Properties

Label 8013.2.a.a
Level $8013$
Weight $2$
Character orbit 8013.a
Self dual yes
Analytic conductor $63.984$
Analytic rank $1$
Dimension $94$
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] = [8013,2,Mod(1,8013)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8013, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8013.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8013 = 3 \cdot 2671 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8013.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: \(63.9841271397\)
Analytic rank: \(1\)
Dimension: \(94\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 94 q - 13 q^{2} + 94 q^{3} + 73 q^{4} - 14 q^{5} - 13 q^{6} - 55 q^{7} - 36 q^{8} + 94 q^{9} - 39 q^{10} - 49 q^{11} + 73 q^{12} - 52 q^{13} - 7 q^{14} - 14 q^{15} + 43 q^{16} - 22 q^{17} - 13 q^{18} - 89 q^{19} - 22 q^{20} - 55 q^{21} - 36 q^{22} - 46 q^{23} - 36 q^{24} + 18 q^{25} + q^{26} + 94 q^{27} - 123 q^{28} - 20 q^{29} - 39 q^{30} - 61 q^{31} - 65 q^{32} - 49 q^{33} - 67 q^{34} - 40 q^{35} + 73 q^{36} - 83 q^{37} - 19 q^{38} - 52 q^{39} - 101 q^{40} - 25 q^{41} - 7 q^{42} - 150 q^{43} - 71 q^{44} - 14 q^{45} - 72 q^{46} - 39 q^{47} + 43 q^{48} - q^{49} - 45 q^{50} - 22 q^{51} - 110 q^{52} - 30 q^{53} - 13 q^{54} - 54 q^{55} - 5 q^{56} - 89 q^{57} - 77 q^{58} - 43 q^{59} - 22 q^{60} - 109 q^{61} - 33 q^{62} - 55 q^{63} + 10 q^{64} - 66 q^{65} - 36 q^{66} - 155 q^{67} - 46 q^{68} - 46 q^{69} - 43 q^{70} - 27 q^{71} - 36 q^{72} - 157 q^{73} - 29 q^{74} + 18 q^{75} - 176 q^{76} - 9 q^{77} + q^{78} - 99 q^{79} - 18 q^{80} + 94 q^{81} - 53 q^{82} - 144 q^{83} - 123 q^{84} - 105 q^{85} + 23 q^{86} - 20 q^{87} - 88 q^{88} - 4 q^{89} - 39 q^{90} - 99 q^{91} - 76 q^{92} - 61 q^{93} - 65 q^{94} - 49 q^{95} - 65 q^{96} - 139 q^{97} - 6 q^{98} - 49 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.78585 1.00000 5.76095 3.26911 −2.78585 −3.26627 −10.4774 1.00000 −9.10723
1.2 −2.73477 1.00000 5.47898 −0.238588 −2.73477 1.07409 −9.51422 1.00000 0.652485
1.3 −2.64959 1.00000 5.02034 −2.74590 −2.64959 −5.01160 −8.00268 1.00000 7.27553
1.4 −2.64705 1.00000 5.00689 −1.00090 −2.64705 −3.26297 −7.95941 1.00000 2.64944
1.5 −2.57838 1.00000 4.64804 1.93561 −2.57838 2.52768 −6.82766 1.00000 −4.99075
1.6 −2.56682 1.00000 4.58855 3.72342 −2.56682 −2.49598 −6.64435 1.00000 −9.55734
1.7 −2.55292 1.00000 4.51739 3.29007 −2.55292 1.28784 −6.42669 1.00000 −8.39927
1.8 −2.51426 1.00000 4.32151 −2.69736 −2.51426 −1.28921 −5.83689 1.00000 6.78186
1.9 −2.46856 1.00000 4.09380 0.0867826 −2.46856 0.912467 −5.16868 1.00000 −0.214228
1.10 −2.45728 1.00000 4.03824 0.781238 −2.45728 −3.06583 −5.00853 1.00000 −1.91972
1.11 −2.33315 1.00000 3.44357 −2.85647 −2.33315 −3.40806 −3.36806 1.00000 6.66457
1.12 −2.27625 1.00000 3.18130 1.70680 −2.27625 −0.170344 −2.68893 1.00000 −3.88509
1.13 −2.15734 1.00000 2.65411 1.74315 −2.15734 2.90909 −1.41114 1.00000 −3.76056
1.14 −2.15634 1.00000 2.64980 −1.53192 −2.15634 −0.507117 −1.40120 1.00000 3.30334
1.15 −2.06033 1.00000 2.24497 1.16597 −2.06033 0.252514 −0.504726 1.00000 −2.40228
1.16 −2.05369 1.00000 2.21762 −2.14256 −2.05369 −0.368331 −0.446932 1.00000 4.40015
1.17 −2.05347 1.00000 2.21673 −2.14654 −2.05347 −0.521799 −0.445046 1.00000 4.40785
1.18 −2.03943 1.00000 2.15929 −1.46688 −2.03943 4.57052 −0.324864 1.00000 2.99161
1.19 −1.99976 1.00000 1.99902 −2.89200 −1.99976 2.76984 0.00195305 1.00000 5.78329
1.20 −1.92790 1.00000 1.71679 3.85366 −1.92790 −4.10922 0.545993 1.00000 −7.42947
See all 94 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.94
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2671\) \(-1\)

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 8013.2.a.a 94
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8013.2.a.a 94 1.a even 1 1 trivial