Properties

Label 1600.2.a.n
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{9} + O(q^{10}) \) \( q - 3q^{9} + 6q^{13} - 2q^{17} + 10q^{29} - 2q^{37} + 10q^{41} - 7q^{49} + 14q^{53} + 10q^{61} + 6q^{73} + 9q^{81} + 10q^{89} - 18q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 0 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.n 1
4.b odd 2 1 CM 1600.2.a.n 1
5.b even 2 1 64.2.a.a 1
5.c odd 4 2 1600.2.c.l 2
8.b even 2 1 800.2.a.d 1
8.d odd 2 1 800.2.a.d 1
15.d odd 2 1 576.2.a.c 1
20.d odd 2 1 64.2.a.a 1
20.e even 4 2 1600.2.c.l 2
24.f even 2 1 7200.2.a.v 1
24.h odd 2 1 7200.2.a.v 1
35.c odd 2 1 3136.2.a.m 1
40.e odd 2 1 32.2.a.a 1
40.f even 2 1 32.2.a.a 1
40.i odd 4 2 800.2.c.e 2
40.k even 4 2 800.2.c.e 2
55.d odd 2 1 7744.2.a.v 1
60.h even 2 1 576.2.a.c 1
80.k odd 4 2 256.2.b.b 2
80.q even 4 2 256.2.b.b 2
120.i odd 2 1 288.2.a.d 1
120.m even 2 1 288.2.a.d 1
120.q odd 4 2 7200.2.f.m 2
120.w even 4 2 7200.2.f.m 2
140.c even 2 1 3136.2.a.m 1
160.y odd 8 4 1024.2.e.j 4
160.z even 8 4 1024.2.e.j 4
220.g even 2 1 7744.2.a.v 1
240.t even 4 2 2304.2.d.j 2
240.bm odd 4 2 2304.2.d.j 2
280.c odd 2 1 1568.2.a.e 1
280.n even 2 1 1568.2.a.e 1
280.ba even 6 2 1568.2.i.f 2
280.bf even 6 2 1568.2.i.g 2
280.bi odd 6 2 1568.2.i.g 2
280.bk odd 6 2 1568.2.i.f 2
360.z odd 6 2 2592.2.i.t 2
360.bd even 6 2 2592.2.i.e 2
360.bh odd 6 2 2592.2.i.e 2
360.bk even 6 2 2592.2.i.t 2
440.c even 2 1 3872.2.a.f 1
440.o odd 2 1 3872.2.a.f 1
520.b odd 2 1 5408.2.a.g 1
520.p even 2 1 5408.2.a.g 1
680.h even 2 1 9248.2.a.f 1
680.k odd 2 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 40.e odd 2 1
32.2.a.a 1 40.f even 2 1
64.2.a.a 1 5.b even 2 1
64.2.a.a 1 20.d odd 2 1
256.2.b.b 2 80.k odd 4 2
256.2.b.b 2 80.q even 4 2
288.2.a.d 1 120.i odd 2 1
288.2.a.d 1 120.m even 2 1
576.2.a.c 1 15.d odd 2 1
576.2.a.c 1 60.h even 2 1
800.2.a.d 1 8.b even 2 1
800.2.a.d 1 8.d odd 2 1
800.2.c.e 2 40.i odd 4 2
800.2.c.e 2 40.k even 4 2
1024.2.e.j 4 160.y odd 8 4
1024.2.e.j 4 160.z even 8 4
1568.2.a.e 1 280.c odd 2 1
1568.2.a.e 1 280.n even 2 1
1568.2.i.f 2 280.ba even 6 2
1568.2.i.f 2 280.bk odd 6 2
1568.2.i.g 2 280.bf even 6 2
1568.2.i.g 2 280.bi odd 6 2
1600.2.a.n 1 1.a even 1 1 trivial
1600.2.a.n 1 4.b odd 2 1 CM
1600.2.c.l 2 5.c odd 4 2
1600.2.c.l 2 20.e even 4 2
2304.2.d.j 2 240.t even 4 2
2304.2.d.j 2 240.bm odd 4 2
2592.2.i.e 2 360.bd even 6 2
2592.2.i.e 2 360.bh odd 6 2
2592.2.i.t 2 360.z odd 6 2
2592.2.i.t 2 360.bk even 6 2
3136.2.a.m 1 35.c odd 2 1
3136.2.a.m 1 140.c even 2 1
3872.2.a.f 1 440.c even 2 1
3872.2.a.f 1 440.o odd 2 1
5408.2.a.g 1 520.b odd 2 1
5408.2.a.g 1 520.p even 2 1
7200.2.a.v 1 24.f even 2 1
7200.2.a.v 1 24.h odd 2 1
7200.2.f.m 2 120.q odd 4 2
7200.2.f.m 2 120.w even 4 2
7744.2.a.v 1 55.d odd 2 1
7744.2.a.v 1 220.g even 2 1
9248.2.a.f 1 680.h even 2 1
9248.2.a.f 1 680.k odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3} \)
\( T_{7} \)
\( T_{11} \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( T \)
$23$ \( T \)
$29$ \( -10 + T \)
$31$ \( T \)
$37$ \( 2 + T \)
$41$ \( -10 + T \)
$43$ \( T \)
$47$ \( T \)
$53$ \( -14 + T \)
$59$ \( T \)
$61$ \( -10 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( -6 + T \)
$79$ \( T \)
$83$ \( T \)
$89$ \( -10 + T \)
$97$ \( 18 + T \)
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