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Properties

Label 8042.2.a.d
Level $8042$
Weight $2$
Character orbit 8042.a
Self dual yes
Analytic conductor $64.216$
Analytic rank $0$
Dimension $101$
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: \(0\)
Dimension: \(101\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 101 q + 101 q^{2} + 10 q^{3} + 101 q^{4} + 19 q^{5} + 10 q^{6} + 42 q^{7} + 101 q^{8} + 147 q^{9} + 19 q^{10} + 4 q^{11} + 10 q^{12} + 58 q^{13} + 42 q^{14} + 27 q^{15} + 101 q^{16} + 34 q^{17} + 147 q^{18} + 36 q^{19} + 19 q^{20} + 45 q^{21} + 4 q^{22} + 47 q^{23} + 10 q^{24} + 174 q^{25} + 58 q^{26} + 31 q^{27} + 42 q^{28} + 62 q^{29} + 27 q^{30} + 47 q^{31} + 101 q^{32} + 55 q^{33} + 34 q^{34} + 16 q^{35} + 147 q^{36} + 90 q^{37} + 36 q^{38} + 50 q^{39} + 19 q^{40} + 54 q^{41} + 45 q^{42} + 65 q^{43} + 4 q^{44} + 47 q^{45} + 47 q^{46} + 54 q^{47} + 10 q^{48} + 189 q^{49} + 174 q^{50} + 36 q^{51} + 58 q^{52} + 94 q^{53} + 31 q^{54} + 68 q^{55} + 42 q^{56} + 79 q^{57} + 62 q^{58} - 6 q^{59} + 27 q^{60} + 58 q^{61} + 47 q^{62} + 117 q^{63} + 101 q^{64} + 89 q^{65} + 55 q^{66} + 127 q^{67} + 34 q^{68} + 45 q^{69} + 16 q^{70} + 87 q^{71} + 147 q^{72} + 83 q^{73} + 90 q^{74} - 4 q^{75} + 36 q^{76} + 53 q^{77} + 50 q^{78} + 74 q^{79} + 19 q^{80} + 241 q^{81} + 54 q^{82} + 11 q^{83} + 45 q^{84} + 120 q^{85} + 65 q^{86} + 37 q^{87} + 4 q^{88} + 89 q^{89} + 47 q^{90} + 31 q^{91} + 47 q^{92} + 123 q^{93} + 54 q^{94} + 61 q^{95} + 10 q^{96} + 85 q^{97} + 189 q^{98} - 55 q^{99}+O(q^{100}) \) 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 −3.43699 1.00000 −1.87422 −3.43699 −3.00439 1.00000 8.81289 −1.87422
1.2 1.00000 −3.38233 1.00000 3.47698 −3.38233 4.68177 1.00000 8.44012 3.47698
1.3 1.00000 −3.33232 1.00000 −1.78644 −3.33232 −0.0456798 1.00000 8.10435 −1.78644
1.4 1.00000 −3.21463 1.00000 −0.910313 −3.21463 2.06915 1.00000 7.33384 −0.910313
1.5 1.00000 −3.07090 1.00000 3.36444 −3.07090 2.10143 1.00000 6.43042 3.36444
1.6 1.00000 −3.06519 1.00000 −3.42202 −3.06519 2.44027 1.00000 6.39539 −3.42202
1.7 1.00000 −3.05772 1.00000 1.21871 −3.05772 −4.97978 1.00000 6.34964 1.21871
1.8 1.00000 −3.03040 1.00000 −4.23996 −3.03040 3.85031 1.00000 6.18334 −4.23996
1.9 1.00000 −3.00872 1.00000 1.27297 −3.00872 −1.78578 1.00000 6.05237 1.27297
1.10 1.00000 −2.92582 1.00000 4.30943 −2.92582 −1.07433 1.00000 5.56042 4.30943
1.11 1.00000 −2.78808 1.00000 1.68662 −2.78808 −0.478055 1.00000 4.77338 1.68662
1.12 1.00000 −2.74700 1.00000 1.34490 −2.74700 3.95509 1.00000 4.54602 1.34490
1.13 1.00000 −2.72397 1.00000 −3.81508 −2.72397 −1.37701 1.00000 4.42003 −3.81508
1.14 1.00000 −2.64364 1.00000 4.09219 −2.64364 −2.76746 1.00000 3.98881 4.09219
1.15 1.00000 −2.58579 1.00000 −4.00078 −2.58579 −2.06018 1.00000 3.68634 −4.00078
1.16 1.00000 −2.42928 1.00000 2.55543 −2.42928 4.80532 1.00000 2.90141 2.55543
1.17 1.00000 −2.42720 1.00000 2.10259 −2.42720 2.07633 1.00000 2.89131 2.10259
1.18 1.00000 −2.39327 1.00000 3.11355 −2.39327 −5.17926 1.00000 2.72774 3.11355
1.19 1.00000 −2.31266 1.00000 −2.37478 −2.31266 −3.49882 1.00000 2.34838 −2.37478
1.20 1.00000 −2.18140 1.00000 −1.90416 −2.18140 1.71599 1.00000 1.75849 −1.90416
See next 80 embeddings (of 101 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.101
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.d ✓ 101
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8042.2.a.d ✓ 101 1.a even 1 1 trivial