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Properties

Label 8007.2.a.j
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $0$
Dimension $64$
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] = [8007,2,Mod(1,8007)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8007, 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("8007.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8007 = 3 \cdot 17 \cdot 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8007.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: \(63.9362168984\)
Analytic rank: \(0\)
Dimension: \(64\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 5 q^{2} - 64 q^{3} + 77 q^{4} - 3 q^{5} - 5 q^{6} + 5 q^{7} + 18 q^{8} + 64 q^{9} + 12 q^{10} - 7 q^{11} - 77 q^{12} + 24 q^{13} - 14 q^{14} + 3 q^{15} + 103 q^{16} + 64 q^{17} + 5 q^{18} + 26 q^{19} - 24 q^{20} - 5 q^{21} + 25 q^{22} + 20 q^{23} - 18 q^{24} + 141 q^{25} + 9 q^{26} - 64 q^{27} + 14 q^{28} + 5 q^{29} - 12 q^{30} + 11 q^{31} + 31 q^{32} + 7 q^{33} + 5 q^{34} - 3 q^{35} + 77 q^{36} + 50 q^{37} + 8 q^{38} - 24 q^{39} + 28 q^{40} - 9 q^{41} + 14 q^{42} + 59 q^{43} - 6 q^{44} - 3 q^{45} + 11 q^{47} - 103 q^{48} + 163 q^{49} + 20 q^{50} - 64 q^{51} + 65 q^{52} + 39 q^{53} - 5 q^{54} + 35 q^{55} - 34 q^{56} - 26 q^{57} - 27 q^{58} - 65 q^{59} + 24 q^{60} + 15 q^{61} + 18 q^{62} + 5 q^{63} + 152 q^{64} + 49 q^{65} - 25 q^{66} + 56 q^{67} + 77 q^{68} - 20 q^{69} + 28 q^{70} - 18 q^{71} + 18 q^{72} + 37 q^{73} - 76 q^{74} - 141 q^{75} + 30 q^{76} + 80 q^{77} - 9 q^{78} + 20 q^{79} - 144 q^{80} + 64 q^{81} + 27 q^{82} + 3 q^{83} - 14 q^{84} - 3 q^{85} + 12 q^{86} - 5 q^{87} + 108 q^{88} + 42 q^{89} + 12 q^{90} + 25 q^{91} + 18 q^{92} - 11 q^{93} + 60 q^{94} + 42 q^{95} - 31 q^{96} + 72 q^{97} + 18 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.81598 −1.00000 5.92975 −4.24598 2.81598 0.0516498 −11.0661 1.00000 11.9566
1.2 −2.69942 −1.00000 5.28690 3.29410 2.69942 4.13717 −8.87273 1.00000 −8.89217
1.3 −2.69904 −1.00000 5.28480 −1.39519 2.69904 −4.02432 −8.86580 1.00000 3.76567
1.4 −2.54273 −1.00000 4.46545 −2.67798 2.54273 2.94599 −6.26898 1.00000 6.80937
1.5 −2.53596 −1.00000 4.43108 −0.923448 2.53596 −0.313569 −6.16510 1.00000 2.34182
1.6 −2.38697 −1.00000 3.69761 2.25593 2.38697 −4.86795 −4.05215 1.00000 −5.38482
1.7 −2.37978 −1.00000 3.66334 0.734525 2.37978 2.72642 −3.95838 1.00000 −1.74801
1.8 −2.37098 −1.00000 3.62156 −2.47104 2.37098 5.22837 −3.84468 1.00000 5.85879
1.9 −2.32739 −1.00000 3.41675 3.32274 2.32739 1.34387 −3.29733 1.00000 −7.73332
1.10 −2.16822 −1.00000 2.70116 −4.31017 2.16822 −1.36362 −1.52027 1.00000 9.34539
1.11 −2.11517 −1.00000 2.47395 2.21169 2.11517 −2.99289 −1.00248 1.00000 −4.67809
1.12 −1.82641 −1.00000 1.33579 0.0479756 1.82641 −0.416922 1.21313 1.00000 −0.0876232
1.13 −1.78864 −1.00000 1.19923 −3.68881 1.78864 4.99843 1.43229 1.00000 6.59796
1.14 −1.70394 −1.00000 0.903397 3.92360 1.70394 −2.84859 1.86854 1.00000 −6.68557
1.15 −1.65285 −1.00000 0.731922 2.84674 1.65285 −0.210210 2.09595 1.00000 −4.70524
1.16 −1.62615 −1.00000 0.644366 0.536374 1.62615 4.53678 2.20447 1.00000 −0.872225
1.17 −1.58531 −1.00000 0.513198 −0.0940015 1.58531 −1.19806 2.35704 1.00000 0.149021
1.18 −1.55552 −1.00000 0.419651 −2.03554 1.55552 −4.57317 2.45827 1.00000 3.16632
1.19 −1.43835 −1.00000 0.0688642 −3.82222 1.43835 1.52533 2.77766 1.00000 5.49771
1.20 −1.28460 −1.00000 −0.349799 −0.580730 1.28460 −1.93204 3.01856 1.00000 0.746007
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.64
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(17\) \(-1\)
\(157\) \(1\)

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 8007.2.a.j ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.j ✓ 64 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 5 T_{2}^{63} - 90 T_{2}^{62} + 479 T_{2}^{61} + 3792 T_{2}^{60} - 21725 T_{2}^{59} + \cdots + 2527 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display