Embedded newform 350.4.a.f.1.1

• Label: 350.4.a.f.1.1
• Level: $350$
• Weight: $4$
• Character: 350.1
• Self dual: yes
• Analytic conductor: $20.651$
• Analytic rank: $0$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(20.6506685020\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 14)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	350.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{2} +2.00000 q^{3} +4.00000 q^{4} -4.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} -23.0000 q^{9} +O(q^{10})\)

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
			\(3\)	2.00000	0.384900	0.192450	−	0.981307i	\(-0.438357\pi\)
0.192450	+	0.981307i				\(0.438357\pi\)
\(5\)	0	0
			\(7\)	−7.00000	−0.377964
\(11\)	48.0000	1.31569				0.657843	−	0.753155i	\(-0.271469\pi\)
			0.657843	+	0.753155i	\(0.271469\pi\)
\(13\)	−56.0000	−1.19474	−0.597369	−	0.801966i	\(-0.703787\pi\)
			−0.597369	+	0.801966i	\(0.703787\pi\)
\(17\)	114.000	1.62642	0.813208	−	0.581974i	\(-0.197719\pi\)
			0.813208	+	0.581974i	\(0.197719\pi\)
\(19\)	2.00000	0.0241490	0.0120745	−	0.999927i	\(-0.496156\pi\)
			0.0120745	+	0.999927i	\(0.496156\pi\)
\(23\)	120.000	1.08790	0.543951	−	0.839117i	\(-0.316928\pi\)
			0.543951	+	0.839117i	\(0.316928\pi\)
\(29\)	−54.0000	−0.345778	−0.172889	−	0.984941i	\(-0.555310\pi\)
			−0.172889	+	0.984941i	\(0.555310\pi\)
\(31\)	236.000	1.36732	0.683659	−	0.729802i	\(-0.260388\pi\)
			0.683659	+	0.729802i	\(0.260388\pi\)
\(37\)	−146.000	−0.648710	−0.324355	−	0.945936i	\(-0.605147\pi\)
			−0.324355	+	0.945936i	\(0.605147\pi\)
\(41\)	126.000	0.479949	0.239974	−	0.970779i	\(-0.422861\pi\)
			0.239974	+	0.970779i	\(0.422861\pi\)
\(43\)	376.000	1.33348	0.666738	−	0.745292i	\(-0.267690\pi\)
			0.666738	+	0.745292i	\(0.267690\pi\)
\(47\)	12.0000	0.0372421	0.0186211	−	0.999827i	\(-0.494072\pi\)
			0.0186211	+	0.999827i	\(0.494072\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.4.a.f.1.1		1
5.2	odd	4		350.4.c.g.99.1		2
5.3	odd	4		350.4.c.g.99.2		2
5.4	even	2		14.4.a.b.1.1	✓	1
7.6	odd	2		2450.4.a.i.1.1		1
15.14	odd	2		126.4.a.d.1.1		1
20.19	odd	2		112.4.a.e.1.1		1
35.4	even	6		98.4.c.c.79.1		2
35.9	even	6		98.4.c.c.67.1		2
35.19	odd	6		98.4.c.b.67.1		2
35.24	odd	6		98.4.c.b.79.1		2
35.34	odd	2		98.4.a.e.1.1		1
40.19	odd	2		448.4.a.g.1.1		1
40.29	even	2		448.4.a.k.1.1		1
55.54	odd	2		1694.4.a.b.1.1		1
60.59	even	2		1008.4.a.r.1.1		1
65.64	even	2		2366.4.a.c.1.1		1
105.44	odd	6		882.4.g.p.361.1		2
105.59	even	6		882.4.g.v.667.1		2
105.74	odd	6		882.4.g.p.667.1		2
105.89	even	6		882.4.g.v.361.1		2
105.104	even	2		882.4.a.b.1.1		1
140.139	even	2		784.4.a.h.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.4.a.b.1.1	✓	1	5.4	even	2
98.4.a.e.1.1		1	35.34	odd	2
98.4.c.b.67.1		2	35.19	odd	6
98.4.c.b.79.1		2	35.24	odd	6
98.4.c.c.67.1		2	35.9	even	6
98.4.c.c.79.1		2	35.4	even	6
112.4.a.e.1.1		1	20.19	odd	2
126.4.a.d.1.1		1	15.14	odd	2
350.4.a.f.1.1		1	1.1	even	1	trivial
350.4.c.g.99.1		2	5.2	odd	4
350.4.c.g.99.2		2	5.3	odd	4
448.4.a.g.1.1		1	40.19	odd	2
448.4.a.k.1.1		1	40.29	even	2
784.4.a.h.1.1		1	140.139	even	2
882.4.a.b.1.1		1	105.104	even	2
882.4.g.p.361.1		2	105.44	odd	6
882.4.g.p.667.1		2	105.74	odd	6
882.4.g.v.361.1		2	105.89	even	6
882.4.g.v.667.1		2	105.59	even	6
1008.4.a.r.1.1		1	60.59	even	2
1694.4.a.b.1.1		1	55.54	odd	2
2366.4.a.c.1.1		1	65.64	even	2
2450.4.a.i.1.1		1	7.6	odd	2