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Properties

Label 8007.2.a.h
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $0$
Dimension $56$
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: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + 7 q^{2} + 56 q^{3} + 61 q^{4} + 17 q^{5} + 7 q^{6} + 5 q^{7} + 18 q^{8} + 56 q^{9} - 2 q^{10} + 35 q^{11} + 61 q^{12} + 8 q^{13} + 36 q^{14} + 17 q^{15} + 71 q^{16} - 56 q^{17} + 7 q^{18} - 2 q^{19} + 58 q^{20} + 5 q^{21} + 27 q^{22} + 40 q^{23} + 18 q^{24} + 85 q^{25} + 15 q^{26} + 56 q^{27} - 4 q^{28} + 41 q^{29} - 2 q^{30} + q^{31} + 43 q^{32} + 35 q^{33} - 7 q^{34} + 57 q^{35} + 61 q^{36} + 34 q^{37} + 52 q^{38} + 8 q^{39} + 14 q^{40} + 49 q^{41} + 36 q^{42} + 27 q^{43} + 66 q^{44} + 17 q^{45} + 10 q^{46} + 43 q^{47} + 71 q^{48} + 51 q^{49} + 30 q^{50} - 56 q^{51} - 7 q^{52} + 73 q^{53} + 7 q^{54} + 15 q^{55} + 118 q^{56} - 2 q^{57} - q^{58} + 53 q^{59} + 58 q^{60} + 15 q^{61} + 16 q^{62} + 5 q^{63} + 124 q^{64} + 107 q^{65} + 27 q^{66} + 20 q^{67} - 61 q^{68} + 40 q^{69} + 16 q^{70} + 56 q^{71} + 18 q^{72} + 49 q^{73} + 28 q^{74} + 85 q^{75} - 38 q^{76} + 50 q^{77} + 15 q^{78} - 4 q^{79} + 74 q^{80} + 56 q^{81} + 59 q^{82} + 35 q^{83} - 4 q^{84} - 17 q^{85} + 38 q^{86} + 41 q^{87} + 64 q^{88} + 66 q^{89} - 2 q^{90} + 5 q^{91} + 96 q^{92} + q^{93} - 12 q^{94} + 70 q^{95} + 43 q^{96} + 60 q^{97} + 26 q^{98} + 35 q^{99}+O(q^{100}) \) 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 −2.74837 1.00000 5.55354 2.55432 −2.74837 −2.04110 −9.76645 1.00000 −7.02023
1.2 −2.65002 1.00000 5.02263 3.74088 −2.65002 −4.42567 −8.01004 1.00000 −9.91341
1.3 −2.59527 1.00000 4.73543 −1.29433 −2.59527 2.82321 −7.09919 1.00000 3.35914
1.4 −2.47538 1.00000 4.12750 2.16981 −2.47538 1.33731 −5.26636 1.00000 −5.37109
1.5 −2.46910 1.00000 4.09645 1.22884 −2.46910 −1.04815 −5.17633 1.00000 −3.03411
1.6 −2.42729 1.00000 3.89172 −3.95283 −2.42729 −4.02222 −4.59173 1.00000 9.59464
1.7 −2.23272 1.00000 2.98503 −1.75437 −2.23272 0.669385 −2.19928 1.00000 3.91702
1.8 −2.14010 1.00000 2.58002 −0.881570 −2.14010 −2.47197 −1.24130 1.00000 1.88665
1.9 −2.04447 1.00000 2.17985 4.17783 −2.04447 2.27767 −0.367694 1.00000 −8.54145
1.10 −1.87368 1.00000 1.51068 −1.39015 −1.87368 0.634123 0.916837 1.00000 2.60469
1.11 −1.81598 1.00000 1.29778 −1.85785 −1.81598 −3.71645 1.27521 1.00000 3.37381
1.12 −1.81301 1.00000 1.28701 1.68912 −1.81301 2.36168 1.29265 1.00000 −3.06240
1.13 −1.59198 1.00000 0.534400 −1.86457 −1.59198 2.61461 2.33321 1.00000 2.96836
1.14 −1.32549 1.00000 −0.243069 4.13081 −1.32549 1.12393 2.97317 1.00000 −5.47535
1.15 −1.30899 1.00000 −0.286535 2.04407 −1.30899 −3.68915 2.99306 1.00000 −2.67568
1.16 −1.30269 1.00000 −0.302986 1.77397 −1.30269 4.96759 3.00009 1.00000 −2.31094
1.17 −1.21361 1.00000 −0.527156 4.28768 −1.21361 1.72310 3.06698 1.00000 −5.20356
1.18 −1.19021 1.00000 −0.583408 2.13314 −1.19021 −4.67655 3.07479 1.00000 −2.53888
1.19 −1.18880 1.00000 −0.586753 −1.07516 −1.18880 −2.55094 3.07513 1.00000 1.27815
1.20 −0.892239 1.00000 −1.20391 −0.989381 −0.892239 3.60802 2.85865 1.00000 0.882764
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.56
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.h ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.h ✓ 56 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 7 T_{2}^{55} - 62 T_{2}^{54} + 533 T_{2}^{53} + 1608 T_{2}^{52} - 18949 T_{2}^{51} + \cdots - 845 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display