Properties

Label 4011.2.a.k
Level $4011$
Weight $2$
Character orbit 4011.a
Self dual yes
Analytic conductor $32.028$
Analytic rank $0$
Dimension $27$
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] = [4011,2,Mod(1,4011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4011.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: \(32.0279962507\)
Analytic rank: \(0\)
Dimension: \(27\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 22 q^{11} + 31 q^{12} + 15 q^{13} + 9 q^{14} + 23 q^{15} + 39 q^{16} + 22 q^{17} + 9 q^{18} + 24 q^{19} + 46 q^{20} + 27 q^{21} - 19 q^{22} + 36 q^{23} + 18 q^{24} + 38 q^{25} + 19 q^{26} + 27 q^{27} + 31 q^{28} + 32 q^{29} + 10 q^{30} + 11 q^{31} + 15 q^{32} + 22 q^{33} - 4 q^{34} + 23 q^{35} + 31 q^{36} + 6 q^{37} + 8 q^{38} + 15 q^{39} + 16 q^{41} + 9 q^{42} - 11 q^{43} + 22 q^{44} + 23 q^{45} - 56 q^{46} + 39 q^{47} + 39 q^{48} + 27 q^{49} - 7 q^{50} + 22 q^{51} + 10 q^{52} + 24 q^{53} + 9 q^{54} + 16 q^{55} + 18 q^{56} + 24 q^{57} - 31 q^{58} + 31 q^{59} + 46 q^{60} + 28 q^{61} - 4 q^{62} + 27 q^{63} + 40 q^{64} + 33 q^{65} - 19 q^{66} - 7 q^{67} + 25 q^{68} + 36 q^{69} + 10 q^{70} + 49 q^{71} + 18 q^{72} - 13 q^{73} + 7 q^{74} + 38 q^{75} + 15 q^{76} + 22 q^{77} + 19 q^{78} - 18 q^{79} + 78 q^{80} + 27 q^{81} + 31 q^{82} + 59 q^{83} + 31 q^{84} - 20 q^{85} + 35 q^{86} + 32 q^{87} - 49 q^{88} + 49 q^{89} + 10 q^{90} + 15 q^{91} + 52 q^{92} + 11 q^{93} + 17 q^{94} + 38 q^{95} + 15 q^{96} - 7 q^{97} + 9 q^{98} + 22 q^{99}+O(q^{100}) \) 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 −2.65342 1.00000 5.04066 4.09935 −2.65342 1.00000 −8.06816 1.00000 −10.8773
1.2 −2.52420 1.00000 4.37157 −0.338133 −2.52420 1.00000 −5.98631 1.00000 0.853514
1.3 −2.42203 1.00000 3.86625 1.50568 −2.42203 1.00000 −4.52011 1.00000 −3.64681
1.4 −1.70789 1.00000 0.916895 2.36693 −1.70789 1.00000 1.84983 1.00000 −4.04246
1.5 −1.69492 1.00000 0.872752 −3.19478 −1.69492 1.00000 1.91059 1.00000 5.41490
1.6 −1.64011 1.00000 0.689959 −1.40502 −1.64011 1.00000 2.14861 1.00000 2.30438
1.7 −1.28891 1.00000 −0.338712 0.716613 −1.28891 1.00000 3.01439 1.00000 −0.923650
1.8 −1.05296 1.00000 −0.891266 4.03535 −1.05296 1.00000 3.04440 1.00000 −4.24908
1.9 −0.768548 1.00000 −1.40933 3.25090 −0.768548 1.00000 2.62024 1.00000 −2.49847
1.10 −0.667457 1.00000 −1.55450 −3.08787 −0.667457 1.00000 2.37248 1.00000 2.06102
1.11 −0.105370 1.00000 −1.98890 −1.28706 −0.105370 1.00000 0.420311 1.00000 0.135618
1.12 −0.0599798 1.00000 −1.99640 3.32796 −0.0599798 1.00000 0.239704 1.00000 −0.199610
1.13 0.267433 1.00000 −1.92848 0.0366447 0.267433 1.00000 −1.05060 1.00000 0.00980000
1.14 0.374739 1.00000 −1.85957 2.21215 0.374739 1.00000 −1.44633 1.00000 0.828982
1.15 0.640778 1.00000 −1.58940 −2.69509 0.640778 1.00000 −2.30001 1.00000 −1.72696
1.16 1.25182 1.00000 −0.432954 2.68317 1.25182 1.00000 −3.04561 1.00000 3.35884
1.17 1.25557 1.00000 −0.423542 3.43059 1.25557 1.00000 −3.04293 1.00000 4.30735
1.18 1.40876 1.00000 −0.0154013 −2.50452 1.40876 1.00000 −2.83921 1.00000 −3.52826
1.19 1.44666 1.00000 0.0928390 2.29705 1.44666 1.00000 −2.75902 1.00000 3.32306
1.20 1.74433 1.00000 1.04268 −1.87140 1.74433 1.00000 −1.66989 1.00000 −3.26434
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(191\) \(-1\)

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 4011.2.a.k 27
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4011.2.a.k 27 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} - 9 T_{2}^{26} - 2 T_{2}^{25} + 243 T_{2}^{24} - 493 T_{2}^{23} - 2496 T_{2}^{22} + 8759 T_{2}^{21} + 10702 T_{2}^{20} - 70333 T_{2}^{19} + 2142 T_{2}^{18} + 318278 T_{2}^{17} - 221040 T_{2}^{16} - 857364 T_{2}^{15} + \cdots - 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\). Copy content Toggle raw display