Properties

Label 1600.2.a.k
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

\(f(q)\) \(=\) \( q - 4q^{7} - 3q^{9} + O(q^{10}) \) \( q - 4q^{7} - 3q^{9} + 4q^{11} - 2q^{13} - 2q^{17} + 4q^{19} + 4q^{23} + 2q^{29} + 8q^{31} + 6q^{37} - 6q^{41} + 8q^{43} + 4q^{47} + 9q^{49} + 6q^{53} - 4q^{59} + 2q^{61} + 12q^{63} - 8q^{67} + 6q^{73} - 16q^{77} + 9q^{81} + 16q^{83} - 6q^{89} + 8q^{91} + 14q^{97} - 12q^{99} + O(q^{100}) \)

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 0 0 0 0 −4.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.k 1
4.b odd 2 1 1600.2.a.o 1
5.b even 2 1 320.2.a.d 1
5.c odd 4 2 1600.2.c.m 2
8.b even 2 1 400.2.a.e 1
8.d odd 2 1 200.2.a.c 1
15.d odd 2 1 2880.2.a.bg 1
20.d odd 2 1 320.2.a.c 1
20.e even 4 2 1600.2.c.k 2
24.f even 2 1 1800.2.a.v 1
24.h odd 2 1 3600.2.a.h 1
40.e odd 2 1 40.2.a.a 1
40.f even 2 1 80.2.a.a 1
40.i odd 4 2 400.2.c.d 2
40.k even 4 2 200.2.c.b 2
56.e even 2 1 9800.2.a.x 1
60.h even 2 1 2880.2.a.t 1
80.k odd 4 2 1280.2.d.j 2
80.q even 4 2 1280.2.d.a 2
120.i odd 2 1 720.2.a.e 1
120.m even 2 1 360.2.a.a 1
120.q odd 4 2 1800.2.f.a 2
120.w even 4 2 3600.2.f.t 2
280.c odd 2 1 3920.2.a.s 1
280.n even 2 1 1960.2.a.g 1
280.ba even 6 2 1960.2.q.i 2
280.bi odd 6 2 1960.2.q.h 2
360.z odd 6 2 3240.2.q.k 2
360.bd even 6 2 3240.2.q.x 2
440.c even 2 1 4840.2.a.f 1
440.o odd 2 1 9680.2.a.q 1
520.b odd 2 1 6760.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.a.a 1 40.e odd 2 1
80.2.a.a 1 40.f even 2 1
200.2.a.c 1 8.d odd 2 1
200.2.c.b 2 40.k even 4 2
320.2.a.c 1 20.d odd 2 1
320.2.a.d 1 5.b even 2 1
360.2.a.a 1 120.m even 2 1
400.2.a.e 1 8.b even 2 1
400.2.c.d 2 40.i odd 4 2
720.2.a.e 1 120.i odd 2 1
1280.2.d.a 2 80.q even 4 2
1280.2.d.j 2 80.k odd 4 2
1600.2.a.k 1 1.a even 1 1 trivial
1600.2.a.o 1 4.b odd 2 1
1600.2.c.k 2 20.e even 4 2
1600.2.c.m 2 5.c odd 4 2
1800.2.a.v 1 24.f even 2 1
1800.2.f.a 2 120.q odd 4 2
1960.2.a.g 1 280.n even 2 1
1960.2.q.h 2 280.bi odd 6 2
1960.2.q.i 2 280.ba even 6 2
2880.2.a.t 1 60.h even 2 1
2880.2.a.bg 1 15.d odd 2 1
3240.2.q.k 2 360.z odd 6 2
3240.2.q.x 2 360.bd even 6 2
3600.2.a.h 1 24.h odd 2 1
3600.2.f.t 2 120.w even 4 2
3920.2.a.s 1 280.c odd 2 1
4840.2.a.f 1 440.c even 2 1
6760.2.a.i 1 520.b odd 2 1
9680.2.a.q 1 440.o odd 2 1
9800.2.a.x 1 56.e even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 + 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 - 2 T + 61 T^{2} \)
$67$ \( 1 + 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 - 16 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 - 14 T + 97 T^{2} \)
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