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Properties

Label 8041.2.a.f
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $0$
Dimension $66$
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: \(0\)
Dimension: \(66\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 12 q^{2} + 66 q^{4} + 6 q^{5} + 7 q^{6} + 13 q^{7} + 30 q^{8} + 58 q^{9} + 7 q^{10} - 66 q^{11} + 12 q^{12} + 12 q^{13} + 13 q^{14} + 35 q^{15} + 58 q^{16} + 66 q^{17} + 37 q^{18} + 24 q^{19} + 17 q^{20} + 16 q^{21} - 12 q^{22} + 25 q^{23} + 22 q^{24} + 56 q^{25} + 36 q^{26} + 17 q^{28} + 29 q^{29} + 28 q^{30} + 37 q^{31} + 62 q^{32} + 12 q^{34} + 40 q^{35} + 107 q^{36} - 34 q^{37} + 22 q^{38} + 61 q^{39} + 37 q^{40} + 41 q^{41} + 19 q^{42} - 66 q^{43} - 66 q^{44} + 10 q^{45} + 43 q^{46} + 61 q^{47} + 29 q^{48} + 33 q^{49} + 59 q^{50} + 51 q^{52} - 35 q^{53} - 37 q^{54} - 6 q^{55} + 37 q^{56} - 7 q^{57} + 17 q^{58} + 48 q^{59} - 56 q^{60} + q^{61} + 37 q^{62} + 43 q^{63} + 68 q^{64} + 41 q^{65} - 7 q^{66} + 10 q^{67} + 66 q^{68} + 18 q^{69} + 77 q^{70} + 84 q^{71} + 83 q^{72} + 5 q^{73} + 36 q^{74} + 14 q^{75} + 14 q^{76} - 13 q^{77} + 41 q^{78} + 58 q^{79} + 25 q^{80} + 78 q^{81} - 28 q^{82} + 47 q^{83} + 44 q^{84} + 6 q^{85} - 12 q^{86} + 101 q^{87} - 30 q^{88} + 53 q^{89} + q^{90} + 2 q^{91} + 34 q^{92} - 3 q^{93} + 17 q^{94} + 91 q^{95} + 27 q^{96} - 28 q^{97} + 87 q^{98} - 58 q^{99}+O(q^{100}) \) 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 −2.69318 −2.72821 5.25323 0.571799 7.34757 0.634478 −8.76155 4.44313 −1.53996
1.2 −2.64866 1.22889 5.01539 −0.305577 −3.25491 −2.34072 −7.98675 −1.48983 0.809369
1.3 −2.63329 −0.751907 4.93424 0.880982 1.97999 −0.474275 −7.72671 −2.43464 −2.31989
1.4 −2.41177 3.28542 3.81663 −1.23247 −7.92367 1.42521 −4.38130 7.79396 2.97244
1.5 −2.28377 1.73431 3.21562 −2.87299 −3.96077 2.79436 −2.77620 0.00783032 6.56126
1.6 −2.25118 2.59454 3.06780 2.27169 −5.84076 2.89150 −2.40380 3.73163 −5.11397
1.7 −2.23777 −0.164695 3.00762 1.12990 0.368549 0.279747 −2.25482 −2.97288 −2.52847
1.8 −2.16346 −1.64599 2.68057 −3.89271 3.56104 −1.32552 −1.47238 −0.290720 8.42173
1.9 −2.13623 −2.11060 2.56346 3.94938 4.50873 −1.45786 −1.20369 1.45464 −8.43678
1.10 −1.99436 −0.822682 1.97746 −2.77484 1.64072 2.78471 0.0449503 −2.32319 5.53402
1.11 −1.92381 −0.292020 1.70106 −2.65493 0.561792 −3.53150 0.575100 −2.91472 5.10760
1.12 −1.88001 −1.81223 1.53444 2.13867 3.40702 1.99941 0.875255 0.284183 −4.02072
1.13 −1.74475 2.06451 1.04415 3.04620 −3.60205 −4.86539 1.66772 1.26220 −5.31485
1.14 −1.74128 −2.85862 1.03206 −1.07282 4.97766 2.85165 1.68545 5.17170 1.86807
1.15 −1.63713 2.43295 0.680210 −0.845755 −3.98306 −3.87420 2.16067 2.91923 1.38461
1.16 −1.36974 0.455204 −0.123817 −1.28468 −0.623510 0.875632 2.90907 −2.79279 1.75968
1.17 −1.30842 −0.505193 −0.288029 1.89375 0.661007 −3.03323 2.99371 −2.74478 −2.47782
1.18 −1.25548 −0.219421 −0.423759 4.17472 0.275480 −0.198048 3.04299 −2.95185 −5.24130
1.19 −1.23666 0.919561 −0.470672 0.683092 −1.13718 3.69740 3.05538 −2.15441 −0.844753
1.20 −1.16270 −2.59083 −0.648140 −0.596152 3.01234 −2.62369 3.07898 3.71239 0.693143
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
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.f ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.f ✓ 66 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} - 12 T_{2}^{65} - 27 T_{2}^{64} + 874 T_{2}^{63} - 1297 T_{2}^{62} - 28828 T_{2}^{61} + \cdots - 85394 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display