Properties

Label 3520.2.a.a
Level $3520$
Weight $2$
Character orbit 3520.a
Self dual yes
Analytic conductor $28.107$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1073415115\)
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

\(f(q)\) \(=\) \( q - 3 q^{3} - q^{5} + q^{7} + 6 q^{9} + O(q^{10}) \) \( q - 3 q^{3} - q^{5} + q^{7} + 6 q^{9} + q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 5 q^{19} - 3 q^{21} - 2 q^{23} + q^{25} - 9 q^{27} + 5 q^{29} + 5 q^{31} - 3 q^{33} - q^{35} + q^{37} - 18 q^{39} - 2 q^{41} - 12 q^{43} - 6 q^{45} - 2 q^{47} - 6 q^{49} - 9 q^{51} + 13 q^{53} - q^{55} - 15 q^{57} - 2 q^{59} - q^{61} + 6 q^{63} - 6 q^{65} - 16 q^{67} + 6 q^{69} + 15 q^{71} + 10 q^{73} - 3 q^{75} + q^{77} + 2 q^{79} + 9 q^{81} + 14 q^{83} - 3 q^{85} - 15 q^{87} + 9 q^{89} + 6 q^{91} - 15 q^{93} - 5 q^{95} - 16 q^{97} + 6 q^{99} + O(q^{100}) \)

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 −3.00000 0 −1.00000 0 1.00000 0 6.00000 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 3520.2.a.a 1
4.b odd 2 1 3520.2.a.bh 1
8.b even 2 1 440.2.a.d 1
8.d odd 2 1 880.2.a.a 1
24.f even 2 1 7920.2.a.e 1
24.h odd 2 1 3960.2.a.f 1
40.e odd 2 1 4400.2.a.be 1
40.f even 2 1 2200.2.a.a 1
40.i odd 4 2 2200.2.b.b 2
40.k even 4 2 4400.2.b.a 2
88.b odd 2 1 4840.2.a.i 1
88.g even 2 1 9680.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.d 1 8.b even 2 1
880.2.a.a 1 8.d odd 2 1
2200.2.a.a 1 40.f even 2 1
2200.2.b.b 2 40.i odd 4 2
3520.2.a.a 1 1.a even 1 1 trivial
3520.2.a.bh 1 4.b odd 2 1
3960.2.a.f 1 24.h odd 2 1
4400.2.a.be 1 40.e odd 2 1
4400.2.b.a 2 40.k even 4 2
4840.2.a.i 1 88.b odd 2 1
7920.2.a.e 1 24.f even 2 1
9680.2.a.a 1 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3520))\):

\( T_{3} + 3 \)
\( T_{7} - 1 \)
\( T_{13} - 6 \)
\( T_{17} - 3 \)
\( T_{19} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( -1 + T \)
$13$ \( -6 + T \)
$17$ \( -3 + T \)
$19$ \( -5 + T \)
$23$ \( 2 + T \)
$29$ \( -5 + T \)
$31$ \( -5 + T \)
$37$ \( -1 + T \)
$41$ \( 2 + T \)
$43$ \( 12 + T \)
$47$ \( 2 + T \)
$53$ \( -13 + T \)
$59$ \( 2 + T \)
$61$ \( 1 + T \)
$67$ \( 16 + T \)
$71$ \( -15 + T \)
$73$ \( -10 + T \)
$79$ \( -2 + T \)
$83$ \( -14 + T \)
$89$ \( -9 + T \)
$97$ \( 16 + T \)
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