Properties

Label 2800.2.a.w
Level $2800$
Weight $2$
Character orbit 2800.a
Self dual yes
Analytic conductor $22.358$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2800 = 2^{4} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581125660\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{7} - 2q^{9} + O(q^{10}) \) \( q + q^{3} - q^{7} - 2q^{9} + 3q^{11} - q^{13} - 7q^{17} - q^{21} + 6q^{23} - 5q^{27} - 5q^{29} - 2q^{31} + 3q^{33} - 2q^{37} - q^{39} + 2q^{41} - 4q^{43} - 3q^{47} + q^{49} - 7q^{51} - 6q^{53} - 10q^{59} - 8q^{61} + 2q^{63} + 2q^{67} + 6q^{69} + 8q^{71} - 6q^{73} - 3q^{77} + 5q^{79} + q^{81} - 4q^{83} - 5q^{87} + q^{91} - 2q^{93} - 7q^{97} - 6q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)

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 2800.2.a.w 1
4.b odd 2 1 175.2.a.a 1
5.b even 2 1 2800.2.a.l 1
5.c odd 4 2 560.2.g.b 2
12.b even 2 1 1575.2.a.k 1
15.e even 4 2 5040.2.t.p 2
20.d odd 2 1 175.2.a.c 1
20.e even 4 2 35.2.b.a 2
28.d even 2 1 1225.2.a.a 1
40.i odd 4 2 2240.2.g.g 2
40.k even 4 2 2240.2.g.h 2
60.h even 2 1 1575.2.a.a 1
60.l odd 4 2 315.2.d.a 2
140.c even 2 1 1225.2.a.i 1
140.j odd 4 2 245.2.b.a 2
140.w even 12 4 245.2.j.e 4
140.x odd 12 4 245.2.j.d 4
420.w even 4 2 2205.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.b.a 2 20.e even 4 2
175.2.a.a 1 4.b odd 2 1
175.2.a.c 1 20.d odd 2 1
245.2.b.a 2 140.j odd 4 2
245.2.j.d 4 140.x odd 12 4
245.2.j.e 4 140.w even 12 4
315.2.d.a 2 60.l odd 4 2
560.2.g.b 2 5.c odd 4 2
1225.2.a.a 1 28.d even 2 1
1225.2.a.i 1 140.c even 2 1
1575.2.a.a 1 60.h even 2 1
1575.2.a.k 1 12.b even 2 1
2205.2.d.b 2 420.w even 4 2
2240.2.g.g 2 40.i odd 4 2
2240.2.g.h 2 40.k even 4 2
2800.2.a.l 1 5.b even 2 1
2800.2.a.w 1 1.a even 1 1 trivial
5040.2.t.p 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2800))\):

\( T_{3} - 1 \)
\( T_{11} - 3 \)
\( T_{13} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( -3 + T \)
$13$ \( 1 + T \)
$17$ \( 7 + T \)
$19$ \( T \)
$23$ \( -6 + T \)
$29$ \( 5 + T \)
$31$ \( 2 + T \)
$37$ \( 2 + T \)
$41$ \( -2 + T \)
$43$ \( 4 + T \)
$47$ \( 3 + T \)
$53$ \( 6 + T \)
$59$ \( 10 + T \)
$61$ \( 8 + T \)
$67$ \( -2 + T \)
$71$ \( -8 + T \)
$73$ \( 6 + T \)
$79$ \( -5 + T \)
$83$ \( 4 + T \)
$89$ \( T \)
$97$ \( 7 + T \)
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