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Properties

Label 8018.2.a.e
Level $8018$
Weight $2$
Character orbit 8018.a
Self dual yes
Analytic conductor $64.024$
Analytic rank $1$
Dimension $32$
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] = [8018,2,Mod(1,8018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8018, 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("8018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8018.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: \(64.0240523407\)
Analytic rank: \(1\)
Dimension: \(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} - 7 q^{3} + 32 q^{4} - 6 q^{5} - 7 q^{6} - 5 q^{7} + 32 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{2} - 7 q^{3} + 32 q^{4} - 6 q^{5} - 7 q^{6} - 5 q^{7} + 32 q^{8} + 15 q^{9} - 6 q^{10} - 21 q^{11} - 7 q^{12} - 19 q^{13} - 5 q^{14} - 14 q^{15} + 32 q^{16} - 14 q^{17} + 15 q^{18} - 32 q^{19} - 6 q^{20} - 26 q^{21} - 21 q^{22} - 13 q^{23} - 7 q^{24} - 10 q^{25} - 19 q^{26} - 25 q^{27} - 5 q^{28} - 42 q^{29} - 14 q^{30} - 15 q^{31} + 32 q^{32} - 32 q^{33} - 14 q^{34} - 22 q^{35} + 15 q^{36} - 54 q^{37} - 32 q^{38} - 32 q^{39} - 6 q^{40} - 16 q^{41} - 26 q^{42} - 37 q^{43} - 21 q^{44} - 46 q^{45} - 13 q^{46} - 9 q^{47} - 7 q^{48} - 7 q^{49} - 10 q^{50} - 32 q^{51} - 19 q^{52} - 53 q^{53} - 25 q^{54} - 17 q^{55} - 5 q^{56} + 7 q^{57} - 42 q^{58} - 34 q^{59} - 14 q^{60} - 33 q^{61} - 15 q^{62} + 18 q^{63} + 32 q^{64} - 50 q^{65} - 32 q^{66} - 53 q^{67} - 14 q^{68} - 40 q^{69} - 22 q^{70} - 27 q^{71} + 15 q^{72} - 43 q^{73} - 54 q^{74} + 5 q^{75} - 32 q^{76} - 56 q^{77} - 32 q^{78} - 11 q^{79} - 6 q^{80} - 8 q^{81} - 16 q^{82} - 17 q^{83} - 26 q^{84} - 30 q^{85} - 37 q^{86} + 3 q^{87} - 21 q^{88} - 21 q^{89} - 46 q^{90} - 67 q^{91} - 13 q^{92} - 31 q^{93} - 9 q^{94} + 6 q^{95} - 7 q^{96} - 51 q^{97} - 7 q^{98} - 32 q^{99}+O(q^{100}) \) 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 −3.42265 1.00000 −0.619065 −3.42265 1.32007 1.00000 8.71451 −0.619065
1.2 1.00000 −2.94867 1.00000 2.45411 −2.94867 1.73401 1.00000 5.69467 2.45411
1.3 1.00000 −2.87951 1.00000 −0.727191 −2.87951 −4.06984 1.00000 5.29156 −0.727191
1.4 1.00000 −2.65255 1.00000 −0.493279 −2.65255 4.58344 1.00000 4.03601 −0.493279
1.5 1.00000 −2.56478 1.00000 −2.25248 −2.56478 1.14049 1.00000 3.57812 −2.25248
1.6 1.00000 −2.29667 1.00000 −2.85656 −2.29667 3.49764 1.00000 2.27470 −2.85656
1.7 1.00000 −1.99099 1.00000 3.02780 −1.99099 −1.37345 1.00000 0.964042 3.02780
1.8 1.00000 −1.92818 1.00000 −0.990094 −1.92818 2.57830 1.00000 0.717864 −0.990094
1.9 1.00000 −1.89970 1.00000 0.525736 −1.89970 −0.379709 1.00000 0.608871 0.525736
1.10 1.00000 −1.64186 1.00000 2.36809 −1.64186 −2.99072 1.00000 −0.304289 2.36809
1.11 1.00000 −1.43969 1.00000 −3.67207 −1.43969 −3.77033 1.00000 −0.927290 −3.67207
1.12 1.00000 −1.26197 1.00000 2.06188 −1.26197 −2.59785 1.00000 −1.40744 2.06188
1.13 1.00000 −0.617240 1.00000 −3.87436 −0.617240 −1.63807 1.00000 −2.61901 −3.87436
1.14 1.00000 −0.547499 1.00000 −0.949125 −0.547499 0.838427 1.00000 −2.70024 −0.949125
1.15 1.00000 −0.474859 1.00000 1.84074 −0.474859 4.63773 1.00000 −2.77451 1.84074
1.16 1.00000 −0.175124 1.00000 3.03871 −0.175124 −1.35318 1.00000 −2.96933 3.03871
1.17 1.00000 −0.0920586 1.00000 2.87697 −0.0920586 2.64672 1.00000 −2.99153 2.87697
1.18 1.00000 0.124655 1.00000 −2.23561 0.124655 3.94220 1.00000 −2.98446 −2.23561
1.19 1.00000 0.137782 1.00000 −1.40913 0.137782 −4.42829 1.00000 −2.98102 −1.40913
1.20 1.00000 0.281409 1.00000 0.922067 0.281409 −0.498686 1.00000 −2.92081 0.922067
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.32
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(1\)
\(211\) \(-1\)

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 8018.2.a.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8018.2.a.e ✓ 32 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{32} + 7 T_{3}^{31} - 31 T_{3}^{30} - 309 T_{3}^{29} + 261 T_{3}^{28} + 5966 T_{3}^{27} + \cdots - 26 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8018))\). Copy content Toggle raw display