Embedded newform 37.5.g.a.14.5

• Label: 37.5.g.a.14.5
• 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			14.5
Character	\(\chi\)	\(=\)	37.14
Dual form			37.5.g.a.8.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.93969 + 0.787689i) q^{2} +(4.37688 - 2.52700i) q^{3} +(-5.83506 + 3.36887i) q^{4} +(10.5460 + 2.82580i) q^{5} +(-10.8762 + 10.8762i) q^{6} +(28.0383 + 48.5638i) q^{7} +(48.9318 - 48.9318i) q^{8} +(-27.7286 + 48.0273i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 10 q^{2} - 6 q^{3} - 24 q^{4} + 98 q^{5} - 36 q^{6} - 2 q^{7} + 240 q^{8} + 306 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.93969	+	0.787689i	−0.734924	+	0.196922i	−0.606821	−	0.794839i	\(-0.707555\pi\)
							−0.128103	+	0.991761i	\(0.540889\pi\)
\(3\)	4.37688	−	2.52700i	0.486320	−	0.280777i	−0.236726	−	0.971576i	\(-0.576074\pi\)
							0.723047	+	0.690799i	\(0.242741\pi\)
\(5\)	10.5460	+	2.82580i	0.421841	+	0.113032i	0.463494	−	0.886100i	\(-0.346596\pi\)
							−0.0416529	+	0.999132i	\(0.513262\pi\)
\(7\)	28.0383	+	48.5638i	0.572211	+	0.991099i	0.996339	+	0.0854961i	\(0.0272475\pi\)
							−0.424127	+	0.905603i	\(0.639419\pi\)
\(11\)			201.391i			1.66439i	0.554483	+	0.832195i	\(0.312916\pi\)
							−0.554483	+	0.832195i	\(0.687084\pi\)
\(13\)	191.309	+	51.2610i	1.13200	+	0.303319i	0.775732	−	0.631063i	\(-0.217381\pi\)
							0.356272	+	0.934382i	\(0.384048\pi\)
\(17\)	−77.4229	−	288.946i	−0.267899	−	0.999813i	−0.960452	−	0.278446i	\(-0.910181\pi\)
							0.692553	−	0.721367i	\(-0.256486\pi\)
\(19\)	−338.036	−	90.5765i	−0.936389	−	0.250905i	−0.241812	−	0.970323i	\(-0.577742\pi\)
							−0.694577	+	0.719419i	\(0.744408\pi\)
\(23\)	548.646	−	548.646i	1.03714	−	1.03714i	0.0378540	−	0.999283i	\(-0.487948\pi\)
							0.999283	−	0.0378540i	\(-0.0120522\pi\)
\(29\)	1014.54	+	1014.54i	1.20635	+	1.20635i	0.972200	+	0.234154i	\(0.0752318\pi\)
							0.234154	+	0.972200i	\(0.424768\pi\)
\(31\)	−381.427	−	381.427i	−0.396906	−	0.396906i	0.480234	−	0.877140i	\(-0.340552\pi\)
							−0.877140	+	0.480234i	\(0.840552\pi\)
\(37\)	−1062.72	−	863.008i	−0.776276	−	0.630393i
							\(41\)	−485.172	+	280.114i	−0.288621	+	0.166635i	−0.637320	−	0.770599i	\(-0.719957\pi\)
0.348699	+	0.937235i	\(0.386624\pi\)
\(43\)	−59.3058	+	59.3058i	−0.0320745	+	0.0320745i	−0.722962	−	0.690888i	\(-0.757220\pi\)
							0.690888	+	0.722962i	\(0.257220\pi\)
\(47\)	3241.60			1.46745			0.733726	−	0.679446i	\(-0.237780\pi\)
							0.733726	+	0.679446i	\(0.237780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.g.a.14.5	yes	44
37.8	odd	12	inner	37.5.g.a.8.5	✓	44

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.g.a.8.5	✓	44	37.8	odd	12	inner
37.5.g.a.14.5	yes	44	1.1	even	1	trivial