Properties

Label 2592.2.i.bc
Level $2592$
Weight $2$
Character orbit 2592.i
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - \beta_{3} - \beta_{2}) q^{7} + ( - \beta_1 + 1) q^{11} - 4 \beta_1 q^{13} - 2 \beta_{3} q^{19} - 6 \beta_1 q^{23} + (8 \beta_1 - 8) q^{25} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{29} - \beta_{2} q^{31} - 13 q^{35} + 10 q^{37} + 2 \beta_{2} q^{41} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{43} + (10 \beta_1 - 10) q^{47} - 6 \beta_1 q^{49} + \beta_{3} q^{53} + \beta_{3} q^{55} - 4 \beta_1 q^{59} + (4 \beta_{3} + 4 \beta_{2}) q^{65} + 2 \beta_{2} q^{67} - 8 q^{71} - 3 q^{73} - \beta_{2} q^{77} + (4 \beta_{3} + 4 \beta_{2}) q^{79} + ( - 9 \beta_1 + 9) q^{83} - 2 \beta_{3} q^{89} + 4 \beta_{3} q^{91} - 26 \beta_1 q^{95} + (7 \beta_1 - 7) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{11} - 8 q^{13} - 12 q^{23} - 16 q^{25} - 52 q^{35} + 40 q^{37} - 20 q^{47} - 12 q^{49} - 8 q^{59} - 32 q^{71} - 12 q^{73} + 18 q^{83} - 52 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
1.15139 + 1.99426i
−0.651388 1.12824i
1.15139 1.99426i
−0.651388 + 1.12824i
0 0 0 −1.80278 3.12250i 0 1.80278 3.12250i 0 0 0
865.2 0 0 0 1.80278 + 3.12250i 0 −1.80278 + 3.12250i 0 0 0
1729.1 0 0 0 −1.80278 + 3.12250i 0 1.80278 + 3.12250i 0 0 0
1729.2 0 0 0 1.80278 3.12250i 0 −1.80278 3.12250i 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
12.b even 2 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.bc 4
3.b odd 2 1 2592.2.i.bb 4
4.b odd 2 1 2592.2.i.bb 4
9.c even 3 1 864.2.a.m 2
9.c even 3 1 inner 2592.2.i.bc 4
9.d odd 6 1 864.2.a.n yes 2
9.d odd 6 1 2592.2.i.bb 4
12.b even 2 1 inner 2592.2.i.bc 4
36.f odd 6 1 864.2.a.n yes 2
36.f odd 6 1 2592.2.i.bb 4
36.h even 6 1 864.2.a.m 2
36.h even 6 1 inner 2592.2.i.bc 4
72.j odd 6 1 1728.2.a.bc 2
72.l even 6 1 1728.2.a.bd 2
72.n even 6 1 1728.2.a.bd 2
72.p odd 6 1 1728.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 9.c even 3 1
864.2.a.m 2 36.h even 6 1
864.2.a.n yes 2 9.d odd 6 1
864.2.a.n yes 2 36.f odd 6 1
1728.2.a.bc 2 72.j odd 6 1
1728.2.a.bc 2 72.p odd 6 1
1728.2.a.bd 2 72.l even 6 1
1728.2.a.bd 2 72.n even 6 1
2592.2.i.bb 4 3.b odd 2 1
2592.2.i.bb 4 4.b odd 2 1
2592.2.i.bb 4 9.d odd 6 1
2592.2.i.bb 4 36.f odd 6 1
2592.2.i.bc 4 1.a even 1 1 trivial
2592.2.i.bc 4 9.c even 3 1 inner
2592.2.i.bc 4 12.b even 2 1 inner
2592.2.i.bc 4 36.h even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{4} + 13T_{5}^{2} + 169 \) Copy content Toggle raw display
\( T_{7}^{4} + 13T_{7}^{2} + 169 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$31$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$37$ \( (T - 10)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$43$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 52T^{2} + 2704 \) Copy content Toggle raw display
$71$ \( (T + 8)^{4} \) Copy content Toggle raw display
$73$ \( (T + 3)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 208 T^{2} + 43264 \) Copy content Toggle raw display
$83$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
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