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Properties

Label 8007.2.a.e
Level $8007$
Weight $2$
Character orbit 8007.a
Self dual yes
Analytic conductor $63.936$
Analytic rank $1$
Dimension $46$
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: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 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 −2.80793 1.00000 5.88448 −2.90815 −2.80793 −1.28847 −10.9074 1.00000 8.16589
1.2 −2.76060 1.00000 5.62090 −0.677149 −2.76060 3.14387 −9.99585 1.00000 1.86934
1.3 −2.60999 1.00000 4.81203 −2.83562 −2.60999 1.53827 −7.33937 1.00000 7.40094
1.4 −2.58291 1.00000 4.67145 2.37262 −2.58291 −1.51271 −6.90012 1.00000 −6.12828
1.5 −2.45922 1.00000 4.04774 2.82362 −2.45922 4.27749 −5.03584 1.00000 −6.94390
1.6 −2.38281 1.00000 3.67776 −2.44898 −2.38281 −2.76989 −3.99778 1.00000 5.83545
1.7 −2.18789 1.00000 2.78687 1.88275 −2.18789 −4.23490 −1.72159 1.00000 −4.11926
1.8 −2.12891 1.00000 2.53226 −4.00748 −2.12891 2.85209 −1.13314 1.00000 8.53157
1.9 −2.00485 1.00000 2.01944 2.86654 −2.00485 −1.37721 −0.0389655 1.00000 −5.74699
1.10 −1.83962 1.00000 1.38420 0.446319 −1.83962 3.21050 1.13283 1.00000 −0.821058
1.11 −1.59838 1.00000 0.554812 1.16995 −1.59838 1.82514 2.30996 1.00000 −1.87002
1.12 −1.58086 1.00000 0.499118 −3.53760 −1.58086 −3.85230 2.37268 1.00000 5.59245
1.13 −1.50188 1.00000 0.255640 −0.915140 −1.50188 −3.11181 2.61982 1.00000 1.37443
1.14 −1.42820 1.00000 0.0397473 −3.43580 −1.42820 0.0177517 2.79963 1.00000 4.90700
1.15 −1.42497 1.00000 0.0305258 0.348852 −1.42497 −0.489415 2.80643 1.00000 −0.497102
1.16 −1.41678 1.00000 0.00726459 2.31073 −1.41678 2.67795 2.82327 1.00000 −3.27380
1.17 −0.943890 1.00000 −1.10907 3.50883 −0.943890 −2.32814 2.93462 1.00000 −3.31195
1.18 −0.938469 1.00000 −1.11928 −3.16176 −0.938469 3.19339 2.92734 1.00000 2.96721
1.19 −0.756159 1.00000 −1.42822 −3.23827 −0.756159 4.69700 2.59228 1.00000 2.44865
1.20 −0.576474 1.00000 −1.66768 0.828352 −0.576474 −0.479856 2.11432 1.00000 −0.477524
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
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.e ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8007.2.a.e ✓ 46 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{46} + 5 T_{2}^{45} - 55 T_{2}^{44} - 304 T_{2}^{43} + 1359 T_{2}^{42} + 8551 T_{2}^{41} + \cdots - 12103 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8007))\). Copy content Toggle raw display