Embedded newform 95.11.m.a.7.18

• Label: 95.11.m.a.7.18
• Level: $95$
• Weight: $11$
• Character: 95.7
• Analytic conductor: $60.359$
• Analytic rank: $0$
• Dimension: $392$
• CM: no
• 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			7.18
Character	\(\chi\)	\(=\)	95.7
Dual form			95.11.m.a.68.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.5840 + 43.2320i) q^{2} +(-420.625 - 112.706i) q^{3} +(-848.005 - 489.596i) q^{4} +(471.452 - 3089.23i) q^{5} +(9745.02 - 16878.9i) q^{6} +(19755.7 + 19755.7i) q^{7} +(-1418.10 + 1418.10i) q^{8} +(113085. + 65289.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 392 q - 2 q^{2} - 2 q^{3} + 2220 q^{5} + 4856 q^{6} + 7780 q^{7} - 142356 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.5840	+	43.2320i	−0.361999	+	1.35100i	0.509445	+	0.860503i	\(0.329851\pi\)
							−0.871444	+	0.490496i	\(0.836816\pi\)
\(3\)	−420.625	−	112.706i	−1.73097	−	0.463812i	−0.750562	−	0.660800i	\(-0.770217\pi\)
							−0.980406	+	0.196988i	\(0.936884\pi\)
\(5\)	471.452	−	3089.23i	0.150865	−	0.988554i
							\(7\)	19755.7	+	19755.7i	1.17545	+	1.17545i	0.980891	+	0.194556i	\(0.0623265\pi\)
0.194556	+	0.980891i	\(0.437674\pi\)
\(11\)	292886.			1.81859			0.909297	−	0.416149i	\(-0.136620\pi\)
							0.909297	+	0.416149i	\(0.136620\pi\)
\(13\)	−131801.	−	491887.i	−0.354978	−	1.32479i	−0.880513	−	0.474023i	\(-0.842801\pi\)
							0.525535	−	0.850772i	\(-0.323865\pi\)
\(17\)	−37255.9	+	139041.i	−0.0262392	+	0.0979261i	−0.977804	−	0.209523i	\(-0.932809\pi\)
							0.951564	+	0.307449i	\(0.0994755\pi\)
\(19\)	1.92368e6	−	1.55901e6i	0.776900	−	0.629624i
							\(23\)	1.25973e6	+	4.70137e6i	0.195721	+	0.730442i	0.992079	+	0.125616i	\(0.0400907\pi\)
−0.796358	+	0.604826i	\(0.793243\pi\)
\(29\)	3.08291e6	+	1.77992e6i	0.150304	+	0.0867781i	0.573266	−	0.819369i	\(-0.305676\pi\)
							−0.422962	+	0.906148i	\(0.639010\pi\)
\(31\)	−3.08156e6			−0.107637			−0.0538186	−	0.998551i	\(-0.517139\pi\)
							−0.0538186	+	0.998551i	\(0.517139\pi\)
\(37\)	−6.37305e6	−	6.37305e6i	−0.0919049	−	0.0919049i	0.659660	−	0.751565i	\(-0.270700\pi\)
							−0.751565	+	0.659660i	\(0.770700\pi\)
\(41\)	−3.03903e7	−	5.26376e7i	−0.262311	−	0.454336i	0.704545	−	0.709660i	\(-0.251151\pi\)
							−0.966856	+	0.255324i	\(0.917818\pi\)
\(43\)	1.62750e8	+	4.36088e7i	1.10708	+	0.296641i	0.765645	−	0.643264i	\(-0.222420\pi\)
							0.341436	+	0.939905i	\(0.389087\pi\)
\(47\)	−3.01252e8	+	8.07203e7i	−1.31353	+	0.351960i	−0.846552	−	0.532306i	\(-0.821325\pi\)
							−0.466982	+	0.884267i	\(0.654659\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	95.11.m.a.7.18	✓	392
5.3	odd	4	inner	95.11.m.a.83.81	yes	392
19.11	even	3	inner	95.11.m.a.87.81	yes	392
95.68	odd	12	inner	95.11.m.a.68.18	yes	392

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.11.m.a.7.18	✓	392	1.1	even	1	trivial
95.11.m.a.68.18	yes	392	95.68	odd	12	inner
95.11.m.a.83.81	yes	392	5.3	odd	4	inner
95.11.m.a.87.81	yes	392	19.11	even	3	inner