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Properties

Label 8013.2.a.d
Level $8013$
Weight $2$
Character orbit 8013.a
Self dual yes
Analytic conductor $63.984$
Analytic rank $0$
Dimension $129$
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: \(0\)
Dimension: \(129\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 129 q + 15 q^{2} + 129 q^{3} + 151 q^{4} + 16 q^{5} + 15 q^{6} + 61 q^{7} + 42 q^{8} + 129 q^{9} + 41 q^{10} + 51 q^{11} + 151 q^{12} + 56 q^{13} + 5 q^{14} + 16 q^{15} + 195 q^{16} + 18 q^{17} + 15 q^{18} + 93 q^{19} + 44 q^{20} + 61 q^{21} + 46 q^{22} + 50 q^{23} + 42 q^{24} + 193 q^{25} + q^{26} + 129 q^{27} + 145 q^{28} + 24 q^{29} + 41 q^{30} + 67 q^{31} + 89 q^{32} + 51 q^{33} + 73 q^{34} + 56 q^{35} + 151 q^{36} + 95 q^{37} + 9 q^{38} + 56 q^{39} + 103 q^{40} + 7 q^{41} + 5 q^{42} + 150 q^{43} + 69 q^{44} + 16 q^{45} + 72 q^{46} + 53 q^{47} + 195 q^{48} + 240 q^{49} + 17 q^{50} + 18 q^{51} + 124 q^{52} + 34 q^{53} + 15 q^{54} + 66 q^{55} - 17 q^{56} + 93 q^{57} + 57 q^{58} + 49 q^{59} + 44 q^{60} + 113 q^{61} + 27 q^{62} + 61 q^{63} + 262 q^{64} + 22 q^{65} + 46 q^{66} + 185 q^{67} + 2 q^{68} + 50 q^{69} + 25 q^{70} + 41 q^{71} + 42 q^{72} + 153 q^{73} - q^{74} + 193 q^{75} + 190 q^{76} + 39 q^{77} + q^{78} + 101 q^{79} + 48 q^{80} + 129 q^{81} + 15 q^{82} + 162 q^{83} + 145 q^{84} + 99 q^{85} + 13 q^{86} + 24 q^{87} + 86 q^{88} - 4 q^{89} + 41 q^{90} + 117 q^{91} + 56 q^{92} + 67 q^{93} + 49 q^{94} + 71 q^{95} + 89 q^{96} + 159 q^{97} + 40 q^{98} + 51 q^{99}+O(q^{100}) \) 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 −2.80990 1.00000 5.89553 3.04419 −2.80990 3.35815 −10.9460 1.00000 −8.55386
1.2 −2.75607 1.00000 5.59595 0.110089 −2.75607 1.43985 −9.91070 1.00000 −0.303413
1.3 −2.69445 1.00000 5.26005 −1.49008 −2.69445 1.50692 −8.78405 1.00000 4.01496
1.4 −2.69326 1.00000 5.25363 −3.14797 −2.69326 0.0430508 −8.76286 1.00000 8.47829
1.5 −2.62323 1.00000 4.88134 −3.97507 −2.62323 3.25996 −7.55842 1.00000 10.4275
1.6 −2.59970 1.00000 4.75846 1.02648 −2.59970 −4.07018 −7.17119 1.00000 −2.66854
1.7 −2.59380 1.00000 4.72779 3.19035 −2.59380 −1.98667 −7.07533 1.00000 −8.27513
1.8 −2.58900 1.00000 4.70290 −3.13677 −2.58900 5.18243 −6.99780 1.00000 8.12109
1.9 −2.52915 1.00000 4.39659 0.815749 −2.52915 5.25378 −6.06134 1.00000 −2.06315
1.10 −2.52474 1.00000 4.37429 −3.85140 −2.52474 −2.38848 −5.99445 1.00000 9.72375
1.11 −2.51884 1.00000 4.34456 3.26876 −2.51884 4.07622 −5.90558 1.00000 −8.23349
1.12 −2.41705 1.00000 3.84214 −2.77464 −2.41705 0.900586 −4.45255 1.00000 6.70646
1.13 −2.41059 1.00000 3.81094 −0.0291843 −2.41059 −4.28110 −4.36543 1.00000 0.0703514
1.14 −2.34305 1.00000 3.48991 4.17579 −2.34305 2.00162 −3.49093 1.00000 −9.78411
1.15 −2.27654 1.00000 3.18262 −2.00184 −2.27654 2.08885 −2.69227 1.00000 4.55727
1.16 −2.21092 1.00000 2.88816 1.51753 −2.21092 0.0486600 −1.96364 1.00000 −3.35514
1.17 −2.18651 1.00000 2.78083 −0.652555 −2.18651 2.68769 −1.70730 1.00000 1.42682
1.18 −2.17979 1.00000 2.75150 1.63083 −2.17979 4.53329 −1.63812 1.00000 −3.55488
1.19 −2.08790 1.00000 2.35934 0.999516 −2.08790 −2.01365 −0.750272 1.00000 −2.08689
1.20 −2.06267 1.00000 2.25459 −1.75345 −2.06267 −2.34820 −0.525142 1.00000 3.61679
See next 80 embeddings (of 129 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.129
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.d ✓ 129
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8013.2.a.d ✓ 129 1.a even 1 1 trivial