Embedded newform 126.3.p.a.103.9

• Label: 126.3.p.a.103.9
• Level: $126$
• Weight: $3$
• Character: 126.103
• 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.p (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			103.9
Character	\(\chi\)	\(=\)	126.103
Dual form			126.3.p.a.115.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +(-2.99238 - 0.213630i) q^{3} +2.00000 q^{4} +(-6.01208 + 3.47108i) q^{5} +(-4.23187 - 0.302118i) q^{6} +(-6.97949 + 0.535409i) q^{7} +2.82843 q^{8} +(8.90872 + 1.27852i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} - 2 q^{7} + 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(73\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41421			0.707107
							\(3\)	−2.99238	−	0.213630i	−0.997461	−	0.0712098i
\(5\)	−6.01208	+	3.47108i	−1.20242	+	0.694216i								−0.961092	−	0.276229i	\(-0.910915\pi\)
							−0.241325	+	0.970444i	\(0.577582\pi\)
\(7\)	−6.97949	+	0.535409i	−0.997071	+	0.0764870i
							\(11\)	−8.72855	+	15.1183i	−0.793504	+	1.37439i	0.130280	+	0.991477i	\(0.458412\pi\)
−0.923785	+	0.382913i	\(0.874921\pi\)
\(13\)	−13.3362	−	7.69965i	−1.02586	−	0.592281i	−0.110065	−	0.993924i	\(-0.535106\pi\)
							−0.915796	+	0.401643i	\(0.868439\pi\)
\(17\)	12.3517	−	7.13128i	0.726573	−	0.419487i	−0.0905943	−	0.995888i	\(-0.528877\pi\)
							0.817167	+	0.576401i	\(0.195543\pi\)
\(19\)	1.25450	+	0.724286i	0.0660264	+	0.0381203i	0.532650	−	0.846336i	\(-0.321196\pi\)
							−0.466623	+	0.884456i	\(0.654530\pi\)
\(23\)	6.43077	+	11.1384i	0.279599	+	0.484279i	0.971285	−	0.237919i	\(-0.0764653\pi\)
							−0.691686	+	0.722198i	\(0.743132\pi\)
\(29\)	−2.98967	−	5.17826i	−0.103092	−	0.178561i	0.809865	−	0.586616i	\(-0.199540\pi\)
							−0.912957	+	0.408056i	\(0.866207\pi\)
\(31\)			16.4679i			0.531223i	0.964080	+	0.265611i	\(0.0855738\pi\)
							−0.964080	+	0.265611i	\(0.914426\pi\)
\(37\)	−0.732115	+	1.26806i	−0.0197869	+	0.0342719i	−0.875749	−	0.482766i	\(-0.839632\pi\)
							0.855962	+	0.517038i	\(0.172965\pi\)
\(41\)	10.5856	+	6.11162i	0.258186	+	0.149064i	0.623507	−	0.781818i	\(-0.285707\pi\)
							−0.365321	+	0.930882i	\(0.619041\pi\)
\(43\)	−40.1499	−	69.5416i	−0.933718	−	1.61725i	−0.776904	−	0.629619i	\(-0.783211\pi\)
							−0.156814	−	0.987628i	\(-0.550122\pi\)
\(47\)			83.6474i			1.77973i	0.456223	+	0.889866i	\(0.349202\pi\)
							−0.456223	+	0.889866i	\(0.650798\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.j.a.31.3	✓	32	7.3	odd	6
126.3.j.a.61.3	yes	32	9.7	even	3
126.3.p.a.103.9	yes	32	1.1	even	1	trivial
126.3.p.a.115.9	yes	32	63.52	odd	6	inner
378.3.j.a.19.15		32	9.2	odd	6
378.3.j.a.199.10		32	21.17	even	6
378.3.p.a.73.7		32	63.38	even	6
378.3.p.a.145.7		32	3.2	odd	2