Properties

Label 1014.2.i.g
Level $1014$
Weight $2$
Character orbit 1014.i
Analytic conductor $8.097$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.i (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - \beta_{7} q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + 2 \beta_{8} + \beta_{2}) q^{5} + \beta_{6} q^{6} + ( - 2 \beta_{6} + 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} + \beta_{6}) q^{8} + (\beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} - \beta_{7} q^{3} + ( - \beta_{7} + 1) q^{4} + (\beta_{11} + 2 \beta_{8} + \beta_{2}) q^{5} + \beta_{6} q^{6} + ( - 2 \beta_{6} + 2 \beta_{2} + \beta_1) q^{7} + ( - \beta_{10} + \beta_{6}) q^{8} + (\beta_{7} - 1) q^{9} + ( - \beta_{9} - \beta_{5} - 2 \beta_{4} + \beta_{3} - 1) q^{10} + ( - \beta_{11} - \beta_{10} - 3 \beta_{8} + 3 \beta_1) q^{11} - q^{12} + (\beta_{5} + 2 \beta_{3}) q^{14} + ( - \beta_{11} - 2 \beta_{8} + 2 \beta_1) q^{15} - \beta_{7} q^{16} + ( - 4 \beta_{9} - 2 \beta_{7} - 2 \beta_{4} + 2) q^{17} + (\beta_{10} - \beta_{6}) q^{18} + 4 \beta_{2} q^{19} + (\beta_{2} + 2 \beta_1) q^{20} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} + 2 \beta_{6} - 2 \beta_{2}) q^{21} + (\beta_{9} - \beta_{7} + 3 \beta_{4} + 1) q^{22} + ( - 2 \beta_{9} + 6 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 2) q^{23} + \beta_{10} q^{24} + ( - 3 \beta_{5} + 4 \beta_{3} - 1) q^{25} + q^{27} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_1) q^{28} + (3 \beta_{9} - 7 \beta_{7} + 2 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + 3) q^{29} + (\beta_{9} + 2 \beta_{4}) q^{30} + ( - 5 \beta_{11} - 5 \beta_{10} - \beta_{8} + 5 \beta_{6} - 5 \beta_{2}) q^{31} + \beta_{6} q^{32} + (\beta_{6} - \beta_{2} - 3 \beta_1) q^{33} + (4 \beta_{11} - 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{6} + 4 \beta_{2}) q^{34} + (\beta_{7} + \beta_{4} - 1) q^{35} + \beta_{7} q^{36} + (2 \beta_{11} - 4 \beta_{10} - 2 \beta_{8} + 2 \beta_1) q^{37} + (4 \beta_{5} + 4 \beta_{3} - 4) q^{38} + ( - \beta_{5} + \beta_{3} - 1) q^{40} + (2 \beta_{11} + 6 \beta_{10} + 2 \beta_{8} - 2 \beta_1) q^{41} + (2 \beta_{9} - 2 \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{3} + 2) q^{42} + ( - 2 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2) q^{43} + ( - \beta_{11} - \beta_{10} - 3 \beta_{8} + \beta_{6} - \beta_{2}) q^{44} + ( - \beta_{2} - 2 \beta_1) q^{45} + ( - 6 \beta_{6} + 2 \beta_{2}) q^{46} + 4 \beta_{8} q^{47} + (\beta_{7} - 1) q^{48} + ( - 7 \beta_{9} + 2 \beta_{7} + 4 \beta_{5} - 3 \beta_{4} + 7 \beta_{3} - 7) q^{49} + ( - 4 \beta_{11} - 3 \beta_{10} - 7 \beta_{8} + 7 \beta_1) q^{50} + (2 \beta_{5} + 4 \beta_{3} - 6) q^{51} + (9 \beta_{5} + 3 \beta_{3} + 1) q^{53} - \beta_{10} q^{54} + (5 \beta_{9} + 3 \beta_{7} + 3 \beta_{5} + 8 \beta_{4} - 5 \beta_{3} + 5) q^{55} + (2 \beta_{9} - 2 \beta_{7} + \beta_{4} + 2) q^{56} + ( - 4 \beta_{11} - 4 \beta_{2}) q^{57} + (7 \beta_{6} - 3 \beta_{2} - 5 \beta_1) q^{58} + (4 \beta_{6} - 2 \beta_{2} - \beta_1) q^{59} + ( - \beta_{11} - 2 \beta_{8} - \beta_{2}) q^{60} + ( - 4 \beta_{7} + 2 \beta_{4} + 4) q^{61} + (5 \beta_{9} - 5 \beta_{7} - 4 \beta_{5} + \beta_{4} - 5 \beta_{3} + 5) q^{62} + (2 \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_1) q^{63} - q^{64} + (2 \beta_{5} - \beta_{3}) q^{66} + ( - 10 \beta_{11} - 2 \beta_{10} - 2 \beta_{8} + 2 \beta_1) q^{67} + ( - 4 \beta_{9} - 2 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4) q^{68} + (2 \beta_{9} - 6 \beta_{7} + 6) q^{69} + (\beta_{10} - \beta_{8} - \beta_{6}) q^{70} + (4 \beta_{6} - 6 \beta_{2}) q^{71} - \beta_{6} q^{72} + (\beta_{11} + 4 \beta_{8} + \beta_{2}) q^{73} + ( - 2 \beta_{9} - 4 \beta_{7} + 2 \beta_{4} + 4) q^{74} + (4 \beta_{9} - 3 \beta_{7} + 3 \beta_{5} + 7 \beta_{4} - 4 \beta_{3} + 4) q^{75} - 4 \beta_{11} q^{76} + (2 \beta_{5} + \beta_{3} + 1) q^{77} + ( - 11 \beta_{5} - 6 \beta_{3} + 4) q^{79} + ( - \beta_{11} - 2 \beta_{8} + 2 \beta_1) q^{80} - \beta_{7} q^{81} + ( - 2 \beta_{9} + 6 \beta_{7} - 2 \beta_{4} - 6) q^{82} + ( - 2 \beta_{11} + 9 \beta_{8} - 2 \beta_{2}) q^{83} + (2 \beta_{6} - 2 \beta_{2} - \beta_1) q^{84} + (2 \beta_{6} + 6 \beta_{2} + 14 \beta_1) q^{85} + (2 \beta_{11} + 2 \beta_{10} - 4 \beta_{8} - 2 \beta_{6} + 2 \beta_{2}) q^{86} + ( - 3 \beta_{9} + 7 \beta_{7} - 5 \beta_{4} - 7) q^{87} + (\beta_{9} - \beta_{7} + 2 \beta_{5} + 3 \beta_{4} - \beta_{3} + 1) q^{88} + ( - 2 \beta_{11} - 4 \beta_{10}) q^{89} + (\beta_{5} - \beta_{3} + 1) q^{90} + (2 \beta_{5} + 2 \beta_{3} + 4) q^{92} + (5 \beta_{11} + 5 \beta_{10} + \beta_{8} - \beta_1) q^{93} + ( - 4 \beta_{5} - 4 \beta_{4}) q^{94} + 4 \beta_{4} q^{95} + (\beta_{10} - \beta_{6}) q^{96} + (3 \beta_{2} + 4 \beta_1) q^{97} + ( - 2 \beta_{6} + 7 \beta_{2} + 3 \beta_1) q^{98} + (\beta_{11} + \beta_{10} + 3 \beta_{8} - \beta_{6} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 6 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 6 q^{4} - 6 q^{9} - 6 q^{10} - 12 q^{12} + 12 q^{14} - 6 q^{16} + 24 q^{17} - 2 q^{22} + 32 q^{23} - 8 q^{25} + 12 q^{27} - 26 q^{29} - 6 q^{30} - 8 q^{35} + 6 q^{36} - 16 q^{38} - 12 q^{40} - 6 q^{42} - 16 q^{43} - 6 q^{48} - 8 q^{49} - 48 q^{51} + 60 q^{53} + 44 q^{55} + 6 q^{56} + 20 q^{61} - 18 q^{62} - 12 q^{64} + 4 q^{66} - 24 q^{68} + 32 q^{69} + 24 q^{74} + 4 q^{75} + 24 q^{77} - 20 q^{79} - 6 q^{81} - 28 q^{82} - 26 q^{87} + 2 q^{88} + 12 q^{90} + 64 q^{92} - 8 q^{94} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(1 - \beta_{7}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−1.07992 0.623490i
0.385418 + 0.222521i
1.56052 + 0.900969i
−1.56052 0.900969i
−0.385418 0.222521i
1.07992 + 0.623490i
1.56052 0.900969i
0.385418 0.222521i
−1.07992 + 0.623490i
1.07992 0.623490i
−0.385418 + 0.222521i
−1.56052 + 0.900969i
−0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.692021i 0.866025 + 0.500000i 0.309081 + 0.178448i 1.00000i −0.500000 + 0.866025i 0.346011 + 0.599308i
361.2 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.356896i 0.866025 + 0.500000i −3.50647 2.02446i 1.00000i −0.500000 + 0.866025i 0.178448 + 0.309081i
361.3 −0.866025 + 0.500000i −0.500000 0.866025i 0.500000 0.866025i 4.04892i 0.866025 + 0.500000i 0.599308 + 0.346011i 1.00000i −0.500000 + 0.866025i −2.02446 3.50647i
361.4 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 4.04892i −0.866025 0.500000i −0.599308 0.346011i 1.00000i −0.500000 + 0.866025i −2.02446 3.50647i
361.5 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.356896i −0.866025 0.500000i 3.50647 + 2.02446i 1.00000i −0.500000 + 0.866025i 0.178448 + 0.309081i
361.6 0.866025 0.500000i −0.500000 0.866025i 0.500000 0.866025i 0.692021i −0.866025 0.500000i −0.309081 0.178448i 1.00000i −0.500000 + 0.866025i 0.346011 + 0.599308i
823.1 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 4.04892i 0.866025 0.500000i 0.599308 0.346011i 1.00000i −0.500000 0.866025i −2.02446 + 3.50647i
823.2 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.356896i 0.866025 0.500000i −3.50647 + 2.02446i 1.00000i −0.500000 0.866025i 0.178448 0.309081i
823.3 −0.866025 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.692021i 0.866025 0.500000i 0.309081 0.178448i 1.00000i −0.500000 0.866025i 0.346011 0.599308i
823.4 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.692021i −0.866025 + 0.500000i −0.309081 + 0.178448i 1.00000i −0.500000 0.866025i 0.346011 0.599308i
823.5 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.356896i −0.866025 + 0.500000i 3.50647 2.02446i 1.00000i −0.500000 0.866025i 0.178448 0.309081i
823.6 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 4.04892i −0.866025 + 0.500000i −0.599308 + 0.346011i 1.00000i −0.500000 0.866025i −2.02446 + 3.50647i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 823.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1014.2.i.g 12
13.b even 2 1 inner 1014.2.i.g 12
13.c even 3 1 1014.2.b.g 6
13.c even 3 1 inner 1014.2.i.g 12
13.d odd 4 1 1014.2.e.k 6
13.d odd 4 1 1014.2.e.m 6
13.e even 6 1 1014.2.b.g 6
13.e even 6 1 inner 1014.2.i.g 12
13.f odd 12 1 1014.2.a.m 3
13.f odd 12 1 1014.2.a.o yes 3
13.f odd 12 1 1014.2.e.k 6
13.f odd 12 1 1014.2.e.m 6
39.h odd 6 1 3042.2.b.r 6
39.i odd 6 1 3042.2.b.r 6
39.k even 12 1 3042.2.a.bd 3
39.k even 12 1 3042.2.a.be 3
52.l even 12 1 8112.2.a.bz 3
52.l even 12 1 8112.2.a.ce 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.m 3 13.f odd 12 1
1014.2.a.o yes 3 13.f odd 12 1
1014.2.b.g 6 13.c even 3 1
1014.2.b.g 6 13.e even 6 1
1014.2.e.k 6 13.d odd 4 1
1014.2.e.k 6 13.f odd 12 1
1014.2.e.m 6 13.d odd 4 1
1014.2.e.m 6 13.f odd 12 1
1014.2.i.g 12 1.a even 1 1 trivial
1014.2.i.g 12 13.b even 2 1 inner
1014.2.i.g 12 13.c even 3 1 inner
1014.2.i.g 12 13.e even 6 1 inner
3042.2.a.bd 3 39.k even 12 1
3042.2.a.be 3 39.k even 12 1
3042.2.b.r 6 39.h odd 6 1
3042.2.b.r 6 39.i odd 6 1
8112.2.a.bz 3 52.l even 12 1
8112.2.a.ce 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1014, [\chi])\):

\( T_{5}^{6} + 17T_{5}^{4} + 10T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} - 17T_{7}^{10} + 279T_{7}^{8} - 168T_{7}^{6} + 83T_{7}^{4} - 10T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 17 T^{4} + 10 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 17 T^{10} + 279 T^{8} - 168 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 33 T^{10} + 859 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} - 12 T^{5} + 124 T^{4} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 80 T^{10} + 4864 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$23$ \( (T^{6} - 16 T^{5} + 180 T^{4} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 13 T^{5} + 157 T^{4} + \cdots + 49729)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 125 T^{4} + 1006 T^{2} + \cdots + 841)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 104 T^{10} + 10608 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( T^{12} - 84 T^{10} + 5488 T^{8} + \cdots + 9834496 \) Copy content Toggle raw display
$43$ \( (T^{6} + 8 T^{5} + 108 T^{4} + \cdots + 118336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 15 T^{2} - 72 T + 1247)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} - 41 T^{10} + 1515 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$61$ \( (T^{6} - 10 T^{5} + 76 T^{4} - 224 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 404 T^{10} + \cdots + 1529041063936 \) Copy content Toggle raw display
$71$ \( T^{12} - 180 T^{10} + \cdots + 116985856 \) Copy content Toggle raw display
$73$ \( (T^{6} + 69 T^{4} + 614 T^{2} + 169)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 5 T^{2} - 204 T - 1469)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 497 T^{4} + 70854 T^{2} + \cdots + 2181529)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 52 T^{10} + 2288 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{12} - 77 T^{10} + 5635 T^{8} + \cdots + 2401 \) Copy content Toggle raw display
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