Properties

Label 4416.2.a.z
Level $4416$
Weight $2$
Character orbit 4416.a
Self dual yes
Analytic conductor $35.262$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4416.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2619375326\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 2q^{5} - 2q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 2q^{5} - 2q^{7} + q^{9} + 6q^{11} + 2q^{13} + 2q^{15} - 2q^{21} - q^{23} - q^{25} + q^{27} - 6q^{29} + 8q^{31} + 6q^{33} - 4q^{35} + 2q^{39} + 10q^{41} + 12q^{43} + 2q^{45} - 8q^{47} - 3q^{49} - 2q^{53} + 12q^{55} + 12q^{59} - 4q^{61} - 2q^{63} + 4q^{65} + 12q^{67} - q^{69} - 10q^{73} - q^{75} - 12q^{77} - 6q^{79} + q^{81} - 14q^{83} - 6q^{87} - 4q^{91} + 8q^{93} - 6q^{97} + 6q^{99} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(1\)

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 4416.2.a.z 1
4.b odd 2 1 4416.2.a.m 1
8.b even 2 1 138.2.a.a 1
8.d odd 2 1 1104.2.a.e 1
24.f even 2 1 3312.2.a.n 1
24.h odd 2 1 414.2.a.d 1
40.f even 2 1 3450.2.a.y 1
40.i odd 4 2 3450.2.d.j 2
56.h odd 2 1 6762.2.a.q 1
184.e odd 2 1 3174.2.a.b 1
552.b even 2 1 9522.2.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.a 1 8.b even 2 1
414.2.a.d 1 24.h odd 2 1
1104.2.a.e 1 8.d odd 2 1
3174.2.a.b 1 184.e odd 2 1
3312.2.a.n 1 24.f even 2 1
3450.2.a.y 1 40.f even 2 1
3450.2.d.j 2 40.i odd 4 2
4416.2.a.m 1 4.b odd 2 1
4416.2.a.z 1 1.a even 1 1 trivial
6762.2.a.q 1 56.h odd 2 1
9522.2.a.i 1 552.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4416))\):

\( T_{5} - 2 \)
\( T_{7} + 2 \)
\( T_{11} - 6 \)
\( T_{17} \)
\( T_{19} \)

Hecke characteristic polynomials

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