Embedded newform 37.8.f.a.7.9

• Label: 37.8.f.a.7.9
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.9
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.76396 + 1.73394i) q^{2} +(65.1424 + 23.7099i) q^{3} +(-78.3649 + 65.7560i) q^{4} +(-43.0745 - 244.287i) q^{5} -351.448 q^{6} +(-294.271 - 1668.89i) q^{7} +(583.772 - 1011.12i) q^{8} +(2006.04 + 1683.27i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.76396	+	1.73394i	−0.421079	+	0.153260i	−0.543864	−	0.839173i	\(-0.683039\pi\)
							0.122786	+	0.992433i	\(0.460817\pi\)
\(3\)	65.1424	+	23.7099i	1.39296	+	0.506997i	0.926082	−	0.377323i	\(-0.123155\pi\)
							0.466881	+	0.884320i	\(0.345378\pi\)
\(5\)	−43.0745	−	244.287i	−0.154108	−	0.873989i	−0.959597	−	0.281379i	\(-0.909208\pi\)
							0.805489	−	0.592611i	\(-0.201903\pi\)
\(7\)	−294.271	−	1668.89i	−0.324268	−	1.83901i	−0.514770	−	0.857328i	\(-0.672123\pi\)
							0.190502	−	0.981687i	\(-0.438988\pi\)
\(11\)	−162.972	+	282.276i	−0.0369180	+	0.0639439i	−0.883894	−	0.467687i	\(-0.845087\pi\)
							0.846976	+	0.531631i	\(0.178421\pi\)
\(13\)	3072.22	−	2577.90i	0.387839	−	0.325435i	−0.427932	−	0.903811i	\(-0.640758\pi\)
							0.815771	+	0.578376i	\(0.196313\pi\)
\(17\)	3815.96	+	3201.97i	0.188379	+	0.158069i	0.732100	−	0.681197i	\(-0.238540\pi\)
							−0.543721	+	0.839266i	\(0.682985\pi\)
\(19\)	26092.1	+	9496.74i	0.872712	+	0.317641i	0.739265	−	0.673415i	\(-0.235173\pi\)
							0.133447	+	0.991056i	\(0.457395\pi\)
\(23\)	−39780.3	−	68901.5i	−0.681743	−	1.18081i	−0.974449	−	0.224610i	\(-0.927889\pi\)
							0.292706	−	0.956202i	\(-0.405444\pi\)
\(29\)	18367.4	−	31813.3i	0.139847	−	0.242223i	−0.787591	−	0.616198i	\(-0.788672\pi\)
							0.927439	+	0.373975i	\(0.122005\pi\)
\(31\)	−319642.			−1.92707			−0.963537	−	0.267577i	\(-0.913777\pi\)
							−0.963537	+	0.267577i	\(0.913777\pi\)
\(37\)	256612.	+	170535.i	0.832859	+	0.553485i
							\(41\)	185871.	−	155964.i	0.421180	−	0.353412i	−0.407432	−	0.913236i	\(-0.633576\pi\)
0.828612	+	0.559824i	\(0.189131\pi\)
\(43\)	424350.			0.813926			0.406963	−	0.913445i	\(-0.366588\pi\)
							0.406963	+	0.913445i	\(0.366588\pi\)
\(47\)	−80852.5	−	140041.i	−0.113593	−	0.196749i	0.803623	−	0.595138i	\(-0.202903\pi\)
							−0.917216	+	0.398389i	\(0.869569\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.9	✓	132
37.16	even	9	inner	37.8.f.a.16.9	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.9	✓	132	1.1	even	1	trivial
37.8.f.a.16.9	yes	132	37.16	even	9	inner