Embedded newform 37.6.e.a.11.6

• Label: 37.6.e.a.11.6
• Level: $37$
• Weight: $6$
• Character: 37.11
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.6
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.47160 + 2.00433i) q^{2} +(-13.3195 + 23.0700i) q^{3} +(-7.96531 + 13.7963i) q^{4} +(85.3532 + 49.2787i) q^{5} -106.786i q^{6} +(-91.8980 + 159.172i) q^{7} -192.138i q^{8} +(-233.316 - 404.115i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.47160	+	2.00433i	−0.613699	+	0.354319i	−0.774412	−	0.632682i	\(-0.781954\pi\)
							0.160713	+	0.987001i	\(0.448621\pi\)
\(3\)	−13.3195	+	23.0700i	−0.854443	+	1.47994i	0.0227173	+	0.999742i	\(0.492768\pi\)
							−0.877161	+	0.480197i	\(0.840565\pi\)
\(5\)	85.3532	+	49.2787i	1.52684	+	0.881524i	0.999492	+	0.0318778i	\(0.0101487\pi\)
							0.527353	+	0.849646i	\(0.323185\pi\)
\(7\)	−91.8980	+	159.172i	−0.708861	+	1.22778i	0.256419	+	0.966566i	\(0.417457\pi\)
							−0.965280	+	0.261218i	\(0.915876\pi\)
\(11\)	64.5270			0.160790			0.0803952	−	0.996763i	\(-0.474382\pi\)
							0.0803952	+	0.996763i	\(0.474382\pi\)
\(13\)	582.223	+	336.146i	0.955500	+	0.551658i	0.894785	−	0.446497i	\(-0.147329\pi\)
							0.0607150	+	0.998155i	\(0.480662\pi\)
\(17\)	1578.36	−	911.269i	1.32460	−	0.764758i	0.340142	−	0.940374i	\(-0.389525\pi\)
							0.984459	+	0.175616i	\(0.0561917\pi\)
\(19\)	−262.334	−	151.458i	−0.166713	−	0.0962520i	0.414322	−	0.910130i	\(-0.364019\pi\)
							−0.581035	+	0.813878i	\(0.697352\pi\)
\(23\)			378.939i			0.149365i	0.997207	+	0.0746826i	\(0.0237944\pi\)
							−0.997207	+	0.0746826i	\(0.976206\pi\)
\(29\)			2893.21i			0.638829i	0.947615	+	0.319414i	\(0.103486\pi\)
							−0.947615	+	0.319414i	\(0.896514\pi\)
\(31\)		−	1342.49i		−	0.250904i	−0.992100	−	0.125452i	\(-0.959962\pi\)
							0.992100	−	0.125452i	\(-0.0400382\pi\)
\(37\)	−5655.00	−	6112.68i	−0.679092	−	0.734053i
							\(41\)	−1241.86	+	2150.97i	−0.115376	+	0.199836i	−0.917930	−	0.396743i	\(-0.870140\pi\)
0.802554	+	0.596579i	\(0.203474\pi\)
\(43\)		−	3200.26i		−	0.263945i	−0.991253	−	0.131973i	\(-0.957869\pi\)
							0.991253	−	0.131973i	\(-0.0421311\pi\)
\(47\)	1429.85			0.0944162			0.0472081	−	0.998885i	\(-0.484968\pi\)
							0.0472081	+	0.998885i	\(0.484968\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.6	✓	32
37.27	even	6	inner	37.6.e.a.27.6	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.6	✓	32	1.1	even	1	trivial
37.6.e.a.27.6	yes	32	37.27	even	6	inner