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Properties

Label 8031.2.a.a
Level $8031$
Weight $2$
Character orbit 8031.a
Self dual yes
Analytic conductor $64.128$
Analytic rank $1$
Dimension $92$
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: \(1\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q - 6 q^{2} + 92 q^{3} + 70 q^{4} - 18 q^{5} - 6 q^{6} - 42 q^{7} - 15 q^{8} + 92 q^{9} - 44 q^{10} - 24 q^{11} + 70 q^{12} - 48 q^{13} - 29 q^{14} - 18 q^{15} + 26 q^{16} - 69 q^{17} - 6 q^{18} - 74 q^{19} - 42 q^{20} - 42 q^{21} - 62 q^{22} - 19 q^{23} - 15 q^{24} + 16 q^{25} - 27 q^{26} + 92 q^{27} - 101 q^{28} - 54 q^{29} - 44 q^{30} - 67 q^{31} - 36 q^{32} - 24 q^{33} - 63 q^{34} - 31 q^{35} + 70 q^{36} - 70 q^{37} - 18 q^{38} - 48 q^{39} - 125 q^{40} - 98 q^{41} - 29 q^{42} - 159 q^{43} - 52 q^{44} - 18 q^{45} - 68 q^{46} - 15 q^{47} + 26 q^{48} - 28 q^{49} - 7 q^{50} - 69 q^{51} - 98 q^{52} - 23 q^{53} - 6 q^{54} - 93 q^{55} - 48 q^{56} - 74 q^{57} - 37 q^{58} - 36 q^{59} - 42 q^{60} - 172 q^{61} - 26 q^{62} - 42 q^{63} - 23 q^{64} - 66 q^{65} - 62 q^{66} - 143 q^{67} - 74 q^{68} - 19 q^{69} - 30 q^{70} - 9 q^{71} - 15 q^{72} - 134 q^{73} - 19 q^{74} + 16 q^{75} - 157 q^{76} - 25 q^{77} - 27 q^{78} - 138 q^{79} - 29 q^{80} + 92 q^{81} - 61 q^{82} - 24 q^{83} - 101 q^{84} - 84 q^{85} + 14 q^{86} - 54 q^{87} - 140 q^{88} - 148 q^{89} - 44 q^{90} - 115 q^{91} - 12 q^{92} - 67 q^{93} - 79 q^{94} - 10 q^{95} - 36 q^{96} - 165 q^{97} + 36 q^{98} - 24 q^{99}+O(q^{100}) \) 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 −2.68500 1.00000 5.20922 −2.40260 −2.68500 −1.66915 −8.61677 1.00000 6.45098
1.2 −2.67330 1.00000 5.14651 3.81418 −2.67330 −0.936528 −8.41154 1.00000 −10.1964
1.3 −2.66818 1.00000 5.11918 2.50193 −2.66818 2.17887 −8.32253 1.00000 −6.67559
1.4 −2.62702 1.00000 4.90122 1.25106 −2.62702 −3.99448 −7.62155 1.00000 −3.28657
1.5 −2.55172 1.00000 4.51128 1.01019 −2.55172 1.76064 −6.40810 1.00000 −2.57772
1.6 −2.49908 1.00000 4.24541 −0.718338 −2.49908 −1.10283 −5.61146 1.00000 1.79519
1.7 −2.48341 1.00000 4.16735 1.79948 −2.48341 −1.15988 −5.38242 1.00000 −4.46885
1.8 −2.42805 1.00000 3.89545 −1.43452 −2.42805 −1.67888 −4.60225 1.00000 3.48309
1.9 −2.38249 1.00000 3.67627 −1.60383 −2.38249 −3.97189 −3.99371 1.00000 3.82111
1.10 −2.36612 1.00000 3.59855 −2.41373 −2.36612 −0.423916 −3.78236 1.00000 5.71118
1.11 −2.23528 1.00000 2.99648 3.03614 −2.23528 −1.46620 −2.22741 1.00000 −6.78662
1.12 −2.18167 1.00000 2.75970 −1.98716 −2.18167 3.32615 −1.65742 1.00000 4.33533
1.13 −2.14091 1.00000 2.58350 1.26139 −2.14091 1.10366 −1.24922 1.00000 −2.70053
1.14 −2.07850 1.00000 2.32016 −0.257221 −2.07850 4.13615 −0.665460 1.00000 0.534635
1.15 −2.04285 1.00000 2.17322 −1.55350 −2.04285 −3.43191 −0.353867 1.00000 3.17355
1.16 −1.98646 1.00000 1.94604 −2.53807 −1.98646 2.04373 0.107184 1.00000 5.04179
1.17 −1.94029 1.00000 1.76473 1.85416 −1.94029 2.13881 0.456483 1.00000 −3.59761
1.18 −1.93878 1.00000 1.75885 3.19668 −1.93878 −4.95854 0.467527 1.00000 −6.19764
1.19 −1.91510 1.00000 1.66760 −3.82966 −1.91510 0.699529 0.636574 1.00000 7.33417
1.20 −1.77964 1.00000 1.16710 2.79221 −1.77964 0.728008 1.48226 1.00000 −4.96911
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
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.a ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8031.2.a.a ✓ 92 1.a even 1 1 trivial