Properties

Label 414.2.e.e
Level $414$
Weight $2$
Character orbit 414.e
Analytic conductor $3.306$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \( x^{12} + 4x^{10} - 3x^{9} + 22x^{8} - 9x^{7} + 69x^{6} - 27x^{5} + 198x^{4} - 81x^{3} + 324x^{2} + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{2} + 1) q^{5} + ( - \beta_{10} - \beta_1) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{7} - q^{8} + ( - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_{6} - \beta_{2} + 1) q^{5} + ( - \beta_{10} - \beta_1) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{7} - q^{8} + ( - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{9} + ( - \beta_{6} + 1) q^{10} + (\beta_{11} - \beta_{10} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{11} - \beta_{10} q^{12} + (\beta_{11} - \beta_{9} - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2) q^{13} + ( - \beta_{11} + \beta_{9} + \beta_{3} - \beta_{2} + 1) q^{14} + (\beta_{10} - \beta_{8} - \beta_{6} - \beta_{4} + 1) q^{15} - \beta_{2} q^{16} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{17} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{4} - 1) q^{18} + (\beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{19} + (\beta_{7} + \beta_{2}) q^{20} + (2 \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{21} + (\beta_{11} - \beta_{10} - \beta_{5} + \beta_{4} + 2 \beta_{2} - 2) q^{22} + ( - \beta_{2} + 1) q^{23} + \beta_1 q^{24} + ( - \beta_{9} - \beta_{7} + \beta_1) q^{25} + (\beta_{8} - \beta_{5} - \beta_{3} - \beta_1 - 2) q^{26} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{27} + ( - \beta_{11} + \beta_{10} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{28} + ( - \beta_{11} + \beta_{10} - \beta_{4} - \beta_{3} + \beta_{2}) q^{29} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{3} + \beta_{2} - \beta_1) q^{30} + ( - \beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{5} - \beta_{4} + 2 \beta_1) q^{31} + ( - \beta_{2} + 1) q^{32} + ( - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{7} + \beta_{6} - \beta_{5} - \beta_1) q^{33} + ( - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - \beta_1) q^{34} + (\beta_{6} + \beta_{5} + \beta_{3} - 2) q^{35} + ( - \beta_{9} - \beta_{6} - \beta_{3} - \beta_{2}) q^{36} + ( - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{6} + \beta_{4} - 2 \beta_1) q^{37} + ( - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_1) q^{38} + ( - 2 \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{3} + \cdots + 3) q^{39}+ \cdots + (\beta_{10} - 5 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1 - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} + 5 q^{5} - 3 q^{7} - 12 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{2} - 6 q^{4} + 5 q^{5} - 3 q^{7} - 12 q^{8} - 8 q^{9} + 10 q^{10} + 6 q^{11} - 6 q^{13} + 3 q^{14} + 7 q^{15} - 6 q^{16} - 8 q^{17} - 10 q^{18} + 4 q^{19} + 5 q^{20} + 17 q^{21} - 6 q^{22} + 6 q^{23} + q^{25} - 12 q^{26} - 9 q^{27} + 6 q^{28} + 12 q^{29} - q^{30} - 6 q^{31} + 6 q^{32} + 9 q^{33} - 4 q^{34} - 34 q^{35} - 2 q^{36} + 8 q^{37} + 2 q^{38} + 23 q^{39} - 5 q^{40} + 15 q^{41} + 7 q^{42} - 14 q^{43} - 12 q^{44} - 37 q^{45} + 12 q^{46} + 9 q^{47} - 5 q^{49} - q^{50} + 9 q^{51} - 6 q^{52} - 10 q^{53} - 9 q^{54} + 16 q^{55} + 3 q^{56} + 37 q^{57} - 12 q^{58} + 18 q^{59} - 8 q^{60} - 3 q^{61} - 12 q^{62} - 42 q^{63} + 12 q^{64} + 9 q^{65} + 3 q^{66} + 8 q^{67} + 4 q^{68} - 17 q^{70} - 18 q^{71} + 8 q^{72} - 32 q^{73} + 4 q^{74} + 34 q^{75} - 2 q^{76} - q^{77} + 22 q^{78} - 7 q^{79} - 10 q^{80} - 56 q^{81} + 30 q^{82} + 3 q^{83} - 10 q^{84} + 7 q^{85} + 14 q^{86} - 9 q^{87} - 6 q^{88} - 42 q^{89} - 17 q^{90} + 18 q^{91} + 6 q^{92} + 69 q^{93} - 9 q^{94} + 11 q^{95} + 13 q^{97} - 10 q^{98} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 4x^{10} - 3x^{9} + 22x^{8} - 9x^{7} + 69x^{6} - 27x^{5} + 198x^{4} - 81x^{3} + 324x^{2} + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10 \nu^{11} - 501 \nu^{10} + 572 \nu^{9} - 2514 \nu^{8} + 2678 \nu^{7} - 7881 \nu^{6} + 11532 \nu^{5} - 25200 \nu^{4} + 18216 \nu^{3} - 38205 \nu^{2} + 45765 \nu - 8262 ) / 78003 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11 \nu^{11} + 444 \nu^{10} - 719 \nu^{9} + 1215 \nu^{8} - 3410 \nu^{7} + 7734 \nu^{6} - 11775 \nu^{5} + 17541 \nu^{4} - 21564 \nu^{3} + 43200 \nu^{2} - 49329 \nu + 22113 ) / 26001 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 52 \nu^{11} - 669 \nu^{10} + 1648 \nu^{9} - 24 \nu^{8} + 7453 \nu^{7} - 8517 \nu^{6} + 22659 \nu^{5} - 22077 \nu^{4} + 66483 \nu^{3} - 18009 \nu^{2} + 126036 \nu - 19440 ) / 78003 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 140 \nu^{11} - 690 \nu^{10} + 659 \nu^{9} - 4287 \nu^{8} + 2954 \nu^{7} - 15819 \nu^{6} + 3225 \nu^{5} - 31437 \nu^{4} + 13653 \nu^{3} - 106488 \nu^{2} - 68688 \nu - 118341 ) / 78003 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{11} - 3 \nu^{10} + \nu^{9} - 33 \nu^{8} + \nu^{7} - 183 \nu^{6} + 168 \nu^{5} - 585 \nu^{4} + 306 \nu^{3} - 1323 \nu^{2} + 891 \nu - 2673 ) / 729 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 104 \nu^{11} + 54 \nu^{10} - 407 \nu^{9} + 690 \nu^{8} - 2387 \nu^{7} + 3231 \nu^{6} - 7761 \nu^{5} + 4671 \nu^{4} - 14517 \nu^{3} + 7128 \nu^{2} - 26730 \nu - 13122 ) / 26001 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{11} + 4\nu^{9} - 3\nu^{8} + 22\nu^{7} - 9\nu^{6} + 69\nu^{5} - 27\nu^{4} + 198\nu^{3} - 81\nu^{2} + 324\nu ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 355 \nu^{11} + 30 \nu^{10} + 83 \nu^{9} - 651 \nu^{8} - 268 \nu^{7} - 4839 \nu^{6} - 852 \nu^{5} - 25011 \nu^{4} + 5310 \nu^{3} - 25893 \nu^{2} - 405 \nu - 137295 ) / 78003 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 167 \nu^{11} + 204 \nu^{10} - 848 \nu^{9} + 966 \nu^{8} - 2657 \nu^{7} + 4074 \nu^{6} - 8490 \nu^{5} + 6732 \nu^{4} - 13005 \nu^{3} + 16335 \nu^{2} - 28755 \nu + 2430 ) / 26001 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 877 \nu^{11} + 1227 \nu^{10} - 3571 \nu^{9} + 3516 \nu^{8} - 17638 \nu^{7} + 15582 \nu^{6} - 47751 \nu^{5} + 31824 \nu^{4} - 80388 \nu^{3} + 51219 \nu^{2} + \cdots - 116154 ) / 78003 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{9} + \beta_{8} + \beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{11} - \beta_{10} - 2\beta_{9} + \beta_{8} - 2\beta_{7} - 2\beta_{6} - \beta_{4} - 3\beta_{3} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} - 3 \beta_{9} - \beta_{7} + \beta_{6} - 5 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 3 \beta_{2} - 8 \beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{11} + \beta_{10} + 2 \beta_{9} + 9 \beta_{8} + 12 \beta_{7} - \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - \beta_{3} + 8 \beta_{2} - 4 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{11} + 16 \beta_{10} + 16 \beta_{9} + 2 \beta_{8} - 5 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} - 11 \beta_{4} - 7 \beta_{3} - 13 \beta_{2} + 8 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 15 \beta_{11} + 6 \beta_{10} - 16 \beta_{8} + 8 \beta_{7} + 15 \beta_{6} - 3 \beta_{5} + 13 \beta_{4} + \beta_{3} - 47 \beta_{2} - 12 \beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{11} - 75 \beta_{10} + 32 \beta_{9} + 13 \beta_{8} + 33 \beta_{7} - 15 \beta_{6} - 11 \beta_{5} - \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 3 \beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3 \beta_{11} - 19 \beta_{10} + 88 \beta_{9} + 28 \beta_{8} - 65 \beta_{7} + 7 \beta_{6} - 9 \beta_{5} - 55 \beta_{4} + 6 \beta_{3} - 90 \beta_{2} + 19 \beta _1 + 112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 116 \beta_{11} + 17 \beta_{10} - 111 \beta_{9} - 72 \beta_{8} + 44 \beta_{7} + 127 \beta_{6} + 67 \beta_{5} + 96 \beta_{4} + 177 \beta_{3} - 87 \beta_{2} + 19 \beta _1 + 84 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
1.44505 + 0.954904i
1.20109 1.24795i
0.416383 + 1.68126i
−0.706634 1.58135i
−1.07449 + 1.35848i
−1.28140 1.16534i
1.44505 0.954904i
1.20109 + 1.24795i
0.416383 1.68126i
−0.706634 + 1.58135i
−1.07449 1.35848i
−1.28140 + 1.16534i
0.500000 + 0.866025i −1.44505 0.954904i −0.500000 + 0.866025i −0.286426 + 0.496105i 0.104448 1.72890i −1.38989 2.40736i −1.00000 1.17632 + 2.75976i −0.572852
139.2 0.500000 + 0.866025i −1.20109 + 1.24795i −0.500000 + 0.866025i 1.39132 2.40983i −1.68130 0.416203i −1.77657 3.07710i −1.00000 −0.114750 2.99780i 2.78263
139.3 0.500000 + 0.866025i −0.416383 1.68126i −0.500000 + 0.866025i −0.229999 + 0.398369i 1.24782 1.20123i 2.38325 + 4.12791i −1.00000 −2.65325 + 1.40009i −0.459997
139.4 0.500000 + 0.866025i 0.706634 + 1.58135i −0.500000 + 0.866025i 1.63705 2.83545i −1.01617 + 1.40264i −0.135712 0.235061i −1.00000 −2.00134 + 2.23487i 3.27410
139.5 0.500000 + 0.866025i 1.07449 1.35848i −0.500000 + 0.866025i 1.11191 1.92588i 1.71372 + 0.251297i −0.920971 1.59517i −1.00000 −0.690936 2.91935i 2.22381
139.6 0.500000 + 0.866025i 1.28140 + 1.16534i −0.500000 + 0.866025i −1.12385 + 1.94656i −0.368518 + 1.69239i 0.339893 + 0.588712i −1.00000 0.283955 + 2.98653i −2.24770
277.1 0.500000 0.866025i −1.44505 + 0.954904i −0.500000 0.866025i −0.286426 0.496105i 0.104448 + 1.72890i −1.38989 + 2.40736i −1.00000 1.17632 2.75976i −0.572852
277.2 0.500000 0.866025i −1.20109 1.24795i −0.500000 0.866025i 1.39132 + 2.40983i −1.68130 + 0.416203i −1.77657 + 3.07710i −1.00000 −0.114750 + 2.99780i 2.78263
277.3 0.500000 0.866025i −0.416383 + 1.68126i −0.500000 0.866025i −0.229999 0.398369i 1.24782 + 1.20123i 2.38325 4.12791i −1.00000 −2.65325 1.40009i −0.459997
277.4 0.500000 0.866025i 0.706634 1.58135i −0.500000 0.866025i 1.63705 + 2.83545i −1.01617 1.40264i −0.135712 + 0.235061i −1.00000 −2.00134 2.23487i 3.27410
277.5 0.500000 0.866025i 1.07449 + 1.35848i −0.500000 0.866025i 1.11191 + 1.92588i 1.71372 0.251297i −0.920971 + 1.59517i −1.00000 −0.690936 + 2.91935i 2.22381
277.6 0.500000 0.866025i 1.28140 1.16534i −0.500000 0.866025i −1.12385 1.94656i −0.368518 1.69239i 0.339893 0.588712i −1.00000 0.283955 2.98653i −2.24770
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.e.e 12
3.b odd 2 1 1242.2.e.e 12
9.c even 3 1 inner 414.2.e.e 12
9.c even 3 1 3726.2.a.w 6
9.d odd 6 1 1242.2.e.e 12
9.d odd 6 1 3726.2.a.x 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.e.e 12 1.a even 1 1 trivial
414.2.e.e 12 9.c even 3 1 inner
1242.2.e.e 12 3.b odd 2 1
1242.2.e.e 12 9.d odd 6 1
3726.2.a.w 6 9.c even 3 1
3726.2.a.x 6 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 5 T_{5}^{11} + 27 T_{5}^{10} - 56 T_{5}^{9} + 182 T_{5}^{8} - 235 T_{5}^{7} + 844 T_{5}^{6} - 525 T_{5}^{5} + 1432 T_{5}^{4} + 1299 T_{5}^{3} + 1365 T_{5}^{2} + 468 T_{5} + 144 \) acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{10} + 3 T^{9} + 22 T^{8} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{12} - 5 T^{11} + 27 T^{10} - 56 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{12} + 3 T^{11} + 28 T^{10} + 99 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + 50 T^{10} - 126 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + 58 T^{10} + \cdots + 35344 \) Copy content Toggle raw display
$17$ \( (T^{6} + 4 T^{5} - 14 T^{4} - 57 T^{3} + \cdots + 81)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} - 42 T^{4} + 130 T^{3} + \cdots + 765)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$29$ \( T^{12} - 12 T^{11} + 113 T^{10} + \cdots + 104976 \) Copy content Toggle raw display
$31$ \( T^{12} + 6 T^{11} + 104 T^{10} + \cdots + 1144900 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} - 91 T^{4} + 65 T^{3} + \cdots + 4008)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 15 T^{11} + 227 T^{10} + \cdots + 1846881 \) Copy content Toggle raw display
$43$ \( T^{12} + 14 T^{11} + \cdots + 7951110561 \) Copy content Toggle raw display
$47$ \( T^{12} - 9 T^{11} + \cdots + 28242146916 \) Copy content Toggle raw display
$53$ \( (T^{6} + 5 T^{5} - 97 T^{4} - 763 T^{3} + \cdots - 870)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 18 T^{11} + \cdots + 1322704161 \) Copy content Toggle raw display
$61$ \( T^{12} + 3 T^{11} + 208 T^{10} + \cdots + 18164644 \) Copy content Toggle raw display
$67$ \( T^{12} - 8 T^{11} + 252 T^{10} + \cdots + 140493609 \) Copy content Toggle raw display
$71$ \( (T^{6} + 9 T^{5} - 326 T^{4} + \cdots - 271728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 16 T^{5} - 98 T^{4} - 1702 T^{3} + \cdots - 32237)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 7 T^{11} + 179 T^{10} + \cdots + 324 \) Copy content Toggle raw display
$83$ \( T^{12} - 3 T^{11} + 277 T^{10} + \cdots + 150111504 \) Copy content Toggle raw display
$89$ \( (T^{6} + 21 T^{5} - 113 T^{4} + \cdots + 644958)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} - 13 T^{11} + \cdots + 1043321287761 \) Copy content Toggle raw display
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