Properties

Label 4009.2.a.e
Level $4009$
Weight $2$
Character orbit 4009.a
Self dual yes
Analytic conductor $32.012$
Analytic rank $0$
Dimension $82$
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] = [4009,2,Mod(1,4009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4009, 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("4009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4009 = 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4009.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: \(32.0120261703\)
Analytic rank: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 82 q + 15 q^{2} + 12 q^{3} + 89 q^{4} + 9 q^{5} + 9 q^{6} + 14 q^{7} + 42 q^{8} + 92 q^{9} + 4 q^{10} + 41 q^{11} + 26 q^{12} + 13 q^{13} + 22 q^{14} + 41 q^{15} + 87 q^{16} + 12 q^{17} + 24 q^{18} + 82 q^{19} + 26 q^{20} + 29 q^{21} + 2 q^{22} + 59 q^{23} + 16 q^{24} + 67 q^{25} + 24 q^{26} + 42 q^{27} - 2 q^{28} + 101 q^{29} - 22 q^{30} + 48 q^{31} + 69 q^{32} + 3 q^{33} + q^{34} + 38 q^{35} + 82 q^{36} + 16 q^{37} + 15 q^{38} + 82 q^{39} + 20 q^{40} + 86 q^{41} - q^{42} + 9 q^{43} + 82 q^{44} - 8 q^{45} + 43 q^{46} + 24 q^{47} + 34 q^{48} + 76 q^{49} + 82 q^{50} + 57 q^{51} - 22 q^{52} + 39 q^{53} + 17 q^{54} - 21 q^{55} + 50 q^{56} + 12 q^{57} + 33 q^{58} + 79 q^{59} + 87 q^{60} + 4 q^{61} + 40 q^{62} + 44 q^{63} + 90 q^{64} + 66 q^{65} - 39 q^{66} + 33 q^{67} - 9 q^{68} + 60 q^{69} + 30 q^{70} + 168 q^{71} + 15 q^{72} - 28 q^{73} + 35 q^{74} + 55 q^{75} + 89 q^{76} + 19 q^{77} - 41 q^{78} + 121 q^{79} + 64 q^{80} + 110 q^{81} + 41 q^{82} + 28 q^{84} + 17 q^{85} + 80 q^{86} + 29 q^{87} + 49 q^{88} + 83 q^{89} - 42 q^{90} + 38 q^{91} + 71 q^{92} - q^{93} + 89 q^{94} + 9 q^{95} + 35 q^{96} - 23 q^{97} + 135 q^{98} + 93 q^{99}+O(q^{100}) \) 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 −2.71228 −0.799829 5.35647 2.45685 2.16936 −1.47623 −9.10370 −2.36027 −6.66368
1.2 −2.64355 −0.244924 4.98837 −2.38993 0.647469 −3.32754 −7.89992 −2.94001 6.31792
1.3 −2.62936 3.14041 4.91352 2.46626 −8.25726 1.15889 −7.66068 6.86218 −6.48468
1.4 −2.62201 −2.46509 4.87496 −1.74208 6.46350 −2.61324 −7.53817 3.07667 4.56776
1.5 −2.40525 0.482964 3.78524 1.80217 −1.16165 −1.90696 −4.29395 −2.76675 −4.33466
1.6 −2.39399 1.52449 3.73121 −2.51919 −3.64962 −1.00997 −4.14450 −0.675931 6.03093
1.7 −2.39206 −0.112128 3.72195 0.840987 0.268217 2.66760 −4.11902 −2.98743 −2.01169
1.8 −2.27080 −1.94834 3.15654 0.0772315 4.42430 2.91283 −2.62627 0.796044 −0.175377
1.9 −2.22938 2.96636 2.97012 −0.893171 −6.61314 4.38195 −2.16277 5.79930 1.99121
1.10 −2.21501 1.93892 2.90626 3.78369 −4.29473 0.133386 −2.00737 0.759427 −8.38090
1.11 −2.16486 3.17577 2.68660 0.570979 −6.87507 −2.83752 −1.48639 7.08548 −1.23609
1.12 −2.09995 −2.73690 2.40979 2.29135 5.74735 −1.74327 −0.860544 4.49062 −4.81171
1.13 −1.93317 −1.52394 1.73714 −2.99074 2.94604 −0.667472 0.508147 −0.677595 5.78161
1.14 −1.90886 1.35744 1.64375 −1.48428 −2.59115 2.77982 0.680038 −1.15737 2.83328
1.15 −1.89977 −2.93667 1.60912 −0.296578 5.57898 2.59029 0.742589 5.62400 0.563430
1.16 −1.69115 −1.97765 0.859975 −4.28235 3.34450 4.00531 1.92795 0.911117 7.24208
1.17 −1.66851 −1.79206 0.783934 2.47432 2.99007 0.566931 2.02902 0.211475 −4.12843
1.18 −1.53574 2.00602 0.358512 1.47205 −3.08073 −4.88163 2.52091 1.02410 −2.26069
1.19 −1.53259 1.29314 0.348820 3.28950 −1.98185 4.48074 2.53058 −1.32779 −5.04144
1.20 −1.51298 0.189655 0.289107 −0.503823 −0.286944 −1.74123 2.58855 −2.96403 0.762274
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \(-1\)
\(211\) \(1\)

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 4009.2.a.e 82
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.e 82 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{82} - 15 T_{2}^{81} - 14 T_{2}^{80} + 1301 T_{2}^{79} - 3930 T_{2}^{78} - 50065 T_{2}^{77} + 272262 T_{2}^{76} + 1062653 T_{2}^{75} - 9435251 T_{2}^{74} - 10815784 T_{2}^{73} + 215226565 T_{2}^{72} + \cdots - 1538721 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\). Copy content Toggle raw display