Properties

Label 6031.2.a.b
Level $6031$
Weight $2$
Character orbit 6031.a
Self dual yes
Analytic conductor $48.158$
Analytic rank $1$
Dimension $109$
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] = [6031,2,Mod(1,6031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6031, 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("6031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6031 = 37 \cdot 163 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6031.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: \(48.1577774590\)
Analytic rank: \(1\)
Dimension: \(109\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 109 q - 11 q^{2} - 14 q^{3} + 99 q^{4} - 28 q^{5} - 14 q^{6} - 16 q^{7} - 27 q^{8} + 65 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 109 q - 11 q^{2} - 14 q^{3} + 99 q^{4} - 28 q^{5} - 14 q^{6} - 16 q^{7} - 27 q^{8} + 65 q^{9} - 21 q^{10} - 35 q^{11} - 34 q^{12} - 15 q^{13} - 19 q^{14} - 9 q^{15} + 67 q^{16} - 82 q^{17} - 7 q^{18} - 21 q^{19} - 49 q^{20} - 38 q^{21} + 8 q^{22} - 28 q^{23} - 45 q^{24} + 63 q^{25} - 59 q^{26} - 32 q^{27} - 44 q^{28} - 69 q^{29} - 10 q^{31} - 45 q^{32} - 53 q^{33} - 35 q^{34} - 40 q^{35} + 5 q^{36} + 109 q^{37} - 34 q^{38} - 18 q^{39} - 61 q^{40} - 158 q^{41} + 5 q^{42} - q^{43} - 89 q^{44} - 49 q^{45} - 28 q^{46} - 50 q^{47} - 39 q^{48} + 13 q^{49} - 56 q^{50} - 33 q^{51} - 35 q^{52} - 79 q^{53} - 57 q^{54} - 33 q^{55} - 21 q^{56} - 57 q^{57} + 3 q^{58} - 105 q^{59} - 10 q^{60} - 51 q^{61} - 100 q^{62} - 61 q^{63} + 63 q^{64} - 120 q^{65} - 37 q^{66} - 9 q^{67} - 109 q^{68} - 80 q^{69} + q^{70} - 46 q^{71} + 36 q^{72} - 81 q^{73} - 11 q^{74} - 37 q^{75} - 22 q^{76} - 111 q^{77} - 46 q^{78} - 22 q^{79} - 116 q^{80} - 59 q^{81} - 82 q^{83} - 113 q^{84} - 26 q^{85} - 70 q^{86} - 56 q^{87} - 9 q^{88} - 171 q^{89} - 84 q^{90} + 11 q^{91} - 32 q^{92} + 42 q^{93} - 123 q^{94} - 42 q^{95} - 99 q^{96} - 28 q^{97} - 81 q^{98} - 45 q^{99}+O(q^{100}) \) 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 −2.78763 −0.692231 5.77089 −0.511620 1.92969 1.79055 −10.5119 −2.52082 1.42621
1.2 −2.71013 0.417619 5.34478 −3.36649 −1.13180 −0.322957 −9.06478 −2.82559 9.12361
1.3 −2.70989 −1.09683 5.34348 −2.68411 2.97228 −1.74258 −9.06044 −1.79697 7.27363
1.4 −2.69025 1.94949 5.23745 2.83503 −5.24461 −1.45115 −8.70956 0.800495 −7.62693
1.5 −2.68695 0.942644 5.21970 3.35356 −2.53284 −4.89514 −8.65118 −2.11142 −9.01084
1.6 −2.61098 3.10227 4.81723 −0.856662 −8.09998 −1.38221 −7.35575 6.62408 2.23673
1.7 −2.58083 0.172052 4.66069 0.569694 −0.444038 4.52515 −6.86679 −2.97040 −1.47028
1.8 −2.49637 −1.90525 4.23188 1.13152 4.75621 −4.79924 −5.57162 0.629973 −2.82469
1.9 −2.45790 −2.51992 4.04129 3.33458 6.19373 1.77040 −5.01729 3.35002 −8.19607
1.10 −2.45032 −2.60360 4.00407 −3.01282 6.37966 3.77378 −4.91062 3.77875 7.38237
1.11 −2.40987 −2.85971 3.80748 1.79940 6.89154 2.51983 −4.35580 5.17794 −4.33633
1.12 −2.39817 2.15006 3.75121 −1.82413 −5.15620 0.948079 −4.19968 1.62275 4.37458
1.13 −2.35175 −1.69234 3.53073 −3.31411 3.97997 −2.65791 −3.59990 −0.135977 7.79397
1.14 −2.30236 0.992330 3.30086 2.64832 −2.28470 −0.416453 −2.99506 −2.01528 −6.09738
1.15 −2.20321 −0.260570 2.85413 0.532828 0.574090 4.62952 −1.88183 −2.93210 −1.17393
1.16 −2.18825 1.71398 2.78846 0.914186 −3.75063 0.835020 −1.72535 −0.0622604 −2.00047
1.17 −2.14221 1.83370 2.58905 −3.64172 −3.92816 3.71202 −1.26186 0.362447 7.80132
1.18 −2.08654 −0.221531 2.35365 3.05486 0.462234 −0.0600719 −0.737910 −2.95092 −6.37409
1.19 −2.05458 0.484378 2.22130 −1.81809 −0.995193 −4.27361 −0.454677 −2.76538 3.73540
1.20 −2.05386 −1.85254 2.21834 −0.958510 3.80486 0.0563033 −0.448438 0.431918 1.96864
See next 80 embeddings (of 109 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.109
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(37\) \(-1\)
\(163\) \(-1\)

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 6031.2.a.b 109
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6031.2.a.b 109 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{109} + 11 T_{2}^{108} - 98 T_{2}^{107} - 1498 T_{2}^{106} + 3459 T_{2}^{105} + 97915 T_{2}^{104} + 3512 T_{2}^{103} - 4085340 T_{2}^{102} - 5536316 T_{2}^{101} + 122004621 T_{2}^{100} + 281881097 T_{2}^{99} + \cdots - 1077199680 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6031))\). Copy content Toggle raw display