Properties

Label 1216.3.d.d
Level $1216$
Weight $3$
Character orbit 1216.d
Analytic conductor $33.134$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1216.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.1336001462\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 76)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} -\beta_{7} q^{5} + \beta_{9} q^{7} + ( -5 - \beta_{7} - \beta_{11} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} -\beta_{7} q^{5} + \beta_{9} q^{7} + ( -5 - \beta_{7} - \beta_{11} ) q^{9} -\beta_{8} q^{11} + ( -4 + 2 \beta_{7} + \beta_{13} ) q^{13} + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{9} ) q^{15} + ( 2 + \beta_{4} + \beta_{7} + \beta_{10} + \beta_{13} ) q^{17} + \beta_{1} q^{19} + ( 3 - \beta_{7} + \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{21} + ( 4 \beta_{1} - \beta_{3} + \beta_{5} ) q^{23} + ( -6 + \beta_{7} + \beta_{11} ) q^{25} + ( -\beta_{1} + 6 \beta_{3} + 2 \beta_{5} + \beta_{8} ) q^{27} + ( -4 + \beta_{4} - 2 \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{29} + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{9} ) q^{31} + ( 1 + \beta_{4} + \beta_{11} + 3 \beta_{13} ) q^{33} + ( 4 \beta_{1} + 2 \beta_{3} + \beta_{6} + \beta_{8} ) q^{35} + ( -8 + \beta_{4} - 2 \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{37} + ( 6 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{39} + ( 16 + 2 \beta_{4} + 5 \beta_{7} - \beta_{10} - \beta_{11} - 2 \beta_{13} ) q^{41} + ( 6 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{43} + ( 11 + 3 \beta_{4} + 12 \beta_{7} + \beta_{10} - 2 \beta_{11} + \beta_{13} ) q^{45} + ( 4 \beta_{1} - 7 \beta_{3} - \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{47} + ( -15 - \beta_{4} - \beta_{7} - 3 \beta_{10} + 4 \beta_{11} + 3 \beta_{13} ) q^{49} + ( 10 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{51} + ( -1 + 5 \beta_{7} - 3 \beta_{11} + \beta_{12} ) q^{53} + ( 4 \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{8} - 3 \beta_{9} ) q^{55} + ( 3 - \beta_{4} - \beta_{7} - \beta_{13} ) q^{57} + ( \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} + \beta_{6} + \beta_{8} ) q^{59} + ( -3 + 3 \beta_{4} + 6 \beta_{7} + \beta_{10} - 2 \beta_{12} - 3 \beta_{13} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} - \beta_{9} ) q^{63} + ( -35 + \beta_{4} + 3 \beta_{7} + \beta_{10} - 4 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} ) q^{65} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + 2 \beta_{8} ) q^{67} + ( -7 - 3 \beta_{4} - 5 \beta_{7} + \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{69} + ( 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 4 \beta_{8} ) q^{71} + ( 5 - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} ) q^{73} + ( \beta_{1} - 4 \beta_{3} - 2 \beta_{5} - \beta_{8} ) q^{75} + ( -15 + \beta_{4} - 12 \beta_{7} - 5 \beta_{10} + 2 \beta_{12} + \beta_{13} ) q^{77} + ( 5 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{8} - 5 \beta_{9} ) q^{79} + ( 25 + 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{11} + 4 \beta_{12} + 2 \beta_{13} ) q^{81} + ( 8 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} - \beta_{6} - 4 \beta_{8} ) q^{83} + ( -4 - 4 \beta_{4} - 7 \beta_{7} - 4 \beta_{11} - 2 \beta_{12} - 2 \beta_{13} ) q^{85} + ( 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{8} + 4 \beta_{9} ) q^{87} + ( 2 - 2 \beta_{4} - \beta_{7} + \beta_{10} + \beta_{11} + 4 \beta_{12} ) q^{89} + ( \beta_{1} - 4 \beta_{2} + 6 \beta_{3} + \beta_{6} - 3 \beta_{8} - 10 \beta_{9} ) q^{91} + ( 14 - 2 \beta_{4} - 10 \beta_{7} - 2 \beta_{10} + 6 \beta_{11} + 2 \beta_{12} + 6 \beta_{13} ) q^{93} + ( \beta_{3} + \beta_{5} + \beta_{9} ) q^{95} + ( 19 + \beta_{4} + 5 \beta_{7} + 5 \beta_{10} - 2 \beta_{11} - 4 \beta_{12} + \beta_{13} ) q^{97} + ( 20 \beta_{1} - 4 \beta_{3} + 2 \beta_{6} - 3 \beta_{8} - 6 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 68q^{9} + O(q^{10}) \) \( 14q - 68q^{9} - 54q^{13} + 34q^{17} + 38q^{21} - 86q^{25} - 54q^{29} + 20q^{33} - 100q^{37} + 224q^{41} + 168q^{45} - 220q^{49} - 14q^{53} + 38q^{57} - 28q^{61} - 472q^{65} - 122q^{69} + 70q^{73} - 228q^{77} + 334q^{81} - 48q^{85} + 176q^{93} + 308q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 2 x^{13} + x^{12} + 14 x^{11} - 42 x^{10} + 28 x^{9} + 132 x^{8} - 440 x^{7} + 528 x^{6} + 448 x^{5} - 2688 x^{4} + 3584 x^{3} + 1024 x^{2} - 8192 x + 16384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{13} + 14 \nu^{12} - 9 \nu^{11} + 94 \nu^{10} - 286 \nu^{9} + 844 \nu^{8} - 980 \nu^{7} - 1880 \nu^{6} + 8592 \nu^{5} - 8832 \nu^{4} - 3968 \nu^{3} + 59392 \nu^{2} - 119808 \nu + 114688 \)\()/20480\)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{12} + 2 \nu^{11} + 27 \nu^{10} - 142 \nu^{9} + 242 \nu^{8} + 4 \nu^{7} - 1172 \nu^{6} + 2552 \nu^{5} - 1872 \nu^{4} - 6208 \nu^{3} + 15744 \nu^{2} - 16896 \nu - 1024 \)\()/1024\)
\(\beta_{3}\)\(=\)\((\)\( -21 \nu^{13} - 6 \nu^{12} + 91 \nu^{11} - 246 \nu^{10} + 354 \nu^{9} + 244 \nu^{8} - 2100 \nu^{7} + 1560 \nu^{6} + 112 \nu^{5} - 3392 \nu^{4} + 10112 \nu^{3} - 4608 \nu^{2} + 13312 \nu - 172032 \)\()/20480\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{13} + 10 \nu^{12} + 11 \nu^{11} - 102 \nu^{10} + 98 \nu^{9} + 84 \nu^{8} - 948 \nu^{7} + 1624 \nu^{6} + 1264 \nu^{5} - 5184 \nu^{4} + 10112 \nu^{3} + 5632 \nu^{2} - 44032 \nu + 32768 \)\()/4096\)
\(\beta_{5}\)\(=\)\((\)\( 31 \nu^{13} + 86 \nu^{12} - 281 \nu^{11} + 166 \nu^{10} + 626 \nu^{9} - 1924 \nu^{8} + 3660 \nu^{7} + 7880 \nu^{6} - 18352 \nu^{5} + 23552 \nu^{4} - 12672 \nu^{3} - 79872 \nu^{2} + 109568 \nu + 315392 \)\()/20480\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{13} + 3 \nu^{11} - 16 \nu^{10} + 14 \nu^{9} + 56 \nu^{8} - 188 \nu^{7} + 176 \nu^{6} + 352 \nu^{5} - 1504 \nu^{4} + 1792 \nu^{3} - 256 \nu^{2} - 8192 \nu + 6144 \)\()/512\)
\(\beta_{7}\)\(=\)\((\)\( -3 \nu^{13} + 9 \nu^{12} - \nu^{11} - 39 \nu^{10} + 112 \nu^{9} - 82 \nu^{8} - 328 \nu^{7} + 1012 \nu^{6} - 952 \nu^{5} - 1168 \nu^{4} + 4672 \nu^{3} - 4736 \nu^{2} - 7680 \nu + 12288 \)\()/1024\)
\(\beta_{8}\)\(=\)\((\)\( -39 \nu^{13} - 84 \nu^{12} + 269 \nu^{11} - 164 \nu^{10} - 454 \nu^{9} + 2416 \nu^{8} - 2420 \nu^{7} - 6720 \nu^{6} + 20288 \nu^{5} - 14048 \nu^{4} - 15872 \nu^{3} + 98048 \nu^{2} - 59392 \nu - 442368 \)\()/10240\)
\(\beta_{9}\)\(=\)\((\)\( 39 \nu^{13} + 94 \nu^{12} - 209 \nu^{11} - 146 \nu^{10} + 1314 \nu^{9} - 3156 \nu^{8} + 1260 \nu^{7} + 13160 \nu^{6} - 28208 \nu^{5} + 11648 \nu^{4} + 60032 \nu^{3} - 188928 \nu^{2} + 166912 \nu + 411648 \)\()/10240\)
\(\beta_{10}\)\(=\)\((\)\( 9 \nu^{13} - 8 \nu^{12} - 75 \nu^{11} + 232 \nu^{10} - 174 \nu^{9} - 840 \nu^{8} + 3004 \nu^{7} - 3344 \nu^{6} - 4256 \nu^{5} + 21472 \nu^{4} - 30464 \nu^{3} - 8448 \nu^{2} + 96256 \nu - 112640 \)\()/2048\)
\(\beta_{11}\)\(=\)\((\)\( -11 \nu^{13} + 30 \nu^{12} - 11 \nu^{11} - 114 \nu^{10} + 334 \nu^{9} - 356 \nu^{8} - 620 \nu^{7} + 2632 \nu^{6} - 3632 \nu^{5} + 704 \nu^{4} + 9856 \nu^{3} - 15872 \nu^{2} - 9216 \nu + 6144 \)\()/2048\)
\(\beta_{12}\)\(=\)\((\)\( 25 \nu^{13} - 62 \nu^{12} + \nu^{11} + 370 \nu^{10} - 882 \nu^{9} + 468 \nu^{8} + 3316 \nu^{7} - 9192 \nu^{6} + 6768 \nu^{5} + 13440 \nu^{4} - 50304 \nu^{3} + 41984 \nu^{2} + 50176 \nu - 139264 \)\()/4096\)
\(\beta_{13}\)\(=\)\((\)\( 27 \nu^{13} - 62 \nu^{12} - 133 \nu^{11} + 594 \nu^{10} - 782 \nu^{9} - 988 \nu^{8} + 6764 \nu^{7} - 10952 \nu^{6} - 3152 \nu^{5} + 40768 \nu^{4} - 75392 \nu^{3} + 27136 \nu^{2} + 201728 \nu - 262144 \)\()/4096\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{12} + \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} + 2\)\()/16\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{4} + 1\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{13} - \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} - 4 \beta_{7} - 2 \beta_{5} - 6 \beta_{3} - \beta_{2} + 4 \beta_{1} - 23\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-\beta_{13} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} - 9 \beta_{7} + \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 10 \beta_{3} - 4 \beta_{2} + 12 \beta_{1} + 33\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{13} - 4 \beta_{12} - 6 \beta_{11} + 9 \beta_{10} + 10 \beta_{9} + 14 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + 7 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 16 \beta_{1} + 37\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-9 \beta_{13} - 8 \beta_{12} - 18 \beta_{11} + 9 \beta_{10} - 14 \beta_{9} + 2 \beta_{8} + 7 \beta_{7} - 7 \beta_{6} + 22 \beta_{5} + 11 \beta_{4} - 18 \beta_{3} - 4 \beta_{2} - 28 \beta_{1} - 75\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(15 \beta_{13} + 12 \beta_{12} + 26 \beta_{11} - 19 \beta_{10} + 14 \beta_{9} + 42 \beta_{8} + 11 \beta_{7} + \beta_{6} + 10 \beta_{5} + 11 \beta_{4} - 182 \beta_{3} - 72 \beta_{1} - 15\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(19 \beta_{13} + 8 \beta_{12} + 86 \beta_{11} - 51 \beta_{10} - 22 \beta_{9} - 6 \beta_{8} - 141 \beta_{7} + 29 \beta_{6} + 78 \beta_{5} - 89 \beta_{4} + 6 \beta_{3} - 36 \beta_{2} + 84 \beta_{1} - 551\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(83 \beta_{13} + 88 \beta_{12} - 46 \beta_{11} - 39 \beta_{10} + 122 \beta_{9} + 134 \beta_{8} + 283 \beta_{7} + 89 \beta_{6} + 66 \beta_{5} - 5 \beta_{4} + 194 \beta_{3} - 28 \beta_{2} + 376 \beta_{1} + 1237\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(-129 \beta_{13} - 120 \beta_{12} - 386 \beta_{11} + 193 \beta_{10} - 126 \beta_{9} + 50 \beta_{8} + 447 \beta_{7} - 15 \beta_{6} + 22 \beta_{5} - 317 \beta_{4} - 210 \beta_{3} - 116 \beta_{2} + 484 \beta_{1} - 915\)\()/8\)
\(\nu^{11}\)\(=\)\((\)\(-225 \beta_{13} + 384 \beta_{12} - 230 \beta_{11} + 61 \beta_{10} - 86 \beta_{9} + 230 \beta_{8} + 1007 \beta_{7} - 491 \beta_{6} - 150 \beta_{5} + 15 \beta_{4} - 1014 \beta_{3} - 132 \beta_{2} - 744 \beta_{1} + 7609\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-373 \beta_{13} + 168 \beta_{12} + 1238 \beta_{11} - 107 \beta_{10} + 970 \beta_{9} + 794 \beta_{8} - 1013 \beta_{7} - 1083 \beta_{6} - 434 \beta_{5} - 1153 \beta_{4} - 2042 \beta_{3} + 764 \beta_{2} + 916 \beta_{1} - 12911\)\()/8\)
\(\nu^{13}\)\(=\)\((\)\(491 \beta_{13} + 1984 \beta_{12} + 66 \beta_{11} - 1023 \beta_{10} + 1362 \beta_{9} - 482 \beta_{8} + 955 \beta_{7} - 679 \beta_{6} + 594 \beta_{5} + 1723 \beta_{4} + 6514 \beta_{3} - 116 \beta_{2} + 3352 \beta_{1} - 23571\)\()/8\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.0607713 + 1.99908i
−1.89728 0.632718i
0.711746 1.86907i
1.94929 0.447510i
1.57398 + 1.23393i
0.645572 1.89294i
−1.92254 + 0.551226i
−1.92254 0.551226i
0.645572 + 1.89294i
1.57398 1.23393i
1.94929 + 0.447510i
0.711746 + 1.86907i
−1.89728 + 0.632718i
−0.0607713 1.99908i
0 5.37609i 0 5.82257 0 5.45132i 0 −19.9023 0
191.2 0 5.34370i 0 −5.79268 0 5.87536i 0 −19.5551 0
191.3 0 4.44946i 0 −4.97973 0 12.2628i 0 −10.7977 0
191.4 0 3.19988i 0 3.90374 0 2.64664i 0 −1.23921 0
191.5 0 2.90118i 0 −3.66290 0 1.93414i 0 0.583162 0
191.6 0 0.820457i 0 2.38184 0 12.3764i 0 8.32685 0
191.7 0 0.644704i 0 2.32715 0 8.62924i 0 8.58436 0
191.8 0 0.644704i 0 2.32715 0 8.62924i 0 8.58436 0
191.9 0 0.820457i 0 2.38184 0 12.3764i 0 8.32685 0
191.10 0 2.90118i 0 −3.66290 0 1.93414i 0 0.583162 0
191.11 0 3.19988i 0 3.90374 0 2.64664i 0 −1.23921 0
191.12 0 4.44946i 0 −4.97973 0 12.2628i 0 −10.7977 0
191.13 0 5.34370i 0 −5.79268 0 5.87536i 0 −19.5551 0
191.14 0 5.37609i 0 5.82257 0 5.45132i 0 −19.9023 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.3.d.d 14
4.b odd 2 1 inner 1216.3.d.d 14
8.b even 2 1 76.3.b.b 14
8.d odd 2 1 76.3.b.b 14
24.f even 2 1 684.3.g.b 14
24.h odd 2 1 684.3.g.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.b 14 8.b even 2 1
76.3.b.b 14 8.d odd 2 1
684.3.g.b 14 24.f even 2 1
684.3.g.b 14 24.h odd 2 1
1216.3.d.d 14 1.a even 1 1 trivial
1216.3.d.d 14 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 97 T_{3}^{12} + 3595 T_{3}^{10} + 63443 T_{3}^{8} + 539872 T_{3}^{6} + 1940896 T_{3}^{4} + 1665792 T_{3}^{2} + 393984 \) acting on \(S_{3}^{\mathrm{new}}(1216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \)
$3$ \( 393984 + 1665792 T^{2} + 1940896 T^{4} + 539872 T^{6} + 63443 T^{8} + 3595 T^{10} + 97 T^{12} + T^{14} \)
$5$ \( ( 13312 - 8400 T - 1348 T^{2} + 1337 T^{3} + 28 T^{4} - 66 T^{5} + T^{7} )^{2} \)
$7$ \( 46106276299 + 23021007551 T^{2} + 3578469343 T^{4} + 212956211 T^{6} + 5808593 T^{8} + 75725 T^{10} + 453 T^{12} + T^{14} \)
$11$ \( 94488337600 + 207279189936 T^{2} + 72889781124 T^{4} + 2781706433 T^{6} + 41107428 T^{8} + 280822 T^{10} + 880 T^{12} + T^{14} \)
$13$ \( ( 1265920 + 2184704 T + 907488 T^{2} + 7824 T^{3} - 9959 T^{4} - 293 T^{5} + 27 T^{6} + T^{7} )^{2} \)
$17$ \( ( -69101055 - 18411921 T + 417581 T^{2} + 355811 T^{3} + 9955 T^{4} - 1107 T^{5} - 17 T^{6} + T^{7} )^{2} \)
$19$ \( ( 19 + T^{2} )^{7} \)
$23$ \( 683970816311296 + 864035000025088 T^{2} + 39218066795008 T^{4} + 551803020800 T^{6} + 2798046987 T^{8} + 5204683 T^{10} + 3881 T^{12} + T^{14} \)
$29$ \( ( 112127472 - 91750704 T + 10060400 T^{2} + 839992 T^{3} - 75791 T^{4} - 2901 T^{5} + 27 T^{6} + T^{7} )^{2} \)
$31$ \( 112006799137177600 + 7005784227446784 T^{2} + 136835964338176 T^{4} + 1042205249536 T^{6} + 3570928640 T^{8} + 5704128 T^{10} + 4048 T^{12} + T^{14} \)
$37$ \( ( -914894720 + 490075456 T + 48142944 T^{2} - 85712 T^{3} - 100072 T^{4} - 1508 T^{5} + 50 T^{6} + T^{7} )^{2} \)
$41$ \( ( 18882293760 + 1089553152 T - 442902400 T^{2} - 1190928 T^{3} + 433472 T^{4} - 1776 T^{5} - 112 T^{6} + T^{7} )^{2} \)
$43$ \( \)\(58\!\cdots\!24\)\( + 9894601275600773376 T^{2} + 51208545244013840 T^{4} + 109716719259377 T^{6} + 112918336940 T^{8} + 57363094 T^{10} + 13244 T^{12} + T^{14} \)
$47$ \( 6486650127800008704 + 2593987636485123072 T^{2} + 16602143603532784 T^{4} + 38825334689281 T^{6} + 44446706748 T^{8} + 26838726 T^{10} + 8204 T^{12} + T^{14} \)
$53$ \( ( -18228603200 + 4696986240 T - 390622104 T^{2} + 10269704 T^{3} + 111981 T^{4} - 6645 T^{5} + 7 T^{6} + T^{7} )^{2} \)
$59$ \( \)\(50\!\cdots\!00\)\( + \)\(25\!\cdots\!04\)\( T^{2} + 4608193003005303072 T^{4} + 3925759870481904 T^{6} + 1663620366787 T^{8} + 336602299 T^{10} + 30513 T^{12} + T^{14} \)
$61$ \( ( -522558358600 - 16277769076 T + 1000269562 T^{2} + 29414873 T^{3} - 485080 T^{4} - 14158 T^{5} + 14 T^{6} + T^{7} )^{2} \)
$67$ \( \)\(29\!\cdots\!76\)\( + \)\(25\!\cdots\!00\)\( T^{2} + 5120555266235997504 T^{4} + 3973974955017136 T^{6} + 1503100062371 T^{8} + 293178683 T^{10} + 27889 T^{12} + T^{14} \)
$71$ \( \)\(49\!\cdots\!00\)\( + \)\(57\!\cdots\!76\)\( T^{2} + 15805923995875204096 T^{4} + 11221518796559104 T^{6} + 3467793570368 T^{8} + 526066128 T^{10} + 37812 T^{12} + T^{14} \)
$73$ \( ( -77971926925 - 6681670753 T + 6909415 T^{2} + 11130931 T^{3} + 95497 T^{4} - 5907 T^{5} - 35 T^{6} + T^{7} )^{2} \)
$79$ \( \)\(18\!\cdots\!00\)\( + \)\(22\!\cdots\!64\)\( T^{2} + 7079503618001244160 T^{4} + 6799300948930560 T^{6} + 2745532366528 T^{8} + 498656432 T^{10} + 38740 T^{12} + T^{14} \)
$83$ \( \)\(34\!\cdots\!76\)\( + \)\(85\!\cdots\!08\)\( T^{2} + \)\(17\!\cdots\!88\)\( T^{4} + 81697848347602944 T^{6} + 15885346272448 T^{8} + 1466300592 T^{10} + 62788 T^{12} + T^{14} \)
$89$ \( ( -88310345728 + 32804608000 T - 2083382784 T^{2} + 23700672 T^{3} + 1100800 T^{4} - 23640 T^{5} + T^{7} )^{2} \)
$97$ \( ( 4410892984320 - 293552295936 T - 20455196416 T^{2} + 115812736 T^{3} + 4374632 T^{4} - 27396 T^{5} - 154 T^{6} + T^{7} )^{2} \)
show more
show less