Properties

Label 1620.1.v.b
Level $1620$
Weight $1$
Character orbit 1620.v
Analytic conductor $0.808$
Analytic rank $0$
Dimension $4$
Projective image $S_{4}$
CM/RM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1620.v (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.808485320465\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.0.40500.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{5} q^{5} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{7} +O(q^{10})\) \( q -\zeta_{12}^{5} q^{5} + ( -\zeta_{12} - \zeta_{12}^{4} ) q^{7} -\zeta_{12}^{4} q^{11} + ( -1 + \zeta_{12}^{3} ) q^{17} -\zeta_{12}^{3} q^{19} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{23} -\zeta_{12}^{4} q^{25} -\zeta_{12} q^{29} -\zeta_{12}^{2} q^{31} + ( -1 - \zeta_{12}^{3} ) q^{35} + \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{49} + ( 1 + \zeta_{12}^{3} ) q^{53} -\zeta_{12}^{3} q^{55} -\zeta_{12}^{5} q^{59} + ( \zeta_{12}^{2} - \zeta_{12}^{5} ) q^{67} + q^{71} + ( 1 + \zeta_{12}^{3} ) q^{73} + ( -\zeta_{12}^{2} + \zeta_{12}^{5} ) q^{77} + ( \zeta_{12}^{2} + \zeta_{12}^{5} ) q^{85} -\zeta_{12}^{3} q^{89} -\zeta_{12}^{2} q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{7} + O(q^{10}) \) \( 4 q + 2 q^{7} + 2 q^{11} - 4 q^{17} + 2 q^{23} + 2 q^{25} - 2 q^{31} - 4 q^{35} + 2 q^{41} + 4 q^{53} + 2 q^{67} + 4 q^{71} + 4 q^{73} - 2 q^{77} + 2 q^{85} - 2 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{3}\) \(-\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
0 0 0 −0.866025 + 0.500000i 0 1.36603 0.366025i 0 0 0
433.1 0 0 0 −0.866025 0.500000i 0 1.36603 + 0.366025i 0 0 0
757.1 0 0 0 0.866025 + 0.500000i 0 −0.366025 + 1.36603i 0 0 0
1513.1 0 0 0 0.866025 0.500000i 0 −0.366025 1.36603i 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.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.1.v.b 4
3.b odd 2 1 1620.1.v.a 4
5.c odd 4 1 inner 1620.1.v.b 4
9.c even 3 1 1620.1.l.a 2
9.c even 3 1 inner 1620.1.v.b 4
9.d odd 6 1 1620.1.l.b yes 2
9.d odd 6 1 1620.1.v.a 4
15.e even 4 1 1620.1.v.a 4
45.k odd 12 1 1620.1.l.a 2
45.k odd 12 1 inner 1620.1.v.b 4
45.l even 12 1 1620.1.l.b yes 2
45.l even 12 1 1620.1.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.1.l.a 2 9.c even 3 1
1620.1.l.a 2 45.k odd 12 1
1620.1.l.b yes 2 9.d odd 6 1
1620.1.l.b yes 2 45.l even 12 1
1620.1.v.a 4 3.b odd 2 1
1620.1.v.a 4 9.d odd 6 1
1620.1.v.a 4 15.e even 4 1
1620.1.v.a 4 45.l even 12 1
1620.1.v.b 4 1.a even 1 1 trivial
1620.1.v.b 4 5.c odd 4 1 inner
1620.1.v.b 4 9.c even 3 1 inner
1620.1.v.b 4 45.k odd 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} - T_{11} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1620, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 1 - T^{2} + T^{4} \)
$7$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 2 + 2 T + T^{2} )^{2} \)
$19$ \( ( 1 + T^{2} )^{2} \)
$23$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$29$ \( 1 - T^{2} + T^{4} \)
$31$ \( ( 1 + T + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 2 - 2 T + T^{2} )^{2} \)
$59$ \( 1 - T^{2} + T^{4} \)
$61$ \( T^{4} \)
$67$ \( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$71$ \( ( -1 + T )^{4} \)
$73$ \( ( 2 - 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( T^{4} \)
$89$ \( ( 1 + T^{2} )^{2} \)
$97$ \( T^{4} \)
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