Properties

Label 8045.2.a.e
Level $8045$
Weight $2$
Character orbit 8045.a
Self dual yes
Analytic conductor $64.240$
Analytic rank $0$
Dimension $142$
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] = [8045,2,Mod(1,8045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8045, 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("8045.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8045 = 5 \cdot 1609 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8045.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: \(64.2396484261\)
Analytic rank: \(0\)
Dimension: \(142\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 142 q + 21 q^{2} + 33 q^{3} + 157 q^{4} + 142 q^{5} + 15 q^{6} + 63 q^{7} + 60 q^{8} + 157 q^{9} + 21 q^{10} + 36 q^{11} + 55 q^{12} + 57 q^{13} + 2 q^{14} + 33 q^{15} + 179 q^{16} + 55 q^{17} + 65 q^{18} + 130 q^{19} + 157 q^{20} + 28 q^{21} + 30 q^{22} + 117 q^{23} + 21 q^{24} + 142 q^{25} + 21 q^{26} + 120 q^{27} + 135 q^{28} + 12 q^{29} + 15 q^{30} + 74 q^{31} + 126 q^{32} + 55 q^{33} + 35 q^{34} + 63 q^{35} + 186 q^{36} + 75 q^{37} + 65 q^{38} + 23 q^{39} + 60 q^{40} + 22 q^{41} + 10 q^{42} + 190 q^{43} + 22 q^{44} + 157 q^{45} + 56 q^{46} + 102 q^{47} + 78 q^{48} + 197 q^{49} + 21 q^{50} + 30 q^{51} + 120 q^{52} + 56 q^{53} - 6 q^{54} + 36 q^{55} + 3 q^{56} + 68 q^{57} + 31 q^{58} + 55 q^{59} + 55 q^{60} + 90 q^{61} + 68 q^{62} + 167 q^{63} + 180 q^{64} + 57 q^{65} + 17 q^{66} + 151 q^{67} + 119 q^{68} + 21 q^{69} + 2 q^{70} + 4 q^{71} + 130 q^{72} + 143 q^{73} - 46 q^{74} + 33 q^{75} + 213 q^{76} + 75 q^{77} - 24 q^{78} + 47 q^{79} + 179 q^{80} + 150 q^{81} + 69 q^{82} + 201 q^{83} - 31 q^{84} + 55 q^{85} - 4 q^{86} + 153 q^{87} + 37 q^{88} + 25 q^{89} + 65 q^{90} + 132 q^{91} + 194 q^{92} + 52 q^{93} + 18 q^{94} + 130 q^{95} + 13 q^{96} + 80 q^{97} + 58 q^{98} + 103 q^{99}+O(q^{100}) \) 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 −2.77343 2.65510 5.69191 1.00000 −7.36373 0.492807 −10.2393 4.04956 −2.77343
1.2 −2.68924 −0.0494506 5.23199 1.00000 0.132984 2.47489 −8.69158 −2.99755 −2.68924
1.3 −2.64935 2.86876 5.01907 1.00000 −7.60037 4.90549 −7.99858 5.22980 −2.64935
1.4 −2.64700 −1.25206 5.00659 1.00000 3.31419 4.95452 −7.95843 −1.43235 −2.64700
1.5 −2.63029 −2.20415 4.91841 1.00000 5.79754 3.10194 −7.67624 1.85826 −2.63029
1.6 −2.60196 1.36677 4.77022 1.00000 −3.55629 2.24153 −7.20801 −1.13194 −2.60196
1.7 −2.59468 −0.861703 4.73239 1.00000 2.23585 −3.11370 −7.08968 −2.25747 −2.59468
1.8 −2.58612 1.58438 4.68803 1.00000 −4.09739 −2.71567 −6.95157 −0.489749 −2.58612
1.9 −2.51445 −1.12018 4.32245 1.00000 2.81662 0.310792 −5.83967 −1.74521 −2.51445
1.10 −2.48735 −2.65295 4.18689 1.00000 6.59881 2.41015 −5.43957 4.03816 −2.48735
1.11 −2.46500 2.31843 4.07621 1.00000 −5.71491 −0.590235 −5.11785 2.37510 −2.46500
1.12 −2.39887 −2.11291 3.75456 1.00000 5.06858 1.17746 −4.20895 1.46438 −2.39887
1.13 −2.28247 0.951173 3.20967 1.00000 −2.17102 −3.35163 −2.76104 −2.09527 −2.28247
1.14 −2.20465 −2.96481 2.86047 1.00000 6.53637 0.166097 −1.89703 5.79013 −2.20465
1.15 −2.19407 0.184631 2.81393 1.00000 −0.405094 1.35112 −1.78582 −2.96591 −2.19407
1.16 −2.18359 0.444286 2.76808 1.00000 −0.970139 −0.684124 −1.67716 −2.80261 −2.18359
1.17 −2.16169 −1.95272 2.67292 1.00000 4.22118 −2.55513 −1.45466 0.813116 −2.16169
1.18 −2.15729 −0.777397 2.65392 1.00000 1.67707 3.51600 −1.41069 −2.39565 −2.15729
1.19 −2.15373 3.04079 2.63855 1.00000 −6.54904 −4.47994 −1.37525 6.24641 −2.15373
1.20 −2.15214 2.27833 2.63171 1.00000 −4.90329 4.73674 −1.35954 2.19079 −2.15214
See next 80 embeddings (of 142 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.142
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(1609\) \(1\)

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 8045.2.a.e 142
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8045.2.a.e 142 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{142} - 21 T_{2}^{141} + 3039 T_{2}^{139} - 15763 T_{2}^{138} - 192561 T_{2}^{137} + 1760982 T_{2}^{136} + 6231649 T_{2}^{135} - 105899723 T_{2}^{134} - 40642006 T_{2}^{133} + 4306862922 T_{2}^{132} + \cdots + 124007569699 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\). Copy content Toggle raw display