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Properties

Label 8005.2.a.h
Level $8005$
Weight $2$
Character orbit 8005.a
Self dual yes
Analytic conductor $63.920$
Analytic rank $0$
Dimension $137$
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] = [8005,2,Mod(1,8005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8005, 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("8005.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8005 = 5 \cdot 1601 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8005.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: \(63.9202468180\)
Analytic rank: \(0\)
Dimension: \(137\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 137 q + 17 q^{2} + 46 q^{3} + 147 q^{4} + 137 q^{5} + 11 q^{6} + 53 q^{7} + 42 q^{8} + 165 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 137 q + 17 q^{2} + 46 q^{3} + 147 q^{4} + 137 q^{5} + 11 q^{6} + 53 q^{7} + 42 q^{8} + 165 q^{9} + 17 q^{10} + 47 q^{11} + 87 q^{12} + 46 q^{13} + 39 q^{14} + 46 q^{15} + 155 q^{16} + 46 q^{17} + 44 q^{18} + 42 q^{19} + 147 q^{20} + 21 q^{21} + 48 q^{22} + 110 q^{23} + 33 q^{24} + 137 q^{25} + 24 q^{26} + 145 q^{27} + 83 q^{28} + 31 q^{29} + 11 q^{30} + 23 q^{31} + 71 q^{32} + 32 q^{33} + 23 q^{34} + 53 q^{35} + 178 q^{36} + 104 q^{37} + 88 q^{38} + 4 q^{39} + 42 q^{40} + 62 q^{41} + 70 q^{42} + 94 q^{43} + 50 q^{44} + 165 q^{45} - 30 q^{46} + 122 q^{47} + 157 q^{48} + 168 q^{49} + 17 q^{50} + 72 q^{51} + 74 q^{52} + 116 q^{53} + 32 q^{54} + 47 q^{55} + 119 q^{56} + 51 q^{57} + 78 q^{58} + 213 q^{59} + 87 q^{60} + 14 q^{61} + 41 q^{62} + 147 q^{63} + 114 q^{64} + 46 q^{65} + 15 q^{66} + 184 q^{67} + 95 q^{68} + 32 q^{69} + 39 q^{70} + 99 q^{71} + 92 q^{72} + 51 q^{73} + 21 q^{74} + 46 q^{75} + 18 q^{76} + 78 q^{77} + 102 q^{78} + 6 q^{79} + 155 q^{80} + 201 q^{81} + 38 q^{82} + 228 q^{83} - 11 q^{84} + 46 q^{85} + 71 q^{86} + 113 q^{87} + 103 q^{88} + 111 q^{89} + 44 q^{90} + 41 q^{91} + 136 q^{92} + 11 q^{93} - 39 q^{94} + 42 q^{95} + 33 q^{96} + 91 q^{97} + 91 q^{98} + 65 q^{99}+O(q^{100}) \) 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 −2.75038 3.12885 5.56462 1.00000 −8.60553 −3.88896 −9.80406 6.78967 −2.75038
1.2 −2.71052 −1.80452 5.34694 1.00000 4.89119 0.944695 −9.07196 0.256280 −2.71052
1.3 −2.68908 0.643829 5.23116 1.00000 −1.73131 −0.523387 −8.68884 −2.58548 −2.68908
1.4 −2.64257 −0.753814 4.98317 1.00000 1.99201 3.63640 −7.88322 −2.43176 −2.64257
1.5 −2.63937 0.837461 4.96628 1.00000 −2.21037 −3.03210 −7.82911 −2.29866 −2.63937
1.6 −2.61871 1.95470 4.85764 1.00000 −5.11880 −0.654107 −7.48334 0.820865 −2.61871
1.7 −2.56739 1.00389 4.59147 1.00000 −2.57738 3.78307 −6.65330 −1.99220 −2.56739
1.8 −2.54538 3.33209 4.47895 1.00000 −8.48144 0.378697 −6.30987 8.10284 −2.54538
1.9 −2.42172 −1.68642 3.86472 1.00000 4.08405 −0.348612 −4.51584 −0.155972 −2.42172
1.10 −2.41370 −2.40190 3.82593 1.00000 5.79746 1.40406 −4.40725 2.76911 −2.41370
1.11 −2.41176 −1.16732 3.81661 1.00000 2.81529 −3.99886 −4.38122 −1.63737 −2.41176
1.12 −2.39758 −1.67027 3.74838 1.00000 4.00460 1.58923 −4.19189 −0.210206 −2.39758
1.13 −2.35668 1.97811 3.55392 1.00000 −4.66176 −1.91021 −3.66208 0.912906 −2.35668
1.14 −2.34704 1.27964 3.50858 1.00000 −3.00336 2.79970 −3.54069 −1.36252 −2.34704
1.15 −2.28711 2.25351 3.23088 1.00000 −5.15404 4.95844 −2.81517 2.07833 −2.28711
1.16 −2.26411 2.82922 3.12619 1.00000 −6.40567 1.05231 −2.54983 5.00450 −2.26411
1.17 −2.23692 0.783252 3.00383 1.00000 −1.75208 −5.00917 −2.24548 −2.38652 −2.23692
1.18 −2.23333 3.33368 2.98778 1.00000 −7.44521 3.40081 −2.20605 8.11340 −2.23333
1.19 −2.20391 −1.08121 2.85724 1.00000 2.38289 −0.132571 −1.88928 −1.83099 −2.20391
1.20 −2.15941 −0.293846 2.66304 1.00000 0.634532 −0.0377799 −1.43176 −2.91365 −2.15941
See next 80 embeddings (of 137 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.137
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1601\) \(1\)

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 8005.2.a.h ✓ 137
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8005.2.a.h ✓ 137 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):

\( T_{2}^{137} - 17 T_{2}^{136} - 66 T_{2}^{135} + 2723 T_{2}^{134} - 4692 T_{2}^{133} - 204769 T_{2}^{132} + \cdots - 278111013 \) Copy content Toggle raw display
\( T_{3}^{137} - 46 T_{3}^{136} + 770 T_{3}^{135} - 3115 T_{3}^{134} - 71254 T_{3}^{133} + \cdots - 54\!\cdots\!12 \) Copy content Toggle raw display