Embedded newform 37.8.c.a.10.16

• Label: 37.8.c.a.10.16
• Level: $37$
• Weight: $8$
• Character: 37.10
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.16
Character	\(\chi\)	\(=\)	37.10
Dual form			37.8.c.a.26.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(4.93583 - 8.54911i) q^{2} +(-6.61210 - 11.4525i) q^{3} +(15.2752 + 26.4574i) q^{4} +(56.3382 + 97.5806i) q^{5} -130.545 q^{6} +(-783.646 - 1357.32i) q^{7} +1565.16 q^{8} +(1006.06 - 1742.55i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 16 q^{2} - 68 q^{3} - 1126 q^{4} - 337 q^{5} + 252 q^{6} - 158 q^{7} + 6264 q^{8} - 12773 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	4.93583	−	8.54911i	0.436270	−	0.755641i	−0.561129	−	0.827729i	\(-0.689633\pi\)
							0.997398	+	0.0720873i	\(0.0229660\pi\)
\(3\)	−6.61210	−	11.4525i	−0.141389	−	0.244893i	0.786631	−	0.617423i	\(-0.211823\pi\)
							−0.928020	+	0.372531i	\(0.878490\pi\)
\(5\)	56.3382	+	97.5806i	0.201562	+	0.349115i	0.949032	−	0.315180i	\(-0.102065\pi\)
							−0.747470	+	0.664295i	\(0.768732\pi\)
\(7\)	−783.646	−	1357.32i	−0.863529	−	1.49568i	−0.868500	−	0.495689i	\(-0.834916\pi\)
							0.00497098	−	0.999988i	\(-0.498418\pi\)
\(11\)	−163.497			−0.0370370			−0.0185185	−	0.999829i	\(-0.505895\pi\)
							−0.0185185	+	0.999829i	\(0.505895\pi\)
\(13\)	−5397.29	−	9348.38i	−0.681356	−	1.18014i	−0.974567	−	0.224095i	\(-0.928057\pi\)
							0.293211	−	0.956048i	\(-0.405276\pi\)
\(17\)	−3758.50	+	6509.91i	−0.185542	+	0.321369i	−0.943759	−	0.330634i	\(-0.892738\pi\)
							0.758217	+	0.652003i	\(0.226071\pi\)
\(19\)	−1006.68	−	1743.61i	−0.0336707	−	0.0583194i	0.848699	−	0.528876i	\(-0.177386\pi\)
							−0.882370	+	0.470557i	\(0.844053\pi\)
\(23\)	−11630.0			−0.199311			−0.0996555	−	0.995022i	\(-0.531774\pi\)
							−0.0996555	+	0.995022i	\(0.531774\pi\)
\(29\)	3055.75			0.0232662			0.0116331	−	0.999932i	\(-0.496297\pi\)
							0.0116331	+	0.999932i	\(0.496297\pi\)
\(31\)	230142.			1.38749			0.693746	−	0.720220i	\(-0.255959\pi\)
							0.693746	+	0.720220i	\(0.255959\pi\)
\(37\)	−300186.	−	69427.7i	−0.974282	−	0.225334i
							\(41\)	286024.	+	495408.i	0.648125	+	1.12258i	0.983570	+	0.180525i	\(0.0577798\pi\)
−0.335446	+	0.942060i	\(0.608887\pi\)
\(43\)	152488.			0.292479			0.146240	−	0.989249i	\(-0.453283\pi\)
							0.146240	+	0.989249i	\(0.453283\pi\)
\(47\)	1.07566e6			1.51124			0.755620	−	0.655010i	\(-0.227336\pi\)
							0.755620	+	0.655010i	\(0.227336\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.c.a.10.16	✓	42
37.26	even	3	inner	37.8.c.a.26.16	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.c.a.10.16	✓	42	1.1	even	1	trivial
37.8.c.a.26.16	yes	42	37.26	even	3	inner