Embedded newform 37.8.f.a.7.14

• Label: 37.8.f.a.7.14
• Level: $37$
• Weight: $8$
• Character: 37.7
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $132$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.f (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(132\)
Relative dimension:	\(22\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			7.14
Character	\(\chi\)	\(=\)	37.7
Dual form			37.8.f.a.16.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.79145 - 1.37998i) q^{2} +(-69.9883 - 25.4737i) q^{3} +(-85.5829 + 71.8126i) q^{4} +(27.7384 + 157.312i) q^{5} -300.510 q^{6} +(-147.454 - 836.251i) q^{7} +(-483.610 + 837.637i) q^{8} +(2574.12 + 2159.94i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 132 q + 15 q^{2} + 33 q^{3} - 279 q^{4} + 576 q^{5} - 12 q^{6} - 597 q^{7} - 3750 q^{8} - 3243 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.79145	−	1.37998i	0.335120	−	0.121974i	−0.168979	−	0.985620i	\(-0.554047\pi\)
							0.504099	+	0.863646i	\(0.331825\pi\)
\(3\)	−69.9883	−	25.4737i	−1.49658	−	0.544712i	−0.541410	−	0.840759i	\(-0.682109\pi\)
							−0.955173	+	0.296047i	\(0.904332\pi\)
\(5\)	27.7384	+	157.312i	0.0992397	+	0.562817i	0.993365	+	0.115000i	\(0.0366870\pi\)
							−0.894126	+	0.447816i	\(0.852202\pi\)
\(7\)	−147.454	−	836.251i	−0.162485	−	0.921496i	−0.951620	−	0.307277i	\(-0.900582\pi\)
							0.789136	−	0.614219i	\(-0.210529\pi\)
\(11\)	−446.947	+	774.134i	−0.101247	+	0.175365i	−0.912199	−	0.409748i	\(-0.865617\pi\)
							0.810952	+	0.585113i	\(0.198950\pi\)
\(13\)	9468.05	−	7944.64i	1.19525	−	1.00293i	0.195497	−	0.980704i	\(-0.437368\pi\)
							0.999753	−	0.0222297i	\(-0.00707651\pi\)
\(17\)	24770.9	+	20785.2i	1.22284	+	1.02608i	0.998671	+	0.0515347i	\(0.0164113\pi\)
							0.224169	+	0.974550i	\(0.428033\pi\)
\(19\)	−22913.9	−	8339.99i	−0.766412	−	0.278951i	−0.0709170	−	0.997482i	\(-0.522593\pi\)
							−0.695495	+	0.718531i	\(0.744815\pi\)
\(23\)	1408.87	+	2440.23i	0.0241448	+	0.0418200i	0.877845	−	0.478944i	\(-0.158980\pi\)
							−0.853701	+	0.520764i	\(0.825647\pi\)
\(29\)	−104608.	+	181186.i	−0.796473	+	1.37953i	0.125427	+	0.992103i	\(0.459970\pi\)
							−0.921900	+	0.387428i	\(0.873363\pi\)
\(31\)	103351.			0.623086			0.311543	−	0.950232i	\(-0.399154\pi\)
							0.311543	+	0.950232i	\(0.399154\pi\)
\(37\)	304833.	−	44817.8i	0.989364	−	0.145460i
							\(41\)	4415.50	−	3705.05i	0.0100054	−	0.00839557i	−0.637771	−	0.770226i	\(-0.720143\pi\)
0.647777	+	0.761830i	\(0.275699\pi\)
\(43\)	−216255.			−0.414789			−0.207394	−	0.978257i	\(-0.566498\pi\)
							−0.207394	+	0.978257i	\(0.566498\pi\)
\(47\)	165739.	+	287068.i	0.232853	+	0.403313i	0.958647	−	0.284599i	\(-0.0918606\pi\)
							−0.725794	+	0.687913i	\(0.758527\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.f.a.7.14	✓	132
37.16	even	9	inner	37.8.f.a.16.14	yes	132

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.f.a.7.14	✓	132	1.1	even	1	trivial
37.8.f.a.16.14	yes	132	37.16	even	9	inner