Properties

Label 8041.2.a.g
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $1$
Dimension $69$
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: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q - 11 q^{2} - 3 q^{3} + 65 q^{4} - 6 q^{5} - 10 q^{6} - 11 q^{7} - 33 q^{8} + 56 q^{9} - q^{10} - 69 q^{11} - 3 q^{12} - 28 q^{13} - 15 q^{14} - 45 q^{15} + 53 q^{16} + 69 q^{17} - 17 q^{18} - 32 q^{19} - 21 q^{20} - 38 q^{21} + 11 q^{22} - 41 q^{23} - 11 q^{24} + 67 q^{25} - 6 q^{26} - 3 q^{27} - 21 q^{28} - 22 q^{29} - 22 q^{30} - 27 q^{31} - 87 q^{32} + 3 q^{33} - 11 q^{34} - 44 q^{35} + 59 q^{36} + 24 q^{37} - 22 q^{38} - 59 q^{39} + q^{40} - 43 q^{41} - 15 q^{42} + 69 q^{43} - 65 q^{44} - 12 q^{45} - 21 q^{46} - 99 q^{47} + 2 q^{48} + 64 q^{49} - 78 q^{50} - 3 q^{51} - 57 q^{52} - 50 q^{53} + 20 q^{54} + 6 q^{55} - 59 q^{56} - 15 q^{57} + 22 q^{58} - 82 q^{59} - 86 q^{60} - 24 q^{61} + 15 q^{62} - 63 q^{63} + 63 q^{64} - 23 q^{65} + 10 q^{66} - 54 q^{67} + 65 q^{68} + 36 q^{69} + 9 q^{70} - 128 q^{71} - 69 q^{72} + 2 q^{73} - 58 q^{74} - 31 q^{75} - 76 q^{76} + 11 q^{77} - 19 q^{78} - 43 q^{79} - 19 q^{80} + 49 q^{81} - 2 q^{82} - 62 q^{83} - 82 q^{84} - 6 q^{85} - 11 q^{86} - 62 q^{87} + 33 q^{88} - 49 q^{89} - 37 q^{90} - 2 q^{91} - 96 q^{92} - 29 q^{93} - 75 q^{94} - 133 q^{95} - 86 q^{96} + 5 q^{97} - 72 q^{98} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.81130 −0.519316 5.90341 −3.87319 1.45995 3.50752 −10.9737 −2.73031 10.8887
1.2 −2.75821 0.843025 5.60771 3.43649 −2.32524 −0.344850 −9.95082 −2.28931 −9.47854
1.3 −2.75592 2.83072 5.59510 −1.76173 −7.80125 2.47832 −9.90779 5.01300 4.85519
1.4 −2.67122 2.62074 5.13539 0.949400 −7.00056 −4.03564 −8.37532 3.86827 −2.53605
1.5 −2.56568 −3.40788 4.58273 3.86072 8.74353 2.70727 −6.62646 8.61362 −9.90537
1.6 −2.53420 −1.82986 4.42217 −1.56039 4.63723 −0.999486 −6.13827 0.348383 3.95435
1.7 −2.47238 −0.879695 4.11267 −2.32585 2.17494 1.92542 −5.22332 −2.22614 5.75038
1.8 −2.45687 −0.886506 4.03623 −0.806805 2.17803 −3.77933 −5.00275 −2.21411 1.98222
1.9 −2.42698 0.265461 3.89025 3.56663 −0.644270 −2.27175 −4.58762 −2.92953 −8.65616
1.10 −2.20984 0.750857 2.88340 0.114086 −1.65928 4.65666 −1.95217 −2.43621 −0.252113
1.11 −2.20191 −3.02493 2.84839 −2.51341 6.66061 0.0943373 −1.86807 6.15021 5.53430
1.12 −2.12129 1.35005 2.49986 −1.60395 −2.86383 −4.53308 −1.06034 −1.17738 3.40244
1.13 −2.10431 −0.209589 2.42810 1.85985 0.441039 3.06120 −0.900860 −2.95607 −3.91370
1.14 −2.02215 2.02595 2.08909 −3.90857 −4.09678 −0.616656 −0.180153 1.10448 7.90371
1.15 −1.87945 −2.61165 1.53233 −1.23543 4.90847 −3.71692 0.878969 3.82074 2.32194
1.16 −1.85886 3.31260 1.45535 0.943187 −6.15766 −1.28029 1.01243 7.97335 −1.75325
1.17 −1.71913 3.18493 0.955421 −4.00741 −5.47533 −0.822562 1.79577 7.14381 6.88927
1.18 −1.69256 −2.67363 0.864767 3.80511 4.52529 −4.13916 1.92145 4.14830 −6.44038
1.19 −1.67732 −1.15146 0.813414 0.864739 1.93138 4.27290 1.99029 −1.67413 −1.45045
1.20 −1.66399 1.84747 0.768874 1.24336 −3.07417 0.0311850 2.04859 0.413132 −2.06894
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
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.g 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.g 69 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{69} + 11 T_{2}^{68} - 41 T_{2}^{67} - 869 T_{2}^{66} - 346 T_{2}^{65} + 31814 T_{2}^{64} + 65777 T_{2}^{63} - 711205 T_{2}^{62} - 2329125 T_{2}^{61} + 10709178 T_{2}^{60} + 48838922 T_{2}^{59} - 111493168 T_{2}^{58} + \cdots + 5248 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display