Embedded newform 126.3.j.a.31.3

• Label: 126.3.j.a.31.3
• Level: $126$
• Weight: $3$
• Character: 126.31
• Analytic conductor: $3.433$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	126.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.43325133094\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	126.31
Dual form			126.3.j.a.61.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.22474i) q^{2} +(-1.31118 - 2.69830i) q^{3} +(-1.00000 - 1.73205i) q^{4} +6.94216i q^{5} +(4.23187 + 0.302118i) q^{6} +(3.95343 - 5.77671i) q^{7} +2.82843 q^{8} +(-5.56160 + 7.07592i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{4} - 2 q^{7} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(73\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.707107	+	1.22474i	−0.353553	+	0.612372i
							\(3\)	−1.31118	−	2.69830i	−0.437061	−	0.899432i
\(5\)			6.94216i			1.38843i								0.719767	+	0.694216i	\(0.244249\pi\)
							−0.719767	+	0.694216i	\(0.755751\pi\)
\(7\)	3.95343	−	5.77671i	0.564775	−	0.825245i
							\(11\)	17.4571			1.58701			0.793504	−	0.608565i	\(-0.208254\pi\)
0.793504	+	0.608565i	\(0.208254\pi\)
\(13\)	13.3362	+	7.69965i	1.02586	+	0.592281i	0.915796	−	0.401643i	\(-0.131561\pi\)
							0.110065	+	0.993924i	\(0.464894\pi\)
\(17\)	12.3517	+	7.13128i	0.726573	+	0.419487i	0.817167	−	0.576401i	\(-0.195543\pi\)
							−0.0905943	+	0.995888i	\(0.528877\pi\)
\(19\)	1.25450	−	0.724286i	0.0660264	−	0.0381203i	−0.466623	−	0.884456i	\(-0.654530\pi\)
							0.532650	+	0.846336i	\(0.321196\pi\)
\(23\)	−12.8615			−0.559197			−0.279599	−	0.960117i	\(-0.590201\pi\)
							−0.279599	+	0.960117i	\(0.590201\pi\)
\(29\)	−2.98967	−	5.17826i	−0.103092	−	0.178561i	0.809865	−	0.586616i	\(-0.199540\pi\)
							−0.912957	+	0.408056i	\(0.866207\pi\)
\(31\)	−14.2616	+	8.23395i	−0.460052	+	0.265611i	−0.712066	−	0.702112i	\(-0.752240\pi\)
							0.252014	+	0.967724i	\(0.418907\pi\)
\(37\)	−0.732115	−	1.26806i	−0.0197869	−	0.0342719i	0.855962	−	0.517038i	\(-0.172965\pi\)
							−0.875749	+	0.482766i	\(0.839632\pi\)
\(41\)	−10.5856	−	6.11162i	−0.258186	−	0.149064i	0.365321	−	0.930882i	\(-0.380959\pi\)
							−0.623507	+	0.781818i	\(0.714293\pi\)
\(43\)	−40.1499	−	69.5416i	−0.933718	−	1.61725i	−0.776904	−	0.629619i	\(-0.783211\pi\)
							−0.156814	−	0.987628i	\(-0.550122\pi\)
\(47\)	72.4407	+	41.8237i	1.54129	+	0.889866i	0.998758	+	0.0498323i	\(0.0158687\pi\)
							0.542535	+	0.840033i	\(0.317465\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.3.j.a.31.3	✓	32
3.2	odd	2		378.3.j.a.199.10		32
7.5	odd	6		126.3.p.a.103.9	yes	32
9.2	odd	6		378.3.p.a.73.7		32
9.7	even	3		126.3.p.a.115.9	yes	32
21.5	even	6		378.3.p.a.145.7		32
63.47	even	6		378.3.j.a.19.15		32
63.61	odd	6	inner	126.3.j.a.61.3	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.j.a.31.3	✓	32	1.1	even	1	trivial
126.3.j.a.61.3	yes	32	63.61	odd	6	inner
126.3.p.a.103.9	yes	32	7.5	odd	6
126.3.p.a.115.9	yes	32	9.7	even	3
378.3.j.a.19.15		32	63.47	even	6
378.3.j.a.199.10		32	3.2	odd	2
378.3.p.a.73.7		32	9.2	odd	6
378.3.p.a.145.7		32	21.5	even	6