Properties

Label 2592.2.p.f
Level $2592$
Weight $2$
Character orbit 2592.p
Analytic conductor $20.697$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2592,2,Mod(431,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.534694406811304329216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{5} + (\beta_{15} + \beta_{9}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{5} + (\beta_{15} + \beta_{9}) q^{7} + ( - \beta_{10} - \beta_{4}) q^{11} + (\beta_{13} + \beta_{9} - \beta_{6}) q^{13} + ( - \beta_{14} + \beta_{10} - \beta_{5}) q^{17} + ( - \beta_{11} - 2) q^{19} + ( - \beta_{12} + \beta_{7}) q^{23} + (2 \beta_{3} - \beta_{2} - 1) q^{25} + 2 \beta_{8} q^{29} + ( - 2 \beta_{13} + 2 \beta_{6}) q^{31} + ( - 2 \beta_{14} + \beta_{10} - 2 \beta_{5}) q^{35} + (\beta_{15} + \beta_{13}) q^{37} + 2 \beta_{4} q^{41} + (2 \beta_{2} + 2) q^{43} + (\beta_{8} - 3 \beta_1) q^{47} + (4 \beta_{11} + 4 \beta_{3} - 2 \beta_{2}) q^{49} + (2 \beta_{12} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_1) q^{53} - 2 \beta_{13} q^{55} + \beta_{4} q^{59} + (\beta_{15} + \beta_{9} - \beta_{6}) q^{61} + ( - \beta_{14} - \beta_{10} - \beta_{4}) q^{65} + ( - 3 \beta_{11} - 3 \beta_{3} + 4 \beta_{2}) q^{67} + (2 \beta_{12} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_1) q^{71} + (4 \beta_{11} - 1) q^{73} + (\beta_{12} + 2 \beta_{7}) q^{77} + ( - 3 \beta_{15} - 3 \beta_{9} + 2 \beta_{6}) q^{79} - 2 \beta_{14} q^{83} + (2 \beta_{13} - 4 \beta_{9} - 2 \beta_{6}) q^{85} + ( - 3 \beta_{14} - \beta_{10} - 3 \beta_{5}) q^{89} + ( - \beta_{11} + 6) q^{91} + (3 \beta_{12} + \beta_{7}) q^{95} + ( - 6 \beta_{3} + \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{19} - 8 q^{25} + 16 q^{43} + 16 q^{49} - 32 q^{67} - 16 q^{73} + 96 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 2\nu^{11} - 4\nu^{7} + 88\nu^{5} + 32\nu^{3} - 128\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 2\nu^{10} - 4\nu^{6} - 8\nu^{4} + 32\nu^{2} - 128 ) / 192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} + 6\nu^{10} - 16\nu^{8} - 12\nu^{6} - 24\nu^{4} - 64\nu^{2} + 192 ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} + 2\nu^{13} - 2\nu^{11} - 4\nu^{9} - 60\nu^{7} + 32\nu^{5} + 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 6\nu^{13} - 2\nu^{11} - 4\nu^{9} + 4\nu^{7} + 32\nu^{5} - 640\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{14} + 2\nu^{10} - 8\nu^{8} - 4\nu^{6} - 8\nu^{4} + 144\nu^{2} + 64 ) / 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 6\nu^{13} - 2\nu^{11} - 4\nu^{9} + 4\nu^{7} + 32\nu^{5} + 128\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{15} + 6\nu^{13} + 10\nu^{11} - 12\nu^{9} - 20\nu^{7} - 64\nu^{5} + 160\nu^{3} + 128\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{14} + \nu^{12} + 8\nu^{10} - 2\nu^{8} + 28\nu^{4} + 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{15} - 6\nu^{11} + 12\nu^{7} + 24\nu^{5} + 64\nu^{3} - 192\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{14} + 2\nu^{12} + 6\nu^{10} + 12\nu^{8} + 20\nu^{6} - 96\nu^{4} + 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + 2\nu^{11} + 4\nu^{9} - 4\nu^{7} - 32\nu^{5} + 112\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{14} - 10\nu^{12} + 2\nu^{10} + 4\nu^{8} + 28\nu^{6} - 32\nu^{4} ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -5\nu^{15} + 6\nu^{13} + 2\nu^{11} - 12\nu^{9} - 4\nu^{7} - 32\nu^{5} + 32\nu^{3} + 640\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -5\nu^{14} - 2\nu^{12} + 10\nu^{10} + 20\nu^{8} - 52\nu^{6} - 160\nu^{4} + 576 ) / 192 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{12} + \beta_{10} + \beta_{8} + \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{9} - \beta_{3} - 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{14} - \beta_{8} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{15} + \beta_{13} + 3\beta_{11} + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{12} - \beta_{5} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{15} + 2\beta_{9} - 2\beta_{6} - 8\beta_{3} + 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3\beta_{14} + 6\beta_{12} + 2\beta_{10} - 3\beta_{8} - 3\beta_{7} - 3\beta_{5} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2\beta_{13} + 2\beta_{11} + 8\beta_{9} - 2\beta_{6} + 2\beta_{3} + 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -4\beta_{14} - 6\beta_{10} + 6\beta_{8} - 6\beta_{4} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -4\beta_{15} - 16\beta_{13} + 12\beta_{11} - 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -8\beta_{12} - 22\beta_{7} - 10\beta_{5} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -8\beta_{15} - 8\beta_{9} - 12\beta_{6} + 4\beta_{3} + 116\beta_{2} + 116 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -36\beta_{14} - 20\beta_{12} - 4\beta_{10} + 16\beta_{8} + 16\beta_{7} - 36\beta_{5} - 20\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−0.841995 + 1.13624i
1.40501 0.161069i
−1.32661 0.490008i
0.238945 1.39388i
1.32661 + 0.490008i
−0.238945 + 1.39388i
0.841995 1.13624i
−1.40501 + 0.161069i
−0.841995 1.13624i
1.40501 + 0.161069i
−1.32661 + 0.490008i
0.238945 + 1.39388i
1.32661 0.490008i
−0.238945 1.39388i
0.841995 + 1.13624i
−1.40501 0.161069i
0 0 0 −1.53819 2.66422i 0 −3.45632 1.99551i 0 0 0
431.2 0 0 0 −1.53819 2.66422i 0 3.45632 + 1.99551i 0 0 0
431.3 0 0 0 −0.796225 1.37910i 0 −1.24653 0.719687i 0 0 0
431.4 0 0 0 −0.796225 1.37910i 0 1.24653 + 0.719687i 0 0 0
431.5 0 0 0 0.796225 + 1.37910i 0 −1.24653 0.719687i 0 0 0
431.6 0 0 0 0.796225 + 1.37910i 0 1.24653 + 0.719687i 0 0 0
431.7 0 0 0 1.53819 + 2.66422i 0 −3.45632 1.99551i 0 0 0
431.8 0 0 0 1.53819 + 2.66422i 0 3.45632 + 1.99551i 0 0 0
2159.1 0 0 0 −1.53819 + 2.66422i 0 −3.45632 + 1.99551i 0 0 0
2159.2 0 0 0 −1.53819 + 2.66422i 0 3.45632 1.99551i 0 0 0
2159.3 0 0 0 −0.796225 + 1.37910i 0 −1.24653 + 0.719687i 0 0 0
2159.4 0 0 0 −0.796225 + 1.37910i 0 1.24653 0.719687i 0 0 0
2159.5 0 0 0 0.796225 1.37910i 0 −1.24653 + 0.719687i 0 0 0
2159.6 0 0 0 0.796225 1.37910i 0 1.24653 0.719687i 0 0 0
2159.7 0 0 0 1.53819 2.66422i 0 −3.45632 + 1.99551i 0 0 0
2159.8 0 0 0 1.53819 2.66422i 0 3.45632 1.99551i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.f even 2 1 inner
72.l even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.p.f 16
3.b odd 2 1 inner 2592.2.p.f 16
4.b odd 2 1 648.2.l.f 16
8.b even 2 1 648.2.l.f 16
8.d odd 2 1 inner 2592.2.p.f 16
9.c even 3 1 864.2.f.a 8
9.c even 3 1 inner 2592.2.p.f 16
9.d odd 6 1 864.2.f.a 8
9.d odd 6 1 inner 2592.2.p.f 16
12.b even 2 1 648.2.l.f 16
24.f even 2 1 inner 2592.2.p.f 16
24.h odd 2 1 648.2.l.f 16
36.f odd 6 1 216.2.f.a 8
36.f odd 6 1 648.2.l.f 16
36.h even 6 1 216.2.f.a 8
36.h even 6 1 648.2.l.f 16
72.j odd 6 1 216.2.f.a 8
72.j odd 6 1 648.2.l.f 16
72.l even 6 1 864.2.f.a 8
72.l even 6 1 inner 2592.2.p.f 16
72.n even 6 1 216.2.f.a 8
72.n even 6 1 648.2.l.f 16
72.p odd 6 1 864.2.f.a 8
72.p odd 6 1 inner 2592.2.p.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.f.a 8 36.f odd 6 1
216.2.f.a 8 36.h even 6 1
216.2.f.a 8 72.j odd 6 1
216.2.f.a 8 72.n even 6 1
648.2.l.f 16 4.b odd 2 1
648.2.l.f 16 8.b even 2 1
648.2.l.f 16 12.b even 2 1
648.2.l.f 16 24.h odd 2 1
648.2.l.f 16 36.f odd 6 1
648.2.l.f 16 36.h even 6 1
648.2.l.f 16 72.j odd 6 1
648.2.l.f 16 72.n even 6 1
864.2.f.a 8 9.c even 3 1
864.2.f.a 8 9.d odd 6 1
864.2.f.a 8 72.l even 6 1
864.2.f.a 8 72.p odd 6 1
2592.2.p.f 16 1.a even 1 1 trivial
2592.2.p.f 16 3.b odd 2 1 inner
2592.2.p.f 16 8.d odd 2 1 inner
2592.2.p.f 16 9.c even 3 1 inner
2592.2.p.f 16 9.d odd 6 1 inner
2592.2.p.f 16 24.f even 2 1 inner
2592.2.p.f 16 72.l even 6 1 inner
2592.2.p.f 16 72.p odd 6 1 inner

Hecke kernels

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

\( T_{5}^{8} + 12T_{5}^{6} + 120T_{5}^{4} + 288T_{5}^{2} + 576 \) Copy content Toggle raw display
\( T_{7}^{8} - 18T_{7}^{6} + 291T_{7}^{4} - 594T_{7}^{2} + 1089 \) Copy content Toggle raw display
\( T_{41}^{8} - 112T_{41}^{6} + 11136T_{41}^{4} - 157696T_{41}^{2} + 1982464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 12 T^{6} + 120 T^{4} + 288 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 18 T^{6} + 291 T^{4} - 594 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + 696 T^{4} - 2464 T^{2} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 30 T^{6} + 867 T^{4} - 990 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 88)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 1)^{8} \) Copy content Toggle raw display
$23$ \( (T^{8} + 36 T^{6} + 1272 T^{4} + 864 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 96 T^{6} + 7680 T^{4} + \cdots + 2359296)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 96 T^{6} + 7104 T^{4} + \cdots + 4460544)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 54 T^{2} + 297)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 112 T^{6} + 11136 T^{4} + \cdots + 1982464)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 4)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 132 T^{6} + 13368 T^{4} + \cdots + 16451136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 144 T^{2} + 3456)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 28 T^{6} + 696 T^{4} - 2464 T^{2} + \cdots + 7744)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 30 T^{6} + 867 T^{4} - 990 T^{2} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + 75 T^{2} - 88 T + 121)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 3456)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2 T - 47)^{8} \) Copy content Toggle raw display
$79$ \( (T^{8} - 186 T^{6} + 29019 T^{4} + \cdots + 31102929)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 160 T^{6} + 19968 T^{4} + \cdots + 31719424)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 436 T^{2} + 46552)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 111 T^{2} + 214 T + 11449)^{4} \) Copy content Toggle raw display
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