Properties

Label 4410.2.a.w
Level 4410
Weight 2
Character orbit 4410.a
Self dual yes
Analytic conductor 35.214
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4410.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(35.2140272914\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - q^{5} + q^{8} + O(q^{10}) \) \( q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 3q^{11} - 5q^{13} + q^{16} - 5q^{19} - q^{20} - 3q^{22} + 9q^{23} + q^{25} - 5q^{26} + 10q^{31} + q^{32} - q^{37} - 5q^{38} - q^{40} + 9q^{41} + 8q^{43} - 3q^{44} + 9q^{46} + 3q^{47} + q^{50} - 5q^{52} + 3q^{53} + 3q^{55} + 12q^{59} - 8q^{61} + 10q^{62} + q^{64} + 5q^{65} + 8q^{67} + 6q^{71} - 2q^{73} - q^{74} - 5q^{76} + 8q^{79} - q^{80} + 9q^{82} + 8q^{86} - 3q^{88} + 6q^{89} + 9q^{92} + 3q^{94} + 5q^{95} - 8q^{97} + O(q^{100}) \)

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 4410.2.a.w 1
3.b odd 2 1 1470.2.a.e 1
7.b odd 2 1 4410.2.a.bh 1
7.d odd 6 2 630.2.k.a 2
15.d odd 2 1 7350.2.a.cx 1
21.c even 2 1 1470.2.a.f 1
21.g even 6 2 210.2.i.c 2
21.h odd 6 2 1470.2.i.p 2
84.j odd 6 2 1680.2.bg.n 2
105.g even 2 1 7350.2.a.cd 1
105.p even 6 2 1050.2.i.i 2
105.w odd 12 4 1050.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 21.g even 6 2
630.2.k.a 2 7.d odd 6 2
1050.2.i.i 2 105.p even 6 2
1050.2.o.c 4 105.w odd 12 4
1470.2.a.e 1 3.b odd 2 1
1470.2.a.f 1 21.c even 2 1
1470.2.i.p 2 21.h odd 6 2
1680.2.bg.n 2 84.j odd 6 2
4410.2.a.w 1 1.a even 1 1 trivial
4410.2.a.bh 1 7.b odd 2 1
7350.2.a.cd 1 105.g even 2 1
7350.2.a.cx 1 15.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4410))\):

\( T_{11} + 3 \)
\( T_{13} + 5 \)
\( T_{17} \)
\( T_{19} + 5 \)
\( T_{29} \)
\( T_{31} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ 1
$5$ \( 1 + T \)
$7$ 1
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 + 5 T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 5 T + 19 T^{2} \)
$23$ \( 1 - 9 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - 10 T + 31 T^{2} \)
$37$ \( 1 + T + 37 T^{2} \)
$41$ \( 1 - 9 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 - 3 T + 47 T^{2} \)
$53$ \( 1 - 3 T + 53 T^{2} \)
$59$ \( 1 - 12 T + 59 T^{2} \)
$61$ \( 1 + 8 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 - 6 T + 89 T^{2} \)
$97$ \( 1 + 8 T + 97 T^{2} \)
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