Properties

Label 3360.2.a.bd
Level $3360$
Weight $2$
Character orbit 3360.a
Self dual yes
Analytic conductor $26.830$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3360.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.8297350792\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{5} - q^{7} + q^{9} + \beta q^{13} - q^{15} + \beta q^{17} + q^{21} + ( -2 + \beta ) q^{23} + q^{25} - q^{27} + ( 4 - \beta ) q^{29} + ( 2 - \beta ) q^{31} - q^{35} + ( 4 - \beta ) q^{37} -\beta q^{39} + ( 2 - 2 \beta ) q^{41} + ( -6 + \beta ) q^{43} + q^{45} + ( -2 - \beta ) q^{47} + q^{49} -\beta q^{51} + 2 q^{53} + 4 q^{59} + ( 8 - \beta ) q^{61} - q^{63} + \beta q^{65} + ( 2 + \beta ) q^{67} + ( 2 - \beta ) q^{69} + ( -2 + 3 \beta ) q^{71} + ( 4 - \beta ) q^{73} - q^{75} + ( -4 + 2 \beta ) q^{79} + q^{81} + ( 8 - 2 \beta ) q^{83} + \beta q^{85} + ( -4 + \beta ) q^{87} + ( 2 + 2 \beta ) q^{89} -\beta q^{91} + ( -2 + \beta ) q^{93} + \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{15} + 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} + 8 q^{29} + 4 q^{31} - 2 q^{35} + 8 q^{37} + 4 q^{41} - 12 q^{43} + 2 q^{45} - 4 q^{47} + 2 q^{49} + 4 q^{53} + 8 q^{59} + 16 q^{61} - 2 q^{63} + 4 q^{67} + 4 q^{69} - 4 q^{71} + 8 q^{73} - 2 q^{75} - 8 q^{79} + 2 q^{81} + 16 q^{83} - 8 q^{87} + 4 q^{89} - 4 q^{93} + O(q^{100}) \)

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
−0.618034
1.61803
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(1\)

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 3360.2.a.bd 2
4.b odd 2 1 3360.2.a.bh yes 2
8.b even 2 1 6720.2.a.cv 2
8.d odd 2 1 6720.2.a.cp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.2.a.bd 2 1.a even 1 1 trivial
3360.2.a.bh yes 2 4.b odd 2 1
6720.2.a.cp 2 8.d odd 2 1
6720.2.a.cv 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3360))\):

\( T_{11} \)
\( T_{13}^{2} - 20 \)
\( T_{17}^{2} - 20 \)
\( T_{19} \)
\( T_{23}^{2} + 4 T_{23} - 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( T^{2} \)
$13$ \( -20 + T^{2} \)
$17$ \( -20 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -16 + 4 T + T^{2} \)
$29$ \( -4 - 8 T + T^{2} \)
$31$ \( -16 - 4 T + T^{2} \)
$37$ \( -4 - 8 T + T^{2} \)
$41$ \( -76 - 4 T + T^{2} \)
$43$ \( 16 + 12 T + T^{2} \)
$47$ \( -16 + 4 T + T^{2} \)
$53$ \( ( -2 + T )^{2} \)
$59$ \( ( -4 + T )^{2} \)
$61$ \( 44 - 16 T + T^{2} \)
$67$ \( -16 - 4 T + T^{2} \)
$71$ \( -176 + 4 T + T^{2} \)
$73$ \( -4 - 8 T + T^{2} \)
$79$ \( -64 + 8 T + T^{2} \)
$83$ \( -16 - 16 T + T^{2} \)
$89$ \( -76 - 4 T + T^{2} \)
$97$ \( -20 + T^{2} \)
show more
show less