Embedded newform 37.11.g.a.8.9

• Label: 37.11.g.a.8.9
• Level: $37$
• Weight: $11$
• Character: 37.8
• Analytic conductor: $23.508$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	37.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(23.5082183489\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(30\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			8.9
Character	\(\chi\)	\(=\)	37.8
Dual form			37.11.g.a.14.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-27.9992 - 7.50237i) q^{2} +(-35.7432 - 20.6364i) q^{3} +(-159.139 - 91.8788i) q^{4} +(-1410.80 + 378.024i) q^{5} +(845.961 + 845.961i) q^{6} +(7821.46 - 13547.2i) q^{7} +(24755.2 + 24755.2i) q^{8} +(-28672.8 - 49662.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} - 6 q^{3} - 3768 q^{4} + 2922 q^{5} - 2052 q^{6} - 2 q^{7} + 20880 q^{8} + 1052964 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−27.9992	−	7.50237i	−0.874976	−	0.234449i	−0.206738	−	0.978396i	\(-0.566285\pi\)
							−0.668238	+	0.743947i	\(0.732951\pi\)
\(3\)	−35.7432	−	20.6364i	−0.147092	−	0.0849233i	0.424648	−	0.905359i	\(-0.360398\pi\)
							−0.571739	+	0.820435i	\(0.693731\pi\)
\(5\)	−1410.80	+	378.024i	−0.451457	+	0.120968i	−0.477381	−	0.878696i	\(-0.658414\pi\)
							0.0259239	+	0.999664i	\(0.491747\pi\)
\(7\)	7821.46	−	13547.2i	0.465369	−	0.806043i	−0.533849	−	0.845580i	\(-0.679255\pi\)
							0.999218	+	0.0395371i	\(0.0125883\pi\)
\(11\)		−	97689.2i		−	0.606573i	−0.952899	−	0.303287i	\(-0.901916\pi\)
							0.952899	−	0.303287i	\(-0.0980839\pi\)
\(13\)	204433.	−	54777.6i	0.550597	−	0.147532i	0.0272143	−	0.999630i	\(-0.491336\pi\)
							0.523383	+	0.852098i	\(0.324670\pi\)
\(17\)	508188.	−	1.89658e6i	0.357915	−	1.33576i	−0.518861	−	0.854859i	\(-0.673644\pi\)
							0.876776	−	0.480899i	\(-0.159690\pi\)
\(19\)	2.02481e6	−	542546.i	0.817742	−	0.219113i	0.174383	−	0.984678i	\(-0.444207\pi\)
							0.643359	+	0.765565i	\(0.277540\pi\)
\(23\)	−2.10108e6	−	2.10108e6i	−0.326440	−	0.326440i	0.524791	−	0.851231i	\(-0.324143\pi\)
							−0.851231	+	0.524791i	\(0.824143\pi\)
\(29\)	−1.55989e7	+	1.55989e7i	−0.760506	+	0.760506i	−0.976414	−	0.215908i	\(-0.930729\pi\)
							0.215908	+	0.976414i	\(0.430729\pi\)
\(31\)	−1.79307e7	+	1.79307e7i	−0.626310	+	0.626310i	−0.947138	−	0.320827i	\(-0.896039\pi\)
							0.320827	+	0.947138i	\(0.396039\pi\)
\(37\)	−2.02045e7	+	6.63352e7i	−0.291367	+	0.956611i
							\(41\)	−3.95001e7	−	2.28054e7i	−0.340941	−	0.196842i	0.319747	−	0.947503i	\(-0.396402\pi\)
−0.660688	+	0.750661i	\(0.729735\pi\)
\(43\)	−8.51064e7	−	8.51064e7i	−0.578922	−	0.578922i	0.355684	−	0.934606i	\(-0.384248\pi\)
							−0.934606	+	0.355684i	\(0.884248\pi\)
\(47\)	8.55577e7			0.373052			0.186526	−	0.982450i	\(-0.440277\pi\)
							0.186526	+	0.982450i	\(0.440277\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.11.g.a.8.9	✓	120
37.14	odd	12	inner	37.11.g.a.14.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.11.g.a.8.9	✓	120	1.1	even	1	trivial
37.11.g.a.14.9	yes	120	37.14	odd	12	inner