Embedded newform 37.10.e.a.11.5

• Label: 37.10.e.a.11.5
• Level: $37$
• Weight: $10$
• Character: 37.11
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.5
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.7974 + 17.2035i) q^{2} +(14.1784 - 24.5577i) q^{3} +(335.923 - 581.835i) q^{4} +(-83.3280 - 48.1094i) q^{5} +975.673i q^{6} +(4286.41 - 7424.29i) q^{7} +5499.82i q^{8} +(9439.45 + 16349.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.7974	+	17.2035i	−1.31687	+	0.760296i	−0.983224	−	0.182402i	\(-0.941613\pi\)
							−0.333647	+	0.942698i	\(0.608279\pi\)
\(3\)	14.1784	−	24.5577i	0.101060	−	0.175042i	−0.811061	−	0.584961i	\(-0.801110\pi\)
							0.912122	+	0.409919i	\(0.134443\pi\)
\(5\)	−83.3280	−	48.1094i	−0.0596246	−	0.0344243i	0.469891	−	0.882724i	\(-0.344293\pi\)
							−0.529516	+	0.848300i	\(0.677626\pi\)
\(7\)	4286.41	−	7424.29i	0.674766	−	1.16873i	−0.301772	−	0.953380i	\(-0.597578\pi\)
							0.976537	−	0.215348i	\(-0.0690886\pi\)
\(11\)	−60409.3			−1.24405			−0.622024	−	0.782999i	\(-0.713689\pi\)
							−0.622024	+	0.782999i	\(0.713689\pi\)
\(13\)	64980.0	+	37516.2i	0.631008	+	0.364313i	0.781142	−	0.624353i	\(-0.214637\pi\)
							−0.150134	+	0.988666i	\(0.547971\pi\)
\(17\)	−422691.	+	244041.i	−1.22745	+	0.708668i	−0.966495	−	0.256684i	\(-0.917370\pi\)
							−0.260953	+	0.965352i	\(0.584037\pi\)
\(19\)	291966.	+	168567.i	0.513974	+	0.296743i	0.734466	−	0.678646i	\(-0.237433\pi\)
							−0.220492	+	0.975389i	\(0.570766\pi\)
\(23\)		−	2.26886e6i		−	1.69057i	−0.534315	−	0.845285i	\(-0.679431\pi\)
							0.534315	−	0.845285i	\(-0.320569\pi\)
\(29\)		−	4.70524e6i		−	1.23535i	−0.786433	−	0.617676i	\(-0.788074\pi\)
							0.786433	−	0.617676i	\(-0.211926\pi\)
\(31\)			7.22770e6i			1.40564i	0.711370	+	0.702818i	\(0.248075\pi\)
							−0.711370	+	0.702818i	\(0.751925\pi\)
\(37\)	−1.11229e7	−	2.49849e6i	−0.975688	−	0.219164i
							\(41\)	5.69381e6	−	9.86197e6i	0.314685	−	0.545050i	−0.664686	−	0.747123i	\(-0.731435\pi\)
0.979370	+	0.202073i	\(0.0647679\pi\)
\(43\)			1.31687e7i			0.587400i	0.955898	+	0.293700i	\(0.0948866\pi\)
							−0.955898	+	0.293700i	\(0.905113\pi\)
\(47\)	−5.25092e7			−1.56962			−0.784811	−	0.619735i	\(-0.787240\pi\)
							−0.784811	+	0.619735i	\(0.787240\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.5	✓	56
37.27	even	6	inner	37.10.e.a.27.5	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.5	✓	56	1.1	even	1	trivial
37.10.e.a.27.5	yes	56	37.27	even	6	inner