Properties

Label 3240.2.q.a
Level $3240$
Weight $2$
Character orbit 3240.q
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8715302549\)
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 120)
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} +O(q^{10})\) \( q -\zeta_{6} q^{5} + ( -4 + 4 \zeta_{6} ) q^{7} + 6 \zeta_{6} q^{13} + 2 q^{17} + 4 q^{19} -8 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + ( -6 + 6 \zeta_{6} ) q^{29} + 4 q^{35} -6 q^{37} + 10 \zeta_{6} q^{41} + ( 4 - 4 \zeta_{6} ) q^{43} + ( 8 - 8 \zeta_{6} ) q^{47} -9 \zeta_{6} q^{49} -10 q^{53} + ( -6 + 6 \zeta_{6} ) q^{61} + ( 6 - 6 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} -14 q^{73} + ( -16 + 16 \zeta_{6} ) q^{79} + ( 12 - 12 \zeta_{6} ) q^{83} -2 \zeta_{6} q^{85} -2 q^{89} -24 q^{91} -4 \zeta_{6} q^{95} + ( -2 + 2 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} - 4q^{7} + O(q^{10}) \) \( 2q - q^{5} - 4q^{7} + 6q^{13} + 4q^{17} + 8q^{19} - 8q^{23} - q^{25} - 6q^{29} + 8q^{35} - 12q^{37} + 10q^{41} + 4q^{43} + 8q^{47} - 9q^{49} - 20q^{53} - 6q^{61} + 6q^{65} + 4q^{67} - 28q^{73} - 16q^{79} + 12q^{83} - 2q^{85} - 4q^{89} - 48q^{91} - 4q^{95} - 2q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1081.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 + 0.866025i 0 −2.00000 3.46410i 0 0 0
2161.1 0 0 0 −0.500000 0.866025i 0 −2.00000 + 3.46410i 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 3240.2.q.a 2
3.b odd 2 1 3240.2.q.m 2
9.c even 3 1 360.2.a.e 1
9.c even 3 1 inner 3240.2.q.a 2
9.d odd 6 1 120.2.a.a 1
9.d odd 6 1 3240.2.q.m 2
36.f odd 6 1 720.2.a.f 1
36.h even 6 1 240.2.a.a 1
45.h odd 6 1 600.2.a.a 1
45.j even 6 1 1800.2.a.c 1
45.k odd 12 2 1800.2.f.g 2
45.l even 12 2 600.2.f.c 2
63.o even 6 1 5880.2.a.p 1
72.j odd 6 1 960.2.a.g 1
72.l even 6 1 960.2.a.n 1
72.n even 6 1 2880.2.a.r 1
72.p odd 6 1 2880.2.a.b 1
144.u even 12 2 3840.2.k.z 2
144.w odd 12 2 3840.2.k.a 2
180.n even 6 1 1200.2.a.r 1
180.p odd 6 1 3600.2.a.bo 1
180.v odd 12 2 1200.2.f.f 2
180.x even 12 2 3600.2.f.l 2
360.bd even 6 1 4800.2.a.bh 1
360.bh odd 6 1 4800.2.a.bl 1
360.br even 12 2 4800.2.f.u 2
360.bt odd 12 2 4800.2.f.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.a.a 1 9.d odd 6 1
240.2.a.a 1 36.h even 6 1
360.2.a.e 1 9.c even 3 1
600.2.a.a 1 45.h odd 6 1
600.2.f.c 2 45.l even 12 2
720.2.a.f 1 36.f odd 6 1
960.2.a.g 1 72.j odd 6 1
960.2.a.n 1 72.l even 6 1
1200.2.a.r 1 180.n even 6 1
1200.2.f.f 2 180.v odd 12 2
1800.2.a.c 1 45.j even 6 1
1800.2.f.g 2 45.k odd 12 2
2880.2.a.b 1 72.p odd 6 1
2880.2.a.r 1 72.n even 6 1
3240.2.q.a 2 1.a even 1 1 trivial
3240.2.q.a 2 9.c even 3 1 inner
3240.2.q.m 2 3.b odd 2 1
3240.2.q.m 2 9.d odd 6 1
3600.2.a.bo 1 180.p odd 6 1
3600.2.f.l 2 180.x even 12 2
3840.2.k.a 2 144.w odd 12 2
3840.2.k.z 2 144.u even 12 2
4800.2.a.bh 1 360.bd even 6 1
4800.2.a.bl 1 360.bh odd 6 1
4800.2.f.n 2 360.bt odd 12 2
4800.2.f.u 2 360.br even 12 2
5880.2.a.p 1 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3240, [\chi])\):

\( T_{7}^{2} + 4 T_{7} + 16 \)
\( T_{11} \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( 16 + 4 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 36 - 6 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 64 + 8 T + T^{2} \)
$29$ \( 36 + 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( 6 + T )^{2} \)
$41$ \( 100 - 10 T + T^{2} \)
$43$ \( 16 - 4 T + T^{2} \)
$47$ \( 64 - 8 T + T^{2} \)
$53$ \( ( 10 + T )^{2} \)
$59$ \( T^{2} \)
$61$ \( 36 + 6 T + T^{2} \)
$67$ \( 16 - 4 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 14 + T )^{2} \)
$79$ \( 256 + 16 T + T^{2} \)
$83$ \( 144 - 12 T + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( 4 + 2 T + T^{2} \)
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