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Properties

Label 2583.2.a.t
Level $2583$
Weight $2$
Character orbit 2583.a
Self dual yes
Analytic conductor $20.625$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2583,2,Mod(1,2583)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2583, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2583.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2583 = 3^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2583.a (trivial)

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: yes
Analytic conductor: \(20.6253588421\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.185257757.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + 1) q^{4} + (\beta_{4} - \beta_{3}) q^{5} - q^{7} + (\beta_{2} + \beta_1 - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + \beta_{2} + 1) q^{4} + (\beta_{4} - \beta_{3}) q^{5} - q^{7} + (\beta_{2} + \beta_1 - 1) q^{8} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + \cdots + 1) q^{10}+ \cdots + \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 9 q^{4} + q^{5} - 6 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} + 9 q^{4} + q^{5} - 6 q^{7} - 3 q^{8} + 10 q^{10} - 6 q^{11} + 7 q^{13} - q^{14} + 7 q^{16} - 7 q^{17} + 2 q^{19} - 11 q^{20} + 15 q^{22} - 20 q^{23} + 29 q^{25} + 43 q^{26} - 9 q^{28} + 9 q^{29} - 27 q^{31} + 10 q^{32} + 6 q^{34} - q^{35} + 19 q^{37} + 23 q^{38} + 23 q^{40} - 6 q^{41} + 19 q^{43} - 21 q^{44} - 8 q^{46} + 19 q^{47} + 6 q^{49} + 58 q^{50} - 5 q^{53} + 3 q^{55} + 3 q^{56} + 13 q^{58} + 7 q^{59} - 12 q^{61} - 37 q^{64} + 13 q^{65} + 27 q^{67} - 31 q^{68} - 10 q^{70} + 6 q^{71} + 52 q^{73} + 14 q^{74} + 13 q^{76} + 6 q^{77} + 26 q^{80} - q^{82} - 12 q^{83} + 25 q^{85} + 10 q^{86} - 2 q^{88} + 38 q^{89} - 7 q^{91} - 45 q^{92} - 8 q^{94} - q^{95} + 8 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 10x^{3} + 23x^{2} - 24x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 5\nu + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 5\nu - 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 10\nu^{3} + 2\nu^{2} + 24\nu - 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} + 10\nu^{3} - 8\nu^{2} - 24\nu + 13 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + 6\beta_{3} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{4} - 2\beta_{3} + 8\beta_{2} + 26\beta _1 - 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.46179
−2.01956
0.306800
0.644787
2.05073
2.47904
−2.46179 0 4.06039 −2.85638 0 −1.00000 −5.07224 0 7.03179
1.2 −2.01956 0 2.07864 0.244521 0 −1.00000 −0.158810 0 −0.493825
1.3 0.306800 0 −1.90587 −0.333855 0 −1.00000 −1.19832 0 −0.102427
1.4 0.644787 0 −1.58425 4.36552 0 −1.00000 −2.31108 0 2.81483
1.5 2.05073 0 2.20548 −4.18004 0 −1.00000 0.421375 0 −8.57212
1.6 2.47904 0 4.14562 3.76023 0 −1.00000 5.31907 0 9.32175
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2583.2.a.t 6
3.b odd 2 1 287.2.a.f ✓ 6
12.b even 2 1 4592.2.a.bg 6
15.d odd 2 1 7175.2.a.p 6
21.c even 2 1 2009.2.a.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.a.f ✓ 6 3.b odd 2 1
2009.2.a.o 6 21.c even 2 1
2583.2.a.t 6 1.a even 1 1 trivial
4592.2.a.bg 6 12.b even 2 1
7175.2.a.p 6 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2583))\):

\( T_{2}^{6} - T_{2}^{5} - 10T_{2}^{4} + 10T_{2}^{3} + 23T_{2}^{2} - 24T_{2} + 5 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} - 29T_{5}^{4} + 16T_{5}^{3} + 200T_{5}^{2} + 16T_{5} - 16 \) Copy content Toggle raw display
\( T_{11}^{6} + 6T_{11}^{5} - 29T_{11}^{4} - 218T_{11}^{3} + 28T_{11}^{2} + 1928T_{11} + 2720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} - 10 T^{4} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + \cdots - 16 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 6 T^{5} + \cdots + 2720 \) Copy content Toggle raw display
$13$ \( T^{6} - 7 T^{5} + \cdots - 1546 \) Copy content Toggle raw display
$17$ \( T^{6} + 7 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} + \cdots - 3212 \) Copy content Toggle raw display
$23$ \( T^{6} + 20 T^{5} + \cdots + 344 \) Copy content Toggle raw display
$29$ \( T^{6} - 9 T^{5} + \cdots + 10448 \) Copy content Toggle raw display
$31$ \( T^{6} + 27 T^{5} + \cdots - 1280 \) Copy content Toggle raw display
$37$ \( T^{6} - 19 T^{5} + \cdots - 376 \) Copy content Toggle raw display
$41$ \( (T + 1)^{6} \) Copy content Toggle raw display
$43$ \( T^{6} - 19 T^{5} + \cdots + 29756 \) Copy content Toggle raw display
$47$ \( T^{6} - 19 T^{5} + \cdots - 512 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + \cdots - 28432 \) Copy content Toggle raw display
$59$ \( T^{6} - 7 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + \cdots - 55952 \) Copy content Toggle raw display
$67$ \( T^{6} - 27 T^{5} + \cdots - 26848 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} + \cdots + 4672 \) Copy content Toggle raw display
$73$ \( T^{6} - 52 T^{5} + \cdots + 656 \) Copy content Toggle raw display
$79$ \( T^{6} - 152 T^{4} + \cdots + 2048 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + \cdots - 19744 \) Copy content Toggle raw display
$89$ \( T^{6} - 38 T^{5} + \cdots - 345082 \) Copy content Toggle raw display
$97$ \( T^{6} - 8 T^{5} + \cdots + 303494 \) Copy content Toggle raw display
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