Embedded newform 37.10.f.a.7.18

• Label: 37.10.f.a.7.18
• Level: $37$
• Weight: $10$
• Character: 37.7
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $162$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(162\)
Relative dimension:	\(27\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.18
Character	\(\chi\)	\(=\)	37.7
Dual form			37.10.f.a.16.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.8129 - 5.39147i) q^{2} +(-119.764 - 43.5906i) q^{3} +(-201.860 + 169.380i) q^{4} +(-288.707 - 1637.34i) q^{5} -2009.08 q^{6} +(1626.30 + 9223.21i) q^{7} +(-6112.41 + 10587.0i) q^{8} +(-2634.73 - 2210.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.8129	−	5.39147i	0.654645	−	0.238271i	0.00672253	−	0.999977i	\(-0.497860\pi\)
							0.647923	+	0.761706i	\(0.275638\pi\)
\(3\)	−119.764	−	43.5906i	−0.853653	−	0.310704i	−0.122124	−	0.992515i	\(-0.538971\pi\)
							−0.731529	+	0.681811i	\(0.761193\pi\)
\(5\)	−288.707	−	1637.34i	−0.206582	−	1.17158i	−0.894931	−	0.446205i	\(-0.852775\pi\)
							0.688349	−	0.725380i	\(-0.258336\pi\)
\(7\)	1626.30	+	9223.21i	0.256011	+	1.45191i	0.793463	+	0.608619i	\(0.208276\pi\)
							−0.537451	+	0.843295i	\(0.680613\pi\)
\(11\)	10148.5	−	17577.8i	0.208995	−	0.361990i	−0.742403	−	0.669954i	\(-0.766314\pi\)
							0.951398	+	0.307963i	\(0.0996473\pi\)
\(13\)	90756.5	−	76153.7i	0.881318	−	0.739513i	−0.0851319	−	0.996370i	\(-0.527131\pi\)
							0.966450	+	0.256856i	\(0.0826867\pi\)
\(17\)	121743.	+	102155.i	0.353529	+	0.296646i	0.802205	−	0.597048i	\(-0.203660\pi\)
							−0.448676	+	0.893694i	\(0.648104\pi\)
\(19\)	1.00328e6	+	365163.i	1.76616	+	0.642829i	1.00000		0.000176049i	\(5.60381e-5\pi\)
							0.766158	+	0.642653i	\(0.222166\pi\)
\(23\)	50695.1	+	87806.4i	0.0377738	+	0.0654261i	0.884294	−	0.466930i	\(-0.154640\pi\)
							−0.846520	+	0.532356i	\(0.821307\pi\)
\(29\)	−2.80880e6	+	4.86499e6i	−0.737446	+	1.27729i	0.216195	+	0.976350i	\(0.430635\pi\)
							−0.953642	+	0.300944i	\(0.902698\pi\)
\(31\)	4.24569e6			0.825697			0.412849	−	0.910800i	\(-0.364534\pi\)
							0.412849	+	0.910800i	\(0.364534\pi\)
\(37\)	−9.57027e6	+	6.19449e6i	−0.839491	+	0.543373i
							\(41\)	2.16910e7	−	1.82009e7i	1.19882	−	1.00593i	0.199152	−	0.979969i	\(-0.436181\pi\)
0.999663	−	0.0259568i	\(-0.00826323\pi\)
\(43\)	−1.92603e7			−0.859123			−0.429561	−	0.903038i	\(-0.641332\pi\)
							−0.429561	+	0.903038i	\(0.641332\pi\)
\(47\)	1.68826e7	+	2.92415e7i	0.504660	+	0.874096i	0.999985	+	0.00538897i	\(0.00171537\pi\)
							−0.495326	+	0.868707i	\(0.664951\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.f.a.7.18	✓	162
37.16	even	9	inner	37.10.f.a.16.18	yes	162

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.f.a.7.18	✓	162	1.1	even	1	trivial
37.10.f.a.16.18	yes	162	37.16	even	9	inner