Properties

Label 975.2.bb.k
Level $975$
Weight $2$
Character orbit 975.bb
Analytic conductor $7.785$
Analytic rank $0$
Dimension $12$
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] = [975,2,Mod(724,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bb (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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 195)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{10} - 83\nu^{8} + 906\nu^{6} - 3874\nu^{4} + 5950\nu^{2} + 1505 ) / 2947 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{10} + 406\nu^{8} - 2063\nu^{6} + 2536\nu^{4} + 3931\nu^{2} + 3290 ) / 5894 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{10} + 151\nu^{8} + 335\nu^{6} - 3609\nu^{4} + 2977\nu^{2} + 3227 ) / 5894 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{11} - 83\nu^{9} + 906\nu^{7} - 3874\nu^{5} + 5950\nu^{3} - 1442\nu ) / 2947 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -40\nu^{11} + 397\nu^{9} + 404\nu^{7} - 17245\nu^{5} + 61188\nu^{3} - 56623\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -58\nu^{10} + 723\nu^{8} - 3540\nu^{6} + 6675\nu^{4} + 902\nu^{2} - 14175 ) / 5894 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -55\nu^{11} + 1651\nu^{9} - 15653\nu^{7} + 69487\nu^{5} - 156047\nu^{3} + 151641\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -89\nu^{10} + 1378\nu^{8} - 8321\nu^{6} + 23780\nu^{4} - 30699\nu^{2} + 8176 ) / 5894 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -501\nu^{11} + 7109\nu^{9} - 40029\nu^{7} + 98967\nu^{5} - 88545\nu^{3} - 45465\nu ) / 41258 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -332\nu^{11} + 5358\nu^{9} - 33779\nu^{7} + 94100\nu^{5} - 82129\nu^{3} - 75957\nu ) / 20629 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 4\beta_{7} - 6\beta_{4} - 8\beta_{3} + 9\beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{11} - 8\beta_{10} - 4\beta_{8} - 18\beta_{6} + 13\beta_{5} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 17\beta_{9} + 13\beta_{7} - 33\beta_{4} - 42\beta_{3} + 65\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50\beta_{11} - 53\beta_{10} - 5\beta_{8} - 90\beta_{6} + 99\beta_{5} + 69\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 104\beta_{9} + 24\beta_{7} - 172\beta_{4} - 168\beta_{3} + 365\beta_{2} - 85 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 276\beta_{11} - 320\beta_{10} + 84\beta_{8} - 288\beta_{6} + 573\beta_{5} + 240\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 489\beta_{9} - 120\beta_{7} - 836\beta_{4} - 467\beta_{3} + 1634\beta_{2} - 1083 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 1325\beta_{11} - 1792\beta_{10} + 978\beta_{8} - 98\beta_{6} + 2612\beta_{5} + 453\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
−0.385124 0.500000i
2.23871 + 0.500000i
−1.75780 + 0.500000i
1.75780 0.500000i
−2.23871 0.500000i
0.385124 + 0.500000i
−0.385124 + 0.500000i
2.23871 0.500000i
−1.75780 0.500000i
1.75780 + 0.500000i
−2.23871 + 0.500000i
0.385124 0.500000i
−2.25312 1.30084i −0.866025 0.500000i 2.38437 + 4.12985i 0 1.30084 + 2.25312i 3.11915 1.80084i 7.20336i 0.500000 + 0.866025i 0
724.2 −1.95878 1.13090i 0.866025 + 0.500000i 1.55787 + 2.69832i 0 −1.13090 1.95878i 1.09275 0.630901i 2.52360i 0.500000 + 0.866025i 0
724.3 −0.294342 0.169938i 0.866025 + 0.500000i −0.942242 1.63201i 0 −0.169938 0.294342i −0.571683 + 0.330062i 1.32025i 0.500000 + 0.866025i 0
724.4 0.294342 + 0.169938i −0.866025 0.500000i −0.942242 1.63201i 0 −0.169938 0.294342i 0.571683 0.330062i 1.32025i 0.500000 + 0.866025i 0
724.5 1.95878 + 1.13090i −0.866025 0.500000i 1.55787 + 2.69832i 0 −1.13090 1.95878i −1.09275 + 0.630901i 2.52360i 0.500000 + 0.866025i 0
724.6 2.25312 + 1.30084i 0.866025 + 0.500000i 2.38437 + 4.12985i 0 1.30084 + 2.25312i −3.11915 + 1.80084i 7.20336i 0.500000 + 0.866025i 0
874.1 −2.25312 + 1.30084i −0.866025 + 0.500000i 2.38437 4.12985i 0 1.30084 2.25312i 3.11915 + 1.80084i 7.20336i 0.500000 0.866025i 0
874.2 −1.95878 + 1.13090i 0.866025 0.500000i 1.55787 2.69832i 0 −1.13090 + 1.95878i 1.09275 + 0.630901i 2.52360i 0.500000 0.866025i 0
874.3 −0.294342 + 0.169938i 0.866025 0.500000i −0.942242 + 1.63201i 0 −0.169938 + 0.294342i −0.571683 0.330062i 1.32025i 0.500000 0.866025i 0
874.4 0.294342 0.169938i −0.866025 + 0.500000i −0.942242 + 1.63201i 0 −0.169938 + 0.294342i 0.571683 + 0.330062i 1.32025i 0.500000 0.866025i 0
874.5 1.95878 1.13090i −0.866025 + 0.500000i 1.55787 2.69832i 0 −1.13090 + 1.95878i −1.09275 0.630901i 2.52360i 0.500000 0.866025i 0
874.6 2.25312 1.30084i 0.866025 0.500000i 2.38437 4.12985i 0 1.30084 2.25312i −3.11915 1.80084i 7.20336i 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bb.k 12
5.b even 2 1 inner 975.2.bb.k 12
5.c odd 4 1 195.2.i.d 6
5.c odd 4 1 975.2.i.l 6
13.c even 3 1 inner 975.2.bb.k 12
15.e even 4 1 585.2.j.f 6
65.n even 6 1 inner 975.2.bb.k 12
65.q odd 12 1 195.2.i.d 6
65.q odd 12 1 975.2.i.l 6
65.q odd 12 1 2535.2.a.bb 3
65.r odd 12 1 2535.2.a.ba 3
195.bf even 12 1 7605.2.a.bw 3
195.bl even 12 1 585.2.j.f 6
195.bl even 12 1 7605.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.d 6 5.c odd 4 1
195.2.i.d 6 65.q odd 12 1
585.2.j.f 6 15.e even 4 1
585.2.j.f 6 195.bl even 12 1
975.2.i.l 6 5.c odd 4 1
975.2.i.l 6 65.q odd 12 1
975.2.bb.k 12 1.a even 1 1 trivial
975.2.bb.k 12 5.b even 2 1 inner
975.2.bb.k 12 13.c even 3 1 inner
975.2.bb.k 12 65.n even 6 1 inner
2535.2.a.ba 3 65.r odd 12 1
2535.2.a.bb 3 65.q odd 12 1
7605.2.a.bv 3 195.bl even 12 1
7605.2.a.bw 3 195.bf even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{12} - 12T_{2}^{10} + 108T_{2}^{8} - 424T_{2}^{6} + 1248T_{2}^{4} - 144T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{12} - 15T_{7}^{10} + 198T_{7}^{8} - 387T_{7}^{6} + 594T_{7}^{4} - 243T_{7}^{2} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 12 T^{10} + 108 T^{8} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 15 T^{10} + 198 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{6} + 24 T^{4} + 32 T^{3} + 576 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 21 T^{10} + 378 T^{8} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} - 84 T^{10} + 5292 T^{8} + \cdots + 92236816 \) Copy content Toggle raw display
$19$ \( (T^{6} - 12 T^{5} + 108 T^{4} - 424 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 96 T^{10} + 6912 T^{8} + \cdots + 84934656 \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + 36 T^{4} - 28 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 3 T^{2} - 93 T + 363)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} - 108 T^{10} + 10464 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{6} + 24 T^{4} + 52 T^{3} + 576 T^{2} + \cdots + 676)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} - 51 T^{10} + 2178 T^{8} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( (T^{6} + 156 T^{4} + 4980 T^{2} + \cdots + 42436)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 48 T^{4} + 576 T^{2} + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 6 T^{5} + 48 T^{4} - 44 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + 30 T^{4} - 71 T^{3} + \cdots + 4489)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 243 T^{10} + \cdots + 15178486401 \) Copy content Toggle raw display
$71$ \( (T^{6} - 12 T^{5} + 192 T^{4} + \cdots + 465124)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 255 T^{4} + 12387 T^{2} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 93 T + 363)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 204 T^{4} + 9648 T^{2} + \cdots + 28224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 30 T^{5} + 612 T^{4} + \cdots + 777924)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 243 T^{10} + \cdots + 120354180241 \) Copy content Toggle raw display
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