Properties

Label 3420.2.a.i
Level $3420$
Weight $2$
Character orbit 3420.a
Self dual yes
Analytic conductor $27.309$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1140)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + (\beta + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + (\beta + 1) q^{7} + ( - \beta + 1) q^{11} + ( - \beta + 1) q^{13} - 2 q^{17} - q^{19} + 2 q^{23} + q^{25} + (\beta + 1) q^{29} + 4 q^{31} + (\beta + 1) q^{35} + (\beta + 7) q^{37} + (\beta - 3) q^{41} + ( - \beta + 7) q^{43} - 6 q^{47} + (2 \beta + 7) q^{49} + ( - \beta + 1) q^{55} + (2 \beta + 2) q^{59} + 2 \beta q^{61} + ( - \beta + 1) q^{65} + 4 q^{67} + ( - 2 \beta + 2) q^{71} + 6 q^{73} - 12 q^{77} + 8 q^{79} + ( - 2 \beta + 4) q^{83} - 2 q^{85} + (\beta - 3) q^{89} - 12 q^{91} - q^{95} + ( - \beta + 13) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} + 14 q^{37} - 6 q^{41} + 14 q^{43} - 12 q^{47} + 14 q^{49} + 2 q^{55} + 4 q^{59} + 2 q^{65} + 8 q^{67} + 4 q^{71} + 12 q^{73} - 24 q^{77} + 16 q^{79} + 8 q^{83} - 4 q^{85} - 6 q^{89} - 24 q^{91} - 2 q^{95} + 26 q^{97}+O(q^{100}) \) 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
−1.30278
2.30278
0 0 0 1.00000 0 −2.60555 0 0 0
1.2 0 0 0 1.00000 0 4.60555 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.i 2
3.b odd 2 1 1140.2.a.e 2
12.b even 2 1 4560.2.a.bl 2
15.d odd 2 1 5700.2.a.u 2
15.e even 4 2 5700.2.f.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1140.2.a.e 2 3.b odd 2 1
3420.2.a.i 2 1.a even 1 1 trivial
4560.2.a.bl 2 12.b even 2 1
5700.2.a.u 2 15.d odd 2 1
5700.2.f.n 4 15.e even 4 2

Hecke kernels

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

\( T_{7}^{2} - 2T_{7} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 2T - 12 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 14T + 36 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$61$ \( T^{2} - 52 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$73$ \( (T - 6)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 26T + 156 \) Copy content Toggle raw display
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