Embedded newform 37.5.i.a.2.7

• Label: 37.5.i.a.2.7
• 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			2.7
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.818396 - 0.0716004i) q^{2} +(2.52396 - 3.00793i) q^{3} +(-15.0923 - 2.66118i) q^{4} +(-18.6068 + 39.9023i) q^{5} +(-2.28096 + 2.28096i) q^{6} +(8.45114 - 3.07597i) q^{7} +(24.8574 + 6.66052i) q^{8} +(11.3882 + 64.5856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.818396	−	0.0716004i	−0.204599	−	0.0179001i	−0.0156045	−	0.999878i	\(-0.504967\pi\)
							−0.188994	+	0.981978i	\(0.560523\pi\)
\(3\)	2.52396	−	3.00793i	0.280440	−	0.334215i	−0.607376	−	0.794415i	\(-0.707778\pi\)
							0.887815	+	0.460200i	\(0.152222\pi\)
\(5\)	−18.6068	+	39.9023i	−0.744270	+	1.59609i	0.0574179	+	0.998350i	\(0.481713\pi\)
							−0.801688	+	0.597742i	\(0.796065\pi\)
\(7\)	8.45114	−	3.07597i	0.172472	−	0.0627748i	−0.254341	−	0.967115i	\(-0.581858\pi\)
							0.426813	+	0.904340i	\(0.359636\pi\)
\(11\)	−144.479	+	83.4149i	−1.19404	+	0.689380i	−0.959220	−	0.282659i	\(-0.908783\pi\)
							−0.234820	+	0.972039i	\(0.575450\pi\)
\(13\)	−130.385	−	186.209i	−0.771509	−	1.10183i	−0.991975	−	0.126435i	\(-0.959647\pi\)
							0.220466	−	0.975395i	\(-0.429242\pi\)
\(17\)	−54.4089	−	38.0975i	−0.188266	−	0.131825i	0.475644	−	0.879638i	\(-0.342215\pi\)
							−0.663910	+	0.747813i	\(0.731104\pi\)
\(19\)	240.019	−	20.9989i	0.664871	−	0.0581687i	0.250276	−	0.968175i	\(-0.419479\pi\)
							0.414595	+	0.910006i	\(0.363923\pi\)
\(23\)	6.40113	+	23.8893i	0.0121004	+	0.0451594i	0.971712	−	0.236169i	\(-0.0758918\pi\)
							−0.959612	+	0.281328i	\(0.909225\pi\)
\(29\)	−162.074	+	604.868i	−0.192716	+	0.719225i	0.800131	+	0.599826i	\(0.204763\pi\)
							−0.992846	+	0.119399i	\(0.961903\pi\)
\(31\)	840.320	+	840.320i	0.874422	+	0.874422i	0.992951	−	0.118529i	\(-0.0378177\pi\)
							−0.118529	+	0.992951i	\(0.537818\pi\)
\(37\)	−1360.02	+	156.536i	−0.993441	+	0.114343i
							\(41\)	2420.04	+	426.718i	1.43964	+	0.253847i	0.838328	−	0.545166i	\(-0.183533\pi\)
0.601313	+	0.799014i	\(0.294645\pi\)
\(43\)	−1086.99	+	1086.99i	−0.587880	+	0.587880i	−0.937057	−	0.349177i	\(-0.886461\pi\)
							0.349177	+	0.937057i	\(0.386461\pi\)
\(47\)	−53.0567	+	91.8970i	−0.0240184	+	0.0416012i	−0.877785	−	0.479055i	\(-0.840979\pi\)
							0.853766	+	0.520656i	\(0.174313\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.7	✓	144
37.19	odd	36	inner	37.5.i.a.19.7	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.7	✓	144	1.1	even	1	trivial
37.5.i.a.19.7	yes	144	37.19	odd	36	inner