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Properties

Label 4006.2.a.f
Level $4006$
Weight $2$
Character orbit 4006.a
Self dual yes
Analytic conductor $31.988$
Analytic rank $1$
Dimension $31$
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] = [4006,2,Mod(1,4006)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4006, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4006.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4006 = 2 \cdot 2003 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4006.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: \(31.9880710497\)
Analytic rank: \(1\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} - 13 q^{3} + 31 q^{4} - 23 q^{5} - 13 q^{6} - 18 q^{7} + 31 q^{8} + 20 q^{9} - 23 q^{10} - 32 q^{11} - 13 q^{12} - 8 q^{13} - 18 q^{14} - 14 q^{15} + 31 q^{16} - 30 q^{17} + 20 q^{18} - 38 q^{19} - 23 q^{20} - 16 q^{21} - 32 q^{22} - 24 q^{23} - 13 q^{24} + 40 q^{25} - 8 q^{26} - 28 q^{27} - 18 q^{28} - 7 q^{29} - 14 q^{30} - 32 q^{31} + 31 q^{32} - 9 q^{33} - 30 q^{34} - 14 q^{35} + 20 q^{36} + 11 q^{37} - 38 q^{38} - 9 q^{39} - 23 q^{40} - 76 q^{41} - 16 q^{42} - 33 q^{43} - 32 q^{44} - 40 q^{45} - 24 q^{46} - 96 q^{47} - 13 q^{48} + 15 q^{49} + 40 q^{50} - 55 q^{51} - 8 q^{52} - 28 q^{53} - 28 q^{54} - 52 q^{55} - 18 q^{56} - 21 q^{57} - 7 q^{58} - 72 q^{59} - 14 q^{60} - 9 q^{61} - 32 q^{62} - 54 q^{63} + 31 q^{64} - 38 q^{65} - 9 q^{66} - 4 q^{67} - 30 q^{68} - 17 q^{69} - 14 q^{70} - 61 q^{71} + 20 q^{72} - 62 q^{73} + 11 q^{74} - 63 q^{75} - 38 q^{76} - 9 q^{77} - 9 q^{78} - 30 q^{79} - 23 q^{80} - 13 q^{81} - 76 q^{82} - 90 q^{83} - 16 q^{84} + 26 q^{85} - 33 q^{86} - 34 q^{87} - 32 q^{88} - 99 q^{89} - 40 q^{90} - 47 q^{91} - 24 q^{92} - 6 q^{93} - 96 q^{94} - 24 q^{95} - 13 q^{96} - 46 q^{97} + 15 q^{98} - 48 q^{99}+O(q^{100}) \) 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 1.00000 −3.10750 1.00000 −0.905636 −3.10750 3.10017 1.00000 6.65658 −0.905636
1.2 1.00000 −2.96975 1.00000 3.59055 −2.96975 −0.0161745 1.00000 5.81940 3.59055
1.3 1.00000 −2.94307 1.00000 −2.92103 −2.94307 −1.64254 1.00000 5.66165 −2.92103
1.4 1.00000 −2.69988 1.00000 2.56179 −2.69988 −4.70163 1.00000 4.28934 2.56179
1.5 1.00000 −2.68137 1.00000 0.326909 −2.68137 2.00522 1.00000 4.18972 0.326909
1.6 1.00000 −2.64840 1.00000 −3.78126 −2.64840 −4.25052 1.00000 4.01402 −3.78126
1.7 1.00000 −2.59151 1.00000 −4.34351 −2.59151 −1.56317 1.00000 3.71590 −4.34351
1.8 1.00000 −2.43870 1.00000 1.19348 −2.43870 2.99932 1.00000 2.94725 1.19348
1.9 1.00000 −1.77148 1.00000 −2.30505 −1.77148 0.622882 1.00000 0.138156 −2.30505
1.10 1.00000 −1.74675 1.00000 −3.10700 −1.74675 3.74168 1.00000 0.0511396 −3.10700
1.11 1.00000 −1.70491 1.00000 −0.390426 −1.70491 −1.43635 1.00000 −0.0932983 −0.390426
1.12 1.00000 −1.50294 1.00000 1.71703 −1.50294 −3.21564 1.00000 −0.741171 1.71703
1.13 1.00000 −0.876341 1.00000 2.54110 −0.876341 −3.25278 1.00000 −2.23203 2.54110
1.14 1.00000 −0.788172 1.00000 4.07685 −0.788172 −1.81597 1.00000 −2.37879 4.07685
1.15 1.00000 −0.724653 1.00000 −3.73538 −0.724653 −3.63449 1.00000 −2.47488 −3.73538
1.16 1.00000 −0.466489 1.00000 1.06478 −0.466489 2.04789 1.00000 −2.78239 1.06478
1.17 1.00000 −0.453821 1.00000 −0.630094 −0.453821 1.98767 1.00000 −2.79405 −0.630094
1.18 1.00000 0.105946 1.00000 1.78036 0.105946 0.955066 1.00000 −2.98878 1.78036
1.19 1.00000 0.115469 1.00000 −3.86988 0.115469 2.67770 1.00000 −2.98667 −3.86988
1.20 1.00000 0.232858 1.00000 −1.02641 0.232858 −0.709779 1.00000 −2.94578 −1.02641
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(2003\) \(-1\)

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 4006.2.a.f ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4006.2.a.f ✓ 31 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{31} + 13 T_{3}^{30} + 28 T_{3}^{29} - 333 T_{3}^{28} - 1710 T_{3}^{27} + 1963 T_{3}^{26} + \cdots + 613 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\). Copy content Toggle raw display