Properties

Label 931.2.f.n
Level $931$
Weight $2$
Character orbit 931.f
Analytic conductor $7.434$
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] = [931,2,Mod(324,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.f (of order \(3\), degree \(2\), 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: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.2672476416.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} - 4x^{5} + 24x^{4} - 10x^{3} + 9x^{2} + 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} - 4x^{5} + 24x^{4} - 10x^{3} + 9x^{2} + 2x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -60\nu^{7} - 25\nu^{6} - 288\nu^{5} + 120\nu^{4} - 1390\nu^{3} - 24\nu^{2} - 550\nu - 125 ) / 538 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -103\nu^{7} - 155\nu^{6} - 602\nu^{5} - 332\nu^{4} - 2162\nu^{3} - 1978\nu^{2} - 989\nu - 237 ) / 538 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -113\nu^{7} + 65\nu^{6} - 650\nu^{5} + 764\nu^{4} - 3380\nu^{3} + 2860\nu^{2} - 3681\nu + 325 ) / 538 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -112\nu^{7} + 43\nu^{6} - 430\nu^{5} + 762\nu^{4} - 2236\nu^{3} + 1892\nu^{2} + 946\nu + 215 ) / 538 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -125\nu^{7} + 60\nu^{6} - 600\nu^{5} + 788\nu^{4} - 3120\nu^{3} + 2640\nu^{2} - 1101\nu + 300 ) / 538 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -300\nu^{7} - 125\nu^{6} - 1440\nu^{5} + 600\nu^{4} - 6412\nu^{3} - 120\nu^{2} - 60\nu - 1163 ) / 538 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 5\beta_{2} - 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{6} + 5\beta_{5} + 5\beta_{4} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} + 9\beta_{6} - 7\beta_{4} + 2\beta_{3} + 26\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26\beta_{7} - 24\beta_{5} - 24\beta_{3} - 44\beta_{2} - 44\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -62\beta_{6} + 20\beta_{5} + 44\beta_{4} + 139\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
−1.18994 + 2.06103i
−0.145683 + 0.252331i
0.375512 0.650406i
0.960110 1.66296i
−1.18994 2.06103i
−0.145683 0.252331i
0.375512 + 0.650406i
0.960110 + 1.66296i
−1.18994 + 2.06103i −0.289905 0.502131i −1.83191 3.17296i −1.83787 + 3.18329i 1.37988 0 3.95969 1.33191 2.30694i −4.37391 7.57584i
324.2 −0.145683 + 0.252331i 1.21605 + 2.10626i 0.957553 + 1.65853i −1.03287 + 1.78898i −0.708633 0 −1.14073 −1.45755 + 2.52456i −0.300944 0.521251i
324.3 0.375512 0.650406i −1.16576 2.01915i 0.717981 + 1.24358i −2.13271 + 3.69397i −1.75102 0 2.58049 −1.21798 + 2.10961i 1.60172 + 2.77426i
324.4 0.960110 1.66296i −0.760387 1.31703i −0.843624 1.46120i 1.00346 1.73804i −2.92022 0 0.600553 0.343624 0.595174i −1.92686 3.33742i
704.1 −1.18994 2.06103i −0.289905 + 0.502131i −1.83191 + 3.17296i −1.83787 3.18329i 1.37988 0 3.95969 1.33191 + 2.30694i −4.37391 + 7.57584i
704.2 −0.145683 0.252331i 1.21605 2.10626i 0.957553 1.65853i −1.03287 1.78898i −0.708633 0 −1.14073 −1.45755 2.52456i −0.300944 + 0.521251i
704.3 0.375512 + 0.650406i −1.16576 + 2.01915i 0.717981 1.24358i −2.13271 3.69397i −1.75102 0 2.58049 −1.21798 2.10961i 1.60172 2.77426i
704.4 0.960110 + 1.66296i −0.760387 + 1.31703i −0.843624 + 1.46120i 1.00346 + 1.73804i −2.92022 0 0.600553 0.343624 + 0.595174i −1.92686 + 3.33742i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.f.n 8
7.b odd 2 1 931.2.f.o 8
7.c even 3 1 931.2.a.m yes 4
7.c even 3 1 inner 931.2.f.n 8
7.d odd 6 1 931.2.a.l 4
7.d odd 6 1 931.2.f.o 8
21.g even 6 1 8379.2.a.bv 4
21.h odd 6 1 8379.2.a.bu 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.a.l 4 7.d odd 6 1
931.2.a.m yes 4 7.c even 3 1
931.2.f.n 8 1.a even 1 1 trivial
931.2.f.n 8 7.c even 3 1 inner
931.2.f.o 8 7.b odd 2 1
931.2.f.o 8 7.d odd 6 1
8379.2.a.bu 4 21.h odd 6 1
8379.2.a.bv 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{8} + 5T_{2}^{6} - 4T_{2}^{5} + 24T_{2}^{4} - 10T_{2}^{3} + 9T_{2}^{2} + 2T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 2T_{3}^{7} + 9T_{3}^{6} + 14T_{3}^{5} + 54T_{3}^{4} + 80T_{3}^{3} + 119T_{3}^{2} + 60T_{3} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} - 4 T^{5} + 24 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + 9 T^{6} + 14 T^{5} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{8} + 8 T^{7} + 52 T^{6} + \cdots + 4225 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + 49 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} - 20 T^{2} - 26 T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + 87 T^{6} + \cdots + 15625 \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + 78 T^{6} + \cdots + 3481 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} - 41 T^{2} + 180 T - 191)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 27 T^{6} + 44 T^{5} + \cdots + 1225 \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{7} + 122 T^{6} + \cdots + 477481 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 9 T^{2} + 158 T + 155)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} - 64 T^{2} + 352 T - 448)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + 340 T^{6} + \cdots + 63123025 \) Copy content Toggle raw display
$53$ \( T^{8} + 12 T^{7} + 137 T^{6} + \cdots + 4225 \) Copy content Toggle raw display
$59$ \( T^{8} + 14 T^{7} + 180 T^{6} + \cdots + 93025 \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + 304 T^{6} + \cdots + 469225 \) Copy content Toggle raw display
$67$ \( T^{8} + 2 T^{7} + 165 T^{6} + \cdots + 5257849 \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} - 130 T^{2} - 62 T + 3569)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 16 T^{7} + 201 T^{6} + \cdots + 93025 \) Copy content Toggle raw display
$79$ \( T^{8} - 8 T^{7} + 172 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$83$ \( (T^{4} - 20 T^{3} - 37 T^{2} + 1352 T + 4555)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + 248 T^{6} + \cdots + 5017600 \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} - 224 T^{2} + 86 T + 815)^{2} \) Copy content Toggle raw display
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