Properties

Label 9.48.a.c
Level $9$
Weight $48$
Character orbit 9.a
Self dual yes
Analytic conductor $125.917$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(125.916896390\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{11}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1446390 - \beta_{1} ) q^{2} + ( 101201874790288 + 2732420 \beta_{1} + 6 \beta_{2} + \beta_{3} ) q^{4} + ( 7778670060568050 - 214856224 \beta_{1} - 1796 \beta_{2} + 96 \beta_{3} ) q^{5} + ( -9792304681472105800 + 3698202764048 \beta_{1} - 4878534 \beta_{2} - 133760 \beta_{3} ) q^{7} + ( -598179247018446084480 - 185574830780256 \beta_{1} - 105238288 \beta_{2} - 2897880 \beta_{3} ) q^{8} +O(q^{10})\) \( q +(-1446390 - \beta_{1}) q^{2} +(101201874790288 + 2732420 \beta_{1} + 6 \beta_{2} + \beta_{3}) q^{4} +(7778670060568050 - 214856224 \beta_{1} - 1796 \beta_{2} + 96 \beta_{3}) q^{5} +(-9792304681472105800 + 3698202764048 \beta_{1} - 4878534 \beta_{2} - 133760 \beta_{3}) q^{7} +(-\)\(59\!\cdots\!80\)\( - 185574830780256 \beta_{1} - 105238288 \beta_{2} - 2897880 \beta_{3}) q^{8} +(\)\(40\!\cdots\!00\)\( - 20577222944857586 \beta_{1} + 2263126656 \beta_{2} + 641207744 \beta_{3}) q^{10} +(\)\(47\!\cdots\!28\)\( + 80173046708155720 \beta_{1} - 65412162475 \beta_{2} + 6954435840 \beta_{3}) q^{11} +(\)\(31\!\cdots\!30\)\( + 2187545461397076512 \beta_{1} + 4897389979140 \beta_{2} - 467536231520 \beta_{3}) q^{13} +(-\)\(87\!\cdots\!76\)\( + 48764929041038400840 \beta_{1} + 6024181979072 \beta_{2} - 2916146241888 \beta_{3}) q^{14} +(\)\(31\!\cdots\!96\)\( + \)\(13\!\cdots\!20\)\( \beta_{1} + 878441883331968 \beta_{2} + 61695767581248 \beta_{3}) q^{16} +(-\)\(52\!\cdots\!70\)\( + \)\(16\!\cdots\!88\)\( \beta_{1} + 2513535368752552 \beta_{2} + 187599812159040 \beta_{3}) q^{17} +(-\)\(26\!\cdots\!60\)\( + \)\(32\!\cdots\!20\)\( \beta_{1} + 73337396648714469 \beta_{2} + 676931170946304 \beta_{3}) q^{19} +(\)\(37\!\cdots\!00\)\( - \)\(12\!\cdots\!32\)\( \beta_{1} + 325607045576761772 \beta_{2} + 7255673126427378 \beta_{3}) q^{20} +(-\)\(19\!\cdots\!20\)\( - \)\(17\!\cdots\!68\)\( \beta_{1} - 669664337454884640 \beta_{2} - 61348773982345520 \beta_{3}) q^{22} +(-\)\(34\!\cdots\!20\)\( - \)\(10\!\cdots\!40\)\( \beta_{1} + 824862716416275506 \beta_{2} + 4835096198025600 \beta_{3}) q^{23} +(\)\(25\!\cdots\!75\)\( - \)\(21\!\cdots\!00\)\( \beta_{1} + 9827455518879730800 \beta_{2} - 2144521474142940800 \beta_{3}) q^{25} +(-\)\(57\!\cdots\!72\)\( + \)\(42\!\cdots\!30\)\( \beta_{1} - 2446702235232694912 \beta_{2} - 3546261034268171712 \beta_{3}) q^{26} +(-\)\(90\!\cdots\!20\)\( + \)\(85\!\cdots\!80\)\( \beta_{1} + \)\(55\!\cdots\!96\)\( \beta_{2} - 33842643361650715080 \beta_{3}) q^{28} +(\)\(56\!\cdots\!90\)\( + \)\(72\!\cdots\!00\)\( \beta_{1} + \)\(37\!\cdots\!64\)\( \beta_{2} - 85281536978719677216 \beta_{3}) q^{29} +(\)\(18\!\cdots\!12\)\( - \)\(72\!\cdots\!60\)\( \beta_{1} + \)\(36\!\cdots\!00\)\( \beta_{2} - \)\(31\!\cdots\!20\)\( \beta_{3}) q^{31} +(-\)\(29\!\cdots\!40\)\( - \)\(22\!\cdots\!36\)\( \beta_{1} - \)\(10\!\cdots\!20\)\( \beta_{2} - \)\(10\!\cdots\!60\)\( \beta_{3}) q^{32} +(-\)\(31\!\cdots\!04\)\( + \)\(41\!\cdots\!30\)\( \beta_{1} - \)\(32\!\cdots\!88\)\( \beta_{2} - \)\(19\!\cdots\!68\)\( \beta_{3}) q^{34} +(\)\(32\!\cdots\!00\)\( + \)\(99\!\cdots\!56\)\( \beta_{1} - \)\(91\!\cdots\!76\)\( \beta_{2} - \)\(89\!\cdots\!24\)\( \beta_{3}) q^{35} +(-\)\(28\!\cdots\!90\)\( + \)\(21\!\cdots\!80\)\( \beta_{1} - \)\(50\!\cdots\!28\)\( \beta_{2} - \)\(78\!\cdots\!60\)\( \beta_{3}) q^{37} +(-\)\(74\!\cdots\!20\)\( - \)\(16\!\cdots\!24\)\( \beta_{1} - \)\(53\!\cdots\!92\)\( \beta_{2} - \)\(45\!\cdots\!60\)\( \beta_{3}) q^{38} +(\)\(17\!\cdots\!00\)\( - \)\(33\!\cdots\!40\)\( \beta_{1} - \)\(13\!\cdots\!60\)\( \beta_{2} - \)\(23\!\cdots\!40\)\( \beta_{3}) q^{40} +(-\)\(32\!\cdots\!82\)\( - \)\(37\!\cdots\!40\)\( \beta_{1} + \)\(21\!\cdots\!00\)\( \beta_{2} - \)\(77\!\cdots\!80\)\( \beta_{3}) q^{41} +(-\)\(11\!\cdots\!00\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + \)\(73\!\cdots\!87\)\( \beta_{2} - \)\(19\!\cdots\!00\)\( \beta_{3}) q^{43} +(\)\(38\!\cdots\!64\)\( + \)\(25\!\cdots\!20\)\( \beta_{1} + \)\(26\!\cdots\!68\)\( \beta_{2} + \)\(83\!\cdots\!48\)\( \beta_{3}) q^{44} +(\)\(30\!\cdots\!92\)\( + \)\(31\!\cdots\!40\)\( \beta_{1} + \)\(28\!\cdots\!20\)\( \beta_{2} + \)\(92\!\cdots\!00\)\( \beta_{3}) q^{46} +(-\)\(50\!\cdots\!80\)\( - \)\(55\!\cdots\!84\)\( \beta_{1} + \)\(72\!\cdots\!36\)\( \beta_{2} + \)\(73\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(22\!\cdots\!93\)\( - \)\(32\!\cdots\!60\)\( \beta_{1} - \)\(35\!\cdots\!40\)\( \beta_{2} + \)\(89\!\cdots\!40\)\( \beta_{3}) q^{49} +(\)\(48\!\cdots\!50\)\( + \)\(17\!\cdots\!25\)\( \beta_{1} + \)\(22\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!00\)\( \beta_{3}) q^{50} +(-\)\(13\!\cdots\!00\)\( + \)\(92\!\cdots\!88\)\( \beta_{1} - \)\(70\!\cdots\!84\)\( \beta_{2} + \)\(20\!\cdots\!50\)\( \beta_{3}) q^{52} +(-\)\(72\!\cdots\!90\)\( - \)\(11\!\cdots\!24\)\( \beta_{1} + \)\(33\!\cdots\!24\)\( \beta_{2} - \)\(80\!\cdots\!80\)\( \beta_{3}) q^{53} +(\)\(48\!\cdots\!00\)\( + \)\(46\!\cdots\!28\)\( \beta_{1} + \)\(15\!\cdots\!62\)\( \beta_{2} - \)\(15\!\cdots\!12\)\( \beta_{3}) q^{55} +(-\)\(69\!\cdots\!60\)\( + \)\(59\!\cdots\!00\)\( \beta_{1} - \)\(60\!\cdots\!76\)\( \beta_{2} - \)\(58\!\cdots\!56\)\( \beta_{3}) q^{56} +(-\)\(18\!\cdots\!20\)\( + \)\(91\!\cdots\!46\)\( \beta_{1} - \)\(36\!\cdots\!12\)\( \beta_{2} - \)\(88\!\cdots\!40\)\( \beta_{3}) q^{58} +(-\)\(11\!\cdots\!20\)\( - \)\(91\!\cdots\!40\)\( \beta_{1} - \)\(23\!\cdots\!67\)\( \beta_{2} - \)\(27\!\cdots\!32\)\( \beta_{3}) q^{59} +(\)\(15\!\cdots\!22\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} + \)\(82\!\cdots\!00\)\( \beta_{3}) q^{61} +(\)\(17\!\cdots\!20\)\( + \)\(41\!\cdots\!08\)\( \beta_{1} + \)\(49\!\cdots\!20\)\( \beta_{2} + \)\(62\!\cdots\!60\)\( \beta_{3}) q^{62} +(\)\(13\!\cdots\!68\)\( + \)\(35\!\cdots\!20\)\( \beta_{1} + \)\(85\!\cdots\!92\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3}) q^{64} +(-\)\(32\!\cdots\!00\)\( + \)\(11\!\cdots\!12\)\( \beta_{1} - \)\(21\!\cdots\!52\)\( \beta_{2} + \)\(11\!\cdots\!52\)\( \beta_{3}) q^{65} +(\)\(46\!\cdots\!20\)\( + \)\(71\!\cdots\!32\)\( \beta_{1} - \)\(38\!\cdots\!67\)\( \beta_{2} + \)\(34\!\cdots\!60\)\( \beta_{3}) q^{67} +(\)\(68\!\cdots\!60\)\( + \)\(57\!\cdots\!68\)\( \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2} - \)\(26\!\cdots\!70\)\( \beta_{3}) q^{68} +(-\)\(24\!\cdots\!00\)\( + \)\(16\!\cdots\!84\)\( \beta_{1} + \)\(35\!\cdots\!36\)\( \beta_{2} - \)\(90\!\cdots\!36\)\( \beta_{3}) q^{70} +(-\)\(55\!\cdots\!92\)\( - \)\(41\!\cdots\!00\)\( \beta_{1} + \)\(14\!\cdots\!50\)\( \beta_{2} - \)\(65\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(26\!\cdots\!70\)\( + \)\(30\!\cdots\!00\)\( \beta_{1} + \)\(16\!\cdots\!84\)\( \beta_{2} - \)\(55\!\cdots\!40\)\( \beta_{3}) q^{73} +(-\)\(46\!\cdots\!36\)\( + \)\(58\!\cdots\!90\)\( \beta_{1} + \)\(12\!\cdots\!60\)\( \beta_{2} - \)\(12\!\cdots\!60\)\( \beta_{3}) q^{74} +(\)\(87\!\cdots\!20\)\( + \)\(14\!\cdots\!00\)\( \beta_{1} - \)\(42\!\cdots\!68\)\( \beta_{2} + \)\(12\!\cdots\!92\)\( \beta_{3}) q^{76} +(\)\(65\!\cdots\!00\)\( + \)\(17\!\cdots\!84\)\( \beta_{1} - \)\(18\!\cdots\!12\)\( \beta_{2} - \)\(21\!\cdots\!60\)\( \beta_{3}) q^{77} +(-\)\(33\!\cdots\!40\)\( - \)\(91\!\cdots\!40\)\( \beta_{1} - \)\(14\!\cdots\!04\)\( \beta_{2} + \)\(34\!\cdots\!96\)\( \beta_{3}) q^{79} +(\)\(23\!\cdots\!00\)\( + \)\(13\!\cdots\!36\)\( \beta_{1} - \)\(18\!\cdots\!56\)\( \beta_{2} + \)\(25\!\cdots\!56\)\( \beta_{3}) q^{80} +(\)\(95\!\cdots\!80\)\( + \)\(45\!\cdots\!62\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2} + \)\(33\!\cdots\!40\)\( \beta_{3}) q^{82} +(\)\(35\!\cdots\!40\)\( - \)\(22\!\cdots\!28\)\( \beta_{1} + \)\(80\!\cdots\!67\)\( \beta_{2} + \)\(34\!\cdots\!00\)\( \beta_{3}) q^{83} +(-\)\(25\!\cdots\!00\)\( + \)\(16\!\cdots\!04\)\( \beta_{1} + \)\(20\!\cdots\!16\)\( \beta_{2} - \)\(46\!\cdots\!16\)\( \beta_{3}) q^{85} +(-\)\(23\!\cdots\!32\)\( + \)\(76\!\cdots\!60\)\( \beta_{1} - \)\(91\!\cdots\!16\)\( \beta_{2} - \)\(12\!\cdots\!76\)\( \beta_{3}) q^{86} +(-\)\(38\!\cdots\!40\)\( - \)\(42\!\cdots\!68\)\( \beta_{1} - \)\(21\!\cdots\!64\)\( \beta_{2} - \)\(21\!\cdots\!40\)\( \beta_{3}) q^{88} +(\)\(19\!\cdots\!70\)\( - \)\(45\!\cdots\!00\)\( \beta_{1} - \)\(23\!\cdots\!68\)\( \beta_{2} - \)\(79\!\cdots\!08\)\( \beta_{3}) q^{89} +(\)\(13\!\cdots\!92\)\( - \)\(29\!\cdots\!00\)\( \beta_{1} - \)\(14\!\cdots\!64\)\( \beta_{2} + \)\(30\!\cdots\!16\)\( \beta_{3}) q^{91} +(-\)\(32\!\cdots\!20\)\( - \)\(39\!\cdots\!72\)\( \beta_{1} - \)\(37\!\cdots\!08\)\( \beta_{2} - \)\(32\!\cdots\!40\)\( \beta_{3}) q^{92} +(\)\(86\!\cdots\!56\)\( - \)\(12\!\cdots\!80\)\( \beta_{1} - \)\(76\!\cdots\!16\)\( \beta_{2} - \)\(58\!\cdots\!56\)\( \beta_{3}) q^{94} +(-\)\(18\!\cdots\!00\)\( + \)\(64\!\cdots\!60\)\( \beta_{1} + \)\(15\!\cdots\!90\)\( \beta_{2} + \)\(37\!\cdots\!60\)\( \beta_{3}) q^{95} +(\)\(23\!\cdots\!30\)\( - \)\(49\!\cdots\!56\)\( \beta_{1} + \)\(16\!\cdots\!64\)\( \beta_{2} + \)\(75\!\cdots\!20\)\( \beta_{3}) q^{97} +(\)\(74\!\cdots\!30\)\( - \)\(24\!\cdots\!33\)\( \beta_{1} + \)\(27\!\cdots\!40\)\( \beta_{2} + \)\(39\!\cdots\!60\)\( \beta_{3}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} - 2392716988073784337920q^{8} + O(q^{10}) \) \( 4q - 5785560q^{2} + 404807499161152q^{4} + 31114680242272200q^{5} - 39169218725888423200q^{7} - \)\(23\!\cdots\!20\)\(q^{8} + \)\(16\!\cdots\!00\)\(q^{10} + \)\(19\!\cdots\!12\)\(q^{11} + \)\(12\!\cdots\!20\)\(q^{13} - \)\(34\!\cdots\!04\)\(q^{14} + \)\(12\!\cdots\!84\)\(q^{16} - \)\(21\!\cdots\!80\)\(q^{17} - \)\(10\!\cdots\!40\)\(q^{19} + \)\(15\!\cdots\!00\)\(q^{20} - \)\(79\!\cdots\!80\)\(q^{22} - \)\(13\!\cdots\!80\)\(q^{23} + \)\(10\!\cdots\!00\)\(q^{25} - \)\(22\!\cdots\!88\)\(q^{26} - \)\(36\!\cdots\!80\)\(q^{28} + \)\(22\!\cdots\!60\)\(q^{29} + \)\(75\!\cdots\!48\)\(q^{31} - \)\(11\!\cdots\!60\)\(q^{32} - \)\(12\!\cdots\!16\)\(q^{34} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(11\!\cdots\!60\)\(q^{37} - \)\(29\!\cdots\!80\)\(q^{38} + \)\(70\!\cdots\!00\)\(q^{40} - \)\(13\!\cdots\!28\)\(q^{41} - \)\(44\!\cdots\!00\)\(q^{43} + \)\(15\!\cdots\!56\)\(q^{44} + \)\(12\!\cdots\!68\)\(q^{46} - \)\(20\!\cdots\!20\)\(q^{47} + \)\(91\!\cdots\!72\)\(q^{49} + \)\(19\!\cdots\!00\)\(q^{50} - \)\(55\!\cdots\!00\)\(q^{52} - \)\(29\!\cdots\!60\)\(q^{53} + \)\(19\!\cdots\!00\)\(q^{55} - \)\(27\!\cdots\!40\)\(q^{56} - \)\(73\!\cdots\!80\)\(q^{58} - \)\(47\!\cdots\!80\)\(q^{59} + \)\(62\!\cdots\!88\)\(q^{61} + \)\(68\!\cdots\!80\)\(q^{62} + \)\(55\!\cdots\!72\)\(q^{64} - \)\(12\!\cdots\!00\)\(q^{65} + \)\(18\!\cdots\!80\)\(q^{67} + \)\(27\!\cdots\!40\)\(q^{68} - \)\(97\!\cdots\!00\)\(q^{70} - \)\(22\!\cdots\!68\)\(q^{71} + \)\(10\!\cdots\!80\)\(q^{73} - \)\(18\!\cdots\!44\)\(q^{74} + \)\(35\!\cdots\!80\)\(q^{76} + \)\(26\!\cdots\!00\)\(q^{77} - \)\(13\!\cdots\!60\)\(q^{79} + \)\(94\!\cdots\!00\)\(q^{80} + \)\(38\!\cdots\!20\)\(q^{82} + \)\(14\!\cdots\!60\)\(q^{83} - \)\(10\!\cdots\!00\)\(q^{85} - \)\(95\!\cdots\!28\)\(q^{86} - \)\(15\!\cdots\!60\)\(q^{88} + \)\(79\!\cdots\!80\)\(q^{89} + \)\(53\!\cdots\!68\)\(q^{91} - \)\(13\!\cdots\!80\)\(q^{92} + \)\(34\!\cdots\!24\)\(q^{94} - \)\(75\!\cdots\!00\)\(q^{95} + \)\(95\!\cdots\!20\)\(q^{97} + \)\(29\!\cdots\!20\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 24 \nu - 6 \)
\(\beta_{2}\)\(=\)\((\)\( 27 \nu^{3} + 1621431 \nu^{2} - 21620549846568 \nu - 600047676703864212 \)\()/171584\)
\(\beta_{3}\)\(=\)\((\)\( -81 \nu^{3} + 44551899 \nu^{2} + 65191807354488 \nu - 18776838253817714244 \)\()/85792\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 6\)\()/24\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6 \beta_{2} - 160348 \beta_{1} + 239847319113552\)\()/576\)
\(\nu^{3}\)\(=\)\((\)\(-180159 \beta_{3} + 9900422 \beta_{2} + 57683687726180 \beta_{1} - 4807255926333560112\)\()/1728\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
901372.
129356.
−124721.
−906006.
−2.30793e7 0 3.91917e14 1.15086e16 0 1.54211e19 −5.79706e21 0 −2.65611e23
1.2 −4.55092e6 0 −1.20027e14 3.08341e16 0 1.11073e20 1.18672e21 0 −1.40324e23
1.3 1.54692e6 0 −1.38345e14 −4.23962e16 0 −3.90714e19 −4.31719e20 0 −6.55837e22
1.4 2.02978e7 0 2.71261e14 3.11682e16 0 −1.26592e20 2.64934e21 0 6.32644e23
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.48.a.c 4
3.b odd 2 1 1.48.a.a 4
12.b even 2 1 16.48.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.48.a.a 4 3.b odd 2 1
9.48.a.c 4 1.a even 1 1 trivial
16.48.a.d 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5785560 T_{2}^{3} - \)\(46\!\cdots\!32\)\( T_{2}^{2} - \)\(14\!\cdots\!60\)\( T_{2} + \)\(32\!\cdots\!56\)\( \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( \)\(32\!\cdots\!56\)\( - \)\(14\!\cdots\!60\)\( T - 467142374034432 T^{2} + 5785560 T^{3} + T^{4} \)
$3$ \( T^{4} \)
$5$ \( -\)\(46\!\cdots\!00\)\( + \)\(59\!\cdots\!00\)\( T - \)\(14\!\cdots\!00\)\( T^{2} - 31114680242272200 T^{3} + T^{4} \)
$7$ \( \)\(84\!\cdots\!96\)\( - \)\(34\!\cdots\!00\)\( T - \)\(14\!\cdots\!72\)\( T^{2} + 39169218725888423200 T^{3} + T^{4} \)
$11$ \( -\)\(18\!\cdots\!44\)\( + \)\(11\!\cdots\!92\)\( T - \)\(75\!\cdots\!96\)\( T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + T^{4} \)
$13$ \( \)\(10\!\cdots\!76\)\( + \)\(33\!\cdots\!80\)\( T - \)\(23\!\cdots\!48\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + T^{4} \)
$17$ \( -\)\(14\!\cdots\!24\)\( - \)\(21\!\cdots\!80\)\( T + \)\(10\!\cdots\!48\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + T^{4} \)
$19$ \( \)\(37\!\cdots\!00\)\( - \)\(81\!\cdots\!00\)\( T - \)\(16\!\cdots\!00\)\( T^{2} + \)\(10\!\cdots\!40\)\( T^{3} + T^{4} \)
$23$ \( \)\(56\!\cdots\!16\)\( + \)\(10\!\cdots\!80\)\( T + \)\(63\!\cdots\!92\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + T^{4} \)
$29$ \( -\)\(69\!\cdots\!00\)\( + \)\(16\!\cdots\!00\)\( T - \)\(80\!\cdots\!00\)\( T^{2} - \)\(22\!\cdots\!60\)\( T^{3} + T^{4} \)
$31$ \( \)\(13\!\cdots\!36\)\( + \)\(24\!\cdots\!88\)\( T - \)\(36\!\cdots\!36\)\( T^{2} - \)\(75\!\cdots\!48\)\( T^{3} + T^{4} \)
$37$ \( -\)\(15\!\cdots\!64\)\( - \)\(78\!\cdots\!60\)\( T - \)\(49\!\cdots\!12\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + T^{4} \)
$41$ \( \)\(89\!\cdots\!76\)\( - \)\(37\!\cdots\!28\)\( T - \)\(23\!\cdots\!56\)\( T^{2} + \)\(13\!\cdots\!28\)\( T^{3} + T^{4} \)
$43$ \( -\)\(57\!\cdots\!04\)\( - \)\(12\!\cdots\!00\)\( T + \)\(50\!\cdots\!72\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + T^{4} \)
$47$ \( \)\(84\!\cdots\!16\)\( - \)\(55\!\cdots\!20\)\( T - \)\(50\!\cdots\!92\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + T^{4} \)
$53$ \( \)\(10\!\cdots\!36\)\( - \)\(86\!\cdots\!40\)\( T - \)\(10\!\cdots\!88\)\( T^{2} + \)\(29\!\cdots\!60\)\( T^{3} + T^{4} \)
$59$ \( -\)\(14\!\cdots\!00\)\( + \)\(36\!\cdots\!00\)\( T + \)\(36\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + T^{4} \)
$61$ \( -\)\(47\!\cdots\!44\)\( + \)\(26\!\cdots\!08\)\( T - \)\(28\!\cdots\!96\)\( T^{2} - \)\(62\!\cdots\!88\)\( T^{3} + T^{4} \)
$67$ \( -\)\(23\!\cdots\!24\)\( - \)\(46\!\cdots\!20\)\( T + \)\(86\!\cdots\!48\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + T^{4} \)
$71$ \( -\)\(15\!\cdots\!04\)\( - \)\(25\!\cdots\!48\)\( T - \)\(86\!\cdots\!16\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + T^{4} \)
$73$ \( -\)\(60\!\cdots\!84\)\( + \)\(28\!\cdots\!20\)\( T - \)\(66\!\cdots\!08\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + T^{4} \)
$79$ \( \)\(29\!\cdots\!00\)\( + \)\(74\!\cdots\!00\)\( T + \)\(55\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + T^{4} \)
$83$ \( -\)\(10\!\cdots\!44\)\( + \)\(38\!\cdots\!40\)\( T - \)\(23\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + T^{4} \)
$89$ \( -\)\(18\!\cdots\!00\)\( + \)\(94\!\cdots\!00\)\( T - \)\(94\!\cdots\!00\)\( T^{2} - \)\(79\!\cdots\!80\)\( T^{3} + T^{4} \)
$97$ \( -\)\(75\!\cdots\!84\)\( + \)\(26\!\cdots\!20\)\( T + \)\(19\!\cdots\!08\)\( T^{2} - \)\(95\!\cdots\!20\)\( T^{3} + T^{4} \)
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