Embedded newform 95.11.m.a.7.3

• Label: 95.11.m.a.7.3
• 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			7.3
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-15.7051 + 58.6121i) q^{2} +(-383.064 - 102.642i) q^{3} +(-2301.92 - 1329.02i) q^{4} +(896.038 - 2993.78i) q^{5} +(12032.1 - 20840.2i) q^{6} +(-11158.5 - 11158.5i) q^{7} +(70111.4 - 70111.4i) q^{8} +(85064.6 + 49112.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−15.7051	+	58.6121i	−0.490783	+	1.83163i	0.0616859	+	0.998096i	\(0.480352\pi\)
							−0.552469	+	0.833533i	\(0.686314\pi\)
\(3\)	−383.064	−	102.642i	−1.57639	−	0.422393i	−0.638587	−	0.769550i	\(-0.720481\pi\)
							−0.937807	+	0.347156i	\(0.887147\pi\)
\(5\)	896.038	−	2993.78i	0.286732	−	0.958011i
							\(7\)	−11158.5	−	11158.5i	−0.663921	−	0.663921i	0.292381	−	0.956302i	\(-0.405553\pi\)
−0.956302	+	0.292381i	\(0.905553\pi\)
\(11\)	−259551.			−1.61161			−0.805805	−	0.592181i	\(-0.798267\pi\)
							−0.805805	+	0.592181i	\(0.798267\pi\)
\(13\)	67955.8	+	253614.i	0.183025	+	0.683057i	0.995045	+	0.0994279i	\(0.0317013\pi\)
							−0.812020	+	0.583629i	\(0.801632\pi\)
\(17\)	316433.	−	1.18094e6i	0.222863	−	0.831735i	−0.760387	−	0.649470i	\(-0.774991\pi\)
							0.983250	−	0.182264i	\(-0.0583427\pi\)
\(19\)	2.43773e6	+	434227.i	0.984503	+	0.175367i
							\(23\)	−515463.	−	1.92373e6i	−0.0800863	−	0.298886i	0.914252	−	0.405146i	\(-0.132779\pi\)
−0.994338	+	0.106260i	\(0.966113\pi\)
\(29\)	5.84343e6	+	3.37371e6i	0.284891	+	0.164482i	0.635635	−	0.771989i	\(-0.280738\pi\)
							−0.350745	+	0.936471i	\(0.614072\pi\)
\(31\)	3.41163e7			1.19166			0.595832	−	0.803109i	\(-0.296822\pi\)
							0.595832	+	0.803109i	\(0.296822\pi\)
\(37\)	9.21890e7	+	9.21890e7i	1.32945	+	1.32945i	0.905853	+	0.423593i	\(0.139231\pi\)
							0.423593	+	0.905853i	\(0.360769\pi\)
\(41\)	−8.12198e7	−	1.40677e8i	−0.701040	−	1.21424i	−0.968102	−	0.250557i	\(-0.919386\pi\)
							0.267062	−	0.963679i	\(-0.413947\pi\)
\(43\)	−5.42530e7	−	1.45370e7i	−0.369047	−	0.0988858i	0.0695292	−	0.997580i	\(-0.477850\pi\)
							−0.438576	+	0.898694i	\(0.644517\pi\)
\(47\)	−3.89435e7	+	1.04349e7i	−0.169803	+	0.0454986i	−0.342719	−	0.939438i	\(-0.611348\pi\)
							0.172916	+	0.984937i	\(0.444681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.3	✓	392
5.3	odd	4	inner	95.11.m.a.83.96	yes	392
19.11	even	3	inner	95.11.m.a.87.96	yes	392
95.68	odd	12	inner	95.11.m.a.68.3	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.3	✓	392	1.1	even	1	trivial
95.11.m.a.68.3	yes	392	95.68	odd	12	inner
95.11.m.a.83.96	yes	392	5.3	odd	4	inner
95.11.m.a.87.96	yes	392	19.11	even	3	inner