Embedded newform 37.5.i.a.2.11

• Label: 37.5.i.a.2.11
• Level: $37$
• Weight: $5$
• Character: 37.2
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			2.11
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(5.93109 + 0.518903i) q^{2} +(-7.67181 + 9.14291i) q^{3} +(19.1516 + 3.37695i) q^{4} +(-13.8164 + 29.6295i) q^{5} +(-50.2465 + 50.2465i) q^{6} +(80.3866 - 29.2583i) q^{7} +(19.8237 + 5.31175i) q^{8} +(-10.6706 - 60.5159i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	5.93109	+	0.518903i	1.48277	+	0.129726i	0.799601	−	0.600532i	\(-0.205045\pi\)
							0.683172	+	0.730258i	\(0.260600\pi\)
\(3\)	−7.67181	+	9.14291i	−0.852423	+	1.01588i	0.147218	+	0.989104i	\(0.452968\pi\)
							−0.999641	+	0.0267745i	\(0.991476\pi\)
\(5\)	−13.8164	+	29.6295i	−0.552658	+	1.18518i	0.409405	+	0.912353i	\(0.365736\pi\)
							−0.962063	+	0.272826i	\(0.912042\pi\)
\(7\)	80.3866	−	29.2583i	1.64054	−	0.597109i	0.653409	−	0.757005i	\(-0.273338\pi\)
							0.987134	+	0.159896i	\(0.0511160\pi\)
\(11\)	105.996	−	61.1969i	0.876001	−	0.505759i	0.00666310	−	0.999978i	\(-0.497879\pi\)
							0.869338	+	0.494218i	\(0.164546\pi\)
\(13\)	−92.1319	−	131.578i	−0.545159	−	0.778568i	0.448162	−	0.893952i	\(-0.352079\pi\)
							−0.993321	+	0.115385i	\(0.963190\pi\)
\(17\)	335.788	+	235.121i	1.16190	+	0.813568i	0.985320	−	0.170719i	\(-0.0546092\pi\)
							0.176575	+	0.984287i	\(0.443498\pi\)
\(19\)	−102.344	+	8.95398i	−0.283503	+	0.0248033i	−0.228020	−	0.973656i	\(-0.573225\pi\)
							−0.0554825	+	0.998460i	\(0.517670\pi\)
\(23\)	60.1210	+	224.375i	0.113650	+	0.424149i	0.999182	−	0.0404296i	\(-0.0128726\pi\)
							−0.885532	+	0.464578i	\(0.846206\pi\)
\(29\)	351.816	−	1312.99i	0.418330	−	1.56123i	−0.359740	−	0.933053i	\(-0.617135\pi\)
							0.778070	−	0.628177i	\(-0.216199\pi\)
\(31\)	270.628	+	270.628i	0.281611	+	0.281611i	0.833751	−	0.552140i	\(-0.186189\pi\)
							−0.552140	+	0.833751i	\(0.686189\pi\)
\(37\)	−1023.04	−	909.703i	−0.747286	−	0.664502i
							\(41\)	250.920	+	44.2439i	0.149268	+	0.0263200i	0.247783	−	0.968816i	\(-0.420298\pi\)
−0.0985147	+	0.995136i	\(0.531409\pi\)
\(43\)	−1117.88	+	1117.88i	−0.604584	+	0.604584i	−0.941526	−	0.336942i	\(-0.890608\pi\)
							0.336942	+	0.941526i	\(0.390608\pi\)
\(47\)	−805.739	+	1395.58i	−0.364753	+	0.631770i	−0.988736	−	0.149667i	\(-0.952180\pi\)
							0.623984	+	0.781437i	\(0.285513\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.11	✓	144
37.19	odd	36	inner	37.5.i.a.19.11	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.11	✓	144	1.1	even	1	trivial
37.5.i.a.19.11	yes	144	37.19	odd	36	inner