Properties

Label 3750.2.a.v
Level 3750
Weight 2
Character orbit 3750.a
Self dual yes
Analytic conductor 29.944
Analytic rank 0
Dimension 8
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3750.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.9439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.71684000000.1
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + ( 1 - \beta_{6} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + ( 1 - \beta_{6} ) q^{7} + q^{8} + q^{9} + ( 1 - \beta_{3} - \beta_{5} ) q^{11} + q^{12} -\beta_{1} q^{13} + ( 1 - \beta_{6} ) q^{14} + q^{16} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{17} + q^{18} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{19} + ( 1 - \beta_{6} ) q^{21} + ( 1 - \beta_{3} - \beta_{5} ) q^{22} + ( 2 + \beta_{4} + \beta_{5} ) q^{23} + q^{24} -\beta_{1} q^{26} + q^{27} + ( 1 - \beta_{6} ) q^{28} + ( 1 + \beta_{3} + \beta_{4} + \beta_{7} ) q^{29} + ( 2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{31} + q^{32} + ( 1 - \beta_{3} - \beta_{5} ) q^{33} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{34} + q^{36} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{37} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{38} -\beta_{1} q^{39} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{41} + ( 1 - \beta_{6} ) q^{42} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{43} + ( 1 - \beta_{3} - \beta_{5} ) q^{44} + ( 2 + \beta_{4} + \beta_{5} ) q^{46} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{47} + q^{48} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{49} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{51} -\beta_{1} q^{52} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{7} ) q^{53} + q^{54} + ( 1 - \beta_{6} ) q^{56} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{57} + ( 1 + \beta_{3} + \beta_{4} + \beta_{7} ) q^{58} + ( -2 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{5} + \beta_{6} ) q^{59} + ( 3 + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{61} + ( 2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{62} + ( 1 - \beta_{6} ) q^{63} + q^{64} + ( 1 - \beta_{3} - \beta_{5} ) q^{66} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{67} + ( 1 + \beta_{3} - \beta_{4} - \beta_{7} ) q^{68} + ( 2 + \beta_{4} + \beta_{5} ) q^{69} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{6} ) q^{71} + q^{72} + ( -1 - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{73} + ( -1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{74} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{76} + ( 2 + \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{77} -\beta_{1} q^{78} + ( -1 + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{79} + q^{81} + ( 2 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{82} + ( 3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{83} + ( 1 - \beta_{6} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} ) q^{86} + ( 1 + \beta_{3} + \beta_{4} + \beta_{7} ) q^{87} + ( 1 - \beta_{3} - \beta_{5} ) q^{88} + ( \beta_{1} + \beta_{2} + 5 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{89} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{91} + ( 2 + \beta_{4} + \beta_{5} ) q^{92} + ( 2 - \beta_{2} - \beta_{3} + \beta_{6} ) q^{93} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{94} + q^{96} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{97} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{7} ) q^{98} + ( 1 - \beta_{3} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{3} + 8q^{4} + 8q^{6} + 4q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{3} + 8q^{4} + 8q^{6} + 4q^{7} + 8q^{8} + 8q^{9} + 6q^{11} + 8q^{12} + 2q^{13} + 4q^{14} + 8q^{16} + 14q^{17} + 8q^{18} + 10q^{19} + 4q^{21} + 6q^{22} + 12q^{23} + 8q^{24} + 2q^{26} + 8q^{27} + 4q^{28} + 10q^{29} + 16q^{31} + 8q^{32} + 6q^{33} + 14q^{34} + 8q^{36} - 6q^{37} + 10q^{38} + 2q^{39} + 6q^{41} + 4q^{42} + 2q^{43} + 6q^{44} + 12q^{46} + 14q^{47} + 8q^{48} + 26q^{49} + 14q^{51} + 2q^{52} + 12q^{53} + 8q^{54} + 4q^{56} + 10q^{57} + 10q^{58} + 16q^{61} + 16q^{62} + 4q^{63} + 8q^{64} + 6q^{66} - 6q^{67} + 14q^{68} + 12q^{69} + 6q^{71} + 8q^{72} - 8q^{73} - 6q^{74} + 10q^{76} + 8q^{77} + 2q^{78} + 10q^{79} + 8q^{81} + 6q^{82} + 22q^{83} + 4q^{84} + 2q^{86} + 10q^{87} + 6q^{88} + 20q^{89} + 6q^{91} + 12q^{92} + 16q^{93} + 14q^{94} + 8q^{96} - 16q^{97} + 26q^{98} + 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 18 x^{6} + 10 x^{5} + 101 x^{4} + 40 x^{3} - 132 x^{2} - 96 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -3 \nu^{7} + 11 \nu^{6} + 32 \nu^{5} - 70 \nu^{4} - 145 \nu^{3} + 21 \nu^{2} + 208 \nu + 106 \)\()/20\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{7} - 19 \nu^{6} - 26 \nu^{5} + 144 \nu^{4} + 48 \nu^{3} - 247 \nu^{2} + 10 \nu + 86 \)\()/20\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{7} - 27 \nu^{6} - 78 \nu^{5} + 222 \nu^{4} + 319 \nu^{3} - 361 \nu^{2} - 340 \nu + 18 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\( -14 \nu^{7} + 47 \nu^{6} + 186 \nu^{5} - 386 \nu^{4} - 862 \nu^{3} + 569 \nu^{2} + 998 \nu + 22 \)\()/20\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{7} - 57 \nu^{6} - 194 \nu^{5} + 450 \nu^{4} + 840 \nu^{3} - 587 \nu^{2} - 866 \nu - 162 \)\()/20\)
\(\beta_{6}\)\(=\)\((\)\( 9 \nu^{7} - 32 \nu^{6} - 111 \nu^{5} + 256 \nu^{4} + 502 \nu^{3} - 364 \nu^{2} - 578 \nu - 42 \)\()/10\)
\(\beta_{7}\)\(=\)\((\)\( 28 \nu^{7} - 103 \nu^{6} - 332 \nu^{5} + 838 \nu^{4} + 1446 \nu^{3} - 1309 \nu^{2} - 1650 \nu - 18 \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{1} + 4\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(5 \beta_{7} - 2 \beta_{6} + 3 \beta_{5} + 6 \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + \beta_{1} + 32\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(27 \beta_{7} - 3 \beta_{6} + 12 \beta_{5} + 29 \beta_{4} - 58 \beta_{3} - 10 \beta_{2} + 14 \beta_{1} + 98\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(119 \beta_{7} - 50 \beta_{6} + 55 \beta_{5} + 110 \beta_{4} - 205 \beta_{3} - 58 \beta_{2} + 35 \beta_{1} + 470\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(549 \beta_{7} - 199 \beta_{6} + 236 \beta_{5} + 487 \beta_{4} - 1054 \beta_{3} - 206 \beta_{2} + 182 \beta_{1} + 1904\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(2411 \beta_{7} - 1082 \beta_{6} + 1063 \beta_{5} + 2016 \beta_{4} - 4417 \beta_{3} - 956 \beta_{2} + 691 \beta_{1} + 8442\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(10719 \beta_{7} - 4723 \beta_{6} + 4642 \beta_{5} + 8799 \beta_{4} - 20178 \beta_{3} - 3894 \beta_{2} + 3094 \beta_{1} + 36238\)\()/5\)

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
2.75978
−0.852282
−1.74919
1.37243
4.37243
−0.0444111
−1.65651
−2.20224
1.00000 1.00000 1.00000 0 1.00000 −4.80694 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 −3.52206 1.00000 1.00000 0
1.3 1.00000 1.00000 1.00000 0 1.00000 −0.533559 1.00000 1.00000 0
1.4 1.00000 1.00000 1.00000 0 1.00000 −0.329315 1.00000 1.00000 0
1.5 1.00000 1.00000 1.00000 0 1.00000 2.61995 1.00000 1.00000 0
1.6 1.00000 1.00000 1.00000 0 1.00000 2.70913 1.00000 1.00000 0
1.7 1.00000 1.00000 1.00000 0 1.00000 3.23143 1.00000 1.00000 0
1.8 1.00000 1.00000 1.00000 0 1.00000 4.63137 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 3750.2.a.v 8
5.b even 2 1 3750.2.a.u 8
5.c odd 4 2 3750.2.c.k 16
25.d even 5 2 750.2.g.f 16
25.e even 10 2 750.2.g.g 16
25.f odd 20 2 150.2.h.b 16
25.f odd 20 2 750.2.h.d 16
75.l even 20 2 450.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b 16 25.f odd 20 2
450.2.l.c 16 75.l even 20 2
750.2.g.f 16 25.d even 5 2
750.2.g.g 16 25.e even 10 2
750.2.h.d 16 25.f odd 20 2
3750.2.a.u 8 5.b even 2 1
3750.2.a.v 8 1.a even 1 1 trivial
3750.2.c.k 16 5.c odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3750))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{8} \)
$3$ \( ( 1 - T )^{8} \)
$5$ \( \)
$7$ \( 1 - 4 T + 23 T^{2} - 48 T^{3} + 191 T^{4} - 264 T^{5} + 1285 T^{6} - 2084 T^{7} + 10564 T^{8} - 14588 T^{9} + 62965 T^{10} - 90552 T^{11} + 458591 T^{12} - 806736 T^{13} + 2705927 T^{14} - 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - 6 T + 45 T^{2} - 240 T^{3} + 1205 T^{4} - 4658 T^{5} + 20073 T^{6} - 68300 T^{7} + 239780 T^{8} - 751300 T^{9} + 2428833 T^{10} - 6199798 T^{11} + 17642405 T^{12} - 38652240 T^{13} + 79720245 T^{14} - 116923026 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 2 T + 47 T^{2} - 176 T^{3} + 1191 T^{4} - 5212 T^{5} + 25325 T^{6} - 85818 T^{7} + 405224 T^{8} - 1115634 T^{9} + 4279925 T^{10} - 11450764 T^{11} + 34016151 T^{12} - 65347568 T^{13} + 226860023 T^{14} - 125497034 T^{15} + 815730721 T^{16} \)
$17$ \( 1 - 14 T + 153 T^{2} - 1158 T^{3} + 7331 T^{4} - 38214 T^{5} + 179335 T^{6} - 760774 T^{7} + 3198344 T^{8} - 12933158 T^{9} + 51827815 T^{10} - 187745382 T^{11} + 612292451 T^{12} - 1644194406 T^{13} + 3693048057 T^{14} - 5744741422 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 10 T + 102 T^{2} - 670 T^{3} + 4268 T^{4} - 22710 T^{5} + 117514 T^{6} - 560450 T^{7} + 2521030 T^{8} - 10648550 T^{9} + 42422554 T^{10} - 155767890 T^{11} + 556210028 T^{12} - 1658986330 T^{13} + 4798679862 T^{14} - 8938717390 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 - 12 T + 132 T^{2} - 716 T^{3} + 4016 T^{4} - 10172 T^{5} + 50860 T^{6} - 53308 T^{7} + 886334 T^{8} - 1226084 T^{9} + 26904940 T^{10} - 123762724 T^{11} + 1123841456 T^{12} - 4608421588 T^{13} + 19540737348 T^{14} - 40857905364 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 10 T + 197 T^{2} - 1510 T^{3} + 17343 T^{4} - 107610 T^{5} + 913319 T^{6} - 4694150 T^{7} + 32002680 T^{8} - 136130350 T^{9} + 768101279 T^{10} - 2624500290 T^{11} + 12266374383 T^{12} - 30971834990 T^{13} + 117180194237 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 16 T + 245 T^{2} - 2700 T^{3} + 25605 T^{4} - 210758 T^{5} + 1537653 T^{6} - 9946750 T^{7} + 58857900 T^{8} - 308349250 T^{9} + 1477684533 T^{10} - 6278691578 T^{11} + 23646755205 T^{12} - 77298707700 T^{13} + 217438401845 T^{14} - 440201825776 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 6 T + 243 T^{2} + 1412 T^{3} + 27071 T^{4} + 147396 T^{5} + 1822865 T^{6} + 8822506 T^{7} + 81640584 T^{8} + 326432722 T^{9} + 2495502185 T^{10} + 7466049588 T^{11} + 50735412431 T^{12} + 97913667284 T^{13} + 623471517387 T^{14} + 569591262798 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 6 T + 145 T^{2} - 850 T^{3} + 10255 T^{4} - 69718 T^{5} + 526063 T^{6} - 4134450 T^{7} + 23274800 T^{8} - 169512450 T^{9} + 884311903 T^{10} - 4805034278 T^{11} + 28978179055 T^{12} - 98477770850 T^{13} + 688765114945 T^{14} - 1168525643286 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 2 T + 182 T^{2} - 166 T^{3} + 15756 T^{4} + 5778 T^{5} + 902810 T^{6} + 1211542 T^{7} + 41601734 T^{8} + 52096306 T^{9} + 1669295690 T^{10} + 459391446 T^{11} + 53866628556 T^{12} - 24403401538 T^{13} + 1150488074918 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 14 T + 158 T^{2} - 1418 T^{3} + 16156 T^{4} - 125194 T^{5} + 1006370 T^{6} - 7336014 T^{7} + 59392774 T^{8} - 344792658 T^{9} + 2223071330 T^{10} - 12998016662 T^{11} + 78836126236 T^{12} - 325211219926 T^{13} + 1703116021982 T^{14} - 7092723686482 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 12 T + 352 T^{2} - 3946 T^{3} + 57861 T^{4} - 577532 T^{5} + 5731520 T^{6} - 48919988 T^{7} + 371517029 T^{8} - 2592759364 T^{9} + 16099839680 T^{10} - 85981231564 T^{11} + 456551121141 T^{12} - 1650199415378 T^{13} + 7801855117408 T^{14} - 14096533678044 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 247 T^{2} + 500 T^{3} + 28643 T^{4} + 123000 T^{5} + 2182549 T^{6} + 13248500 T^{7} + 135310720 T^{8} + 781661500 T^{9} + 7597453069 T^{10} + 25261617000 T^{11} + 347077571123 T^{12} + 357462149500 T^{13} + 10418591809327 T^{14} + 146830437604321 T^{16} \)
$61$ \( 1 - 16 T + 335 T^{2} - 3580 T^{3} + 40995 T^{4} - 303668 T^{5} + 2560793 T^{6} - 14411280 T^{7} + 131714360 T^{8} - 879088080 T^{9} + 9528710753 T^{10} - 68926866308 T^{11} + 567610251795 T^{12} - 3023654757580 T^{13} + 17259325410935 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 6 T + 238 T^{2} + 1202 T^{3} + 35076 T^{4} + 154266 T^{5} + 3495250 T^{6} + 13666446 T^{7} + 270524214 T^{8} + 915651882 T^{9} + 15690177250 T^{10} + 46397504958 T^{11} + 706820720196 T^{12} + 1622850378614 T^{13} + 21529094956222 T^{14} + 36364269631938 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 6 T + 390 T^{2} - 2050 T^{3} + 74900 T^{4} - 339378 T^{5} + 9085578 T^{6} - 35263990 T^{7} + 765200950 T^{8} - 2503743290 T^{9} + 45800398698 T^{10} - 121467119358 T^{11} + 1903334906900 T^{12} - 3698670169550 T^{13} + 49959110729190 T^{14} - 54570720950346 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 8 T + 327 T^{2} + 3024 T^{3} + 60051 T^{4} + 506008 T^{5} + 7383005 T^{6} + 54348912 T^{7} + 633329824 T^{8} + 3967470576 T^{9} + 39344033645 T^{10} + 196845714136 T^{11} + 1705342770291 T^{12} + 6268968497232 T^{13} + 49486291996503 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 10 T + 227 T^{2} - 1720 T^{3} + 32163 T^{4} - 256360 T^{5} + 3728769 T^{6} - 26100650 T^{7} + 323096880 T^{8} - 2061951350 T^{9} + 23271247329 T^{10} - 126395478040 T^{11} + 1252751455203 T^{12} - 5292537006280 T^{13} + 55180852403267 T^{14} - 192039089861590 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 22 T + 487 T^{2} - 7796 T^{3} + 106551 T^{4} - 1298832 T^{5} + 14259445 T^{6} - 141748378 T^{7} + 1361172004 T^{8} - 11765115374 T^{9} + 98233316605 T^{10} - 742655252784 T^{11} + 5056731560871 T^{12} - 30708760852828 T^{13} + 159219961830703 T^{14} - 596993121771794 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 20 T + 537 T^{2} - 8210 T^{3} + 124663 T^{4} - 1568070 T^{5} + 17732339 T^{6} - 192143300 T^{7} + 1812907320 T^{8} - 17100753700 T^{9} + 140457857219 T^{10} - 1105440739830 T^{11} + 7821635989783 T^{12} - 45845128076290 T^{13} + 266878953246057 T^{14} - 884626697910580 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 16 T + 458 T^{2} + 4612 T^{3} + 72331 T^{4} + 417776 T^{5} + 4642080 T^{6} + 3332736 T^{7} + 188812609 T^{8} + 323275392 T^{9} + 43677330720 T^{10} + 381292875248 T^{11} + 6403411424011 T^{12} + 39604813265284 T^{13} + 381501178257482 T^{14} + 1292772551649808 T^{15} + 7837433594376961 T^{16} \)
show more
show less