Properties

Label 3420.2.a.h
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: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 380)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + 2 q^{7} +O(q^{10})\) \( q - q^{5} + 2 q^{7} -2 \beta q^{11} + ( -1 + \beta ) q^{13} -2 \beta q^{17} + q^{19} -2 \beta q^{23} + q^{25} + 2 \beta q^{29} + ( 2 + 2 \beta ) q^{31} -2 q^{35} + ( 5 - \beta ) q^{37} + 6 q^{41} + ( 2 + 4 \beta ) q^{43} + ( -6 + 4 \beta ) q^{47} -3 q^{49} + ( 9 - \beta ) q^{53} + 2 \beta q^{55} + 4 \beta q^{59} + ( 2 - 6 \beta ) q^{61} + ( 1 - \beta ) q^{65} + ( 5 - \beta ) q^{67} + ( 6 + 2 \beta ) q^{71} + ( -4 - 2 \beta ) q^{73} -4 \beta q^{77} + ( -4 - 4 \beta ) q^{79} + 2 \beta q^{83} + 2 \beta q^{85} + ( 12 - 2 \beta ) q^{89} + ( -2 + 2 \beta ) q^{91} - q^{95} + ( -1 + 9 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{5} + 4q^{7} + O(q^{10}) \) \( 2q - 2q^{5} + 4q^{7} - 2q^{13} + 2q^{19} + 2q^{25} + 4q^{31} - 4q^{35} + 10q^{37} + 12q^{41} + 4q^{43} - 12q^{47} - 6q^{49} + 18q^{53} + 4q^{61} + 2q^{65} + 10q^{67} + 12q^{71} - 8q^{73} - 8q^{79} + 24q^{89} - 4q^{91} - 2q^{95} - 2q^{97} + O(q^{100}) \)

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
1.73205
−1.73205
0 0 0 −1.00000 0 2.00000 0 0 0
1.2 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.h 2
3.b odd 2 1 380.2.a.d 2
12.b even 2 1 1520.2.a.l 2
15.d odd 2 1 1900.2.a.d 2
15.e even 4 2 1900.2.c.e 4
24.f even 2 1 6080.2.a.bj 2
24.h odd 2 1 6080.2.a.z 2
57.d even 2 1 7220.2.a.h 2
60.h even 2 1 7600.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.a.d 2 3.b odd 2 1
1520.2.a.l 2 12.b even 2 1
1900.2.a.d 2 15.d odd 2 1
1900.2.c.e 4 15.e even 4 2
3420.2.a.h 2 1.a even 1 1 trivial
6080.2.a.z 2 24.h odd 2 1
6080.2.a.bj 2 24.f even 2 1
7220.2.a.h 2 57.d even 2 1
7600.2.a.bf 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3420))\):

\( T_{7} - 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} + 2 T_{13} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -2 + 2 T + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -12 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( 22 - 10 T + T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -44 - 4 T + T^{2} \)
$47$ \( -12 + 12 T + T^{2} \)
$53$ \( 78 - 18 T + T^{2} \)
$59$ \( -48 + T^{2} \)
$61$ \( -104 - 4 T + T^{2} \)
$67$ \( 22 - 10 T + T^{2} \)
$71$ \( 24 - 12 T + T^{2} \)
$73$ \( 4 + 8 T + T^{2} \)
$79$ \( -32 + 8 T + T^{2} \)
$83$ \( -12 + T^{2} \)
$89$ \( 132 - 24 T + T^{2} \)
$97$ \( -242 + 2 T + T^{2} \)
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