Properties

Label 3200.2.a.x
Level $3200$
Weight $2$
Character orbit 3200.a
Self dual yes
Analytic conductor $25.552$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{3} + 4q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} + 4q^{7} + q^{9} + 2q^{11} + 2q^{13} + 2q^{17} - 2q^{19} + 8q^{21} - 4q^{23} - 4q^{27} + 6q^{29} + 4q^{33} + 10q^{37} + 4q^{39} - 6q^{41} + 6q^{43} + 8q^{47} + 9q^{49} + 4q^{51} - 6q^{53} - 4q^{57} - 14q^{59} - 2q^{61} + 4q^{63} + 10q^{67} - 8q^{69} + 12q^{71} - 14q^{73} + 8q^{77} - 8q^{79} - 11q^{81} - 6q^{83} + 12q^{87} - 2q^{89} + 8q^{91} + 2q^{97} + 2q^{99} + O(q^{100}) \)

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 4.00000 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 3200.2.a.x 1
4.b odd 2 1 3200.2.a.e 1
5.b even 2 1 128.2.a.a 1
5.c odd 4 2 3200.2.c.l 2
8.b even 2 1 3200.2.a.h 1
8.d odd 2 1 3200.2.a.u 1
15.d odd 2 1 1152.2.a.m 1
20.d odd 2 1 128.2.a.c yes 1
20.e even 4 2 3200.2.c.f 2
35.c odd 2 1 6272.2.a.h 1
40.e odd 2 1 128.2.a.b yes 1
40.f even 2 1 128.2.a.d yes 1
40.i odd 4 2 3200.2.c.e 2
40.k even 4 2 3200.2.c.k 2
60.h even 2 1 1152.2.a.r 1
80.k odd 4 2 256.2.b.a 2
80.q even 4 2 256.2.b.c 2
120.i odd 2 1 1152.2.a.c 1
120.m even 2 1 1152.2.a.h 1
140.c even 2 1 6272.2.a.b 1
160.y odd 8 4 1024.2.e.m 4
160.z even 8 4 1024.2.e.i 4
240.t even 4 2 2304.2.d.b 2
240.bm odd 4 2 2304.2.d.r 2
280.c odd 2 1 6272.2.a.a 1
280.n even 2 1 6272.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.a.a 1 5.b even 2 1
128.2.a.b yes 1 40.e odd 2 1
128.2.a.c yes 1 20.d odd 2 1
128.2.a.d yes 1 40.f even 2 1
256.2.b.a 2 80.k odd 4 2
256.2.b.c 2 80.q even 4 2
1024.2.e.i 4 160.z even 8 4
1024.2.e.m 4 160.y odd 8 4
1152.2.a.c 1 120.i odd 2 1
1152.2.a.h 1 120.m even 2 1
1152.2.a.m 1 15.d odd 2 1
1152.2.a.r 1 60.h even 2 1
2304.2.d.b 2 240.t even 4 2
2304.2.d.r 2 240.bm odd 4 2
3200.2.a.e 1 4.b odd 2 1
3200.2.a.h 1 8.b even 2 1
3200.2.a.u 1 8.d odd 2 1
3200.2.a.x 1 1.a even 1 1 trivial
3200.2.c.e 2 40.i odd 4 2
3200.2.c.f 2 20.e even 4 2
3200.2.c.k 2 40.k even 4 2
3200.2.c.l 2 5.c odd 4 2
6272.2.a.a 1 280.c odd 2 1
6272.2.a.b 1 140.c even 2 1
6272.2.a.g 1 280.n even 2 1
6272.2.a.h 1 35.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\):

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

Hecke characteristic polynomials

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