Properties

Label 189.2.f.b
Level 189
Weight 2
Character orbit 189.f
Analytic conductor 1.509
Analytic rank 0
Dimension 6
CM no
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.50917259820\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{5} -\zeta_{18}^{3} q^{7} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{5} -\zeta_{18}^{3} q^{7} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{10} + ( \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( 1 - 2 \zeta_{18} + 4 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{13} + ( 1 - \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{14} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} + ( -2 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} ) q^{17} + ( -1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{19} + ( -\zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{20} + ( -3 + 4 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{22} + ( 4 + \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{23} + ( 2 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - 6 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{26} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{28} + ( -4 \zeta_{18} + 5 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( 1 - 3 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 6 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{31} + ( -3 \zeta_{18} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{32} + ( \zeta_{18} + \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{34} + ( -1 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{35} + ( -1 + 6 \zeta_{18} + 6 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{38} + ( 3 - 2 \zeta_{18} - 2 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{40} + ( \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{41} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{43} + ( -5 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{44} + ( -4 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{46} + ( 3 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{47} + ( -1 + \zeta_{18}^{3} ) q^{49} + ( -2 + 5 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{50} + ( 5 \zeta_{18} - 10 \zeta_{18}^{2} + 7 \zeta_{18}^{3} - 10 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{52} + ( -2 + \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{53} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{55} + ( 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{56} + ( 3 - 6 \zeta_{18} + 9 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{58} + ( -1 - 5 \zeta_{18} + \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{59} + ( -3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{61} + ( 10 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{62} + ( 4 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{64} + ( -5 \zeta_{18} - \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( 4 - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{67} + ( 2 - 3 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{68} + ( 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{70} + ( -3 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( -7 + 5 \zeta_{18} + 5 \zeta_{18}^{2} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{73} + ( \zeta_{18}^{2} - 10 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{74} + ( -5 + 3 \zeta_{18} - 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} ) q^{76} + ( 2 - \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} ) q^{77} + ( -3 \zeta_{18} + 7 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{79} + ( 5 - 3 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{80} -3 q^{82} + ( -\zeta_{18} - 4 \zeta_{18}^{2} - 6 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{83} + ( -3 - 2 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{85} + ( 2 + \zeta_{18} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{86} + ( -7 \zeta_{18} + 8 \zeta_{18}^{2} - 9 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{88} + ( -4 - 4 \zeta_{18} - 4 \zeta_{18}^{2} + 7 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{89} + ( -1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{91} + ( 2 \zeta_{18} - 8 \zeta_{18}^{2} + \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{92} + ( 6 + 2 \zeta_{18} - \zeta_{18}^{2} - 6 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{94} + ( -4 - \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{95} + ( -8 \zeta_{18} + 7 \zeta_{18}^{2} + \zeta_{18}^{3} + 7 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{97} + ( -1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{2} - 3q^{4} + 3q^{5} - 3q^{7} - 12q^{8} + O(q^{10}) \) \( 6q + 3q^{2} - 3q^{4} + 3q^{5} - 3q^{7} - 12q^{8} + 6q^{11} + 3q^{13} + 3q^{14} - 3q^{16} - 12q^{17} - 6q^{19} - 6q^{20} - 9q^{22} + 12q^{23} + 6q^{25} + 6q^{26} + 6q^{28} + 9q^{29} + 3q^{31} - 9q^{34} - 6q^{35} - 6q^{37} + 6q^{38} + 9q^{40} + 3q^{43} - 30q^{44} + 3q^{47} - 3q^{49} - 6q^{50} + 21q^{52} - 12q^{53} + 6q^{56} + 9q^{58} - 3q^{59} - 6q^{61} + 60q^{62} + 24q^{64} + 15q^{65} + 12q^{67} + 6q^{68} - 18q^{71} - 42q^{73} - 30q^{74} - 15q^{76} + 6q^{77} + 21q^{79} + 30q^{80} - 18q^{82} - 18q^{83} - 9q^{85} + 6q^{86} - 27q^{88} - 24q^{89} - 6q^{91} + 3q^{92} + 18q^{94} - 12q^{95} + 3q^{97} - 6q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1 + \zeta_{18}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i 0.673648 1.16679i 0 −0.500000 0.866025i −2.83750 0 −1.18479
64.2 0.673648 + 1.16679i 0 0.0923963 0.160035i 1.26604 2.19285i 0 −0.500000 0.866025i 2.94356 0 3.41147
64.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i −0.439693 + 0.761570i 0 −0.500000 0.866025i −6.10607 0 −2.22668
127.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i 0.673648 + 1.16679i 0 −0.500000 + 0.866025i −2.83750 0 −1.18479
127.2 0.673648 1.16679i 0 0.0923963 + 0.160035i 1.26604 + 2.19285i 0 −0.500000 + 0.866025i 2.94356 0 3.41147
127.3 1.26604 2.19285i 0 −2.20574 3.82045i −0.439693 0.761570i 0 −0.500000 + 0.866025i −6.10607 0 −2.22668
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.f.b 6
3.b odd 2 1 63.2.f.a 6
4.b odd 2 1 3024.2.r.k 6
7.b odd 2 1 1323.2.f.d 6
7.c even 3 1 1323.2.g.d 6
7.c even 3 1 1323.2.h.c 6
7.d odd 6 1 1323.2.g.e 6
7.d odd 6 1 1323.2.h.b 6
9.c even 3 1 inner 189.2.f.b 6
9.c even 3 1 567.2.a.c 3
9.d odd 6 1 63.2.f.a 6
9.d odd 6 1 567.2.a.h 3
12.b even 2 1 1008.2.r.h 6
21.c even 2 1 441.2.f.c 6
21.g even 6 1 441.2.g.b 6
21.g even 6 1 441.2.h.e 6
21.h odd 6 1 441.2.g.c 6
21.h odd 6 1 441.2.h.d 6
36.f odd 6 1 3024.2.r.k 6
36.f odd 6 1 9072.2.a.bs 3
36.h even 6 1 1008.2.r.h 6
36.h even 6 1 9072.2.a.ca 3
63.g even 3 1 1323.2.h.c 6
63.h even 3 1 1323.2.g.d 6
63.i even 6 1 441.2.g.b 6
63.j odd 6 1 441.2.g.c 6
63.k odd 6 1 1323.2.h.b 6
63.l odd 6 1 1323.2.f.d 6
63.l odd 6 1 3969.2.a.l 3
63.n odd 6 1 441.2.h.d 6
63.o even 6 1 441.2.f.c 6
63.o even 6 1 3969.2.a.q 3
63.s even 6 1 441.2.h.e 6
63.t odd 6 1 1323.2.g.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 3.b odd 2 1
63.2.f.a 6 9.d odd 6 1
189.2.f.b 6 1.a even 1 1 trivial
189.2.f.b 6 9.c even 3 1 inner
441.2.f.c 6 21.c even 2 1
441.2.f.c 6 63.o even 6 1
441.2.g.b 6 21.g even 6 1
441.2.g.b 6 63.i even 6 1
441.2.g.c 6 21.h odd 6 1
441.2.g.c 6 63.j odd 6 1
441.2.h.d 6 21.h odd 6 1
441.2.h.d 6 63.n odd 6 1
441.2.h.e 6 21.g even 6 1
441.2.h.e 6 63.s even 6 1
567.2.a.c 3 9.c even 3 1
567.2.a.h 3 9.d odd 6 1
1008.2.r.h 6 12.b even 2 1
1008.2.r.h 6 36.h even 6 1
1323.2.f.d 6 7.b odd 2 1
1323.2.f.d 6 63.l odd 6 1
1323.2.g.d 6 7.c even 3 1
1323.2.g.d 6 63.h even 3 1
1323.2.g.e 6 7.d odd 6 1
1323.2.g.e 6 63.t odd 6 1
1323.2.h.b 6 7.d odd 6 1
1323.2.h.b 6 63.k odd 6 1
1323.2.h.c 6 7.c even 3 1
1323.2.h.c 6 63.g even 3 1
3024.2.r.k 6 4.b odd 2 1
3024.2.r.k 6 36.f odd 6 1
3969.2.a.l 3 63.l odd 6 1
3969.2.a.q 3 63.o even 6 1
9072.2.a.bs 3 36.f odd 6 1
9072.2.a.ca 3 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 6 T_{2}^{3} + 9 T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(189, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 3 T^{2} - 3 T^{4} + 6 T^{5} - 11 T^{6} + 12 T^{7} - 12 T^{8} + 48 T^{10} - 96 T^{11} + 64 T^{12} \)
$3$ 1
$5$ \( 1 - 3 T - 6 T^{2} + 9 T^{3} + 69 T^{4} - 30 T^{5} - 371 T^{6} - 150 T^{7} + 1725 T^{8} + 1125 T^{9} - 3750 T^{10} - 9375 T^{11} + 15625 T^{12} \)
$7$ \( ( 1 + T + T^{2} )^{3} \)
$11$ \( 1 - 6 T - 6 T^{2} + 18 T^{3} + 492 T^{4} - 852 T^{5} - 2873 T^{6} - 9372 T^{7} + 59532 T^{8} + 23958 T^{9} - 87846 T^{10} - 966306 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 3 T + 3 T^{2} - 76 T^{3} + 45 T^{4} + 135 T^{5} + 3246 T^{6} + 1755 T^{7} + 7605 T^{8} - 166972 T^{9} + 85683 T^{10} - 1113879 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 + 6 T + 60 T^{2} + 207 T^{3} + 1020 T^{4} + 1734 T^{5} + 4913 T^{6} )^{2} \)
$19$ \( ( 1 + 3 T + 51 T^{2} + 97 T^{3} + 969 T^{4} + 1083 T^{5} + 6859 T^{6} )^{2} \)
$23$ \( 1 - 12 T + 48 T^{2} - 54 T^{3} + 420 T^{4} - 6060 T^{5} + 37591 T^{6} - 139380 T^{7} + 222180 T^{8} - 657018 T^{9} + 13432368 T^{10} - 77236116 T^{11} + 148035889 T^{12} \)
$29$ \( 1 - 9 T + 30 T^{2} - 81 T^{3} - 579 T^{4} + 9414 T^{5} - 59051 T^{6} + 273006 T^{7} - 486939 T^{8} - 1975509 T^{9} + 21218430 T^{10} - 184600341 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 3 T - 6 T^{2} - 319 T^{3} + 171 T^{4} + 1962 T^{5} + 62727 T^{6} + 60822 T^{7} + 164331 T^{8} - 9503329 T^{9} - 5541126 T^{10} - 85887453 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 3 T + 33 T^{2} - 101 T^{3} + 1221 T^{4} + 4107 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 114 T^{2} + 18 T^{3} + 8322 T^{4} - 1026 T^{5} - 394913 T^{6} - 42066 T^{7} + 13989282 T^{8} + 1240578 T^{9} - 322136754 T^{10} + 4750104241 T^{12} \)
$43$ \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 9063 T^{4} - 5670 T^{5} - 441093 T^{6} - 243810 T^{7} + 16757487 T^{8} + 11846543 T^{9} - 389743314 T^{10} - 441025329 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 3 T - 78 T^{2} + 405 T^{3} + 2481 T^{4} - 11064 T^{5} - 57089 T^{6} - 520008 T^{7} + 5480529 T^{8} + 42048315 T^{9} - 380615118 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \)
$53$ \( ( 1 + 6 T + 150 T^{2} + 639 T^{3} + 7950 T^{4} + 16854 T^{5} + 148877 T^{6} )^{2} \)
$59$ \( 1 + 3 T - 96 T^{2} - 495 T^{3} + 3615 T^{4} + 15798 T^{5} - 107021 T^{6} + 932082 T^{7} + 12583815 T^{8} - 101662605 T^{9} - 1163266656 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 + 6 T - 132 T^{2} - 418 T^{3} + 13698 T^{4} + 19134 T^{5} - 893289 T^{6} + 1167174 T^{7} + 50970258 T^{8} - 94878058 T^{9} - 1827651012 T^{10} + 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 12 T - 78 T^{2} + 518 T^{3} + 15318 T^{4} - 50094 T^{5} - 815637 T^{6} - 3356298 T^{7} + 68762502 T^{8} + 155795234 T^{9} - 1571787438 T^{10} - 16201501284 T^{11} + 90458382169 T^{12} \)
$71$ \( ( 1 + 9 T + 159 T^{2} + 1305 T^{3} + 11289 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( ( 1 + 21 T + 303 T^{2} + 2797 T^{3} + 22119 T^{4} + 111909 T^{5} + 389017 T^{6} )^{2} \)
$79$ \( 1 - 21 T + 84 T^{2} - 499 T^{3} + 25767 T^{4} - 195678 T^{5} + 408327 T^{6} - 15458562 T^{7} + 160811847 T^{8} - 246026461 T^{9} + 3271806804 T^{10} - 64618184379 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 18 T + 30 T^{2} - 702 T^{3} + 8088 T^{4} + 126648 T^{5} + 719359 T^{6} + 10511784 T^{7} + 55718232 T^{8} - 401394474 T^{9} + 1423749630 T^{10} + 70902731574 T^{11} + 326940373369 T^{12} \)
$89$ \( ( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 18156 T^{4} + 95052 T^{5} + 704969 T^{6} )^{2} \)
$97$ \( 1 - 3 T - 114 T^{2} + 149 T^{3} + 2421 T^{4} + 11502 T^{5} + 340233 T^{6} + 1115694 T^{7} + 22779189 T^{8} + 135988277 T^{9} - 10092338034 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \)
show more
show less