Newform orbit 1650.2.a.z

• Label: 1650.2.a.z
• Level: $1650$
• Weight: $2$
• Character orbit: 1650.a
• Self dual: yes
• Analytic conductor: $13.175$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.1753163335\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 330)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

1.00000

1.00000

1.00000

0

1.00000

0.585786

1.00000

1.00000

0

1.2

1.00000

1.00000

1.00000

0

1.00000

3.41421

1.00000

1.00000

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(3\)	\(-1\)
\(5\)	\(-1\)
\(11\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1650.2.a.z		2
3.b	odd	2	1		4950.2.a.ca		2
5.b	even	2	1		1650.2.a.u		2
5.c	odd	4	2		330.2.c.b	✓	4
15.d	odd	2	1		4950.2.a.cb		2
15.e	even	4	2		990.2.c.h		4
20.e	even	4	2		2640.2.d.e		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
330.2.c.b	✓	4	5.c	odd	4	2
990.2.c.h		4	15.e	even	4	2
1650.2.a.u		2	5.b	even	2	1
1650.2.a.z		2	1.a	even	1	1	trivial
2640.2.d.e		4	20.e	even	4	2
4950.2.a.ca		2	3.b	odd	2	1
4950.2.a.cb		2	15.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1650))\):

\( T_{7}^{2} - 4T_{7} + 2 \)

\( T_{13}^{2} - 8T_{13} + 14 \)

\( T_{17}^{2} - 32 \)

Hecke characteristic polynomials

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