Properties

Label 2520.2.bi.h
Level $2520$
Weight $2$
Character orbit 2520.bi
Analytic conductor $20.122$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( -2 + 3 \zeta_{6} ) q^{7} + ( 4 - 4 \zeta_{6} ) q^{11} -5 q^{13} + ( 6 - 6 \zeta_{6} ) q^{17} + 5 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 2 q^{29} + ( 9 - 9 \zeta_{6} ) q^{31} + ( -3 + \zeta_{6} ) q^{35} + 11 \zeta_{6} q^{37} + 8 q^{41} + q^{43} + 6 \zeta_{6} q^{47} + ( -5 - 3 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{53} + 4 q^{55} + ( 6 - 6 \zeta_{6} ) q^{59} + 14 \zeta_{6} q^{61} -5 \zeta_{6} q^{65} + ( 1 - \zeta_{6} ) q^{67} -12 q^{71} + ( -11 + 11 \zeta_{6} ) q^{73} + ( 4 + 8 \zeta_{6} ) q^{77} -\zeta_{6} q^{79} + 4 q^{83} + 6 q^{85} + 18 \zeta_{6} q^{89} + ( 10 - 15 \zeta_{6} ) q^{91} + ( -5 + 5 \zeta_{6} ) q^{95} -18 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - q^{7} + O(q^{10}) \) \( 2q + q^{5} - q^{7} + 4q^{11} - 10q^{13} + 6q^{17} + 5q^{19} + 2q^{23} - q^{25} + 4q^{29} + 9q^{31} - 5q^{35} + 11q^{37} + 16q^{41} + 2q^{43} + 6q^{47} - 13q^{49} - 4q^{53} + 8q^{55} + 6q^{59} + 14q^{61} - 5q^{65} + q^{67} - 24q^{71} - 11q^{73} + 16q^{77} - q^{79} + 8q^{83} + 12q^{85} + 18q^{89} + 5q^{91} - 5q^{95} - 36q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 −0.500000 + 2.59808i 0 0 0
1801.1 0 0 0 0.500000 0.866025i 0 −0.500000 2.59808i 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.h 2
3.b odd 2 1 840.2.bg.b 2
7.c even 3 1 inner 2520.2.bi.h 2
12.b even 2 1 1680.2.bg.c 2
21.g even 6 1 5880.2.a.ba 1
21.h odd 6 1 840.2.bg.b 2
21.h odd 6 1 5880.2.a.q 1
84.n even 6 1 1680.2.bg.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bg.b 2 3.b odd 2 1
840.2.bg.b 2 21.h odd 6 1
1680.2.bg.c 2 12.b even 2 1
1680.2.bg.c 2 84.n even 6 1
2520.2.bi.h 2 1.a even 1 1 trivial
2520.2.bi.h 2 7.c even 3 1 inner
5880.2.a.q 1 21.h odd 6 1
5880.2.a.ba 1 21.g even 6 1

Hecke kernels

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

\( T_{11}^{2} - 4 T_{11} + 16 \)
\( T_{13} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 7 + T + T^{2} \)
$11$ \( 16 - 4 T + T^{2} \)
$13$ \( ( 5 + T )^{2} \)
$17$ \( 36 - 6 T + T^{2} \)
$19$ \( 25 - 5 T + T^{2} \)
$23$ \( 4 - 2 T + T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( 81 - 9 T + T^{2} \)
$37$ \( 121 - 11 T + T^{2} \)
$41$ \( ( -8 + T )^{2} \)
$43$ \( ( -1 + T )^{2} \)
$47$ \( 36 - 6 T + T^{2} \)
$53$ \( 16 + 4 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 196 - 14 T + T^{2} \)
$67$ \( 1 - T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 121 + 11 T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( -4 + T )^{2} \)
$89$ \( 324 - 18 T + T^{2} \)
$97$ \( ( 18 + T )^{2} \)
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