Embedded newform 37.10.c.a.10.2

• Label: 37.10.c.a.10.2
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.2
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-20.2191 + 35.0205i) q^{2} +(-84.5687 - 146.477i) q^{3} +(-561.626 - 972.764i) q^{4} +(-198.799 - 344.329i) q^{5} +6839.62 q^{6} +(3839.21 + 6649.70i) q^{7} +24717.9 q^{8} +(-4462.23 + 7728.80i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.2191	+	35.0205i	−0.893567	+	1.54770i	−0.0579994	+	0.998317i	\(0.518472\pi\)
							−0.835568	+	0.549387i	\(0.814861\pi\)
\(3\)	−84.5687	−	146.477i	−0.602787	−	1.04406i	−0.992397	−	0.123078i	\(-0.960723\pi\)
							0.389610	−	0.920980i	\(-0.372610\pi\)
\(5\)	−198.799	−	344.329i	−0.142249	−	0.246382i	0.786094	−	0.618106i	\(-0.212100\pi\)
							−0.928343	+	0.371725i	\(0.878767\pi\)
\(7\)	3839.21	+	6649.70i	0.604367	+	1.04679i	0.992151	+	0.125044i	\(0.0399071\pi\)
							−0.387785	+	0.921750i	\(0.626760\pi\)
\(11\)	31272.4			0.644013			0.322007	−	0.946737i	\(-0.395643\pi\)
							0.322007	+	0.946737i	\(0.395643\pi\)
\(13\)	−51004.3	−	88342.0i	−0.495292	−	0.857871i	0.504693	−	0.863299i	\(-0.331606\pi\)
							−0.999985	+	0.00542787i	\(0.998272\pi\)
\(17\)	27719.0	−	48010.7i	0.0804929	−	0.139418i	−0.822969	−	0.568086i	\(-0.807684\pi\)
							0.903462	+	0.428669i	\(0.141017\pi\)
\(19\)	258109.	+	447058.i	0.454372	+	0.786996i	0.998652	−	0.0519082i	\(-0.0165303\pi\)
							−0.544280	+	0.838904i	\(0.683197\pi\)
\(23\)	−165117.			−0.123032			−0.0615160	−	0.998106i	\(-0.519594\pi\)
							−0.0615160	+	0.998106i	\(0.519594\pi\)
\(29\)	−6.67804e6			−1.75331			−0.876654	−	0.481121i	\(-0.840230\pi\)
							−0.876654	+	0.481121i	\(0.840230\pi\)
\(31\)	−2.01062e6			−0.391024			−0.195512	−	0.980701i	\(-0.562637\pi\)
							−0.195512	+	0.980701i	\(0.562637\pi\)
\(37\)	1.09582e7	+	3.14323e6i	0.961238	+	0.275720i
							\(41\)	−1.65416e6	−	2.86508e6i	−0.0914217	−	0.158347i	0.816688	−	0.577080i	\(-0.195808\pi\)
−0.908110	+	0.418733i	\(0.862474\pi\)
\(43\)	−3.18794e7			−1.42201			−0.711003	−	0.703189i	\(-0.751759\pi\)
							−0.711003	+	0.703189i	\(0.751759\pi\)
\(47\)	−3.95329e7			−1.18173			−0.590865	−	0.806770i	\(-0.701214\pi\)
							−0.590865	+	0.806770i	\(0.701214\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.2	✓	54
37.26	even	3	inner	37.10.c.a.26.2	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.2	✓	54	1.1	even	1	trivial
37.10.c.a.26.2	yes	54	37.26	even	3	inner