Properties

Label 936.2.m.h
Level $936$
Weight $2$
Character orbit 936.m
Analytic conductor $7.474$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} - \beta_{5} q^{7} + (\beta_{14} + \beta_{12} - \beta_{10} - \beta_{9}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + \beta_{3} q^{4} + \beta_{14} q^{5} - \beta_{5} q^{7} + (\beta_{14} + \beta_{12} - \beta_{10} - \beta_{9}) q^{8} + (\beta_{4} - \beta_{2} + 1) q^{10} + ( - \beta_{14} - \beta_{12} - \beta_{10}) q^{11} + (\beta_{15} + \beta_{4} + \beta_{2}) q^{13} + (\beta_{11} - \beta_{6} - \beta_1) q^{14} - 2 \beta_{2} q^{16} + ( - \beta_{11} + \beta_{7} - 2 \beta_1) q^{17} + ( - \beta_{15} + \beta_{13}) q^{19} + ( - 2 \beta_{12} - 2 \beta_{9}) q^{20} + ( - \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{22} + (\beta_{11} - 2 \beta_{6}) q^{23} + ( - \beta_{4} - \beta_{3}) q^{25} + ( - 2 \beta_{12} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_1) q^{26} + ( - \beta_{15} + 2 \beta_{13} - \beta_{8}) q^{28} - \beta_{7} q^{29} + (2 \beta_{8} - \beta_{5}) q^{31} - 4 \beta_{9} q^{32} + ( - 2 \beta_{15} - \beta_{13} - \beta_{8} + \beta_{5}) q^{34} + ( - \beta_{11} - 2 \beta_{7}) q^{35} + ( - 2 \beta_{15} - 2 \beta_{13}) q^{37} + ( - \beta_{11} - \beta_{6} - \beta_1) q^{38} + ( - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 6) q^{40} + ( - 3 \beta_{12} + 3 \beta_{10} - 3 \beta_{9}) q^{41} + (\beta_{4} - \beta_{3} + 1) q^{43} + (\beta_{14} + 3 \beta_{12} - 3 \beta_{10} + \beta_{9}) q^{44} + (\beta_{15} + 3 \beta_{13} + \beta_{5}) q^{46} + (4 \beta_{12} - 4 \beta_{10} - 3 \beta_{9}) q^{47} + (\beta_{4} + \beta_{3} - 8) q^{49} + ( - \beta_{14} + \beta_{12} + \beta_{9}) q^{50} + (\beta_{13} + \beta_{8} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{52} + ( - \beta_{11} + 2 \beta_{7}) q^{53} + (3 \beta_{4} + 3 \beta_{3} - 5) q^{55} + ( - 2 \beta_{7} - 2 \beta_{6}) q^{56} + (\beta_{15} - \beta_{8}) q^{58} + (\beta_{14} - 3 \beta_{12} - 3 \beta_{10}) q^{59} + (3 \beta_{4} + 3 \beta_{3} + 6 \beta_{2} - 3) q^{61} + ( - \beta_{11} + 4 \beta_{7} - \beta_{6} - 3 \beta_1) q^{62} + ( - 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 4) q^{64} + ( - \beta_{12} + \beta_{10} + 3 \beta_{9} + \beta_{7} - 2 \beta_{6} - 2 \beta_1) q^{65} + (\beta_{15} - 3 \beta_{13}) q^{67} + ( - 2 \beta_{11} - 2 \beta_{7} + 2 \beta_{6}) q^{68} + (3 \beta_{15} - \beta_{13} - 2 \beta_{8} + \beta_{5}) q^{70} + (4 \beta_{12} - 4 \beta_{10} - 5 \beta_{9}) q^{71} + (2 \beta_{8} - 2 \beta_{5}) q^{73} + ( - 2 \beta_{11} + 2 \beta_{6} - 2 \beta_1) q^{74} + (\beta_{15} - \beta_{8} + 2 \beta_{5}) q^{76} + (3 \beta_{11} + \beta_{7}) q^{77} + ( - 4 \beta_{4} - 4 \beta_{3} + 2) q^{79} + ( - 2 \beta_{14} + 2 \beta_{12} - 6 \beta_{10} - 2 \beta_{9}) q^{80} + ( - 3 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} + 9) q^{82} + (3 \beta_{14} + \beta_{12} + \beta_{10}) q^{83} + (2 \beta_{15} - 4 \beta_{13}) q^{85} + ( - \beta_{14} - 3 \beta_{12} + \beta_{10} + \beta_{9}) q^{86} + (2 \beta_{4} + 4 \beta_{3} - 6) q^{88} + (\beta_{12} - \beta_{10} + 7 \beta_{9}) q^{89} + ( - \beta_{15} - 3 \beta_{13} + \beta_{4} + 5 \beta_{3} + 6 \beta_{2} - 5) q^{91} + ( - 2 \beta_{6} + 2 \beta_1) q^{92} + ( - 3 \beta_{4} + \beta_{3} - 3 \beta_{2} - 5) q^{94} + (\beta_{11} - 2 \beta_{7} + 2 \beta_{6} + 4 \beta_1) q^{95} + 2 \beta_{8} q^{97} + (\beta_{14} - \beta_{12} + 8 \beta_{10} - \beta_{9}) 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 \( 16 q + 8 q^{4} - 16 q^{16} + 40 q^{22} + 80 q^{40} - 128 q^{49} + 40 q^{52} - 80 q^{55} + 32 q^{64} + 32 q^{79} + 96 q^{82} - 80 q^{88} - 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 16x^{12} - 72x^{10} + 26x^{8} + 360x^{6} + 725x^{4} + 1000x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 310 \nu^{14} - 2816 \nu^{12} - 9222 \nu^{10} + 18376 \nu^{8} + 130292 \nu^{6} - 53906 \nu^{4} + 290050 \nu^{2} + 217000 ) / 176275 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11992 \nu^{14} + 9614 \nu^{12} - 190612 \nu^{10} - 554504 \nu^{8} + 1001932 \nu^{6} + 2217000 \nu^{4} + 3496000 \nu^{2} + 5574000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16948 \nu^{14} + 5486 \nu^{12} - 299188 \nu^{10} - 779596 \nu^{8} + 1911868 \nu^{6} + 4019270 \nu^{4} + 6074000 \nu^{2} + 6351000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17344 \nu^{14} - 7298 \nu^{12} + 306684 \nu^{10} + 805078 \nu^{8} - 1876924 \nu^{6} - 3981500 \nu^{4} - 6068600 \nu^{2} - 8342875 ) / 881375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10661 \nu^{15} - 14207 \nu^{13} + 225956 \nu^{11} + 888177 \nu^{9} - 521116 \nu^{7} - 5727895 \nu^{5} - 13225300 \nu^{3} - 26244000 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 25832 \nu^{14} - 21274 \nu^{12} + 450392 \nu^{10} + 1381764 \nu^{8} - 2270262 \nu^{6} - 7392230 \nu^{4} - 12703750 \nu^{2} - 13499250 ) / 881375 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28806 \nu^{14} - 29832 \nu^{12} + 448156 \nu^{10} + 1496702 \nu^{8} - 2047066 \nu^{6} - 6175530 \nu^{4} - 12354350 \nu^{2} - 13465750 ) / 881375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15392 \nu^{15} + 52754 \nu^{13} - 145257 \nu^{11} - 987344 \nu^{9} - 868348 \nu^{7} + 1700315 \nu^{5} + 6288975 \nu^{3} + 15325500 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu ) / 400625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 22194 \nu^{15} - 11523 \nu^{13} + 339409 \nu^{11} + 1024903 \nu^{9} - 1669899 \nu^{7} - 3769000 \nu^{5} - 11127550 \nu^{3} - 9814875 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000 ) / 881375 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 46447 \nu^{15} - 41694 \nu^{13} + 802127 \nu^{11} + 2508384 \nu^{9} - 4035597 \nu^{7} - 13232745 \nu^{5} - 19604750 \nu^{3} - 21119125 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 78923 \nu^{15} + 34611 \nu^{13} - 1339113 \nu^{11} - 3553821 \nu^{9} + 8128668 \nu^{7} + 15701920 \nu^{5} + 26400825 \nu^{3} + 32586750 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 108813 \nu^{15} - 85331 \nu^{13} + 1829673 \nu^{11} + 5646091 \nu^{9} - 9386828 \nu^{7} - 27713410 \nu^{5} - 49911275 \nu^{3} - 50144500 \nu ) / 4406875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 128244 \nu^{15} - 47208 \nu^{13} + 2180589 \nu^{11} + 5658588 \nu^{9} - 13459254 \nu^{7} - 25923185 \nu^{5} - 43314975 \nu^{3} - 53041500 \nu ) / 4406875 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{13} + 2\beta_{12} - 2\beta_{9} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{14} - \beta_{13} + 5\beta_{12} + 7\beta_{10} + 5\beta_{9} + \beta_{8} + \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} + 8\beta_{4} + 5\beta_{3} + 5\beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{15} - 12\beta_{13} - 3\beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11\beta_{11} - 11\beta_{7} - 6\beta_{6} + 11\beta_{3} - 15\beta_{2} + 5\beta _1 + 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -5\beta_{15} - 55\beta_{14} - 4\beta_{13} + 127\beta_{12} + 55\beta_{10} - 55\beta_{9} - 9\beta_{8} - 4\beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 32\beta_{11} - 52\beta_{6} + 53\beta_{4} + 18\beta_{3} + 53\beta_{2} + 32\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 137 \beta_{15} - 57 \beta_{14} - 224 \beta_{13} - 57 \beta_{12} + 247 \beta_{10} + 247 \beta_{9} + 87 \beta_{8} + 50 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 21\beta_{11} - 36\beta_{7} + 275\beta_{4} + 275\beta_{3} + 607 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 384 \beta_{15} - 341 \beta_{14} - 623 \beta_{13} + 1421 \beta_{12} - 341 \beta_{10} - 1421 \beta_{9} - 239 \beta_{8} - 145 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 752\beta_{11} - 462\beta_{7} - 752\beta_{6} - 110\beta_{4} - 177\beta_{2} + 462\beta _1 + 110 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 395 \beta_{15} - 2315 \beta_{14} - 642 \beta_{13} + 2315 \beta_{12} + 5199 \beta_{10} + 2315 \beta_{9} + 1037 \beta_{8} + 642 \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 639\beta_{7} - 639\beta_{6} + 5431\beta_{4} + 3360\beta_{3} + 3360\beta_{2} + 400\beta _1 + 5431 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -4399\beta_{15} - 7109\beta_{13} + 2475\beta_{12} - 2475\beta_{10} - 3996\beta_{9} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.556839 1.81878i
1.90184 0.0324487i
1.90184 + 0.0324487i
−0.556839 + 1.81878i
0.752864 0.902863i
0.0783900 + 1.17295i
0.0783900 1.17295i
0.752864 + 0.902863i
−0.752864 0.902863i
−0.0783900 + 1.17295i
−0.0783900 1.17295i
−0.752864 + 0.902863i
0.556839 1.81878i
−1.90184 0.0324487i
−1.90184 + 0.0324487i
0.556839 + 1.81878i
−1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.2 −1.34500 0.437016i 0 1.61803 + 1.17557i −1.66251 0 3.57266i −1.66251 2.28825i 0 2.23607 + 0.726543i
181.3 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.4 −1.34500 + 0.437016i 0 1.61803 1.17557i −1.66251 0 3.57266i −1.66251 + 2.28825i 0 2.23607 0.726543i
181.5 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.6 −0.831254 1.14412i 0 −0.618034 + 1.90211i 2.68999 0 4.15163i 2.68999 0.874032i 0 −2.23607 3.07768i
181.7 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.8 −0.831254 + 1.14412i 0 −0.618034 1.90211i 2.68999 0 4.15163i 2.68999 + 0.874032i 0 −2.23607 + 3.07768i
181.9 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.10 0.831254 1.14412i 0 −0.618034 1.90211i −2.68999 0 4.15163i −2.68999 0.874032i 0 −2.23607 + 3.07768i
181.11 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.12 0.831254 + 1.14412i 0 −0.618034 + 1.90211i −2.68999 0 4.15163i −2.68999 + 0.874032i 0 −2.23607 3.07768i
181.13 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.14 1.34500 0.437016i 0 1.61803 1.17557i 1.66251 0 3.57266i 1.66251 2.28825i 0 2.23607 0.726543i
181.15 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
181.16 1.34500 + 0.437016i 0 1.61803 + 1.17557i 1.66251 0 3.57266i 1.66251 + 2.28825i 0 2.23607 + 0.726543i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.m.h 16
3.b odd 2 1 inner 936.2.m.h 16
4.b odd 2 1 3744.2.m.h 16
8.b even 2 1 inner 936.2.m.h 16
8.d odd 2 1 3744.2.m.h 16
12.b even 2 1 3744.2.m.h 16
13.b even 2 1 inner 936.2.m.h 16
24.f even 2 1 3744.2.m.h 16
24.h odd 2 1 inner 936.2.m.h 16
39.d odd 2 1 inner 936.2.m.h 16
52.b odd 2 1 3744.2.m.h 16
104.e even 2 1 inner 936.2.m.h 16
104.h odd 2 1 3744.2.m.h 16
156.h even 2 1 3744.2.m.h 16
312.b odd 2 1 inner 936.2.m.h 16
312.h even 2 1 3744.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 1.a even 1 1 trivial
936.2.m.h 16 3.b odd 2 1 inner
936.2.m.h 16 8.b even 2 1 inner
936.2.m.h 16 13.b even 2 1 inner
936.2.m.h 16 24.h odd 2 1 inner
936.2.m.h 16 39.d odd 2 1 inner
936.2.m.h 16 104.e even 2 1 inner
936.2.m.h 16 312.b odd 2 1 inner
3744.2.m.h 16 4.b odd 2 1
3744.2.m.h 16 8.d odd 2 1
3744.2.m.h 16 12.b even 2 1
3744.2.m.h 16 24.f even 2 1
3744.2.m.h 16 52.b odd 2 1
3744.2.m.h 16 104.h odd 2 1
3744.2.m.h 16 156.h even 2 1
3744.2.m.h 16 312.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10T_{5}^{2} + 20 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{6} + 4 T^{4} - 8 T^{2} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} - 10 T^{2} + 20)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 30 T^{2} + 220)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 20)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 12 T^{6} + 54 T^{4} - 2028 T^{2} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 60 T^{2} + 880)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 34 T^{2} + 44)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 880)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 28 T^{2} + 176)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 150 T^{2} + 5500)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 104 T^{2} + 704)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 126 T^{2} + 324)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 20 T^{2} + 80)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 84 T^{2} + 1444)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 128 T^{2} + 176)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 100 T^{2} + 500)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 180 T^{2} + 6480)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 154 T^{2} + 5324)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 116 T^{2} + 484)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 200 T^{2} + 3520)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 76)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 100 T^{2} + 500)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 230 T^{2} + 12100)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 160 T^{2} + 3520)^{4} \) Copy content Toggle raw display
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