Embedded newform 37.10.e.a.11.11

• Label: 37.10.e.a.11.11
• 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			11.11
Character	\(\chi\)	\(=\)	37.11
Dual form			37.10.e.a.27.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.7800 + 6.80116i) q^{2} +(32.8904 - 56.9678i) q^{3} +(-163.488 + 283.170i) q^{4} +(1285.87 + 742.395i) q^{5} +894.771i q^{6} +(2908.10 - 5036.98i) q^{7} -11412.0i q^{8} +(7677.95 + 13298.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.7800	+	6.80116i	−0.520606	+	0.300572i	−0.737182	−	0.675694i	\(-0.763844\pi\)
							0.216577	+	0.976266i	\(0.430511\pi\)
\(3\)	32.8904	−	56.9678i	0.234435	−	0.406054i	−0.724673	−	0.689093i	\(-0.758009\pi\)
							0.959108	+	0.283039i	\(0.0913425\pi\)
\(5\)	1285.87	+	742.395i	0.920091	+	0.531215i	0.883664	−	0.468122i	\(-0.155069\pi\)
							0.0364267	+	0.999336i	\(0.488402\pi\)
\(7\)	2908.10	−	5036.98i	0.457792	−	0.792920i	−0.541052	−	0.840989i	\(-0.681974\pi\)
							0.998844	+	0.0480698i	\(0.0153070\pi\)
\(11\)	−27288.2			−0.561963			−0.280982	−	0.959713i	\(-0.590660\pi\)
							−0.280982	+	0.959713i	\(0.590660\pi\)
\(13\)	−63570.5	−	36702.4i	−0.617320	−	0.356410i	0.158505	−	0.987358i	\(-0.449333\pi\)
							−0.775825	+	0.630948i	\(0.782666\pi\)
\(17\)	414618.	−	239380.i	1.20400	−	0.695132i	0.242561	−	0.970136i	\(-0.422013\pi\)
							0.961443	+	0.275004i	\(0.0886793\pi\)
\(19\)	760030.	+	438804.i	1.33795	+	0.772466i	0.986503	−	0.163742i	\(-0.0523566\pi\)
							0.351446	+	0.936208i	\(0.385690\pi\)
\(23\)			1.98803e6i			1.48131i	0.671883	+	0.740657i	\(0.265486\pi\)
							−0.671883	+	0.740657i	\(0.734514\pi\)
\(29\)			6.11568e6i			1.60566i	0.596208	+	0.802830i	\(0.296673\pi\)
							−0.596208	+	0.802830i	\(0.703327\pi\)
\(31\)			3.58759e6i			0.697711i	0.937177	+	0.348855i	\(0.113430\pi\)
							−0.937177	+	0.348855i	\(0.886570\pi\)
\(37\)	1.05815e7	+	4.24188e6i	0.928196	+	0.372092i
							\(41\)	1.11519e7	−	1.93156e7i	0.616340	−	1.06753i	−0.373808	−	0.927506i	\(-0.621948\pi\)
0.990148	−	0.140026i	\(-0.0447187\pi\)
\(43\)		−	1.30758e7i		−	0.583258i	−0.956532	−	0.291629i	\(-0.905803\pi\)
							0.956532	−	0.291629i	\(-0.0941972\pi\)
\(47\)	−5.04197e7			−1.50716			−0.753581	−	0.657355i	\(-0.771675\pi\)
							−0.753581	+	0.657355i	\(0.771675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.e.a.11.11	✓	56
37.27	even	6	inner	37.10.e.a.27.11	yes	56

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.e.a.11.11	✓	56	1.1	even	1	trivial
37.10.e.a.27.11	yes	56	37.27	even	6	inner