Properties

Label 8015.2.a.l
Level $8015$
Weight $2$
Character orbit 8015.a
Self dual yes
Analytic conductor $64.000$
Analytic rank $0$
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] = [8015,2,Mod(1,8015)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, 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("8015.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8015 = 5 \cdot 7 \cdot 229 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8015.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: \(64.0000972201\)
Analytic rank: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 62 q + 2 q^{2} + 11 q^{3} + 64 q^{4} - 62 q^{5} + 3 q^{6} - 62 q^{7} + 15 q^{8} + 69 q^{9} - 2 q^{10} - 13 q^{11} + 37 q^{12} + 31 q^{13} - 2 q^{14} - 11 q^{15} + 64 q^{16} + 30 q^{17} + 18 q^{18} + 20 q^{19} - 64 q^{20} - 11 q^{21} + 7 q^{22} + 29 q^{24} + 62 q^{25} + 59 q^{27} - 64 q^{28} - 29 q^{29} - 3 q^{30} + 20 q^{31} + 22 q^{32} + 72 q^{33} + 13 q^{34} + 62 q^{35} + 53 q^{36} + 35 q^{37} + 34 q^{38} - 6 q^{39} - 15 q^{40} + 13 q^{41} - 3 q^{42} - 4 q^{43} - 44 q^{44} - 69 q^{45} - 19 q^{46} + 58 q^{47} + 64 q^{48} + 62 q^{49} + 2 q^{50} - 30 q^{51} + 82 q^{52} + 18 q^{53} + 22 q^{54} + 13 q^{55} - 15 q^{56} + 21 q^{57} + 18 q^{58} - 11 q^{59} - 37 q^{60} + 24 q^{61} + 48 q^{62} - 69 q^{63} + 65 q^{64} - 31 q^{65} + 25 q^{66} - 6 q^{67} + 65 q^{68} + 27 q^{69} + 2 q^{70} - 35 q^{71} + 53 q^{72} + 116 q^{73} - 69 q^{74} + 11 q^{75} + 65 q^{76} + 13 q^{77} + 102 q^{78} - 83 q^{79} - 64 q^{80} + 126 q^{81} + 71 q^{82} + 84 q^{83} - 37 q^{84} - 30 q^{85} + 24 q^{86} + 49 q^{87} + 20 q^{88} - 16 q^{89} - 18 q^{90} - 31 q^{91} + 19 q^{92} + 65 q^{93} + 54 q^{94} - 20 q^{95} + 17 q^{96} + 155 q^{97} + 2 q^{98} + 6 q^{99}+O(q^{100}) \) 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 −2.78055 0.298033 5.73146 −1.00000 −0.828697 −1.00000 −10.3755 −2.91118 2.78055
1.2 −2.65622 −1.51381 5.05553 −1.00000 4.02102 −1.00000 −8.11618 −0.708384 2.65622
1.3 −2.61699 0.399813 4.84863 −1.00000 −1.04630 −1.00000 −7.45482 −2.84015 2.61699
1.4 −2.48664 3.28438 4.18340 −1.00000 −8.16709 −1.00000 −5.42933 7.78717 2.48664
1.5 −2.41590 1.84179 3.83658 −1.00000 −4.44959 −1.00000 −4.43701 0.392199 2.41590
1.6 −2.35302 1.42181 3.53669 −1.00000 −3.34553 −1.00000 −3.61585 −0.978466 2.35302
1.7 −2.28892 −1.23476 3.23914 −1.00000 2.82625 −1.00000 −2.83628 −1.47538 2.28892
1.8 −2.23328 1.54114 2.98754 −1.00000 −3.44181 −1.00000 −2.20546 −0.624877 2.23328
1.9 −2.20228 −3.01220 2.85005 −1.00000 6.63371 −1.00000 −1.87206 6.07333 2.20228
1.10 −2.17211 3.11306 2.71804 −1.00000 −6.76191 −1.00000 −1.55967 6.69117 2.17211
1.11 −2.12264 −2.04208 2.50559 −1.00000 4.33460 −1.00000 −1.07319 1.17010 2.12264
1.12 −1.90469 −0.920285 1.62783 −1.00000 1.75286 −1.00000 0.708859 −2.15308 1.90469
1.13 −1.87703 −2.63781 1.52324 −1.00000 4.95125 −1.00000 0.894900 3.95806 1.87703
1.14 −1.76681 2.93332 1.12163 −1.00000 −5.18264 −1.00000 1.55191 5.60439 1.76681
1.15 −1.54368 0.772565 0.382940 −1.00000 −1.19259 −1.00000 2.49622 −2.40314 1.54368
1.16 −1.46418 −1.88953 0.143813 −1.00000 2.76661 −1.00000 2.71779 0.570340 1.46418
1.17 −1.44897 −0.334420 0.0995284 −1.00000 0.484566 −1.00000 2.75374 −2.88816 1.44897
1.18 −1.34348 −2.34976 −0.195065 −1.00000 3.15685 −1.00000 2.94902 2.52136 1.34348
1.19 −1.28523 −0.829686 −0.348175 −1.00000 1.06634 −1.00000 3.01795 −2.31162 1.28523
1.20 −1.21148 1.67937 −0.532324 −1.00000 −2.03452 −1.00000 3.06785 −0.179714 1.21148
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
\(5\) \(1\)
\(7\) \(1\)
\(229\) \(-1\)

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 8015.2.a.l 62
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8015.2.a.l 62 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8015))\):

\( T_{2}^{62} - 2 T_{2}^{61} - 92 T_{2}^{60} + 179 T_{2}^{59} + 4010 T_{2}^{58} - 7570 T_{2}^{57} - 110189 T_{2}^{56} + 201232 T_{2}^{55} + 2142834 T_{2}^{54} - 3773350 T_{2}^{53} - 31384856 T_{2}^{52} + 53093437 T_{2}^{51} + \cdots - 42758 \) Copy content Toggle raw display
\( T_{3}^{62} - 11 T_{3}^{61} - 67 T_{3}^{60} + 1139 T_{3}^{59} + 1017 T_{3}^{58} - 54828 T_{3}^{57} + 54202 T_{3}^{56} + 1627229 T_{3}^{55} - 3404316 T_{3}^{54} - 33266881 T_{3}^{53} + 98204795 T_{3}^{52} + \cdots - 82512 \) Copy content Toggle raw display