Properties

Label 630.2.bf.c
Level $630$
Weight $2$
Character orbit 630.bf
Analytic conductor $5.031$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [630,2,Mod(209,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 630.bf (of order \(6\), degree \(2\), 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: \(5.03057532734\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.856615824.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{4} + \beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{5} + \beta_{6} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{7} - q^{8} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{4} + \beta_{2} - 1) q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{4} - \beta_{2} + \beta_1) q^{5} + \beta_{6} q^{6} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \beta_1) q^{7} - q^{8} + (\beta_{7} + 2 \beta_{5} - \beta_{3} + \beta_1 + 1) q^{9} + (\beta_{6} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{10} + (\beta_{3} - \beta_{2} + \beta_1 + 2) q^{11} + (\beta_{6} + \beta_{4} - \beta_{2} + 1) q^{12} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{13} + (\beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{14} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{15} - \beta_{2} q^{16} + ( - \beta_{5} + \beta_{4} - 2 \beta_{2} - \beta_1 + 1) q^{17} + (2 \beta_{7} + \beta_{5} + \beta_{3} + 2 \beta_1 + 2) q^{18} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 4 \beta_{2} + 2) q^{19} + (\beta_{6} + \beta_{4} + \beta_{3} + 2) q^{20} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{21} + (\beta_{3} + \beta_{2} + 1) q^{22} + (\beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{2} - 4) q^{23} + (\beta_{4} - \beta_{2} + 1) q^{24} + (2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_1 - 3) q^{25} + ( - \beta_{5} - \beta_{4}) q^{26} + ( - \beta_{6} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{27} + (\beta_{7} - \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{28} + (2 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - \beta_{2} + 4 \beta_1 + 3) q^{29} + (2 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + \beta_1) q^{30} + (\beta_{7} + \beta_{5} + \beta_{4} - 3 \beta_{3} + 3 \beta_{2} + 4) q^{31} + ( - \beta_{2} + 1) q^{32} + ( - \beta_{4} + \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{33} + ( - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{34} + ( - 2 \beta_{7} + 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_1 - 4) q^{35} + (\beta_{7} - \beta_{5} + 2 \beta_{3} + \beta_1 + 1) q^{36} + ( - \beta_{5} + \beta_{4}) q^{37} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - 2 \beta_{2} + 1) q^{38} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{39} + (\beta_{4} + \beta_{2} - \beta_1) q^{40} + (\beta_{7} - \beta_{6} - \beta_{4} + 7 \beta_{2} - 7) q^{41} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_1 - 1) q^{42} + ( - \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 7) q^{43} + (2 \beta_{2} - \beta_1 - 1) q^{44} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 5) q^{45} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 5) q^{46} + ( - 3 \beta_{7} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 3) q^{47} - \beta_{6} q^{48} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} + 4 \beta_{2} + \beta_1 + 1) q^{49} + (3 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{50} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{51} + ( - \beta_{7} + \beta_{6} - \beta_{5}) q^{52} + (\beta_{7} + \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_1 + 3) q^{53} + (\beta_{6} - \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2) q^{54} + (\beta_{7} - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{55} + (\beta_{6} - \beta_{5} + \beta_{3} - \beta_1) q^{56} + ( - 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{2} - 3 \beta_1 + 2) q^{57} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - \beta_{4} + 4 \beta_{3} + \beta_{2} + 2) q^{58} + (2 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + 1) q^{59} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{60} + ( - 3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - \beta_{2} - 1) q^{61} + (\beta_{7} - \beta_{6} + 6 \beta_{2} + 3 \beta_1 - 3) q^{62} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - \beta_{3} + 5 \beta_{2} - 2 \beta_1 - 1) q^{63} + q^{64} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{65} + (\beta_{6} - \beta_{3} + \beta_{2} - 2 \beta_1 - 2) q^{66} + (2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - \beta_{2} + 3) q^{67} + ( - \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{68} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 + 6) q^{69} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 1) q^{70} + ( - 5 \beta_{7} + 5 \beta_{6} - 3 \beta_{5} + 3 \beta_{4} - 6 \beta_{2} - \beta_1 + 3) q^{71} + ( - \beta_{7} - 2 \beta_{5} + \beta_{3} - \beta_1 - 1) q^{72} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + 10) q^{73} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2) q^{74} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 7 \beta_{2} + 2 \beta_1) q^{75} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 1) q^{76} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 2) q^{77} + (\beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{78} + (2 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} + 5 \beta_{4} - \beta_{3} - 7 \beta_{2} + \beta_1 + 5) q^{79} + ( - \beta_{6} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{80} + (3 \beta_{5} + 3 \beta_{4} + 3 \beta_{2} - 3 \beta_1) q^{81} + ( - \beta_{5} - \beta_{4} - 7) q^{82} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{83} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{84} + ( - \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{3} - \beta_{2} - 3 \beta_1) q^{85} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{2} - 5) q^{86} + (\beta_{7} + \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} - 3 \beta_1 + 4) q^{87} + ( - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{88} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} + 10 \beta_{3} + 5 \beta_1 - 6) q^{89} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 2) q^{90} + ( - 3 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots - 7) q^{91}+ \cdots + (\beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} - 3 q^{6} + 2 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} - 3 q^{6} + 2 q^{7} - 8 q^{8} + 6 q^{10} + 9 q^{11} - 3 q^{12} + 4 q^{13} + q^{14} + 9 q^{15} - 4 q^{16} + 3 q^{18} + 6 q^{20} - 12 q^{21} + 9 q^{22} - 9 q^{23} - 16 q^{25} + 8 q^{26} - 18 q^{27} - q^{28} + 21 q^{29} + 42 q^{31} + 4 q^{32} - 3 q^{33} + 9 q^{34} - 33 q^{35} + 3 q^{36} + 21 q^{38} + 12 q^{39} - 24 q^{41} - 15 q^{42} - 27 q^{43} - 39 q^{45} - 18 q^{46} + 15 q^{47} + 3 q^{48} - 4 q^{49} - 20 q^{50} + 21 q^{51} + 4 q^{52} - 12 q^{53} - 20 q^{55} - 2 q^{56} + 39 q^{57} + 21 q^{58} - 3 q^{59} - 9 q^{60} + 21 q^{61} - 18 q^{63} + 8 q^{64} + 18 q^{65} - 12 q^{66} + 9 q^{68} + 21 q^{69} - 24 q^{70} + 82 q^{73} - 6 q^{74} - 39 q^{75} + 21 q^{76} - 9 q^{77} + 24 q^{78} - 8 q^{79} - 6 q^{80} - 12 q^{81} - 48 q^{82} + 15 q^{83} - 3 q^{84} + 19 q^{85} - 27 q^{86} + 30 q^{87} - 9 q^{88} - 42 q^{89} - 18 q^{90} - 8 q^{91} - 9 q^{92} + 6 q^{93} + 15 q^{94} + 9 q^{95} + 3 q^{96} - 11 q^{97} - 29 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 7\nu^{3} + 10\nu + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 10\nu^{2} - 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 9\nu^{4} + 2\nu^{3} + 22\nu^{2} + 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 9\nu^{4} - 2\nu^{3} + 22\nu^{2} - 10\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 10\nu^{5} - 9\nu^{4} - 29\nu^{3} - 20\nu^{2} - 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - \nu^{6} + 10\nu^{5} - 9\nu^{4} + 29\nu^{3} - 20\nu^{2} + 20\nu - 6 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{7} - 6\beta_{6} - 5\beta_{5} - 5\beta_{4} - 2\beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} - 7\beta_{4} + 4\beta_{2} + 25\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{7} + 32\beta_{6} + 25\beta_{5} + 25\beta_{4} + 18\beta_{3} + 9\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 41\beta_{5} + 41\beta_{4} - 40\beta_{2} - 125\beta _1 + 20 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0.385731i
2.06288i
2.33086i
1.07834i
0.385731i
2.06288i
2.33086i
1.07834i
0.500000 0.866025i −1.73065 + 0.0696054i −0.500000 0.866025i −1.73065 + 1.41593i −0.805046 + 1.53359i 2.36975 1.17656i −1.00000 2.99031 0.240925i 0.360902 + 2.20675i
209.2 0.500000 0.866025i −0.574618 1.63396i −0.500000 0.866025i −0.574618 + 2.16098i −1.70236 0.319344i −0.00953166 + 2.64573i −1.00000 −2.33963 + 1.87780i 1.58415 + 1.57812i
209.3 0.500000 0.866025i 1.05903 1.37057i −0.500000 0.866025i 1.05903 1.96938i −0.657430 1.60243i 1.11699 2.39840i −1.00000 −0.756906 2.90295i −1.17602 1.90184i
209.4 0.500000 0.866025i 1.24624 + 1.20287i −0.500000 0.866025i 1.24624 + 1.85658i 1.66483 0.477841i −2.47720 + 0.929227i −1.00000 0.106223 + 2.99812i 2.23096 0.150985i
419.1 0.500000 + 0.866025i −1.73065 0.0696054i −0.500000 + 0.866025i −1.73065 1.41593i −0.805046 1.53359i 2.36975 + 1.17656i −1.00000 2.99031 + 0.240925i 0.360902 2.20675i
419.2 0.500000 + 0.866025i −0.574618 + 1.63396i −0.500000 + 0.866025i −0.574618 2.16098i −1.70236 + 0.319344i −0.00953166 2.64573i −1.00000 −2.33963 1.87780i 1.58415 1.57812i
419.3 0.500000 + 0.866025i 1.05903 + 1.37057i −0.500000 + 0.866025i 1.05903 + 1.96938i −0.657430 + 1.60243i 1.11699 + 2.39840i −1.00000 −0.756906 + 2.90295i −1.17602 + 1.90184i
419.4 0.500000 + 0.866025i 1.24624 1.20287i −0.500000 + 0.866025i 1.24624 1.85658i 1.66483 + 0.477841i −2.47720 0.929227i −1.00000 0.106223 2.99812i 2.23096 + 0.150985i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
315.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.bf.c yes 8
3.b odd 2 1 1890.2.bf.b 8
5.b even 2 1 630.2.bf.b yes 8
7.b odd 2 1 630.2.bf.d yes 8
9.c even 3 1 1890.2.bf.d 8
9.d odd 6 1 630.2.bf.a 8
15.d odd 2 1 1890.2.bf.c 8
21.c even 2 1 1890.2.bf.a 8
35.c odd 2 1 630.2.bf.a 8
45.h odd 6 1 630.2.bf.d yes 8
45.j even 6 1 1890.2.bf.a 8
63.l odd 6 1 1890.2.bf.c 8
63.o even 6 1 630.2.bf.b yes 8
105.g even 2 1 1890.2.bf.d 8
315.z even 6 1 inner 630.2.bf.c yes 8
315.bg odd 6 1 1890.2.bf.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.bf.a 8 9.d odd 6 1
630.2.bf.a 8 35.c odd 2 1
630.2.bf.b yes 8 5.b even 2 1
630.2.bf.b yes 8 63.o even 6 1
630.2.bf.c yes 8 1.a even 1 1 trivial
630.2.bf.c yes 8 315.z even 6 1 inner
630.2.bf.d yes 8 7.b odd 2 1
630.2.bf.d yes 8 45.h odd 6 1
1890.2.bf.a 8 21.c even 2 1
1890.2.bf.a 8 45.j even 6 1
1890.2.bf.b 8 3.b odd 2 1
1890.2.bf.b 8 315.bg odd 6 1
1890.2.bf.c 8 15.d odd 2 1
1890.2.bf.c 8 63.l odd 6 1
1890.2.bf.d 8 9.c even 3 1
1890.2.bf.d 8 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{8} - 9T_{11}^{7} + 32T_{11}^{6} - 45T_{11}^{5} + 18T_{11}^{4} + 15T_{11}^{3} - 7T_{11}^{2} - 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} + 22T_{13}^{6} - 28T_{13}^{5} + 136T_{13}^{4} - 124T_{13}^{3} + 700T_{13}^{2} + 104T_{13} + 16 \) Copy content Toggle raw display
\( T_{23}^{8} + 9T_{23}^{7} + 81T_{23}^{6} + 108T_{23}^{5} + 540T_{23}^{4} + 972T_{23}^{3} + 2916T_{23}^{2} + 2916T_{23} + 2916 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{5} + 3 T^{4} + 18 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{6} + 6 T^{5} + 51 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 2 T^{7} + 4 T^{6} + 10 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 9 T^{7} + 32 T^{6} - 45 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + 22 T^{6} - 28 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$17$ \( T^{8} + 29 T^{6} + 87 T^{4} + 71 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$19$ \( T^{8} + 93 T^{6} + 2079 T^{4} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( T^{8} + 9 T^{7} + 81 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$29$ \( T^{8} - 21 T^{7} + 131 T^{6} + 336 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$31$ \( T^{8} - 42 T^{7} + 768 T^{6} + \cdots + 1065024 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{6} + 132 T^{4} + \cdots + 144 \) Copy content Toggle raw display
$41$ \( T^{8} + 24 T^{7} + 372 T^{6} + \cdots + 848241 \) Copy content Toggle raw display
$43$ \( T^{8} + 27 T^{7} + 306 T^{6} + \cdots + 2916 \) Copy content Toggle raw display
$47$ \( T^{8} - 15 T^{7} + 23 T^{6} + \cdots + 868624 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6 T^{3} - 108 T^{2} - 306 T + 2064)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 3 T^{7} + 198 T^{6} + \cdots + 46457856 \) Copy content Toggle raw display
$61$ \( T^{8} - 21 T^{7} + 111 T^{6} + \cdots + 2022084 \) Copy content Toggle raw display
$67$ \( T^{8} - 42 T^{6} + 1527 T^{4} + \cdots + 56169 \) Copy content Toggle raw display
$71$ \( T^{8} + 404 T^{6} + \cdots + 17272336 \) Copy content Toggle raw display
$73$ \( (T^{4} - 41 T^{3} + 591 T^{2} - 3551 T + 7588)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + 304 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$83$ \( T^{8} - 15 T^{7} + 83 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$89$ \( (T^{4} + 21 T^{3} - 204 T^{2} + \cdots - 40272)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 11 T^{7} + 310 T^{6} + \cdots + 78074896 \) Copy content Toggle raw display
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