Properties

Label 8043.2.a.l
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + q^{3} + (\beta + 2) q^{4} + ( - \beta + 1) q^{5} - \beta q^{6} + q^{7} + ( - \beta - 4) q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + q^{3} + (\beta + 2) q^{4} + ( - \beta + 1) q^{5} - \beta q^{6} + q^{7} + ( - \beta - 4) q^{8} + q^{9} + 4 q^{10} + (\beta - 5) q^{11} + (\beta + 2) q^{12} + 2 q^{13} - \beta q^{14} + ( - \beta + 1) q^{15} + 3 \beta q^{16} - 2 q^{17} - \beta q^{18} + (\beta - 5) q^{19} + ( - 2 \beta - 2) q^{20} + q^{21} + (4 \beta - 4) q^{22} + ( - 2 \beta + 2) q^{23} + ( - \beta - 4) q^{24} - \beta q^{25} - 2 \beta q^{26} + q^{27} + (\beta + 2) q^{28} + (2 \beta - 4) q^{29} + 4 q^{30} + (\beta - 1) q^{31} + ( - \beta - 4) q^{32} + (\beta - 5) q^{33} + 2 \beta q^{34} + ( - \beta + 1) q^{35} + (\beta + 2) q^{36} + (4 \beta - 2) q^{37} + (4 \beta - 4) q^{38} + 2 q^{39} + 4 \beta q^{40} + ( - 3 \beta - 1) q^{41} - \beta q^{42} + (3 \beta - 3) q^{43} + ( - 2 \beta - 6) q^{44} + ( - \beta + 1) q^{45} + 8 q^{46} + (\beta + 7) q^{47} + 3 \beta q^{48} + q^{49} + (\beta + 4) q^{50} - 2 q^{51} + (2 \beta + 4) q^{52} + (2 \beta + 2) q^{53} - \beta q^{54} + (5 \beta - 9) q^{55} + ( - \beta - 4) q^{56} + (\beta - 5) q^{57} + (2 \beta - 8) q^{58} + ( - 3 \beta - 1) q^{59} + ( - 2 \beta - 2) q^{60} + (4 \beta - 6) q^{61} - 4 q^{62} + q^{63} + ( - \beta + 4) q^{64} + ( - 2 \beta + 2) q^{65} + (4 \beta - 4) q^{66} + (3 \beta - 3) q^{67} + ( - 2 \beta - 4) q^{68} + ( - 2 \beta + 2) q^{69} + 4 q^{70} + 8 q^{71} + ( - \beta - 4) q^{72} + ( - \beta + 3) q^{73} + ( - 2 \beta - 16) q^{74} - \beta q^{75} + ( - 2 \beta - 6) q^{76} + (\beta - 5) q^{77} - 2 \beta q^{78} - 2 \beta q^{79} - 12 q^{80} + q^{81} + (4 \beta + 12) q^{82} + 4 q^{83} + (\beta + 2) q^{84} + (2 \beta - 2) q^{85} - 12 q^{86} + (2 \beta - 4) q^{87} + 16 q^{88} + (\beta - 13) q^{89} + 4 q^{90} + 2 q^{91} + ( - 4 \beta - 4) q^{92} + (\beta - 1) q^{93} + ( - 8 \beta - 4) q^{94} + (5 \beta - 9) q^{95} + ( - \beta - 4) q^{96} + (2 \beta - 8) q^{97} - \beta q^{98} + (\beta - 5) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 2 q^{3} + 5 q^{4} + q^{5} - q^{6} + 2 q^{7} - 9 q^{8} + 2 q^{9} + 8 q^{10} - 9 q^{11} + 5 q^{12} + 4 q^{13} - q^{14} + q^{15} + 3 q^{16} - 4 q^{17} - q^{18} - 9 q^{19} - 6 q^{20} + 2 q^{21} - 4 q^{22} + 2 q^{23} - 9 q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + 5 q^{28} - 6 q^{29} + 8 q^{30} - q^{31} - 9 q^{32} - 9 q^{33} + 2 q^{34} + q^{35} + 5 q^{36} - 4 q^{38} + 4 q^{39} + 4 q^{40} - 5 q^{41} - q^{42} - 3 q^{43} - 14 q^{44} + q^{45} + 16 q^{46} + 15 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 4 q^{51} + 10 q^{52} + 6 q^{53} - q^{54} - 13 q^{55} - 9 q^{56} - 9 q^{57} - 14 q^{58} - 5 q^{59} - 6 q^{60} - 8 q^{61} - 8 q^{62} + 2 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 3 q^{67} - 10 q^{68} + 2 q^{69} + 8 q^{70} + 16 q^{71} - 9 q^{72} + 5 q^{73} - 34 q^{74} - q^{75} - 14 q^{76} - 9 q^{77} - 2 q^{78} - 2 q^{79} - 24 q^{80} + 2 q^{81} + 28 q^{82} + 8 q^{83} + 5 q^{84} - 2 q^{85} - 24 q^{86} - 6 q^{87} + 32 q^{88} - 25 q^{89} + 8 q^{90} + 4 q^{91} - 12 q^{92} - q^{93} - 16 q^{94} - 13 q^{95} - 9 q^{96} - 14 q^{97} - q^{98} - 9 q^{99}+O(q^{100}) \) 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
2.56155
−1.56155
−2.56155 1.00000 4.56155 −1.56155 −2.56155 1.00000 −6.56155 1.00000 4.00000
1.2 1.56155 1.00000 0.438447 2.56155 1.56155 1.00000 −2.43845 1.00000 4.00000
\(n\): 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.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.l 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{5}^{2} - T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 9T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( (T + 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$31$ \( T^{2} + T - 4 \) Copy content Toggle raw display
$37$ \( T^{2} - 68 \) Copy content Toggle raw display
$41$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$47$ \( T^{2} - 15T + 52 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$59$ \( T^{2} + 5T - 32 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 52 \) Copy content Toggle raw display
$67$ \( T^{2} + 3T - 36 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 25T + 152 \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 32 \) Copy content Toggle raw display
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