Properties

Label 833.2.v.a
Level $833$
Weight $2$
Character orbit 833.v
Analytic conductor $6.652$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(128,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(24))
 
chi = DirichletCharacter(H, H._module([8, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.128");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.v (of order \(24\), degree \(8\), not minimal)

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: no
Analytic conductor: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{24}^{7} - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24}^{2} + 1) q^{2} + ( - \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{4} - \zeta_{24}^{2} - \zeta_{24}) q^{3} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} - 2 \zeta_{24}) q^{4} + ( - \zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{5} + (\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} - 1) q^{6} + ( - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 3 \zeta_{24} + 1) q^{8} + (\zeta_{24}^{5} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{24}^{7} - \zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24}^{2} + 1) q^{2} + ( - \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{4} - \zeta_{24}^{2} - \zeta_{24}) q^{3} + (2 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} - 2 \zeta_{24}) q^{4} + ( - \zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24}^{2}) q^{5} + (\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} - 1) q^{6} + ( - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 3 \zeta_{24} + 1) q^{8} + (\zeta_{24}^{5} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2) q^{9} + ( - \zeta_{24}^{7} + \zeta_{24}^{4}) q^{10} + (\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{4} + \zeta_{24}^{2} - \zeta_{24}) q^{11} + ( - 3 \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{4} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{2} - 1) q^{12} + ( - \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}) q^{13} + ( - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{3} - 2) q^{15} + ( - 3 \zeta_{24}^{4} + 3) q^{16} + ( - 2 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2} - 2 \zeta_{24}) q^{17} + ( - \zeta_{24}^{7} + 3 \zeta_{24}^{4} - \zeta_{24}) q^{18} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 2) q^{19} + (2 \zeta_{24}^{6} + \zeta_{24}^{5} - 2 \zeta_{24}^{3} - \zeta_{24} - 1) q^{20} + ( - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - \zeta_{24}^{3} - 3 \zeta_{24} + 3) q^{22} + (\zeta_{24}^{7} + 3 \zeta_{24}^{5} + \zeta_{24}^{4} - \zeta_{24}^{3} - 3 \zeta_{24}^{2} - 1) q^{23} + ( - 3 \zeta_{24}^{7} - \zeta_{24}^{6} + 3 \zeta_{24}^{4} + \zeta_{24}^{2} + \zeta_{24}) q^{24} + ( - \zeta_{24}^{6} + \zeta_{24}^{4} + \zeta_{24}^{2} - 3 \zeta_{24}) q^{25} + ( - 2 \zeta_{24}^{5} + \zeta_{24}^{4} + \zeta_{24}^{2} - 1) q^{26} + ( - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24} - 2) q^{27} + ( - 2 \zeta_{24}^{6} - \zeta_{24}^{5} - 2 \zeta_{24}^{3} + \zeta_{24} - 1) q^{29} - 2 \zeta_{24}^{2} q^{30} + (3 \zeta_{24}^{7} - 3 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{2} - 3 \zeta_{24}) q^{31} + ( - 3 \zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{32} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3}) q^{33} + ( - \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{3} - 3 \zeta_{24} + 5) q^{34} - 7 \zeta_{24}^{3} q^{36} + ( - 5 \zeta_{24}^{7} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{2}) q^{37} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{2}) q^{38} + (2 \zeta_{24}^{7} + 2 \zeta_{24}^{6} - 2 \zeta_{24}^{2}) q^{39} + (2 \zeta_{24}^{7} - \zeta_{24}^{5} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{3} + \zeta_{24}^{2} - 2) q^{40} + ( - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{5} - \zeta_{24}^{3} + 4 \zeta_{24} + 1) q^{41} + (2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 2 \zeta_{24} - 2) q^{43} + (5 \zeta_{24}^{7} + \zeta_{24}^{5} - 5 \zeta_{24}^{4} - 5 \zeta_{24}^{3} + \zeta_{24}^{2} + 5) q^{44} + (3 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{2} + 2 \zeta_{24}) q^{45} + (3 \zeta_{24}^{7} + 3 \zeta_{24}^{6} - 5 \zeta_{24}^{4} - 3 \zeta_{24}^{2} + 5 \zeta_{24}) q^{46} + (2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} - 8 \zeta_{24}^{2}) q^{47} + (3 \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{3} - 3 \zeta_{24} - 3) q^{48} + (4 \zeta_{24}^{5} - 4 \zeta_{24}^{3} - 4 \zeta_{24} + 5) q^{50} + (3 \zeta_{24}^{7} + \zeta_{24}^{5} + 7 \zeta_{24}^{4} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{2} - 7) q^{51} + ( - \zeta_{24}^{7} + 4 \zeta_{24}^{4} - \zeta_{24}) q^{52} + (\zeta_{24}^{6} + \zeta_{24}^{4} - \zeta_{24}^{2}) q^{53} + ( - 6 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{4} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{2} - 2) q^{54} + 2 \zeta_{24}^{6} q^{55} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{3} - 2 \zeta_{24} - 6) q^{57} + ( - \zeta_{24}^{5} - \zeta_{24}^{2}) q^{58} - 6 \zeta_{24} q^{59} + ( - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 6 \zeta_{24}) q^{60} + ( - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{2}) q^{61} + ( - 9 \zeta_{24}^{6} + 9 \zeta_{24}^{5} + 3 \zeta_{24}^{3} - 9 \zeta_{24} + 3) q^{62} + (7 \zeta_{24}^{6} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24}) q^{64} + ( - \zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24}^{2} + 1) q^{65} + ( - 4 \zeta_{24}^{7} + 2 \zeta_{24}^{6} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{2}) q^{66} + ( - 2 \zeta_{24}^{7} - 4 \zeta_{24}^{4} - 2 \zeta_{24}) q^{67} + (8 \zeta_{24}^{7} + 4 \zeta_{24}^{5} - 3 \zeta_{24}^{4} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{2} + 3) q^{68} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24} + 8) q^{69} + ( - 5 \zeta_{24}^{6} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{3} + 5 \zeta_{24} + 5) q^{71} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{5} - 5 \zeta_{24}^{3} + \zeta_{24}^{2}) q^{72} + ( - 7 \zeta_{24}^{4} + 7 \zeta_{24}) q^{73} + ( - 5 \zeta_{24}^{7} + 10 \zeta_{24}^{6} - 5 \zeta_{24}^{4} - 10 \zeta_{24}^{2} + 10 \zeta_{24}) q^{74} + ( - 3 \zeta_{24}^{7} + \zeta_{24}^{5} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{3} + \zeta_{24}^{2} - 3) q^{75} + (2 \zeta_{24}^{6} + 6 \zeta_{24}^{5} - 6 \zeta_{24} - 2) q^{76} + (2 \zeta_{24}^{3} - 2) q^{78} + ( - \zeta_{24}^{7} - 3 \zeta_{24}^{5} - \zeta_{24}^{4} + \zeta_{24}^{3} + 3 \zeta_{24}^{2} + 1) q^{79} + ( - 3 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2}) q^{80} + ( - 2 \zeta_{24}^{7} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{2} + 2 \zeta_{24}) q^{81} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 1) q^{82} + ( - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{3} - 4) q^{83} + ( - 2 \zeta_{24}^{6} - 5 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 5 \zeta_{24} + 1) q^{85} + (2 \zeta_{24}^{4} - 2) q^{86} + (6 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 4 \zeta_{24}^{4} - 4 \zeta_{24}^{2}) q^{87} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{6} - 3 \zeta_{24}^{4} + 5 \zeta_{24}^{2} - 3 \zeta_{24}) q^{88} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + 8 \zeta_{24}^{2}) q^{89} + (2 \zeta_{24}^{6} - \zeta_{24}^{5} + 2 \zeta_{24}^{3} + \zeta_{24} - 1) q^{90} + ( - 7 \zeta_{24}^{6} - 7 \zeta_{24}^{5} + 7 \zeta_{24}^{3} + 7 \zeta_{24} - 7) q^{92} + (6 \zeta_{24}^{4} + 6 \zeta_{24}^{2} - 6) q^{93} + (6 \zeta_{24}^{6} - 6 \zeta_{24}^{4} - 6 \zeta_{24}^{2} + 4 \zeta_{24}) q^{94} + ( - 2 \zeta_{24}^{7} - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{2} + 4 \zeta_{24}) q^{95} + (5 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{3} - 3 \zeta_{24}^{2} - 5) q^{96} + ( - \zeta_{24}^{6} - 6 \zeta_{24}^{5} - \zeta_{24}^{3} + 6 \zeta_{24} - 6) q^{97} + (3 \zeta_{24}^{6} + \zeta_{24}^{5} - 3 \zeta_{24}^{3} - \zeta_{24} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{3} - 8 q^{6} + 8 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{3} - 8 q^{6} + 8 q^{8} - 8 q^{9} + 4 q^{10} + 4 q^{11} - 4 q^{12} - 16 q^{15} + 12 q^{16} + 12 q^{18} + 8 q^{19} - 8 q^{20} + 24 q^{22} - 4 q^{23} + 12 q^{24} + 4 q^{25} - 4 q^{26} - 16 q^{27} - 8 q^{29} - 12 q^{31} - 4 q^{32} + 40 q^{34} - 8 q^{40} + 8 q^{41} - 16 q^{43} + 20 q^{44} + 12 q^{45} - 20 q^{46} - 24 q^{48} + 40 q^{50} - 28 q^{51} + 16 q^{52} + 4 q^{53} - 8 q^{54} - 48 q^{57} + 8 q^{60} + 24 q^{62} + 4 q^{65} + 8 q^{66} - 16 q^{67} + 12 q^{68} + 64 q^{69} + 40 q^{71} - 28 q^{73} - 20 q^{74} - 12 q^{75} - 16 q^{76} - 16 q^{78} + 4 q^{79} + 4 q^{82} - 32 q^{83} + 8 q^{85} - 8 q^{86} + 16 q^{87} - 12 q^{88} - 8 q^{90} - 56 q^{92} - 24 q^{93} - 24 q^{94} + 8 q^{95} - 20 q^{96} - 48 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/833\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(785\)
\(\chi(n)\) \(-1 + \zeta_{24}^{4}\) \(\zeta_{24} - \zeta_{24}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
128.1
−0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0.258819 0.965926i
0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 0.258819i
−0.107206 0.400100i −0.341081 2.59077i 1.58346 0.914214i −1.12484 1.46593i −1.00000 + 0.414214i 0 −1.12132 1.12132i −3.69798 + 0.990870i −0.465926 + 0.607206i
263.1 2.33195 + 0.624844i −0.658919 0.858719i 3.31552 + 1.91421i −0.0999004 0.758819i −1.00000 2.41421i 0 3.12132 + 3.12132i 0.473232 1.76612i 0.241181 1.83195i
410.1 −0.107206 + 0.400100i −0.341081 + 2.59077i 1.58346 + 0.914214i −1.12484 + 1.46593i −1.00000 0.414214i 0 −1.12132 + 1.12132i −3.69798 0.990870i −0.465926 0.607206i
508.1 −0.624844 + 2.33195i 1.07313 + 0.141281i −3.31552 1.91421i −0.607206 0.465926i −1.00000 + 2.41421i 0 3.12132 3.12132i −1.76612 0.473232i 1.46593 1.12484i
569.1 −0.624844 2.33195i 1.07313 0.141281i −3.31552 + 1.91421i −0.607206 + 0.465926i −1.00000 2.41421i 0 3.12132 + 3.12132i −1.76612 + 0.473232i 1.46593 + 1.12484i
655.1 0.400100 + 0.107206i −2.07313 + 1.59077i −1.58346 0.914214i 1.83195 0.241181i −1.00000 + 0.414214i 0 −1.12132 1.12132i 0.990870 3.69798i 0.758819 + 0.0999004i
716.1 0.400100 0.107206i −2.07313 1.59077i −1.58346 + 0.914214i 1.83195 + 0.241181i −1.00000 0.414214i 0 −1.12132 + 1.12132i 0.990870 + 3.69798i 0.758819 0.0999004i
814.1 2.33195 0.624844i −0.658919 + 0.858719i 3.31552 1.91421i −0.0999004 + 0.758819i −1.00000 + 2.41421i 0 3.12132 3.12132i 0.473232 + 1.76612i 0.241181 + 1.83195i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 128.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.d even 8 1 inner
119.q even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.v.a 8
7.b odd 2 1 833.2.v.b 8
7.c even 3 1 833.2.l.a 4
7.c even 3 1 inner 833.2.v.a 8
7.d odd 6 1 17.2.d.a 4
7.d odd 6 1 833.2.v.b 8
17.d even 8 1 inner 833.2.v.a 8
21.g even 6 1 153.2.l.c 4
28.f even 6 1 272.2.v.d 4
35.i odd 6 1 425.2.m.a 4
35.k even 12 1 425.2.n.a 4
35.k even 12 1 425.2.n.b 4
119.h odd 6 1 289.2.d.a 4
119.l odd 8 1 833.2.v.b 8
119.m odd 12 1 289.2.d.b 4
119.m odd 12 1 289.2.d.c 4
119.q even 24 1 833.2.l.a 4
119.q even 24 1 inner 833.2.v.a 8
119.r odd 24 1 17.2.d.a 4
119.r odd 24 1 289.2.d.a 4
119.r odd 24 1 289.2.d.b 4
119.r odd 24 1 289.2.d.c 4
119.r odd 24 1 833.2.v.b 8
119.s even 48 2 289.2.a.f 4
119.s even 48 2 289.2.b.b 4
119.s even 48 4 289.2.c.c 8
357.bj even 24 1 153.2.l.c 4
357.bn odd 48 2 2601.2.a.bb 4
476.bj even 24 1 272.2.v.d 4
476.bk odd 48 2 4624.2.a.bp 4
595.cf even 24 1 425.2.n.a 4
595.ch odd 24 1 425.2.m.a 4
595.cl even 24 1 425.2.n.b 4
595.cr even 48 2 7225.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.2.d.a 4 7.d odd 6 1
17.2.d.a 4 119.r odd 24 1
153.2.l.c 4 21.g even 6 1
153.2.l.c 4 357.bj even 24 1
272.2.v.d 4 28.f even 6 1
272.2.v.d 4 476.bj even 24 1
289.2.a.f 4 119.s even 48 2
289.2.b.b 4 119.s even 48 2
289.2.c.c 8 119.s even 48 4
289.2.d.a 4 119.h odd 6 1
289.2.d.a 4 119.r odd 24 1
289.2.d.b 4 119.m odd 12 1
289.2.d.b 4 119.r odd 24 1
289.2.d.c 4 119.m odd 12 1
289.2.d.c 4 119.r odd 24 1
425.2.m.a 4 35.i odd 6 1
425.2.m.a 4 595.ch odd 24 1
425.2.n.a 4 35.k even 12 1
425.2.n.a 4 595.cf even 24 1
425.2.n.b 4 35.k even 12 1
425.2.n.b 4 595.cl even 24 1
833.2.l.a 4 7.c even 3 1
833.2.l.a 4 119.q even 24 1
833.2.v.a 8 1.a even 1 1 trivial
833.2.v.a 8 7.c even 3 1 inner
833.2.v.a 8 17.d even 8 1 inner
833.2.v.a 8 119.q even 24 1 inner
833.2.v.b 8 7.b odd 2 1
833.2.v.b 8 7.d odd 6 1
833.2.v.b 8 119.l odd 8 1
833.2.v.b 8 119.r odd 24 1
2601.2.a.bb 4 357.bn odd 48 2
4624.2.a.bp 4 476.bk odd 48 2
7225.2.a.u 4 595.cr even 48 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{8} - 4T_{2}^{7} + 8T_{2}^{6} - 24T_{2}^{5} + 47T_{2}^{4} - 24T_{2}^{3} + 8T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 4T_{3}^{7} + 12T_{3}^{6} + 16T_{3}^{5} + 8T_{3}^{4} - 64T_{3}^{3} - 32T_{3}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + 8 T^{6} - 24 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + 12 T^{6} + 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{6} - 8 T^{5} + 2 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + 4 T^{6} - 16 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{6} - 285 T^{4} + \cdots + 83521 \) Copy content Toggle raw display
$19$ \( T^{8} - 8 T^{7} + 32 T^{6} - 192 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{7} + 4 T^{6} + \cdots + 153664 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + 22 T^{2} + 12 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 12 T^{7} + 36 T^{6} + \cdots + 419904 \) Copy content Toggle raw display
$37$ \( T^{8} - 50 T^{6} + 1000 T^{5} + \cdots + 1562500 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + 54 T^{2} + 140 T + 98)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + 32 T^{2} + 32 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 144 T^{6} + 17600 T^{4} + \cdots + 9834496 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 1296 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$61$ \( T^{8} - 50 T^{6} - 1000 T^{5} + \cdots + 1562500 \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + 56 T^{2} + 64 T + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 20 T^{3} + 100 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 28 T^{7} + 490 T^{6} + \cdots + 23059204 \) Copy content Toggle raw display
$79$ \( T^{8} - 4 T^{7} + 4 T^{6} + \cdots + 153664 \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} + 128 T^{2} - 64 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 132 T^{6} + \cdots + 14776336 \) Copy content Toggle raw display
$97$ \( (T^{4} + 24 T^{3} + 242 T^{2} + 1316 T + 4418)^{2} \) Copy content Toggle raw display
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