Embedded newform 37.9.g.a.8.19

• Label: 37.9.g.a.8.19
• 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			8.19
Character	\(\chi\)	\(=\)	37.8
Dual form			37.9.g.a.14.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.0121 + 3.75453i) q^{2} +(-97.4491 - 56.2622i) q^{3} +(-39.4602 - 22.7824i) q^{4} +(1028.99 - 275.718i) q^{5} +(-1154.23 - 1154.23i) q^{6} +(-1032.28 + 1787.96i) q^{7} +(-3093.32 - 3093.32i) q^{8} +(3050.38 + 5283.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q - 10 q^{2} - 6 q^{3} + 552 q^{4} - 1642 q^{5} - 516 q^{6} - 2 q^{7} - 7440 q^{8} + 117258 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.0121	+	3.75453i	0.875756	+	0.234658i	0.668575	−	0.743645i	\(-0.266905\pi\)
							0.207181	+	0.978303i	\(0.433571\pi\)
\(3\)	−97.4491	−	56.2622i	−1.20307	−	0.694596i	−0.241837	−	0.970317i	\(-0.577750\pi\)
							−0.961238	+	0.275721i	\(0.911083\pi\)
\(5\)	1028.99	−	275.718i	1.64639	−	0.441148i	0.687789	−	0.725911i	\(-0.258581\pi\)
							0.958598	+	0.284763i	\(0.0919148\pi\)
\(7\)	−1032.28	+	1787.96i	−0.429937	+	0.744673i	−0.996867	−	0.0790927i	\(-0.974798\pi\)
							0.566930	+	0.823766i	\(0.308131\pi\)
\(11\)		−	8817.06i		−	0.602217i	−0.953590	−	0.301109i	\(-0.902643\pi\)
							0.953590	−	0.301109i	\(-0.0973567\pi\)
\(13\)	−53488.6	+	14332.2i	−1.87278	+	0.501811i	−0.872880	+	0.487935i	\(0.837750\pi\)
							−0.999904	+	0.0138760i	\(0.995583\pi\)
\(17\)	−28210.4	+	105283.i	−0.337765	+	1.26055i	0.563076	+	0.826405i	\(0.309618\pi\)
							−0.900841	+	0.434150i	\(0.857049\pi\)
\(19\)	−82399.9	+	22079.0i	−0.632284	+	0.169420i	−0.560706	−	0.828015i	\(-0.689470\pi\)
							−0.0715782	+	0.997435i	\(0.522804\pi\)
\(23\)	−193442.	−	193442.i	−0.691256	−	0.691256i	0.271252	−	0.962508i	\(-0.412562\pi\)
							−0.962508	+	0.271252i	\(0.912562\pi\)
\(29\)	279512.	−	279512.i	0.395193	−	0.395193i	−0.481341	−	0.876534i	\(-0.659850\pi\)
							0.876534	+	0.481341i	\(0.159850\pi\)
\(31\)	49586.5	−	49586.5i	0.0536929	−	0.0536929i	−0.679750	−	0.733443i	\(-0.737912\pi\)
							0.733443	+	0.679750i	\(0.237912\pi\)
\(37\)	−449537.	−	1.81945e6i	−0.239861	−	0.970807i
							\(41\)	1.75047e6	+	1.01063e6i	0.619468	+	0.357650i	0.776662	−	0.629918i	\(-0.216911\pi\)
−0.157194	+	0.987568i	\(0.550245\pi\)
\(43\)	−4.04417e6	−	4.04417e6i	−1.18292	−	1.18292i	−0.978984	−	0.203937i	\(-0.934626\pi\)
							−0.203937	−	0.978984i	\(-0.565374\pi\)
\(47\)	−3.42468e6			−0.701825			−0.350912	−	0.936408i	\(-0.614128\pi\)
							−0.350912	+	0.936408i	\(0.614128\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.9.g.a.8.19	✓	100
37.14	odd	12	inner	37.9.g.a.14.19	yes	100

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.9.g.a.8.19	✓	100	1.1	even	1	trivial
37.9.g.a.14.19	yes	100	37.14	odd	12	inner