Embedded newform 37.10.c.a.10.4

• Label: 37.10.c.a.10.4
• 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			10.4
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-18.3336 + 31.7547i) q^{2} +(82.9316 + 143.642i) q^{3} +(-416.240 - 720.949i) q^{4} +(-705.543 - 1222.04i) q^{5} -6081.73 q^{6} +(-777.430 - 1346.55i) q^{7} +11751.1 q^{8} +(-3913.79 + 6778.88i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−18.3336	+	31.7547i	−0.810237	+	1.40337i	0.102461	+	0.994737i	\(0.467328\pi\)
							−0.912698	+	0.408635i	\(0.866005\pi\)
\(3\)	82.9316	+	143.642i	0.591118	+	1.02385i	0.994082	+	0.108631i	\(0.0346466\pi\)
							−0.402964	+	0.915216i	\(0.632020\pi\)
\(5\)	−705.543	−	1222.04i	−0.504845	−	0.874417i	−0.999984	−	0.00560379i	\(-0.998216\pi\)
							0.495139	−	0.868814i	\(-0.335117\pi\)
\(7\)	−777.430	−	1346.55i	−0.122383	−	0.211973i	0.798324	−	0.602228i	\(-0.205720\pi\)
							−0.920707	+	0.390255i	\(0.872387\pi\)
\(11\)	41837.9			0.861595			0.430797	−	0.902449i	\(-0.358232\pi\)
							0.430797	+	0.902449i	\(0.358232\pi\)
\(13\)	−12870.3	−	22292.0i	−0.124981	−	0.216473i	0.796745	−	0.604316i	\(-0.206554\pi\)
							−0.921725	+	0.387843i	\(0.873220\pi\)
\(17\)	96686.7	−	167466.i	0.280767	−	0.486303i	−0.690807	−	0.723040i	\(-0.742744\pi\)
							0.971574	+	0.236736i	\(0.0760777\pi\)
\(19\)	81090.4	+	140453.i	0.142751	+	0.247252i	0.928532	−	0.371254i	\(-0.121072\pi\)
							−0.785781	+	0.618505i	\(0.787739\pi\)
\(23\)	52232.2			0.0389191			0.0194596	−	0.999811i	\(-0.493805\pi\)
							0.0194596	+	0.999811i	\(0.493805\pi\)
\(29\)	−1.33546e6			−0.350623			−0.175312	−	0.984513i	\(-0.556093\pi\)
							−0.175312	+	0.984513i	\(0.556093\pi\)
\(31\)	789472.			0.153536			0.0767678	−	0.997049i	\(-0.475540\pi\)
							0.0767678	+	0.997049i	\(0.475540\pi\)
\(37\)	−1.11538e7	−	2.35675e6i	−0.978398	−	0.206731i
							\(41\)	−1.22044e7	−	2.11387e7i	−0.674512	−	1.16829i	−0.976611	−	0.215013i	\(-0.931021\pi\)
0.302099	−	0.953277i	\(-0.402313\pi\)
\(43\)	−3.44385e6			−0.153616			−0.0768078	−	0.997046i	\(-0.524473\pi\)
							−0.0768078	+	0.997046i	\(0.524473\pi\)
\(47\)	4.48410e7			1.34040			0.670200	−	0.742180i	\(-0.266208\pi\)
							0.670200	+	0.742180i	\(0.266208\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.4	✓	54
37.26	even	3	inner	37.10.c.a.26.4	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.4	✓	54	1.1	even	1	trivial
37.10.c.a.26.4	yes	54	37.26	even	3	inner