Properties

Label 4400.2.a.w
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 110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 5 q^{7} - 2 q^{9} + O(q^{10}) \) \( q + q^{3} + 5 q^{7} - 2 q^{9} - q^{11} - 2 q^{13} - 3 q^{17} + 7 q^{19} + 5 q^{21} - 6 q^{23} - 5 q^{27} - 3 q^{29} + 7 q^{31} - q^{33} + 7 q^{37} - 2 q^{39} + 6 q^{41} + 8 q^{43} + 6 q^{47} + 18 q^{49} - 3 q^{51} + 3 q^{53} + 7 q^{57} + 6 q^{59} - q^{61} - 10 q^{63} + 8 q^{67} - 6 q^{69} - 3 q^{71} - 2 q^{73} - 5 q^{77} + 10 q^{79} + q^{81} - 6 q^{83} - 3 q^{87} + 9 q^{89} - 10 q^{91} + 7 q^{93} + 4 q^{97} + 2 q^{99} + O(q^{100}) \)

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 5.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.w 1
4.b odd 2 1 550.2.a.i 1
5.b even 2 1 880.2.a.c 1
5.c odd 4 2 4400.2.b.g 2
12.b even 2 1 4950.2.a.a 1
15.d odd 2 1 7920.2.a.s 1
20.d odd 2 1 110.2.a.a 1
20.e even 4 2 550.2.b.b 2
40.e odd 2 1 3520.2.a.l 1
40.f even 2 1 3520.2.a.z 1
44.c even 2 1 6050.2.a.i 1
55.d odd 2 1 9680.2.a.j 1
60.h even 2 1 990.2.a.l 1
60.l odd 4 2 4950.2.c.a 2
140.c even 2 1 5390.2.a.h 1
220.g even 2 1 1210.2.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.a 1 20.d odd 2 1
550.2.a.i 1 4.b odd 2 1
550.2.b.b 2 20.e even 4 2
880.2.a.c 1 5.b even 2 1
990.2.a.l 1 60.h even 2 1
1210.2.a.k 1 220.g even 2 1
3520.2.a.l 1 40.e odd 2 1
3520.2.a.z 1 40.f even 2 1
4400.2.a.w 1 1.a even 1 1 trivial
4400.2.b.g 2 5.c odd 4 2
4950.2.a.a 1 12.b even 2 1
4950.2.c.a 2 60.l odd 4 2
5390.2.a.h 1 140.c even 2 1
6050.2.a.i 1 44.c even 2 1
7920.2.a.s 1 15.d odd 2 1
9680.2.a.j 1 55.d odd 2 1

Hecke kernels

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

\( T_{3} - 1 \)
\( T_{7} - 5 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -5 + T \)
$11$ \( 1 + T \)
$13$ \( 2 + T \)
$17$ \( 3 + T \)
$19$ \( -7 + T \)
$23$ \( 6 + T \)
$29$ \( 3 + T \)
$31$ \( -7 + T \)
$37$ \( -7 + T \)
$41$ \( -6 + T \)
$43$ \( -8 + T \)
$47$ \( -6 + T \)
$53$ \( -3 + T \)
$59$ \( -6 + T \)
$61$ \( 1 + T \)
$67$ \( -8 + T \)
$71$ \( 3 + T \)
$73$ \( 2 + T \)
$79$ \( -10 + T \)
$83$ \( 6 + T \)
$89$ \( -9 + T \)
$97$ \( -4 + T \)
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