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Properties

Label 8619.2.a.bv
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
Dimension $24$
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: \(0\)
Dimension: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 24 q^{3} + 17 q^{4} + 11 q^{5} - q^{6} + 2 q^{7} + 9 q^{8} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + q^{2} - 24 q^{3} + 17 q^{4} + 11 q^{5} - q^{6} + 2 q^{7} + 9 q^{8} + 24 q^{9} - 4 q^{10} - q^{11} - 17 q^{12} + 12 q^{14} - 11 q^{15} - 5 q^{16} + 24 q^{17} + q^{18} + 28 q^{19} + 18 q^{20} - 2 q^{21} - 4 q^{22} + 7 q^{23} - 9 q^{24} + 7 q^{25} - 24 q^{27} + 14 q^{28} - 9 q^{29} + 4 q^{30} + 8 q^{31} + 24 q^{32} + q^{33} + q^{34} + 11 q^{35} + 17 q^{36} + 28 q^{37} - 11 q^{38} + 47 q^{40} + 25 q^{41} - 12 q^{42} + 27 q^{43} - 21 q^{44} + 11 q^{45} + 26 q^{46} - 5 q^{47} + 5 q^{48} + 10 q^{49} - 5 q^{50} - 24 q^{51} + 11 q^{53} - q^{54} - 22 q^{55} + 4 q^{56} - 28 q^{57} + 19 q^{58} + 17 q^{59} - 18 q^{60} - 26 q^{61} + 2 q^{63} - 47 q^{64} + 4 q^{66} + 56 q^{67} + 17 q^{68} - 7 q^{69} - 2 q^{70} - 31 q^{71} + 9 q^{72} + 28 q^{73} + 46 q^{74} - 7 q^{75} + 74 q^{76} - 39 q^{77} + 18 q^{79} + 34 q^{80} + 24 q^{81} + 23 q^{82} + 8 q^{83} - 14 q^{84} + 11 q^{85} - 14 q^{86} + 9 q^{87} - 6 q^{88} + 46 q^{89} - 4 q^{90} + 38 q^{92} - 8 q^{93} - 65 q^{94} + 74 q^{95} - 24 q^{96} + 36 q^{97} + 33 q^{98} - q^{99}+O(q^{100}) \) 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 −2.47129 −1.00000 4.10727 −0.571649 2.47129 2.89293 −5.20767 1.00000 1.41271
1.2 −2.25452 −1.00000 3.08288 0.0750544 2.25452 −2.08802 −2.44137 1.00000 −0.169212
1.3 −2.10927 −1.00000 2.44902 4.41670 2.10927 0.449151 −0.947100 1.00000 −9.31601
1.4 −1.94744 −1.00000 1.79254 −2.23167 1.94744 −1.87668 0.404022 1.00000 4.34606
1.5 −1.84967 −1.00000 1.42128 1.21829 1.84967 −3.19073 1.07044 1.00000 −2.25343
1.6 −1.61650 −1.00000 0.613084 −1.13116 1.61650 −2.59609 2.24195 1.00000 1.82852
1.7 −1.49260 −1.00000 0.227845 0.260076 1.49260 1.75114 2.64511 1.00000 −0.388188
1.8 −1.29914 −1.00000 −0.312230 4.16905 1.29914 3.46206 3.00392 1.00000 −5.41618
1.9 −0.931077 −1.00000 −1.13309 2.04292 0.931077 2.58098 2.91715 1.00000 −1.90212
1.10 −0.407664 −1.00000 −1.83381 −3.10899 0.407664 −0.433130 1.56291 1.00000 1.26742
1.11 −0.387748 −1.00000 −1.84965 0.0320700 0.387748 1.47789 1.49270 1.00000 −0.0124351
1.12 −0.122270 −1.00000 −1.98505 4.08345 0.122270 −4.88900 0.487252 1.00000 −0.499283
1.13 0.110694 −1.00000 −1.98775 −0.0419640 −0.110694 −3.06459 −0.441419 1.00000 −0.00464516
1.14 0.565143 −1.00000 −1.68061 0.0334053 −0.565143 4.89694 −2.08007 1.00000 0.0188788
1.15 0.600928 −1.00000 −1.63889 −2.28783 −0.600928 −1.05285 −2.18671 1.00000 −1.37482
1.16 0.909777 −1.00000 −1.17231 0.959575 −0.909777 −1.03281 −2.88609 1.00000 0.872999
1.17 0.978973 −1.00000 −1.04161 2.77984 −0.978973 −1.51983 −2.97766 1.00000 2.72139
1.18 1.58588 −1.00000 0.515005 −2.48285 −1.58588 4.66783 −2.35502 1.00000 −3.93750
1.19 1.74499 −1.00000 1.04498 −3.15525 −1.74499 0.691511 −1.66649 1.00000 −5.50587
1.20 1.76203 −1.00000 1.10475 1.29294 −1.76203 −0.000478372 0 −1.57745 1.00000 2.27821
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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.bv yes 24
13.b even 2 1 8619.2.a.bu ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8619.2.a.bu ✓ 24 13.b even 2 1
8619.2.a.bv yes 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{24} - T_{2}^{23} - 32 T_{2}^{22} + 28 T_{2}^{21} + 444 T_{2}^{20} - 335 T_{2}^{19} - 3500 T_{2}^{18} + \cdots + 13 \) Copy content Toggle raw display
\( T_{5}^{24} - 11 T_{5}^{23} - 3 T_{5}^{22} + 414 T_{5}^{21} - 885 T_{5}^{20} - 6028 T_{5}^{19} + 20737 T_{5}^{18} + \cdots - 1 \) Copy content Toggle raw display
\( T_{7}^{24} - 2 T_{7}^{23} - 87 T_{7}^{22} + 152 T_{7}^{21} + 3151 T_{7}^{20} - 4610 T_{7}^{19} + \cdots - 1183 \) Copy content Toggle raw display