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Properties

Label 8029.2.a.e
Level $8029$
Weight $2$
Character orbit 8029.a
Self dual yes
Analytic conductor $64.112$
Analytic rank $0$
Dimension $69$
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] = [8029,2,Mod(1,8029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8029, 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("8029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8029 = 7 \cdot 31 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8029.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: \(64.1118877829\)
Analytic rank: \(0\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q + 6 q^{2} + 6 q^{3} + 72 q^{4} + 11 q^{6} - 69 q^{7} + 18 q^{8} + 77 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q + 6 q^{2} + 6 q^{3} + 72 q^{4} + 11 q^{6} - 69 q^{7} + 18 q^{8} + 77 q^{9} + 4 q^{10} + 45 q^{11} + 3 q^{12} - 10 q^{13} - 6 q^{14} + 16 q^{15} + 90 q^{16} + q^{17} + 41 q^{18} + 23 q^{19} + 3 q^{20} - 6 q^{21} + 20 q^{22} + 30 q^{23} + 33 q^{24} + 67 q^{25} + 33 q^{26} + 18 q^{27} - 72 q^{28} + 46 q^{29} + 2 q^{30} + 69 q^{31} + 37 q^{32} + 8 q^{33} + 2 q^{34} + 95 q^{36} - 69 q^{37} - 7 q^{38} + 20 q^{39} - 25 q^{40} + 37 q^{41} - 11 q^{42} + 24 q^{43} + 98 q^{44} - q^{45} + 20 q^{46} - 24 q^{47} + q^{48} + 69 q^{49} + 47 q^{50} + 89 q^{51} - 3 q^{52} + 43 q^{53} + 55 q^{54} + 39 q^{55} - 18 q^{56} + 35 q^{57} + 50 q^{58} + 97 q^{59} + 31 q^{60} + 3 q^{61} + 6 q^{62} - 77 q^{63} + 102 q^{64} + 68 q^{65} - 49 q^{66} + 35 q^{67} + 17 q^{68} + 10 q^{69} - 4 q^{70} + 83 q^{71} + 69 q^{72} - 13 q^{73} - 6 q^{74} + 52 q^{75} + 41 q^{76} - 45 q^{77} - 23 q^{78} + 38 q^{79} + 44 q^{80} + 89 q^{81} + 14 q^{82} - 8 q^{83} - 3 q^{84} + 3 q^{85} + 64 q^{86} - 29 q^{87} + 141 q^{88} + 70 q^{89} - 11 q^{90} + 10 q^{91} + 48 q^{92} + 6 q^{93} - 9 q^{94} + 12 q^{95} + 12 q^{96} - 17 q^{97} + 6 q^{98} + 164 q^{99}+O(q^{100}) \) 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 −2.74790 −2.77023 5.55096 2.51263 7.61233 −1.00000 −9.75769 4.67419 −6.90446
1.2 −2.72439 2.93581 5.42232 2.30642 −7.99829 −1.00000 −9.32374 5.61896 −6.28359
1.3 −2.63384 −0.440743 4.93710 −2.30591 1.16085 −1.00000 −7.73583 −2.80575 6.07340
1.4 −2.57794 −0.202180 4.64578 0.528676 0.521207 −1.00000 −6.82066 −2.95912 −1.36290
1.5 −2.53036 1.19896 4.40274 2.21645 −3.03381 −1.00000 −6.07981 −1.56249 −5.60842
1.6 −2.41746 −2.90923 3.84410 −3.42697 7.03294 −1.00000 −4.45803 5.46362 8.28456
1.7 −2.31869 −1.99799 3.37631 −0.655219 4.63272 −1.00000 −3.19123 0.991981 1.51925
1.8 −2.27937 −0.146043 3.19554 −4.14410 0.332887 −1.00000 −2.72508 −2.97867 9.44595
1.9 −2.21183 1.21496 2.89218 −1.78899 −2.68729 −1.00000 −1.97335 −1.52387 3.95694
1.10 −2.20302 0.911547 2.85328 3.20458 −2.00815 −1.00000 −1.87979 −2.16908 −7.05973
1.11 −2.04116 1.10763 2.16635 −1.94120 −2.26085 −1.00000 −0.339541 −1.77316 3.96231
1.12 −1.95297 −1.58711 1.81410 3.53259 3.09958 −1.00000 0.363050 −0.481092 −6.89905
1.13 −1.93505 2.52265 1.74443 −2.95034 −4.88147 −1.00000 0.494535 3.36378 5.70906
1.14 −1.79821 0.748872 1.23355 0.832366 −1.34663 −1.00000 1.37824 −2.43919 −1.49676
1.15 −1.75548 −2.49596 1.08173 −0.0382408 4.38163 −1.00000 1.61202 3.22983 0.0671311
1.16 −1.70146 3.23997 0.894957 −1.00710 −5.51266 −1.00000 1.88018 7.49737 1.71354
1.17 −1.63063 −2.88443 0.658946 1.48735 4.70344 −1.00000 2.18676 5.31995 −2.42532
1.18 −1.61963 0.534433 0.623195 3.20836 −0.865582 −1.00000 2.22991 −2.71438 −5.19635
1.19 −1.30150 −1.62706 −0.306090 1.44263 2.11762 −1.00000 3.00138 −0.352687 −1.87759
1.20 −1.11933 2.01365 −0.747091 −3.36606 −2.25395 −1.00000 3.07491 1.05478 3.76774
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(31\) \(-1\)
\(37\) \(1\)

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 8029.2.a.e ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8029.2.a.e ✓ 69 1.a even 1 1 trivial