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Properties

Label 8007.2.a.c
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $1$
Dimension $39$
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: \(1\)
Dimension: \(39\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 39 q - 4 q^{2} - 39 q^{3} + 30 q^{4} - 3 q^{5} + 4 q^{6} - 5 q^{7} - 3 q^{8} + 39 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 39 q - 4 q^{2} - 39 q^{3} + 30 q^{4} - 3 q^{5} + 4 q^{6} - 5 q^{7} - 3 q^{8} + 39 q^{9} + 4 q^{10} + q^{11} - 30 q^{12} - 26 q^{13} - 4 q^{14} + 3 q^{15} + 8 q^{16} + 39 q^{17} - 4 q^{18} - 14 q^{19} - 14 q^{20} + 5 q^{21} - 17 q^{22} + 2 q^{23} + 3 q^{24} - 6 q^{25} - 17 q^{26} - 39 q^{27} - 14 q^{28} - 7 q^{29} - 4 q^{30} - q^{31} - 30 q^{32} - q^{33} - 4 q^{34} + q^{35} + 30 q^{36} - 24 q^{37} - 20 q^{38} + 26 q^{39} + 12 q^{40} + q^{41} + 4 q^{42} - 41 q^{43} - 2 q^{44} - 3 q^{45} - 6 q^{46} - 9 q^{47} - 8 q^{48} - 10 q^{49} - 9 q^{50} - 39 q^{51} - 37 q^{52} - 47 q^{53} + 4 q^{54} - 39 q^{55} + 8 q^{56} + 14 q^{57} - 27 q^{58} + 41 q^{59} + 14 q^{60} - 41 q^{61} + 36 q^{62} - 5 q^{63} - 47 q^{64} - 39 q^{65} + 17 q^{66} - 36 q^{67} + 30 q^{68} - 2 q^{69} - 52 q^{70} - 2 q^{71} - 3 q^{72} - 63 q^{73} - 6 q^{74} + 6 q^{75} - 34 q^{76} - 64 q^{77} + 17 q^{78} + 20 q^{79} - 28 q^{80} + 39 q^{81} - 37 q^{82} + 45 q^{83} + 14 q^{84} - 3 q^{85} + 32 q^{86} + 7 q^{87} + 6 q^{88} - 32 q^{89} + 4 q^{90} - 11 q^{91} + 28 q^{92} + q^{93} - 44 q^{94} + 22 q^{95} + 30 q^{96} - 20 q^{97} + 63 q^{98} + q^{99}+O(q^{100}) \) 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 −2.68305 −1.00000 5.19875 −0.525183 2.68305 1.96908 −8.58242 1.00000 1.40909
1.2 −2.62708 −1.00000 4.90155 1.61007 2.62708 −0.284525 −7.62261 1.00000 −4.22979
1.3 −2.53786 −1.00000 4.44073 −3.08666 2.53786 −3.86465 −6.19424 1.00000 7.83351
1.4 −2.23101 −1.00000 2.97739 0.320461 2.23101 0.645663 −2.18055 1.00000 −0.714950
1.5 −2.20952 −1.00000 2.88199 −2.30853 2.20952 −2.22002 −1.94877 1.00000 5.10076
1.6 −2.11675 −1.00000 2.48063 2.70234 2.11675 2.58592 −1.01738 1.00000 −5.72017
1.7 −2.06382 −1.00000 2.25934 −2.15851 2.06382 0.293019 −0.535232 1.00000 4.45478
1.8 −1.97468 −1.00000 1.89938 1.34463 1.97468 −4.50545 0.198692 1.00000 −2.65521
1.9 −1.86416 −1.00000 1.47510 −3.19016 1.86416 1.30918 0.978494 1.00000 5.94698
1.10 −1.63095 −1.00000 0.660014 1.38859 1.63095 2.37558 2.18546 1.00000 −2.26473
1.11 −1.60951 −1.00000 0.590533 2.17104 1.60951 3.27452 2.26856 1.00000 −3.49431
1.12 −1.19134 −1.00000 −0.580715 −1.39828 1.19134 −2.85096 3.07450 1.00000 1.66582
1.13 −1.11359 −1.00000 −0.759925 3.93850 1.11359 −0.664710 3.07342 1.00000 −4.38586
1.14 −1.03615 −1.00000 −0.926391 −2.19284 1.03615 0.525649 3.03218 1.00000 2.27211
1.15 −0.833201 −1.00000 −1.30578 −2.56183 0.833201 1.99624 2.75438 1.00000 2.13452
1.16 −0.744255 −1.00000 −1.44608 1.71466 0.744255 −3.69118 2.56477 1.00000 −1.27614
1.17 −0.679888 −1.00000 −1.53775 2.83101 0.679888 −0.718741 2.40527 1.00000 −1.92477
1.18 −0.642908 −1.00000 −1.58667 1.17491 0.642908 −0.0115143 2.30590 1.00000 −0.755361
1.19 −0.554461 −1.00000 −1.69257 −2.60082 0.554461 2.71630 2.04739 1.00000 1.44205
1.20 −0.177236 −1.00000 −1.96859 −2.36208 0.177236 −1.71756 0.703377 1.00000 0.418646
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
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.c ✓ 39
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.c ✓ 39 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{39} + 4 T_{2}^{38} - 46 T_{2}^{37} - 199 T_{2}^{36} + 937 T_{2}^{35} + 4500 T_{2}^{34} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display