Properties

Label 441.4.e.z
Level $441$
Weight $4$
Character orbit 441.e
Analytic conductor $26.020$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.0198423125\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{4} q^{2} + ( -9 + 9 \beta_{2} + \beta_{7} ) q^{4} -\beta_{6} q^{5} + ( 11 \beta_{4} - 11 \beta_{12} + 2 \beta_{15} ) q^{8} +O(q^{10})\) \( q -\beta_{4} q^{2} + ( -9 + 9 \beta_{2} + \beta_{7} ) q^{4} -\beta_{6} q^{5} + ( 11 \beta_{4} - 11 \beta_{12} + 2 \beta_{15} ) q^{8} + ( -2 \beta_{13} - 3 \beta_{14} ) q^{10} + ( \beta_{9} + 7 \beta_{12} + \beta_{15} ) q^{11} + ( 3 \beta_{8} - 2 \beta_{10} - 3 \beta_{13} - 2 \beta_{14} ) q^{13} + ( 9 \beta_{1} - 105 \beta_{2} - 9 \beta_{7} ) q^{16} + ( \beta_{3} + 2 \beta_{5} + \beta_{6} - 2 \beta_{11} ) q^{17} + ( 2 \beta_{8} + 6 \beta_{10} ) q^{19} + ( -14 \beta_{3} + 5 \beta_{5} ) q^{20} + ( 124 - 4 \beta_{1} ) q^{22} + ( 7 \beta_{4} - 3 \beta_{9} ) q^{23} + ( -69 + 69 \beta_{2} + 4 \beta_{7} ) q^{25} + ( 18 \beta_{6} + 5 \beta_{11} ) q^{26} + ( 13 \beta_{4} - 13 \beta_{12} - 9 \beta_{15} ) q^{29} + ( 10 \beta_{13} - 2 \beta_{14} ) q^{31} + ( -2 \beta_{9} + 107 \beta_{12} - 2 \beta_{15} ) q^{32} + ( 18 \beta_{8} + \beta_{10} - 18 \beta_{13} + \beta_{14} ) q^{34} + ( -24 \beta_{1} + 4 \beta_{2} + 24 \beta_{7} ) q^{37} + ( -32 \beta_{3} + 4 \beta_{5} - 32 \beta_{6} - 4 \beta_{11} ) q^{38} + ( 62 \beta_{8} - 23 \beta_{10} ) q^{40} + ( -\beta_{3} + 10 \beta_{5} ) q^{41} + ( 260 + 20 \beta_{1} ) q^{43} + ( -108 \beta_{4} + 16 \beta_{9} ) q^{44} + ( 104 - 104 \beta_{2} - 16 \beta_{7} ) q^{46} + ( -30 \beta_{6} - 2 \beta_{11} ) q^{47} + ( 109 \beta_{4} - 109 \beta_{12} + 8 \beta_{15} ) q^{50} + ( 62 \beta_{13} + 43 \beta_{14} ) q^{52} + ( -14 \beta_{9} + 34 \beta_{12} - 14 \beta_{15} ) q^{53} + ( 6 \beta_{8} + 26 \beta_{10} - 6 \beta_{13} + 26 \beta_{14} ) q^{55} + ( -14 \beta_{1} - 266 \beta_{2} + 14 \beta_{7} ) q^{58} + ( -2 \beta_{3} + 10 \beta_{5} - 2 \beta_{6} - 10 \beta_{11} ) q^{59} + ( 55 \beta_{8} + 22 \beta_{10} ) q^{61} + ( 8 \beta_{3} - 8 \beta_{5} ) q^{62} + ( 969 - 41 \beta_{1} ) q^{64} + ( 191 \beta_{4} - 17 \beta_{9} ) q^{65} + ( 20 - 20 \beta_{2} - 40 \beta_{7} ) q^{67} + ( 38 \beta_{6} + \beta_{11} ) q^{68} + ( -163 \beta_{4} + 163 \beta_{12} + 9 \beta_{15} ) q^{71} + ( 23 \beta_{13} - 28 \beta_{14} ) q^{73} + ( 48 \beta_{9} - 244 \beta_{12} + 48 \beta_{15} ) q^{74} + ( 120 \beta_{8} - 52 \beta_{10} - 120 \beta_{13} - 52 \beta_{14} ) q^{76} + ( 28 \beta_{1} - 688 \beta_{2} - 28 \beta_{7} ) q^{79} + ( 150 \beta_{3} - 45 \beta_{5} + 150 \beta_{6} + 45 \beta_{11} ) q^{80} + ( 102 \beta_{8} - 13 \beta_{10} ) q^{82} + ( -52 \beta_{3} - 16 \beta_{5} ) q^{83} + ( 326 + 64 \beta_{1} ) q^{85} + ( -60 \beta_{4} - 40 \beta_{9} ) q^{86} + ( -764 + 764 \beta_{2} + 124 \beta_{7} ) q^{88} + ( -7 \beta_{6} - 10 \beta_{11} ) q^{89} + ( -208 \beta_{4} + 208 \beta_{12} - 8 \beta_{15} ) q^{92} + ( -80 \beta_{13} - 92 \beta_{14} ) q^{94} + ( -26 \beta_{9} - 342 \beta_{12} - 26 \beta_{15} ) q^{95} + ( -57 \beta_{8} + 28 \beta_{10} + 57 \beta_{13} + 28 \beta_{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 68q^{4} + O(q^{10}) \) \( 16q - 68q^{4} - 804q^{16} + 1952q^{22} - 536q^{25} - 64q^{37} + 4320q^{43} + 768q^{46} - 2184q^{58} + 15176q^{64} - 5392q^{79} + 5728q^{85} - 5616q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 74 x^{14} + 4007 x^{12} + 91050 x^{10} + 1502189 x^{8} + 12598332 x^{6} + 74261084 x^{4} + 22070000 x^{2} + 6250000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(5955326 \nu^{14} + 420756261 \nu^{12} + 21224761594 \nu^{10} + 399577053316 \nu^{8} + 4077769899992 \nu^{6} + 22188556375664 \nu^{4} + 6595574220000 \nu^{2} + 2060425458941185\)\()/ 156438029994849 \)
\(\beta_{2}\)\(=\)\((\)\(2249798084357 \nu^{14} + 165523687535043 \nu^{12} + 8950261034338249 \nu^{10} + 201417793651814725 \nu^{8} + 3315118047702508223 \nu^{6} + 27333084737355121649 \nu^{4} + 163490536811521715988 \nu^{2} + 48588317482855240000\)\()/ 46253068953138052500 \)
\(\beta_{3}\)\(=\)\((\)\(5596355476439 \nu^{14} + 378608216872794 \nu^{12} + 19945391869847941 \nu^{10} + 375491657481782674 \nu^{8} + 5545670366939508311 \nu^{6} + 20851091776877154296 \nu^{4} + 6198011310607830000 \nu^{2} - 1244227061134685320568\)\()/ \)\(10\!\cdots\!44\)\( \)
\(\beta_{4}\)\(=\)\((\)\(11310865459517 \nu^{15} + 773166243923574 \nu^{13} + 40311885999197623 \nu^{11} + 758910979981201222 \nu^{9} + 10456380827878555493 \nu^{7} + 42142407637473355688 \nu^{5} + 12526879742717490000 \nu^{3} - 4701473085998115583976 \nu\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(81945585249437 \nu^{14} + 5641408644608886 \nu^{12} + 292053786912774103 \nu^{10} + 5498200348095991942 \nu^{8} + 71663640280722515861 \nu^{6} + 305315651577077926568 \nu^{4} + 90755432954299890000 \nu^{2} - 11287233145670329435880\)\()/ \)\(26\!\cdots\!60\)\( \)
\(\beta_{6}\)\(=\)\((\)\(36549057905883239 \nu^{14} + 2716062717018068436 \nu^{12} + 147193283270968061173 \nu^{10} + 3368536871004748877200 \nu^{8} + 55670660323382905812671 \nu^{6} + 474549060694699086636098 \nu^{4} + 2756768003598295094681076 \nu^{2} + 819299223684428072230000\)\()/ \)\(65\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-1035048282591857 \nu^{14} - 75366198338390343 \nu^{12} - 4075230308964056149 \nu^{10} - 90616156314394050925 \nu^{8} - 1509438606757895941523 \nu^{6} - 12445292369887636979549 \nu^{4} - 76073073971134978813188 \nu^{2} - 1063284696474873437500\)\()/ \)\(16\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(79176058216619 \nu^{15} + 5412163707465018 \nu^{13} + 282183201994383361 \nu^{11} + 5312376859868408554 \nu^{9} + 73194665795149888451 \nu^{7} + 294996853462313489816 \nu^{5} + 87688158199022430000 \nu^{3} - 23741103212716721560232 \nu\)\()/ \)\(13\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(38872184755339 \nu^{15} + 2625734874661842 \nu^{13} + 138540334159707041 \nu^{11} + 2608158317528460074 \nu^{9} + 39176637776066606827 \nu^{7} + 144831309468040223896 \nu^{5} + 43051275387340830000 \nu^{3} - 21780492554335386764440 \nu\)\()/ \)\(52\!\cdots\!20\)\( \)
\(\beta_{10}\)\(=\)\((\)\(4417491293917 \nu^{15} + 300026796677634 \nu^{13} + 15743923935811223 \nu^{11} + 296394883214191622 \nu^{9} + 4277010047478401233 \nu^{7} + 16458839468079877288 \nu^{5} + 4892409197285490000 \nu^{3} - 1566391769961532269496 \nu\)\()/ 59540314216039529400 \)
\(\beta_{11}\)\(=\)\((\)\(-63275195548744273 \nu^{14} - 4765069144825024452 \nu^{12} - 259282498216755770411 \nu^{10} - 6055965732126021373400 \nu^{8} - 100600434716506560273097 \nu^{6} - 865142366320131816159886 \nu^{4} - 5007028502862878323309932 \nu^{2} - 1488079696206944407610000\)\()/ \)\(32\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-139708174970895173 \nu^{15} - 10345479671980724052 \nu^{13} - 560339201436145870711 \nu^{11} - 12745643623835228145400 \nu^{9} - 210342767559956131671197 \nu^{7} - 1761815992670299711407686 \nu^{5} - 10401239765058335770397532 \nu^{3} - 3091194738786765780610000 \nu\)\()/ \)\(81\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(-699207242144433911 \nu^{15} - 51909972818273240664 \nu^{13} - 2813437067898512043877 \nu^{11} - 64263840733386752590300 \nu^{9} - 1061656246809352152868679 \nu^{7} - 8946142749733798407542702 \nu^{5} - 52552200844460809781869524 \nu^{3} - 15618270635381596228270000 \nu\)\()/ \)\(81\!\cdots\!00\)\( \)
\(\beta_{14}\)\(=\)\((\)\(46524892852974133 \nu^{15} + 3445959372421567092 \nu^{13} + 186630777641743075931 \nu^{11} + 4247201545491490453400 \nu^{9} + 70098339932879564514637 \nu^{7} + 589712610118527035764906 \nu^{5} + 3466603523571558801590172 \nu^{3} + 1030256822982238427810000 \nu\)\()/ \)\(37\!\cdots\!00\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-560405796289847 \nu^{15} - 41256773136579303 \nu^{13} - 2230852239373189429 \nu^{11} - 50245648889503936225 \nu^{9} - 826293053591415900083 \nu^{7} - 6812770383049970560829 \nu^{5} - 39835642226376958682898 \nu^{3} - 582060611643123437500 \nu\)\()/ \)\(27\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{8} - 7 \beta_{4}\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 7 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + 133 \beta_{2} - 133\)\()/7\)
\(\nu^{3}\)\(=\)\((\)\(14 \beta_{15} + 21 \beta_{14} + 44 \beta_{13} - 231 \beta_{12} + 21 \beta_{10} - 44 \beta_{8} + 231 \beta_{4}\)\()/7\)
\(\nu^{4}\)\(=\)\((\)\(-92 \beta_{11} - 315 \beta_{7} - 296 \beta_{6} - 4599 \beta_{2} + 315 \beta_{1}\)\()/7\)
\(\nu^{5}\)\(=\)\((\)\(-742 \beta_{15} - 1295 \beta_{14} - 2034 \beta_{13} + 9373 \beta_{12} - 742 \beta_{9}\)\()/7\)
\(\nu^{6}\)\(=\)\((\)\(4350 \beta_{5} - 15364 \beta_{3} - 14189 \beta_{1} + 191877\)\()/7\)
\(\nu^{7}\)\(=\)\((\)\(-64631 \beta_{10} + 35042 \beta_{9} + 95558 \beta_{8} - 414659 \beta_{4}\)\()/7\)
\(\nu^{8}\)\(=\)\((\)\(204408 \beta_{11} + 649047 \beta_{7} + 737424 \beta_{6} - 204408 \beta_{5} + 737424 \beta_{3} + 8599591 \beta_{2} - 8599591\)\()/7\)
\(\nu^{9}\)\(=\)\((\)\(1626702 \beta_{15} + 3065727 \beta_{14} + 4469278 \beta_{13} - 18937597 \beta_{12} + 3065727 \beta_{10} - 4469278 \beta_{8} + 18937597 \beta_{4}\)\()/7\)
\(\nu^{10}\)\(=\)\((\)\(-9543218 \beta_{11} - 29949157 \beta_{7} - 34602748 \beta_{6} - 394769893 \beta_{2} + 29949157 \beta_{1}\)\()/7\)
\(\nu^{11}\)\(=\)\((\)\(-75414626 \beta_{15} - 143300619 \beta_{14} - 208198022 \beta_{13} + 874415451 \beta_{12} - 75414626 \beta_{9}\)\()/7\)
\(\nu^{12}\)\(=\)\((\)\(444094580 \beta_{5} - 1612220888 \beta_{3} - 1387260567 \beta_{1} + 18262565559\)\()/7\)
\(\nu^{13}\)\(=\)\((\)\(-6668017811 \beta_{10} + 3498552862 \beta_{9} + 9679785270 \beta_{8} - 40524590533 \beta_{4}\)\()/7\)
\(\nu^{14}\)\(=\)\(2948235786 \beta_{11} + 9193754963 \beta_{7} + 10706353996 \beta_{6} - 2948235786 \beta_{5} + 10706353996 \beta_{3} + 120992910771 \beta_{2} - 120992910771\)
\(\nu^{15}\)\(=\)\((\)\(162381746450 \beta_{15} + 309827369159 \beta_{14} + 449677106846 \beta_{13} - 1880447860739 \beta_{12} + 309827369159 \beta_{10} - 449677106846 \beta_{8} + 1880447860739 \beta_{4}\)\()/7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−3.40706 + 5.90120i
−1.99285 + 3.45171i
−0.272818 + 0.472535i
−1.68703 + 2.92202i
0.272818 0.472535i
1.68703 2.92202i
3.40706 5.90120i
1.99285 3.45171i
−3.40706 5.90120i
−1.99285 3.45171i
−0.272818 0.472535i
−1.68703 2.92202i
0.272818 + 0.472535i
1.68703 + 2.92202i
3.40706 + 5.90120i
1.99285 + 3.45171i
−2.69995 + 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 71.0573 0 −42.0566 72.8441i
226.2 −2.69995 + 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 71.0573 0 42.0566 + 72.8441i
226.3 −0.979925 + 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 −23.8298 0 −11.6510 20.1801i
226.4 −0.979925 + 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 −23.8298 0 11.6510 + 20.1801i
226.5 0.979925 1.69728i 0 2.07949 + 3.60179i −5.94483 + 10.2967i 0 0 23.8298 0 11.6510 + 20.1801i
226.6 0.979925 1.69728i 0 2.07949 + 3.60179i 5.94483 10.2967i 0 0 23.8298 0 −11.6510 20.1801i
226.7 2.69995 4.67646i 0 −10.5795 18.3242i −7.78839 + 13.4899i 0 0 −71.0573 0 42.0566 + 72.8441i
226.8 2.69995 4.67646i 0 −10.5795 18.3242i 7.78839 13.4899i 0 0 −71.0573 0 −42.0566 72.8441i
361.1 −2.69995 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 71.0573 0 −42.0566 + 72.8441i
361.2 −2.69995 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 71.0573 0 42.0566 72.8441i
361.3 −0.979925 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 −23.8298 0 −11.6510 + 20.1801i
361.4 −0.979925 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 −23.8298 0 11.6510 20.1801i
361.5 0.979925 + 1.69728i 0 2.07949 3.60179i −5.94483 10.2967i 0 0 23.8298 0 11.6510 20.1801i
361.6 0.979925 + 1.69728i 0 2.07949 3.60179i 5.94483 + 10.2967i 0 0 23.8298 0 −11.6510 + 20.1801i
361.7 2.69995 + 4.67646i 0 −10.5795 + 18.3242i −7.78839 13.4899i 0 0 −71.0573 0 42.0566 72.8441i
361.8 2.69995 + 4.67646i 0 −10.5795 + 18.3242i 7.78839 + 13.4899i 0 0 −71.0573 0 −42.0566 + 72.8441i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.4.e.z 16
3.b odd 2 1 inner 441.4.e.z 16
7.b odd 2 1 inner 441.4.e.z 16
7.c even 3 1 441.4.a.x 8
7.c even 3 1 inner 441.4.e.z 16
7.d odd 6 1 441.4.a.x 8
7.d odd 6 1 inner 441.4.e.z 16
21.c even 2 1 inner 441.4.e.z 16
21.g even 6 1 441.4.a.x 8
21.g even 6 1 inner 441.4.e.z 16
21.h odd 6 1 441.4.a.x 8
21.h odd 6 1 inner 441.4.e.z 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.4.a.x 8 7.c even 3 1
441.4.a.x 8 7.d odd 6 1
441.4.a.x 8 21.g even 6 1
441.4.a.x 8 21.h odd 6 1
441.4.e.z 16 1.a even 1 1 trivial
441.4.e.z 16 3.b odd 2 1 inner
441.4.e.z 16 7.b odd 2 1 inner
441.4.e.z 16 7.c even 3 1 inner
441.4.e.z 16 7.d odd 6 1 inner
441.4.e.z 16 21.c even 2 1 inner
441.4.e.z 16 21.g even 6 1 inner
441.4.e.z 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{8} + 33 T_{2}^{6} + 977 T_{2}^{4} + 3696 T_{2}^{2} + 12544 \)
\( T_{5}^{8} + 384 T_{5}^{6} + 113156 T_{5}^{4} + 13171200 T_{5}^{2} + 1176490000 \)
\( T_{13}^{4} - 5268 T_{13}^{2} + 4900 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 12544 + 3696 T^{2} + 977 T^{4} + 33 T^{6} + T^{8} )^{2} \)
$3$ \( T^{16} \)
$5$ \( ( 1176490000 + 13171200 T^{2} + 113156 T^{4} + 384 T^{6} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( ( 1836567040000 + 3182009600 T^{2} + 4157904 T^{4} + 2348 T^{6} + T^{8} )^{2} \)
$13$ \( ( 4900 - 5268 T^{2} + T^{4} )^{4} \)
$17$ \( ( 66171521929744 + 85380635648 T^{2} + 102031428 T^{4} + 10496 T^{6} + T^{8} )^{2} \)
$19$ \( ( 17710099023855616 + 3071470151680 T^{2} + 399607104 T^{4} + 23080 T^{6} + T^{8} )^{2} \)
$23$ \( ( 81603616374784 + 54309233664 T^{2} + 27110672 T^{4} + 6012 T^{6} + T^{8} )^{2} \)
$29$ \( ( 16601200 - 53088 T^{2} + T^{4} )^{4} \)
$31$ \( ( 2639927418290176 + 1000064679936 T^{2} + 327467072 T^{4} + 19464 T^{6} + T^{8} )^{2} \)
$37$ \( ( 8508217600 - 1475840 T + 92496 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$41$ \( ( 10038958300 - 233264 T^{2} + T^{4} )^{4} \)
$43$ \( ( 8800 - 540 T + T^{2} )^{8} \)
$47$ \( ( \)\(36\!\cdots\!44\)\( + 6417687430260736 T^{2} + 93160817472 T^{4} + 335128 T^{6} + T^{8} )^{2} \)
$53$ \( ( 19660334433029914624 + 591388859629568 T^{2} + 13355159808 T^{4} + 133376 T^{6} + T^{8} )^{2} \)
$59$ \( ( \)\(11\!\cdots\!00\)\( + 2485779502080000 T^{2} + 42648881216 T^{4} + 231096 T^{6} + T^{8} )^{2} \)
$61$ \( ( 15433361192191875856 + 2930071785321296 T^{2} + 552354740652 T^{4} + 745844 T^{6} + T^{8} )^{2} \)
$67$ \( ( 65740960000 + 256400 T^{2} + T^{4} )^{4} \)
$71$ \( ( 187904819200 - 959388 T^{2} + T^{4} )^{4} \)
$73$ \( ( \)\(30\!\cdots\!00\)\( + 29509298801278800 T^{2} + 231851422124 T^{4} + 535668 T^{6} + T^{8} )^{2} \)
$79$ \( ( 108004249600 + 443006720 T + 1488464 T^{2} + 1348 T^{3} + T^{4} )^{4} \)
$83$ \( ( 351934815232 - 1919232 T^{2} + T^{4} )^{4} \)
$89$ \( ( \)\(17\!\cdots\!00\)\( + 3105436250000000 T^{2} + 40321676676 T^{4} + 231776 T^{6} + T^{8} )^{2} \)
$97$ \( ( 35588822500 - 1382388 T^{2} + T^{4} )^{4} \)
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