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Properties

Label 4031.2.a.d
Level $4031$
Weight $2$
Character orbit 4031.a
Self dual yes
Analytic conductor $32.188$
Analytic rank $0$
Dimension $98$
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] = [4031,2,Mod(1,4031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4031, 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("4031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4031 = 29 \cdot 139 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4031.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: \(32.1876970548\)
Analytic rank: \(0\)
Dimension: \(98\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 98 q + 6 q^{2} + 6 q^{3} + 116 q^{4} + q^{5} + 7 q^{6} + 12 q^{7} + 15 q^{8} + 136 q^{9} + 16 q^{10} + 25 q^{11} + 8 q^{12} + 18 q^{13} + 34 q^{14} + 14 q^{15} + 136 q^{16} + 35 q^{17} + 20 q^{18} + 48 q^{19} - 20 q^{20} + 62 q^{21} + 32 q^{22} + q^{23} + 8 q^{24} + 173 q^{25} + 15 q^{26} + 18 q^{27} + 37 q^{28} + 98 q^{29} - 6 q^{30} + 20 q^{31} + 16 q^{32} + 24 q^{33} - 6 q^{34} + 11 q^{35} + 155 q^{36} + 62 q^{37} - 5 q^{38} + 45 q^{39} + 38 q^{40} + 50 q^{41} + 7 q^{42} + 72 q^{43} + 92 q^{44} - 13 q^{45} + 32 q^{46} - 16 q^{47} - 13 q^{48} + 194 q^{49} + 50 q^{50} + 2 q^{51} + 83 q^{52} - 11 q^{53} + 58 q^{54} + 32 q^{55} + 106 q^{56} + 84 q^{57} + 6 q^{58} + 20 q^{59} + 62 q^{60} + 142 q^{61} - 27 q^{62} + 26 q^{63} + 213 q^{64} + 13 q^{65} - 35 q^{66} + 5 q^{67} + 69 q^{68} + 68 q^{69} - 2 q^{70} - 10 q^{71} - 9 q^{72} + 74 q^{73} + 21 q^{74} + 19 q^{75} + 116 q^{76} + 41 q^{77} - 162 q^{78} + 104 q^{79} - 86 q^{80} + 230 q^{81} + 53 q^{82} - 19 q^{83} + 76 q^{84} + 125 q^{85} + 9 q^{86} + 6 q^{87} + 68 q^{88} + 120 q^{89} + 122 q^{90} + 30 q^{91} + 3 q^{92} - 56 q^{93} + 22 q^{94} + 32 q^{95} - 55 q^{96} + 98 q^{97} - 15 q^{98} + 69 q^{99}+O(q^{100}) \) 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 −2.78351 2.95190 5.74795 −3.19044 −8.21665 −4.40114 −10.4325 5.71371 8.88064
1.2 −2.76750 0.603367 5.65907 3.54828 −1.66982 1.63115 −10.1265 −2.63595 −9.81988
1.3 −2.75511 −2.40315 5.59062 −0.846757 6.62092 3.63959 −9.89254 2.77511 2.33291
1.4 −2.74332 −0.716173 5.52580 −3.18087 1.96469 1.27957 −9.67240 −2.48710 8.72614
1.5 −2.61634 −3.34235 4.84525 −3.10216 8.74473 −4.97241 −7.44414 8.17130 8.11632
1.6 −2.59643 0.964956 4.74146 3.18498 −2.50544 −0.304804 −7.11802 −2.06886 −8.26958
1.7 −2.58392 3.19893 4.67665 1.67511 −8.26577 1.22424 −6.91625 7.23312 −4.32836
1.8 −2.57508 1.48970 4.63103 −2.75202 −3.83609 2.46323 −6.77510 −0.780799 7.08668
1.9 −2.48847 −3.00036 4.19251 0.297405 7.46633 −2.60645 −5.45599 6.00219 −0.740084
1.10 −2.44621 0.244458 3.98396 −3.48270 −0.597997 1.44143 −4.85320 −2.94024 8.51943
1.11 −2.43943 −2.23361 3.95082 1.05551 5.44873 −1.84487 −4.75890 1.98900 −2.57484
1.12 −2.22708 1.27662 2.95990 0.197788 −2.84313 −3.89763 −2.13778 −1.37025 −0.440491
1.13 −2.11438 2.42067 2.47061 1.02780 −5.11822 0.853130 −0.995051 2.85965 −2.17316
1.14 −2.08450 0.675318 2.34516 −1.05546 −1.40770 −4.46025 −0.719483 −2.54395 2.20012
1.15 −2.03151 −1.34287 2.12703 3.37426 2.72805 0.525143 −0.258068 −1.19670 −6.85484
1.16 −1.99223 −2.07417 1.96900 0.414014 4.13224 4.11338 0.0617619 1.30220 −0.824814
1.17 −1.97068 0.422540 1.88358 2.63171 −0.832693 −1.79791 0.229417 −2.82146 −5.18627
1.18 −1.96419 −0.913998 1.85806 −4.29424 1.79527 −2.01487 0.278799 −2.16461 8.43472
1.19 −1.93664 0.437969 1.75056 3.14696 −0.848187 5.11562 0.483081 −2.80818 −6.09452
1.20 −1.86328 0.307270 1.47182 0.176749 −0.572530 −0.402020 0.984149 −2.90559 −0.329334
See all 98 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.98
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(29\) \(-1\)
\(139\) \(1\)

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 4031.2.a.d ✓ 98
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4031.2.a.d ✓ 98 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{98} - 6 T_{2}^{97} - 138 T_{2}^{96} + 887 T_{2}^{95} + 9088 T_{2}^{94} - 63290 T_{2}^{93} + \cdots - 142987209 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\). Copy content Toggle raw display