Properties

Label 1620.2.i.n
Level $1620$
Weight $2$
Character orbit 1620.i
Analytic conductor $12.936$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(12.9357651274\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} +O(q^{10})\) \( q + ( 1 - \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{7} + ( \zeta_{12} + \zeta_{12}^{3} ) q^{11} + ( -2 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{13} + ( -3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{17} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{19} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{2} q^{25} + ( -\zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + ( 1 + 4 \zeta_{12} - \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{31} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{35} + ( -1 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{37} + ( 6 - 3 \zeta_{12} - 6 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{41} + ( -\zeta_{12} - 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{43} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{47} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + ( -9 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{53} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} + ( 6 - \zeta_{12} - 6 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{59} + 4 \zeta_{12}^{2} q^{61} + ( -2 \zeta_{12} + 2 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( 4 + 6 \zeta_{12} - 4 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{67} + ( -6 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{71} + ( 5 + 6 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{73} + ( 3 - \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{77} + ( -6 \zeta_{12} + 4 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{79} + ( -7 \zeta_{12} + 3 \zeta_{12}^{2} - 7 \zeta_{12}^{3} ) q^{83} + ( -3 + \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{85} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{89} + ( -8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{91} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{95} + ( \zeta_{12} + \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{5} + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{5} + 2q^{7} - 4q^{13} - 12q^{17} - 4q^{19} - 2q^{25} + 12q^{29} + 2q^{31} + 4q^{35} - 4q^{37} + 12q^{41} - 10q^{43} + 6q^{47} + 6q^{49} - 36q^{53} + 12q^{59} + 8q^{61} + 4q^{65} + 8q^{67} - 24q^{71} + 20q^{73} + 6q^{77} + 8q^{79} + 6q^{83} - 6q^{85} - 32q^{91} - 2q^{95} + 2q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \zeta_{12}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
541.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0 0 0 0.500000 + 0.866025i 0 −0.366025 + 0.633975i 0 0 0
541.2 0 0 0 0.500000 + 0.866025i 0 1.36603 2.36603i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 −0.366025 0.633975i 0 0 0
1081.2 0 0 0 0.500000 0.866025i 0 1.36603 + 2.36603i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1620.2.i.n 4
3.b odd 2 1 1620.2.i.m 4
9.c even 3 1 1620.2.a.g 2
9.c even 3 1 inner 1620.2.i.n 4
9.d odd 6 1 1620.2.a.h yes 2
9.d odd 6 1 1620.2.i.m 4
36.f odd 6 1 6480.2.a.bh 2
36.h even 6 1 6480.2.a.bp 2
45.h odd 6 1 8100.2.a.s 2
45.j even 6 1 8100.2.a.t 2
45.k odd 12 2 8100.2.d.m 4
45.l even 12 2 8100.2.d.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 9.c even 3 1
1620.2.a.h yes 2 9.d odd 6 1
1620.2.i.m 4 3.b odd 2 1
1620.2.i.m 4 9.d odd 6 1
1620.2.i.n 4 1.a even 1 1 trivial
1620.2.i.n 4 9.c even 3 1 inner
6480.2.a.bh 2 36.f odd 6 1
6480.2.a.bp 2 36.h even 6 1
8100.2.a.s 2 45.h odd 6 1
8100.2.a.t 2 45.j even 6 1
8100.2.d.l 4 45.l even 12 2
8100.2.d.m 4 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1620, [\chi])\):

\( T_{7}^{4} - 2 T_{7}^{3} + 6 T_{7}^{2} + 4 T_{7} + 4 \)
\( T_{11}^{4} + 3 T_{11}^{2} + 9 \)
\( T_{17}^{2} + 6 T_{17} + 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
$11$ \( 9 + 3 T^{2} + T^{4} \)
$13$ \( 64 - 32 T + 24 T^{2} + 4 T^{3} + T^{4} \)
$17$ \( ( 6 + 6 T + T^{2} )^{2} \)
$19$ \( ( -11 + 2 T + T^{2} )^{2} \)
$23$ \( 144 + 12 T^{2} + T^{4} \)
$29$ \( 1089 - 396 T + 111 T^{2} - 12 T^{3} + T^{4} \)
$31$ \( 2209 + 94 T + 51 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( ( -26 + 2 T + T^{2} )^{2} \)
$41$ \( 81 - 108 T + 135 T^{2} - 12 T^{3} + T^{4} \)
$43$ \( 484 + 220 T + 78 T^{2} + 10 T^{3} + T^{4} \)
$47$ \( 36 - 36 T + 30 T^{2} - 6 T^{3} + T^{4} \)
$53$ \( ( 78 + 18 T + T^{2} )^{2} \)
$59$ \( 1089 - 396 T + 111 T^{2} - 12 T^{3} + T^{4} \)
$61$ \( ( 16 - 4 T + T^{2} )^{2} \)
$67$ \( 8464 + 736 T + 156 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( ( 9 + 12 T + T^{2} )^{2} \)
$73$ \( ( -2 - 10 T + T^{2} )^{2} \)
$79$ \( 8464 + 736 T + 156 T^{2} - 8 T^{3} + T^{4} \)
$83$ \( 19044 + 828 T + 174 T^{2} - 6 T^{3} + T^{4} \)
$89$ \( ( -27 + T^{2} )^{2} \)
$97$ \( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} \)
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