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Properties

Label 8042.2.a.a
Level $8042$
Weight $2$
Character orbit 8042.a
Self dual yes
Analytic conductor $64.216$
Analytic rank $1$
Dimension $67$
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] = [8042,2,Mod(1,8042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8042 = 2 \cdot 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8042.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: \(64.2156933055\)
Analytic rank: \(1\)
Dimension: \(67\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 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 1.00000 −3.09236 1.00000 1.36808 −3.09236 −2.10219 1.00000 6.56268 1.36808
1.2 1.00000 −3.08937 1.00000 −1.29634 −3.08937 4.06006 1.00000 6.54422 −1.29634
1.3 1.00000 −3.06256 1.00000 2.88453 −3.06256 −1.43370 1.00000 6.37925 2.88453
1.4 1.00000 −3.05107 1.00000 −1.41145 −3.05107 −1.76739 1.00000 6.30900 −1.41145
1.5 1.00000 −3.01737 1.00000 −1.22094 −3.01737 2.53637 1.00000 6.10451 −1.22094
1.6 1.00000 −2.60859 1.00000 0.145596 −2.60859 0.982128 1.00000 3.80473 0.145596
1.7 1.00000 −2.60646 1.00000 −0.616826 −2.60646 −3.04419 1.00000 3.79362 −0.616826
1.8 1.00000 −2.51956 1.00000 −0.909270 −2.51956 0.00482385 1.00000 3.34819 −0.909270
1.9 1.00000 −2.46688 1.00000 0.597421 −2.46688 −5.04196 1.00000 3.08551 0.597421
1.10 1.00000 −2.36999 1.00000 1.57394 −2.36999 1.59393 1.00000 2.61687 1.57394
1.11 1.00000 −2.32940 1.00000 −2.43485 −2.32940 0.839554 1.00000 2.42611 −2.43485
1.12 1.00000 −2.22608 1.00000 1.44593 −2.22608 1.32066 1.00000 1.95543 1.44593
1.13 1.00000 −2.16681 1.00000 −2.63325 −2.16681 −3.58663 1.00000 1.69508 −2.63325
1.14 1.00000 −2.07802 1.00000 −3.66993 −2.07802 −3.45056 1.00000 1.31817 −3.66993
1.15 1.00000 −2.04025 1.00000 2.67295 −2.04025 4.19458 1.00000 1.16263 2.67295
1.16 1.00000 −1.98933 1.00000 4.18312 −1.98933 −0.474573 1.00000 0.957423 4.18312
1.17 1.00000 −1.67548 1.00000 −3.27092 −1.67548 1.23025 1.00000 −0.192762 −3.27092
1.18 1.00000 −1.59531 1.00000 2.52318 −1.59531 −1.75596 1.00000 −0.454973 2.52318
1.19 1.00000 −1.55257 1.00000 2.63171 −1.55257 −0.882040 1.00000 −0.589531 2.63171
1.20 1.00000 −1.50743 1.00000 −2.46037 −1.50743 −5.08447 1.00000 −0.727642 −2.46037
See all 67 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.67
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4021\) \(-1\)

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 8042.2.a.a ✓ 67
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8042.2.a.a ✓ 67 1.a even 1 1 trivial