Newform orbit 350.4.g.c

• Label: 350.4.g.c
• Level: $350$
• Weight: $4$
• Character orbit: 350.g
• Analytic conductor: $20.651$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(20.6506685020\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 80 q^{11} - 512 q^{16} + 1144 q^{21} - 1088 q^{36} - 4064 q^{46} + 352 q^{51} + 448 q^{56} + 4576 q^{71} + 2176 q^{81} - 3520 q^{86} + 4864 q^{91}+O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
293.1	−1.41421	+	1.41421i	−4.63728	+	4.63728i		−	4.00000i		0			−	13.1162i	−13.7309	+	12.4283i	5.65685	+	5.65685i		−	16.0088i		0
293.2	−1.41421	+	1.41421i	−6.26739	+	6.26739i		−	4.00000i		0			−	17.7269i	−13.1131	−	13.0785i	5.65685	+	5.65685i		−	51.5604i		0
293.3	−1.41421	+	1.41421i	−1.36793	+	1.36793i		−	4.00000i		0			−	3.86909i	−0.863384	−	18.5001i	5.65685	+	5.65685i			23.2575i		0
293.4	−1.41421	+	1.41421i	1.36793	−	1.36793i		−	4.00000i		0				3.86909i	18.5001	+	0.863384i	5.65685	+	5.65685i			23.2575i		0
293.5	−1.41421	+	1.41421i	6.26739	−	6.26739i		−	4.00000i		0				17.7269i	13.0785	+	13.1131i	5.65685	+	5.65685i		−	51.5604i		0
293.6	−1.41421	+	1.41421i	2.88863	−	2.88863i		−	4.00000i		0				8.17027i	−0.0636818	−	18.5201i	5.65685	+	5.65685i			10.3117i		0
293.7	−1.41421	+	1.41421i	−2.88863	+	2.88863i		−	4.00000i		0			−	8.17027i	18.5201	+	0.0636818i	5.65685	+	5.65685i			10.3117i		0
293.8	−1.41421	+	1.41421i	4.63728	−	4.63728i		−	4.00000i		0				13.1162i	−12.4283	+	13.7309i	5.65685	+	5.65685i		−	16.0088i		0
293.9	1.41421	−	1.41421i	1.36793	−	1.36793i		−	4.00000i		0			−	3.86909i	0.863384	+	18.5001i	−5.65685	−	5.65685i			23.2575i		0
293.10	1.41421	−	1.41421i	4.63728	−	4.63728i		−	4.00000i		0			−	13.1162i	13.7309	−	12.4283i	−5.65685	−	5.65685i		−	16.0088i		0
293.11	1.41421	−	1.41421i	−6.26739	+	6.26739i		−	4.00000i		0				17.7269i	−13.0785	−	13.1131i	−5.65685	−	5.65685i		−	51.5604i		0
293.12	1.41421	−	1.41421i	−4.63728	+	4.63728i		−	4.00000i		0				13.1162i	12.4283	−	13.7309i	−5.65685	−	5.65685i		−	16.0088i		0
293.13	1.41421	−	1.41421i	6.26739	−	6.26739i		−	4.00000i		0			−	17.7269i	13.1131	+	13.0785i	−5.65685	−	5.65685i		−	51.5604i		0
293.14	1.41421	−	1.41421i	−2.88863	+	2.88863i		−	4.00000i		0				8.17027i	0.0636818	+	18.5201i	−5.65685	−	5.65685i			10.3117i		0
293.15	1.41421	−	1.41421i	2.88863	−	2.88863i		−	4.00000i		0			−	8.17027i	−18.5201	−	0.0636818i	−5.65685	−	5.65685i			10.3117i		0
293.16	1.41421	−	1.41421i	−1.36793	+	1.36793i		−	4.00000i		0				3.86909i	−18.5001	−	0.863384i	−5.65685	−	5.65685i			23.2575i		0
307.1	−1.41421	−	1.41421i	−4.63728	−	4.63728i			4.00000i		0				13.1162i	−13.7309	−	12.4283i	5.65685	−	5.65685i			16.0088i		0
307.2	−1.41421	−	1.41421i	−6.26739	−	6.26739i			4.00000i		0				17.7269i	−13.1131	+	13.0785i	5.65685	−	5.65685i			51.5604i		0
307.3	−1.41421	−	1.41421i	−1.36793	−	1.36793i			4.00000i		0				3.86909i	−0.863384	+	18.5001i	5.65685	−	5.65685i		−	23.2575i		0
307.4	−1.41421	−	1.41421i	1.36793	+	1.36793i			4.00000i		0			−	3.86909i	18.5001	−	0.863384i	5.65685	−	5.65685i		−	23.2575i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner
35.f	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.4.g.c	✓	32
5.b	even	2	1	inner	350.4.g.c	✓	32
5.c	odd	4	2	inner	350.4.g.c	✓	32
7.b	odd	2	1	inner	350.4.g.c	✓	32
35.c	odd	2	1	inner	350.4.g.c	✓	32
35.f	even	4	2	inner	350.4.g.c	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
350.4.g.c	✓	32	1.a	even	1	1	trivial
350.4.g.c	✓	32	5.b	even	2	1	inner
350.4.g.c	✓	32	5.c	odd	4	2	inner
350.4.g.c	✓	32	7.b	odd	2	1	inner
350.4.g.c	✓	32	35.c	odd	2	1	inner
350.4.g.c	✓	32	35.f	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 8314T_{3}^{12} + 13766457T_{3}^{8} + 3370602976T_{3}^{4} + 44531128576 \) acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\).