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Properties

Label 4006.2.a.i
Level $4006$
Weight $2$
Character orbit 4006.a
Self dual yes
Analytic conductor $31.988$
Analytic rank $0$
Dimension $46$
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: \(0\)
Dimension: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q + 46 q^{2} + 21 q^{3} + 46 q^{4} + 23 q^{5} + 21 q^{6} + 26 q^{7} + 46 q^{8} + 59 q^{9} + 23 q^{10} + 39 q^{11} + 21 q^{12} + 8 q^{13} + 26 q^{14} + 14 q^{15} + 46 q^{16} + 36 q^{17} + 59 q^{18} + 37 q^{19} + 23 q^{20} + 20 q^{21} + 39 q^{22} + 38 q^{23} + 21 q^{24} + 57 q^{25} + 8 q^{26} + 63 q^{27} + 26 q^{28} + 23 q^{29} + 14 q^{30} + 44 q^{31} + 46 q^{32} + 25 q^{33} + 36 q^{34} + 26 q^{35} + 59 q^{36} + 9 q^{37} + 37 q^{38} - 2 q^{39} + 23 q^{40} + 50 q^{41} + 20 q^{42} + 46 q^{43} + 39 q^{44} + 30 q^{45} + 38 q^{46} + 57 q^{47} + 21 q^{48} + 62 q^{49} + 57 q^{50} + 5 q^{51} + 8 q^{52} + 21 q^{53} + 63 q^{54} + 40 q^{55} + 26 q^{56} + 3 q^{57} + 23 q^{58} + 68 q^{59} + 14 q^{60} - q^{61} + 44 q^{62} + 40 q^{63} + 46 q^{64} + 18 q^{65} + 25 q^{66} + 42 q^{67} + 36 q^{68} - 7 q^{69} + 26 q^{70} + 67 q^{71} + 59 q^{72} + 48 q^{73} + 9 q^{74} + 71 q^{75} + 37 q^{76} - 5 q^{77} - 2 q^{78} + 40 q^{79} + 23 q^{80} + 82 q^{81} + 50 q^{82} + 48 q^{83} + 20 q^{84} - 68 q^{85} + 46 q^{86} + 18 q^{87} + 39 q^{88} + 90 q^{89} + 30 q^{90} + 9 q^{91} + 38 q^{92} - 42 q^{93} + 57 q^{94} + 6 q^{95} + 21 q^{96} + 46 q^{97} + 62 q^{98} + 61 q^{99}+O(q^{100}) \) 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 −3.34123 1.00000 −1.21966 −3.34123 1.21220 1.00000 8.16382 −1.21966
1.2 1.00000 −2.83861 1.00000 −0.543432 −2.83861 −3.69136 1.00000 5.05769 −0.543432
1.3 1.00000 −2.79379 1.00000 1.09256 −2.79379 0.952354 1.00000 4.80523 1.09256
1.4 1.00000 −2.67899 1.00000 3.78338 −2.67899 0.625122 1.00000 4.17700 3.78338
1.5 1.00000 −2.41935 1.00000 −2.86721 −2.41935 3.82612 1.00000 2.85324 −2.86721
1.6 1.00000 −2.24859 1.00000 3.12554 −2.24859 2.41821 1.00000 2.05614 3.12554
1.7 1.00000 −2.23424 1.00000 1.35082 −2.23424 1.10993 1.00000 1.99184 1.35082
1.8 1.00000 −1.96762 1.00000 −0.579208 −1.96762 −3.04888 1.00000 0.871523 −0.579208
1.9 1.00000 −1.75438 1.00000 1.48781 −1.75438 4.69044 1.00000 0.0778341 1.48781
1.10 1.00000 −1.64474 1.00000 −0.535091 −1.64474 −1.74380 1.00000 −0.294845 −0.535091
1.11 1.00000 −1.53516 1.00000 3.32650 −1.53516 2.69032 1.00000 −0.643295 3.32650
1.12 1.00000 −1.38097 1.00000 −1.46012 −1.38097 0.773639 1.00000 −1.09292 −1.46012
1.13 1.00000 −1.20020 1.00000 4.05453 −1.20020 −4.32579 1.00000 −1.55952 4.05453
1.14 1.00000 −0.857869 1.00000 −2.26962 −0.857869 −3.52222 1.00000 −2.26406 −2.26962
1.15 1.00000 −0.676583 1.00000 −1.65458 −0.676583 3.29858 1.00000 −2.54224 −1.65458
1.16 1.00000 −0.506267 1.00000 2.89870 −0.506267 0.00816549 1.00000 −2.74369 2.89870
1.17 1.00000 −0.354737 1.00000 2.50160 −0.354737 4.23908 1.00000 −2.87416 2.50160
1.18 1.00000 −0.317846 1.00000 −2.51845 −0.317846 4.77839 1.00000 −2.89897 −2.51845
1.19 1.00000 −0.151617 1.00000 −4.25887 −0.151617 0.316610 1.00000 −2.97701 −4.25887
1.20 1.00000 0.0490190 1.00000 0.547018 0.0490190 −4.51424 1.00000 −2.99760 0.547018
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
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.i ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4006.2.a.i ✓ 46 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{46} - 21 T_{3}^{45} + 122 T_{3}^{44} + 462 T_{3}^{43} - 7802 T_{3}^{42} + 15645 T_{3}^{41} + \cdots - 87552 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4006))\). Copy content Toggle raw display