Embedded newform 37.7.g.a.8.10

• Label: 37.7.g.a.8.10
• Level: $37$
• Weight: $7$
• Character: 37.8
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(18\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.10
Character	\(\chi\)	\(=\)	37.8
Dual form			37.7.g.a.14.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.46393 + 0.392259i) q^{2} +(-38.8847 - 22.4501i) q^{3} +(-53.4364 - 30.8515i) q^{4} +(78.2708 - 20.9726i) q^{5} +(-48.1182 - 48.1182i) q^{6} +(-50.3376 + 87.1872i) q^{7} +(-134.712 - 134.712i) q^{8} +(643.512 + 1114.60i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q - 2 q^{2} - 6 q^{3} + 264 q^{4} + 52 q^{5} - 132 q^{6} - 2 q^{7} - 1968 q^{8} + 8094 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.46393	+	0.392259i	0.182991	+	0.0490323i	0.349151	−	0.937067i	\(-0.386470\pi\)
							−0.166160	+	0.986099i	\(0.553137\pi\)
\(3\)	−38.8847	−	22.4501i	−1.44017	−	0.831485i	−0.442313	−	0.896861i	\(-0.645842\pi\)
							−0.997861	+	0.0653763i	\(0.979175\pi\)
\(5\)	78.2708	−	20.9726i	0.626166	−	0.167781i	0.0682368	−	0.997669i	\(-0.478263\pi\)
							0.557929	+	0.829888i	\(0.311596\pi\)
\(7\)	−50.3376	+	87.1872i	−0.146757	+	0.254190i	−0.930027	−	0.367491i	\(-0.880217\pi\)
							0.783270	+	0.621681i	\(0.213550\pi\)
\(11\)			1450.98i			1.09014i	0.838390	+	0.545070i	\(0.183497\pi\)
							−0.838390	+	0.545070i	\(0.816503\pi\)
\(13\)	19.9852	−	5.35501i	0.00909658	−	0.00243742i	−0.254268	−	0.967134i	\(-0.581834\pi\)
							0.263364	+	0.964696i	\(0.415168\pi\)
\(17\)	1410.37	−	5263.57i	0.287069	−	1.07136i	−0.660245	−	0.751050i	\(-0.729548\pi\)
							0.947314	−	0.320306i	\(-0.103786\pi\)
\(19\)	−5470.53	+	1465.82i	−0.797570	+	0.213708i	−0.634517	−	0.772909i	\(-0.718801\pi\)
							−0.163053	+	0.986617i	\(0.552134\pi\)
\(23\)	11150.6	+	11150.6i	0.916464	+	0.916464i	0.996770	−	0.0803060i	\(-0.0255898\pi\)
							−0.0803060	+	0.996770i	\(0.525590\pi\)
\(29\)	12353.7	−	12353.7i	0.506526	−	0.506526i	−0.406932	−	0.913458i	\(-0.633402\pi\)
							0.913458	+	0.406932i	\(0.133402\pi\)
\(31\)	−27722.2	+	27722.2i	−0.930556	+	0.930556i	−0.997741	−	0.0671850i	\(-0.978598\pi\)
							0.0671850	+	0.997741i	\(0.478598\pi\)
\(37\)	−49644.8	+	10055.9i	−0.980096	+	0.198526i
							\(41\)	−80418.2	−	46429.5i	−1.16682	−	0.673662i	−0.213889	−	0.976858i	\(-0.568613\pi\)
−0.952928	+	0.303196i	\(0.901946\pi\)
\(43\)	56993.6	+	56993.6i	0.716838	+	0.716838i	0.967956	−	0.251118i	\(-0.0807983\pi\)
							−0.251118	+	0.967956i	\(0.580798\pi\)
\(47\)	−59772.6			−0.575717			−0.287858	−	0.957673i	\(-0.592943\pi\)
							−0.287858	+	0.957673i	\(0.592943\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.g.a.8.10	✓	72
37.14	odd	12	inner	37.7.g.a.14.10	yes	72

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.g.a.8.10	✓	72	1.1	even	1	trivial
37.7.g.a.14.10	yes	72	37.14	odd	12	inner