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Properties

Label 8281.2.a.ct
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 1183)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} + 23 q^{4} - 13 q^{5} - 14 q^{6} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} + 23 q^{4} - 13 q^{5} - 14 q^{6} + 26 q^{9} - 5 q^{10} - q^{11} - 5 q^{12} + 5 q^{15} + 17 q^{16} + 5 q^{17} - 24 q^{19} - 34 q^{20} - 14 q^{22} + 11 q^{23} - 32 q^{24} + 33 q^{25} + 21 q^{27} + 4 q^{29} - 22 q^{30} - 40 q^{31} - 6 q^{32} - 24 q^{33} - 36 q^{34} - 15 q^{36} - 4 q^{37} + 29 q^{38} + 4 q^{40} - 49 q^{41} + 13 q^{43} + 10 q^{44} - 58 q^{45} - 10 q^{46} - 62 q^{47} - 89 q^{48} - 23 q^{50} - 21 q^{51} - 18 q^{53} - 12 q^{54} + 14 q^{55} - 13 q^{57} + 56 q^{58} - 79 q^{59} + 22 q^{60} - 13 q^{61} - 12 q^{62} + 18 q^{64} + 38 q^{66} - 2 q^{67} + 12 q^{68} + 28 q^{69} - 19 q^{71} + 81 q^{72} - 17 q^{73} + 17 q^{74} - 24 q^{75} - 58 q^{76} - 9 q^{79} - 63 q^{80} + 16 q^{81} + 22 q^{82} - 81 q^{83} - 34 q^{85} + 22 q^{86} - 70 q^{87} - 33 q^{88} - 72 q^{89} - q^{90} - 4 q^{92} + 19 q^{93} + 30 q^{94} + 13 q^{95} - 11 q^{96} - 45 q^{97} - 39 q^{99}+O(q^{100}) \) 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 −2.63978 −1.49073 4.96846 −2.80490 3.93520 0 −7.83610 −0.777726 7.40433
1.2 −2.54105 −0.312918 4.45692 −3.08154 0.795139 0 −6.24314 −2.90208 7.83034
1.3 −2.37543 0.696489 3.64268 2.62567 −1.65446 0 −3.90209 −2.51490 −6.23710
1.4 −2.26331 3.32695 3.12257 1.16195 −7.52992 0 −2.54072 8.06859 −2.62985
1.5 −1.94710 1.73455 1.79121 −3.71933 −3.37734 0 0.406541 0.00865975 7.24191
1.6 −1.75182 −0.671818 1.06888 2.67750 1.17690 0 1.63116 −2.54866 −4.69050
1.7 −1.36142 1.31293 −0.146524 3.05867 −1.78745 0 2.92233 −1.27622 −4.16415
1.8 −1.18747 2.93451 −0.589922 −3.40390 −3.48463 0 3.07545 5.61133 4.04203
1.9 −1.02884 −2.66326 −0.941498 −1.49113 2.74006 0 3.02632 4.09295 1.53412
1.10 −0.617518 −3.07329 −1.61867 −1.39605 1.89781 0 2.23459 6.44510 0.862085
1.11 −0.588471 −0.773677 −1.65370 2.69602 0.455287 0 2.15010 −2.40142 −1.58653
1.12 −0.254753 1.57524 −1.93510 −0.518004 −0.401297 0 1.00248 −0.518622 0.131963
1.13 −0.136454 −2.96681 −1.98138 −3.29030 0.404833 0 0.543274 5.80198 0.448974
1.14 0.00236726 3.09829 −1.99999 1.13142 0.00733446 0 −0.00946903 6.59939 0.00267836
1.15 0.877203 0.755024 −1.23052 0.265839 0.662309 0 −2.83382 −2.42994 0.233195
1.16 1.09197 −1.39541 −0.807612 1.62650 −1.52374 0 −3.06581 −1.05283 1.77608
1.17 1.20612 −1.03207 −0.545276 −3.02929 −1.24480 0 −3.06991 −1.93483 −3.65368
1.18 1.54383 1.35319 0.383396 −0.341965 2.08910 0 −2.49575 −1.16886 −0.527934
1.19 1.57441 −2.20221 0.478777 3.76060 −3.46719 0 −2.39503 1.84974 5.92074
1.20 1.76732 2.32802 1.12344 −4.06760 4.11437 0 −1.54917 2.41970 −7.18877
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
\(7\) \(-1\)
\(13\) \(-1\)

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 8281.2.a.ct 24
7.b odd 2 1 8281.2.a.cu 24
7.d odd 6 2 1183.2.e.l yes 48
13.b even 2 1 8281.2.a.cw 24
91.b odd 2 1 8281.2.a.cv 24
91.s odd 6 2 1183.2.e.k ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.e.k ✓ 48 91.s odd 6 2
1183.2.e.l yes 48 7.d odd 6 2
8281.2.a.ct 24 1.a even 1 1 trivial
8281.2.a.cu 24 7.b odd 2 1
8281.2.a.cv 24 91.b odd 2 1
8281.2.a.cw 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{24} + T_{2}^{23} - 35 T_{2}^{22} - 34 T_{2}^{21} + 527 T_{2}^{20} + 496 T_{2}^{19} - 4477 T_{2}^{18} + \cdots - 1 \) Copy content Toggle raw display
\( T_{3}^{24} - 49 T_{3}^{22} - 7 T_{3}^{21} + 1030 T_{3}^{20} + 282 T_{3}^{19} - 12168 T_{3}^{18} + \cdots - 28469 \) Copy content Toggle raw display
\( T_{5}^{24} + 13 T_{5}^{23} + 8 T_{5}^{22} - 560 T_{5}^{21} - 1887 T_{5}^{20} + 8750 T_{5}^{19} + \cdots + 346087 \) Copy content Toggle raw display
\( T_{11}^{24} + T_{11}^{23} - 124 T_{11}^{22} - 117 T_{11}^{21} + 6433 T_{11}^{20} + 5835 T_{11}^{19} + \cdots + 64936579 \) Copy content Toggle raw display
\( T_{17}^{24} - 5 T_{17}^{23} - 178 T_{17}^{22} + 992 T_{17}^{21} + 12871 T_{17}^{20} - 81819 T_{17}^{19} + \cdots - 85617259 \) Copy content Toggle raw display