Embedded newform 37.10.h.a.3.10

• Label: 37.10.h.a.3.10
• Level: $37$
• Weight: $10$
• Character: 37.3
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(28\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.10
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.94057 + 11.8467i) q^{2} +(-168.752 + 141.600i) q^{3} +(47.3783 + 268.696i) q^{4} +(-172.317 + 473.436i) q^{5} -3406.75i q^{6} +(-6406.43 - 2331.75i) q^{7} +(-10511.3 - 6068.69i) q^{8} +(5008.88 - 28406.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.94057	+	11.8467i	−0.439315	+	0.523555i	−0.939586	−	0.342314i	\(-0.888789\pi\)
							0.500271	+	0.865869i	\(0.333234\pi\)
\(3\)	−168.752	+	141.600i	−1.20283	+	1.00929i	−0.203286	+	0.979119i	\(0.565162\pi\)
							−0.999545	+	0.0301752i	\(0.990393\pi\)
\(5\)	−172.317	+	473.436i	−0.123300	+	0.338763i	−0.985951	−	0.167036i	\(-0.946580\pi\)
							0.862651	+	0.505800i	\(0.168803\pi\)
\(7\)	−6406.43	−	2331.75i	−1.00850	−	0.367063i	−0.215642	−	0.976473i	\(-0.569184\pi\)
							−0.792855	+	0.609410i	\(0.791406\pi\)
\(11\)	−37916.9	+	65674.1i	−0.780847	+	1.35247i	0.150601	+	0.988595i	\(0.451879\pi\)
							−0.931449	+	0.363873i	\(0.881454\pi\)
\(13\)	−108996.	+	19219.0i	−1.05844	+	0.186632i	−0.675665	−	0.737209i	\(-0.736143\pi\)
							−0.382777	+	0.923841i	\(0.625032\pi\)
\(17\)	554902.	+	97844.2i	1.61137	+	0.284129i	0.905544	−	0.424253i	\(-0.139463\pi\)
							0.705830	+	0.708382i	\(0.250574\pi\)
\(19\)	−194512.	−	231810.i	−0.342417	−	0.408077i	0.567163	−	0.823606i	\(-0.308041\pi\)
							−0.909580	+	0.415529i	\(0.863597\pi\)
\(23\)	−1.77298e6	+	1.02363e6i	−1.32108	+	0.762723i	−0.983900	−	0.178719i	\(-0.942805\pi\)
							−0.337175	+	0.941442i	\(0.609471\pi\)
\(29\)	1.78511e6	+	1.03063e6i	0.468677	+	0.270591i	0.715686	−	0.698423i	\(-0.246114\pi\)
							−0.247009	+	0.969013i	\(0.579448\pi\)
\(31\)			7.13405e6i			1.38742i	0.720254	+	0.693710i	\(0.244025\pi\)
							−0.720254	+	0.693710i	\(0.755975\pi\)
\(37\)	4.74359e6	+	1.03663e7i	0.416101	+	0.909318i
							\(41\)	6.02003e6	+	3.41413e7i	0.332714	+	1.88692i	0.448729	+	0.893668i	\(0.351877\pi\)
−0.116015	+	0.993247i	\(0.537012\pi\)
\(43\)		−	1.78425e7i		−	0.795880i	−0.917411	−	0.397940i	\(-0.869725\pi\)
							0.917411	−	0.397940i	\(-0.130275\pi\)
\(47\)	−1.92410e7	−	3.33264e7i	−0.575158	−	0.996203i	−0.996024	−	0.0890805i	\(-0.971607\pi\)
							0.420866	−	0.907123i	\(-0.361726\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.10	✓	168
37.25	even	18	inner	37.10.h.a.25.10	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.10	✓	168	1.1	even	1	trivial
37.10.h.a.25.10	yes	168	37.25	even	18	inner