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Properties

Label 8041.2.a.c
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $1$
Dimension $60$
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] = [8041,2,Mod(1,8041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8041.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: \(64.2077082653\)
Analytic rank: \(1\)
Dimension: \(60\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9} + q^{10} + 60 q^{11} - 13 q^{12} - 2 q^{13} - 15 q^{14} - 32 q^{15} + 21 q^{16} + 60 q^{17} - 41 q^{18} - 24 q^{19} - 33 q^{20} - 12 q^{21} - 9 q^{22} - 20 q^{23} + 9 q^{24} + 25 q^{25} - 40 q^{26} - 18 q^{27} - 3 q^{28} - 54 q^{29} - 18 q^{30} - 50 q^{31} - 51 q^{32} - 6 q^{33} - 9 q^{34} - 30 q^{35} + 27 q^{36} - 63 q^{37} - 38 q^{38} - 55 q^{39} - 5 q^{40} - 53 q^{41} - 47 q^{42} - 60 q^{43} + 43 q^{44} - 36 q^{45} - 27 q^{46} - 47 q^{47} - 38 q^{48} + 5 q^{49} - 4 q^{50} - 6 q^{51} - 13 q^{52} - 24 q^{53} - 58 q^{54} - 15 q^{55} - 45 q^{56} + 23 q^{57} + 16 q^{58} - 57 q^{59} - 4 q^{60} - 8 q^{61} - 3 q^{62} - 57 q^{63} - 5 q^{64} - 27 q^{65} - 4 q^{66} - 49 q^{67} + 43 q^{68} - 57 q^{69} - 7 q^{70} - 151 q^{71} - 29 q^{72} - 12 q^{73} + 18 q^{74} - 23 q^{75} - 38 q^{76} - 17 q^{77} + 37 q^{78} - 11 q^{79} - 21 q^{80} + 28 q^{81} + 44 q^{82} - 42 q^{83} + 16 q^{84} - 15 q^{85} + 9 q^{86} - 56 q^{87} - 21 q^{88} - 88 q^{89} + 47 q^{90} - 60 q^{91} + 26 q^{92} - 16 q^{93} + 37 q^{94} - 57 q^{95} + 108 q^{96} - 22 q^{97} + 8 q^{98} + 40 q^{99}+O(q^{100}) \) 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 −2.78000 −1.81181 5.72839 1.28422 5.03683 3.89203 −10.3649 0.282655 −3.57012
1.2 −2.66007 −2.34749 5.07599 −2.77927 6.24449 0.942946 −8.18236 2.51070 7.39305
1.3 −2.54628 1.17861 4.48352 1.63315 −3.00107 −0.708864 −6.32373 −1.61088 −4.15845
1.4 −2.53062 2.95421 4.40406 −1.92556 −7.47600 2.94104 −6.08377 5.72738 4.87287
1.5 −2.51152 −1.72847 4.30774 −3.07277 4.34109 −4.07006 −5.79593 −0.0123937 7.71734
1.6 −2.42660 2.76366 3.88839 −0.843592 −6.70630 −2.86536 −4.58236 4.63781 2.04706
1.7 −2.38636 −2.80769 3.69472 2.77009 6.70017 −2.49366 −4.04422 4.88313 −6.61043
1.8 −2.36766 0.323518 3.60581 −3.56257 −0.765980 3.82937 −3.80202 −2.89534 8.43496
1.9 −2.17121 0.274303 2.71415 −0.625848 −0.595570 −3.36796 −1.55058 −2.92476 1.35885
1.10 −2.12431 0.433471 2.51268 3.47058 −0.920826 2.89316 −1.08909 −2.81210 −7.37258
1.11 −2.07976 −1.74571 2.32541 −0.111413 3.63066 −0.659970 −0.676772 0.0475049 0.231713
1.12 −1.94215 1.51854 1.77194 3.53502 −2.94923 −1.41025 0.442922 −0.694041 −6.86554
1.13 −1.83493 1.92990 1.36698 0.407070 −3.54123 0.437202 1.16155 0.724496 −0.746946
1.14 −1.57436 0.0662834 0.478605 −1.53153 −0.104354 0.748849 2.39522 −2.99561 2.41117
1.15 −1.55010 2.25408 0.402812 −2.41752 −3.49405 2.04645 2.47580 2.08087 3.74740
1.16 −1.50893 −3.04602 0.276872 −1.87360 4.59624 −2.31151 2.60008 6.27825 2.82713
1.17 −1.49915 −0.148971 0.247445 −2.56328 0.223329 2.63900 2.62734 −2.97781 3.84274
1.18 −1.33974 3.18054 −0.205105 0.454728 −4.26109 −3.38234 2.95426 7.11585 −0.609216
1.19 −1.32373 −3.06819 −0.247749 3.29067 4.06144 −2.61144 2.97540 6.41379 −4.35594
1.20 −1.23062 −0.724033 −0.485567 2.55129 0.891011 −2.06260 3.05880 −2.47578 −3.13968
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.60
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(17\) \(-1\)
\(43\) \(1\)

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 8041.2.a.c ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.c ✓ 60 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + 9 T_{2}^{59} - 41 T_{2}^{58} - 593 T_{2}^{57} + 275 T_{2}^{56} + 18073 T_{2}^{55} + \cdots - 2858 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display