Properties

Label 2583.2.a.v
Level $2583$
Weight $2$
Character orbit 2583.a
Self dual yes
Analytic conductor $20.625$
Analytic rank $1$
Dimension $8$
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 16x^{4} - 58x^{3} - 28x^{2} + 26x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{6} - 1) q^{5} + q^{7} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{6} - 1) q^{5} + q^{7} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_1 - 2) q^{8} + ( - \beta_{7} - \beta_{2} - \beta_1 + 1) q^{10} + (\beta_{3} - \beta_{2}) q^{11} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_1) q^{13} + (\beta_1 - 1) q^{14} + (\beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{16} + ( - \beta_{3} - \beta_1 - 3) q^{17} + (\beta_{7} + \beta_{4} - \beta_{3} + \beta_{2}) q^{19} + (\beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{20} + ( - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{22} + ( - \beta_{6} + \beta_{3} - 2) q^{23} + ( - 2 \beta_{6} - \beta_{5} - \beta_{2} + \beta_1) q^{25} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 3) q^{26} + (\beta_{2} - \beta_1 + 2) q^{28} + (\beta_{4} + \beta_{3} + \beta_{2}) q^{29} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 1) q^{31} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 4) q^{32} + ( - \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{34} + (\beta_{6} - 1) q^{35} + ( - 2 \beta_{6} - \beta_{2} + \beta_1 - 2) q^{37} + ( - 2 \beta_{7} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{38} + (\beta_{6} + \beta_{5} - 3 \beta_1) q^{40} + q^{41} + ( - \beta_{6} - \beta_{5} + \beta_{4} + \beta_1) q^{43} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + \cdots - 1) q^{44}+ \cdots + (\beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 8 q^{7} - 12 q^{8} + 6 q^{10} - 2 q^{11} - 2 q^{13} - 4 q^{14} + 16 q^{16} - 26 q^{17} - 22 q^{20} - 18 q^{22} - 18 q^{23} + 4 q^{25} - 20 q^{26} + 12 q^{28} - 2 q^{29} + 6 q^{31} - 18 q^{32} - 6 q^{34} - 8 q^{35} - 12 q^{37} + 8 q^{38} - 12 q^{40} + 8 q^{41} + 4 q^{43} - 20 q^{44} - 40 q^{47} + 8 q^{49} + 14 q^{50} + 30 q^{52} - 12 q^{53} + 8 q^{55} - 12 q^{56} - 2 q^{58} - 6 q^{59} + 22 q^{61} - 26 q^{62} + 40 q^{64} - 14 q^{65} - 16 q^{67} - 48 q^{68} + 6 q^{70} + 4 q^{71} - 28 q^{73} + 16 q^{74} - 6 q^{76} - 2 q^{77} + 2 q^{79} - 32 q^{80} - 4 q^{82} - 28 q^{83} + 26 q^{85} + 20 q^{86} - 42 q^{88} + 2 q^{89} - 2 q^{91} + 6 q^{92} + 18 q^{94} + 6 q^{95} + 4 q^{97} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 6x^{6} + 28x^{5} + 16x^{4} - 58x^{3} - 28x^{2} + 26x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 8\nu^{3} + 13\nu^{2} + 5\nu - 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 5\nu^{6} + \nu^{5} + 21\nu^{4} - 18\nu^{3} - 15\nu^{2} + 19\nu - 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 6\nu^{5} + 13\nu^{4} - 28\nu^{3} - 29\nu^{2} + 20\nu + 11 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 5\nu^{5} + 25\nu^{4} + 8\nu^{3} - 41\nu^{2} - 6\nu + 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 5\nu^{6} - 2\nu^{5} + 33\nu^{4} - 9\nu^{3} - 66\nu^{2} + 13\nu + 29 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + 3\beta_{2} + 5\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 4\beta_{6} - 3\beta_{5} - 2\beta_{3} + 13\beta_{2} + 11\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{7} - 19\beta_{6} - 15\beta_{5} + \beta_{4} - 11\beta_{3} + 44\beta_{2} + 40\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 66\beta_{7} - 69\beta_{6} - 52\beta_{5} + 3\beta_{4} - 32\beta_{3} + 160\beta_{2} + 117\beta _1 + 189 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 246\beta_{7} - 260\beta_{6} - 200\beta_{5} + 17\beta_{4} - 125\beta_{3} + 552\beta_{2} + 400\beta _1 + 592 \) Copy content Toggle raw display

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
−1.79360
−1.41978
−0.940000
−0.351752
0.740623
2.04374
2.28120
3.43957
−2.79360 0 5.80420 −0.548043 0 1.00000 −10.6274 0 1.53101
1.2 −2.41978 0 3.85532 −1.32904 0 1.00000 −4.48946 0 3.21599
1.3 −1.94000 0 1.76360 −3.48399 0 1.00000 0.458617 0 6.75895
1.4 −1.35175 0 −0.172766 1.36361 0 1.00000 2.93704 0 −1.84326
1.5 −0.259377 0 −1.93272 −3.60445 0 1.00000 1.02006 0 0.934912
1.6 1.04374 0 −0.910616 2.36717 0 1.00000 −3.03791 0 2.47070
1.7 1.28120 0 −0.358525 0.278215 0 1.00000 −3.02174 0 0.356450
1.8 2.43957 0 3.95150 −3.04347 0 1.00000 4.76083 0 −7.42475
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.v 8
3.b odd 2 1 2583.2.a.w yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2583.2.a.v 8 1.a even 1 1 trivial
2583.2.a.w yes 8 3.b odd 2 1

Hecke kernels

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

\( T_{2}^{8} + 4T_{2}^{7} - 6T_{2}^{6} - 36T_{2}^{5} - 4T_{2}^{4} + 82T_{2}^{3} + 28T_{2}^{2} - 56T_{2} - 15 \) Copy content Toggle raw display
\( T_{5}^{8} + 8T_{5}^{7} + 10T_{5}^{6} - 56T_{5}^{5} - 124T_{5}^{4} + 60T_{5}^{3} + 202T_{5}^{2} + 32T_{5} - 25 \) Copy content Toggle raw display
\( T_{11}^{8} + 2T_{11}^{7} - 44T_{11}^{6} - 80T_{11}^{5} + 560T_{11}^{4} + 896T_{11}^{3} - 1920T_{11}^{2} - 2880T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} - 6 T^{6} - 36 T^{5} + \cdots - 15 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + 10 T^{6} - 56 T^{5} + \cdots - 25 \) Copy content Toggle raw display
$7$ \( (T - 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 2 T^{7} - 44 T^{6} - 80 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} - 64 T^{6} + \cdots - 19683 \) Copy content Toggle raw display
$17$ \( T^{8} + 26 T^{7} + 254 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$19$ \( T^{8} - 72 T^{6} - 46 T^{5} + \cdots + 5508 \) Copy content Toggle raw display
$23$ \( T^{8} + 18 T^{7} + 86 T^{6} + \cdots - 5431 \) Copy content Toggle raw display
$29$ \( T^{8} + 2 T^{7} - 96 T^{6} + \cdots + 23391 \) Copy content Toggle raw display
$31$ \( T^{8} - 6 T^{7} - 122 T^{6} + \cdots + 33444 \) Copy content Toggle raw display
$37$ \( T^{8} + 12 T^{7} - 16 T^{6} + \cdots - 3875 \) Copy content Toggle raw display
$41$ \( (T - 1)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} - 4 T^{7} - 72 T^{6} + \cdots - 64756 \) Copy content Toggle raw display
$47$ \( T^{8} + 40 T^{7} + 602 T^{6} + \cdots - 1741983 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} - 228 T^{6} + \cdots - 1211601 \) Copy content Toggle raw display
$59$ \( T^{8} + 6 T^{7} - 236 T^{6} + \cdots + 625052 \) Copy content Toggle raw display
$61$ \( T^{8} - 22 T^{7} + 82 T^{6} + \cdots + 8468 \) Copy content Toggle raw display
$67$ \( T^{8} + 16 T^{7} - 268 T^{6} + \cdots - 127213 \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{7} - 284 T^{6} + \cdots + 1722300 \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + 98 T^{6} + \cdots - 976572 \) Copy content Toggle raw display
$79$ \( T^{8} - 2 T^{7} - 380 T^{6} + \cdots - 4110025 \) Copy content Toggle raw display
$83$ \( T^{8} + 28 T^{7} + 172 T^{6} + \cdots + 192832 \) Copy content Toggle raw display
$89$ \( T^{8} - 2 T^{7} - 244 T^{6} + \cdots + 89136 \) Copy content Toggle raw display
$97$ \( T^{8} - 4 T^{7} - 416 T^{6} + \cdots - 201023 \) Copy content Toggle raw display
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