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Properties

Label 2005.2.a.e
Level $2005$
Weight $2$
Character orbit 2005.a
Self dual yes
Analytic conductor $16.010$
Analytic rank $1$
Dimension $29$
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] = [2005,2,Mod(1,2005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2005, 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("2005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2005 = 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2005.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: \(16.0100056053\)
Analytic rank: \(1\)
Dimension: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 5 q^{2} - 3 q^{3} + 19 q^{4} - 29 q^{5} - 6 q^{6} + 12 q^{7} - 15 q^{8} + 14 q^{9} + 5 q^{10} - 38 q^{11} - 6 q^{12} + 5 q^{13} - 18 q^{14} + 3 q^{15} + 7 q^{16} - 16 q^{17} - 2 q^{18} - 18 q^{19} - 19 q^{20} - 20 q^{21} - 2 q^{22} - 19 q^{23} - 19 q^{24} + 29 q^{25} - 21 q^{26} - 21 q^{27} + 26 q^{28} - 31 q^{29} + 6 q^{30} - 13 q^{31} - 30 q^{32} + 2 q^{33} - 14 q^{34} - 12 q^{35} - 29 q^{36} - q^{37} - 23 q^{38} - 39 q^{39} + 15 q^{40} - 24 q^{41} - 20 q^{42} - 27 q^{43} - 76 q^{44} - 14 q^{45} - 11 q^{46} - 5 q^{47} - 2 q^{48} - 11 q^{49} - 5 q^{50} - 58 q^{51} + 11 q^{52} - 37 q^{53} - 18 q^{54} + 38 q^{55} - 50 q^{56} - 6 q^{57} + 31 q^{58} - 67 q^{59} + 6 q^{60} - 31 q^{61} - 19 q^{62} - 2 q^{63} - 13 q^{64} - 5 q^{65} + 6 q^{66} - 17 q^{67} - 16 q^{68} - 48 q^{69} + 18 q^{70} - 53 q^{71} + 9 q^{72} + 29 q^{73} - 59 q^{74} - 3 q^{75} - 21 q^{76} - 62 q^{77} - 12 q^{78} - 13 q^{79} - 7 q^{80} - 11 q^{81} + 32 q^{82} - 72 q^{83} - 58 q^{84} + 16 q^{85} - 43 q^{86} + 4 q^{87} + 12 q^{88} - 38 q^{89} + 2 q^{90} - 45 q^{91} - 37 q^{92} - 27 q^{93} - 44 q^{94} + 18 q^{95} - 21 q^{96} + 32 q^{97} - 32 q^{98} - 107 q^{99}+O(q^{100}) \) 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 −2.67547 2.10999 5.15811 −1.00000 −5.64522 2.87574 −8.44943 1.45208 2.67547
1.2 −2.65420 −1.14990 5.04475 −1.00000 3.05206 2.92882 −8.08137 −1.67773 2.65420
1.3 −2.47126 −1.08024 4.10715 −1.00000 2.66955 0.388284 −5.20731 −1.83308 2.47126
1.4 −2.27234 1.15962 3.16351 −1.00000 −2.63504 −2.94936 −2.64390 −1.65529 2.27234
1.5 −2.13554 1.65605 2.56051 −1.00000 −3.53655 1.78297 −1.19700 −0.257503 2.13554
1.6 −1.86230 −1.84094 1.46818 −1.00000 3.42839 3.02254 0.990420 0.389053 1.86230
1.7 −1.77645 −3.27205 1.15578 −1.00000 5.81263 2.00064 1.49972 7.70629 1.77645
1.8 −1.71603 −0.663123 0.944755 −1.00000 1.13794 −2.54726 1.81083 −2.56027 1.71603
1.9 −1.29166 0.182887 −0.331605 −1.00000 −0.236228 2.69438 3.01165 −2.96655 1.29166
1.10 −1.19798 2.37610 −0.564833 −1.00000 −2.84653 −3.68594 3.07263 2.64584 1.19798
1.11 −1.13376 2.41050 −0.714583 −1.00000 −2.73294 4.26135 3.07769 2.81053 1.13376
1.12 −0.706999 2.56202 −1.50015 −1.00000 −1.81135 −2.27480 2.47460 3.56397 0.706999
1.13 −0.524480 −2.05313 −1.72492 −1.00000 1.07683 −2.06215 1.95365 1.21534 0.524480
1.14 −0.475745 −1.88724 −1.77367 −1.00000 0.897847 −2.32576 1.79530 0.561689 0.475745
1.15 −0.370815 −1.61059 −1.86250 −1.00000 0.597230 4.02410 1.43227 −0.406010 0.370815
1.16 0.154915 0.533773 −1.97600 −1.00000 0.0826896 2.26312 −0.615943 −2.71509 −0.154915
1.17 0.282613 −0.0447289 −1.92013 −1.00000 −0.0126409 0.565532 −1.10788 −2.99800 −0.282613
1.18 0.562660 −3.02550 −1.68341 −1.00000 −1.70233 −2.33072 −2.07251 6.15366 −0.562660
1.19 0.609831 −1.81759 −1.62811 −1.00000 −1.10842 −0.599157 −2.21253 0.303632 −0.609831
1.20 0.631551 1.59060 −1.60114 −1.00000 1.00455 2.69469 −2.27430 −0.469979 −0.631551
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(401\) \(1\)

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 2005.2.a.e ✓ 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.2.a.e ✓ 29 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2005))\):

\( T_{2}^{29} + 5 T_{2}^{28} - 26 T_{2}^{27} - 160 T_{2}^{26} + 256 T_{2}^{25} + 2235 T_{2}^{24} + \cdots + 63 \) Copy content Toggle raw display
\( T_{11}^{29} + 38 T_{11}^{28} + 561 T_{11}^{27} + 3324 T_{11}^{26} - 6848 T_{11}^{25} - 213562 T_{11}^{24} + \cdots - 172526976 \) Copy content Toggle raw display