Embedded newform 37.6.c.a.10.4

• Label: 37.6.c.a.10.4
• Level: $37$
• Weight: $6$
• Character: 37.10
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) 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			37.6.c.a.26.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.51823 + 6.09376i) q^{2} +(-0.728242 - 1.26135i) q^{3} +(-8.75591 - 15.1657i) q^{4} +(-24.4115 - 42.2819i) q^{5} +10.2485 q^{6} +(-37.5928 - 65.1127i) q^{7} -101.945 q^{8} +(120.439 - 208.607i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.51823	+	6.09376i	−0.621941	+	1.07723i	0.367182	+	0.930149i	\(0.380322\pi\)
							−0.989124	+	0.147085i	\(0.953011\pi\)
\(3\)	−0.728242	−	1.26135i	−0.0467167	−	0.0809157i	0.841721	−	0.539912i	\(-0.181542\pi\)
							−0.888438	+	0.458996i	\(0.848209\pi\)
\(5\)	−24.4115	−	42.2819i	−0.436686	−	0.756362i	0.560746	−	0.827988i	\(-0.310514\pi\)
							−0.997432	+	0.0716261i	\(0.977181\pi\)
\(7\)	−37.5928	−	65.1127i	−0.289975	−	0.502251i	0.683829	−	0.729643i	\(-0.260314\pi\)
							−0.973803	+	0.227392i	\(0.926980\pi\)
\(11\)	394.862			0.983928			0.491964	−	0.870616i	\(-0.336279\pi\)
							0.491964	+	0.870616i	\(0.336279\pi\)
\(13\)	36.2026	+	62.7047i	0.0594129	+	0.102906i	0.894202	−	0.447664i	\(-0.147744\pi\)
							−0.834789	+	0.550570i	\(0.814410\pi\)
\(17\)	201.247	−	348.570i	0.168891	−	0.292528i	−0.769139	−	0.639081i	\(-0.779315\pi\)
							0.938030	+	0.346553i	\(0.112648\pi\)
\(19\)	−1014.18	−	1756.60i	−0.644509	−	1.11632i	−0.984415	−	0.175863i	\(-0.943728\pi\)
							0.339906	−	0.940460i	\(-0.389605\pi\)
\(23\)	−3998.91			−1.57624			−0.788120	−	0.615522i	\(-0.788945\pi\)
							−0.788120	+	0.615522i	\(0.788945\pi\)
\(29\)	−4555.69			−1.00591			−0.502955	−	0.864312i	\(-0.667754\pi\)
							−0.502955	+	0.864312i	\(0.667754\pi\)
\(31\)	1795.88			0.335639			0.167820	−	0.985818i	\(-0.446327\pi\)
							0.167820	+	0.985818i	\(0.446327\pi\)
\(37\)	3146.00	−	7710.16i	0.377793	−	0.925890i
							\(41\)	2063.36	+	3573.84i	0.191697	+	0.332029i	0.945813	−	0.324713i	\(-0.105268\pi\)
−0.754116	+	0.656741i	\(0.771934\pi\)
\(43\)	7463.57			0.615567			0.307783	−	0.951456i	\(-0.400413\pi\)
							0.307783	+	0.951456i	\(0.400413\pi\)
\(47\)	−12659.6			−0.835941			−0.417971	−	0.908461i	\(-0.637258\pi\)
							−0.417971	+	0.908461i	\(0.637258\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.10.4	✓	30
37.26	even	3	inner	37.6.c.a.26.4	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.4	✓	30	1.1	even	1	trivial
37.6.c.a.26.4	yes	30	37.26	even	3	inner