Properties

Label 8619.2.a.bp
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $1$
Dimension $21$
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] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, 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("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(1\)
Dimension: \(21\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 21 q - 9 q^{2} + 21 q^{3} + 29 q^{4} - 26 q^{5} - 9 q^{6} + 5 q^{7} - 24 q^{8} + 21 q^{9} + 12 q^{10} - 37 q^{11} + 29 q^{12} - 6 q^{14} - 26 q^{15} + 33 q^{16} - 21 q^{17} - 9 q^{18} + 6 q^{19} - 53 q^{20} + 5 q^{21} + 17 q^{22} - 8 q^{23} - 24 q^{24} + 57 q^{25} + 21 q^{27} - 5 q^{28} - 6 q^{29} + 12 q^{30} - 9 q^{31} - 37 q^{32} - 37 q^{33} + 9 q^{34} - 8 q^{35} + 29 q^{36} - 9 q^{37} + 11 q^{38} - 8 q^{40} - 50 q^{41} - 6 q^{42} + 18 q^{43} - 67 q^{44} - 26 q^{45} - 23 q^{46} - 71 q^{47} + 33 q^{48} + 46 q^{49} - 2 q^{50} - 21 q^{51} - 14 q^{53} - 9 q^{54} + 29 q^{55} - 17 q^{56} + 6 q^{57} - 37 q^{58} - 59 q^{59} - 53 q^{60} + 44 q^{61} + 2 q^{62} + 5 q^{63} + 44 q^{64} + 17 q^{66} - 8 q^{67} - 29 q^{68} - 8 q^{69} - 15 q^{70} - 60 q^{71} - 24 q^{72} - 13 q^{73} - 14 q^{74} + 57 q^{75} - 15 q^{76} - 28 q^{77} + 51 q^{79} - 131 q^{80} + 21 q^{81} + 18 q^{82} - 30 q^{83} - 5 q^{84} + 26 q^{85} + 15 q^{86} - 6 q^{87} + 71 q^{88} - 62 q^{89} + 12 q^{90} + 20 q^{92} - 9 q^{93} - 72 q^{94} - 16 q^{95} - 37 q^{96} + 6 q^{97} - 58 q^{98} - 37 q^{99}+O(q^{100}) \) 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 −2.72531 1.00000 5.42733 −1.92039 −2.72531 1.85372 −9.34055 1.00000 5.23366
1.2 −2.71200 1.00000 5.35495 −1.71865 −2.71200 −2.50513 −9.09863 1.00000 4.66097
1.3 −2.61367 1.00000 4.83127 2.60872 −2.61367 2.00384 −7.40001 1.00000 −6.81834
1.4 −2.55115 1.00000 4.50837 −4.40886 −2.55115 −0.225599 −6.39922 1.00000 11.2477
1.5 −2.15254 1.00000 2.63345 −4.25374 −2.15254 5.21772 −1.36353 1.00000 9.15637
1.6 −2.02507 1.00000 2.10093 −2.79919 −2.02507 −4.41587 −0.204384 1.00000 5.66856
1.7 −1.89317 1.00000 1.58410 2.58913 −1.89317 −0.953007 0.787374 1.00000 −4.90167
1.8 −1.70893 1.00000 0.920426 −1.13927 −1.70893 −3.51731 1.84491 1.00000 1.94693
1.9 −1.66488 1.00000 0.771826 0.938526 −1.66488 2.88538 2.04476 1.00000 −1.56253
1.10 −0.634677 1.00000 −1.59718 −2.98066 −0.634677 1.67204 2.28305 1.00000 1.89176
1.11 −0.575048 1.00000 −1.66932 −1.38919 −0.575048 4.30260 2.11004 1.00000 0.798850
1.12 −0.296463 1.00000 −1.91211 −3.60484 −0.296463 −4.47069 1.15980 1.00000 1.06870
1.13 0.0282265 1.00000 −1.99920 0.933793 0.0282265 4.39781 −0.112883 1.00000 0.0263577
1.14 0.0929476 1.00000 −1.99136 −3.17289 0.0929476 0.513856 −0.370988 1.00000 −0.294913
1.15 0.886807 1.00000 −1.21357 1.36817 0.886807 −4.28444 −2.84982 1.00000 1.21331
1.16 1.49792 1.00000 0.243764 −3.93316 1.49792 4.20773 −2.63070 1.00000 −5.89156
1.17 1.66522 1.00000 0.772953 2.12863 1.66522 1.32842 −2.04330 1.00000 3.54464
1.18 1.66910 1.00000 0.785901 2.71566 1.66910 −2.36594 −2.02645 1.00000 4.53271
1.19 1.76323 1.00000 1.10898 −0.184346 1.76323 1.70842 −1.57107 1.00000 −0.325045
1.20 2.26272 1.00000 3.11992 −4.00833 2.26272 −1.84015 2.53407 1.00000 −9.06973
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(17\) \(1\)

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 8619.2.a.bp 21
13.b even 2 1 8619.2.a.bq yes 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8619.2.a.bp 21 1.a even 1 1 trivial
8619.2.a.bq yes 21 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{21} + 9 T_{2}^{20} + 5 T_{2}^{19} - 178 T_{2}^{18} - 448 T_{2}^{17} + 1188 T_{2}^{16} + 5228 T_{2}^{15} - 1844 T_{2}^{14} - 27758 T_{2}^{13} - 14479 T_{2}^{12} + 77160 T_{2}^{11} + 81690 T_{2}^{10} - 107543 T_{2}^{9} + \cdots - 13 \) Copy content Toggle raw display
\( T_{5}^{21} + 26 T_{5}^{20} + 257 T_{5}^{19} + 962 T_{5}^{18} - 1955 T_{5}^{17} - 29076 T_{5}^{16} - 68477 T_{5}^{15} + 179294 T_{5}^{14} + 1113521 T_{5}^{13} + 731744 T_{5}^{12} - 6017014 T_{5}^{11} + \cdots - 4794259 \) Copy content Toggle raw display
\( T_{7}^{21} - 5 T_{7}^{20} - 84 T_{7}^{19} + 429 T_{7}^{18} + 2829 T_{7}^{17} - 15091 T_{7}^{16} - 48262 T_{7}^{15} + 280993 T_{7}^{14} + 426193 T_{7}^{13} - 2991801 T_{7}^{12} - 1605566 T_{7}^{11} + 18446283 T_{7}^{10} + \cdots - 3109499 \) Copy content Toggle raw display