Embedded newform 37.9.g.a.8.16

• Label: 37.9.g.a.8.16
• Level: $37$
• Weight: $9$
• Character: 37.8
• Analytic conductor: $15.073$
• Analytic rank: $0$
• Dimension: $100$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0730085723\)
Analytic rank:	\(0\)
Dimension:	\(100\)
Relative dimension:	\(25\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.16
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.3785 + 2.78090i) q^{2} +(-134.250 - 77.5093i) q^{3} +(-121.724 - 70.2771i) q^{4} +(-903.574 + 242.112i) q^{5} +(-1177.76 - 1177.76i) q^{6} +(1653.64 - 2864.19i) q^{7} +(-3012.84 - 3012.84i) q^{8} +(8734.88 + 15129.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.3785	+	2.78090i	0.648653	+	0.173806i	0.568120	−	0.822946i	\(-0.307671\pi\)
							0.0805334	+	0.996752i	\(0.474338\pi\)
\(3\)	−134.250	−	77.5093i	−1.65741	−	0.956905i	−0.973905	−	0.226957i	\(-0.927122\pi\)
							−0.683503	−	0.729948i	\(-0.739544\pi\)
\(5\)	−903.574	+	242.112i	−1.44572	+	0.387379i	−0.894532	−	0.447004i	\(-0.852491\pi\)
							−0.551186	+	0.834383i	\(0.685824\pi\)
\(7\)	1653.64	−	2864.19i	0.688730	−	1.19292i	−0.283519	−	0.958967i	\(-0.591502\pi\)
							0.972249	−	0.233949i	\(-0.0751648\pi\)
\(11\)			870.940i			0.0594864i	0.999558	+	0.0297432i	\(0.00946894\pi\)
							−0.999558	+	0.0297432i	\(0.990531\pi\)
\(13\)	−29669.9	+	7950.02i	−1.03883	+	0.278352i	−0.737627	−	0.675209i	\(-0.764053\pi\)
							−0.301199	+	0.953561i	\(0.597387\pi\)
\(17\)	14667.5	−	54739.9i	0.175615	−	0.655402i	−0.820832	−	0.571170i	\(-0.806490\pi\)
							0.996446	−	0.0842321i	\(-0.0268437\pi\)
\(19\)	102149.	−	27370.8i	0.783828	−	0.210026i	0.155356	−	0.987858i	\(-0.450347\pi\)
							0.628472	+	0.777832i	\(0.283681\pi\)
\(23\)	−155312.	−	155312.i	−0.555001	−	0.555001i	0.372879	−	0.927880i	\(-0.378371\pi\)
							−0.927880	+	0.372879i	\(0.878371\pi\)
\(29\)	−753256.	+	753256.i	−1.06500	+	1.06500i	−0.0672672	+	0.997735i	\(0.521428\pi\)
							−0.997735	+	0.0672672i	\(0.978572\pi\)
\(31\)	164537.	−	164537.i	0.178163	−	0.178163i	−0.612392	−	0.790555i	\(-0.709792\pi\)
							0.790555	+	0.612392i	\(0.209792\pi\)
\(37\)	1.79261e6	+	546840.i	0.956486	+	0.291779i
							\(41\)	−4.11039e6	−	2.37313e6i	−1.45461	−	0.839821i	−0.455875	−	0.890044i	\(-0.650674\pi\)
−0.998738	+	0.0502222i	\(0.984007\pi\)
\(43\)	173502.	+	173502.i	0.0507493	+	0.0507493i	0.732026	−	0.681277i	\(-0.238575\pi\)
							−0.681277	+	0.732026i	\(0.738575\pi\)
\(47\)	3.47419e6			0.711970			0.355985	−	0.934492i	\(-0.384145\pi\)
							0.355985	+	0.934492i	\(0.384145\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.16	✓	100
37.14	odd	12	inner	37.9.g.a.14.16	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.16	✓	100	1.1	even	1	trivial
37.9.g.a.14.16	yes	100	37.14	odd	12	inner