Properties

Label 378.2.m.a
Level $378$
Weight $2$
Character orbit 378.m
Analytic conductor $3.018$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.01834519640\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{15} + \beta_{8} + \beta_{6}) q^{5} + (\beta_{13} - \beta_{10} + \beta_{2} - 1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + (\beta_{5} + 1) q^{4} + ( - \beta_{15} + \beta_{8} + \beta_{6}) q^{5} + (\beta_{13} - \beta_{10} + \beta_{2} - 1) q^{7} + \beta_{3} q^{8} + ( - \beta_{15} + \beta_{8}) q^{10} + ( - \beta_{12} + \beta_{10} - \beta_{7} + \beta_{5} + \beta_{3} - \beta_1 + 2) q^{11} + ( - \beta_{15} + \beta_{4}) q^{13} + (\beta_{11} - \beta_{9} + \beta_{3} - \beta_{2} + 1) q^{14} + \beta_{5} q^{16} + ( - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{17} + (\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{19} + ( - \beta_{15} + \beta_{8} + \beta_{4}) q^{20} + ( - \beta_{12} + \beta_{10} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{22} + (\beta_{12} - \beta_{10} + 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{3} + \beta_{2} + 1) q^{23} + ( - \beta_{14} - \beta_{13} + \beta_{12} - 2 \beta_{7} + 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{25} + (\beta_{15} + \beta_{8} - \beta_{6} + \beta_{4}) q^{26} + ( - \beta_{10} + \beta_{7}) q^{28} + (\beta_{14} + \beta_{13} - \beta_{10} - 2 \beta_{7} + \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - \beta_{3} + \cdots + 1) q^{31}+ \cdots + ( - 2 \beta_{15} + 2 \beta_{14} - \beta_{12} + \beta_{11} - 2 \beta_{8} - \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 2 q^{7} + 12 q^{11} + 6 q^{14} - 8 q^{16} + 48 q^{23} - 8 q^{25} + 4 q^{28} + 12 q^{29} - 8 q^{37} + 4 q^{43} + 24 q^{46} - 8 q^{49} - 60 q^{50} + 6 q^{56} - 12 q^{58} - 16 q^{64} - 84 q^{65} - 28 q^{67} - 36 q^{74} - 78 q^{77} - 4 q^{79} - 12 q^{85} + 24 q^{86} + 24 q^{91} + 48 q^{92} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 72\nu^{6} + 234\nu^{4} + 729\nu^{2} - 243 ) / 1944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{14} + \nu^{12} - 6\nu^{10} + 36\nu^{8} - 180\nu^{6} - 396\nu^{4} + 972\nu^{2} - 4131 ) / 1944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} - 3\nu^{12} - 9\nu^{10} + 81\nu^{8} - 126\nu^{6} - 135\nu^{4} + 1458\nu^{2} - 2187 ) / 1458 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{15} - 9\nu^{13} + 18\nu^{11} - 396\nu^{7} + 216\nu^{5} - 324\nu^{3} - 9477\nu ) / 5832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{14} - 21\nu^{12} + 18\nu^{10} + 108\nu^{8} - 576\nu^{6} + 648\nu^{4} + 972\nu^{2} - 9477 ) / 5832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{15} + 24\nu^{13} - 36\nu^{11} - 540\nu^{9} + 1044\nu^{7} - 1134\nu^{5} - 8019\nu^{3} + 13122\nu ) / 17496 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{14} - 10\nu^{12} - 12\nu^{10} + 180\nu^{8} - 432\nu^{6} - 198\nu^{4} + 3483\nu^{2} - 5832 ) / 1944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -4\nu^{15} + 9\nu^{13} + 18\nu^{11} - 216\nu^{9} + 504\nu^{7} + 432\nu^{5} - 2754\nu^{3} + 9477\nu ) / 5832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8 \nu^{15} - 3 \nu^{14} + 48 \nu^{13} - 36 \nu^{12} - 18 \nu^{11} + 216 \nu^{10} - 270 \nu^{9} - 162 \nu^{8} + 1332 \nu^{7} - 594 \nu^{6} - 2430 \nu^{5} + 5346 \nu^{4} + 486 \nu^{3} + \cdots - 8748 ) / 17496 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8 \nu^{15} - 27 \nu^{14} + 12 \nu^{13} + 72 \nu^{12} + 90 \nu^{11} + 54 \nu^{10} - 432 \nu^{9} - 810 \nu^{8} - 126 \nu^{7} + 2916 \nu^{6} + 2106 \nu^{5} - 3240 \nu^{4} - 7776 \nu^{3} + \cdots + 21870 ) / 17496 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4 \nu^{15} - 39 \nu^{14} + 24 \nu^{13} + 108 \nu^{12} - 90 \nu^{11} + 162 \nu^{10} + 108 \nu^{9} - 1782 \nu^{8} + 666 \nu^{7} + 4428 \nu^{6} - 3402 \nu^{5} - 1620 \nu^{4} + 3888 \nu^{3} + \cdots + 48114 ) / 17496 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8 \nu^{15} + 39 \nu^{14} + 12 \nu^{13} - 108 \nu^{12} + 90 \nu^{11} - 162 \nu^{10} - 432 \nu^{9} + 1782 \nu^{8} - 126 \nu^{7} - 4428 \nu^{6} + 2106 \nu^{5} + 1620 \nu^{4} - 7776 \nu^{3} + \cdots - 48114 ) / 17496 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14 \nu^{15} - 6 \nu^{14} - 66 \nu^{13} + 99 \nu^{12} + 72 \nu^{11} - 108 \nu^{10} + 594 \nu^{9} - 972 \nu^{8} - 2574 \nu^{7} + 5130 \nu^{6} + 1620 \nu^{5} - 5022 \nu^{4} + 7776 \nu^{3} + \cdots + 85293 ) / 17496 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22 \nu^{15} - 6 \nu^{14} + 78 \nu^{13} + 27 \nu^{12} + 18 \nu^{11} - 162 \nu^{10} - 1026 \nu^{9} - 162 \nu^{8} + 2448 \nu^{7} + 1242 \nu^{6} + 486 \nu^{5} - 3240 \nu^{4} - 15552 \nu^{3} + \cdots + 6561 ) / 17496 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -35\nu^{15} + 138\nu^{13} + 36\nu^{11} - 2052\nu^{9} + 6192\nu^{7} - 2106\nu^{5} - 37179\nu^{3} + 87480\nu ) / 17496 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + 2\beta_{12} + 2\beta_{11} - \beta_{9} + \beta_{8} - \beta_{4} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + \beta_{6} + \beta_{2} - \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{13} + 2\beta_{12} - \beta_{10} - 4\beta_{7} - 3\beta_{5} + 3\beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{15} - 5 \beta_{13} + \beta_{11} + 4 \beta_{10} - 5 \beta_{9} + 3 \beta_{8} - 2 \beta_{6} - 3 \beta_{4} + \beta_{3} - 5 \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{12} + 3\beta_{10} - 6\beta_{7} + 6\beta_{5} + 18\beta_{3} - 6\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3 \beta_{15} + 3 \beta_{14} - 18 \beta_{12} - 12 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} - 9 \beta_{6} + 6 \beta_{4} + 6 \beta_{3} - 3 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{14} - 12 \beta_{13} - 3 \beta_{12} + 15 \beta_{10} - 33 \beta_{7} + 3 \beta_{5} + 75 \beta_{3} - 12 \beta_{2} - 15 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18 \beta_{15} - 15 \beta_{14} - 27 \beta_{13} + 21 \beta_{12} + 39 \beta_{11} + 9 \beta_{10} - 42 \beta_{9} - 21 \beta_{8} - 72 \beta_{6} + 3 \beta_{4} + 18 \beta_{3} - 27 \beta_{2} + 15 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 45 \beta_{14} - 45 \beta_{13} - 45 \beta_{12} + 90 \beta_{10} + 90 \beta_{7} - 27 \beta_{5} - 27 \beta_{3} - 72 \beta_{2} - 45 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 63 \beta_{15} + 36 \beta_{14} + 153 \beta_{13} - 54 \beta_{12} - 135 \beta_{11} - 72 \beta_{10} + 189 \beta_{9} - 36 \beta_{8} - 117 \beta_{6} + 72 \beta_{4} - 81 \beta_{3} + 153 \beta_{2} - 36 \beta _1 - 153 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -126\beta_{14} - 126\beta_{13} + 90\beta_{12} + 36\beta_{10} + 90\beta_{7} - 648\beta_{5} - 216\beta_{3} - 243 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 36 \beta_{15} + 189 \beta_{14} - 36 \beta_{13} + 216 \beta_{12} + 18 \beta_{11} + 234 \beta_{10} + 153 \beta_{9} - 459 \beta_{8} - 144 \beta_{6} - 297 \beta_{4} - 198 \beta_{3} - 36 \beta_{2} - 189 \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 54 \beta_{14} + 54 \beta_{13} - 54 \beta_{10} + 999 \beta_{7} - 621 \beta_{5} - 1377 \beta_{3} - 432 \beta_{2} - 243 \beta _1 - 1242 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 81 \beta_{15} + 297 \beta_{14} + 837 \beta_{13} - 891 \beta_{12} - 1809 \beta_{11} + 81 \beta_{10} + 1134 \beta_{9} - 1026 \beta_{8} + 999 \beta_{6} - 864 \beta_{4} - 918 \beta_{3} + 837 \beta_{2} - 297 \beta _1 - 837 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1 + \beta_{5}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
−1.69547 0.354107i
0.0967785 1.72934i
−0.0967785 + 1.72934i
1.69547 + 0.354107i
−1.62181 + 0.608059i
1.40917 + 1.00709i
−1.40917 1.00709i
1.62181 0.608059i
−1.69547 + 0.354107i
0.0967785 + 1.72934i
−0.0967785 1.72934i
1.69547 0.354107i
−1.62181 0.608059i
1.40917 1.00709i
−1.40917 + 1.00709i
1.62181 + 0.608059i
−0.866025 + 0.500000i 0 0.500000 0.866025i −0.895175 + 1.55049i 0 0.0213944 2.64566i 1.00000i 0 1.79035i
125.2 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.183299 + 0.317483i 0 −0.624224 + 2.57106i 1.00000i 0 0.366598i
125.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.183299 0.317483i 0 2.53871 + 0.744936i 1.00000i 0 0.366598i
125.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 0.895175 1.55049i 0 −2.30191 1.30430i 1.00000i 0 1.79035i
125.5 0.866025 0.500000i 0 0.500000 0.866025i −1.94556 + 3.36980i 0 0.343982 + 2.62329i 1.00000i 0 3.89111i
125.6 0.866025 0.500000i 0 0.500000 0.866025i −1.17468 + 2.03460i 0 1.55364 2.14154i 1.00000i 0 2.34936i
125.7 0.866025 0.500000i 0 0.500000 0.866025i 1.17468 2.03460i 0 −2.63145 + 0.274725i 1.00000i 0 2.34936i
125.8 0.866025 0.500000i 0 0.500000 0.866025i 1.94556 3.36980i 0 2.09985 + 1.60954i 1.00000i 0 3.89111i
251.1 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.895175 1.55049i 0 0.0213944 + 2.64566i 1.00000i 0 1.79035i
251.2 −0.866025 0.500000i 0 0.500000 + 0.866025i −0.183299 0.317483i 0 −0.624224 2.57106i 1.00000i 0 0.366598i
251.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.183299 + 0.317483i 0 2.53871 0.744936i 1.00000i 0 0.366598i
251.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 0.895175 + 1.55049i 0 −2.30191 + 1.30430i 1.00000i 0 1.79035i
251.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.94556 3.36980i 0 0.343982 2.62329i 1.00000i 0 3.89111i
251.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −1.17468 2.03460i 0 1.55364 + 2.14154i 1.00000i 0 2.34936i
251.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.17468 + 2.03460i 0 −2.63145 0.274725i 1.00000i 0 2.34936i
251.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.94556 + 3.36980i 0 2.09985 1.60954i 1.00000i 0 3.89111i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
9.d odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.m.a 16
3.b odd 2 1 126.2.m.a 16
4.b odd 2 1 3024.2.cc.b 16
7.b odd 2 1 inner 378.2.m.a 16
7.c even 3 1 2646.2.l.b 16
7.c even 3 1 2646.2.t.a 16
7.d odd 6 1 2646.2.l.b 16
7.d odd 6 1 2646.2.t.a 16
9.c even 3 1 126.2.m.a 16
9.c even 3 1 1134.2.d.a 16
9.d odd 6 1 inner 378.2.m.a 16
9.d odd 6 1 1134.2.d.a 16
12.b even 2 1 1008.2.cc.b 16
21.c even 2 1 126.2.m.a 16
21.g even 6 1 882.2.l.a 16
21.g even 6 1 882.2.t.b 16
21.h odd 6 1 882.2.l.a 16
21.h odd 6 1 882.2.t.b 16
28.d even 2 1 3024.2.cc.b 16
36.f odd 6 1 1008.2.cc.b 16
36.h even 6 1 3024.2.cc.b 16
63.g even 3 1 882.2.l.a 16
63.h even 3 1 882.2.t.b 16
63.i even 6 1 2646.2.t.a 16
63.j odd 6 1 2646.2.t.a 16
63.k odd 6 1 882.2.l.a 16
63.l odd 6 1 126.2.m.a 16
63.l odd 6 1 1134.2.d.a 16
63.n odd 6 1 2646.2.l.b 16
63.o even 6 1 inner 378.2.m.a 16
63.o even 6 1 1134.2.d.a 16
63.s even 6 1 2646.2.l.b 16
63.t odd 6 1 882.2.t.b 16
84.h odd 2 1 1008.2.cc.b 16
252.s odd 6 1 3024.2.cc.b 16
252.bi even 6 1 1008.2.cc.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a 16 3.b odd 2 1
126.2.m.a 16 9.c even 3 1
126.2.m.a 16 21.c even 2 1
126.2.m.a 16 63.l odd 6 1
378.2.m.a 16 1.a even 1 1 trivial
378.2.m.a 16 7.b odd 2 1 inner
378.2.m.a 16 9.d odd 6 1 inner
378.2.m.a 16 63.o even 6 1 inner
882.2.l.a 16 21.g even 6 1
882.2.l.a 16 21.h odd 6 1
882.2.l.a 16 63.g even 3 1
882.2.l.a 16 63.k odd 6 1
882.2.t.b 16 21.g even 6 1
882.2.t.b 16 21.h odd 6 1
882.2.t.b 16 63.h even 3 1
882.2.t.b 16 63.t odd 6 1
1008.2.cc.b 16 12.b even 2 1
1008.2.cc.b 16 36.f odd 6 1
1008.2.cc.b 16 84.h odd 2 1
1008.2.cc.b 16 252.bi even 6 1
1134.2.d.a 16 9.c even 3 1
1134.2.d.a 16 9.d odd 6 1
1134.2.d.a 16 63.l odd 6 1
1134.2.d.a 16 63.o even 6 1
2646.2.l.b 16 7.c even 3 1
2646.2.l.b 16 7.d odd 6 1
2646.2.l.b 16 63.n odd 6 1
2646.2.l.b 16 63.s even 6 1
2646.2.t.a 16 7.c even 3 1
2646.2.t.a 16 7.d odd 6 1
2646.2.t.a 16 63.i even 6 1
2646.2.t.a 16 63.j odd 6 1
3024.2.cc.b 16 4.b odd 2 1
3024.2.cc.b 16 28.d even 2 1
3024.2.cc.b 16 36.h even 6 1
3024.2.cc.b 16 252.s odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(378, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 24 T^{14} + 423 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + 6 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} - 9 T^{6} + 126 T^{5} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 36 T^{14} + 972 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$17$ \( (T^{8} - 42 T^{6} + 477 T^{4} - 1296 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 54 T^{6} + 594 T^{4} + 1854 T^{2} + \cdots + 1521)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 24 T^{7} + 225 T^{6} + \cdots + 443556)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 6 T^{7} - 18 T^{6} + 180 T^{5} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 144 T^{14} + \cdots + 557256278016 \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} - 102 T^{2} - 184 T + 1336)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + 258 T^{14} + \cdots + 73499483897856 \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + 43 T^{6} - 218 T^{5} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 240 T^{14} + \cdots + 1485512441856 \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} + 294 T^{14} + 68787 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$61$ \( T^{16} - 240 T^{14} + \cdots + 2425818710016 \) Copy content Toggle raw display
$67$ \( (T^{8} + 14 T^{7} + 307 T^{6} + \cdots + 824464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 90 T^{6} + 2745 T^{4} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 222 T^{6} + 12069 T^{4} + \cdots + 1710864)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 2 T^{7} + 133 T^{6} + \cdots + 1444804)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 708 T^{14} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{8} - 216 T^{6} + 12960 T^{4} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 702 T^{14} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
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