Embedded newform 37.5.i.a.2.9

• Label: 37.5.i.a.2.9
• 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			2.9
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.08231 + 0.182179i) q^{2} +(2.85756 - 3.40550i) q^{3} +(-11.4541 - 2.01967i) q^{4} +(15.1773 - 32.5477i) q^{5} +(6.57073 - 6.57073i) q^{6} +(66.1524 - 24.0775i) q^{7} +(-55.7877 - 14.9483i) q^{8} +(10.6337 + 60.3066i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.08231	+	0.182179i	0.520578	+	0.0455446i	0.344417	−	0.938817i	\(-0.388077\pi\)
							0.176161	+	0.984361i	\(0.443632\pi\)
\(3\)	2.85756	−	3.40550i	0.317506	−	0.378389i	−0.583561	−	0.812070i	\(-0.698341\pi\)
							0.901067	+	0.433681i	\(0.142785\pi\)
\(5\)	15.1773	−	32.5477i	0.607090	−	1.30191i	−0.327051	−	0.945007i	\(-0.606055\pi\)
							0.934142	−	0.356903i	\(-0.116167\pi\)
\(7\)	66.1524	−	24.0775i	1.35005	−	0.491377i	0.437085	−	0.899420i	\(-0.356011\pi\)
							0.912963	+	0.408043i	\(0.133789\pi\)
\(11\)	−35.1588	+	20.2989i	−0.290568	+	0.167760i	−0.638198	−	0.769872i	\(-0.720320\pi\)
							0.347630	+	0.937632i	\(0.386987\pi\)
\(13\)	−17.5022	−	24.9957i	−0.103563	−	0.147904i	0.764025	−	0.645186i	\(-0.223220\pi\)
							−0.867589	+	0.497282i	\(0.834331\pi\)
\(17\)	165.739	+	116.051i	0.573490	+	0.401562i	0.824032	−	0.566544i	\(-0.191720\pi\)
							−0.250542	+	0.968106i	\(0.580609\pi\)
\(19\)	−389.629	+	34.0881i	−1.07930	+	0.0944269i	−0.612912	−	0.790151i	\(-0.710002\pi\)
							−0.466392	+	0.884578i	\(0.654446\pi\)
\(23\)	159.991	+	597.094i	0.302440	+	1.12872i	0.935126	+	0.354314i	\(0.115286\pi\)
							−0.632686	+	0.774408i	\(0.718048\pi\)
\(29\)	9.73128	−	36.3176i	0.0115711	−	0.0431839i	−0.959899	−	0.280346i	\(-0.909551\pi\)
							0.971470	+	0.237162i	\(0.0762174\pi\)
\(31\)	604.457	+	604.457i	0.628988	+	0.628988i	0.947813	−	0.318826i	\(-0.103288\pi\)
							−0.318826	+	0.947813i	\(0.603288\pi\)
\(37\)	1367.99	−	52.6759i	0.999259	−	0.0384777i
							\(41\)	−3163.86	−	557.874i	−1.88213	−	0.331870i	−0.889886	−	0.456183i	\(-0.849216\pi\)
−0.992243	+	0.124312i	\(0.960327\pi\)
\(43\)	1544.67	−	1544.67i	0.835411	−	0.835411i	−0.152840	−	0.988251i	\(-0.548842\pi\)
							0.988251	+	0.152840i	\(0.0488419\pi\)
\(47\)	871.786	−	1509.98i	0.394652	−	0.683557i	−0.598405	−	0.801194i	\(-0.704199\pi\)
							0.993057	+	0.117637i	\(0.0375319\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.9	✓	144
37.19	odd	36	inner	37.5.i.a.19.9	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.9	✓	144	1.1	even	1	trivial
37.5.i.a.19.9	yes	144	37.19	odd	36	inner