Embedded newform 37.5.g.a.14.4

• Label: 37.5.g.a.14.4
• Level: $37$
• Weight: $5$
• Character: 37.14
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(11\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			14.4
Character	\(\chi\)	\(=\)	37.14
Dual form			37.5.g.a.8.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.02030 + 0.809288i) q^{2} +(-8.80383 + 5.08290i) q^{3} +(-5.38912 + 3.11141i) q^{4} +(28.4771 + 7.63041i) q^{5} +(22.4767 - 22.4767i) q^{6} +(-48.9523 - 84.7878i) q^{7} +(49.1350 - 49.1350i) q^{8} +(11.1717 - 19.3499i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.02030	+	0.809288i	−0.755076	+	0.202322i	−0.615768	−	0.787927i	\(-0.711154\pi\)
							−0.139307	+	0.990249i	\(0.544488\pi\)
\(3\)	−8.80383	+	5.08290i	−0.978204	+	0.564766i	−0.901727	−	0.432305i	\(-0.857700\pi\)
							−0.0764765	+	0.997071i	\(0.524367\pi\)
\(5\)	28.4771	+	7.63041i	1.13908	+	0.305217i	0.778583	−	0.627542i	\(-0.215939\pi\)
							0.360501	+	0.932759i	\(0.382606\pi\)
\(7\)	−48.9523	−	84.7878i	−0.999026	−	1.73036i	−0.537734	−	0.843115i	\(-0.680720\pi\)
							−0.461292	−	0.887249i	\(-0.652614\pi\)
\(11\)		−	40.3511i		−	0.333480i	−0.986001	−	0.166740i	\(-0.946676\pi\)
							0.986001	−	0.166740i	\(-0.0533241\pi\)
\(13\)	92.7389	+	24.8493i	0.548751	+	0.147037i	0.522533	−	0.852619i	\(-0.324987\pi\)
							0.0262176	+	0.999656i	\(0.491654\pi\)
\(17\)	−102.092	−	381.013i	−0.353260	−	1.31839i	−0.882660	−	0.470013i	\(-0.844249\pi\)
							0.529399	−	0.848373i	\(-0.322417\pi\)
\(19\)	−88.0795	−	23.6008i	−0.243988	−	0.0653763i	0.134753	−	0.990879i	\(-0.456976\pi\)
							−0.378740	+	0.925503i	\(0.623643\pi\)
\(23\)	−396.767	+	396.767i	−0.750032	+	0.750032i	−0.974485	−	0.224453i	\(-0.927940\pi\)
							0.224453	+	0.974485i	\(0.427940\pi\)
\(29\)	−842.453	−	842.453i	−1.00173	−	1.00173i	−0.999999	−	0.00172933i	\(-0.999450\pi\)
							−0.00172933	−	0.999999i	\(-0.500550\pi\)
\(31\)	−724.606	−	724.606i	−0.754013	−	0.754013i	0.221213	−	0.975226i	\(-0.428998\pi\)
							−0.975226	+	0.221213i	\(0.928998\pi\)
\(37\)	1325.37	+	342.869i	0.968129	+	0.250452i
							\(41\)	−173.869	+	100.383i	−0.103432	+	0.0597165i	−0.550824	−	0.834622i	\(-0.685686\pi\)
0.447392	+	0.894338i	\(0.352353\pi\)
\(43\)	1269.44	−	1269.44i	0.686554	−	0.686554i	−0.274915	−	0.961469i	\(-0.588650\pi\)
							0.961469	+	0.274915i	\(0.0886497\pi\)
\(47\)	−40.6134			−0.0183854			−0.00919270	−	0.999958i	\(-0.502926\pi\)
							−0.00919270	+	0.999958i	\(0.502926\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.14.4	yes	44
37.8	odd	12	inner	37.5.g.a.8.4	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.4	✓	44	37.8	odd	12	inner
37.5.g.a.14.4	yes	44	1.1	even	1	trivial