Properties

Label 891.2.n.d
Level 891
Weight 2
Character orbit 891.n
Analytic conductor 7.115
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 891.n (of order \(15\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{15}^{2} - \zeta_{15}^{5} ) q^{2} + ( -1 + \zeta_{15}^{4} - \zeta_{15}^{5} ) q^{4} + ( \zeta_{15} - \zeta_{15}^{4} + \zeta_{15}^{7} ) q^{5} + 3 \zeta_{15}^{2} q^{7} + ( -2 - \zeta_{15}^{3} - 2 \zeta_{15}^{6} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{15}^{2} - \zeta_{15}^{5} ) q^{2} + ( -1 + \zeta_{15}^{4} - \zeta_{15}^{5} ) q^{4} + ( \zeta_{15} - \zeta_{15}^{4} + \zeta_{15}^{7} ) q^{5} + 3 \zeta_{15}^{2} q^{7} + ( -2 - \zeta_{15}^{3} - 2 \zeta_{15}^{6} ) q^{8} + ( \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{7} ) q^{10} + ( 1 + 2 \zeta_{15} - \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 3 \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{11} + ( 2 + 3 \zeta_{15} + 2 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{13} + ( -3 \zeta_{15}^{4} - 3 \zeta_{15}^{7} ) q^{14} + ( -3 + 3 \zeta_{15} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{4} + 3 \zeta_{15}^{7} ) q^{16} + ( \zeta_{15}^{2} - \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{17} + ( -3 + \zeta_{15}^{3} - 3 \zeta_{15}^{6} ) q^{19} + ( -3 \zeta_{15} + 2 \zeta_{15}^{2} + 2 \zeta_{15}^{5} - 3 \zeta_{15}^{6} ) q^{20} + ( 4 - \zeta_{15} - 3 \zeta_{15}^{4} + 4 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{22} + ( -1 + 4 \zeta_{15} + 4 \zeta_{15}^{4} - \zeta_{15}^{5} ) q^{23} + ( 3 + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{4} + 3 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{25} + ( -2 \zeta_{15}^{2} - 5 \zeta_{15}^{3} - 5 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{26} + ( -3 \zeta_{15}^{2} + 3 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{28} + ( -4 + 2 \zeta_{15}^{2} - 4 \zeta_{15}^{3} + 4 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 4 \zeta_{15}^{6} + 4 \zeta_{15}^{7} ) q^{29} + ( -3 + \zeta_{15} - 3 \zeta_{15}^{5} + 3 \zeta_{15}^{7} ) q^{31} + ( 5 - \zeta_{15} - \zeta_{15}^{4} + 5 \zeta_{15}^{5} ) q^{32} -\zeta_{15}^{5} q^{34} + ( -3 + 3 \zeta_{15}^{2} - 6 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{35} + ( \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{37} + ( -2 - \zeta_{15} + 3 \zeta_{15}^{2} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} - 3 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{38} + ( -3 + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{40} + ( -7 + 7 \zeta_{15} - \zeta_{15}^{4} - 7 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{41} + ( 2 - 2 \zeta_{15} - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} + 5 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{43} + ( -3 + 3 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - 5 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{44} + ( 3 - 3 \zeta_{15}^{2} - 4 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{46} + ( \zeta_{15} - \zeta_{15}^{5} + \zeta_{15}^{6} ) q^{47} + 2 \zeta_{15}^{4} q^{49} + ( -3 \zeta_{15} - 6 \zeta_{15}^{4} - 3 \zeta_{15}^{7} ) q^{50} + ( -2 - \zeta_{15} - 2 \zeta_{15}^{3} + 2 \zeta_{15}^{4} + \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{52} + ( -1 + \zeta_{15}^{2} - 9 \zeta_{15}^{6} + \zeta_{15}^{7} ) q^{53} + ( 5 - 2 \zeta_{15}^{2} - \zeta_{15}^{3} + 5 \zeta_{15}^{6} - 2 \zeta_{15}^{7} ) q^{55} + ( 6 - 6 \zeta_{15} - 6 \zeta_{15}^{2} + 6 \zeta_{15}^{3} - 6 \zeta_{15}^{4} + 3 \zeta_{15}^{5} - 6 \zeta_{15}^{7} ) q^{56} + ( -4 + 4 \zeta_{15} + 6 \zeta_{15}^{4} - 4 \zeta_{15}^{5} + 6 \zeta_{15}^{7} ) q^{58} + ( 7 - 7 \zeta_{15}^{4} + 7 \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{59} + ( 3 + 3 \zeta_{15} - 6 \zeta_{15}^{2} + 3 \zeta_{15}^{3} - 3 \zeta_{15}^{4} + 6 \zeta_{15}^{6} - 3 \zeta_{15}^{7} ) q^{61} + ( 3 \zeta_{15}^{2} + 2 \zeta_{15}^{3} + 2 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{62} + ( -2 + 2 \zeta_{15}^{2} + \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{64} + ( -1 + \zeta_{15} + \zeta_{15}^{2} - \zeta_{15}^{3} + \zeta_{15}^{4} - 4 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{65} + ( 4 - 9 \zeta_{15} - 9 \zeta_{15}^{4} + 4 \zeta_{15}^{5} ) q^{67} + ( 1 - 2 \zeta_{15} + \zeta_{15}^{5} - \zeta_{15}^{7} ) q^{68} + ( -3 + 3 \zeta_{15}^{2} - 3 \zeta_{15}^{3} + 3 \zeta_{15}^{4} - 3 \zeta_{15}^{5} - 3 \zeta_{15}^{6} + 3 \zeta_{15}^{7} ) q^{70} + ( -9 - 9 \zeta_{15}^{3} ) q^{71} + ( 2 \zeta_{15}^{2} - 2 \zeta_{15}^{3} - 2 \zeta_{15}^{6} + 2 \zeta_{15}^{7} ) q^{73} + ( 5 - 2 \zeta_{15} + 5 \zeta_{15}^{3} - 5 \zeta_{15}^{4} + 2 \zeta_{15}^{5} + 3 \zeta_{15}^{6} - 5 \zeta_{15}^{7} ) q^{74} + ( 7 - 4 \zeta_{15} - 4 \zeta_{15}^{4} + 7 \zeta_{15}^{5} ) q^{76} + ( -6 + 9 \zeta_{15} + 6 \zeta_{15}^{4} - 6 \zeta_{15}^{5} + 12 \zeta_{15}^{7} ) q^{77} + ( -7 \zeta_{15} + 4 \zeta_{15}^{2} + 4 \zeta_{15}^{5} - 7 \zeta_{15}^{6} ) q^{79} + ( 3 + 3 \zeta_{15}^{6} ) q^{80} + ( -9 + 8 \zeta_{15}^{2} - 9 \zeta_{15}^{3} - 8 \zeta_{15}^{6} + 8 \zeta_{15}^{7} ) q^{82} + ( -9 + 6 \zeta_{15} + 3 \zeta_{15}^{2} - 9 \zeta_{15}^{3} + 9 \zeta_{15}^{4} - 3 \zeta_{15}^{6} + 9 \zeta_{15}^{7} ) q^{83} + ( -3 + 3 \zeta_{15} + \zeta_{15}^{4} - 3 \zeta_{15}^{5} + \zeta_{15}^{7} ) q^{85} + ( 3 - 2 \zeta_{15} + 3 \zeta_{15}^{5} - 3 \zeta_{15}^{7} ) q^{86} + ( 2 - 2 \zeta_{15} + 3 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{4} - 4 \zeta_{15}^{5} - 2 \zeta_{15}^{7} ) q^{88} + ( 7 - 4 \zeta_{15}^{2} + 4 \zeta_{15}^{3} - 4 \zeta_{15}^{7} ) q^{89} + ( 6 + 15 \zeta_{15}^{3} + 6 \zeta_{15}^{6} ) q^{91} + ( 1 - 5 \zeta_{15}^{2} + \zeta_{15}^{3} - \zeta_{15}^{4} + \zeta_{15}^{5} + \zeta_{15}^{6} - \zeta_{15}^{7} ) q^{92} -\zeta_{15}^{4} q^{94} + ( -7 + 7 \zeta_{15}^{4} - 7 \zeta_{15}^{5} - 4 \zeta_{15}^{7} ) q^{95} + ( 6 \zeta_{15} - 13 \zeta_{15}^{2} - 13 \zeta_{15}^{5} + 6 \zeta_{15}^{6} ) q^{97} + ( 2 - 2 \zeta_{15}^{2} + 2 \zeta_{15}^{3} - 2 \zeta_{15}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 3q^{2} - 3q^{4} + q^{5} + 3q^{7} - 10q^{8} + O(q^{10}) \) \( 8q + 3q^{2} - 3q^{4} + q^{5} + 3q^{7} - 10q^{8} + 4q^{10} - 9q^{11} + 9q^{13} - 6q^{14} - 9q^{16} + 4q^{17} - 20q^{19} - 3q^{20} + 8q^{22} + 4q^{23} + 6q^{25} + 16q^{26} - 12q^{28} + 10q^{29} - 8q^{31} + 18q^{32} + 4q^{34} - 6q^{35} - 6q^{37} - 10q^{40} - 23q^{41} - 16q^{43} - 4q^{44} + 26q^{46} + 3q^{47} + 2q^{49} - 12q^{50} - 7q^{52} + 12q^{53} + 28q^{55} + 20q^{59} - 3q^{61} - 2q^{62} - 14q^{64} + 14q^{65} - 2q^{67} + q^{68} + 9q^{70} - 54q^{71} + 12q^{73} + 4q^{74} + 20q^{76} + 3q^{77} - 5q^{79} + 18q^{80} - 22q^{82} - 21q^{83} - 7q^{85} + 7q^{86} + 25q^{88} + 40q^{89} + 6q^{91} - 7q^{92} - q^{94} - 25q^{95} + 33q^{97} + 8q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-\zeta_{15}^{2} - \zeta_{15}^{7}\) \(-1 - \zeta_{15}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
−0.104528 + 0.994522i
−0.104528 0.994522i
0.669131 0.743145i
0.913545 0.406737i
−0.978148 + 0.207912i
−0.978148 0.207912i
0.669131 + 0.743145i
0.913545 + 0.406737i
1.47815 0.658114i 0 0.413545 0.459289i −0.348943 0.155360i 0 −2.93444 0.623735i −0.690983 + 2.12663i 0 −0.618034
190.1 1.47815 + 0.658114i 0 0.413545 + 0.459289i −0.348943 + 0.155360i 0 −2.93444 + 0.623735i −0.690983 2.12663i 0 −0.618034
379.1 0.604528 + 0.128496i 0 −1.47815 0.658114i 2.56082 0.544320i 0 −0.313585 2.98357i −1.80902 1.31433i 0 1.61803
433.1 −0.169131 + 1.60917i 0 −0.604528 0.128496i 0.0399263 + 0.379874i 0 2.00739 2.22943i −0.690983 + 2.12663i 0 −0.618034
460.1 −0.413545 0.459289i 0 0.169131 1.60917i −1.75181 + 1.94558i 0 2.74064 1.22021i −1.80902 + 1.31433i 0 1.61803
676.1 −0.413545 + 0.459289i 0 0.169131 + 1.60917i −1.75181 1.94558i 0 2.74064 + 1.22021i −1.80902 1.31433i 0 1.61803
757.1 0.604528 0.128496i 0 −1.47815 + 0.658114i 2.56082 + 0.544320i 0 −0.313585 + 2.98357i −1.80902 + 1.31433i 0 1.61803
784.1 −0.169131 1.60917i 0 −0.604528 + 0.128496i 0.0399263 0.379874i 0 2.00739 + 2.22943i −0.690983 2.12663i 0 −0.618034
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 784.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 891.2.n.d 8
3.b odd 2 1 891.2.n.a 8
9.c even 3 1 33.2.e.a 4
9.c even 3 1 inner 891.2.n.d 8
9.d odd 6 1 99.2.f.b 4
9.d odd 6 1 891.2.n.a 8
11.c even 5 1 inner 891.2.n.d 8
33.h odd 10 1 891.2.n.a 8
36.f odd 6 1 528.2.y.f 4
45.j even 6 1 825.2.n.f 4
45.k odd 12 2 825.2.bx.b 8
99.h odd 6 1 363.2.e.j 4
99.m even 15 1 33.2.e.a 4
99.m even 15 1 363.2.a.h 2
99.m even 15 2 363.2.e.h 4
99.m even 15 1 inner 891.2.n.d 8
99.n odd 30 1 99.2.f.b 4
99.n odd 30 1 891.2.n.a 8
99.n odd 30 1 1089.2.a.m 2
99.o odd 30 1 363.2.a.e 2
99.o odd 30 2 363.2.e.c 4
99.o odd 30 1 363.2.e.j 4
99.p even 30 1 1089.2.a.s 2
396.be odd 30 1 528.2.y.f 4
396.be odd 30 1 5808.2.a.bl 2
396.bf even 30 1 5808.2.a.bm 2
495.bl even 30 1 825.2.n.f 4
495.bl even 30 1 9075.2.a.x 2
495.br odd 30 1 9075.2.a.bv 2
495.bt odd 60 2 825.2.bx.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.e.a 4 9.c even 3 1
33.2.e.a 4 99.m even 15 1
99.2.f.b 4 9.d odd 6 1
99.2.f.b 4 99.n odd 30 1
363.2.a.e 2 99.o odd 30 1
363.2.a.h 2 99.m even 15 1
363.2.e.c 4 99.o odd 30 2
363.2.e.h 4 99.m even 15 2
363.2.e.j 4 99.h odd 6 1
363.2.e.j 4 99.o odd 30 1
528.2.y.f 4 36.f odd 6 1
528.2.y.f 4 396.be odd 30 1
825.2.n.f 4 45.j even 6 1
825.2.n.f 4 495.bl even 30 1
825.2.bx.b 8 45.k odd 12 2
825.2.bx.b 8 495.bt odd 60 2
891.2.n.a 8 3.b odd 2 1
891.2.n.a 8 9.d odd 6 1
891.2.n.a 8 33.h odd 10 1
891.2.n.a 8 99.n odd 30 1
891.2.n.d 8 1.a even 1 1 trivial
891.2.n.d 8 9.c even 3 1 inner
891.2.n.d 8 11.c even 5 1 inner
891.2.n.d 8 99.m even 15 1 inner
1089.2.a.m 2 99.n odd 30 1
1089.2.a.s 2 99.p even 30 1
5808.2.a.bl 2 396.be odd 30 1
5808.2.a.bm 2 396.bf even 30 1
9075.2.a.x 2 495.bl even 30 1
9075.2.a.bv 2 495.br odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 3 T_{2}^{7} + 5 T_{2}^{6} - 8 T_{2}^{5} + 9 T_{2}^{4} - 2 T_{2}^{3} - 2 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(891, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 7 T^{2} - 6 T^{3} + 3 T^{4} + 6 T^{5} + 14 T^{6} - 48 T^{7} + 113 T^{8} - 96 T^{9} + 56 T^{10} + 48 T^{11} + 48 T^{12} - 192 T^{13} + 448 T^{14} - 384 T^{15} + 256 T^{16} \)
$3$ 1
$5$ \( 1 - T + 21 T^{3} - 46 T^{4} + 6 T^{5} + 80 T^{6} - 436 T^{7} + 291 T^{8} - 2180 T^{9} + 2000 T^{10} + 750 T^{11} - 28750 T^{12} + 65625 T^{13} - 78125 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 3 T + 7 T^{2} - 36 T^{3} + 108 T^{4} - 219 T^{5} + 854 T^{6} - 2628 T^{7} + 5483 T^{8} - 18396 T^{9} + 41846 T^{10} - 75117 T^{11} + 259308 T^{12} - 605052 T^{13} + 823543 T^{14} - 2470629 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 9 T + 40 T^{2} + 171 T^{3} + 669 T^{4} + 1881 T^{5} + 4840 T^{6} + 11979 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 9 T + 63 T^{2} - 392 T^{3} + 2148 T^{4} - 10637 T^{5} + 46966 T^{6} - 191706 T^{7} + 723193 T^{8} - 2492178 T^{9} + 7937254 T^{10} - 23369489 T^{11} + 61349028 T^{12} - 145546856 T^{13} + 304088967 T^{14} - 564736653 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 - 2 T - 13 T^{2} - 20 T^{3} + 341 T^{4} - 340 T^{5} - 3757 T^{6} - 9826 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 1330 T^{5} + 7581 T^{6} + 68590 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 - 2 T - 23 T^{2} + 38 T^{3} + 108 T^{4} + 874 T^{5} - 12167 T^{6} - 24334 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( 1 - 10 T + 69 T^{2} + 90 T^{3} - 2860 T^{4} + 24420 T^{5} - 22309 T^{6} - 509200 T^{7} + 5518359 T^{8} - 14766800 T^{9} - 18761869 T^{10} + 595579380 T^{11} - 2022823660 T^{12} + 1846003410 T^{13} + 41042809149 T^{14} - 172498763090 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 8 T + 61 T^{2} - 68 T^{3} - 1534 T^{4} - 17236 T^{5} - 7051 T^{6} + 306886 T^{7} + 4196107 T^{8} + 9513466 T^{9} - 6776011 T^{10} - 513477676 T^{11} - 1416681214 T^{12} - 1946782268 T^{13} + 54137724541 T^{14} + 220100912888 T^{15} + 852891037441 T^{16} \)
$37$ \( ( 1 + 3 T - 18 T^{2} + 155 T^{3} + 1851 T^{4} + 5735 T^{5} - 24642 T^{6} + 151959 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 + 23 T + 321 T^{2} + 2862 T^{3} + 17186 T^{4} + 56439 T^{5} - 110206 T^{6} - 2943754 T^{7} - 24697863 T^{8} - 120693914 T^{9} - 185256286 T^{10} + 3889832319 T^{11} + 48563528546 T^{12} + 331580447262 T^{13} + 1524783461361 T^{14} + 4479348299263 T^{15} + 7984925229121 T^{16} \)
$43$ \( ( 1 + 8 T - 33 T^{2} + 88 T^{3} + 4808 T^{4} + 3784 T^{5} - 61017 T^{6} + 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 3 T + 52 T^{2} + 39 T^{3} - 132 T^{4} + 12876 T^{5} - 90616 T^{6} + 250602 T^{7} - 2713027 T^{8} + 11778294 T^{9} - 200170744 T^{10} + 1336824948 T^{11} - 644117892 T^{12} + 8944455273 T^{13} + 560519197108 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 6 T + 23 T^{2} + 120 T^{3} - 1319 T^{4} + 6360 T^{5} + 64607 T^{6} - 893262 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 20 T + 269 T^{2} - 1560 T^{3} + 2710 T^{4} + 53340 T^{5} + 63041 T^{6} - 7076090 T^{7} + 90705499 T^{8} - 417489310 T^{9} + 219445721 T^{10} + 10954915860 T^{11} + 32838048310 T^{12} - 1115281906440 T^{13} + 11346563549429 T^{14} - 49773029696380 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 3 T + 16 T^{2} - 903 T^{3} - 6174 T^{4} - 5586 T^{5} + 95984 T^{6} + 3140316 T^{7} + 2204147 T^{8} + 191559276 T^{9} + 357156464 T^{10} - 1267915866 T^{11} - 85484222334 T^{12} - 762670459803 T^{13} + 824325989776 T^{14} + 9428228508063 T^{15} + 191707312997281 T^{16} \)
$67$ \( ( 1 + T - 32 T^{2} - 101 T^{3} - 3467 T^{4} - 6767 T^{5} - 143648 T^{6} + 300763 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 27 T + 253 T^{2} + 819 T^{3} + 100 T^{4} + 58149 T^{5} + 1275373 T^{6} + 9663597 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 6 T - 57 T^{2} + 130 T^{3} + 4761 T^{4} + 9490 T^{5} - 303753 T^{6} - 2334102 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( 1 + 5 T + 19 T^{2} - 1400 T^{3} - 13360 T^{4} - 37435 T^{5} + 244766 T^{6} + 8996620 T^{7} + 41542459 T^{8} + 710732980 T^{9} + 1527584606 T^{10} - 18456914965 T^{11} - 520373082160 T^{12} - 4307878958600 T^{13} + 4618661654899 T^{14} + 96019544930795 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 21 T + 353 T^{2} + 3678 T^{3} + 40158 T^{4} + 397893 T^{5} + 4806046 T^{6} + 50181804 T^{7} + 508233323 T^{8} + 4165089732 T^{9} + 33108850894 T^{10} + 227510044791 T^{11} + 1905831254718 T^{12} + 14487791484954 T^{13} + 115409951799257 T^{14} + 569857070782167 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 - 10 T + 183 T^{2} - 890 T^{3} + 7921 T^{4} )^{4} \)
$97$ \( 1 - 33 T + 552 T^{2} - 4411 T^{3} - 2442 T^{4} + 479666 T^{5} - 5140616 T^{6} + 22640112 T^{7} - 41156417 T^{8} + 2196090864 T^{9} - 48368055944 T^{10} + 437778207218 T^{11} - 216188504202 T^{12} - 37878757873627 T^{13} + 459800546720808 T^{14} - 2666343387777729 T^{15} + 7837433594376961 T^{16} \)
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