Properties

Label 224.3.o.d
Level 224
Weight 3
Character orbit 224.o
Analytic conductor 6.104
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 224.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{8} ) q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{7} + ( \beta_{3} + 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{10} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{3} + \beta_{6} + \beta_{7} - \beta_{10} ) q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{8} ) q^{5} + ( 2 \beta_{2} - \beta_{4} - \beta_{8} + \beta_{9} - 2 \beta_{11} ) q^{7} + ( \beta_{3} + 4 \beta_{5} + 5 \beta_{6} - 4 \beta_{10} ) q^{9} + ( -6 - 6 \beta_{6} - 3 \beta_{10} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 5 \beta_{8} - 2 \beta_{9} - 5 \beta_{11} ) q^{13} + ( -3 \beta_{4} - 2 \beta_{8} - 3 \beta_{9} + 2 \beta_{11} ) q^{15} + ( 6 - \beta_{3} + 6 \beta_{6} + \beta_{7} + 2 \beta_{10} ) q^{17} + ( 4 \beta_{3} - \beta_{5} + 12 \beta_{6} + \beta_{10} ) q^{19} + ( 2 \beta_{1} - \beta_{2} - 8 \beta_{4} - 7 \beta_{8} - 3 \beta_{9} + 7 \beta_{11} ) q^{21} + ( -7 \beta_{4} + 5 \beta_{8} ) q^{23} + ( -14 - 2 \beta_{3} - 14 \beta_{6} + 2 \beta_{7} + 2 \beta_{10} ) q^{25} + ( -8 + 13 \beta_{5} + 2 \beta_{7} ) q^{27} + ( 8 \beta_{1} + 8 \beta_{2} + 2 \beta_{4} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{29} + ( -9 \beta_{9} + 6 \beta_{11} ) q^{31} + ( 6 \beta_{3} + 6 \beta_{5} + 9 \beta_{6} - 6 \beta_{10} ) q^{33} + ( 23 - 2 \beta_{3} + 3 \beta_{5} + 6 \beta_{6} + 3 \beta_{7} - \beta_{10} ) q^{35} + ( -9 \beta_{1} + \beta_{4} - 4 \beta_{8} ) q^{37} + ( -14 \beta_{2} + 8 \beta_{9} - 17 \beta_{11} ) q^{39} + ( -25 - 18 \beta_{5} + \beta_{7} ) q^{41} + ( 10 - 10 \beta_{5} - 10 \beta_{7} ) q^{43} + ( 4 \beta_{2} + 2 \beta_{9} + 9 \beta_{11} ) q^{45} + ( -16 \beta_{1} + 5 \beta_{4} - 2 \beta_{8} ) q^{47} + ( 20 - 6 \beta_{3} + 2 \beta_{5} - 24 \beta_{6} + 9 \beta_{7} - 10 \beta_{10} ) q^{49} + ( -4 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} + 3 \beta_{10} ) q^{51} + ( 3 \beta_{2} + 7 \beta_{9} - 20 \beta_{11} ) q^{53} + ( 12 \beta_{1} + 12 \beta_{2} + 9 \beta_{4} + 3 \beta_{8} + 9 \beta_{9} - 3 \beta_{11} ) q^{55} + ( 25 + 12 \beta_{5} - 20 \beta_{7} ) q^{57} + ( 17 - 5 \beta_{3} + 17 \beta_{6} + 5 \beta_{7} - 9 \beta_{10} ) q^{59} + ( 15 \beta_{1} + 25 \beta_{4} + 10 \beta_{8} ) q^{61} + ( 6 \beta_{2} + 4 \beta_{4} - 17 \beta_{8} - 4 \beta_{9} + 36 \beta_{11} ) q^{63} + ( -11 \beta_{3} - 18 \beta_{5} + 15 \beta_{6} + 18 \beta_{10} ) q^{65} + ( -73 + 3 \beta_{3} - 73 \beta_{6} - 3 \beta_{7} + \beta_{10} ) q^{67} + ( -31 \beta_{1} - 31 \beta_{2} + 3 \beta_{4} - 13 \beta_{8} + 3 \beta_{9} + 13 \beta_{11} ) q^{69} + ( 6 \beta_{1} + 6 \beta_{2} + 8 \beta_{4} + 22 \beta_{8} + 8 \beta_{9} - 22 \beta_{11} ) q^{71} + ( 27 - 12 \beta_{3} + 27 \beta_{6} + 12 \beta_{7} + 18 \beta_{10} ) q^{73} + ( 18 \beta_{3} - 24 \beta_{5} - 8 \beta_{6} + 24 \beta_{10} ) q^{75} + ( 24 \beta_{1} + 3 \beta_{2} + 12 \beta_{4} + 3 \beta_{8} + 3 \beta_{9} + 6 \beta_{11} ) q^{77} + ( 16 \beta_{1} + 25 \beta_{4} - 5 \beta_{8} ) q^{79} + ( -12 + 21 \beta_{3} - 12 \beta_{6} - 21 \beta_{7} + 26 \beta_{10} ) q^{81} + ( 36 - 12 \beta_{5} - 14 \beta_{7} ) q^{83} + ( -11 \beta_{1} - 11 \beta_{2} - 11 \beta_{4} - 4 \beta_{8} - 11 \beta_{9} + 4 \beta_{11} ) q^{85} + ( -8 \beta_{2} + 8 \beta_{9} + 23 \beta_{11} ) q^{87} + ( 36 \beta_{3} + 4 \beta_{5} - 49 \beta_{6} - 4 \beta_{10} ) q^{89} + ( -1 - 20 \beta_{3} - 26 \beta_{5} - 10 \beta_{6} + 23 \beta_{7} + 46 \beta_{10} ) q^{91} + ( 15 \beta_{1} + 21 \beta_{4} + 54 \beta_{8} ) q^{93} + ( -10 \beta_{2} - \beta_{9} + 11 \beta_{11} ) q^{95} + ( -15 + 8 \beta_{5} - 15 \beta_{7} ) q^{97} + ( -45 + 12 \beta_{5} - 21 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} - 40q^{9} + O(q^{10}) \) \( 12q + 6q^{3} - 40q^{9} - 30q^{11} + 30q^{17} - 78q^{19} - 92q^{25} - 156q^{27} - 78q^{33} + 222q^{35} - 232q^{41} + 200q^{43} + 372q^{49} - 10q^{51} + 332q^{57} + 110q^{59} - 32q^{65} - 434q^{67} + 102q^{73} + 60q^{75} - 82q^{81} + 536q^{83} + 214q^{89} + 8q^{91} - 152q^{97} - 504q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 3 x^{11} - 4 x^{10} + 3 x^{9} + 86 x^{8} - 163 x^{7} + 155 x^{6} - 166 x^{5} + 164 x^{4} - 116 x^{3} + 60 x^{2} - 20 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(2848753 \nu^{11} + 128409 \nu^{10} - 36949178 \nu^{9} - 33927641 \nu^{8} + 280693408 \nu^{7} + 324275527 \nu^{6} - 906643427 \nu^{5} + 266302456 \nu^{4} - 492489836 \nu^{3} + 675410750 \nu^{2} - 298885728 \nu + 114476732\)\()/20513668\)
\(\beta_{2}\)\(=\)\((\)\(908144 \nu^{11} - 6494043 \nu^{10} + 3252536 \nu^{9} + 28087343 \nu^{8} + 91567337 \nu^{7} - 469374709 \nu^{6} + 376525973 \nu^{5} - 267919561 \nu^{4} + 416913921 \nu^{3} - 270088448 \nu^{2} + 111915639 \nu - 26684466\)\()/5128417\)
\(\beta_{3}\)\(=\)\((\)\(775023 \nu^{11} - 2790290 \nu^{10} - 2275565 \nu^{9} + 5234055 \nu^{8} + 68919723 \nu^{7} - 163955841 \nu^{6} + 148312850 \nu^{5} - 164484671 \nu^{4} + 171431608 \nu^{3} - 108634476 \nu^{2} + 67339246 \nu - 23024344\)\()/2930524\)
\(\beta_{4}\)\(=\)\((\)\(-894552 \nu^{11} + 880921 \nu^{10} + 8410783 \nu^{9} + 6680758 \nu^{8} - 80688819 \nu^{7} - 13990590 \nu^{6} + 103354841 \nu^{5} - 1783409 \nu^{4} + 46480180 \nu^{3} - 87465566 \nu^{2} + 40145506 \nu - 16276324\)\()/2930524\)
\(\beta_{5}\)\(=\)\((\)\(8673537 \nu^{11} - 20159299 \nu^{10} - 48705482 \nu^{9} - 5607853 \nu^{8} + 742433540 \nu^{7} - 912321093 \nu^{6} + 701452097 \nu^{5} - 889048300 \nu^{4} + 667643040 \nu^{3} - 426711138 \nu^{2} + 154926652 \nu - 29235872\)\()/20513668\)
\(\beta_{6}\)\(=\)\((\)\(8864783 \nu^{11} - 20455402 \nu^{10} - 48834261 \nu^{9} - 8785933 \nu^{8} + 751559539 \nu^{7} - 927115697 \nu^{6} + 797312818 \nu^{5} - 981317799 \nu^{4} + 836256216 \nu^{3} - 526622432 \nu^{2} + 236999870 \nu - 70375800\)\()/20513668\)
\(\beta_{7}\)\(=\)\((\)\(-4885386 \nu^{11} + 11354967 \nu^{10} + 27428562 \nu^{9} + 3157071 \nu^{8} - 418178829 \nu^{7} + 513874517 \nu^{6} - 395095635 \nu^{5} + 500760792 \nu^{4} - 376052347 \nu^{3} + 240347478 \nu^{2} - 87263352 \nu - 20957728\)\()/10256834\)
\(\beta_{8}\)\(=\)\((\)\(5752921 \nu^{11} - 9678616 \nu^{10} - 40188533 \nu^{9} - 25630209 \nu^{8} + 486470615 \nu^{7} - 292655047 \nu^{6} + 125254932 \nu^{5} - 345038603 \nu^{4} + 172274176 \nu^{3} + 19638430 \nu^{2} - 24222874 \nu + 18987776\)\()/10256834\)
\(\beta_{9}\)\(=\)\((\)\(-1785337 \nu^{11} + 6633082 \nu^{10} + 5117731 \nu^{9} - 14794777 \nu^{8} - 159689841 \nu^{7} + 400659219 \nu^{6} - 329059530 \nu^{5} + 293285149 \nu^{4} - 357701700 \nu^{3} + 229568032 \nu^{2} - 91947870 \nu + 22371428\)\()/2930524\)
\(\beta_{10}\)\(=\)\((\)\(-15221487 \nu^{11} + 28755150 \nu^{10} + 95322325 \nu^{9} + 55362465 \nu^{8} - 1263522623 \nu^{7} + 1071722261 \nu^{6} - 959373046 \nu^{5} + 1267262003 \nu^{4} - 975668172 \nu^{3} + 603936480 \nu^{2} - 194264578 \nu + 55637788\)\()/20513668\)
\(\beta_{11}\)\(=\)\((\)\(-14292759 \nu^{11} + 36030049 \nu^{10} + 73017973 \nu^{9} - 5911956 \nu^{8} - 1221850366 \nu^{7} + 1754115116 \nu^{6} - 1489733269 \nu^{5} + 1698131925 \nu^{4} - 1577732703 \nu^{3} + 998782176 \nu^{2} - 379578618 \nu + 95337586\)\()/10256834\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} + \beta_{6} + \beta_{2} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{10} - 7 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{1}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{11} + 9 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 3 \beta_{5} + 9 \beta_{4} + 8 \beta_{2} + 8 \beta_{1} + 11\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{11} - 16 \beta_{10} + 19 \beta_{9} - 14 \beta_{7} - 51 \beta_{6} + 14 \beta_{3} + 17 \beta_{2} - 51\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-45 \beta_{10} + 20 \beta_{8} - 146 \beta_{6} + 45 \beta_{5} + 61 \beta_{4} + 40 \beta_{3} + 54 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-75 \beta_{11} + 231 \beta_{9} + 75 \beta_{8} - 85 \beta_{7} - 96 \beta_{5} + 231 \beta_{4} + 205 \beta_{2} + 205 \beta_{1} - 310\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(101 \beta_{11} - 497 \beta_{10} - 311 \beta_{9} - 441 \beta_{7} - 1609 \beta_{6} + 441 \beta_{3} - 276 \beta_{2} - 1609\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(356 \beta_{10} + 768 \beta_{8} + 1153 \beta_{6} - 356 \beta_{5} + 2367 \beta_{4} - 316 \beta_{3} + 2101 \beta_{1}\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(130 \beta_{11} - 401 \beta_{9} - 130 \beta_{8} - 4242 \beta_{7} - 4779 \beta_{5} - 401 \beta_{4} - 356 \beta_{2} - 356 \beta_{1} - 15478\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(6965 \beta_{11} - 1610 \beta_{10} - 21471 \beta_{9} - 1429 \beta_{7} - 5214 \beta_{6} + 1429 \beta_{3} - 19059 \beta_{2} - 5214\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(40931 \beta_{10} + 5807 \beta_{8} + 132571 \beta_{6} - 40931 \beta_{5} + 17901 \beta_{4} - 36333 \beta_{3} + 15890 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\beta_{6}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.378279 + 0.358951i
0.121721 + 0.507075i
−2.29733 + 1.90372i
2.79733 1.03769i
0.907369 + 0.0534805i
−0.407369 + 0.812545i
0.378279 0.358951i
0.121721 0.507075i
−2.29733 1.90372i
2.79733 + 1.03769i
0.907369 0.0534805i
−0.407369 0.812545i
0 −1.99052 3.44767i 0 −1.63031 0.941260i 0 −5.14749 + 4.74377i 0 −3.42430 + 5.93106i 0
79.2 0 −1.99052 3.44767i 0 1.63031 + 0.941260i 0 5.14749 4.74377i 0 −3.42430 + 5.93106i 0
79.3 0 0.824388 + 1.42788i 0 −3.95004 2.28056i 0 −6.75545 + 1.83408i 0 3.14077 5.43997i 0
79.4 0 0.824388 + 1.42788i 0 3.95004 + 2.28056i 0 6.75545 1.83408i 0 3.14077 5.43997i 0
79.5 0 2.66613 + 4.61787i 0 −1.86796 1.07847i 0 −6.91861 1.06433i 0 −9.71647 + 16.8294i 0
79.6 0 2.66613 + 4.61787i 0 1.86796 + 1.07847i 0 6.91861 + 1.06433i 0 −9.71647 + 16.8294i 0
207.1 0 −1.99052 + 3.44767i 0 −1.63031 + 0.941260i 0 −5.14749 4.74377i 0 −3.42430 5.93106i 0
207.2 0 −1.99052 + 3.44767i 0 1.63031 0.941260i 0 5.14749 + 4.74377i 0 −3.42430 5.93106i 0
207.3 0 0.824388 1.42788i 0 −3.95004 + 2.28056i 0 −6.75545 1.83408i 0 3.14077 + 5.43997i 0
207.4 0 0.824388 1.42788i 0 3.95004 2.28056i 0 6.75545 + 1.83408i 0 3.14077 + 5.43997i 0
207.5 0 2.66613 4.61787i 0 −1.86796 + 1.07847i 0 −6.91861 + 1.06433i 0 −9.71647 16.8294i 0
207.6 0 2.66613 4.61787i 0 1.86796 1.07847i 0 6.91861 1.06433i 0 −9.71647 16.8294i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.3.o.d 12
4.b odd 2 1 56.3.k.d 12
7.c even 3 1 inner 224.3.o.d 12
7.c even 3 1 1568.3.g.j 6
7.d odd 6 1 1568.3.g.l 6
8.b even 2 1 56.3.k.d 12
8.d odd 2 1 inner 224.3.o.d 12
28.d even 2 1 392.3.k.l 12
28.f even 6 1 392.3.g.i 6
28.f even 6 1 392.3.k.l 12
28.g odd 6 1 56.3.k.d 12
28.g odd 6 1 392.3.g.j 6
56.h odd 2 1 392.3.k.l 12
56.j odd 6 1 392.3.g.i 6
56.j odd 6 1 392.3.k.l 12
56.k odd 6 1 inner 224.3.o.d 12
56.k odd 6 1 1568.3.g.j 6
56.m even 6 1 1568.3.g.l 6
56.p even 6 1 56.3.k.d 12
56.p even 6 1 392.3.g.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d 12 4.b odd 2 1
56.3.k.d 12 8.b even 2 1
56.3.k.d 12 28.g odd 6 1
56.3.k.d 12 56.p even 6 1
224.3.o.d 12 1.a even 1 1 trivial
224.3.o.d 12 7.c even 3 1 inner
224.3.o.d 12 8.d odd 2 1 inner
224.3.o.d 12 56.k odd 6 1 inner
392.3.g.i 6 28.f even 6 1
392.3.g.i 6 56.j odd 6 1
392.3.g.j 6 28.g odd 6 1
392.3.g.j 6 56.p even 6 1
392.3.k.l 12 28.d even 2 1
392.3.k.l 12 28.f even 6 1
392.3.k.l 12 56.h odd 2 1
392.3.k.l 12 56.j odd 6 1
1568.3.g.j 6 7.c even 3 1
1568.3.g.j 6 56.k odd 6 1
1568.3.g.l 6 7.d odd 6 1
1568.3.g.l 6 56.m even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(224, [\chi])\):

\( T_{3}^{6} - 3 T_{3}^{5} + 28 T_{3}^{4} - 13 T_{3}^{3} + 466 T_{3}^{2} - 665 T_{3} + 1225 \)
\( T_{5}^{12} - 29 T_{5}^{10} + 654 T_{5}^{8} - 4737 T_{5}^{6} + 25022 T_{5}^{4} - 64141 T_{5}^{2} + 117649 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - 3 T + T^{2} + 14 T^{3} - 65 T^{4} + 37 T^{5} + 514 T^{6} + 333 T^{7} - 5265 T^{8} + 10206 T^{9} + 6561 T^{10} - 177147 T^{11} + 531441 T^{12} )^{2} \)
$5$ \( 1 + 121 T^{2} + 7979 T^{4} + 380588 T^{6} + 14474897 T^{8} + 457025259 T^{10} + 12295779174 T^{12} + 285640786875 T^{14} + 5654256640625 T^{16} + 92916992187500 T^{18} + 1217498779296875 T^{20} + 11539459228515625 T^{22} + 59604644775390625 T^{24} \)
$7$ \( 1 - 186 T^{2} + 16527 T^{4} - 956284 T^{6} + 39681327 T^{8} - 1072252986 T^{10} + 13841287201 T^{12} \)
$11$ \( ( 1 + 15 T - 129 T^{2} - 924 T^{3} + 35727 T^{4} + 38013 T^{5} - 5198650 T^{6} + 4599573 T^{7} + 523079007 T^{8} - 1636922364 T^{9} - 27652295649 T^{10} + 389061369015 T^{11} + 3138428376721 T^{12} )^{2} \)
$13$ \( ( 1 - 86 T^{2} + 60895 T^{4} - 4902788 T^{6} + 1739222095 T^{8} - 70152842006 T^{10} + 23298085122481 T^{12} )^{2} \)
$17$ \( ( 1 - 15 T - 665 T^{2} + 3920 T^{3} + 405365 T^{4} - 1221665 T^{5} - 124870762 T^{6} - 353061185 T^{7} + 33856490165 T^{8} + 94619270480 T^{9} - 4638878698265 T^{10} - 30239908506735 T^{11} + 582622237229761 T^{12} )^{2} \)
$19$ \( ( 1 + 39 T + 151 T^{2} - 3964 T^{3} + 263147 T^{4} + 5820949 T^{5} + 41830430 T^{6} + 2101362589 T^{7} + 34293580187 T^{8} - 186489872284 T^{9} + 2564518019191 T^{10} + 239111584054239 T^{11} + 2213314919066161 T^{12} )^{2} \)
$23$ \( 1 + 693 T^{2} - 15581 T^{4} - 315659500 T^{6} - 121343121715 T^{8} + 25399919636503 T^{10} + 46081428374834798 T^{12} + 7107938910998636023 T^{14} - \)\(95\!\cdots\!15\)\( T^{16} - \)\(69\!\cdots\!00\)\( T^{18} - \)\(95\!\cdots\!41\)\( T^{20} + \)\(11\!\cdots\!93\)\( T^{22} + \)\(48\!\cdots\!41\)\( T^{24} \)
$29$ \( ( 1 - 3662 T^{2} + 6471151 T^{4} - 6858243380 T^{6} + 4576922150431 T^{8} - 1831902364263182 T^{10} + 353814783205469041 T^{12} )^{2} \)
$31$ \( 1 + 3561 T^{2} + 5838879 T^{4} + 8280992264 T^{6} + 11856779149293 T^{8} + 13054196823637455 T^{10} + 12193464685853753718 T^{12} + \)\(12\!\cdots\!55\)\( T^{14} + \)\(10\!\cdots\!13\)\( T^{16} + \)\(65\!\cdots\!04\)\( T^{18} + \)\(42\!\cdots\!99\)\( T^{20} + \)\(23\!\cdots\!61\)\( T^{22} + \)\(62\!\cdots\!21\)\( T^{24} \)
$37$ \( 1 + 5785 T^{2} + 17889827 T^{4} + 38047414052 T^{6} + 62842022068961 T^{8} + 88030006321881747 T^{10} + \)\(11\!\cdots\!14\)\( T^{12} + \)\(16\!\cdots\!67\)\( T^{14} + \)\(22\!\cdots\!81\)\( T^{16} + \)\(25\!\cdots\!12\)\( T^{18} + \)\(22\!\cdots\!07\)\( T^{20} + \)\(13\!\cdots\!85\)\( T^{22} + \)\(43\!\cdots\!61\)\( T^{24} \)
$41$ \( ( 1 + 58 T + 3139 T^{2} + 88960 T^{3} + 5276659 T^{4} + 163894138 T^{5} + 4750104241 T^{6} )^{4} \)
$43$ \( ( 1 - 50 T + 4047 T^{2} - 107900 T^{3} + 7482903 T^{4} - 170940050 T^{5} + 6321363049 T^{6} )^{4} \)
$47$ \( 1 + 3905 T^{2} + 6537087 T^{4} + 2079696952 T^{6} - 28977286497379 T^{8} - 99900087904130553 T^{10} - \)\(24\!\cdots\!46\)\( T^{12} - \)\(48\!\cdots\!93\)\( T^{14} - \)\(68\!\cdots\!19\)\( T^{16} + \)\(24\!\cdots\!32\)\( T^{18} + \)\(37\!\cdots\!27\)\( T^{20} + \)\(10\!\cdots\!05\)\( T^{22} + \)\(13\!\cdots\!81\)\( T^{24} \)
$53$ \( 1 + 10561 T^{2} + 59733731 T^{4} + 209884512884 T^{6} + 494098524300977 T^{8} + 757507316642438811 T^{10} + \)\(12\!\cdots\!10\)\( T^{12} + \)\(59\!\cdots\!91\)\( T^{14} + \)\(30\!\cdots\!97\)\( T^{16} + \)\(10\!\cdots\!44\)\( T^{18} + \)\(23\!\cdots\!51\)\( T^{20} + \)\(32\!\cdots\!61\)\( T^{22} + \)\(24\!\cdots\!81\)\( T^{24} \)
$59$ \( ( 1 - 55 T - 7367 T^{2} + 181142 T^{3} + 51119807 T^{4} - 598293727 T^{5} - 186579818926 T^{6} - 2082660463687 T^{7} + 619437155669327 T^{8} + 7640666224798022 T^{9} - 1081699833831032807 T^{10} - 28111421431535277055 T^{11} + \)\(17\!\cdots\!81\)\( T^{12} )^{2} \)
$61$ \( 1 + 9201 T^{2} + 56941411 T^{4} + 154999649300 T^{6} - 75695642240335 T^{8} - 3911957905117164149 T^{10} - \)\(19\!\cdots\!14\)\( T^{12} - \)\(54\!\cdots\!09\)\( T^{14} - \)\(14\!\cdots\!35\)\( T^{16} + \)\(41\!\cdots\!00\)\( T^{18} + \)\(20\!\cdots\!71\)\( T^{20} + \)\(46\!\cdots\!01\)\( T^{22} + \)\(70\!\cdots\!41\)\( T^{24} \)
$67$ \( ( 1 + 217 T + 18053 T^{2} + 1666970 T^{3} + 209974835 T^{4} + 15078221277 T^{5} + 815515698066 T^{6} + 67686135312453 T^{7} + 4231228307040035 T^{8} + 150791409324257930 T^{9} + 7330739782930039973 T^{10} + \)\(39\!\cdots\!33\)\( T^{11} + \)\(81\!\cdots\!61\)\( T^{12} )^{2} \)
$71$ \( ( 1 - 23062 T^{2} + 245626031 T^{4} - 1559141837940 T^{6} + 6241770345068111 T^{8} - 14892367937589740182 T^{10} + \)\(16\!\cdots\!41\)\( T^{12} )^{2} \)
$73$ \( ( 1 - 51 T - 9165 T^{2} + 579660 T^{3} + 45654729 T^{4} - 1993134921 T^{5} - 160997144218 T^{6} - 10621415994009 T^{7} + 1296513996931689 T^{8} + 87722397610681740 T^{9} - 7391206742209252365 T^{10} - \)\(21\!\cdots\!99\)\( T^{11} + \)\(22\!\cdots\!21\)\( T^{12} )^{2} \)
$79$ \( 1 + 16693 T^{2} + 149251283 T^{4} + 640487711012 T^{6} - 978609215845699 T^{8} - 44625107831251066425 T^{10} - \)\(37\!\cdots\!74\)\( T^{12} - \)\(17\!\cdots\!25\)\( T^{14} - \)\(14\!\cdots\!39\)\( T^{16} + \)\(37\!\cdots\!92\)\( T^{18} + \)\(34\!\cdots\!43\)\( T^{20} + \)\(14\!\cdots\!93\)\( T^{22} + \)\(34\!\cdots\!81\)\( T^{24} \)
$83$ \( ( 1 - 134 T + 22583 T^{2} - 1661172 T^{3} + 155574287 T^{4} - 6359415014 T^{5} + 326940373369 T^{6} )^{4} \)
$89$ \( ( 1 - 107 T + 1395 T^{2} + 497084 T^{3} - 62702935 T^{4} + 2422278015 T^{5} + 93422604838 T^{6} + 19186864156815 T^{7} - 3934122659177335 T^{8} + 247041448036057724 T^{9} + 5491541383954402995 T^{10} - \)\(33\!\cdots\!07\)\( T^{11} + \)\(24\!\cdots\!21\)\( T^{12} )^{2} \)
$97$ \( ( 1 + 38 T + 25191 T^{2} + 597344 T^{3} + 237022119 T^{4} + 3364112678 T^{5} + 832972004929 T^{6} )^{4} \)
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