Embedded newform 37.10.f.a.7.16

• Label: 37.10.f.a.7.16
• 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			7.16
Character	\(\chi\)	\(=\)	37.7
Dual form			37.10.f.a.16.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.62864 - 3.14057i) q^{2} +(125.460 + 45.6637i) q^{3} +(-327.624 + 274.910i) q^{4} +(-365.091 - 2070.54i) q^{5} +1225.96 q^{6} +(119.762 + 679.206i) q^{7} +(-4314.28 + 7472.55i) q^{8} +(-1423.02 - 1194.05i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.62864	−	3.14057i	0.381336	−	0.138795i	−0.144238	−	0.989543i	\(-0.546073\pi\)
							0.525574	+	0.850748i	\(0.323851\pi\)
\(3\)	125.460	+	45.6637i	0.894251	+	0.325481i	0.747947	−	0.663759i	\(-0.231040\pi\)
							0.146305	+	0.989240i	\(0.453262\pi\)
\(5\)	−365.091	−	2070.54i	−0.261238	−	1.48155i	−0.779538	−	0.626356i	\(-0.784546\pi\)
							0.518299	−	0.855199i	\(-0.326565\pi\)
\(7\)	119.762	+	679.206i	0.0188529	+	0.106920i	0.992782	−	0.119932i	\(-0.0382676\pi\)
							−0.973929	+	0.226852i	\(0.927157\pi\)
\(11\)	−20865.8	+	36140.6i	−0.429702	+	0.744265i	−0.996847	−	0.0793527i	\(-0.974715\pi\)
							0.567145	+	0.823618i	\(0.308048\pi\)
\(13\)	−99005.7	+	83075.6i	−0.961424	+	0.806730i	−0.981184	−	0.193074i	\(-0.938154\pi\)
							0.0197603	+	0.999805i	\(0.493710\pi\)
\(17\)	−435878.	−	365745.i	−1.26574	−	1.06208i	−0.995046	−	0.0994166i	\(-0.968302\pi\)
							−0.270694	−	0.962665i	\(-0.587253\pi\)
\(19\)	190122.	+	69198.6i	0.334688	+	0.121816i	0.503897	−	0.863764i	\(-0.331899\pi\)
							−0.169209	+	0.985580i	\(0.554121\pi\)
\(23\)	−436862.	−	756667.i	−0.325513	−	0.563806i	0.656103	−	0.754672i	\(-0.272204\pi\)
							−0.981616	+	0.190866i	\(0.938870\pi\)
\(29\)	−1.54400e6	+	2.67429e6i	−0.405374	+	0.702129i	−0.994365	−	0.106011i	\(-0.966192\pi\)
							0.588991	+	0.808140i	\(0.299526\pi\)
\(31\)	−3.51171e6			−0.682953			−0.341476	−	0.939890i	\(-0.610927\pi\)
							−0.341476	+	0.939890i	\(0.610927\pi\)
\(37\)	9.95040e6	−	5.56339e6i	0.872836	−	0.488013i
							\(41\)	−1.50683e7	+	1.26438e7i	−0.832791	+	0.698794i	−0.955930	−	0.293595i	\(-0.905148\pi\)
0.123139	+	0.992389i	\(0.460704\pi\)
\(43\)	7.72131e6			0.344416			0.172208	−	0.985061i	\(-0.444910\pi\)
							0.172208	+	0.985061i	\(0.444910\pi\)
\(47\)	2.09144e7	+	3.62248e7i	0.625179	+	1.08284i	0.988506	+	0.151181i	\(0.0483075\pi\)
							−0.363327	+	0.931662i	\(0.618359\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.f.a.7.16	✓	162
37.16	even	9	inner	37.10.f.a.16.16	yes	162

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.f.a.7.16	✓	162	1.1	even	1	trivial
37.10.f.a.16.16	yes	162	37.16	even	9	inner