Properties

Label 4031.2.a.b
Level $4031$
Weight $2$
Character orbit 4031.a
Self dual yes
Analytic conductor $32.188$
Analytic rank $1$
Dimension $59$
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: \(1\)
Dimension: \(59\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 59 q - 5 q^{2} - 6 q^{3} + 41 q^{4} - 5 q^{5} - 7 q^{6} - 10 q^{7} - 12 q^{8} + 23 q^{9} - 18 q^{10} - 27 q^{11} - 8 q^{12} - 22 q^{13} - 24 q^{14} - 18 q^{15} + 5 q^{16} - 23 q^{17} + q^{18} - 32 q^{19} - 14 q^{20} - 36 q^{21} - 6 q^{22} - 3 q^{23} - 18 q^{24} - 8 q^{25} - q^{26} - 12 q^{27} - 9 q^{28} + 59 q^{29} - 18 q^{30} - 32 q^{31} - 39 q^{32} - 12 q^{33} - 18 q^{34} - 9 q^{35} + 10 q^{36} - 44 q^{37} + 5 q^{38} - 27 q^{39} - 68 q^{40} - 44 q^{41} - 25 q^{42} - 40 q^{43} - 56 q^{44} - 39 q^{45} - 40 q^{46} - 20 q^{47} - 9 q^{48} - 39 q^{49} - 21 q^{50} - 28 q^{51} - 49 q^{52} - 31 q^{53} - 32 q^{54} - 32 q^{55} - 48 q^{56} - 58 q^{57} - 5 q^{58} + 6 q^{59} - 44 q^{60} - 88 q^{61} + 35 q^{62} - 22 q^{63} - 10 q^{64} - 43 q^{65} - 31 q^{66} - 45 q^{67} - 29 q^{68} - 60 q^{69} - 14 q^{70} - 20 q^{71} - 4 q^{72} - 90 q^{73} - 25 q^{74} + 15 q^{75} - 64 q^{76} - 39 q^{77} - 28 q^{78} - 120 q^{79} + 24 q^{80} - 77 q^{81} - 71 q^{82} - 33 q^{83} - 14 q^{84} - 71 q^{85} - 61 q^{86} - 6 q^{87} - 34 q^{88} - 78 q^{89} - 88 q^{90} - 28 q^{91} - 31 q^{92} - 36 q^{93} - 4 q^{94} - 12 q^{95} - 29 q^{96} - 48 q^{97} - 4 q^{98} - 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.77566 2.36943 5.70431 1.60665 −6.57673 0.747628 −10.2819 2.61418 −4.45952
1.2 −2.69273 −0.478176 5.25077 0.135724 1.28760 −4.02730 −8.75344 −2.77135 −0.365467
1.3 −2.49699 −1.64487 4.23496 3.35969 4.10722 2.65574 −5.58068 −0.294408 −8.38911
1.4 −2.47917 −2.68137 4.14631 1.95848 6.64759 0.410621 −5.32107 4.18976 −4.85541
1.5 −2.43722 1.24735 3.94004 0.277998 −3.04005 4.01805 −4.72830 −1.44413 −0.677543
1.6 −2.32696 −0.944934 3.41472 −1.62617 2.19882 3.30319 −3.29200 −2.10710 3.78402
1.7 −2.24849 2.23227 3.05570 −0.764282 −5.01923 −0.688314 −2.37374 1.98301 1.71848
1.8 −2.19361 −1.71235 2.81195 −2.23633 3.75624 −3.04720 −1.78110 −0.0678453 4.90564
1.9 −2.18971 −0.875363 2.79481 −1.80760 1.91679 −0.359878 −1.74040 −2.23374 3.95812
1.10 −2.00564 2.28170 2.02257 −4.04874 −4.57625 −0.876531 −0.0452719 2.20615 8.12030
1.11 −1.84055 2.78518 1.38763 3.40108 −5.12627 −2.70641 1.12710 4.75723 −6.25987
1.12 −1.78339 0.526387 1.18048 0.564222 −0.938754 1.86157 1.46152 −2.72292 −1.00623
1.13 −1.69299 −2.95831 0.866208 −2.49874 5.00838 1.49318 1.91950 5.75160 4.23033
1.14 −1.67397 −0.759884 0.802190 2.08858 1.27203 −1.20948 2.00510 −2.42258 −3.49623
1.15 −1.54716 −2.20924 0.393702 3.10213 3.41805 −0.403081 2.48520 1.88074 −4.79950
1.16 −1.45467 0.452328 0.116052 −2.57197 −0.657986 3.66955 2.74051 −2.79540 3.74136
1.17 −1.40700 1.38079 −0.0203573 0.687727 −1.94276 −3.68772 2.84264 −1.09343 −0.967630
1.18 −1.33108 2.24902 −0.228227 −0.768051 −2.99362 2.32195 2.96595 2.05807 1.02234
1.19 −1.25853 0.116839 −0.416106 −2.24208 −0.147046 −3.42737 3.04074 −2.98635 2.82172
1.20 −1.17292 −0.112413 −0.624270 4.03608 0.131851 1.76356 3.07805 −2.98736 −4.73397
See all 59 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.59
Significant digits:
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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.b 59
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4031.2.a.b 59 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{59} + 5 T_{2}^{58} - 67 T_{2}^{57} - 366 T_{2}^{56} + 2065 T_{2}^{55} + 12599 T_{2}^{54} - 38552 T_{2}^{53} - 271237 T_{2}^{52} + 480449 T_{2}^{51} + 4096812 T_{2}^{50} - 4095853 T_{2}^{49} - 46165271 T_{2}^{48} + \cdots + 83 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\). Copy content Toggle raw display