Properties

Label 3420.2.a.f
Level $3420$
Weight $2$
Character orbit 3420.a
Self dual yes
Analytic conductor $27.309$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + 2 q^{7} + 6 q^{13} - 2 q^{17} - q^{19} + 2 q^{23} + q^{25} + 2 q^{29} + 4 q^{31} + 2 q^{35} - 10 q^{37} + 10 q^{41} + 6 q^{43} + 6 q^{47} - 3 q^{49} - 6 q^{53} + 4 q^{59} + 2 q^{61} + 6 q^{65} - 2 q^{67} - 12 q^{71} - 6 q^{73} + 8 q^{79} + 2 q^{83} - 2 q^{85} - 2 q^{89} + 12 q^{91} - q^{95} - 18 q^{97}+O(q^{100}) \) 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 0 0 1.00000 0 2.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)
\(19\) \(1\)

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 3420.2.a.f 1
3.b odd 2 1 380.2.a.b 1
12.b even 2 1 1520.2.a.b 1
15.d odd 2 1 1900.2.a.a 1
15.e even 4 2 1900.2.c.a 2
24.f even 2 1 6080.2.a.v 1
24.h odd 2 1 6080.2.a.f 1
57.d even 2 1 7220.2.a.b 1
60.h even 2 1 7600.2.a.r 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.b 1 3.b odd 2 1
1520.2.a.b 1 12.b even 2 1
1900.2.a.a 1 15.d odd 2 1
1900.2.c.a 2 15.e even 4 2
3420.2.a.f 1 1.a even 1 1 trivial
6080.2.a.f 1 24.h odd 2 1
6080.2.a.v 1 24.f even 2 1
7220.2.a.b 1 57.d even 2 1
7600.2.a.r 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3420))\):

\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 2 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T - 6 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T + 6 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 2 \) Copy content Toggle raw display
$89$ \( T + 2 \) Copy content Toggle raw display
$97$ \( T + 18 \) Copy content Toggle raw display
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