Embedded newform 37.10.e.a.11.15

• Label: 37.10.e.a.11.15
• 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			11.15
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.40508 - 1.38857i) q^{2} +(61.5559 - 106.618i) q^{3} +(-252.144 + 436.726i) q^{4} +(-1906.94 - 1100.97i) q^{5} -341.899i q^{6} +(241.891 - 418.967i) q^{7} +2822.38i q^{8} +(2263.23 + 3920.03i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.40508	−	1.38857i	0.106290	−	0.0613668i	−0.445912	−	0.895077i	\(-0.647121\pi\)
							0.552203	+	0.833710i	\(0.313787\pi\)
\(3\)	61.5559	−	106.618i	0.438757	−	0.759950i	−0.558837	−	0.829278i	\(-0.688752\pi\)
							0.997594	+	0.0693279i	\(0.0220855\pi\)
\(5\)	−1906.94	−	1100.97i	−1.36450	−	0.787792i	−0.374277	−	0.927317i	\(-0.622109\pi\)
							−0.990218	+	0.139525i	\(0.955442\pi\)
\(7\)	241.891	−	418.967i	0.0380783	−	0.0659536i	−0.846358	−	0.532614i	\(-0.821210\pi\)
							0.884436	+	0.466661i	\(0.154543\pi\)
\(11\)	6296.41			0.129666			0.0648330	−	0.997896i	\(-0.479349\pi\)
							0.0648330	+	0.997896i	\(0.479349\pi\)
\(13\)	164354.	+	94890.0i	1.59601	+	0.921458i	0.992245	+	0.124297i	\(0.0396675\pi\)
							0.603767	+	0.797161i	\(0.293666\pi\)
\(17\)	−74150.7	+	42810.9i	−0.215325	+	0.124318i	−0.603784	−	0.797148i	\(-0.706341\pi\)
							0.388459	+	0.921466i	\(0.373008\pi\)
\(19\)	295931.	+	170856.i	0.520954	+	0.300773i	0.737325	−	0.675538i	\(-0.236089\pi\)
							−0.216371	+	0.976311i	\(0.569422\pi\)
\(23\)		−	354727.i		−	0.264313i	−0.991229	−	0.132157i	\(-0.957810\pi\)
							0.991229	−	0.132157i	\(-0.0421902\pi\)
\(29\)			5.72576e6i			1.50329i	0.659569	+	0.751644i	\(0.270739\pi\)
							−0.659569	+	0.751644i	\(0.729261\pi\)
\(31\)			4.34848e6i			0.845687i	0.906203	+	0.422844i	\(0.138968\pi\)
							−0.906203	+	0.422844i	\(0.861032\pi\)
\(37\)	8.92239e6	+	7.09597e6i	0.782660	+	0.622449i
							\(41\)	41927.7	−	72621.0i	0.00231726	−	0.00401361i	−0.864864	−	0.502005i	\(-0.832596\pi\)
0.867182	+	0.497992i	\(0.165929\pi\)
\(43\)		−	1.36875e7i		−	0.610544i	−0.952265	−	0.305272i	\(-0.901253\pi\)
							0.952265	−	0.305272i	\(-0.0987475\pi\)
\(47\)	−2.15066e7			−0.642881			−0.321441	−	0.946930i	\(-0.604167\pi\)
							−0.321441	+	0.946930i	\(0.604167\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.15	✓	56
37.27	even	6	inner	37.10.e.a.27.15	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.15	✓	56	1.1	even	1	trivial
37.10.e.a.27.15	yes	56	37.27	even	6	inner