Properties

Label 8022.2.a.k
Level $8022$
Weight $2$
Character orbit 8022.a
Self dual yes
Analytic conductor $64.056$
Analytic rank $0$
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] = [8022,2,Mod(1,8022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8022, 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("8022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8022.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: \(64.0559925015\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} - q^{6} - q^{7} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{3} + q^{4} + (\beta + 1) q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + (\beta + 1) q^{10} + (2 \beta - 2) q^{11} - q^{12} - 2 q^{13} - q^{14} + ( - \beta - 1) q^{15} + q^{16} + (\beta - 3) q^{17} + q^{18} + (2 \beta - 2) q^{19} + (\beta + 1) q^{20} + q^{21} + (2 \beta - 2) q^{22} + (3 \beta + 1) q^{23} - q^{24} + 3 \beta q^{25} - 2 q^{26} - q^{27} - q^{28} + (\beta + 5) q^{29} + ( - \beta - 1) q^{30} + ( - 2 \beta + 2) q^{31} + q^{32} + ( - 2 \beta + 2) q^{33} + (\beta - 3) q^{34} + ( - \beta - 1) q^{35} + q^{36} + ( - 2 \beta + 4) q^{37} + (2 \beta - 2) q^{38} + 2 q^{39} + (\beta + 1) q^{40} + ( - 2 \beta - 4) q^{41} + q^{42} + (2 \beta + 2) q^{43} + (2 \beta - 2) q^{44} + (\beta + 1) q^{45} + (3 \beta + 1) q^{46} + ( - \beta - 3) q^{47} - q^{48} + q^{49} + 3 \beta q^{50} + ( - \beta + 3) q^{51} - 2 q^{52} + ( - \beta + 7) q^{53} - q^{54} + (2 \beta + 6) q^{55} - q^{56} + ( - 2 \beta + 2) q^{57} + (\beta + 5) q^{58} - 4 \beta q^{59} + ( - \beta - 1) q^{60} + (3 \beta - 5) q^{61} + ( - 2 \beta + 2) q^{62} - q^{63} + q^{64} + ( - 2 \beta - 2) q^{65} + ( - 2 \beta + 2) q^{66} - 4 \beta q^{67} + (\beta - 3) q^{68} + ( - 3 \beta - 1) q^{69} + ( - \beta - 1) q^{70} + (2 \beta + 6) q^{71} + q^{72} + (\beta - 11) q^{73} + ( - 2 \beta + 4) q^{74} - 3 \beta q^{75} + (2 \beta - 2) q^{76} + ( - 2 \beta + 2) q^{77} + 2 q^{78} + ( - 6 \beta + 6) q^{79} + (\beta + 1) q^{80} + q^{81} + ( - 2 \beta - 4) q^{82} + ( - 5 \beta + 9) q^{83} + q^{84} + ( - \beta + 1) q^{85} + (2 \beta + 2) q^{86} + ( - \beta - 5) q^{87} + (2 \beta - 2) q^{88} + ( - 6 \beta + 8) q^{89} + (\beta + 1) q^{90} + 2 q^{91} + (3 \beta + 1) q^{92} + (2 \beta - 2) q^{93} + ( - \beta - 3) q^{94} + (2 \beta + 6) q^{95} - q^{96} + (4 \beta - 2) q^{97} + q^{98} + (2 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} - 2 q^{14} - 3 q^{15} + 2 q^{16} - 5 q^{17} + 2 q^{18} - 2 q^{19} + 3 q^{20} + 2 q^{21} - 2 q^{22} + 5 q^{23} - 2 q^{24} + 3 q^{25} - 4 q^{26} - 2 q^{27} - 2 q^{28} + 11 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} + 2 q^{33} - 5 q^{34} - 3 q^{35} + 2 q^{36} + 6 q^{37} - 2 q^{38} + 4 q^{39} + 3 q^{40} - 10 q^{41} + 2 q^{42} + 6 q^{43} - 2 q^{44} + 3 q^{45} + 5 q^{46} - 7 q^{47} - 2 q^{48} + 2 q^{49} + 3 q^{50} + 5 q^{51} - 4 q^{52} + 13 q^{53} - 2 q^{54} + 14 q^{55} - 2 q^{56} + 2 q^{57} + 11 q^{58} - 4 q^{59} - 3 q^{60} - 7 q^{61} + 2 q^{62} - 2 q^{63} + 2 q^{64} - 6 q^{65} + 2 q^{66} - 4 q^{67} - 5 q^{68} - 5 q^{69} - 3 q^{70} + 14 q^{71} + 2 q^{72} - 21 q^{73} + 6 q^{74} - 3 q^{75} - 2 q^{76} + 2 q^{77} + 4 q^{78} + 6 q^{79} + 3 q^{80} + 2 q^{81} - 10 q^{82} + 13 q^{83} + 2 q^{84} + q^{85} + 6 q^{86} - 11 q^{87} - 2 q^{88} + 10 q^{89} + 3 q^{90} + 4 q^{91} + 5 q^{92} - 2 q^{93} - 7 q^{94} + 14 q^{95} - 2 q^{96} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
1.00000 −1.00000 1.00000 −0.561553 −1.00000 −1.00000 1.00000 1.00000 −0.561553
1.2 1.00000 −1.00000 1.00000 3.56155 −1.00000 −1.00000 1.00000 1.00000 3.56155
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)
\(191\) \(1\)

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 8022.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8022.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):

\( T_{5}^{2} - 3T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 16 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 2 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} - 11T + 26 \) Copy content Toggle raw display
$31$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$47$ \( T^{2} + 7T + 8 \) Copy content Toggle raw display
$53$ \( T^{2} - 13T + 38 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$61$ \( T^{2} + 7T - 26 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 21T + 106 \) Copy content Toggle raw display
$79$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$83$ \( T^{2} - 13T - 64 \) Copy content Toggle raw display
$89$ \( T^{2} - 10T - 128 \) Copy content Toggle raw display
$97$ \( T^{2} - 68 \) Copy content Toggle raw display
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