Embedded newform 144.2.u.a.131.8

• Label: 144.2.u.a.131.8
• Level: $144$
• Weight: $2$
• Character: 144.131
• Analytic conductor: $1.150$
• Analytic rank: $0$
• Dimension: $88$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 144 = 2^{4} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	144.u (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.14984578911\)
Analytic rank:	\(0\)
Dimension:	\(88\)
Relative dimension:	\(22\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			131.8
Character	\(\chi\)	\(=\)	144.131
Dual form			144.2.u.a.11.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.700562 + 1.22850i) q^{2} +(1.70352 + 0.313101i) q^{3} +(-1.01842 - 1.72128i) q^{4} +(2.80938 - 0.752772i) q^{5} +(-1.57806 + 1.87342i) q^{6} +(-1.02581 - 1.77675i) q^{7} +(2.82806 - 0.0452695i) q^{8} +(2.80394 + 1.06674i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 6 q^{2} - 4 q^{3} - 2 q^{4} - 6 q^{5} + 2 q^{6} - 4 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\)	\(37\)	\(65\)	\(127\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.700562	+	1.22850i	−0.495372	+	0.868681i
							\(3\)	1.70352	+	0.313101i	0.983526	+	0.180769i
\(5\)	2.80938	−	0.752772i	1.25639	−	0.336650i								0.431591	−	0.902069i	\(-0.357952\pi\)
							0.824804	+	0.565419i	\(0.191286\pi\)
\(7\)	−1.02581	−	1.77675i	−0.387718	−	0.671548i	0.604424	−	0.796663i	\(-0.293403\pi\)
							−0.992142	+	0.125115i	\(0.960070\pi\)
\(11\)	−1.67103	−	0.447752i	−0.503835	−	0.135002i	−0.00205598	−	0.999998i	\(-0.500654\pi\)
							−0.501779	+	0.864996i	\(0.667321\pi\)
\(13\)	−4.86091	+	1.30248i	−1.34817	+	0.361242i	−0.859460	−	0.511203i	\(-0.829200\pi\)
							−0.488713	+	0.872445i	\(0.662534\pi\)
\(17\)			2.90960i			0.705681i	0.935683	+	0.352841i	\(0.114784\pi\)
							−0.935683	+	0.352841i	\(0.885216\pi\)
\(19\)	−3.23424	+	3.23424i	−0.741986	+	0.741986i	−0.972960	−	0.230974i	\(-0.925809\pi\)
							0.230974	+	0.972960i	\(0.425809\pi\)
\(23\)	−0.831633	−	0.480143i	−0.173407	−	0.100117i	0.410784	−	0.911733i	\(-0.365255\pi\)
							−0.584192	+	0.811616i	\(0.698588\pi\)
\(29\)	5.55800	+	1.48926i	1.03209	+	0.276549i	0.734834	−	0.678248i	\(-0.237260\pi\)
							0.297261	+	0.954796i	\(0.403927\pi\)
\(31\)	−8.00513	−	4.62177i	−1.43776	−	0.830094i	−0.440070	−	0.897963i	\(-0.645046\pi\)
							−0.997694	+	0.0678697i	\(0.978380\pi\)
\(37\)	8.18664	−	8.18664i	1.34588	−	1.34588i	0.455787	−	0.890089i	\(-0.349358\pi\)
							0.890089	−	0.455787i	\(-0.150642\pi\)
\(41\)	−4.06472	+	7.04030i	−0.634803	+	1.09951i	0.351754	+	0.936092i	\(0.385585\pi\)
							−0.986557	+	0.163418i	\(0.947748\pi\)
\(43\)	0.907834	−	3.38808i	0.138443	−	0.516678i	−0.861517	−	0.507729i	\(-0.830485\pi\)
							0.999960	−	0.00894834i	\(-0.00284838\pi\)
\(47\)	−1.60711	−	2.78360i	−0.234421	−	0.406030i	0.724683	−	0.689082i	\(-0.241986\pi\)
							−0.959104	+	0.283053i	\(0.908653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	144.2.u.a.131.8	yes	88
3.2	odd	2		432.2.v.a.179.15		88
4.3	odd	2		576.2.y.a.239.1		88
9.2	odd	6	inner	144.2.u.a.83.1	yes	88
9.7	even	3		432.2.v.a.35.22		88
12.11	even	2		1728.2.z.a.1583.3		88
16.5	even	4		576.2.y.a.527.12		88
16.11	odd	4	inner	144.2.u.a.59.1	yes	88
36.7	odd	6		1728.2.z.a.1007.3		88
36.11	even	6		576.2.y.a.47.12		88
48.5	odd	4		1728.2.z.a.719.3		88
48.11	even	4		432.2.v.a.395.22		88
144.11	even	12	inner	144.2.u.a.11.8	✓	88
144.43	odd	12		432.2.v.a.251.15		88
144.101	odd	12		576.2.y.a.335.1		88
144.133	even	12		1728.2.z.a.143.3		88

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.u.a.11.8	✓	88	144.11	even	12	inner
144.2.u.a.59.1	yes	88	16.11	odd	4	inner
144.2.u.a.83.1	yes	88	9.2	odd	6	inner
144.2.u.a.131.8	yes	88	1.1	even	1	trivial
432.2.v.a.35.22		88	9.7	even	3
432.2.v.a.179.15		88	3.2	odd	2
432.2.v.a.251.15		88	144.43	odd	12
432.2.v.a.395.22		88	48.11	even	4
576.2.y.a.47.12		88	36.11	even	6
576.2.y.a.239.1		88	4.3	odd	2
576.2.y.a.335.1		88	144.101	odd	12
576.2.y.a.527.12		88	16.5	even	4
1728.2.z.a.143.3		88	144.133	even	12
1728.2.z.a.719.3		88	48.5	odd	4
1728.2.z.a.1007.3		88	36.7	odd	6
1728.2.z.a.1583.3		88	12.11	even	2