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Properties

Label 4009.2.a.c
Level $4009$
Weight $2$
Character orbit 4009.a
Self dual yes
Analytic conductor $32.012$
Analytic rank $1$
Dimension $71$
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: \(1\)
Dimension: \(71\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 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 −2.82251 2.93474 5.96657 0.420997 −8.28333 −3.44749 −11.1957 5.61269 −1.18827
1.2 −2.70369 1.93926 5.30992 1.68888 −5.24315 4.83108 −8.94900 0.760725 −4.56619
1.3 −2.69837 1.88273 5.28118 −2.05530 −5.08030 0.784756 −8.85381 0.544674 5.54594
1.4 −2.67657 0.00971531 5.16401 0.0508514 −0.0260037 0.312050 −8.46867 −2.99991 −0.136107
1.5 −2.66661 −1.96401 5.11081 −2.76483 5.23725 2.50280 −8.29531 0.857341 7.37272
1.6 −2.61964 −2.82043 4.86252 −3.49446 7.38851 2.54132 −7.49878 4.95481 9.15424
1.7 −2.51184 −2.51408 4.30932 3.86977 6.31496 2.24136 −5.80064 3.32060 −9.72023
1.8 −2.48400 0.221625 4.17025 −2.50863 −0.550515 2.33062 −5.39089 −2.95088 6.23143
1.9 −2.35519 0.815593 3.54692 1.05865 −1.92088 −4.06376 −3.64328 −2.33481 −2.49333
1.10 −2.30773 −2.77619 3.32561 2.46750 6.40670 −0.00709873 −3.05915 4.70724 −5.69433
1.11 −2.28791 −0.433636 3.23453 1.92947 0.992121 3.12765 −2.82450 −2.81196 −4.41445
1.12 −2.25185 −1.69642 3.07082 −1.12007 3.82008 −4.02650 −2.41131 −0.122162 2.52222
1.13 −2.08913 2.53881 2.36446 0.545075 −5.30390 0.502260 −0.761409 3.44555 −1.13873
1.14 −2.08320 2.50442 2.33972 −4.26568 −5.21720 −3.89477 −0.707698 3.27211 8.88625
1.15 −1.98460 0.183292 1.93862 −1.97615 −0.363761 −5.05020 0.121816 −2.96640 3.92186
1.16 −1.93894 −1.61421 1.75950 −0.710068 3.12986 −1.51648 0.466323 −0.394323 1.37678
1.17 −1.89616 2.32800 1.59541 −3.08512 −4.41426 2.27635 0.767159 2.41961 5.84987
1.18 −1.89239 −0.431235 1.58115 3.97235 0.816065 −2.66937 0.792636 −2.81404 −7.51724
1.19 −1.74061 −3.34117 1.02972 −2.66559 5.81567 0.0455688 1.68888 8.16342 4.63975
1.20 −1.69413 −0.649307 0.870061 −0.0469641 1.10001 3.97959 1.91426 −2.57840 0.0795630
See all 71 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.71
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.c ✓ 71
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4009.2.a.c ✓ 71 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{71} + 15 T_{2}^{70} + 7 T_{2}^{69} - 987 T_{2}^{68} - 3958 T_{2}^{67} + 26949 T_{2}^{66} + \cdots - 104841 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4009))\). Copy content Toggle raw display