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Properties

Label 8029.2.a.d
Level $8029$
Weight $2$
Character orbit 8029.a
Self dual yes
Analytic conductor $64.112$
Analytic rank $1$
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] = [8029,2,Mod(1,8029)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8029, 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("8029.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8029 = 7 \cdot 31 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8029.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: \(64.1118877829\)
Analytic rank: \(1\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 5 q^{2} - 12 q^{3} + 63 q^{4} - 26 q^{5} - 19 q^{6} + 66 q^{7} - 15 q^{8} + 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 5 q^{2} - 12 q^{3} + 63 q^{4} - 26 q^{5} - 19 q^{6} + 66 q^{7} - 15 q^{8} + 66 q^{9} - 6 q^{10} - 57 q^{11} - 29 q^{12} - 28 q^{13} - 5 q^{14} - 24 q^{15} + 69 q^{16} - 47 q^{17} + 8 q^{18} - 27 q^{19} - 77 q^{20} - 12 q^{21} - 12 q^{22} - 46 q^{23} - 57 q^{24} + 72 q^{25} - 21 q^{26} - 36 q^{27} + 63 q^{28} - 62 q^{29} + 2 q^{30} + 66 q^{31} - 40 q^{32} + 4 q^{33} - 46 q^{34} - 26 q^{35} + 62 q^{36} - 66 q^{37} - 31 q^{38} - 8 q^{39} - 37 q^{40} - 33 q^{41} - 19 q^{42} - 22 q^{43} - 84 q^{44} - 77 q^{45} - 14 q^{46} - 20 q^{47} - 43 q^{48} + 66 q^{49} - 10 q^{50} - 39 q^{51} - 41 q^{52} - 47 q^{53} - 65 q^{54} - 15 q^{55} - 15 q^{56} + 5 q^{57} + 24 q^{58} - 125 q^{59} - 77 q^{60} - 57 q^{61} - 5 q^{62} + 66 q^{63} + 81 q^{64} - 40 q^{65} + 33 q^{66} - 25 q^{67} - 107 q^{68} - 72 q^{69} - 6 q^{70} - 57 q^{71} + 38 q^{72} + 5 q^{73} + 5 q^{74} - 60 q^{75} - 33 q^{76} - 57 q^{77} - 19 q^{78} - 4 q^{79} - 132 q^{80} + 58 q^{81} + 8 q^{82} - 84 q^{83} - 29 q^{84} - 33 q^{85} - 60 q^{86} - 31 q^{87} + 21 q^{88} - 132 q^{89} - 61 q^{90} - 28 q^{91} - 100 q^{92} - 12 q^{93} - 35 q^{94} + 4 q^{95} - 198 q^{96} - 39 q^{97} - 5 q^{98} - 174 q^{99}+O(q^{100}) \) 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 −2.80590 −0.0132385 5.87308 −3.56068 0.0371460 1.00000 −10.8675 −2.99982 9.99092
1.2 −2.73042 −0.103023 5.45518 1.88404 0.281297 1.00000 −9.43408 −2.98939 −5.14422
1.3 −2.69752 3.32820 5.27662 −2.48867 −8.97789 1.00000 −8.83876 8.07691 6.71324
1.4 −2.59988 −2.59912 4.75936 1.38295 6.75739 1.00000 −7.17399 3.75543 −3.59551
1.5 −2.59210 2.40228 4.71899 −0.0624542 −6.22695 1.00000 −7.04789 2.77094 0.161888
1.6 −2.49545 −2.27678 4.22728 −4.17212 5.68159 1.00000 −5.55806 2.18372 10.4113
1.7 −2.47161 0.604832 4.10887 1.32822 −1.49491 1.00000 −5.21230 −2.63418 −3.28285
1.8 −2.29618 −2.98190 3.27246 2.81992 6.84699 1.00000 −2.92179 5.89173 −6.47505
1.9 −2.28651 1.33514 3.22812 −2.86692 −3.05281 1.00000 −2.80810 −1.21739 6.55524
1.10 −2.26372 −0.696084 3.12444 0.596642 1.57574 1.00000 −2.54542 −2.51547 −1.35063
1.11 −2.08149 0.154001 2.33259 3.22599 −0.320550 1.00000 −0.692278 −2.97628 −6.71486
1.12 −2.04712 0.0917560 2.19072 −3.63810 −0.187836 1.00000 −0.390421 −2.99158 7.44763
1.13 −1.93012 2.98089 1.72535 0.211062 −5.75346 1.00000 0.530105 5.88570 −0.407374
1.14 −1.88567 0.960243 1.55574 −3.42212 −1.81070 1.00000 0.837718 −2.07793 6.45298
1.15 −1.84828 −2.46792 1.41613 −1.91230 4.56140 1.00000 1.07916 3.09064 3.53445
1.16 −1.83302 −2.42596 1.35995 −0.162299 4.44682 1.00000 1.17323 2.88528 0.297496
1.17 −1.77612 −1.63185 1.15461 0.973411 2.89836 1.00000 1.50152 −0.337079 −1.72890
1.18 −1.63972 1.31150 0.688696 3.72445 −2.15050 1.00000 2.15018 −1.27996 −6.10708
1.19 −1.30686 2.83406 −0.292114 −4.20735 −3.70372 1.00000 2.99547 5.03190 5.49842
1.20 −1.29935 0.411617 −0.311677 −1.65503 −0.534837 1.00000 3.00369 −2.83057 2.15047
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
\(7\) \(-1\)
\(31\) \(-1\)
\(37\) \(1\)

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 8029.2.a.d ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8029.2.a.d ✓ 66 1.a even 1 1 trivial