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Properties

Label 675.2.k.c
Level $675$
Weight $2$
Character orbit 675.k
Analytic conductor $5.390$
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] = [675,2,Mod(199,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.k (of order \(6\), degree \(2\), not 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: \(5.38990213644\)
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: \( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 225)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 1) q^{11} + (2 \beta_{14} + \beta_{9}) q^{13} + (\beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{14} + (2 \beta_{10} + \beta_{6} + 3 \beta_{3} + 1) q^{16} + ( - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{8} - 2 \beta_{4} - \beta_{2}) q^{17} + ( - 2 \beta_{11} + \beta_{5} - 2) q^{19} + (\beta_{15} + \beta_{14} + 2 \beta_{9}) q^{22} + ( - \beta_{15} + \beta_{14} + 5 \beta_{13} - \beta_{9}) q^{23} + ( - 2 \beta_{12} - 3 \beta_{11} - \beta_{5} + 1) q^{26} + (2 \beta_{14} + \beta_{13} - 2 \beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_1) q^{28} + ( - \beta_{10} - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} - 2) q^{29} + (2 \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} + 1) q^{31} + (3 \beta_{14} - 2 \beta_{9}) q^{32} + ( - \beta_{10} + \beta_{7} - \beta_{5} - 5 \beta_{3}) q^{34} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} - 4 \beta_{9} + 3 \beta_{8} + \beta_{4} - \beta_{2} + 4 \beta_1) q^{37} + (5 \beta_{8} - 2 \beta_{4} - \beta_{2}) q^{38} + ( - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 1) q^{41} + (\beta_{8} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{12} + \beta_{11} + \beta_{5} + 6) q^{44} + ( - 2 \beta_{12} + \beta_{11} - 4 \beta_{5} + 2) q^{46} + ( - 5 \beta_{8} - \beta_{4} - 2 \beta_{2}) q^{47} + ( - \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{7} - \beta_{5} + \beta_{3} + 1) q^{49} + (3 \beta_{8} - \beta_{4} + \beta_{2} - \beta_1) q^{52} + ( - 3 \beta_{14} - 6 \beta_{13} - \beta_{9} - 6 \beta_{8} - 3 \beta_{4} + \beta_1) q^{53} + ( - 6 \beta_{6} + 3 \beta_{3} - 6) q^{56} + (2 \beta_{15} - 3 \beta_{14} + \beta_{13} + 3 \beta_{9}) q^{58} + ( - 3 \beta_{12} - \beta_{11} - 3 \beta_{10} + 3 \beta_{6} - 3 \beta_{5} - \beta_{3} - 1) q^{59} + ( - \beta_{10} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 1) q^{61} + (3 \beta_{14} + 3 \beta_{4}) q^{62} + (\beta_{12} - 2 \beta_{11} - \beta_{5} - 5) q^{64} + ( - \beta_{15} + 3 \beta_{13} + 3 \beta_{9}) q^{67} + (2 \beta_{15} - 2 \beta_{14} + 5 \beta_{13} + 4 \beta_{9}) q^{68} + (\beta_{11} - 3 \beta_{5} + 4) q^{71} + ( - 2 \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{9} + \beta_{8} - 3 \beta_{4} + \cdots + 3 \beta_1) q^{73}+ \cdots + (3 \beta_{15} - 6 \beta_{13} + 4 \beta_{9} - 6 \beta_{8} + 3 \beta_{2} - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 2 q^{11} - 6 q^{14} - 8 q^{16} - 8 q^{19} + 40 q^{26} - 2 q^{29} + 8 q^{31} + 18 q^{34} - 10 q^{41} + 88 q^{44} - 6 q^{49} - 60 q^{56} - 34 q^{59} + 26 q^{61} - 76 q^{64} + 32 q^{71} - 80 q^{74} - 22 q^{76} - 14 q^{79} - 68 q^{86} - 36 q^{89} - 68 q^{91} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 28 \nu^{15} + 943 \nu^{13} - 2470 \nu^{11} + 8108 \nu^{9} + 208132 \nu^{7} + 15890 \nu^{5} - 3474 \nu^{3} + 4310244 \nu ) / 795285 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\nu^{14} - 74\nu^{12} - 10\nu^{10} + 10439\nu^{8} - 38576\nu^{6} + 94895\nu^{4} - 82467\nu^{2} + 18063 ) / 29025 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1540 \nu^{15} - 16519 \nu^{13} + 135850 \nu^{11} - 445940 \nu^{9} + 1153589 \nu^{7} - 873950 \nu^{5} + 191070 \nu^{3} + 2158308 \nu ) / 795285 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 266 \nu^{14} - 2999 \nu^{12} + 23465 \nu^{10} - 77026 \nu^{8} + 148849 \nu^{6} - 150955 \nu^{4} + 33003 \nu^{2} - 126387 ) / 92475 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11836 \nu^{14} + 138574 \nu^{12} - 1170340 \nu^{10} + 4500371 \nu^{8} - 12811274 \nu^{6} + 19088030 \nu^{4} - 21981813 \nu^{2} + 814212 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12338 \nu^{14} + 121397 \nu^{12} - 1025270 \nu^{10} + 3257143 \nu^{8} - 11223247 \nu^{6} + 16721965 \nu^{4} - 26841054 \nu^{2} + 713286 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} - 2137205 \nu^{5} + 467253 \nu^{3} + 2721243 \nu ) / 1325475 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + 19088030 \nu^{5} - 21981813 \nu^{3} + 4790637 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13873 \nu^{14} - 220432 \nu^{12} + 2012770 \nu^{10} - 10392128 \nu^{8} + 31813907 \nu^{6} - 59345540 \nu^{4} + 58685109 \nu^{2} - 12812391 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + 8453480 \nu^{4} - 1848168 \nu^{2} - 5535918 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18998 \nu^{14} - 206837 \nu^{12} + 1675895 \nu^{10} - 5501278 \nu^{8} + 13209112 \nu^{6} - 10781365 \nu^{4} + 2357109 \nu^{2} + 14978844 ) / 3976425 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} - 44578415 \nu^{5} + 47402299 \nu^{3} - 10338741 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 30232 \nu^{15} - 363428 \nu^{13} + 3087680 \nu^{11} - 12331002 \nu^{9} + 35467228 \nu^{7} - 57068185 \nu^{5} + 61524806 \nu^{3} - 13412214 \nu ) / 3976425 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66701 \nu^{15} - 773434 \nu^{13} + 6504640 \nu^{11} - 24700761 \nu^{9} + 70028009 \nu^{7} - 105967505 \nu^{5} + 119626108 \nu^{3} - 26067942 \nu ) / 3976425 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{7} - 2\beta_{6} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - \beta_{13} - 5\beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} + 6\beta_{7} - 7\beta_{6} - 6\beta_{5} + 9\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} + 11\beta_{14} - 8\beta_{13} - 30\beta_{9} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -19\beta_{12} - 68\beta_{11} - 57\beta_{5} - 101 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 60\beta_{8} - 87\beta_{4} + 57\beta_{2} - 196\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -144\beta_{12} - 487\beta_{11} - 144\beta_{10} - 253\beta_{7} + 191\beta_{6} - 144\beta_{5} - 487\beta_{3} - 487 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 397 \beta_{15} - 631 \beta_{14} + 433 \beta_{13} + 1328 \beta_{9} + 433 \beta_{8} - 631 \beta_{4} + 397 \beta_{2} - 1328 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1028\beta_{10} - 1725\beta_{7} + 1231\beta_{6} + 1725\beta_{5} - 3420\beta_{3} + 1231 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2753\beta_{15} - 4448\beta_{14} + 3059\beta_{13} + 9129\beta_{9} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7201\beta_{12} + 23837\beta_{11} + 19083\beta_{5} + 32141 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -21390\beta_{8} + 31038\beta_{4} - 19083\beta_{2} + 63106\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 50121 \beta_{12} + 165655 \beta_{11} + 50121 \beta_{10} + 82189 \beta_{7} - 57008 \beta_{6} + 50121 \beta_{5} + 165655 \beta_{3} + 165655 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 132310 \beta_{15} + 215776 \beta_{14} - 148879 \beta_{13} - 437162 \beta_{9} - 148879 \beta_{8} + 215776 \beta_{4} - 132310 \beta_{2} + 437162 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−2.28087 + 1.31686i
−1.41485 + 0.816862i
−1.27588 + 0.736627i
−0.409850 + 0.236627i
0.409850 − 0.236627i
1.27588 − 0.736627i
1.41485 − 0.816862i
2.28087 − 1.31686i
−2.28087 − 1.31686i
−1.41485 − 0.816862i
−1.27588 − 0.736627i
−0.409850 − 0.236627i
0.409850 + 0.236627i
1.27588 + 0.736627i
1.41485 + 0.816862i
2.28087 + 1.31686i
−2.28087 + 1.31686i 0 2.46825 − 4.27513i 0 0 −1.55662 + 0.898714i 7.73393i 0 0
199.2 −1.41485 + 0.816862i 0 0.334526 − 0.579416i 0 0 0.437645 − 0.252674i − 2.17440i 0 0
199.3 −1.27588 + 0.736627i 0 0.0852394 − 0.147639i 0 0 3.34791 − 1.93291i − 2.69535i 0 0
199.4 −0.409850 + 0.236627i 0 −0.888015 + 1.53809i 0 0 2.21967 − 1.28153i − 1.78702i 0 0
199.5 0.409850 − 0.236627i 0 −0.888015 + 1.53809i 0 0 −2.21967 + 1.28153i 1.78702i 0 0
199.6 1.27588 − 0.736627i 0 0.0852394 − 0.147639i 0 0 −3.34791 + 1.93291i 2.69535i 0 0
199.7 1.41485 − 0.816862i 0 0.334526 − 0.579416i 0 0 −0.437645 + 0.252674i 2.17440i 0 0
199.8 2.28087 − 1.31686i 0 2.46825 − 4.27513i 0 0 1.55662 − 0.898714i − 7.73393i 0 0
424.1 −2.28087 − 1.31686i 0 2.46825 + 4.27513i 0 0 −1.55662 − 0.898714i − 7.73393i 0 0
424.2 −1.41485 − 0.816862i 0 0.334526 + 0.579416i 0 0 0.437645 + 0.252674i 2.17440i 0 0
424.3 −1.27588 − 0.736627i 0 0.0852394 + 0.147639i 0 0 3.34791 + 1.93291i 2.69535i 0 0
424.4 −0.409850 − 0.236627i 0 −0.888015 − 1.53809i 0 0 2.21967 + 1.28153i 1.78702i 0 0
424.5 0.409850 + 0.236627i 0 −0.888015 − 1.53809i 0 0 −2.21967 − 1.28153i − 1.78702i 0 0
424.6 1.27588 + 0.736627i 0 0.0852394 + 0.147639i 0 0 −3.34791 − 1.93291i − 2.69535i 0 0
424.7 1.41485 + 0.816862i 0 0.334526 + 0.579416i 0 0 −0.437645 − 0.252674i − 2.17440i 0 0
424.8 2.28087 + 1.31686i 0 2.46825 + 4.27513i 0 0 1.55662 + 0.898714i 7.73393i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.k.c 16
3.b odd 2 1 225.2.k.c 16
5.b even 2 1 inner 675.2.k.c 16
5.c odd 4 1 675.2.e.c 8
5.c odd 4 1 675.2.e.e 8
9.c even 3 1 inner 675.2.k.c 16
9.c even 3 1 2025.2.b.o 8
9.d odd 6 1 225.2.k.c 16
9.d odd 6 1 2025.2.b.n 8
15.d odd 2 1 225.2.k.c 16
15.e even 4 1 225.2.e.c ✓ 8
15.e even 4 1 225.2.e.e yes 8
45.h odd 6 1 225.2.k.c 16
45.h odd 6 1 2025.2.b.n 8
45.j even 6 1 inner 675.2.k.c 16
45.j even 6 1 2025.2.b.o 8
45.k odd 12 1 675.2.e.c 8
45.k odd 12 1 675.2.e.e 8
45.k odd 12 1 2025.2.a.p 4
45.k odd 12 1 2025.2.a.z 4
45.l even 12 1 225.2.e.c ✓ 8
45.l even 12 1 225.2.e.e yes 8
45.l even 12 1 2025.2.a.q 4
45.l even 12 1 2025.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c ✓ 8 15.e even 4 1
225.2.e.c ✓ 8 45.l even 12 1
225.2.e.e yes 8 15.e even 4 1
225.2.e.e yes 8 45.l even 12 1
225.2.k.c 16 3.b odd 2 1
225.2.k.c 16 9.d odd 6 1
225.2.k.c 16 15.d odd 2 1
225.2.k.c 16 45.h odd 6 1
675.2.e.c 8 5.c odd 4 1
675.2.e.c 8 45.k odd 12 1
675.2.e.e 8 5.c odd 4 1
675.2.e.e 8 45.k odd 12 1
675.2.k.c 16 1.a even 1 1 trivial
675.2.k.c 16 5.b even 2 1 inner
675.2.k.c 16 9.c even 3 1 inner
675.2.k.c 16 45.j even 6 1 inner
2025.2.a.p 4 45.k odd 12 1
2025.2.a.q 4 45.l even 12 1
2025.2.a.y 4 45.l even 12 1
2025.2.a.z 4 45.k odd 12 1
2025.2.b.n 8 9.d odd 6 1
2025.2.b.n 8 45.h odd 6 1
2025.2.b.o 8 9.c even 3 1
2025.2.b.o 8 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + 102 T^{12} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 25 T^{14} + 451 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( (T^{8} + T^{7} + 26 T^{6} - 107 T^{5} + \cdots + 81)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 64 T^{14} + \cdots + 131079601 \) Copy content Toggle raw display
$17$ \( (T^{8} + 81 T^{6} + 2214 T^{4} + \cdots + 91809)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 27 T^{2} - 80 T - 25)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} - 111 T^{14} + \cdots + 3486784401 \) Copy content Toggle raw display
$29$ \( (T^{8} + T^{7} + 41 T^{6} + 244 T^{5} + \cdots + 16641)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 4 T^{7} + 58 T^{6} + 114 T^{5} + \cdots + 59049)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 199 T^{6} + 9513 T^{4} + \cdots + 418609)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 5 T^{7} + 50 T^{6} + 197 T^{5} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} - 196 T^{14} + \cdots + 205144679041 \) Copy content Toggle raw display
$47$ \( T^{16} - 186 T^{14} + \cdots + 21071715921 \) Copy content Toggle raw display
$53$ \( (T^{8} + 228 T^{6} + 13614 T^{4} + \cdots + 221841)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 17 T^{7} + 287 T^{6} + \cdots + 5349969)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 217 T^{14} + \cdots + 3486784401 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 196 T^{6} + 8478 T^{4} + \cdots + 12769)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} + 82 T^{6} - 93 T^{5} + \cdots + 42849)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} - 324 T^{14} + \cdots + 282429536481 \) Copy content Toggle raw display
$89$ \( (T^{4} + 9 T^{3} - 99 T^{2} - 405 T + 2025)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} - 199 T^{14} + \cdots + 824843587681 \) Copy content Toggle raw display
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