Embedded newform 1050.4.a.g.1.1

• Label: 1050.4.a.g.1.1
• Level: $1050$
• Weight: $4$
• Character: 1050.1
• Self dual: yes
• Analytic conductor: $61.952$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1050.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(61.9520055060\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1050.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)

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)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−0.707107
			\(3\)	3.00000	0.577350
\(5\)	0	0
			\(7\)	−7.00000	−0.377964
\(11\)	−72.0000	−1.97353				−0.986764	−	0.162160i	\(-0.948154\pi\)
			−0.986764	+	0.162160i	\(0.948154\pi\)
\(13\)	34.0000	0.725377	0.362689	−	0.931910i	\(-0.381859\pi\)
			0.362689	+	0.931910i	\(0.381859\pi\)
\(17\)	−6.00000	−0.0856008	−0.0428004	−	0.999084i	\(-0.513628\pi\)
			−0.0428004	+	0.999084i	\(0.513628\pi\)
\(19\)	92.0000	1.11086	0.555428	−	0.831565i	\(-0.312555\pi\)
			0.555428	+	0.831565i	\(0.312555\pi\)
\(23\)	180.000	1.63185	0.815926	−	0.578156i	\(-0.196228\pi\)
			0.815926	+	0.578156i	\(0.196228\pi\)
\(29\)	−114.000	−0.729975	−0.364987	−	0.931012i	\(-0.618927\pi\)
			−0.364987	+	0.931012i	\(0.618927\pi\)
\(31\)	56.0000	0.324448	0.162224	−	0.986754i	\(-0.448133\pi\)
			0.162224	+	0.986754i	\(0.448133\pi\)
\(37\)	34.0000	0.151069	0.0755347	−	0.997143i	\(-0.475934\pi\)
			0.0755347	+	0.997143i	\(0.475934\pi\)
\(41\)	6.00000	0.0228547	0.0114273	−	0.999935i	\(-0.496362\pi\)
			0.0114273	+	0.999935i	\(0.496362\pi\)
\(43\)	−164.000	−0.581622	−0.290811	−	0.956780i	\(-0.593925\pi\)
			−0.290811	+	0.956780i	\(0.593925\pi\)
\(47\)	−168.000	−0.521390	−0.260695	−	0.965421i	\(-0.583952\pi\)
			−0.260695	+	0.965421i	\(0.583952\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.4.a.g.1.1		1
5.2	odd	4		1050.4.g.a.799.1		2
5.3	odd	4		1050.4.g.a.799.2		2
5.4	even	2		42.4.a.a.1.1	✓	1
15.14	odd	2		126.4.a.a.1.1		1
20.19	odd	2		336.4.a.l.1.1		1
35.4	even	6		294.4.e.c.79.1		2
35.9	even	6		294.4.e.c.67.1		2
35.19	odd	6		294.4.e.b.67.1		2
35.24	odd	6		294.4.e.b.79.1		2
35.34	odd	2		294.4.a.i.1.1		1
40.19	odd	2		1344.4.a.a.1.1		1
40.29	even	2		1344.4.a.o.1.1		1
60.59	even	2		1008.4.a.b.1.1		1
105.44	odd	6		882.4.g.w.361.1		2
105.59	even	6		882.4.g.o.667.1		2
105.74	odd	6		882.4.g.w.667.1		2
105.89	even	6		882.4.g.o.361.1		2
105.104	even	2		882.4.a.g.1.1		1
140.139	even	2		2352.4.a.a.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.a.1.1	✓	1	5.4	even	2
126.4.a.a.1.1		1	15.14	odd	2
294.4.a.i.1.1		1	35.34	odd	2
294.4.e.b.67.1		2	35.19	odd	6
294.4.e.b.79.1		2	35.24	odd	6
294.4.e.c.67.1		2	35.9	even	6
294.4.e.c.79.1		2	35.4	even	6
336.4.a.l.1.1		1	20.19	odd	2
882.4.a.g.1.1		1	105.104	even	2
882.4.g.o.361.1		2	105.89	even	6
882.4.g.o.667.1		2	105.59	even	6
882.4.g.w.361.1		2	105.44	odd	6
882.4.g.w.667.1		2	105.74	odd	6
1008.4.a.b.1.1		1	60.59	even	2
1050.4.a.g.1.1		1	1.1	even	1	trivial
1050.4.g.a.799.1		2	5.2	odd	4
1050.4.g.a.799.2		2	5.3	odd	4
1344.4.a.a.1.1		1	40.19	odd	2
1344.4.a.o.1.1		1	40.29	even	2
2352.4.a.a.1.1		1	140.139	even	2