Properties

Label 2520.2.bi.l
Level $2520$
Weight $2$
Character orbit 2520.bi
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.bi (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + \beta_{1} q^{11} + ( -1 + 3 \beta_{3} ) q^{13} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{17} + ( 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{19} + ( -5 \beta_{1} - 5 \beta_{3} ) q^{23} + ( -1 - \beta_{2} ) q^{25} + ( 2 - 5 \beta_{3} ) q^{29} + ( 5 + 5 \beta_{2} ) q^{31} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{35} + ( \beta_{1} - 9 \beta_{2} + \beta_{3} ) q^{37} + ( 4 - 3 \beta_{3} ) q^{41} + ( -1 - 7 \beta_{3} ) q^{43} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{47} + ( 5 + 2 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -4 + 2 \beta_{1} - 4 \beta_{2} ) q^{53} -\beta_{3} q^{55} + ( -10 - 3 \beta_{1} - 10 \beta_{2} ) q^{59} + ( 2 \beta_{1} + 12 \beta_{2} + 2 \beta_{3} ) q^{61} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{65} + ( 1 + 7 \beta_{1} + \beta_{2} ) q^{67} + ( -4 - 5 \beta_{3} ) q^{71} + ( 3 + 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 2 + 4 \beta_{2} + \beta_{3} ) q^{77} + ( 2 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} ) q^{79} + ( -2 - 3 \beta_{3} ) q^{83} + ( -4 + \beta_{3} ) q^{85} + ( 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{89} + ( -12 - 4 \beta_{1} - 7 \beta_{2} - 2 \beta_{3} ) q^{91} + ( -3 + 2 \beta_{1} - 3 \beta_{2} ) q^{95} -6 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} - 2q^{7} + O(q^{10}) \) \( 4q + 2q^{5} - 2q^{7} - 4q^{13} - 8q^{17} + 6q^{19} - 2q^{25} + 8q^{29} + 10q^{31} + 2q^{35} + 18q^{37} + 16q^{41} - 4q^{43} - 4q^{47} + 10q^{49} - 8q^{53} - 20q^{59} - 24q^{61} - 2q^{65} + 2q^{67} - 16q^{71} + 6q^{73} + 18q^{79} - 8q^{83} - 16q^{85} - 4q^{89} - 34q^{91} - 6q^{95} + O(q^{100}) \)

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

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

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
0 0 0 0.500000 + 0.866025i 0 −2.62132 + 0.358719i 0 0 0
361.2 0 0 0 0.500000 + 0.866025i 0 1.62132 2.09077i 0 0 0
1801.1 0 0 0 0.500000 0.866025i 0 −2.62132 0.358719i 0 0 0
1801.2 0 0 0 0.500000 0.866025i 0 1.62132 + 2.09077i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.bi.l 4
3.b odd 2 1 840.2.bg.g 4
7.c even 3 1 inner 2520.2.bi.l 4
12.b even 2 1 1680.2.bg.r 4
21.g even 6 1 5880.2.a.bm 2
21.h odd 6 1 840.2.bg.g 4
21.h odd 6 1 5880.2.a.bs 2
84.n even 6 1 1680.2.bg.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.g 4 3.b odd 2 1
840.2.bg.g 4 21.h odd 6 1
1680.2.bg.r 4 12.b even 2 1
1680.2.bg.r 4 84.n even 6 1
2520.2.bi.l 4 1.a even 1 1 trivial
2520.2.bi.l 4 7.c even 3 1 inner
5880.2.a.bm 2 21.g even 6 1
5880.2.a.bs 2 21.h odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 2 T_{11}^{2} + 4 \)
\( T_{13}^{2} + 2 T_{13} - 17 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 49 + 14 T - 3 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( 4 + 2 T^{2} + T^{4} \)
$13$ \( ( -17 + 2 T + T^{2} )^{2} \)
$17$ \( 196 + 112 T + 50 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 1 - 6 T + 35 T^{2} - 6 T^{3} + T^{4} \)
$23$ \( 2500 + 50 T^{2} + T^{4} \)
$29$ \( ( -46 - 4 T + T^{2} )^{2} \)
$31$ \( ( 25 - 5 T + T^{2} )^{2} \)
$37$ \( 6241 - 1422 T + 245 T^{2} - 18 T^{3} + T^{4} \)
$41$ \( ( -2 - 8 T + T^{2} )^{2} \)
$43$ \( ( -97 + 2 T + T^{2} )^{2} \)
$47$ \( 784 - 112 T + 44 T^{2} + 4 T^{3} + T^{4} \)
$53$ \( 64 + 64 T + 56 T^{2} + 8 T^{3} + T^{4} \)
$59$ \( 6724 + 1640 T + 318 T^{2} + 20 T^{3} + T^{4} \)
$61$ \( 18496 + 3264 T + 440 T^{2} + 24 T^{3} + T^{4} \)
$67$ \( 9409 + 194 T + 101 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( -34 + 8 T + T^{2} )^{2} \)
$73$ \( 81 + 54 T + 45 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 5329 - 1314 T + 251 T^{2} - 18 T^{3} + T^{4} \)
$83$ \( ( -14 + 4 T + T^{2} )^{2} \)
$89$ \( 196 - 56 T + 30 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( ( -72 + T^{2} )^{2} \)
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