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Properties

Label 8041.2.a.d
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $1$
Dimension $62$
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: \(62\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q - 7 q^{2} - 8 q^{3} + 49 q^{4} - 13 q^{5} - 2 q^{6} - 11 q^{7} - 9 q^{8} + 40 q^{9} - 7 q^{10} + 62 q^{11} - 17 q^{12} - 31 q^{14} - 20 q^{15} + 27 q^{16} - 62 q^{17} + 3 q^{18} - 29 q^{20} - 18 q^{21} - 7 q^{22} - 50 q^{23} - 31 q^{24} + 35 q^{25} - 32 q^{26} - 14 q^{27} - 13 q^{28} - 26 q^{29} - 10 q^{30} - 58 q^{31} - 5 q^{32} - 8 q^{33} + 7 q^{34} - 32 q^{35} - 29 q^{36} - 41 q^{37} - 10 q^{38} - 53 q^{39} - 31 q^{40} - 55 q^{41} - 7 q^{42} + 62 q^{43} + 49 q^{44} - 34 q^{45} - 39 q^{46} - 31 q^{47} - 30 q^{48} + 35 q^{49} - 40 q^{50} + 8 q^{51} + 13 q^{52} - 74 q^{53} + 48 q^{54} - 13 q^{55} - 75 q^{56} - 43 q^{57} - 46 q^{58} - 65 q^{59} - 8 q^{60} - 14 q^{61} - 29 q^{62} - 23 q^{63} - 15 q^{64} - 9 q^{65} - 2 q^{66} - q^{67} - 49 q^{68} - 59 q^{69} - 31 q^{70} - 141 q^{71} + 9 q^{72} - 4 q^{73} - 94 q^{74} - 43 q^{75} + 34 q^{76} - 11 q^{77} - 11 q^{78} - 63 q^{79} - 41 q^{80} - 30 q^{81} + 38 q^{82} - 44 q^{83} - 16 q^{84} + 13 q^{85} - 7 q^{86} - 8 q^{87} - 9 q^{88} - 58 q^{89} - 55 q^{90} - 78 q^{91} - 104 q^{92} - 5 q^{94} - 99 q^{95} - 148 q^{96} - 26 q^{97} + 16 q^{98} + 40 q^{99}+O(q^{100}) \) 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 −2.76025 2.03734 5.61901 −1.74676 −5.62359 0.752232 −9.98938 1.15077 4.82150
1.2 −2.70684 −0.791084 5.32698 2.18902 2.14134 2.25606 −9.00562 −2.37419 −5.92532
1.3 −2.51837 −0.407906 4.34217 −4.11514 1.02726 −1.84586 −5.89846 −2.83361 10.3634
1.4 −2.46781 0.627778 4.09007 2.55941 −1.54924 −1.57206 −5.15790 −2.60589 −6.31614
1.5 −2.41829 −2.95622 3.84815 1.73328 7.14900 3.91504 −4.46936 5.73922 −4.19159
1.6 −2.40656 −2.51592 3.79153 0.539420 6.05472 −2.08610 −4.31144 3.32987 −1.29815
1.7 −2.36585 2.45510 3.59727 1.09940 −5.80841 2.69770 −3.77890 3.02752 −2.60102
1.8 −2.25455 2.07382 3.08298 1.31329 −4.67552 −0.325302 −2.44162 1.30073 −2.96087
1.9 −2.21985 0.267340 2.92774 −2.37360 −0.593455 −4.48408 −2.05944 −2.92853 5.26904
1.10 −2.09677 0.607527 2.39646 −1.49692 −1.27385 −0.898210 −0.831287 −2.63091 3.13870
1.11 −1.99324 1.03243 1.97299 −3.57518 −2.05788 3.51904 0.0538292 −1.93409 7.12618
1.12 −1.97447 −2.45298 1.89852 −4.33084 4.84333 2.34067 0.200361 3.01712 8.55111
1.13 −1.89473 −0.654571 1.59002 2.95213 1.24024 −2.97686 0.776803 −2.57154 −5.59350
1.14 −1.81914 −2.58004 1.30927 −0.309884 4.69345 2.19724 1.25653 3.65659 0.563723
1.15 −1.79824 −0.403049 1.23368 −0.0361831 0.724781 4.76370 1.37802 −2.83755 0.0650661
1.16 −1.70186 −1.87988 0.896329 −0.570066 3.19929 −0.204447 1.87829 0.533941 0.970172
1.17 −1.47469 −1.86783 0.174714 4.42666 2.75448 0.892669 2.69173 0.488806 −6.52796
1.18 −1.44193 2.68271 0.0791741 −0.0677310 −3.86829 1.81686 2.76970 4.19693 0.0976637
1.19 −1.34818 −2.19319 −0.182400 −0.782983 2.95682 −4.69818 2.94228 1.81007 1.05561
1.20 −1.29109 −0.909633 −0.333090 −2.82704 1.17442 −1.79091 3.01223 −2.17257 3.64996
See all 62 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.62
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.d ✓ 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.d ✓ 62 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{62} + 7 T_{2}^{61} - 62 T_{2}^{60} - 536 T_{2}^{59} + 1604 T_{2}^{58} + 19232 T_{2}^{57} + \cdots + 18954 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display