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Properties

Label 8005.2.a.g
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.

\(\operatorname{Tr}(f)(q) = \) \( 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 137 q + 4 q^{2} + 20 q^{3} + 152 q^{4} - 137 q^{5} + 14 q^{6} - 30 q^{7} + 24 q^{8} + 163 q^{9} - 4 q^{10} + 51 q^{11} + 32 q^{12} - 6 q^{13} + 49 q^{14} - 20 q^{15} + 170 q^{16} + 46 q^{17} + 4 q^{18} + 47 q^{19} - 152 q^{20} + 28 q^{21} - 19 q^{22} + 29 q^{23} + 39 q^{24} + 137 q^{25} + 67 q^{26} + 77 q^{27} - 58 q^{28} + 27 q^{29} - 14 q^{30} + 23 q^{31} + 42 q^{32} + 40 q^{33} + 38 q^{34} + 30 q^{35} + 222 q^{36} - 56 q^{37} + 87 q^{38} + 44 q^{39} - 24 q^{40} + 66 q^{41} + 34 q^{42} + 15 q^{43} + 87 q^{44} - 163 q^{45} + 37 q^{46} + 52 q^{47} + 56 q^{48} + 195 q^{49} + 4 q^{50} + 106 q^{51} - 31 q^{52} + 45 q^{53} + 83 q^{54} - 51 q^{55} + 148 q^{56} + 4 q^{57} - 101 q^{58} + 239 q^{59} - 32 q^{60} + 46 q^{61} + 63 q^{62} - 59 q^{63} + 200 q^{64} + 6 q^{65} + 108 q^{66} - 18 q^{67} + 152 q^{68} + 63 q^{69} - 49 q^{70} + 110 q^{71} + 6 q^{72} - 19 q^{73} + 81 q^{74} + 20 q^{75} + 94 q^{76} + 43 q^{77} - 3 q^{78} + 40 q^{79} - 170 q^{80} + 229 q^{81} + 3 q^{82} + 235 q^{83} + 94 q^{84} - 46 q^{85} + 110 q^{86} + 31 q^{87} - 105 q^{88} + 150 q^{89} - 4 q^{90} + 110 q^{91} + 76 q^{92} + 11 q^{93} + 56 q^{94} - 47 q^{95} + 146 q^{96} + 17 q^{97} + 75 q^{98} + 125 q^{99}+O(q^{100}) \) 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 −2.80769 1.97654 5.88311 −1.00000 −5.54949 −3.00934 −10.9026 0.906693 2.80769
1.2 −2.78843 −0.336245 5.77535 −1.00000 0.937596 −4.55951 −10.5273 −2.88694 2.78843
1.3 −2.78369 −2.85719 5.74891 −1.00000 7.95352 −1.73481 −10.4358 5.16354 2.78369
1.4 −2.73421 3.20791 5.47593 −1.00000 −8.77111 1.52439 −9.50394 7.29068 2.73421
1.5 −2.57744 0.488387 4.64318 −1.00000 −1.25879 1.78000 −6.81263 −2.76148 2.57744
1.6 −2.54897 0.946332 4.49724 −1.00000 −2.41217 0.0675168 −6.36539 −2.10446 2.54897
1.7 −2.54647 −2.27046 4.48451 −1.00000 5.78165 −1.05836 −6.32674 2.15498 2.54647
1.8 −2.53925 −0.755801 4.44781 −1.00000 1.91917 −1.64604 −6.21562 −2.42876 2.53925
1.9 −2.50295 −2.57024 4.26474 −1.00000 6.43316 2.87384 −5.66852 3.60612 2.50295
1.10 −2.48869 −2.85866 4.19357 −1.00000 7.11432 −4.63571 −5.45912 5.17196 2.48869
1.11 −2.48115 −3.25106 4.15609 −1.00000 8.06635 0.841508 −5.34956 7.56938 2.48115
1.12 −2.46764 1.73940 4.08925 −1.00000 −4.29222 −3.99512 −5.15552 0.0255271 2.46764
1.13 −2.45773 2.52580 4.04044 −1.00000 −6.20774 1.85923 −5.01486 3.37966 2.45773
1.14 −2.43829 −0.0824677 3.94524 −1.00000 0.201080 0.816221 −4.74304 −2.99320 2.43829
1.15 −2.40143 2.43968 3.76685 −1.00000 −5.85871 −3.81708 −4.24295 2.95203 2.40143
1.16 −2.33815 −1.98025 3.46696 −1.00000 4.63012 −4.85711 −3.42998 0.921375 2.33815
1.17 −2.33561 0.566627 3.45506 −1.00000 −1.32342 2.06989 −3.39845 −2.67893 2.33561
1.18 −2.22931 3.17028 2.96982 −1.00000 −7.06754 −4.76730 −2.16203 7.05069 2.22931
1.19 −2.21085 −0.155925 2.88785 −1.00000 0.344727 4.20602 −1.96290 −2.97569 2.21085
1.20 −2.20604 −1.02446 2.86662 −1.00000 2.26001 0.152086 −1.91181 −1.95047 2.20604
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.g ✓ 137
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8005.2.a.g ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8005))\):

\( T_{2}^{137} - 4 T_{2}^{136} - 205 T_{2}^{135} + 828 T_{2}^{134} + 20500 T_{2}^{133} - 83626 T_{2}^{132} + \cdots + 16394986737 \) Copy content Toggle raw display
\( T_{3}^{137} - 20 T_{3}^{136} - 87 T_{3}^{135} + 4341 T_{3}^{134} - 9276 T_{3}^{133} + \cdots - 32\!\cdots\!16 \) Copy content Toggle raw display