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Properties

Label 8045.2.a.d
Level $8045$
Weight $2$
Character orbit 8045.a
Self dual yes
Analytic conductor $64.240$
Analytic rank $0$
Dimension $141$
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: \(141\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 141 q - 8 q^{2} + 11 q^{3} + 158 q^{4} - 141 q^{5} + 23 q^{6} + 29 q^{7} - 21 q^{8} + 160 q^{9} + 8 q^{10} + 32 q^{11} + 17 q^{12} + 35 q^{13} + 18 q^{14} - 11 q^{15} + 188 q^{16} - 13 q^{17} - 16 q^{18} + 152 q^{19} - 158 q^{20} + 40 q^{21} + 14 q^{22} - 77 q^{23} + 69 q^{24} + 141 q^{25} + 27 q^{26} + 38 q^{27} + 67 q^{28} + 22 q^{29} - 23 q^{30} + 86 q^{31} - 65 q^{32} + 51 q^{33} + 79 q^{34} - 29 q^{35} + 191 q^{36} + 45 q^{37} - 9 q^{38} + 55 q^{39} + 21 q^{40} + 36 q^{41} + 6 q^{42} + 132 q^{43} + 74 q^{44} - 160 q^{45} + 72 q^{46} - 16 q^{47} + 22 q^{48} + 212 q^{49} - 8 q^{50} + 82 q^{51} + 106 q^{52} - 28 q^{53} + 86 q^{54} - 32 q^{55} + 27 q^{56} + 10 q^{57} + 23 q^{58} + 71 q^{59} - 17 q^{60} + 116 q^{61} - 36 q^{62} + 45 q^{63} + 237 q^{64} - 35 q^{65} + 69 q^{66} + 99 q^{67} - 7 q^{68} + 45 q^{69} - 18 q^{70} + 34 q^{71} - 53 q^{72} + 125 q^{73} + 50 q^{74} + 11 q^{75} + 271 q^{76} - 31 q^{77} + 2 q^{78} + 101 q^{79} - 188 q^{80} + 221 q^{81} + 67 q^{82} + 67 q^{83} + 141 q^{84} + 13 q^{85} + 48 q^{86} - 21 q^{87} + 71 q^{88} + 79 q^{89} + 16 q^{90} + 228 q^{91} - 198 q^{92} - 12 q^{93} + 114 q^{94} - 152 q^{95} + 129 q^{96} + 98 q^{97} - 31 q^{98} + 195 q^{99}+O(q^{100}) \) 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 −2.79563 0.863485 5.81558 −1.00000 −2.41399 −1.88296 −10.6670 −2.25439 2.79563
1.2 −2.76152 0.640687 5.62598 −1.00000 −1.76927 4.78584 −10.0132 −2.58952 2.76152
1.3 −2.76023 −1.69328 5.61886 −1.00000 4.67383 −3.35455 −9.98888 −0.132818 2.76023
1.4 −2.75850 −2.99511 5.60931 −1.00000 8.26199 0.810789 −9.95629 5.97065 2.75850
1.5 −2.73622 3.29432 5.48693 −1.00000 −9.01399 1.99589 −9.54102 7.85252 2.73622
1.6 −2.73567 −0.941263 5.48388 −1.00000 2.57498 4.04862 −9.53074 −2.11402 2.73567
1.7 −2.69245 1.61209 5.24928 −1.00000 −4.34048 −1.07253 −8.74853 −0.401159 2.69245
1.8 −2.68753 −2.90926 5.22280 −1.00000 7.81872 −1.56079 −8.66137 5.46381 2.68753
1.9 −2.61030 −1.72585 4.81367 −1.00000 4.50499 −4.19514 −7.34453 −0.0214352 2.61030
1.10 −2.59840 0.965890 4.75167 −1.00000 −2.50977 −4.65256 −7.14995 −2.06706 2.59840
1.11 −2.53833 1.49333 4.44314 −1.00000 −3.79058 3.83094 −6.20150 −0.769952 2.53833
1.12 −2.51610 −3.34886 4.33077 −1.00000 8.42608 3.53380 −5.86445 8.21488 2.51610
1.13 −2.50554 −0.175735 4.27775 −1.00000 0.440311 3.32352 −5.70700 −2.96912 2.50554
1.14 −2.49958 −1.75258 4.24791 −1.00000 4.38073 1.57689 −5.61884 0.0715494 2.49958
1.15 −2.46599 −2.26246 4.08113 −1.00000 5.57921 −2.99607 −5.13205 2.11871 2.46599
1.16 −2.41054 1.33575 3.81069 −1.00000 −3.21988 −0.579711 −4.36473 −1.21576 2.41054
1.17 −2.37811 2.67843 3.65539 −1.00000 −6.36960 −2.27861 −3.93671 4.17400 2.37811
1.18 −2.33944 3.10966 3.47296 −1.00000 −7.27485 −2.57808 −3.44589 6.66998 2.33944
1.19 −2.32435 1.69451 3.40258 −1.00000 −3.93862 3.87404 −3.26009 −0.128648 2.32435
1.20 −2.29844 3.20612 3.28283 −1.00000 −7.36907 2.27898 −2.94851 7.27918 2.29844
See next 80 embeddings (of 141 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.141
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.d ✓ 141
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8045.2.a.d ✓ 141 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{141} + 8 T_{2}^{140} - 188 T_{2}^{139} - 1657 T_{2}^{138} + 16906 T_{2}^{137} + 167329 T_{2}^{136} + \cdots - 1020397287 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\). Copy content Toggle raw display