Properties

Label 273.2.cd.c
Level $273$
Weight $2$
Character orbit 273.cd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{7} - \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24}^{2} + 1) q^{3} + \zeta_{24}^{2} q^{4} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{5} + (\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}) q^{6} + (\zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{7} + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{8} + ( - 2 \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} - 2 \zeta_{24}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{7} q^{2} + ( - \zeta_{24}^{7} - \zeta_{24}^{4} + \zeta_{24}^{3} - \zeta_{24}^{2} + 1) q^{3} + \zeta_{24}^{2} q^{4} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{5} + (\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}) q^{6} + (\zeta_{24}^{7} + 2 \zeta_{24}^{3}) q^{7} + (3 \zeta_{24}^{5} - 3 \zeta_{24}) q^{8} + ( - 2 \zeta_{24}^{7} + \zeta_{24}^{6} - \zeta_{24}^{2} - 2 \zeta_{24}) q^{9} + ( - \zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2 \zeta_{24}^{2}) q^{10} + (3 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{2} + 2) q^{11} + ( - \zeta_{24}^{6} - \zeta_{24}^{4} + \zeta_{24}^{2} + \zeta_{24}) q^{12} + ( - 3 \zeta_{24}^{6} + 2) q^{13} + (2 \zeta_{24}^{6} - 3 \zeta_{24}^{2}) q^{14} + ( - 2 \zeta_{24}^{6} + \zeta_{24}^{5} - 3 \zeta_{24}^{3} - \zeta_{24} - 2) q^{15} - \zeta_{24}^{4} q^{16} + ( - 2 \zeta_{24}^{7} - 2 \zeta_{24}^{5} - \zeta_{24}^{4} + 2 \zeta_{24}^{3} + 1) q^{17} + ( - \zeta_{24}^{5} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 2) q^{18} + (4 \zeta_{24}^{6} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{2} + \zeta_{24}) q^{19} + ( - \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 2 \zeta_{24} + 1) q^{20} + ( - 2 \zeta_{24}^{7} + \zeta_{24}^{6} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + \zeta_{24}) q^{21} + ( - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{3} + 2 \zeta_{24} - 3) q^{22} + (\zeta_{24}^{7} + 8 \zeta_{24}^{4} + \zeta_{24}) q^{23} + ( - 3 \zeta_{24}^{7} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{4} + 3 \zeta_{24}^{3} - 3) q^{24} + (4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3} - \zeta_{24}^{2}) q^{25} + (2 \zeta_{24}^{7} + 3 \zeta_{24}) q^{26} + ( - \zeta_{24}^{6} + 5 \zeta_{24}^{5} - 5 \zeta_{24} - 1) q^{27} + (3 \zeta_{24}^{5} - \zeta_{24}) q^{28} - \zeta_{24}^{6} q^{29} + ( - 2 \zeta_{24}^{7} - 3 \zeta_{24}^{6} - \zeta_{24}^{4} + 3 \zeta_{24}^{2} + 2 \zeta_{24}) q^{30} + (2 \zeta_{24}^{5} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2) q^{31} + (5 \zeta_{24}^{7} - 5 \zeta_{24}^{3}) q^{32} + ( - 5 \zeta_{24}^{7} + 4 \zeta_{24}^{6} + 3 \zeta_{24}^{4} - 4 \zeta_{24}^{2} + \zeta_{24}) q^{33} + (2 \zeta_{24}^{6} + \zeta_{24}^{3} + 2) q^{34} + ( - 3 \zeta_{24}^{7} - 4 \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 6 \zeta_{24}^{2} + 2 \zeta_{24}) q^{35} + ( - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} + 2 \zeta_{24} - 1) q^{36} + 4 \zeta_{24}^{7} q^{37} + ( - 4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} + \zeta_{24}^{4} + 4 \zeta_{24}^{3} - 1) q^{38} + ( - 2 \zeta_{24}^{7} - 3 \zeta_{24}^{5} + \zeta_{24}^{4} + 2 \zeta_{24}^{3} - 5 \zeta_{24}^{2} - 1) q^{39} + (3 \zeta_{24}^{7} + 6 \zeta_{24}^{4} + 3 \zeta_{24}) q^{40} + (4 \zeta_{24}^{6} + 4) q^{41} + (3 \zeta_{24}^{6} + 2 \zeta_{24}^{5} + \zeta_{24}^{4} - \zeta_{24}^{2} - 3 \zeta_{24} + 2) q^{42} - 6 \zeta_{24}^{6} q^{43} + (3 \zeta_{24}^{7} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2}) q^{44} + (4 \zeta_{24}^{7} + 2 \zeta_{24}^{5} + 5 \zeta_{24}^{4} - 4 \zeta_{24}^{3} - 3 \zeta_{24}^{2} - 5) q^{45} + (8 \zeta_{24}^{7} + \zeta_{24}^{4} - 8 \zeta_{24}^{3} - \zeta_{24}^{2} - 1) q^{46} + (4 \zeta_{24}^{6} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{2} - 5 \zeta_{24}) q^{47} + (\zeta_{24}^{6} - \zeta_{24}^{3} - 1) q^{48} + (8 \zeta_{24}^{6} - 5 \zeta_{24}^{2}) q^{49} + ( - 4 \zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24} + 4) q^{50} + ( - \zeta_{24}^{7} - \zeta_{24}^{6} - 3 \zeta_{24}^{4} + \zeta_{24}^{2} - 4 \zeta_{24}) q^{51} + ( - 3 \zeta_{24}^{4} + 2 \zeta_{24}^{2} + 3) q^{52} + (4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} - 4 \zeta_{24}^{3} + 3 \zeta_{24}^{2}) q^{53} + ( - \zeta_{24}^{7} - 5 \zeta_{24}^{4} + \zeta_{24}) q^{54} + (\zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24} + 2) q^{55} + ( - 3 \zeta_{24}^{4} - 6) q^{56} + (8 \zeta_{24}^{6} + 3 \zeta_{24}^{5} - 5 \zeta_{24}^{3} - 3 \zeta_{24} + 1) q^{57} + \zeta_{24} q^{58} + ( - \zeta_{24}^{5} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{2} - 6) q^{59} + (\zeta_{24}^{7} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{4} - \zeta_{24}^{3} - 2 \zeta_{24}^{2} + 2) q^{60} + \zeta_{24}^{4} q^{61} + (2 \zeta_{24}^{5} - 2 \zeta_{24}^{3} - 2 \zeta_{24} - 2) q^{62} + ( - 4 \zeta_{24}^{6} - \zeta_{24}^{5} - 6 \zeta_{24}^{4} + 6 \zeta_{24}^{2} - 2 \zeta_{24} + 2) q^{63} - 7 \zeta_{24}^{6} q^{64} + ( - 4 \zeta_{24}^{7} + \zeta_{24}^{6} - 5 \zeta_{24}^{4} - \zeta_{24}^{2} - 6 \zeta_{24}) q^{65} + (3 \zeta_{24}^{7} - 4 \zeta_{24}^{5} + \zeta_{24}^{4} - 3 \zeta_{24}^{3} + 5 \zeta_{24}^{2} - 1) q^{66} + ( - 5 \zeta_{24}^{5} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{2} - 6) q^{67} + ( - 2 \zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{2} + 2 \zeta_{24}) q^{68} + ( - 7 \zeta_{24}^{6} - 2 \zeta_{24}^{5} + 8 \zeta_{24}^{3} + 2 \zeta_{24} + 9) q^{69} + (\zeta_{24}^{6} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{2} - 2 \zeta_{24} - 3) q^{70} + (4 \zeta_{24}^{6} + 5 \zeta_{24}^{3} + 4) q^{71} + ( - 3 \zeta_{24}^{7} - 6 \zeta_{24}^{6} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{2}) q^{72} + (10 \zeta_{24}^{7} - 2 \zeta_{24}^{4} - 10 \zeta_{24}^{3} + 2 \zeta_{24}^{2} + 2) q^{73} - 4 \zeta_{24}^{2} q^{74} + (8 \zeta_{24}^{7} + 5 \zeta_{24}^{6} - 3 \zeta_{24}^{4} - 5 \zeta_{24}^{2} - \zeta_{24}) q^{75} + ( - 4 \zeta_{24}^{6} + \zeta_{24}^{3} - 4) q^{76} + ( - 4 \zeta_{24}^{7} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{4} + 6 \zeta_{24}^{3} + 2 \zeta_{24} - 9) q^{77} + (2 \zeta_{24}^{6} - 5 \zeta_{24}^{5} - \zeta_{24}^{3} + 5 \zeta_{24} + 3) q^{78} + (\zeta_{24}^{7} + \zeta_{24}) q^{79} + (2 \zeta_{24}^{7} + \zeta_{24}^{4} - 2 \zeta_{24}^{3} - \zeta_{24}^{2} - 1) q^{80} + ( - 4 \zeta_{24}^{7} + 4 \zeta_{24}^{5} + 7 \zeta_{24}^{4} + 4 \zeta_{24}^{3} - 7) q^{81} + (4 \zeta_{24}^{7} - 4 \zeta_{24}) q^{82} + ( - 4 \zeta_{24}^{6} - 10 \zeta_{24}^{3} - 4) q^{83} + ( - 3 \zeta_{24}^{7} + \zeta_{24}^{5} + 3 \zeta_{24}^{4} + \zeta_{24}^{3} + 2 \zeta_{24} - 1) q^{84} + ( - 5 \zeta_{24}^{6} - 6 \zeta_{24}^{3} - 5) q^{85} + 6 \zeta_{24} q^{86} + ( - \zeta_{24}^{5} + \zeta_{24}^{4} - \zeta_{24}^{2} - 1) q^{87} + ( - 6 \zeta_{24}^{7} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{3} - 9 \zeta_{24}^{2}) q^{88} + (6 \zeta_{24}^{6} - 6 \zeta_{24}^{4} - 6 \zeta_{24}^{2} + 4 \zeta_{24}) q^{89} + ( - 4 \zeta_{24}^{6} - 3 \zeta_{24}^{5} - 5 \zeta_{24}^{3} + 3 \zeta_{24} - 2) q^{90} + (2 \zeta_{24}^{7} - 6 \zeta_{24}^{5} + 4 \zeta_{24}^{3} + 9 \zeta_{24}) q^{91} + (8 \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} - \zeta_{24}) q^{92} + ( - 4 \zeta_{24}^{6} + 2 \zeta_{24}^{4} + 4 \zeta_{24}^{2} + 4 \zeta_{24}) q^{93} + ( - 4 \zeta_{24}^{7} - 4 \zeta_{24}^{5} - 5 \zeta_{24}^{4} + 4 \zeta_{24}^{3} + 5) q^{94} + (7 \zeta_{24}^{7} + 7 \zeta_{24}^{5} + 6 \zeta_{24}^{4} - 7 \zeta_{24}^{3} - 6) q^{95} + (5 \zeta_{24}^{7} + 5 \zeta_{24}^{6} - 5 \zeta_{24}^{2} + 5 \zeta_{24}) q^{96} + (7 \zeta_{24}^{6} - 7) q^{97} + ( - 5 \zeta_{24}^{5} - 3 \zeta_{24}) q^{98} + ( - 4 \zeta_{24}^{6} + 8 \zeta_{24}^{5} - 3 \zeta_{24}^{3} - 8 \zeta_{24} + 8) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{5} + 8 q^{11} - 4 q^{12} + 16 q^{13} - 16 q^{15} - 4 q^{16} + 4 q^{17} + 8 q^{18} - 16 q^{19} + 8 q^{20} - 24 q^{22} + 32 q^{23} - 12 q^{24} - 8 q^{27} - 4 q^{30} - 8 q^{31} + 12 q^{33} + 16 q^{34} - 8 q^{36} - 4 q^{38} - 4 q^{39} + 24 q^{40} + 32 q^{41} + 20 q^{42} - 8 q^{44} - 20 q^{45} - 4 q^{46} - 16 q^{47} - 8 q^{48} + 32 q^{50} - 12 q^{51} + 12 q^{52} - 20 q^{54} + 16 q^{55} - 60 q^{56} + 8 q^{57} - 24 q^{59} + 8 q^{60} + 4 q^{61} - 16 q^{62} - 8 q^{63} - 20 q^{65} - 4 q^{66} - 24 q^{67} + 72 q^{69} - 16 q^{70} + 32 q^{71} + 24 q^{72} + 8 q^{73} - 12 q^{75} - 32 q^{76} - 48 q^{77} + 24 q^{78} - 4 q^{80} - 28 q^{81} - 32 q^{83} + 4 q^{84} - 40 q^{85} - 4 q^{87} - 24 q^{89} - 16 q^{90} + 8 q^{93} + 20 q^{94} - 24 q^{95} - 56 q^{97} + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{6}\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i 1.62484 0.599900i −0.866025 + 0.500000i 0.565826 0.151613i −1.41421 + 1.00000i −2.38014 1.15539i 2.12132 2.12132i 2.28024 1.94949i −0.507306 + 0.292893i
44.2 0.965926 0.258819i 1.10721 + 1.33195i −0.866025 + 0.500000i −3.29788 + 0.883663i 1.41421 + 1.00000i 2.38014 + 1.15539i −2.12132 + 2.12132i −0.548188 + 2.94949i −2.95680 + 1.70711i
86.1 −0.258819 0.965926i 0.599900 + 1.62484i 0.866025 0.500000i 0.883663 + 3.29788i 1.41421 1.00000i 1.15539 2.38014i −2.12132 2.12132i −2.28024 + 1.94949i 2.95680 1.70711i
86.2 0.258819 + 0.965926i −1.33195 + 1.10721i 0.866025 0.500000i −0.151613 0.565826i −1.41421 1.00000i −1.15539 + 2.38014i 2.12132 + 2.12132i 0.548188 2.94949i 0.507306 0.292893i
200.1 −0.258819 + 0.965926i 0.599900 1.62484i 0.866025 + 0.500000i 0.883663 3.29788i 1.41421 + 1.00000i 1.15539 + 2.38014i −2.12132 + 2.12132i −2.28024 1.94949i 2.95680 + 1.70711i
200.2 0.258819 0.965926i −1.33195 1.10721i 0.866025 + 0.500000i −0.151613 + 0.565826i −1.41421 + 1.00000i −1.15539 2.38014i 2.12132 2.12132i 0.548188 + 2.94949i 0.507306 + 0.292893i
242.1 −0.965926 0.258819i 1.62484 + 0.599900i −0.866025 0.500000i 0.565826 + 0.151613i −1.41421 1.00000i −2.38014 + 1.15539i 2.12132 + 2.12132i 2.28024 + 1.94949i −0.507306 0.292893i
242.2 0.965926 + 0.258819i 1.10721 1.33195i −0.866025 0.500000i −3.29788 0.883663i 1.41421 1.00000i 2.38014 1.15539i −2.12132 2.12132i −0.548188 2.94949i −2.95680 1.70711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 242.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
39.f even 4 1 inner
273.cd even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.c 8
3.b odd 2 1 273.2.cd.d yes 8
7.c even 3 1 inner 273.2.cd.c 8
13.d odd 4 1 273.2.cd.d yes 8
21.h odd 6 1 273.2.cd.d yes 8
39.f even 4 1 inner 273.2.cd.c 8
91.z odd 12 1 273.2.cd.d yes 8
273.cd even 12 1 inner 273.2.cd.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.c 8 1.a even 1 1 trivial
273.2.cd.c 8 7.c even 3 1 inner
273.2.cd.c 8 39.f even 4 1 inner
273.2.cd.c 8 273.cd even 12 1 inner
273.2.cd.d yes 8 3.b odd 2 1
273.2.cd.d yes 8 13.d odd 4 1
273.2.cd.d yes 8 21.h odd 6 1
273.2.cd.d yes 8 91.z odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{8} - T_{2}^{4} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 4T_{5}^{7} + 8T_{5}^{6} + 48T_{5}^{5} + 92T_{5}^{4} - 96T_{5}^{3} + 32T_{5}^{2} - 32T_{5} + 16 \) Copy content Toggle raw display
\( T_{19}^{8} + 16 T_{19}^{7} + 128 T_{19}^{6} + 1056 T_{19}^{5} + 7487 T_{19}^{4} + 32736 T_{19}^{3} + 123008 T_{19}^{2} + 476656 T_{19} + 923521 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + 8 T^{6} + 48 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 23T^{4} + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} - 8 T^{7} + 32 T^{6} - 272 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 13)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} + 11 T^{2} + 14 T + 49)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 923521 \) Copy content Toggle raw display
$23$ \( (T^{4} - 16 T^{3} + 194 T^{2} - 992 T + 3844)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 8 T^{7} + 32 T^{6} + 192 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( T^{8} - 256 T^{4} + 65536 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T + 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$53$ \( T^{8} - 82 T^{6} + 6195 T^{4} + \cdots + 279841 \) Copy content Toggle raw display
$59$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 25411681 \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 4879681 \) Copy content Toggle raw display
$71$ \( (T^{4} - 16 T^{3} + 128 T^{2} - 112 T + 49)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 71639296 \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} + 128 T^{2} - 1088 T + 4624)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 9834496 \) Copy content Toggle raw display
$97$ \( (T^{2} + 14 T + 98)^{4} \) Copy content Toggle raw display
show more
show less