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Properties

Label 8031.2.a.c
Level $8031$
Weight $2$
Character orbit 8031.a
Self dual yes
Analytic conductor $64.128$
Analytic rank $0$
Dimension $121$
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: \(121\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 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 −2.78421 −1.00000 5.75185 0.686525 2.78421 −0.869505 −10.4459 1.00000 −1.91143
1.2 −2.74757 −1.00000 5.54913 2.21252 2.74757 −1.55769 −9.75147 1.00000 −6.07904
1.3 −2.68356 −1.00000 5.20147 −3.95156 2.68356 −4.75750 −8.59132 1.00000 10.6042
1.4 −2.64556 −1.00000 4.99897 −1.40412 2.64556 −0.985442 −7.93394 1.00000 3.71467
1.5 −2.62963 −1.00000 4.91495 3.86074 2.62963 3.78778 −7.66525 1.00000 −10.1523
1.6 −2.60983 −1.00000 4.81123 0.810523 2.60983 −4.83270 −7.33685 1.00000 −2.11533
1.7 −2.47165 −1.00000 4.10906 −0.0868982 2.47165 −4.58387 −5.21287 1.00000 0.214782
1.8 −2.46498 −1.00000 4.07611 3.49515 2.46498 1.81173 −5.11755 1.00000 −8.61546
1.9 −2.44537 −1.00000 3.97985 0.860526 2.44537 −1.02072 −4.84146 1.00000 −2.10431
1.10 −2.41661 −1.00000 3.83999 −2.56069 2.41661 −0.401342 −4.44652 1.00000 6.18818
1.11 −2.39891 −1.00000 3.75476 −1.21800 2.39891 −2.25601 −4.20952 1.00000 2.92186
1.12 −2.36975 −1.00000 3.61573 −1.85498 2.36975 4.52062 −3.82888 1.00000 4.39585
1.13 −2.36162 −1.00000 3.57726 0.566306 2.36162 3.15276 −3.72490 1.00000 −1.33740
1.14 −2.35691 −1.00000 3.55501 3.33677 2.35691 −1.10286 −3.66500 1.00000 −7.86445
1.15 −2.31301 −1.00000 3.35001 −2.44479 2.31301 −2.31166 −3.12259 1.00000 5.65482
1.16 −2.29229 −1.00000 3.25458 1.65455 2.29229 2.03484 −2.87586 1.00000 −3.79270
1.17 −2.27538 −1.00000 3.17734 −2.29093 2.27538 1.46815 −2.67889 1.00000 5.21272
1.18 −2.15810 −1.00000 2.65739 −1.43595 2.15810 1.17187 −1.41872 1.00000 3.09891
1.19 −2.11893 −1.00000 2.48988 3.42435 2.11893 −4.49601 −1.03803 1.00000 −7.25596
1.20 −1.99295 −1.00000 1.97186 0.956056 1.99295 2.24151 0.0560744 1.00000 −1.90538
See next 80 embeddings (of 121 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.121
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.c ✓ 121
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8031.2.a.c ✓ 121 1.a even 1 1 trivial