Embedded newform 37.10.h.a.3.15

• Label: 37.10.h.a.3.15
• 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			3.15
Character	\(\chi\)	\(=\)	37.3
Dual form			37.10.h.a.25.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15965 + 1.38201i) q^{2} +(114.860 - 96.3791i) q^{3} +(88.3427 + 501.016i) q^{4} +(184.041 - 505.649i) q^{5} +270.504i q^{6} +(6332.05 + 2304.68i) q^{7} +(-1594.80 - 920.758i) q^{8} +(486.007 - 2756.29i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 168 q + 45 q^{2} + 216 q^{3} + 249 q^{4} + 177 q^{5} - 5820 q^{7} - 74520 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15965	+	1.38201i	−0.0512496	+	0.0610769i	−0.791060	−	0.611738i	\(-0.790471\pi\)
							0.739811	+	0.672815i	\(0.234915\pi\)
\(3\)	114.860	−	96.3791i	0.818698	−	0.686969i	−0.133969	−	0.990986i	\(-0.542772\pi\)
							0.952667	+	0.304016i	\(0.0983277\pi\)
\(5\)	184.041	−	505.649i	0.131689	−	0.361813i	−0.856270	−	0.516529i	\(-0.827224\pi\)
							0.987959	+	0.154716i	\(0.0494462\pi\)
\(7\)	6332.05	+	2304.68i	0.996789	+	0.362801i	0.788345	−	0.615233i	\(-0.210938\pi\)
							0.208443	+	0.978034i	\(0.433160\pi\)
\(11\)	−22203.1	+	38456.9i	−0.457243	+	0.791968i	−0.998814	−	0.0486868i	\(-0.984496\pi\)
							0.541571	+	0.840655i	\(0.317830\pi\)
\(13\)	70451.7	−	12422.5i	0.684142	−	0.120633i	0.179235	−	0.983806i	\(-0.442638\pi\)
							0.504907	+	0.863174i	\(0.331527\pi\)
\(17\)	77961.1	+	13746.6i	0.226390	+	0.0399187i	0.285692	−	0.958321i	\(-0.407776\pi\)
							−0.0593022	+	0.998240i	\(0.518888\pi\)
\(19\)	−58154.2	−	69305.5i	−0.102374	−	0.122005i	0.712425	−	0.701749i	\(-0.247597\pi\)
							−0.814799	+	0.579744i	\(0.803153\pi\)
\(23\)	1.23033e6	−	710331.i	0.916739	−	0.529280i	0.0341460	−	0.999417i	\(-0.489129\pi\)
							0.882593	+	0.470137i	\(0.155796\pi\)
\(29\)	2.81408e6	+	1.62471e6i	0.738830	+	0.426564i	0.821644	−	0.570001i	\(-0.193057\pi\)
							−0.0828135	+	0.996565i	\(0.526391\pi\)
\(31\)			3.06870e6i			0.596796i	0.954441	+	0.298398i	\(0.0964523\pi\)
							−0.954441	+	0.298398i	\(0.903548\pi\)
\(37\)	1.08832e7	+	3.39388e6i	0.954657	+	0.297707i
							\(41\)	−3.21414e6	−	1.82283e7i	−0.177639	−	1.00744i	−0.935054	−	0.354506i	\(-0.884649\pi\)
0.757415	−	0.652934i	\(-0.226462\pi\)
\(43\)		−	1.61884e7i		−	0.722099i	−0.932547	−	0.361049i	\(-0.882419\pi\)
							0.932547	−	0.361049i	\(-0.117581\pi\)
\(47\)	−4.19823e6	−	7.27154e6i	−0.125495	−	0.217363i	0.796431	−	0.604729i	\(-0.206718\pi\)
							−0.921926	+	0.387365i	\(0.873385\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.h.a.3.15	✓	168
37.25	even	18	inner	37.10.h.a.25.15	yes	168

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.h.a.3.15	✓	168	1.1	even	1	trivial
37.10.h.a.25.15	yes	168	37.25	even	18	inner