Properties

Label 157.2.a.b
Level $157$
Weight $2$
Character orbit 157.a
Self dual yes
Analytic conductor $1.254$
Analytic rank $0$
Dimension $7$
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] = [157,2,Mod(1,157)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(157, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("157.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 157 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 157.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: \(1.25365131173\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 7x^{5} + 9x^{4} + 17x^{3} - 5x^{2} - 10x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 7x^{5} + 9x^{4} + 17x^{3} - 5x^{2} - 10x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 5\nu^{2} + 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 14\nu^{3} + 9\nu^{2} - 11\nu - 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 2\nu^{5} - 7\nu^{4} + 10\nu^{3} + 15\nu^{2} - 8\nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_{2} + 8\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - \beta_{5} + 2\beta_{4} + 6\beta_{3} + 10\beta_{2} + 27\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{6} - 2\beta_{5} + 11\beta_{4} + 9\beta_{3} + 37\beta_{2} + 53\beta _1 + 51 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.57182
2.32861
0.961783
−0.257490
−0.633468
−1.20581
−1.76543
−1.57182 2.80998 0.470605 0.844339 −4.41677 0.249425 2.40393 4.89598 −1.32715
1.2 −1.32861 −1.67897 −0.234804 −0.345097 2.23069 0.934378 2.96918 −0.181064 0.458498
1.3 0.0382168 1.26443 −1.99854 2.23374 0.0483225 2.48880 −0.152811 −1.40121 0.0853665
1.4 1.25749 3.17498 −0.418719 −3.14450 3.99251 0.0719361 −3.04151 7.08051 −3.95418
1.5 1.63347 1.14188 0.668218 1.68642 1.86523 −2.39649 −2.17542 −1.69611 2.75471
1.6 2.20581 −1.81425 2.86562 1.29760 −4.00190 2.99313 1.90939 0.291498 2.86226
1.7 2.76543 0.101944 5.64762 −3.57250 0.281918 −3.34117 10.0873 −2.98961 −9.87951
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(157\) \(-1\)

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 157.2.a.b 7
3.b odd 2 1 1413.2.a.e 7
4.b odd 2 1 2512.2.a.h 7
5.b even 2 1 3925.2.a.g 7
7.b odd 2 1 7693.2.a.d 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
157.2.a.b 7 1.a even 1 1 trivial
1413.2.a.e 7 3.b odd 2 1
2512.2.a.h 7 4.b odd 2 1
3925.2.a.g 7 5.b even 2 1
7693.2.a.d 7 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 5T_{2}^{6} + 2T_{2}^{5} + 21T_{2}^{4} - 22T_{2}^{3} - 21T_{2}^{2} + 27T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 5 T^{6} + 2 T^{5} + 21 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$3$ \( T^{7} - 5 T^{6} - T^{5} + 31 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{7} + T^{6} - 16 T^{5} + 3 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{7} - T^{6} - 16 T^{5} + 19 T^{4} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{7} - 10 T^{6} + 28 T^{5} - 9 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$13$ \( T^{7} + 5 T^{6} - 16 T^{5} - 63 T^{4} + \cdots + 113 \) Copy content Toggle raw display
$17$ \( T^{7} - 5 T^{6} - 44 T^{5} + 152 T^{4} + \cdots + 413 \) Copy content Toggle raw display
$19$ \( T^{7} + 3 T^{6} - 95 T^{5} + \cdots + 5296 \) Copy content Toggle raw display
$23$ \( T^{7} - 15 T^{6} + 54 T^{5} + \cdots - 10073 \) Copy content Toggle raw display
$29$ \( T^{7} - 8 T^{6} - 76 T^{5} + \cdots - 18992 \) Copy content Toggle raw display
$31$ \( T^{7} + 13 T^{6} + 33 T^{5} + \cdots - 436 \) Copy content Toggle raw display
$37$ \( T^{7} + 15 T^{6} - 46 T^{5} + \cdots + 39539 \) Copy content Toggle raw display
$41$ \( T^{7} - 3 T^{6} - 175 T^{5} + \cdots - 47404 \) Copy content Toggle raw display
$43$ \( T^{7} - 11 T^{6} - 100 T^{5} + \cdots + 475171 \) Copy content Toggle raw display
$47$ \( T^{7} - 8 T^{6} - 70 T^{5} + \cdots - 13444 \) Copy content Toggle raw display
$53$ \( T^{7} - 9 T^{6} - 23 T^{5} + \cdots + 8612 \) Copy content Toggle raw display
$59$ \( T^{7} - 31 T^{6} + 176 T^{5} + \cdots + 863917 \) Copy content Toggle raw display
$61$ \( T^{7} + 6 T^{6} - 166 T^{5} + \cdots - 385772 \) Copy content Toggle raw display
$67$ \( T^{7} - 4 T^{6} - 161 T^{5} + \cdots + 317732 \) Copy content Toggle raw display
$71$ \( T^{7} - 14 T^{6} - 220 T^{5} + \cdots + 1683928 \) Copy content Toggle raw display
$73$ \( T^{7} + 3 T^{6} - 178 T^{5} + \cdots - 11564 \) Copy content Toggle raw display
$79$ \( T^{7} - 6 T^{6} - 213 T^{5} + \cdots - 169324 \) Copy content Toggle raw display
$83$ \( T^{7} - 41 T^{6} + 583 T^{5} + \cdots + 1053284 \) Copy content Toggle raw display
$89$ \( T^{7} + 13 T^{6} - 225 T^{5} + \cdots + 4949 \) Copy content Toggle raw display
$97$ \( T^{7} + 12 T^{6} - 421 T^{5} + \cdots - 13926728 \) Copy content Toggle raw display
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