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Properties

Label 8031.2.a.d
Level $8031$
Weight $2$
Character orbit 8031.a
Self dual yes
Analytic conductor $64.128$
Analytic rank $0$
Dimension $132$
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] = [8031,2,Mod(1,8031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8031, 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("8031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8031 = 3 \cdot 2677 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8031.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: \(64.1278578633\)
Analytic rank: \(0\)
Dimension: \(132\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q + 4 q^{2} + 132 q^{3} + 156 q^{4} + 20 q^{5} + 4 q^{6} + 44 q^{7} + 9 q^{8} + 132 q^{9} + 40 q^{10} + 24 q^{11} + 156 q^{12} + 62 q^{13} + 25 q^{14} + 20 q^{15} + 192 q^{16} + 77 q^{17} + 4 q^{18} + 86 q^{19} + 26 q^{20} + 44 q^{21} + 52 q^{22} + 17 q^{23} + 9 q^{24} + 212 q^{25} + 13 q^{26} + 132 q^{27} + 95 q^{28} + 52 q^{29} + 40 q^{30} + 59 q^{31} - 8 q^{32} + 24 q^{33} + 41 q^{34} + 21 q^{35} + 156 q^{36} + 76 q^{37} + 2 q^{38} + 62 q^{39} + 91 q^{40} + 114 q^{41} + 25 q^{42} + 173 q^{43} + 44 q^{44} + 20 q^{45} + 48 q^{46} + 15 q^{47} + 192 q^{48} + 262 q^{49} - 9 q^{50} + 77 q^{51} + 144 q^{52} + 15 q^{53} + 4 q^{54} + 111 q^{55} + 66 q^{56} + 86 q^{57} + 33 q^{58} + 20 q^{59} + 26 q^{60} + 182 q^{61} + 16 q^{62} + 44 q^{63} + 255 q^{64} + 70 q^{65} + 52 q^{66} + 169 q^{67} + 128 q^{68} + 17 q^{69} + 2 q^{70} + 23 q^{71} + 9 q^{72} + 148 q^{73} + 57 q^{74} + 212 q^{75} + 143 q^{76} + 31 q^{77} + 13 q^{78} + 152 q^{79} + 27 q^{80} + 132 q^{81} + 67 q^{82} + 28 q^{83} + 95 q^{84} + 88 q^{85} - 10 q^{86} + 52 q^{87} + 130 q^{88} + 136 q^{89} + 40 q^{90} + 125 q^{91} + 59 q^{93} + 95 q^{94} + 2 q^{95} - 8 q^{96} + 147 q^{97} - 18 q^{98} + 24 q^{99}+O(q^{100}) \) 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 −2.80398 1.00000 5.86231 1.87605 −2.80398 3.15563 −10.8299 1.00000 −5.26041
1.2 −2.77294 1.00000 5.68922 −1.74396 −2.77294 0.0730159 −10.2300 1.00000 4.83589
1.3 −2.75661 1.00000 5.59891 −4.00125 −2.75661 −2.17438 −9.92081 1.00000 11.0299
1.4 −2.72616 1.00000 5.43195 −1.18317 −2.72616 −4.48555 −9.35605 1.00000 3.22552
1.5 −2.69638 1.00000 5.27048 2.40729 −2.69638 −4.24108 −8.81846 1.00000 −6.49098
1.6 −2.68578 1.00000 5.21342 1.21683 −2.68578 2.10686 −8.63055 1.00000 −3.26813
1.7 −2.67095 1.00000 5.13397 −2.93949 −2.67095 4.35242 −8.37068 1.00000 7.85124
1.8 −2.59180 1.00000 4.71742 2.79387 −2.59180 −2.04048 −7.04300 1.00000 −7.24116
1.9 −2.57206 1.00000 4.61548 3.83645 −2.57206 −2.68797 −6.72717 1.00000 −9.86756
1.10 −2.56954 1.00000 4.60252 −1.45340 −2.56954 2.82926 −6.68726 1.00000 3.73456
1.11 −2.55916 1.00000 4.54932 −0.940256 −2.55916 2.24269 −6.52413 1.00000 2.40627
1.12 −2.50203 1.00000 4.26014 −3.95750 −2.50203 1.89022 −5.65494 1.00000 9.90177
1.13 −2.48023 1.00000 4.15155 4.11076 −2.48023 4.68191 −5.33633 1.00000 −10.1956
1.14 −2.46874 1.00000 4.09466 −2.24657 −2.46874 4.89936 −5.17116 1.00000 5.54618
1.15 −2.45456 1.00000 4.02485 −4.17412 −2.45456 −1.11531 −4.97011 1.00000 10.2456
1.16 −2.29731 1.00000 3.27764 −1.75176 −2.29731 −2.80141 −2.93514 1.00000 4.02433
1.17 −2.23840 1.00000 3.01043 3.52664 −2.23840 3.86222 −2.26174 1.00000 −7.89403
1.18 −2.20104 1.00000 2.84456 0.313894 −2.20104 1.59317 −1.85890 1.00000 −0.690891
1.19 −2.19060 1.00000 2.79873 1.38696 −2.19060 −0.332475 −1.74969 1.00000 −3.03827
1.20 −2.17916 1.00000 2.74876 2.82413 −2.17916 4.47038 −1.63167 1.00000 −6.15424
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.132
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(2677\) \(1\)

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 8031.2.a.d ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8031.2.a.d ✓ 132 1.a even 1 1 trivial