Properties

Label 126.2.h.c
Level $126$
Weight $2$
Character orbit 126.h
Analytic conductor $1.006$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.00611506547\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
Defining polynomial: \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( 1 - \beta_{5} ) q^{3} + ( -1 - \beta_{4} ) q^{4} + ( 2 + \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{6} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + q^{8} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{2} + ( 1 - \beta_{5} ) q^{3} + ( -1 - \beta_{4} ) q^{4} + ( 2 + \beta_{2} + \beta_{3} ) q^{5} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{6} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + q^{8} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{9} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{10} + ( 2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} ) q^{12} + ( \beta_{1} + \beta_{4} - \beta_{5} ) q^{13} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{14} + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{15} + \beta_{4} q^{16} + ( -2 - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{17} + ( 2 - 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{18} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{20} + ( -3 - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{21} + ( -2 + 3 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{22} + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{23} + ( 1 - \beta_{5} ) q^{24} + ( 1 + \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{26} + ( -2 + 2 \beta_{1} - 4 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{27} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{28} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{29} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{30} + ( -5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + \beta_{5} ) q^{31} + ( -1 - \beta_{4} ) q^{32} + ( -4 \beta_{1} + \beta_{2} + 4 \beta_{3} - 7 \beta_{4} - \beta_{5} ) q^{33} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} - 4 \beta_{5} ) q^{34} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{35} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{36} + ( -3 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{37} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{38} + ( -2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{39} + ( 2 + \beta_{2} + \beta_{3} ) q^{40} + ( -1 + 2 \beta_{1} - \beta_{3} + 5 \beta_{4} ) q^{41} + ( 1 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{42} + ( 5 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} ) q^{44} + ( -7 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{45} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{46} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{47} + ( -\beta_{1} - \beta_{2} + \beta_{5} ) q^{48} + ( -1 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} ) q^{49} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - 4 \beta_{5} ) q^{50} + ( 10 + 2 \beta_{2} + 4 \beta_{3} + 8 \beta_{4} ) q^{51} + ( 1 + \beta_{2} + \beta_{3} ) q^{52} + ( -1 + 5 \beta_{1} - \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{53} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{54} + ( 4 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{55} + ( -\beta_{3} + \beta_{4} - \beta_{5} ) q^{56} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{57} + ( \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{58} + ( -5 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} + 4 \beta_{5} ) q^{59} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{60} + ( 3 - 8 \beta_{1} + 3 \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{61} + ( 6 - \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{62} + ( 3 + \beta_{1} - \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{63} + q^{64} + ( 1 + \beta_{1} + \beta_{3} + 4 \beta_{4} - 3 \beta_{5} ) q^{65} + ( 4 + 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{66} + ( 3 + \beta_{1} + 4 \beta_{2} + \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{67} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} ) q^{68} + ( 3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + 5 \beta_{4} - 4 \beta_{5} ) q^{69} + ( -2 - 3 \beta_{1} - 4 \beta_{2} - \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{70} + ( 4 - 3 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{71} + ( -2 - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{72} + ( 3 - 4 \beta_{1} + 3 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{73} + ( 1 + 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} ) q^{74} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} + 7 \beta_{4} - 5 \beta_{5} ) q^{75} + ( 1 - 2 \beta_{1} + \beta_{3} ) q^{76} + ( -9 + \beta_{1} + 5 \beta_{2} + \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{77} + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{78} + ( 4 - \beta_{1} + 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} ) q^{79} + ( \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{80} + ( 6 + 5 \beta_{1} + \beta_{2} - 2 \beta_{3} + 8 \beta_{4} - \beta_{5} ) q^{81} + ( -4 - \beta_{1} - \beta_{3} - 4 \beta_{4} ) q^{82} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{83} + ( 2 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{84} + ( -6 + 2 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} + 10 \beta_{5} ) q^{85} + ( -6 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{86} + ( -4 + 3 \beta_{1} + \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{87} + ( 2 - 2 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{88} + ( -4 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{89} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{90} + ( -2 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{91} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{92} + ( -4 + 4 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} ) q^{93} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{94} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{95} + ( -1 + \beta_{1} + \beta_{2} ) q^{96} + ( -10 - 2 \beta_{2} - 10 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -2 + \beta_{1} + 5 \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{98} + ( -7 + 7 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{2} + 4q^{3} - 3q^{4} + 10q^{5} - 2q^{6} - 2q^{7} + 6q^{8} - 4q^{9} + O(q^{10}) \) \( 6q - 3q^{2} + 4q^{3} - 3q^{4} + 10q^{5} - 2q^{6} - 2q^{7} + 6q^{8} - 4q^{9} - 5q^{10} + 2q^{11} - 2q^{12} - 2q^{13} - 2q^{14} - 2q^{15} - 3q^{16} - 4q^{17} + 8q^{18} - 3q^{19} - 5q^{20} - 7q^{21} - q^{22} + 14q^{23} + 4q^{24} + 4q^{25} - 2q^{26} + 7q^{27} + 4q^{28} - 5q^{29} - 5q^{30} - 14q^{31} - 3q^{32} - 4q^{33} - 4q^{34} - 19q^{35} - 4q^{36} - 9q^{37} + 6q^{38} - 3q^{39} + 10q^{40} - 12q^{41} + 2q^{42} + 18q^{43} - q^{44} - 31q^{45} - 7q^{46} + 3q^{47} - 2q^{48} - 2q^{50} + 26q^{51} + 4q^{52} + 9q^{53} + q^{54} + 14q^{55} - 2q^{56} + 2q^{57} + 10q^{58} + 4q^{59} + 7q^{60} + 4q^{61} + 28q^{62} + 28q^{63} + 6q^{64} - 12q^{65} + 23q^{66} + 5q^{67} + 8q^{68} - q^{69} + 2q^{70} + 14q^{71} - 4q^{72} - 25q^{73} + 18q^{74} - 25q^{75} - 3q^{76} - 35q^{77} + 9q^{78} + 7q^{79} - 5q^{80} + 32q^{81} - 12q^{82} + 8q^{83} + 5q^{84} + 14q^{85} - 36q^{86} - 20q^{87} + 2q^{88} - 9q^{89} + 29q^{90} + 4q^{91} - 7q^{92} - 3q^{93} + 3q^{94} + 2q^{95} - 2q^{96} - 28q^{97} - 12q^{98} - 41q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - \nu^{4} + 5 \nu^{3} + \nu^{2} + 6 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 5 \nu^{3} + 2 \nu^{2} - 3 \nu \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 6 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 33 \nu - 9 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 3 \beta_{2} - 6 \beta_{1} + 6\)
\(\nu^{5}\)\(=\)\(-3 \beta_{5} - 2 \beta_{4} - 11 \beta_{3} - 6 \beta_{2} + 8 \beta_{1} + 7\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(\beta_{4}\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 + 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
−0.500000 + 0.866025i −0.349814 + 1.69636i −0.500000 0.866025i 3.69963 −1.29418 1.15113i −1.40545 + 2.24159i 1.00000 −2.75526 1.18682i −1.84981 + 3.20397i
67.2 −0.500000 + 0.866025i 0.619562 1.61745i −0.500000 0.866025i 1.76088 1.09097 + 1.34528i −1.85185 1.88962i 1.00000 −2.23229 2.00422i −0.880438 + 1.52496i
67.3 −0.500000 + 0.866025i 1.73025 0.0789082i −0.500000 0.866025i −0.460505 −0.796790 + 1.53790i 2.25729 + 1.38008i 1.00000 2.98755 0.273062i 0.230252 0.398809i
79.1 −0.500000 0.866025i −0.349814 1.69636i −0.500000 + 0.866025i 3.69963 −1.29418 + 1.15113i −1.40545 2.24159i 1.00000 −2.75526 + 1.18682i −1.84981 3.20397i
79.2 −0.500000 0.866025i 0.619562 + 1.61745i −0.500000 + 0.866025i 1.76088 1.09097 1.34528i −1.85185 + 1.88962i 1.00000 −2.23229 + 2.00422i −0.880438 1.52496i
79.3 −0.500000 0.866025i 1.73025 + 0.0789082i −0.500000 + 0.866025i −0.460505 −0.796790 1.53790i 2.25729 1.38008i 1.00000 2.98755 + 0.273062i 0.230252 + 0.398809i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.2.h.c yes 6
3.b odd 2 1 378.2.h.d 6
4.b odd 2 1 1008.2.t.g 6
7.b odd 2 1 882.2.h.o 6
7.c even 3 1 126.2.e.d 6
7.c even 3 1 882.2.f.l 6
7.d odd 6 1 882.2.e.p 6
7.d odd 6 1 882.2.f.m 6
9.c even 3 1 126.2.e.d 6
9.c even 3 1 1134.2.g.k 6
9.d odd 6 1 378.2.e.c 6
9.d odd 6 1 1134.2.g.n 6
12.b even 2 1 3024.2.t.g 6
21.c even 2 1 2646.2.h.p 6
21.g even 6 1 2646.2.e.o 6
21.g even 6 1 2646.2.f.n 6
21.h odd 6 1 378.2.e.c 6
21.h odd 6 1 2646.2.f.o 6
28.g odd 6 1 1008.2.q.h 6
36.f odd 6 1 1008.2.q.h 6
36.h even 6 1 3024.2.q.h 6
63.g even 3 1 inner 126.2.h.c yes 6
63.g even 3 1 7938.2.a.cb 3
63.h even 3 1 882.2.f.l 6
63.h even 3 1 1134.2.g.k 6
63.i even 6 1 2646.2.f.n 6
63.j odd 6 1 1134.2.g.n 6
63.j odd 6 1 2646.2.f.o 6
63.k odd 6 1 882.2.h.o 6
63.k odd 6 1 7938.2.a.by 3
63.l odd 6 1 882.2.e.p 6
63.n odd 6 1 378.2.h.d 6
63.n odd 6 1 7938.2.a.bu 3
63.o even 6 1 2646.2.e.o 6
63.s even 6 1 2646.2.h.p 6
63.s even 6 1 7938.2.a.bx 3
63.t odd 6 1 882.2.f.m 6
84.n even 6 1 3024.2.q.h 6
252.o even 6 1 3024.2.t.g 6
252.bl odd 6 1 1008.2.t.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 7.c even 3 1
126.2.e.d 6 9.c even 3 1
126.2.h.c yes 6 1.a even 1 1 trivial
126.2.h.c yes 6 63.g even 3 1 inner
378.2.e.c 6 9.d odd 6 1
378.2.e.c 6 21.h odd 6 1
378.2.h.d 6 3.b odd 2 1
378.2.h.d 6 63.n odd 6 1
882.2.e.p 6 7.d odd 6 1
882.2.e.p 6 63.l odd 6 1
882.2.f.l 6 7.c even 3 1
882.2.f.l 6 63.h even 3 1
882.2.f.m 6 7.d odd 6 1
882.2.f.m 6 63.t odd 6 1
882.2.h.o 6 7.b odd 2 1
882.2.h.o 6 63.k odd 6 1
1008.2.q.h 6 28.g odd 6 1
1008.2.q.h 6 36.f odd 6 1
1008.2.t.g 6 4.b odd 2 1
1008.2.t.g 6 252.bl odd 6 1
1134.2.g.k 6 9.c even 3 1
1134.2.g.k 6 63.h even 3 1
1134.2.g.n 6 9.d odd 6 1
1134.2.g.n 6 63.j odd 6 1
2646.2.e.o 6 21.g even 6 1
2646.2.e.o 6 63.o even 6 1
2646.2.f.n 6 21.g even 6 1
2646.2.f.n 6 63.i even 6 1
2646.2.f.o 6 21.h odd 6 1
2646.2.f.o 6 63.j odd 6 1
2646.2.h.p 6 21.c even 2 1
2646.2.h.p 6 63.s even 6 1
3024.2.q.h 6 36.h even 6 1
3024.2.q.h 6 84.n even 6 1
3024.2.t.g 6 12.b even 2 1
3024.2.t.g 6 252.o even 6 1
7938.2.a.bu 3 63.n odd 6 1
7938.2.a.bx 3 63.s even 6 1
7938.2.a.by 3 63.k odd 6 1
7938.2.a.cb 3 63.g even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 5 T_{5}^{2} + 4 T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( 27 - 36 T + 30 T^{2} - 21 T^{3} + 10 T^{4} - 4 T^{5} + T^{6} \)
$5$ \( ( 3 + 4 T - 5 T^{2} + T^{3} )^{2} \)
$7$ \( 343 + 98 T + 14 T^{2} - 19 T^{3} + 2 T^{4} + 2 T^{5} + T^{6} \)
$11$ \( ( -33 - 26 T - T^{2} + T^{3} )^{2} \)
$13$ \( 9 + 9 T + 15 T^{2} + 7 T^{4} + 2 T^{5} + T^{6} \)
$17$ \( 28224 + 7392 T + 2608 T^{2} + 160 T^{3} + 60 T^{4} + 4 T^{5} + T^{6} \)
$19$ \( 49 + 42 T + 57 T^{2} - 4 T^{3} + 15 T^{4} + 3 T^{5} + T^{6} \)
$23$ \( ( 3 + 4 T - 7 T^{2} + T^{3} )^{2} \)
$29$ \( 1089 - 1056 T + 859 T^{2} - 226 T^{3} + 57 T^{4} + 5 T^{5} + T^{6} \)
$31$ \( 729 - 1215 T + 2403 T^{2} + 684 T^{3} + 151 T^{4} + 14 T^{5} + T^{6} \)
$37$ \( 5329 + 657 T + 738 T^{2} + 65 T^{3} + 90 T^{4} + 9 T^{5} + T^{6} \)
$41$ \( 729 + 1053 T + 1197 T^{2} + 414 T^{3} + 105 T^{4} + 12 T^{5} + T^{6} \)
$43$ \( 1 - 81 T + 6543 T^{2} - 1456 T^{3} + 243 T^{4} - 18 T^{5} + T^{6} \)
$47$ \( 729 + 648 T + 495 T^{2} + 126 T^{3} + 33 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 81 + 378 T + 1683 T^{2} + 396 T^{3} + 123 T^{4} - 9 T^{5} + T^{6} \)
$59$ \( 31329 + 17877 T + 9493 T^{2} + 758 T^{3} + 117 T^{4} - 4 T^{5} + T^{6} \)
$61$ \( 514089 - 96795 T + 21093 T^{2} - 894 T^{3} + 151 T^{4} - 4 T^{5} + T^{6} \)
$67$ \( 22201 - 8642 T + 4109 T^{2} - 8 T^{3} + 83 T^{4} - 5 T^{5} + T^{6} \)
$71$ \( ( 99 - 50 T - 7 T^{2} + T^{3} )^{2} \)
$73$ \( 2401 - 7448 T + 24329 T^{2} + 3898 T^{3} + 473 T^{4} + 25 T^{5} + T^{6} \)
$79$ \( 594441 - 111024 T + 26133 T^{2} - 534 T^{3} + 193 T^{4} - 7 T^{5} + T^{6} \)
$83$ \( 8649 - 465 T + 769 T^{2} - 146 T^{3} + 69 T^{4} - 8 T^{5} + T^{6} \)
$89$ \( 3969 + 378 T + 603 T^{2} + 72 T^{3} + 87 T^{4} + 9 T^{5} + T^{6} \)
$97$ \( 287296 + 126496 T + 40688 T^{2} + 5536 T^{3} + 548 T^{4} + 28 T^{5} + T^{6} \)
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