Newform orbit 475.2.g.d

• Label: 475.2.g.d
• Level: $475$
• Weight: $2$
• Character orbit: 475.g
• Analytic conductor: $3.793$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.79289409601\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{6} + 24 q^{11} - 48 q^{16} - 24 q^{26} - 120 q^{36} + 88 q^{61} - 56 q^{66} + 176 q^{76} + 56 q^{81} - 208 q^{96}+O(q^{100}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
18.1	−1.82464	−	1.82464i	−0.857273	+	0.857273i			4.65859i		0		3.12842			−2.60677	+	2.60677i	4.85095	−	4.85095i			1.53017i		0
18.2	−1.82464	−	1.82464i	−0.857273	+	0.857273i			4.65859i		0		3.12842			2.60677	−	2.60677i	4.85095	−	4.85095i			1.53017i		0
18.3	−1.24481	−	1.24481i	0.534241	−	0.534241i			1.09911i		0		−1.33006			−2.85373	+	2.85373i	−1.12143	+	1.12143i			2.42917i		0
18.4	−1.24481	−	1.24481i	0.534241	−	0.534241i			1.09911i		0		−1.33006			2.85373	−	2.85373i	−1.12143	+	1.12143i			2.42917i		0
18.5	−0.348065	−	0.348065i	−1.72617	+	1.72617i		−	1.75770i		0		1.20164			−0.246957	+	0.246957i	−1.30792	+	1.30792i		−	2.95934i		0
18.6	−0.348065	−	0.348065i	−1.72617	+	1.72617i		−	1.75770i		0		1.20164			0.246957	−	0.246957i	−1.30792	+	1.30792i		−	2.95934i		0
18.7	0.348065	+	0.348065i	1.72617	−	1.72617i		−	1.75770i		0		1.20164			−0.246957	+	0.246957i	1.30792	−	1.30792i		−	2.95934i		0
18.8	0.348065	+	0.348065i	1.72617	−	1.72617i		−	1.75770i		0		1.20164			0.246957	−	0.246957i	1.30792	−	1.30792i		−	2.95934i		0
18.9	1.24481	+	1.24481i	−0.534241	+	0.534241i			1.09911i		0		−1.33006			−2.85373	+	2.85373i	1.12143	−	1.12143i			2.42917i		0
18.10	1.24481	+	1.24481i	−0.534241	+	0.534241i			1.09911i		0		−1.33006			2.85373	−	2.85373i	1.12143	−	1.12143i			2.42917i		0
18.11	1.82464	+	1.82464i	0.857273	−	0.857273i			4.65859i		0		3.12842			−2.60677	+	2.60677i	−4.85095	+	4.85095i			1.53017i		0
18.12	1.82464	+	1.82464i	0.857273	−	0.857273i			4.65859i		0		3.12842			2.60677	−	2.60677i	−4.85095	+	4.85095i			1.53017i		0
132.1	−1.82464	+	1.82464i	−0.857273	−	0.857273i		−	4.65859i		0		3.12842			−2.60677	−	2.60677i	4.85095	+	4.85095i		−	1.53017i		0
132.2	−1.82464	+	1.82464i	−0.857273	−	0.857273i		−	4.65859i		0		3.12842			2.60677	+	2.60677i	4.85095	+	4.85095i		−	1.53017i		0
132.3	−1.24481	+	1.24481i	0.534241	+	0.534241i		−	1.09911i		0		−1.33006			−2.85373	−	2.85373i	−1.12143	−	1.12143i		−	2.42917i		0
132.4	−1.24481	+	1.24481i	0.534241	+	0.534241i		−	1.09911i		0		−1.33006			2.85373	+	2.85373i	−1.12143	−	1.12143i		−	2.42917i		0
132.5	−0.348065	+	0.348065i	−1.72617	−	1.72617i			1.75770i		0		1.20164			−0.246957	−	0.246957i	−1.30792	−	1.30792i			2.95934i		0
132.6	−0.348065	+	0.348065i	−1.72617	−	1.72617i			1.75770i		0		1.20164			0.246957	+	0.246957i	−1.30792	−	1.30792i			2.95934i		0
132.7	0.348065	−	0.348065i	1.72617	+	1.72617i			1.75770i		0		1.20164			−0.246957	−	0.246957i	1.30792	+	1.30792i			2.95934i		0
132.8	0.348065	−	0.348065i	1.72617	+	1.72617i			1.75770i		0		1.20164			0.246957	+	0.246957i	1.30792	+	1.30792i			2.95934i		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
19.b	odd	2	1	inner
95.d	odd	2	1	inner
95.g	even	4	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	475.2.g.d	✓	24
5.b	even	2	1	inner	475.2.g.d	✓	24
5.c	odd	4	2	inner	475.2.g.d	✓	24
19.b	odd	2	1	inner	475.2.g.d	✓	24
95.d	odd	2	1	inner	475.2.g.d	✓	24
95.g	even	4	2	inner	475.2.g.d	✓	24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
475.2.g.d	✓	24	1.a	even	1	1	trivial
475.2.g.d	✓	24	5.b	even	2	1	inner
475.2.g.d	✓	24	5.c	odd	4	2	inner
475.2.g.d	✓	24	19.b	odd	2	1	inner
475.2.g.d	✓	24	95.d	odd	2	1	inner
475.2.g.d	✓	24	95.g	even	4	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 54T_{2}^{8} + 429T_{2}^{4} + 25 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).