Properties

Label 1620.2.i.i
Level $1620$
Weight $2$
Character orbit 1620.i
Analytic conductor $12.936$
Analytic rank $0$
Dimension $2$
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: \(2\)
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 540)
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{5} + ( -2 + 2 \zeta_{6} ) q^{7} -2 \zeta_{6} q^{13} + 3 q^{17} + 5 q^{19} + 3 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} -5 \zeta_{6} q^{31} -2 q^{35} + 2 q^{37} + 12 \zeta_{6} q^{41} + ( -8 + 8 \zeta_{6} ) q^{43} + ( -12 + 12 \zeta_{6} ) q^{47} + 3 \zeta_{6} q^{49} + 3 q^{53} + 6 \zeta_{6} q^{59} + ( 7 - 7 \zeta_{6} ) q^{61} + ( 2 - 2 \zeta_{6} ) q^{65} -2 \zeta_{6} q^{67} -12 q^{71} -16 q^{73} + ( 1 - \zeta_{6} ) q^{79} + ( -15 + 15 \zeta_{6} ) q^{83} + 3 \zeta_{6} q^{85} + 12 q^{89} + 4 q^{91} + 5 \zeta_{6} q^{95} + ( 16 - 16 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{5} - 2q^{7} + O(q^{10}) \) \( 2q + q^{5} - 2q^{7} - 2q^{13} + 6q^{17} + 10q^{19} + 3q^{23} - q^{25} - 6q^{29} - 5q^{31} - 4q^{35} + 4q^{37} + 12q^{41} - 8q^{43} - 12q^{47} + 3q^{49} + 6q^{53} + 6q^{59} + 7q^{61} + 2q^{65} - 2q^{67} - 24q^{71} - 32q^{73} + q^{79} - 15q^{83} + 3q^{85} + 24q^{89} + 8q^{91} + 5q^{95} + 16q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 0 0
1081.1 0 0 0 0.500000 0.866025i 0 −1.00000 1.73205i 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.i 2
3.b odd 2 1 1620.2.i.a 2
9.c even 3 1 540.2.a.c 1
9.c even 3 1 inner 1620.2.i.i 2
9.d odd 6 1 540.2.a.f yes 1
9.d odd 6 1 1620.2.i.a 2
36.f odd 6 1 2160.2.a.c 1
36.h even 6 1 2160.2.a.o 1
45.h odd 6 1 2700.2.a.g 1
45.j even 6 1 2700.2.a.f 1
45.k odd 12 2 2700.2.d.e 2
45.l even 12 2 2700.2.d.f 2
72.j odd 6 1 8640.2.a.w 1
72.l even 6 1 8640.2.a.g 1
72.n even 6 1 8640.2.a.bz 1
72.p odd 6 1 8640.2.a.bl 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
540.2.a.c 1 9.c even 3 1
540.2.a.f yes 1 9.d odd 6 1
1620.2.i.a 2 3.b odd 2 1
1620.2.i.a 2 9.d odd 6 1
1620.2.i.i 2 1.a even 1 1 trivial
1620.2.i.i 2 9.c even 3 1 inner
2160.2.a.c 1 36.f odd 6 1
2160.2.a.o 1 36.h even 6 1
2700.2.a.f 1 45.j even 6 1
2700.2.a.g 1 45.h odd 6 1
2700.2.d.e 2 45.k odd 12 2
2700.2.d.f 2 45.l even 12 2
8640.2.a.g 1 72.l even 6 1
8640.2.a.w 1 72.j odd 6 1
8640.2.a.bl 1 72.p odd 6 1
8640.2.a.bz 1 72.n even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4 + 2 T + T^{2} \)
$17$ \( ( -3 + T )^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 9 - 3 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( 25 + 5 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( 144 - 12 T + T^{2} \)
$43$ \( 64 + 8 T + T^{2} \)
$47$ \( 144 + 12 T + T^{2} \)
$53$ \( ( -3 + T )^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 49 - 7 T + T^{2} \)
$67$ \( 4 + 2 T + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( ( 16 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( 225 + 15 T + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( 256 - 16 T + T^{2} \)
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