Embedded newform 95.11.m.a.7.13

• Label: 95.11.m.a.7.13
• 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			7.13
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-12.5417 + 46.8064i) q^{2} +(-107.569 - 28.8232i) q^{3} +(-1146.74 - 662.069i) q^{4} +(2321.76 - 2091.66i) q^{5} +(2698.22 - 4673.45i) q^{6} +(4809.27 + 4809.27i) q^{7} +(10284.1 - 10284.1i) q^{8} +(-40397.5 - 23323.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−12.5417	+	46.8064i	−0.391930	+	1.46270i	0.435018	+	0.900422i	\(0.356742\pi\)
							−0.826947	+	0.562279i	\(0.809925\pi\)
\(3\)	−107.569	−	28.8232i	−0.442673	−	0.118614i	0.0305961	−	0.999532i	\(-0.490259\pi\)
							−0.473269	+	0.880918i	\(0.656926\pi\)
\(5\)	2321.76	−	2091.66i	0.742965	−	0.669331i
							\(7\)	4809.27	+	4809.27i	0.286147	+	0.286147i	0.835554	−	0.549408i	\(-0.185147\pi\)
−0.549408	+	0.835554i	\(0.685147\pi\)
\(11\)	−211557.			−1.31360			−0.656802	−	0.754063i	\(-0.728091\pi\)
							−0.656802	+	0.754063i	\(0.728091\pi\)
\(13\)	−105291.	−	392952.i	−0.283580	−	1.05833i	−0.949871	−	0.312642i	\(-0.898786\pi\)
							0.666292	−	0.745691i	\(-0.267881\pi\)
\(17\)	−606953.	+	2.26518e6i	−0.427475	+	1.59536i	0.330984	+	0.943636i	\(0.392619\pi\)
							−0.758459	+	0.651721i	\(0.774047\pi\)
\(19\)	−2.26533e6	−	999674.i	−0.914878	−	0.403730i
							\(23\)	315237.	+	1.17648e6i	0.0489776	+	0.182787i	0.986081	−	0.166264i	\(-0.0531705\pi\)
−0.937104	+	0.349051i	\(0.886504\pi\)
\(29\)	−6.85010e6	−	3.95490e6i	−0.333969	−	0.192817i	0.323633	−	0.946183i	\(-0.395096\pi\)
							−0.657602	+	0.753365i	\(0.728429\pi\)
\(31\)	8.32434e6			0.290764			0.145382	−	0.989376i	\(-0.453559\pi\)
							0.145382	+	0.989376i	\(0.453559\pi\)
\(37\)	7.33590e7	+	7.33590e7i	1.05790	+	1.05790i	0.998217	+	0.0596836i	\(0.0190092\pi\)
							0.0596836	+	0.998217i	\(0.480991\pi\)
\(41\)	9.66563e7	+	1.67414e8i	0.834278	+	1.44501i	0.894617	+	0.446834i	\(0.147449\pi\)
							−0.0603390	+	0.998178i	\(0.519218\pi\)
\(43\)	7.24936e7	+	1.94246e7i	0.493126	+	0.132133i	0.496808	−	0.867861i	\(-0.334505\pi\)
							−0.00368210	+	0.999993i	\(0.501172\pi\)
\(47\)	1.29754e8	−	3.47676e7i	0.565761	−	0.151595i	0.0354093	−	0.999373i	\(-0.488727\pi\)
							0.530352	+	0.847778i	\(0.322060\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.13	✓	392
5.3	odd	4	inner	95.11.m.a.83.86	yes	392
19.11	even	3	inner	95.11.m.a.87.86	yes	392
95.68	odd	12	inner	95.11.m.a.68.13	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.13	✓	392	1.1	even	1	trivial
95.11.m.a.68.13	yes	392	95.68	odd	12	inner
95.11.m.a.83.86	yes	392	5.3	odd	4	inner
95.11.m.a.87.86	yes	392	19.11	even	3	inner