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Properties

Label 8014.2.a.e
Level $8014$
Weight $2$
Character orbit 8014.a
Self dual yes
Analytic conductor $63.992$
Analytic rank $0$
Dimension $91$
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] = [8014,2,Mod(1,8014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8014.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8014 = 2 \cdot 4007 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8014.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: \(63.9921121799\)
Analytic rank: \(0\)
Dimension: \(91\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 91 q - 91 q^{2} - 2 q^{3} + 91 q^{4} + 22 q^{5} + 2 q^{6} - 14 q^{7} - 91 q^{8} + 123 q^{9} - 22 q^{10} + 59 q^{11} - 2 q^{12} - 15 q^{13} + 14 q^{14} + 23 q^{15} + 91 q^{16} + 25 q^{17} - 123 q^{18} + 2 q^{19} + 22 q^{20} + 20 q^{21} - 59 q^{22} + 36 q^{23} + 2 q^{24} + 121 q^{25} + 15 q^{26} - 8 q^{27} - 14 q^{28} + 71 q^{29} - 23 q^{30} + 13 q^{31} - 91 q^{32} + 24 q^{33} - 25 q^{34} + 27 q^{35} + 123 q^{36} + 5 q^{37} - 2 q^{38} + 48 q^{39} - 22 q^{40} + 77 q^{41} - 20 q^{42} - 36 q^{43} + 59 q^{44} + 62 q^{45} - 36 q^{46} + 41 q^{47} - 2 q^{48} + 137 q^{49} - 121 q^{50} + 13 q^{51} - 15 q^{52} + 33 q^{53} + 8 q^{54} + 12 q^{55} + 14 q^{56} + 52 q^{57} - 71 q^{58} + 76 q^{59} + 23 q^{60} + 18 q^{61} - 13 q^{62} - 18 q^{63} + 91 q^{64} + 84 q^{65} - 24 q^{66} - 59 q^{67} + 25 q^{68} + 64 q^{69} - 27 q^{70} + 124 q^{71} - 123 q^{72} + 43 q^{73} - 5 q^{74} + 9 q^{75} + 2 q^{76} + 50 q^{77} - 48 q^{78} - 20 q^{79} + 22 q^{80} + 227 q^{81} - 77 q^{82} + 29 q^{83} + 20 q^{84} + 3 q^{85} + 36 q^{86} - 25 q^{87} - 59 q^{88} + 148 q^{89} - 62 q^{90} + 27 q^{91} + 36 q^{92} + 65 q^{93} - 41 q^{94} + 54 q^{95} + 2 q^{96} + 38 q^{97} - 137 q^{98} + 157 q^{99}+O(q^{100}) \) 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 −1.00000 −3.44800 1.00000 −2.43856 3.44800 4.30446 −1.00000 8.88870 2.43856
1.2 −1.00000 −3.41500 1.00000 2.43339 3.41500 −1.50978 −1.00000 8.66224 −2.43339
1.3 −1.00000 −3.27811 1.00000 4.00106 3.27811 −4.58549 −1.00000 7.74601 −4.00106
1.4 −1.00000 −3.27677 1.00000 −0.135024 3.27677 −2.05105 −1.00000 7.73722 0.135024
1.5 −1.00000 −3.21498 1.00000 −2.38557 3.21498 −0.918897 −1.00000 7.33611 2.38557
1.6 −1.00000 −3.13974 1.00000 3.30555 3.13974 0.656006 −1.00000 6.85799 −3.30555
1.7 −1.00000 −3.09151 1.00000 −2.61298 3.09151 −3.49285 −1.00000 6.55743 2.61298
1.8 −1.00000 −3.06251 1.00000 1.40510 3.06251 4.44564 −1.00000 6.37896 −1.40510
1.9 −1.00000 −2.93164 1.00000 −4.27136 2.93164 −1.93753 −1.00000 5.59453 4.27136
1.10 −1.00000 −2.86981 1.00000 −0.806947 2.86981 −4.79468 −1.00000 5.23582 0.806947
1.11 −1.00000 −2.86622 1.00000 0.269496 2.86622 2.82265 −1.00000 5.21519 −0.269496
1.12 −1.00000 −2.85455 1.00000 −2.30346 2.85455 3.17509 −1.00000 5.14844 2.30346
1.13 −1.00000 −2.66657 1.00000 3.59792 2.66657 3.19859 −1.00000 4.11059 −3.59792
1.14 −1.00000 −2.62836 1.00000 2.05043 2.62836 −5.05998 −1.00000 3.90825 −2.05043
1.15 −1.00000 −2.35351 1.00000 2.15005 2.35351 1.12674 −1.00000 2.53903 −2.15005
1.16 −1.00000 −2.32734 1.00000 3.86368 2.32734 −1.21196 −1.00000 2.41649 −3.86368
1.17 −1.00000 −2.31549 1.00000 0.682396 2.31549 −3.17347 −1.00000 2.36151 −0.682396
1.18 −1.00000 −2.25449 1.00000 0.777840 2.25449 1.47835 −1.00000 2.08272 −0.777840
1.19 −1.00000 −2.03768 1.00000 −1.30464 2.03768 −1.43874 −1.00000 1.15214 1.30464
1.20 −1.00000 −1.99318 1.00000 0.808905 1.99318 3.91873 −1.00000 0.972766 −0.808905
See all 91 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.91
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4007\) \(-1\)

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 8014.2.a.e ✓ 91
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8014.2.a.e ✓ 91 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{91} + 2 T_{3}^{90} - 196 T_{3}^{89} - 388 T_{3}^{88} + 18477 T_{3}^{87} + 36194 T_{3}^{86} + \cdots + 115514277888 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8014))\). Copy content Toggle raw display