Embedded newform 37.5.i.a.2.3

• Label: 37.5.i.a.2.3
• 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			2.3
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.82370 - 0.509508i) q^{2} +(0.698255 - 0.832148i) q^{3} +(17.8989 + 3.15607i) q^{4} +(10.8573 - 23.2835i) q^{5} +(-4.49041 + 4.49041i) q^{6} +(-65.7606 + 23.9349i) q^{7} +(-12.2821 - 3.29097i) q^{8} +(13.8606 + 78.6073i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.82370	−	0.509508i	−1.45592	−	0.127377i	−0.668465	−	0.743743i	\(-0.733048\pi\)
							−0.787459	+	0.616366i	\(0.788604\pi\)
\(3\)	0.698255	−	0.832148i	0.0775839	−	0.0924608i	−0.725856	−	0.687846i	\(-0.758556\pi\)
							0.803440	+	0.595386i	\(0.203001\pi\)
\(5\)	10.8573	−	23.2835i	0.434292	−	0.931342i	−0.560379	−	0.828237i	\(-0.689344\pi\)
							0.994670	−	0.103105i	\(-0.0328778\pi\)
\(7\)	−65.7606	+	23.9349i	−1.34205	+	0.488467i	−0.910458	−	0.413601i	\(-0.864271\pi\)
							−0.431594	+	0.902068i	\(0.642049\pi\)
\(11\)	−187.237	+	108.101i	−1.54741	+	0.893400i	−0.549077	+	0.835772i	\(0.685020\pi\)
							−0.998338	+	0.0576282i	\(0.981646\pi\)
\(13\)	−34.8528	−	49.7750i	−0.206230	−	0.294527i	0.702691	−	0.711495i	\(-0.251982\pi\)
							−0.908921	+	0.416968i	\(0.863093\pi\)
\(17\)	255.172	+	178.674i	0.882949	+	0.618248i	0.924623	−	0.380884i	\(-0.124380\pi\)
							−0.0416738	+	0.999131i	\(0.513269\pi\)
\(19\)	−341.163	+	29.8479i	−0.945049	+	0.0826811i	−0.549252	−	0.835657i	\(-0.685087\pi\)
							−0.395797	+	0.918338i	\(0.629532\pi\)
\(23\)	32.5845	+	121.607i	0.0615964	+	0.229881i	0.989861	−	0.142040i	\(-0.0453660\pi\)
							−0.928265	+	0.371921i	\(0.878699\pi\)
\(29\)	−54.3457	+	202.821i	−0.0646204	+	0.241167i	−0.990680	−	0.136211i	\(-0.956507\pi\)
							0.926059	+	0.377378i	\(0.123174\pi\)
\(31\)	−1088.90	−	1088.90i	−1.13309	−	1.13309i	−0.989660	−	0.143433i	\(-0.954186\pi\)
							−0.143433	−	0.989660i	\(-0.545814\pi\)
\(37\)	−1339.78	+	281.320i	−0.978659	+	0.205493i
							\(41\)	−181.404	−	31.9864i	−0.107914	−	0.0190282i	0.119430	−	0.992843i	\(-0.461893\pi\)
−0.227344	+	0.973814i	\(0.573004\pi\)
\(43\)	103.871	−	103.871i	0.0561770	−	0.0561770i	−0.678460	−	0.734637i	\(-0.737352\pi\)
							0.734637	+	0.678460i	\(0.237352\pi\)
\(47\)	2115.37	−	3663.92i	0.957613	−	1.65863i	0.229342	−	0.973346i	\(-0.426343\pi\)
							0.728271	−	0.685289i	\(-0.240324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.3	✓	144
37.19	odd	36	inner	37.5.i.a.19.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.3	✓	144	1.1	even	1	trivial
37.5.i.a.19.3	yes	144	37.19	odd	36	inner