↑

Properties

Label 8038.2.a.d
Level $8038$
Weight $2$
Character orbit 8038.a
Self dual yes
Analytic conductor $64.184$
Analytic rank $0$
Dimension $92$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8038,2,Mod(1,8038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8038, 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("8038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8038 = 2 \cdot 4019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8038.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: \(64.1837531447\)
Analytic rank: \(0\)
Dimension: \(92\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 92 q + 92 q^{2} + 31 q^{3} + 92 q^{4} + 28 q^{5} + 31 q^{6} + 29 q^{7} + 92 q^{8} + 113 q^{9} + 28 q^{10} + 37 q^{11} + 31 q^{12} + 20 q^{13} + 29 q^{14} + 30 q^{15} + 92 q^{16} + 52 q^{17} + 113 q^{18} + 61 q^{19} + 28 q^{20} + 5 q^{21} + 37 q^{22} + 71 q^{23} + 31 q^{24} + 118 q^{25} + 20 q^{26} + 112 q^{27} + 29 q^{28} + 30 q^{29} + 30 q^{30} + 89 q^{31} + 92 q^{32} + 52 q^{33} + 52 q^{34} + 58 q^{35} + 113 q^{36} + 15 q^{37} + 61 q^{38} + 43 q^{39} + 28 q^{40} + 75 q^{41} + 5 q^{42} + 46 q^{43} + 37 q^{44} + 63 q^{45} + 71 q^{46} + 92 q^{47} + 31 q^{48} + 131 q^{49} + 118 q^{50} + 45 q^{51} + 20 q^{52} + 72 q^{53} + 112 q^{54} + 86 q^{55} + 29 q^{56} + 44 q^{57} + 30 q^{58} + 95 q^{59} + 30 q^{60} - 4 q^{61} + 89 q^{62} + 67 q^{63} + 92 q^{64} + 55 q^{65} + 52 q^{66} + 40 q^{67} + 52 q^{68} + 25 q^{69} + 58 q^{70} + 84 q^{71} + 113 q^{72} + 87 q^{73} + 15 q^{74} + 132 q^{75} + 61 q^{76} + 96 q^{77} + 43 q^{78} + 68 q^{79} + 28 q^{80} + 156 q^{81} + 75 q^{82} + 120 q^{83} + 5 q^{84} - 14 q^{85} + 46 q^{86} + 73 q^{87} + 37 q^{88} + 86 q^{89} + 63 q^{90} + 93 q^{91} + 71 q^{92} + 29 q^{93} + 92 q^{94} + 67 q^{95} + 31 q^{96} + 65 q^{97} + 131 q^{98} + 94 q^{99}+O(q^{100}) \) 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 −3.24576 1.00000 −1.41185 −3.24576 −1.89864 1.00000 7.53496 −1.41185
1.2 1.00000 −3.23664 1.00000 3.20472 −3.23664 2.34255 1.00000 7.47583 3.20472
1.3 1.00000 −3.14562 1.00000 −0.303255 −3.14562 2.95430 1.00000 6.89495 −0.303255
1.4 1.00000 −2.82708 1.00000 −2.48763 −2.82708 −1.73566 1.00000 4.99240 −2.48763
1.5 1.00000 −2.74439 1.00000 2.72118 −2.74439 4.78598 1.00000 4.53169 2.72118
1.6 1.00000 −2.70157 1.00000 0.143003 −2.70157 −2.92792 1.00000 4.29850 0.143003
1.7 1.00000 −2.67211 1.00000 1.54258 −2.67211 1.53058 1.00000 4.14019 1.54258
1.8 1.00000 −2.65852 1.00000 2.22462 −2.65852 0.870094 1.00000 4.06774 2.22462
1.9 1.00000 −2.58627 1.00000 −0.666816 −2.58627 −1.51739 1.00000 3.68880 −0.666816
1.10 1.00000 −2.54065 1.00000 −2.68248 −2.54065 5.07030 1.00000 3.45490 −2.68248
1.11 1.00000 −2.45077 1.00000 −1.39013 −2.45077 4.27652 1.00000 3.00630 −1.39013
1.12 1.00000 −2.41917 1.00000 2.79360 −2.41917 0.842715 1.00000 2.85238 2.79360
1.13 1.00000 −2.40015 1.00000 −4.19507 −2.40015 1.33387 1.00000 2.76072 −4.19507
1.14 1.00000 −2.39141 1.00000 3.31334 −2.39141 −3.74056 1.00000 2.71882 3.31334
1.15 1.00000 −2.38521 1.00000 −2.77606 −2.38521 −1.22030 1.00000 2.68923 −2.77606
1.16 1.00000 −2.15753 1.00000 −0.384990 −2.15753 −2.87024 1.00000 1.65493 −0.384990
1.17 1.00000 −1.94937 1.00000 2.21034 −1.94937 3.83725 1.00000 0.800030 2.21034
1.18 1.00000 −1.90955 1.00000 4.18412 −1.90955 −2.40151 1.00000 0.646387 4.18412
1.19 1.00000 −1.78655 1.00000 0.381834 −1.78655 1.78538 1.00000 0.191778 0.381834
1.20 1.00000 −1.75958 1.00000 2.55886 −1.75958 −0.837744 1.00000 0.0961261 2.55886
See all 92 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.92
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(4019\) \(1\)

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 8038.2.a.d ✓ 92
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8038.2.a.d ✓ 92 1.a even 1 1 trivial