Properties

Label 2520.2.t.f
Level $2520$
Weight $2$
Character orbit 2520.t
Analytic conductor $20.122$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.t (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + \beta_{3} q^{5} + \beta_{1} q^{7} + 2 q^{11} -2 \beta_{2} q^{13} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{17} -2 q^{19} + 4 \beta_{1} q^{23} + 5 q^{25} + ( -4 + 2 \beta_{3} ) q^{29} + ( 4 + 2 \beta_{3} ) q^{31} + \beta_{2} q^{35} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 8 + 2 \beta_{3} ) q^{41} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{47} - q^{49} + 2 \beta_{1} q^{53} + 2 \beta_{3} q^{55} + ( -8 - 2 \beta_{3} ) q^{61} -10 \beta_{1} q^{65} + ( 6 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -8 + 2 \beta_{3} ) q^{71} + ( -12 \beta_{1} - 2 \beta_{2} ) q^{73} + 2 \beta_{1} q^{77} + 4 \beta_{3} q^{79} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{83} + ( 10 \beta_{1} - 2 \beta_{2} ) q^{85} + ( -4 + 6 \beta_{3} ) q^{89} + 2 \beta_{3} q^{91} -2 \beta_{3} q^{95} + ( 8 \beta_{1} + 2 \beta_{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 8q^{11} - 8q^{19} + 20q^{25} - 16q^{29} + 16q^{31} + 32q^{41} - 4q^{49} - 32q^{61} - 32q^{71} - 16q^{89} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
1.61803i
1.61803i
0.618034i
0.618034i
0 0 0 −2.23607 0 1.00000i 0 0 0
1009.2 0 0 0 −2.23607 0 1.00000i 0 0 0
1009.3 0 0 0 2.23607 0 1.00000i 0 0 0
1009.4 0 0 0 2.23607 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.t.f 4
3.b odd 2 1 840.2.t.c 4
4.b odd 2 1 5040.2.t.u 4
5.b even 2 1 inner 2520.2.t.f 4
12.b even 2 1 1680.2.t.h 4
15.d odd 2 1 840.2.t.c 4
15.e even 4 1 4200.2.a.bj 2
15.e even 4 1 4200.2.a.bk 2
20.d odd 2 1 5040.2.t.u 4
60.h even 2 1 1680.2.t.h 4
60.l odd 4 1 8400.2.a.cz 2
60.l odd 4 1 8400.2.a.db 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.t.c 4 3.b odd 2 1
840.2.t.c 4 15.d odd 2 1
1680.2.t.h 4 12.b even 2 1
1680.2.t.h 4 60.h even 2 1
2520.2.t.f 4 1.a even 1 1 trivial
2520.2.t.f 4 5.b even 2 1 inner
4200.2.a.bj 2 15.e even 4 1
4200.2.a.bk 2 15.e even 4 1
5040.2.t.u 4 4.b odd 2 1
5040.2.t.u 4 20.d odd 2 1
8400.2.a.cz 2 60.l odd 4 1
8400.2.a.db 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11} - 2 \)
\( T_{13}^{2} + 20 \)
\( T_{17}^{4} + 48 T_{17}^{2} + 256 \)
\( T_{19} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -5 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -2 + T )^{4} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( 256 + 48 T^{2} + T^{4} \)
$19$ \( ( 2 + T )^{4} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( -4 + 8 T + T^{2} )^{2} \)
$31$ \( ( -4 - 8 T + T^{2} )^{2} \)
$37$ \( 256 + 48 T^{2} + T^{4} \)
$41$ \( ( 44 - 16 T + T^{2} )^{2} \)
$43$ \( 256 + 48 T^{2} + T^{4} \)
$47$ \( 256 + 48 T^{2} + T^{4} \)
$53$ \( ( 4 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( ( 44 + 16 T + T^{2} )^{2} \)
$67$ \( 256 + 112 T^{2} + T^{4} \)
$71$ \( ( 44 + 16 T + T^{2} )^{2} \)
$73$ \( 15376 + 328 T^{2} + T^{4} \)
$79$ \( ( -80 + T^{2} )^{2} \)
$83$ \( 4096 + 192 T^{2} + T^{4} \)
$89$ \( ( -164 + 8 T + T^{2} )^{2} \)
$97$ \( 1936 + 168 T^{2} + T^{4} \)
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