Properties

Label 3520.2.a.bd
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 220)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{3} - q^{5} - 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} - q^{5} - 4 q^{7} + q^{9} + q^{11} + 4 q^{13} - 2 q^{15} + 4 q^{19} - 8 q^{21} - 6 q^{23} + q^{25} - 4 q^{27} + 6 q^{29} + 8 q^{31} + 2 q^{33} + 4 q^{35} - 2 q^{37} + 8 q^{39} + 6 q^{41} - 8 q^{43} - q^{45} + 6 q^{47} + 9 q^{49} + 6 q^{53} - q^{55} + 8 q^{57} + 12 q^{59} - 2 q^{61} - 4 q^{63} - 4 q^{65} + 10 q^{67} - 12 q^{69} - 12 q^{71} - 16 q^{73} + 2 q^{75} - 4 q^{77} + 8 q^{79} - 11 q^{81} + 12 q^{87} + 6 q^{89} - 16 q^{91} + 16 q^{93} - 4 q^{95} + 14 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −1.00000 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 3520.2.a.bd 1
4.b odd 2 1 3520.2.a.d 1
8.b even 2 1 220.2.a.a 1
8.d odd 2 1 880.2.a.j 1
24.f even 2 1 7920.2.a.o 1
24.h odd 2 1 1980.2.a.a 1
40.e odd 2 1 4400.2.a.e 1
40.f even 2 1 1100.2.a.e 1
40.i odd 4 2 1100.2.b.a 2
40.k even 4 2 4400.2.b.f 2
88.b odd 2 1 2420.2.a.b 1
88.g even 2 1 9680.2.a.bb 1
120.i odd 2 1 9900.2.a.bd 1
120.w even 4 2 9900.2.c.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 8.b even 2 1
880.2.a.j 1 8.d odd 2 1
1100.2.a.e 1 40.f even 2 1
1100.2.b.a 2 40.i odd 4 2
1980.2.a.a 1 24.h odd 2 1
2420.2.a.b 1 88.b odd 2 1
3520.2.a.d 1 4.b odd 2 1
3520.2.a.bd 1 1.a even 1 1 trivial
4400.2.a.e 1 40.e odd 2 1
4400.2.b.f 2 40.k even 4 2
7920.2.a.o 1 24.f even 2 1
9680.2.a.bb 1 88.g even 2 1
9900.2.a.bd 1 120.i odd 2 1
9900.2.c.m 2 120.w even 4 2

Hecke kernels

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

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T - 10 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 16 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 14 \) Copy content Toggle raw display
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