Embedded newform 37.6.c.a.10.12

• Label: 37.6.c.a.10.12
• 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			10.12
Character	\(\chi\)	\(=\)	37.10
Dual form			37.6.c.a.26.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.64161 - 6.30745i) q^{2} +(-11.9080 - 20.6253i) q^{3} +(-10.5226 - 18.2257i) q^{4} +(-4.92186 - 8.52492i) q^{5} -173.457 q^{6} +(-8.42954 - 14.6004i) q^{7} +79.7857 q^{8} +(-162.101 + 280.767i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.64161	−	6.30745i	0.643752	−	1.11501i	−0.340837	−	0.940123i	\(-0.610710\pi\)
							0.984588	−	0.174888i	\(-0.0559563\pi\)
\(3\)	−11.9080	−	20.6253i	−0.763899	−	1.32311i	−0.940827	−	0.338887i	\(-0.889949\pi\)
							0.176928	−	0.984224i	\(-0.443384\pi\)
\(5\)	−4.92186	−	8.52492i	−0.0880449	−	0.152498i	0.818640	−	0.574307i	\(-0.194729\pi\)
							−0.906685	+	0.421809i	\(0.861395\pi\)
\(7\)	−8.42954	−	14.6004i	−0.0650218	−	0.112621i	0.831682	−	0.555252i	\(-0.187378\pi\)
							−0.896704	+	0.442631i	\(0.854045\pi\)
\(11\)	−239.280			−0.596244			−0.298122	−	0.954528i	\(-0.596360\pi\)
							−0.298122	+	0.954528i	\(0.596360\pi\)
\(13\)	47.9132	+	82.9882i	0.0786316	+	0.136194i	0.902660	−	0.430355i	\(-0.141612\pi\)
							−0.824028	+	0.566549i	\(0.808278\pi\)
\(17\)	443.781	−	768.652i	0.372432	−	0.645071i	−0.617507	−	0.786565i	\(-0.711857\pi\)
							0.989939	+	0.141494i	\(0.0451907\pi\)
\(19\)	−873.426	−	1512.82i	−0.555063	−	0.961397i	−0.997899	−	0.0647945i	\(-0.979361\pi\)
							0.442836	−	0.896603i	\(-0.353973\pi\)
\(23\)	−2355.39			−0.928419			−0.464210	−	0.885725i	\(-0.653662\pi\)
							−0.464210	+	0.885725i	\(0.653662\pi\)
\(29\)	5754.27			1.27056			0.635280	−	0.772282i	\(-0.280885\pi\)
							0.635280	+	0.772282i	\(0.280885\pi\)
\(31\)	6186.18			1.15616			0.578081	−	0.815980i	\(-0.303802\pi\)
							0.578081	+	0.815980i	\(0.303802\pi\)
\(37\)	907.510	+	8277.70i	0.108980	+	0.994044i
							\(41\)	2332.53	+	4040.06i	0.216705	+	0.375343i	0.953799	−	0.300447i	\(-0.0971358\pi\)
−0.737094	+	0.675790i	\(0.763803\pi\)
\(43\)	6873.41			0.566893			0.283446	−	0.958988i	\(-0.408522\pi\)
							0.283446	+	0.958988i	\(0.408522\pi\)
\(47\)	−12040.1			−0.795031			−0.397515	−	0.917596i	\(-0.630127\pi\)
							−0.397515	+	0.917596i	\(0.630127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.10.12	✓	30
37.26	even	3	inner	37.6.c.a.26.12	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.12	✓	30	1.1	even	1	trivial
37.6.c.a.26.12	yes	30	37.26	even	3	inner