Properties

Label 4400.2.a.bb
Level $4400$
Weight $2$
Character orbit 4400.a
Self dual yes
Analytic conductor $35.134$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} + 4 q^{7} + q^{9} + O(q^{10}) \) \( q + 2 q^{3} + 4 q^{7} + q^{9} - q^{11} + 6 q^{13} - 2 q^{17} - 4 q^{19} + 8 q^{21} + 6 q^{23} - 4 q^{27} - 2 q^{29} - 8 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 6 q^{41} + 12 q^{43} + 10 q^{47} + 9 q^{49} - 4 q^{51} - 8 q^{57} + 4 q^{59} - 10 q^{61} + 4 q^{63} + 2 q^{67} + 12 q^{69} + 8 q^{71} - 2 q^{73} - 4 q^{77} - 4 q^{79} - 11 q^{81} - 4 q^{83} - 4 q^{87} - 14 q^{89} + 24 q^{91} - 16 q^{93} + 4 q^{97} - q^{99} + O(q^{100}) \)

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 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
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 4400.2.a.bb 1
4.b odd 2 1 2200.2.a.b 1
5.b even 2 1 4400.2.a.c 1
5.c odd 4 2 880.2.b.d 2
20.d odd 2 1 2200.2.a.j 1
20.e even 4 2 440.2.b.b 2
60.l odd 4 2 3960.2.d.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.b.b 2 20.e even 4 2
880.2.b.d 2 5.c odd 4 2
2200.2.a.b 1 4.b odd 2 1
2200.2.a.j 1 20.d odd 2 1
3960.2.d.b 2 60.l odd 4 2
4400.2.a.c 1 5.b even 2 1
4400.2.a.bb 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4400))\):

\( T_{3} - 2 \)
\( T_{7} - 4 \)
\( T_{13} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -2 + T \)
$5$ \( T \)
$7$ \( -4 + T \)
$11$ \( 1 + T \)
$13$ \( -6 + T \)
$17$ \( 2 + T \)
$19$ \( 4 + T \)
$23$ \( -6 + T \)
$29$ \( 2 + T \)
$31$ \( 8 + T \)
$37$ \( -8 + T \)
$41$ \( -6 + T \)
$43$ \( -12 + T \)
$47$ \( -10 + T \)
$53$ \( T \)
$59$ \( -4 + T \)
$61$ \( 10 + T \)
$67$ \( -2 + T \)
$71$ \( -8 + T \)
$73$ \( 2 + T \)
$79$ \( 4 + T \)
$83$ \( 4 + T \)
$89$ \( 14 + T \)
$97$ \( -4 + T \)
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