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Properties

Label 74.2.f.b
Level $74$
Weight $2$
Character orbit 74.f
Analytic conductor $0.591$
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] = [74,2,Mod(7,74)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(74, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("74.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 74 = 2 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 74.f (of order \(9\), degree \(6\), 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: \(0.590892974957\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -12\nu^{11} + 263\nu^{9} - 2568\nu^{7} + 13644\nu^{5} - 40150\nu^{3} + 52500\nu + 3125 ) / 6250 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} + 125\nu - 7500 ) / 250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} - 125\nu - 7500 ) / 250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 45 \nu^{10} - 24 \nu^{9} + 955 \nu^{8} + 264 \nu^{7} - 8880 \nu^{6} - 1687 \nu^{5} + 46040 \nu^{4} + 6600 \nu^{3} - 133000 \nu^{2} - 11875 \nu + 175000 ) / 6250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 45 \nu^{10} - 24 \nu^{9} - 955 \nu^{8} + 264 \nu^{7} + 8880 \nu^{6} - 1687 \nu^{5} - 46040 \nu^{4} + 6600 \nu^{3} + 133000 \nu^{2} - 11875 \nu - 175000 ) / 6250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12 \nu^{11} + 40 \nu^{10} - 188 \nu^{9} - 835 \nu^{8} + 1393 \nu^{7} + 7560 \nu^{6} - 5719 \nu^{5} - 37605 \nu^{4} + 13000 \nu^{3} + 103125 \nu^{2} - 11875 \nu - 125000 ) / 6250 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12 \nu^{11} - 40 \nu^{10} - 188 \nu^{9} + 835 \nu^{8} + 1393 \nu^{7} - 7560 \nu^{6} - 5719 \nu^{5} + 37605 \nu^{4} + 13000 \nu^{3} - 103125 \nu^{2} - 11875 \nu + 125000 ) / 6250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} - 20 \nu^{10} - 24 \nu^{9} + 355 \nu^{8} + 264 \nu^{7} - 2905 \nu^{6} - 1687 \nu^{5} + 13240 \nu^{4} + 5975 \nu^{3} - 33000 \nu^{2} - 10000 \nu + 34375 ) / 1250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} - 20 \nu^{10} + 24 \nu^{9} + 355 \nu^{8} - 264 \nu^{7} - 2905 \nu^{6} + 1687 \nu^{5} + 13240 \nu^{4} - 5975 \nu^{3} - 33000 \nu^{2} + 10000 \nu + 34375 ) / 1250 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28 \nu^{11} + 100 \nu^{10} - 572 \nu^{9} - 1775 \nu^{8} + 4992 \nu^{7} + 14525 \nu^{6} - 23961 \nu^{5} - 66200 \nu^{4} + 62975 \nu^{3} + 168125 \nu^{2} - 72500 \nu - 184375 ) / 6250 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28 \nu^{11} - 100 \nu^{10} - 572 \nu^{9} + 1775 \nu^{8} + 4992 \nu^{7} - 14525 \nu^{6} - 23961 \nu^{5} + 66200 \nu^{4} + 62975 \nu^{3} - 168125 \nu^{2} - 72500 \nu + 184375 ) / 6250 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( -\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + 5\beta_{5} + 5\beta_{4} - 3\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{10} + 11\beta_{9} - 11\beta_{8} + 35\beta_{5} + 35\beta_{4} - \beta_{3} + \beta_{2} - 8\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 26 \beta_{11} + 26 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 50 \beta_{7} + 50 \beta_{6} - 55 \beta_{5} + 55 \beta_{4} + 15 \beta_{3} + 15 \beta_{2} - 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 19 \beta_{11} - 19 \beta_{10} + 65 \beta_{9} - 65 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 125 \beta_{5} + 125 \beta_{4} + 46 \beta_{3} - 46 \beta_{2} - 112 \beta _1 + 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 38 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} + 14 \beta_{8} - 230 \beta_{7} + 230 \beta_{6} - 325 \beta_{5} + 325 \beta_{4} + 121 \beta_{3} + 121 \beta_{2} - 268 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 192 \beta_{11} - 192 \beta_{10} + 218 \beta_{9} - 218 \beta_{8} + 120 \beta_{7} + 120 \beta_{6} + 70 \beta_{5} + 70 \beta_{4} + 282 \beta_{3} - 282 \beta_{2} - 826 \beta _1 + 413 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 118 \beta_{11} - 118 \beta_{10} - 430 \beta_{9} - 430 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} - 1090 \beta_{5} + 1090 \beta_{4} + 631 \beta_{3} + 631 \beta_{2} - 1292 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1279 \beta_{11} - 1279 \beta_{10} + 29 \beta_{9} - 29 \beta_{8} + 1560 \beta_{7} + 1560 \beta_{6} - 2150 \beta_{5} - 2150 \beta_{4} + 862 \beta_{3} - 862 \beta_{2} - 3752 \beta _1 + 1876 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\) \(39\)
\(\chi(n)\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.00752 + 0.984808i
−2.00752 + 0.984808i
−2.14169 + 0.642788i
2.14169 + 0.642788i
−2.14169 − 0.642788i
2.14169 − 0.642788i
−2.20976 + 0.342020i
2.20976 + 0.342020i
2.00752 − 0.984808i
−2.00752 − 0.984808i
−2.20976 − 0.342020i
2.20976 − 0.342020i
−0.939693 + 0.342020i −2.04963 − 0.746005i 0.766044 − 0.642788i −0.326352 − 1.85083i 2.18117 −0.711830 − 4.03699i −0.500000 + 0.866025i 1.34633 + 1.12971i 0.939693 + 1.62760i
7.2 −0.939693 + 0.342020i 1.72328 + 0.627223i 0.766044 − 0.642788i −0.326352 − 1.85083i −1.83388 0.598489 + 3.39420i −0.500000 + 0.866025i 0.278152 + 0.233397i 0.939693 + 1.62760i
9.1 0.173648 + 0.984808i −0.238878 + 1.35474i −0.939693 + 0.342020i 0.266044 + 0.223238i −1.37564 −0.365982 − 0.307095i −0.500000 − 0.866025i 1.04081 + 0.378824i −0.173648 + 0.300767i
9.2 0.173648 + 0.984808i 0.504922 − 2.86356i −0.939693 + 0.342020i 0.266044 + 0.223238i 2.90773 0.773586 + 0.649116i −0.500000 − 0.866025i −5.12593 − 1.86569i −0.173648 + 0.300767i
33.1 0.173648 − 0.984808i −0.238878 − 1.35474i −0.939693 − 0.342020i 0.266044 − 0.223238i −1.37564 −0.365982 + 0.307095i −0.500000 + 0.866025i 1.04081 − 0.378824i −0.173648 − 0.300767i
33.2 0.173648 − 0.984808i 0.504922 + 2.86356i −0.939693 − 0.342020i 0.266044 − 0.223238i 2.90773 0.773586 − 0.649116i −0.500000 + 0.866025i −5.12593 + 1.86569i −0.173648 − 0.300767i
49.1 0.766044 − 0.642788i −2.41262 − 2.02443i 0.173648 − 0.984808i −1.43969 + 0.524005i −3.14945 4.53424 − 1.65033i −0.500000 − 0.866025i 1.20148 + 6.81391i −0.766044 + 1.32683i
49.2 0.766044 − 0.642788i 0.972925 + 0.816381i 0.173648 − 0.984808i −1.43969 + 0.524005i 1.27006 −1.82850 + 0.665520i −0.500000 − 0.866025i −0.240839 − 1.36587i −0.766044 + 1.32683i
53.1 −0.939693 − 0.342020i −2.04963 + 0.746005i 0.766044 + 0.642788i −0.326352 + 1.85083i 2.18117 −0.711830 + 4.03699i −0.500000 − 0.866025i 1.34633 − 1.12971i 0.939693 − 1.62760i
53.2 −0.939693 − 0.342020i 1.72328 − 0.627223i 0.766044 + 0.642788i −0.326352 + 1.85083i −1.83388 0.598489 − 3.39420i −0.500000 − 0.866025i 0.278152 − 0.233397i 0.939693 − 1.62760i
71.1 0.766044 + 0.642788i −2.41262 + 2.02443i 0.173648 + 0.984808i −1.43969 − 0.524005i −3.14945 4.53424 + 1.65033i −0.500000 + 0.866025i 1.20148 − 6.81391i −0.766044 − 1.32683i
71.2 0.766044 + 0.642788i 0.972925 − 0.816381i 0.173648 + 0.984808i −1.43969 − 0.524005i 1.27006 −1.82850 − 0.665520i −0.500000 + 0.866025i −0.240839 + 1.36587i −0.766044 − 1.32683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.f.b ✓ 12
3.b odd 2 1 666.2.x.g 12
4.b odd 2 1 592.2.bc.d 12
37.f even 9 1 inner 74.2.f.b ✓ 12
37.f even 9 1 2738.2.a.t 6
37.h even 18 1 2738.2.a.q 6
111.p odd 18 1 666.2.x.g 12
148.p odd 18 1 592.2.bc.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.b ✓ 12 1.a even 1 1 trivial
74.2.f.b ✓ 12 37.f even 9 1 inner
592.2.bc.d 12 4.b odd 2 1
592.2.bc.d 12 148.p odd 18 1
666.2.x.g 12 3.b odd 2 1
666.2.x.g 12 111.p odd 18 1
2738.2.a.q 6 37.h even 18 1
2738.2.a.t 6 37.f even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 3 T_{3}^{11} + 6 T_{3}^{10} + 8 T_{3}^{9} + 24 T_{3}^{8} - 126 T_{3}^{7} - 151 T_{3}^{6} + 504 T_{3}^{5} + 384 T_{3}^{4} - 512 T_{3}^{3} + 1536 T_{3}^{2} - 3072 T_{3} + 4096 \) acting on \(S_{2}^{\mathrm{new}}(74, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + 6 T^{10} + 8 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + 3 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + 24 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( T^{12} - 3 T^{11} + 33 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 516961 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + 18 T^{10} + \cdots + 26569 \) Copy content Toggle raw display
$19$ \( T^{12} + 3 T^{11} - 9 T^{10} + 228 T^{9} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{12} + 21 T^{11} + 297 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{12} - 6 T^{11} + 60 T^{10} + \cdots + 1369 \) Copy content Toggle raw display
$31$ \( (T^{6} - 21 T^{5} + 24 T^{4} + \cdots + 130112)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 3 T^{11} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( T^{12} + 21 T^{11} + \cdots + 466905664 \) Copy content Toggle raw display
$43$ \( (T^{6} - 18 T^{5} + 51 T^{4} + 639 T^{3} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 6890328064 \) Copy content Toggle raw display
$53$ \( T^{12} + 6 T^{11} + 6 T^{10} + 109 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$59$ \( T^{12} + 6 T^{11} + 12 T^{10} + \cdots + 1183744 \) Copy content Toggle raw display
$61$ \( T^{12} + 18 T^{11} + \cdots + 2801373184 \) Copy content Toggle raw display
$67$ \( T^{12} + 27 T^{11} + 444 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$71$ \( T^{12} + 18 T^{11} + 192 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$73$ \( (T^{6} - 27 T^{5} + 108 T^{4} + \cdots + 216289)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 12 T^{11} - 54 T^{10} + \cdots + 95883264 \) Copy content Toggle raw display
$83$ \( T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 1183744 \) Copy content Toggle raw display
$89$ \( T^{12} + 15 T^{11} + \cdots + 48144697561 \) Copy content Toggle raw display
$97$ \( T^{12} + 42 T^{11} + 1215 T^{10} + \cdots + 7529536 \) Copy content Toggle raw display
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