Newform orbit 350.2.e.b

• Label: 350.2.e.b
• Level: $350$
• Weight: $2$
• Character orbit: 350.e
• Analytic conductor: $2.795$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	350.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.79476407074\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{2} + (2 \zeta_{6} - 2) q^{3} + (\zeta_{6} - 1) q^{4} + 2 q^{6} + ( - 2 \zeta_{6} + 3) q^{7} + q^{8} - \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + 4 q^{6} + 4 q^{7} + 2 q^{8} - q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\)	\(101\)	\(127\)
\(\chi(n)\)	\(-\zeta_{6}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

51.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−0.500000

+

0.866025i

−1.00000

−

1.73205i

−0.500000

−

0.866025i

0

2.00000

2.00000

+

1.73205i

1.00000

−0.500000

+

0.866025i

0

151.1

−0.500000

−

0.866025i

−1.00000

+

1.73205i

−0.500000

+

0.866025i

0

2.00000

2.00000

−

1.73205i

1.00000

−0.500000

−

0.866025i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	350.2.e.b		2
5.b	even	2	1		70.2.e.d	✓	2
5.c	odd	4	2		350.2.j.d		4
7.c	even	3	1	inner	350.2.e.b		2
7.c	even	3	1		2450.2.a.bf		1
7.d	odd	6	1		2450.2.a.v		1
15.d	odd	2	1		630.2.k.d		2
20.d	odd	2	1		560.2.q.b		2
35.c	odd	2	1		490.2.e.g		2
35.i	odd	6	1		490.2.a.d		1
35.i	odd	6	1		490.2.e.g		2
35.j	even	6	1		70.2.e.d	✓	2
35.j	even	6	1		490.2.a.a		1
35.k	even	12	2		2450.2.c.e		2
35.l	odd	12	2		350.2.j.d		4
35.l	odd	12	2		2450.2.c.q		2
105.o	odd	6	1		630.2.k.d		2
105.o	odd	6	1		4410.2.a.x		1
105.p	even	6	1		4410.2.a.bg		1
140.p	odd	6	1		560.2.q.b		2
140.p	odd	6	1		3920.2.a.bh		1
140.s	even	6	1		3920.2.a.e		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.e.d	✓	2	5.b	even	2	1
70.2.e.d	✓	2	35.j	even	6	1
350.2.e.b		2	1.a	even	1	1	trivial
350.2.e.b		2	7.c	even	3	1	inner
350.2.j.d		4	5.c	odd	4	2
350.2.j.d		4	35.l	odd	12	2
490.2.a.a		1	35.j	even	6	1
490.2.a.d		1	35.i	odd	6	1
490.2.e.g		2	35.c	odd	2	1
490.2.e.g		2	35.i	odd	6	1
560.2.q.b		2	20.d	odd	2	1
560.2.q.b		2	140.p	odd	6	1
630.2.k.d		2	15.d	odd	2	1
630.2.k.d		2	105.o	odd	6	1
2450.2.a.v		1	7.d	odd	6	1
2450.2.a.bf		1	7.c	even	3	1
2450.2.c.e		2	35.k	even	12	2
2450.2.c.q		2	35.l	odd	12	2
3920.2.a.e		1	140.s	even	6	1
3920.2.a.bh		1	140.p	odd	6	1
4410.2.a.x		1	105.o	odd	6	1
4410.2.a.bg		1	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{2} + 2T_{3} + 4 \)

\( T_{11}^{2} + 3T_{11} + 9 \)

\( T_{13} - 1 \)

\( T_{17}^{2} + 6T_{17} + 36 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \)
$3$	\( T^{2} + 2T + 4 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - 4T + 7 \)
$11$	\( T^{2} + 3T + 9 \)