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Properties

Label 322.2.g.b
Level $322$
Weight $2$
Character orbit 322.g
Analytic conductor $2.571$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [322,2,Mod(45,322)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(322, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("322.45");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.g (of order \(6\), 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: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} - 1) q^{4} + (\beta_{7} - \beta_{4} + \beta_1) q^{5} + (\beta_{11} - \beta_{8}) q^{6} + ( - \beta_{15} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{13} - 2 \beta_{6} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{11} q^{3} + ( - \beta_{6} - 1) q^{4} + (\beta_{7} - \beta_{4} + \beta_1) q^{5} + (\beta_{11} - \beta_{8}) q^{6} + ( - \beta_{15} + \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{2} + \beta_1) q^{7} - q^{8} + (2 \beta_{13} - 2 \beta_{6} + \beta_{3}) q^{9} + (\beta_{15} - \beta_{9} + \beta_{7} - \beta_{4} - \beta_{2} + \beta_1) q^{10} + (\beta_{10} + 2 \beta_{9} - \beta_{7} + 2 \beta_{4} + \beta_{2}) q^{11} - \beta_{8} q^{12} + ( - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{3}) q^{13} + ( - \beta_{15} + \beta_{10} - \beta_{7} - \beta_{4} + \beta_1) q^{14} + ( - \beta_{15} + \beta_{9} - \beta_{7} - \beta_{5} + \beta_{2} + \beta_1) q^{15} + \beta_{6} q^{16} + ( - 2 \beta_{15} + \beta_{9} - 2 \beta_{7} + \beta_{2}) q^{17} + (\beta_{13} - 2 \beta_{6} + 2 \beta_{3} - 2) q^{18} + (2 \beta_{9} - \beta_{7} + \beta_{4} + 2 \beta_{2} + \beta_1) q^{19} + (\beta_{15} - \beta_{9} - \beta_{2}) q^{20} + (3 \beta_{15} - 2 \beta_{9} + \beta_{7} + \beta_{5} - \beta_{2} - 2 \beta_1) q^{21} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} + 2 \beta_{2} + \beta_1) q^{22} + (2 \beta_{14} - \beta_{10} - \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{23} - \beta_{11} q^{24} + ( - 2 \beta_{12} - \beta_{11} + 2 \beta_{8} + \beta_{6} - 2 \beta_{3} + 1) q^{25} + (\beta_{14} - \beta_{13} + 2 \beta_{12} + \beta_{11} - 2 \beta_{8}) q^{26} + ( - \beta_{14} - 2 \beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} + 2 \beta_{6} - 2 \beta_{3} + \cdots + 1) q^{27}+ \cdots + ( - 5 \beta_{15} + 5 \beta_{9} - 6 \beta_{7} - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 6 q^{3} - 8 q^{4} - 16 q^{8} + 10 q^{9} + 6 q^{12} - 8 q^{16} - 10 q^{18} + 8 q^{23} + 6 q^{24} + 2 q^{25} - 6 q^{26} - 16 q^{29} + 12 q^{31} + 8 q^{32} - 20 q^{36} - 2 q^{39} - 8 q^{46} - 6 q^{47} - 18 q^{49} + 4 q^{50} - 6 q^{52} + 18 q^{54} - 8 q^{58} + 36 q^{59} + 16 q^{64} - 12 q^{70} - 52 q^{71} - 10 q^{72} + 24 q^{73} + 30 q^{77} - 4 q^{78} - 20 q^{81} + 54 q^{82} + 80 q^{85} + 54 q^{87} - 16 q^{92} - 26 q^{93} - 6 q^{94} - 6 q^{95} - 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 12\nu^{13} + 73\nu^{11} + 312\nu^{9} + 1045\nu^{7} + 2808\nu^{5} + 5913\nu^{3} + 6561\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 100 \nu^{14} - 1713 \nu^{12} - 9244 \nu^{10} - 39975 \nu^{8} - 129853 \nu^{6} - 323595 \nu^{4} - 601668 \nu^{2} - 661203 ) / 103518 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 311 \nu^{15} - 3174 \nu^{13} - 16250 \nu^{11} - 59943 \nu^{9} - 179573 \nu^{7} - 452016 \nu^{5} - 856251 \nu^{3} - 769095 \nu ) / 310554 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 452 \nu^{15} - 4650 \nu^{13} - 23303 \nu^{11} - 95700 \nu^{9} - 286580 \nu^{7} - 673740 \nu^{5} - 1285470 \nu^{3} - 1237113 \nu ) / 310554 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 386 \nu^{14} - 3660 \nu^{12} - 19349 \nu^{10} - 74748 \nu^{8} - 227600 \nu^{6} - 541512 \nu^{4} - 985446 \nu^{2} - 1070901 ) / 103518 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 199 \nu^{15} - 2346 \nu^{13} - 12619 \nu^{11} - 49464 \nu^{9} - 149950 \nu^{7} - 367725 \nu^{5} - 668304 \nu^{3} - 737991 \nu ) / 103518 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 148 \nu^{14} + 1155 \nu^{12} + 5323 \nu^{10} + 18906 \nu^{8} + 55462 \nu^{6} + 118908 \nu^{4} + 194751 \nu^{2} + 142155 ) / 34506 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 386 \nu^{15} - 3660 \nu^{13} - 19349 \nu^{11} - 74748 \nu^{9} - 227600 \nu^{7} - 541512 \nu^{5} - 985446 \nu^{3} - 967383 \nu ) / 103518 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 149 \nu^{15} - 1525 \nu^{13} - 7784 \nu^{11} - 30151 \nu^{9} - 92585 \nu^{7} - 221938 \nu^{5} - 403254 \nu^{3} - 433269 \nu ) / 34506 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 361 \nu^{14} - 3498 \nu^{12} - 18316 \nu^{10} - 69813 \nu^{8} - 211591 \nu^{6} - 511680 \nu^{4} - 907875 \nu^{2} - 935793 ) / 34506 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1241 \nu^{14} + 12012 \nu^{12} + 60812 \nu^{10} + 231384 \nu^{8} + 703412 \nu^{6} + 1673847 \nu^{4} + 3027618 \nu^{2} + 3031182 ) / 103518 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1249 \nu^{14} + 11280 \nu^{12} + 56644 \nu^{10} + 216690 \nu^{8} + 651076 \nu^{6} + 1557621 \nu^{4} + 2756916 \nu^{2} + 2703132 ) / 103518 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1493 \nu^{14} + 13236 \nu^{12} + 65627 \nu^{10} + 247134 \nu^{8} + 730286 \nu^{6} + 1700397 \nu^{4} + 2902716 \nu^{2} + 2635335 ) / 103518 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2365 \nu^{15} - 20352 \nu^{13} - 102067 \nu^{11} - 382929 \nu^{9} - 1142809 \nu^{7} - 2673774 \nu^{5} - 4606875 \nu^{3} - 4299642 \nu ) / 310554 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{8} - \beta_{6} + \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{9} + 2\beta_{7} - \beta_{5} - 3\beta_{4} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} + 2\beta_{13} - 2\beta_{12} - 3\beta_{11} + \beta_{8} + 6\beta_{6} - \beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{15} + 2\beta_{10} + 11\beta_{9} - 8\beta_{7} + 2\beta_{5} + 5\beta_{4} + 9\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -6\beta_{14} - \beta_{13} - 5\beta_{12} - 4\beta_{11} + 18\beta_{8} - 8\beta_{6} - 10\beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{15} - 18\beta_{10} - 19\beta_{9} + 20\beta_{7} + 11\beta_{5} + 12\beta_{4} - 13\beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 31\beta_{14} - 3\beta_{13} + 8\beta_{12} + 34\beta_{11} - 56\beta_{8} - 27\beta_{6} + 12\beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -22\beta_{15} + 52\beta_{10} + 34\beta_{9} - 52\beta_{7} - 62\beta_{5} + 16\beta_{4} + 21\beta_{2} + 25\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -18\beta_{14} - 56\beta_{13} + 10\beta_{12} - 72\beta_{11} + 28\beta_{8} + 60\beta_{3} - 57 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 76\beta_{15} + 12\beta_{10} - 228\beta_{9} + 56\beta_{7} + 146\beta_{5} - 168\beta_{4} - 148\beta_{2} + 81\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 60\beta_{14} + 4\beta_{13} + 219\beta_{12} + 119\beta_{11} - 101\beta_{8} + 519\beta_{6} - 77\beta_{3} + 476 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -9\beta_{15} - 414\beta_{10} + 546\beta_{9} + 114\beta_{7} - 183\beta_{5} - 183\beta_{4} - 111\beta_{2} - 257\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -729\beta_{14} + 1134\beta_{13} - 926\beta_{12} - 89\beta_{11} + 935\beta_{8} - 1108\beta_{6} + 145\beta_{3} - 740 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 899 \beta_{15} + 1062 \beta_{10} + 277 \beta_{9} + 506 \beta_{7} - 316 \beta_{5} + 627 \beta_{4} + 1997 \beta_{2} + 1387 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1
0.105715 + 1.72882i
−0.105715 − 1.72882i
−0.452119 − 1.67200i
0.452119 + 1.67200i
1.36749 − 1.06300i
−1.36749 + 1.06300i
−0.956239 − 1.44416i
0.956239 + 1.44416i
0.105715 − 1.72882i
−0.105715 + 1.72882i
−0.452119 + 1.67200i
0.452119 − 1.67200i
1.36749 + 1.06300i
−1.36749 − 1.06300i
−0.956239 + 1.44416i
0.956239 − 1.44416i
0.500000 + 0.866025i −2.64992 − 1.52993i −0.500000 + 0.866025i −0.920495 − 1.59434i − 3.05986i −0.254505 + 2.63348i −1.00000 3.18138 + 5.51031i 0.920495 − 1.59434i
45.2 0.500000 + 0.866025i −2.64992 − 1.52993i −0.500000 + 0.866025i 0.920495 + 1.59434i − 3.05986i 0.254505 − 2.63348i −1.00000 3.18138 + 5.51031i −0.920495 + 1.59434i
45.3 0.500000 + 0.866025i −1.07743 − 0.622057i −0.500000 + 0.866025i −0.668984 − 1.15871i − 1.24411i −2.09536 + 1.61539i −1.00000 −0.726090 − 1.25762i 0.668984 − 1.15871i
45.4 0.500000 + 0.866025i −1.07743 − 0.622057i −0.500000 + 0.866025i 0.668984 + 1.15871i − 1.24411i 2.09536 − 1.61539i −1.00000 −0.726090 − 1.25762i −0.668984 + 1.15871i
45.5 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i −1.78715 − 3.09543i 0.106550i −0.672033 − 2.55898i −1.00000 −1.49432 − 2.58824i 1.78715 − 3.09543i
45.6 0.500000 + 0.866025i 0.0922753 + 0.0532752i −0.500000 + 0.866025i 1.78715 + 3.09543i 0.106550i 0.672033 + 2.55898i −1.00000 −1.49432 − 2.58824i −1.78715 + 3.09543i
45.7 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i −0.511122 − 0.885289i 2.46537i 2.61593 + 0.396147i −1.00000 1.53904 + 2.66569i 0.511122 − 0.885289i
45.8 0.500000 + 0.866025i 2.13508 + 1.23269i −0.500000 + 0.866025i 0.511122 + 0.885289i 2.46537i −2.61593 − 0.396147i −1.00000 1.53904 + 2.66569i −0.511122 + 0.885289i
229.1 0.500000 − 0.866025i −2.64992 + 1.52993i −0.500000 − 0.866025i −0.920495 + 1.59434i 3.05986i −0.254505 − 2.63348i −1.00000 3.18138 − 5.51031i 0.920495 + 1.59434i
229.2 0.500000 − 0.866025i −2.64992 + 1.52993i −0.500000 − 0.866025i 0.920495 − 1.59434i 3.05986i 0.254505 + 2.63348i −1.00000 3.18138 − 5.51031i −0.920495 − 1.59434i
229.3 0.500000 − 0.866025i −1.07743 + 0.622057i −0.500000 − 0.866025i −0.668984 + 1.15871i 1.24411i −2.09536 − 1.61539i −1.00000 −0.726090 + 1.25762i 0.668984 + 1.15871i
229.4 0.500000 − 0.866025i −1.07743 + 0.622057i −0.500000 − 0.866025i 0.668984 − 1.15871i 1.24411i 2.09536 + 1.61539i −1.00000 −0.726090 + 1.25762i −0.668984 − 1.15871i
229.5 0.500000 − 0.866025i 0.0922753 − 0.0532752i −0.500000 − 0.866025i −1.78715 + 3.09543i − 0.106550i −0.672033 + 2.55898i −1.00000 −1.49432 + 2.58824i 1.78715 + 3.09543i
229.6 0.500000 − 0.866025i 0.0922753 − 0.0532752i −0.500000 − 0.866025i 1.78715 − 3.09543i − 0.106550i 0.672033 − 2.55898i −1.00000 −1.49432 + 2.58824i −1.78715 − 3.09543i
229.7 0.500000 − 0.866025i 2.13508 − 1.23269i −0.500000 − 0.866025i −0.511122 + 0.885289i − 2.46537i 2.61593 − 0.396147i −1.00000 1.53904 − 2.66569i 0.511122 + 0.885289i
229.8 0.500000 − 0.866025i 2.13508 − 1.23269i −0.500000 − 0.866025i 0.511122 − 0.885289i − 2.46537i −2.61593 + 0.396147i −1.00000 1.53904 − 2.66569i −0.511122 − 0.885289i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 229.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.b odd 2 1 inner
161.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 322.2.g.b ✓ 16
7.c even 3 1 2254.2.c.b 16
7.d odd 6 1 inner 322.2.g.b ✓ 16
7.d odd 6 1 2254.2.c.b 16
23.b odd 2 1 inner 322.2.g.b ✓ 16
161.f odd 6 1 2254.2.c.b 16
161.g even 6 1 inner 322.2.g.b ✓ 16
161.g even 6 1 2254.2.c.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.b ✓ 16 1.a even 1 1 trivial
322.2.g.b ✓ 16 7.d odd 6 1 inner
322.2.g.b ✓ 16 23.b odd 2 1 inner
322.2.g.b ✓ 16 161.g even 6 1 inner
2254.2.c.b 16 7.c even 3 1
2254.2.c.b 16 7.d odd 6 1
2254.2.c.b 16 161.f odd 6 1
2254.2.c.b 16 161.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 3T_{3}^{7} - 4T_{3}^{6} - 21T_{3}^{5} + 33T_{3}^{4} + 105T_{3}^{3} + 68T_{3}^{2} - 15T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{7} - 4 T^{6} - 21 T^{5} + 33 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + 19 T^{14} + 270 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{16} + 9 T^{14} - 28 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 64 T^{14} + 2928 T^{12} + \cdots + 74805201 \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + 562 T^{4} + 1356 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 49 T^{14} + 2037 T^{12} + \cdots + 194481 \) Copy content Toggle raw display
$19$ \( T^{16} + 83 T^{14} + \cdots + 639128961 \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 11 T^{2} - 48 T - 15)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + 4 T^{6} + 48 T^{5} + \cdots + 225)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 62 T^{14} + 2886 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( (T^{8} + 131 T^{6} + 930 T^{4} + 1820 T^{2} + \cdots + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 80 T^{6} + 1756 T^{4} + 8073 T^{2} + \cdots + 729)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 3 T^{7} - 112 T^{6} - 345 T^{5} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} - 315 T^{14} + 69572 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$59$ \( (T^{8} - 18 T^{7} + 44 T^{6} + 1152 T^{5} + \cdots + 73441)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + 345 T^{14} + \cdots + 10756569837841 \) Copy content Toggle raw display
$67$ \( T^{16} - 405 T^{14} + \cdots + 2311278643521 \) Copy content Toggle raw display
$71$ \( (T^{4} + 13 T^{3} + 46 T^{2} + 36 T - 15)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 12 T^{7} - 80 T^{6} + \cdots + 10595025)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 415557398300625 \) Copy content Toggle raw display
$83$ \( (T^{8} - 196 T^{6} + 11713 T^{4} + \cdots + 1896129)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 121639745379681 \) Copy content Toggle raw display
$97$ \( (T^{8} - 900 T^{6} + 302197 T^{4} + \cdots + 2486119321)^{2} \) Copy content Toggle raw display
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