Properties

Label 8043.2.a.m
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $3$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - q^{7} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 1) q^{4} + (\beta_1 - 1) q^{5} - \beta_1 q^{6} - q^{7} + ( - \beta_{2} - 2 \beta_1 - 2) q^{8} + q^{9} + ( - \beta_{2} - 3) q^{10} + (\beta_{2} + 1) q^{11} + (\beta_{2} + \beta_1 + 1) q^{12} + ( - \beta_1 + 5) q^{13} + \beta_1 q^{14} + (\beta_1 - 1) q^{15} + (4 \beta_1 + 3) q^{16} + (\beta_{2} + \beta_1 - 4) q^{17} - \beta_1 q^{18} + (2 \beta_{2} + 2 \beta_1) q^{19} + (3 \beta_1 + 1) q^{20} - q^{21} + ( - 3 \beta_1 + 1) q^{22} + ( - \beta_{2} + 3 \beta_1 + 2) q^{23} + ( - \beta_{2} - 2 \beta_1 - 2) q^{24} + (\beta_{2} - \beta_1 - 1) q^{25} + (\beta_{2} - 4 \beta_1 + 3) q^{26} + q^{27} + ( - \beta_{2} - \beta_1 - 1) q^{28} + (3 \beta_{2} + \beta_1 + 2) q^{29} + ( - \beta_{2} - 3) q^{30} + ( - \beta_{2} - \beta_1 - 6) q^{31} + ( - 2 \beta_{2} - 3 \beta_1 - 8) q^{32} + (\beta_{2} + 1) q^{33} + ( - \beta_{2} + \beta_1 - 2) q^{34} + ( - \beta_1 + 1) q^{35} + (\beta_{2} + \beta_1 + 1) q^{36} + 10 q^{37} + ( - 2 \beta_{2} - 6 \beta_1 - 4) q^{38} + ( - \beta_1 + 5) q^{39} + ( - \beta_{2} - 4 \beta_1 - 3) q^{40} + ( - 2 \beta_{2} + \beta_1 + 5) q^{41} + \beta_1 q^{42} + (2 \beta_1 + 2) q^{43} + (\beta_{2} + 2 \beta_1 + 7) q^{44} + (\beta_1 - 1) q^{45} + ( - 3 \beta_{2} - 3 \beta_1 - 10) q^{46} + (2 \beta_{2} + 2) q^{47} + (4 \beta_1 + 3) q^{48} + q^{49} + (\beta_{2} + 4) q^{50} + (\beta_{2} + \beta_1 - 4) q^{51} + (4 \beta_{2} + \beta_1 + 3) q^{52} + (\beta_{2} + \beta_1 + 4) q^{53} - \beta_1 q^{54} + ( - \beta_{2} + 3 \beta_1 - 2) q^{55} + (\beta_{2} + 2 \beta_1 + 2) q^{56} + (2 \beta_{2} + 2 \beta_1) q^{57} + ( - \beta_{2} - 9 \beta_1) q^{58} + (2 \beta_{2} + 2 \beta_1 + 4) q^{59} + (3 \beta_1 + 1) q^{60} + (2 \beta_{2} - 3 \beta_1 - 7) q^{61} + (\beta_{2} + 9 \beta_1 + 2) q^{62} - q^{63} + (3 \beta_{2} + 7 \beta_1 + 1) q^{64} + ( - \beta_{2} + 5 \beta_1 - 8) q^{65} + ( - 3 \beta_1 + 1) q^{66} + (2 \beta_{2} + 4 \beta_1 - 10) q^{67} + ( - 3 \beta_{2} + \beta_1 + 4) q^{68} + ( - \beta_{2} + 3 \beta_1 + 2) q^{69} + (\beta_{2} + 3) q^{70} + ( - 3 \beta_{2} - 3 \beta_1 + 2) q^{71} + ( - \beta_{2} - 2 \beta_1 - 2) q^{72} + ( - 4 \beta_1 + 2) q^{73} - 10 \beta_1 q^{74} + (\beta_{2} - \beta_1 - 1) q^{75} + (2 \beta_{2} + 10 \beta_1 + 16) q^{76} + ( - \beta_{2} - 1) q^{77} + (\beta_{2} - 4 \beta_1 + 3) q^{78} + (\beta_{2} + 2 \beta_1 - 5) q^{79} + (4 \beta_{2} + 3 \beta_1 + 9) q^{80} + q^{81} + ( - \beta_{2} - 2 \beta_1 - 5) q^{82} + ( - 4 \beta_1 + 8) q^{83} + ( - \beta_{2} - \beta_1 - 1) q^{84} + ( - 2 \beta_1 + 6) q^{85} + ( - 2 \beta_{2} - 4 \beta_1 - 6) q^{86} + (3 \beta_{2} + \beta_1 + 2) q^{87} + ( - 2 \beta_{2} - 5 \beta_1 - 7) q^{88} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{89} + ( - \beta_{2} - 3) q^{90} + (\beta_1 - 5) q^{91} + (5 \beta_{2} + 13 \beta_1 + 2) q^{92} + ( - \beta_{2} - \beta_1 - 6) q^{93} + ( - 6 \beta_1 + 2) q^{94} + (4 \beta_1 + 4) q^{95} + ( - 2 \beta_{2} - 3 \beta_1 - 8) q^{96} + ( - 2 \beta_{2} - \beta_1 + 7) q^{97} - \beta_1 q^{98} + (\beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{3} + 5 q^{4} - 2 q^{5} - q^{6} - 3 q^{7} - 9 q^{8} + 3 q^{9} - 10 q^{10} + 4 q^{11} + 5 q^{12} + 14 q^{13} + q^{14} - 2 q^{15} + 13 q^{16} - 10 q^{17} - q^{18} + 4 q^{19} + 6 q^{20} - 3 q^{21} + 8 q^{23} - 9 q^{24} - 3 q^{25} + 6 q^{26} + 3 q^{27} - 5 q^{28} + 10 q^{29} - 10 q^{30} - 20 q^{31} - 29 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{35} + 5 q^{36} + 30 q^{37} - 20 q^{38} + 14 q^{39} - 14 q^{40} + 14 q^{41} + q^{42} + 8 q^{43} + 24 q^{44} - 2 q^{45} - 36 q^{46} + 8 q^{47} + 13 q^{48} + 3 q^{49} + 13 q^{50} - 10 q^{51} + 14 q^{52} + 14 q^{53} - q^{54} - 4 q^{55} + 9 q^{56} + 4 q^{57} - 10 q^{58} + 16 q^{59} + 6 q^{60} - 22 q^{61} + 16 q^{62} - 3 q^{63} + 13 q^{64} - 20 q^{65} - 24 q^{67} + 10 q^{68} + 8 q^{69} + 10 q^{70} - 9 q^{72} + 2 q^{73} - 10 q^{74} - 3 q^{75} + 60 q^{76} - 4 q^{77} + 6 q^{78} - 12 q^{79} + 34 q^{80} + 3 q^{81} - 18 q^{82} + 20 q^{83} - 5 q^{84} + 16 q^{85} - 24 q^{86} + 10 q^{87} - 28 q^{88} - 2 q^{89} - 10 q^{90} - 14 q^{91} + 24 q^{92} - 20 q^{93} + 16 q^{95} - 29 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{2} + 2\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 2 \) 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
2.17009
−1.48119
0.311108
−2.70928 1.00000 5.34017 1.70928 −2.70928 −1.00000 −9.04945 1.00000 −4.63090
1.2 −0.193937 1.00000 −1.96239 −0.806063 −0.193937 −1.00000 0.768452 1.00000 0.156325
1.3 1.90321 1.00000 1.62222 −2.90321 1.90321 −1.00000 −0.719004 1.00000 −5.52543
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(383\) \(1\)

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 8043.2.a.m 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.m 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{3} + T_{2}^{2} - 5T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{3} + 2T_{5}^{2} - 4T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 4T_{11} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 5T - 1 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} - 4 T - 4 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 4 T + 20 \) Copy content Toggle raw display
$13$ \( T^{3} - 14 T^{2} + 60 T - 76 \) Copy content Toggle raw display
$17$ \( T^{3} + 10 T^{2} + 20 T - 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 4 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$23$ \( T^{3} - 8 T^{2} - 40 T + 304 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} - 52 T + 536 \) Copy content Toggle raw display
$31$ \( T^{3} + 20 T^{2} + 120 T + 208 \) Copy content Toggle raw display
$37$ \( (T - 10)^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 20 T + 100 \) Copy content Toggle raw display
$43$ \( T^{3} - 8T^{2} + 32 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} - 16 T + 160 \) Copy content Toggle raw display
$53$ \( T^{3} - 14 T^{2} + 52 T - 40 \) Copy content Toggle raw display
$59$ \( T^{3} - 16 T^{2} + 32 T + 128 \) Copy content Toggle raw display
$61$ \( T^{3} + 22 T^{2} + 68 T - 620 \) Copy content Toggle raw display
$67$ \( T^{3} + 24 T^{2} + 80 T - 800 \) Copy content Toggle raw display
$71$ \( T^{3} - 120T + 16 \) Copy content Toggle raw display
$73$ \( T^{3} - 2 T^{2} - 84 T + 104 \) Copy content Toggle raw display
$79$ \( T^{3} + 12 T^{2} + 20 T - 100 \) Copy content Toggle raw display
$83$ \( T^{3} - 20 T^{2} + 48 T + 320 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} - 76 T + 116 \) Copy content Toggle raw display
$97$ \( T^{3} - 18 T^{2} + 68 T - 52 \) Copy content Toggle raw display
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