\)
							0.160323	+	0.987065i	\(0.448747\pi\)
\(41\)	8.82741e7	+	1.52895e8i	0.761928	+	1.31970i	0.941855	+	0.336018i	\(0.109080\pi\)
							−0.179927	+	0.983680i	\(0.557586\pi\)
\(43\)	1.04007e8	+	2.78687e7i	0.707492	+	0.189572i	0.594584	−	0.804033i	\(-0.297317\pi\)
							0.112908	+	0.993605i	\(0.463983\pi\)
\(47\)	3.67755e8	−	9.85396e7i	1.60350	−	0.429657i	0.657403	−	0.753539i	\(-0.271655\pi\)
							0.946097	+	0.323882i	\(0.104988\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.4	✓	392
5.3	odd	4	inner	95.11.m.a.83.95	yes	392
19.11	even	3	inner	95.11.m.a.87.95	yes	392
95.68	odd	12	inner	95.11.m.a.68.4	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.4	✓	392	1.1	even	1	trivial
95.11.m.a.68.4	yes	392	95.68	odd	12	inner
95.11.m.a.83.95	yes	392	5.3	odd	4	inner
95.11.m.a.87.95	yes	392	19.11	even	3	inner