Embedded newform 37.10.f.a.16.4

• Label: 37.10.f.a.16.4
• Level: $37$
• Weight: $10$
• Character: 37.16
• 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			16.4
Character	\(\chi\)	\(=\)	37.16
Dual form			37.10.f.a.7.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-33.3678 - 12.1449i) q^{2} +(12.1843 - 4.43471i) q^{3} +(573.697 + 481.389i) q^{4} +(-434.250 + 2462.75i) q^{5} -460.421 q^{6} +(-1475.57 + 8368.36i) q^{7} +(-4206.21 - 7285.37i) q^{8} +(-14949.3 + 12543.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−33.3678	−	12.1449i	−1.47466	−	0.536733i	−0.525300	−	0.850917i	\(-0.676047\pi\)
							−0.949362	+	0.314184i	\(0.898269\pi\)
\(3\)	12.1843	−	4.43471i	0.0868467	−	0.0316096i	−0.298231	−	0.954494i	\(-0.596397\pi\)
							0.385078	+	0.922884i	\(0.374174\pi\)
\(5\)	−434.250	+	2462.75i	−0.310724	+	1.76220i	0.284531	+	0.958667i	\(0.408162\pi\)
							−0.595255	+	0.803537i	\(0.702949\pi\)
\(7\)	−1475.57	+	8368.36i	−0.232283	+	1.31734i	0.615977	+	0.787764i	\(0.288761\pi\)
							−0.848260	+	0.529580i	\(0.822350\pi\)
\(11\)	6974.32	+	12079.9i	0.143627	+	0.248768i	0.928860	−	0.370432i	\(-0.120790\pi\)
							−0.785233	+	0.619200i	\(0.787457\pi\)
\(13\)	84498.1	+	70902.3i	0.820544	+	0.688518i	0.953099	−	0.302657i	\(-0.0978738\pi\)
							−0.132555	+	0.991176i	\(0.542318\pi\)
\(17\)	−46015.2	+	38611.3i	−0.133623	+	0.112123i	−0.707150	−	0.707064i	\(-0.750019\pi\)
							0.573527	+	0.819187i	\(0.305575\pi\)
\(19\)	168355.	−	61276.2i	0.296370	−	0.107870i	−0.189556	−	0.981870i	\(-0.560705\pi\)
							0.485926	+	0.874000i	\(0.338482\pi\)
\(23\)	−808154.	+	1.39976e6i	−0.602170	+	1.04299i	0.390322	+	0.920678i	\(0.372364\pi\)
							−0.992492	+	0.122310i	\(0.960970\pi\)
\(29\)	526976.	+	912749.i	0.138357	+	0.239641i	0.926875	−	0.375371i	\(-0.122485\pi\)
							−0.788518	+	0.615012i	\(0.789151\pi\)
\(31\)	2.66083e6			0.517476			0.258738	−	0.965948i	\(-0.416693\pi\)
							0.258738	+	0.965948i	\(0.416693\pi\)
\(37\)	1.01665e7	+	5.15792e6i	0.891792	+	0.452447i
							\(41\)	1.76143e7	+	1.47802e7i	0.973506	+	0.816869i	0.983097	−	0.183085i	\(-0.0586084\pi\)
−0.00959077	+	0.999954i	\(0.503053\pi\)
\(43\)	−2.57226e6			−0.114738			−0.0573690	−	0.998353i	\(-0.518271\pi\)
							−0.0573690	+	0.998353i	\(0.518271\pi\)
\(47\)	1.38722e7	−	2.40273e7i	0.414671	−	0.718231i	−0.580723	−	0.814101i	\(-0.697230\pi\)
							0.995394	+	0.0958700i	\(0.0305633\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.f.a.16.4	yes	162
37.7	even	9	inner	37.10.f.a.7.4	✓	162

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.f.a.7.4	✓	162	37.7	even	9	inner
37.10.f.a.16.4	yes	162	1.1	even	1	trivial