Properties

Label 3200.2.a.i
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} - 2 q^{9} + O(q^{10}) \) \( q - q^{3} - 2 q^{9} - q^{11} - 2 q^{13} - 3 q^{17} + 3 q^{19} - 6 q^{23} + 5 q^{27} + 2 q^{29} + 2 q^{31} + q^{33} + 4 q^{37} + 2 q^{39} - 3 q^{41} + 4 q^{43} + 6 q^{47} - 7 q^{49} + 3 q^{51} - 10 q^{53} - 3 q^{57} + 12 q^{59} + 12 q^{61} + q^{67} + 6 q^{69} + 10 q^{71} - q^{73} + 16 q^{79} + q^{81} - 11 q^{83} - 2 q^{87} - 13 q^{89} - 2 q^{93} - 6 q^{97} + 2 q^{99} + O(q^{100}) \)

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 −1.00000 0 0 0 0 0 −2.00000 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.i 1
4.b odd 2 1 3200.2.a.s yes 1
5.b even 2 1 3200.2.a.r yes 1
5.c odd 4 2 3200.2.c.m 2
8.b even 2 1 3200.2.a.t yes 1
8.d odd 2 1 3200.2.a.j yes 1
20.d odd 2 1 3200.2.a.l yes 1
20.e even 4 2 3200.2.c.o 2
40.e odd 2 1 3200.2.a.q yes 1
40.f even 2 1 3200.2.a.k yes 1
40.i odd 4 2 3200.2.c.p 2
40.k even 4 2 3200.2.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.i 1 1.a even 1 1 trivial
3200.2.a.j yes 1 8.d odd 2 1
3200.2.a.k yes 1 40.f even 2 1
3200.2.a.l yes 1 20.d odd 2 1
3200.2.a.q yes 1 40.e odd 2 1
3200.2.a.r yes 1 5.b even 2 1
3200.2.a.s yes 1 4.b odd 2 1
3200.2.a.t yes 1 8.b even 2 1
3200.2.c.m 2 5.c odd 4 2
3200.2.c.n 2 40.k even 4 2
3200.2.c.o 2 20.e even 4 2
3200.2.c.p 2 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3200))\):

\( T_{3} + 1 \)
\( T_{7} \)
\( T_{11} + 1 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 1 + T \)
$5$ \( T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( 2 + T \)
$17$ \( 3 + T \)
$19$ \( -3 + T \)
$23$ \( 6 + T \)
$29$ \( -2 + T \)
$31$ \( -2 + T \)
$37$ \( -4 + T \)
$41$ \( 3 + T \)
$43$ \( -4 + T \)
$47$ \( -6 + T \)
$53$ \( 10 + T \)
$59$ \( -12 + T \)
$61$ \( -12 + T \)
$67$ \( -1 + T \)
$71$ \( -10 + T \)
$73$ \( 1 + T \)
$79$ \( -16 + T \)
$83$ \( 11 + T \)
$89$ \( 13 + T \)
$97$ \( 6 + T \)
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