Properties

Label 4020.2.g.a
Level 4020
Weight 2
Character orbit 4020.g
Analytic conductor 32.100
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4020.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0998616126\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} - q^{9} +O(q^{10})\) \( q + i q^{3} + ( 1 - 2 i ) q^{5} - q^{9} + 2 i q^{13} + ( 2 + i ) q^{15} -2 i q^{17} -8 i q^{23} + ( -3 - 4 i ) q^{25} -i q^{27} -6 q^{29} -2 q^{31} -2 q^{39} -6 q^{41} + 4 i q^{43} + ( -1 + 2 i ) q^{45} + 7 q^{49} + 2 q^{51} -4 i q^{53} -2 q^{59} -10 q^{61} + ( 4 + 2 i ) q^{65} -i q^{67} + 8 q^{69} -10 q^{71} -8 i q^{73} + ( 4 - 3 i ) q^{75} + 14 q^{79} + q^{81} -12 i q^{83} + ( -4 - 2 i ) q^{85} -6 i q^{87} + 2 q^{89} -2 i q^{93} + 14 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} - 2q^{9} + O(q^{10}) \) \( 2q + 2q^{5} - 2q^{9} + 4q^{15} - 6q^{25} - 12q^{29} - 4q^{31} - 4q^{39} - 12q^{41} - 2q^{45} + 14q^{49} + 4q^{51} - 4q^{59} - 20q^{61} + 8q^{65} + 16q^{69} - 20q^{71} + 8q^{75} + 28q^{79} + 2q^{81} - 8q^{85} + 4q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(2011\) \(2681\) \(3217\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1609.1
1.00000i
1.00000i
0 1.00000i 0 1.00000 + 2.00000i 0 0 0 −1.00000 0
1609.2 0 1.00000i 0 1.00000 2.00000i 0 0 0 −1.00000 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 4020.2.g.a 2
5.b even 2 1 inner 4020.2.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4020.2.g.a 2 1.a even 1 1 trivial
4020.2.g.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} \) acting on \(S_{2}^{\mathrm{new}}(4020, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( 1 + T^{2} \)
$5$ \( 1 - 2 T + 5 T^{2} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( 1 - 22 T^{2} + 169 T^{4} \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( 1 + 18 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 2 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{2} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( ( 1 - 47 T^{2} )^{2} \)
$53$ \( ( 1 - 14 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} ) \)
$59$ \( ( 1 + 2 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 10 T + 61 T^{2} )^{2} \)
$67$ \( 1 + T^{2} \)
$71$ \( ( 1 + 10 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 82 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 14 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 22 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 2 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 2 T^{2} + 9409 T^{4} \)
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