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Properties

Label 8038.2.a.a
Level $8038$
Weight $2$
Character orbit 8038.a
Self dual yes
Analytic conductor $64.184$
Analytic rank $1$
Dimension $75$
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] = [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: \(1\)
Dimension: \(75\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 75 q + 75 q^{2} - 30 q^{3} + 75 q^{4} - 29 q^{5} - 30 q^{6} - 31 q^{7} + 75 q^{8} + 55 q^{9} - 29 q^{10} - 30 q^{11} - 30 q^{12} - 23 q^{13} - 31 q^{14} - 22 q^{15} + 75 q^{16} - 48 q^{17} + 55 q^{18} - 58 q^{19} - 29 q^{20} - 5 q^{21} - 30 q^{22} - 79 q^{23} - 30 q^{24} + 42 q^{25} - 23 q^{26} - 108 q^{27} - 31 q^{28} - 39 q^{29} - 22 q^{30} - 95 q^{31} + 75 q^{32} - 44 q^{33} - 48 q^{34} - 60 q^{35} + 55 q^{36} - 16 q^{37} - 58 q^{38} - 57 q^{39} - 29 q^{40} - 85 q^{41} - 5 q^{42} - 55 q^{43} - 30 q^{44} - 48 q^{45} - 79 q^{46} - 88 q^{47} - 30 q^{48} + 26 q^{49} + 42 q^{50} - 39 q^{51} - 23 q^{52} - 79 q^{53} - 108 q^{54} - 78 q^{55} - 31 q^{56} - 40 q^{57} - 39 q^{58} - 74 q^{59} - 22 q^{60} - 21 q^{61} - 95 q^{62} - 97 q^{63} + 75 q^{64} - 63 q^{65} - 44 q^{66} - 59 q^{67} - 48 q^{68} + 3 q^{69} - 60 q^{70} - 72 q^{71} + 55 q^{72} - 91 q^{73} - 16 q^{74} - 95 q^{75} - 58 q^{76} - 60 q^{77} - 57 q^{78} - 64 q^{79} - 29 q^{80} + 47 q^{81} - 85 q^{82} - 105 q^{83} - 5 q^{84} + 14 q^{85} - 55 q^{86} - 75 q^{87} - 30 q^{88} - 78 q^{89} - 48 q^{90} - 89 q^{91} - 79 q^{92} + 27 q^{93} - 88 q^{94} - 53 q^{95} - 30 q^{96} - 79 q^{97} + 26 q^{98} - 57 q^{99}+O(q^{100}) \) 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 −3.39091 1.00000 3.77047 −3.39091 −1.27110 1.00000 8.49828 3.77047
1.2 1.00000 −3.38305 1.00000 1.91300 −3.38305 −4.34480 1.00000 8.44501 1.91300
1.3 1.00000 −3.32095 1.00000 −3.93433 −3.32095 −4.89708 1.00000 8.02870 −3.93433
1.4 1.00000 −3.27949 1.00000 −1.91910 −3.27949 3.63095 1.00000 7.75505 −1.91910
1.5 1.00000 −3.25539 1.00000 −2.85398 −3.25539 2.81527 1.00000 7.59759 −2.85398
1.6 1.00000 −3.20148 1.00000 0.754953 −3.20148 1.04626 1.00000 7.24947 0.754953
1.7 1.00000 −3.17284 1.00000 −1.87632 −3.17284 −0.162262 1.00000 7.06690 −1.87632
1.8 1.00000 −2.93359 1.00000 2.13721 −2.93359 0.512181 1.00000 5.60596 2.13721
1.9 1.00000 −2.85626 1.00000 0.772797 −2.85626 −4.64843 1.00000 5.15824 0.772797
1.10 1.00000 −2.79660 1.00000 −1.17558 −2.79660 1.04192 1.00000 4.82097 −1.17558
1.11 1.00000 −2.59529 1.00000 3.17530 −2.59529 −0.203775 1.00000 3.73551 3.17530
1.12 1.00000 −2.54835 1.00000 −2.91194 −2.54835 −2.80949 1.00000 3.49408 −2.91194
1.13 1.00000 −2.53842 1.00000 2.78579 −2.53842 −2.33236 1.00000 3.44358 2.78579
1.14 1.00000 −2.46931 1.00000 0.696459 −2.46931 −2.94189 1.00000 3.09749 0.696459
1.15 1.00000 −2.32361 1.00000 0.467112 −2.32361 3.12912 1.00000 2.39917 0.467112
1.16 1.00000 −2.28450 1.00000 −2.56431 −2.28450 0.255204 1.00000 2.21893 −2.56431
1.17 1.00000 −2.22920 1.00000 −4.17184 −2.22920 −0.836281 1.00000 1.96935 −4.17184
1.18 1.00000 −2.12788 1.00000 3.19034 −2.12788 2.73143 1.00000 1.52787 3.19034
1.19 1.00000 −2.10614 1.00000 −0.0733427 −2.10614 4.17239 1.00000 1.43582 −0.0733427
1.20 1.00000 −1.99460 1.00000 1.10571 −1.99460 2.73220 1.00000 0.978446 1.10571
See all 75 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.75
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.a ✓ 75
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8038.2.a.a ✓ 75 1.a even 1 1 trivial