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Properties

Label 8045.2.a.b
Level $8045$
Weight $2$
Character orbit 8045.a
Self dual yes
Analytic conductor $64.240$
Analytic rank $1$
Dimension $126$
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: \(1\)
Dimension: \(126\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 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 −2.77015 2.48053 5.67370 −1.00000 −6.87144 2.38067 −10.1767 3.15305 2.77015
1.2 −2.68479 −1.35137 5.20811 −1.00000 3.62815 −0.768058 −8.61310 −1.17380 2.68479
1.3 −2.59235 −0.476573 4.72028 −1.00000 1.23544 −4.74079 −7.05191 −2.77288 2.59235
1.4 −2.58262 2.30051 4.66995 −1.00000 −5.94134 −3.25366 −6.89548 2.29232 2.58262
1.5 −2.57758 3.01279 4.64389 −1.00000 −7.76569 −3.99315 −6.81483 6.07689 2.57758
1.6 −2.55817 −0.0316431 4.54425 −1.00000 0.0809484 0.209260 −6.50863 −2.99900 2.55817
1.7 −2.52674 −0.799891 4.38441 −1.00000 2.02112 −0.269880 −6.02477 −2.36017 2.52674
1.8 −2.51572 −0.0214177 4.32884 −1.00000 0.0538809 1.26771 −5.85870 −2.99954 2.51572
1.9 −2.44573 0.659690 3.98158 −1.00000 −1.61342 −0.125305 −4.84641 −2.56481 2.44573
1.10 −2.44115 2.21626 3.95922 −1.00000 −5.41023 1.20932 −4.78275 1.91181 2.44115
1.11 −2.40759 −3.16484 3.79648 −1.00000 7.61962 −1.46420 −4.32518 7.01619 2.40759
1.12 −2.37056 −2.51302 3.61953 −1.00000 5.95725 3.42592 −3.83919 3.31526 2.37056
1.13 −2.28169 −1.33238 3.20613 −1.00000 3.04008 −1.04082 −2.75203 −1.22477 2.28169
1.14 −2.25161 −2.78948 3.06974 −1.00000 6.28081 1.00140 −2.40864 4.78119 2.25161
1.15 −2.22019 1.33183 2.92925 −1.00000 −2.95691 2.14660 −2.06312 −1.22624 2.22019
1.16 −2.20326 −1.91505 2.85434 −1.00000 4.21934 2.75796 −1.88232 0.667415 2.20326
1.17 −2.11951 0.398701 2.49230 −1.00000 −0.845049 −4.30325 −1.04344 −2.84104 2.11951
1.18 −2.05998 0.444613 2.24351 −1.00000 −0.915893 4.66376 −0.501618 −2.80232 2.05998
1.19 −2.03743 2.30287 2.15111 −1.00000 −4.69193 −3.80447 −0.307877 2.30320 2.03743
1.20 −2.02597 −2.39891 2.10454 −1.00000 4.86012 −3.38637 −0.211794 2.75479 2.02597
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.126
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.b ✓ 126
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8045.2.a.b ✓ 126 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{126} - 5 T_{2}^{125} - 168 T_{2}^{124} + 871 T_{2}^{123} + 13679 T_{2}^{122} - 73797 T_{2}^{121} + \cdots - 35354521 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8045))\). Copy content Toggle raw display