Properties

Label 630.2.i.f
Level $630$
Weight $2$
Character orbit 630.i
Analytic conductor $5.031$
Analytic rank $0$
Dimension $12$
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(121,630)]
 
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([2, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("630.121");
 
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.i (of order \(3\), 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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: 12.0.91830304992969.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + x^{10} + 4x^{9} - 7x^{8} + x^{7} + 7x^{6} + 2x^{5} - 28x^{4} + 32x^{3} + 16x^{2} - 64x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + x^{10} + 4x^{9} - 7x^{8} + x^{7} + 7x^{6} + 2x^{5} - 28x^{4} + 32x^{3} + 16x^{2} - 64x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 17 \nu^{9} + 76 \nu^{8} - 73 \nu^{7} + 15 \nu^{6} + 73 \nu^{5} - 2 \nu^{4} - 100 \nu^{3} - 112 \nu^{2} + 784 \nu - 960 ) / 288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} + 10 \nu^{10} - 43 \nu^{9} + 56 \nu^{8} - 11 \nu^{7} - 3 \nu^{6} - \nu^{5} + 98 \nu^{4} + 16 \nu^{3} - 344 \nu^{2} + 992 \nu - 672 ) / 288 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 4\nu^{9} + 5\nu^{8} - 2\nu^{7} + \nu^{6} - 5\nu^{5} + 13\nu^{4} + 8\nu^{3} - 36\nu^{2} + 60\nu - 56 ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3 \nu^{11} - 4 \nu^{10} - 9 \nu^{9} + 22 \nu^{8} - 21 \nu^{7} - 19 \nu^{6} + 31 \nu^{5} + 36 \nu^{4} - 128 \nu^{3} + 16 \nu^{2} + 208 \nu - 288 ) / 96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{11} + 10 \nu^{10} - 71 \nu^{9} + 104 \nu^{8} - 7 \nu^{7} - 131 \nu^{6} + 51 \nu^{5} + 182 \nu^{4} - 4 \nu^{3} - 488 \nu^{2} + 1264 \nu - 640 ) / 288 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 11 \nu^{10} + 13 \nu^{9} - \nu^{8} - \nu^{7} - 2 \nu^{6} + 24 \nu^{5} + 11 \nu^{4} - 94 \nu^{3} + 172 \nu^{2} - 152 \nu + 128 ) / 144 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{11} + 15 \nu^{10} - 32 \nu^{9} + 27 \nu^{8} + 2 \nu^{7} - 7 \nu^{6} + 5 \nu^{5} + 9 \nu^{4} + 62 \nu^{3} - 300 \nu^{2} + 352 \nu - 320 ) / 144 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10 \nu^{11} + 11 \nu^{10} - 14 \nu^{9} - 5 \nu^{8} - 4 \nu^{7} - 3 \nu^{6} - 5 \nu^{5} - 41 \nu^{4} + 188 \nu^{3} - 160 \nu^{2} + 88 \nu ) / 144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7 \nu^{11} - 20 \nu^{10} + 17 \nu^{9} + 14 \nu^{8} - 35 \nu^{7} - 3 \nu^{6} + 41 \nu^{5} - 10 \nu^{4} - 266 \nu^{3} + 208 \nu^{2} - 16 \nu - 288 ) / 144 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7 \nu^{11} - 16 \nu^{10} + 31 \nu^{9} - 38 \nu^{8} + 11 \nu^{7} + 13 \nu^{6} + 3 \nu^{5} - 36 \nu^{4} - 84 \nu^{3} + 376 \nu^{2} - 496 \nu + 448 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25 \nu^{11} + 54 \nu^{10} - 25 \nu^{9} - 24 \nu^{8} + 55 \nu^{7} + 67 \nu^{6} - 131 \nu^{5} - 150 \nu^{4} + 436 \nu^{3} - 504 \nu^{2} + 32 \nu - 64 ) / 288 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{3} + 2\beta_{2} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{5} - 2\beta_{4} + \beta_{3} - \beta_{2} + 2\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7 \beta_{11} - 3 \beta_{10} - 7 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + 2 \beta_{6} - 5 \beta_{5} - 7 \beta_{3} + 2 \beta_{2} + 3 \beta _1 - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} + 7 \beta_{7} + 7 \beta_{6} - 7 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} + 2 \beta _1 + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6 \beta_{11} + 3 \beta_{9} + 5 \beta_{7} + 23 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} - 4 \beta_{3} + 11 \beta_{2} - 6 \beta _1 - 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{7} + 9 \beta_{6} + 4 \beta_{5} - 3 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 3 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{11} - 19 \beta_{10} - 9 \beta_{9} - 24 \beta_{8} - 23 \beta_{7} - 23 \beta_{6} - 13 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} + 13 \beta _1 - 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2 \beta_{11} - 4 \beta_{10} - 43 \beta_{9} - 38 \beta_{8} - 16 \beta_{7} - 46 \beta_{6} - 5 \beta_{5} + 32 \beta_{4} - 17 \beta_{3} - 27 \beta_{2} + 28 \beta _1 + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 28 \beta_{11} + 21 \beta_{10} - 13 \beta_{9} - 35 \beta_{8} + 26 \beta_{7} - 85 \beta_{6} + 7 \beta_{5} - 21 \beta_{4} - \beta_{3} - 25 \beta_{2} + 30 \beta _1 - 5 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.989378 + 1.01051i
0.683706 1.23796i
1.04029 + 0.958022i
1.36982 0.351572i
−1.41396 0.0268737i
0.309529 + 1.37992i
−0.989378 1.01051i
0.683706 + 1.23796i
1.04029 0.958022i
1.36982 + 0.351572i
−1.41396 + 0.0268737i
0.309529 1.37992i
−1.00000 −1.56899 0.733675i 1.00000 −0.500000 0.866025i 1.56899 + 0.733675i −2.56238 0.658939i −1.00000 1.92344 + 2.30225i 0.500000 + 0.866025i
121.2 −1.00000 −0.554872 1.64077i 1.00000 −0.500000 0.866025i 0.554872 + 1.64077i 2.32383 + 1.26483i −1.00000 −2.38423 + 1.82083i 0.500000 + 0.866025i
121.3 −1.00000 −0.433986 + 1.67680i 1.00000 −0.500000 0.866025i 0.433986 1.67680i 1.23855 2.33795i −1.00000 −2.62331 1.45541i 0.500000 + 0.866025i
121.4 −1.00000 0.478015 + 1.66478i 1.00000 −0.500000 0.866025i −0.478015 1.66478i −2.56238 0.658939i −1.00000 −2.54300 + 1.59158i 0.500000 + 0.866025i
121.5 −1.00000 1.35166 1.08306i 1.00000 −0.500000 0.866025i −1.35166 + 1.08306i 2.32383 + 1.26483i −1.00000 0.653981 2.92785i 0.500000 + 0.866025i
121.6 −1.00000 1.72817 + 0.115916i 1.00000 −0.500000 0.866025i −1.72817 0.115916i 1.23855 2.33795i −1.00000 2.97313 + 0.400645i 0.500000 + 0.866025i
151.1 −1.00000 −1.56899 + 0.733675i 1.00000 −0.500000 + 0.866025i 1.56899 0.733675i −2.56238 + 0.658939i −1.00000 1.92344 2.30225i 0.500000 0.866025i
151.2 −1.00000 −0.554872 + 1.64077i 1.00000 −0.500000 + 0.866025i 0.554872 1.64077i 2.32383 1.26483i −1.00000 −2.38423 1.82083i 0.500000 0.866025i
151.3 −1.00000 −0.433986 1.67680i 1.00000 −0.500000 + 0.866025i 0.433986 + 1.67680i 1.23855 + 2.33795i −1.00000 −2.62331 + 1.45541i 0.500000 0.866025i
151.4 −1.00000 0.478015 1.66478i 1.00000 −0.500000 + 0.866025i −0.478015 + 1.66478i −2.56238 + 0.658939i −1.00000 −2.54300 1.59158i 0.500000 0.866025i
151.5 −1.00000 1.35166 + 1.08306i 1.00000 −0.500000 + 0.866025i −1.35166 1.08306i 2.32383 1.26483i −1.00000 0.653981 + 2.92785i 0.500000 0.866025i
151.6 −1.00000 1.72817 0.115916i 1.00000 −0.500000 + 0.866025i −1.72817 + 0.115916i 1.23855 + 2.33795i −1.00000 2.97313 0.400645i 0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.2.i.f 12
3.b odd 2 1 1890.2.i.h 12
7.c even 3 1 630.2.l.h yes 12
9.c even 3 1 630.2.l.h yes 12
9.d odd 6 1 1890.2.l.f 12
21.h odd 6 1 1890.2.l.f 12
63.h even 3 1 inner 630.2.i.f 12
63.j odd 6 1 1890.2.i.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.i.f 12 1.a even 1 1 trivial
630.2.i.f 12 63.h even 3 1 inner
630.2.l.h yes 12 7.c even 3 1
630.2.l.h yes 12 9.c even 3 1
1890.2.i.h 12 3.b odd 2 1
1890.2.i.h 12 63.j odd 6 1
1890.2.l.f 12 9.d odd 6 1
1890.2.l.f 12 21.h odd 6 1

Hecke kernels

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

\( T_{11}^{12} + 7 T_{11}^{11} + 57 T_{11}^{10} + 208 T_{11}^{9} + 1120 T_{11}^{8} + 3390 T_{11}^{7} + 14424 T_{11}^{6} + 30303 T_{11}^{5} + 87408 T_{11}^{4} + 128304 T_{11}^{3} + 332424 T_{11}^{2} + 354294 T_{11} + 531441 \) Copy content Toggle raw display
\( T_{13}^{12} + 2 T_{13}^{11} + 34 T_{13}^{10} - 28 T_{13}^{9} + 808 T_{13}^{8} - 67 T_{13}^{7} + 4072 T_{13}^{6} - 5050 T_{13}^{5} + 14470 T_{13}^{4} - 6420 T_{13}^{3} + 2973 T_{13}^{2} + 153 T_{13} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 2 T^{11} + 4 T^{10} - 9 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} - 4 T^{4} + 31 T^{3} + \cdots + 343)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 7 T^{11} + 57 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + 34 T^{10} - 28 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{12} - T^{11} + 72 T^{10} + \cdots + 49660209 \) Copy content Toggle raw display
$19$ \( T^{12} + 2 T^{11} + 34 T^{10} - 28 T^{9} + \cdots + 9 \) Copy content Toggle raw display
$23$ \( T^{12} + 9 T^{11} + 90 T^{10} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{12} - 3 T^{11} + 87 T^{10} + \cdots + 531441 \) Copy content Toggle raw display
$31$ \( (T^{6} - 9 T^{5} - 36 T^{4} + 308 T^{3} + \cdots + 109)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 6 T^{11} + 111 T^{10} + \cdots + 32137561 \) Copy content Toggle raw display
$41$ \( T^{12} + 11 T^{11} + \cdots + 44545901481 \) Copy content Toggle raw display
$43$ \( T^{12} - 23 T^{11} + 371 T^{10} + \cdots + 11771761 \) Copy content Toggle raw display
$47$ \( (T^{6} + T^{5} - 62 T^{4} - 159 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 4 T^{11} + 249 T^{10} + \cdots + 503418969 \) Copy content Toggle raw display
$59$ \( (T^{6} + 11 T^{5} - 203 T^{4} + \cdots + 276399)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 25 T^{5} + 134 T^{4} + 674 T^{3} + \cdots - 3821)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 2 T^{5} - 225 T^{4} + 427 T^{3} + \cdots + 21339)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 11 T^{5} - 182 T^{4} + 1860 T^{3} + \cdots + 27)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} - 24 T^{11} + 480 T^{10} - 3596 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{6} - T^{5} - 295 T^{4} + 1310 T^{3} + \cdots + 247267)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 4 T^{11} + 204 T^{10} + \cdots + 130439241 \) Copy content Toggle raw display
$89$ \( T^{12} - 2 T^{11} + 279 T^{10} + \cdots + 4549689 \) Copy content Toggle raw display
$97$ \( (T^{6} + 18 T^{5} + 255 T^{4} + 1256 T^{3} + \cdots + 49)^{2} \) Copy content Toggle raw display
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