Properties

Label 331.2.a.c
Level $331$
Weight $2$
Character orbit 331.a
Self dual yes
Analytic conductor $2.643$
Analytic rank $1$
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] = [331,2,Mod(1,331)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(331, 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("331.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 331 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 331.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: \(2.64304830690\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.30653489.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 7x^{5} - x^{4} + 11x^{3} + 3x^{2} - 3x - 1 \) 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 7x^{5} - x^{4} + 11x^{3} + 3x^{2} - 3x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{6} + 7\nu^{4} + \nu^{3} - 11\nu^{2} - 2\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{6} - 7\nu^{4} - \nu^{3} + 11\nu^{2} + 3\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{6} - \nu^{5} - 13\nu^{4} + 4\nu^{3} + 17\nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{6} - \nu^{5} - 14\nu^{4} + 5\nu^{3} + 22\nu^{2} - 5\nu - 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 3\nu^{6} - \nu^{5} - 20\nu^{4} + 4\nu^{3} + 28\nu^{2} - 2\nu - 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\nu^{6} - 2\nu^{5} - 20\nu^{4} + 10\nu^{3} + 29\nu^{2} - 8\nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - \beta_{3} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{3} + 3\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{6} + \beta_{5} - 6\beta_{4} - 5\beta_{3} - \beta_{2} - 5\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} - \beta_{4} - 7\beta_{3} + 12\beta_{2} + 17\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 24\beta_{6} + 8\beta_{5} - 31\beta_{4} - 25\beta_{3} - 6\beta_{2} - 23\beta _1 + 37 \) 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
−1.23899
−0.527302
1.39627
−2.13101
−0.376996
0.598638
2.27939
−2.62059 0.698381 4.86747 0.835678 −1.83017 −2.05608 −7.51444 −2.51226 −2.18997
1.2 −2.02189 −2.70896 2.08803 −2.51900 5.47722 3.28257 −0.177996 4.33848 5.09315
1.3 −0.821306 3.12572 −1.32546 −3.90438 −2.56717 −5.01387 2.73122 6.77010 3.20669
1.4 0.292130 0.375694 −1.91466 −2.89562 0.109751 3.78194 −1.14359 −2.85885 −0.845898
1.5 0.525598 −0.783973 −1.72375 1.43090 −0.412055 −2.89293 −1.95719 −2.38539 0.752077
1.6 0.819257 0.482667 −1.32882 −1.83510 0.395428 −2.84484 −2.72716 −2.76703 −1.50342
1.7 1.82680 −1.18952 1.33718 −4.11246 −2.17301 −2.25679 −1.21084 −1.58504 −7.51263
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(331\) \(1\)

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 331.2.a.c 7
3.b odd 2 1 2979.2.a.h 7
4.b odd 2 1 5296.2.a.m 7
5.b even 2 1 8275.2.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
331.2.a.c 7 1.a even 1 1 trivial
2979.2.a.h 7 3.b odd 2 1
5296.2.a.m 7 4.b odd 2 1
8275.2.a.c 7 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 2T_{2}^{6} - 6T_{2}^{5} - 8T_{2}^{4} + 11T_{2}^{3} + 3T_{2}^{2} - 5T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(331))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + 2 T^{6} - 6 T^{5} - 8 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{7} - 10 T^{5} - 3 T^{4} + 12 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{7} + 13 T^{6} + 58 T^{5} + \cdots + 257 \) Copy content Toggle raw display
$7$ \( T^{7} + 8 T^{6} - 6 T^{5} - 184 T^{4} + \cdots + 2377 \) Copy content Toggle raw display
$11$ \( T^{7} + 10 T^{6} + 4 T^{5} + \cdots + 1857 \) Copy content Toggle raw display
$13$ \( T^{7} - 5 T^{6} - 30 T^{5} + 189 T^{4} + \cdots - 79 \) Copy content Toggle raw display
$17$ \( T^{7} + 17 T^{6} + 75 T^{5} + \cdots - 13061 \) Copy content Toggle raw display
$19$ \( T^{7} + 19 T^{6} + 76 T^{5} + \cdots - 29763 \) Copy content Toggle raw display
$23$ \( T^{7} - 15 T^{6} + 43 T^{5} + \cdots + 1543 \) Copy content Toggle raw display
$29$ \( T^{7} + 24 T^{6} + 163 T^{5} + \cdots + 2861 \) Copy content Toggle raw display
$31$ \( T^{7} - 4 T^{6} - 90 T^{5} + \cdots - 1399 \) Copy content Toggle raw display
$37$ \( T^{7} - 2 T^{6} - 159 T^{5} + \cdots - 182141 \) Copy content Toggle raw display
$41$ \( T^{7} + 18 T^{6} + 50 T^{5} + \cdots - 5367 \) Copy content Toggle raw display
$43$ \( T^{7} + 8 T^{6} - 105 T^{5} + \cdots - 56737 \) Copy content Toggle raw display
$47$ \( T^{7} - 10 T^{6} - 121 T^{5} + \cdots + 8779 \) Copy content Toggle raw display
$53$ \( T^{7} + 21 T^{6} + 84 T^{5} + \cdots + 2531 \) Copy content Toggle raw display
$59$ \( T^{7} + 17 T^{6} - 59 T^{5} + \cdots + 48049 \) Copy content Toggle raw display
$61$ \( T^{7} + 12 T^{6} - 59 T^{5} + \cdots + 7193 \) Copy content Toggle raw display
$67$ \( T^{7} + 16 T^{6} - 104 T^{5} + \cdots - 15947 \) Copy content Toggle raw display
$71$ \( T^{7} - 4 T^{6} - 82 T^{5} + 296 T^{4} + \cdots + 383 \) Copy content Toggle raw display
$73$ \( T^{7} - 11 T^{6} - 195 T^{5} + \cdots + 13193 \) Copy content Toggle raw display
$79$ \( T^{7} + 2 T^{6} - 438 T^{5} + \cdots - 2733417 \) Copy content Toggle raw display
$83$ \( T^{7} + 4 T^{6} - 94 T^{5} - 67 T^{4} + \cdots - 219 \) Copy content Toggle raw display
$89$ \( T^{7} + 28 T^{6} + 49 T^{5} + \cdots - 363183 \) Copy content Toggle raw display
$97$ \( T^{7} + 32 T^{6} + 160 T^{5} + \cdots + 867563 \) Copy content Toggle raw display
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