Properties

Label 1600.2.a.f
Level $1600$
Weight $2$
Character orbit 1600.a
Self dual yes
Analytic conductor $12.776$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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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 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{3} + 2q^{7} + q^{9} + O(q^{10}) \) \( q - 2q^{3} + 2q^{7} + q^{9} + 4q^{11} - 4q^{13} + 4q^{19} - 4q^{21} - 2q^{23} + 4q^{27} - 2q^{29} - 8q^{33} - 4q^{37} + 8q^{39} + 2q^{41} + 6q^{43} - 6q^{47} - 3q^{49} + 4q^{53} - 8q^{57} + 12q^{59} + 10q^{61} + 2q^{63} - 14q^{67} + 4q^{69} + 8q^{71} + 8q^{73} + 8q^{77} + 16q^{79} - 11q^{81} - 2q^{83} + 4q^{87} + 6q^{89} - 8q^{91} + 16q^{97} + 4q^{99} + O(q^{100}) \)

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 −2.00000 0 0 0 2.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1600.2.a.f 1
4.b odd 2 1 1600.2.a.u 1
5.b even 2 1 1600.2.a.v 1
5.c odd 4 2 320.2.c.c 2
8.b even 2 1 200.2.a.d 1
8.d odd 2 1 400.2.a.b 1
15.e even 4 2 2880.2.f.h 2
20.d odd 2 1 1600.2.a.d 1
20.e even 4 2 320.2.c.b 2
24.f even 2 1 3600.2.a.k 1
24.h odd 2 1 1800.2.a.s 1
40.e odd 2 1 400.2.a.g 1
40.f even 2 1 200.2.a.b 1
40.i odd 4 2 40.2.c.a 2
40.k even 4 2 80.2.c.a 2
56.h odd 2 1 9800.2.a.d 1
60.l odd 4 2 2880.2.f.i 2
80.i odd 4 2 1280.2.f.f 2
80.j even 4 2 1280.2.f.e 2
80.s even 4 2 1280.2.f.b 2
80.t odd 4 2 1280.2.f.a 2
120.i odd 2 1 1800.2.a.j 1
120.m even 2 1 3600.2.a.bb 1
120.q odd 4 2 720.2.f.e 2
120.w even 4 2 360.2.f.c 2
280.c odd 2 1 9800.2.a.bf 1
280.s even 4 2 1960.2.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 40.i odd 4 2
80.2.c.a 2 40.k even 4 2
200.2.a.b 1 40.f even 2 1
200.2.a.d 1 8.b even 2 1
320.2.c.b 2 20.e even 4 2
320.2.c.c 2 5.c odd 4 2
360.2.f.c 2 120.w even 4 2
400.2.a.b 1 8.d odd 2 1
400.2.a.g 1 40.e odd 2 1
720.2.f.e 2 120.q odd 4 2
1280.2.f.a 2 80.t odd 4 2
1280.2.f.b 2 80.s even 4 2
1280.2.f.e 2 80.j even 4 2
1280.2.f.f 2 80.i odd 4 2
1600.2.a.d 1 20.d odd 2 1
1600.2.a.f 1 1.a even 1 1 trivial
1600.2.a.u 1 4.b odd 2 1
1600.2.a.v 1 5.b even 2 1
1800.2.a.j 1 120.i odd 2 1
1800.2.a.s 1 24.h odd 2 1
1960.2.g.b 2 280.s even 4 2
2880.2.f.h 2 15.e even 4 2
2880.2.f.i 2 60.l odd 4 2
3600.2.a.k 1 24.f even 2 1
3600.2.a.bb 1 120.m even 2 1
9800.2.a.d 1 56.h odd 2 1
9800.2.a.bf 1 280.c odd 2 1

Hecke kernels

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

\( T_{3} + 2 \)
\( T_{7} - 2 \)
\( T_{11} - 4 \)
\( T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 2 + T \)
$5$ \( T \)
$7$ \( -2 + T \)
$11$ \( -4 + T \)
$13$ \( 4 + T \)
$17$ \( T \)
$19$ \( -4 + T \)
$23$ \( 2 + T \)
$29$ \( 2 + T \)
$31$ \( T \)
$37$ \( 4 + T \)
$41$ \( -2 + T \)
$43$ \( -6 + T \)
$47$ \( 6 + T \)
$53$ \( -4 + T \)
$59$ \( -12 + T \)
$61$ \( -10 + T \)
$67$ \( 14 + T \)
$71$ \( -8 + T \)
$73$ \( -8 + T \)
$79$ \( -16 + T \)
$83$ \( 2 + T \)
$89$ \( -6 + T \)
$97$ \( -16 + T \)
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