Properties

Label 441.2.h.e
Level $441$
Weight $2$
Character orbit 441.h
Analytic conductor $3.521$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.52140272914\)
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: \(\Z[a_1, a_2, a_3]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(-1 + \zeta_{18}\) \(-1 + \zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.766044 + 0.642788i
−0.879385 −1.70574 0.300767i −1.22668 0.673648 1.16679i 1.50000 + 0.264490i 0 2.83750 2.81908 + 1.02606i −0.592396 + 1.02606i
214.2 1.34730 1.11334 1.32683i −0.184793 1.26604 2.19285i 1.50000 1.78763i 0 −2.94356 −0.520945 2.95442i 1.70574 2.95442i
214.3 2.53209 0.592396 + 1.62760i 4.41147 −0.439693 + 0.761570i 1.50000 + 4.12122i 0 6.10607 −2.29813 + 1.92836i −1.11334 + 1.92836i
373.1 −0.879385 −1.70574 + 0.300767i −1.22668 0.673648 + 1.16679i 1.50000 0.264490i 0 2.83750 2.81908 1.02606i −0.592396 1.02606i
373.2 1.34730 1.11334 + 1.32683i −0.184793 1.26604 + 2.19285i 1.50000 + 1.78763i 0 −2.94356 −0.520945 + 2.95442i 1.70574 + 2.95442i
373.3 2.53209 0.592396 1.62760i 4.41147 −0.439693 0.761570i 1.50000 4.12122i 0 6.10607 −2.29813 1.92836i −1.11334 1.92836i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.h.e 6
3.b odd 2 1 1323.2.h.b 6
7.b odd 2 1 441.2.h.d 6
7.c even 3 1 441.2.f.c 6
7.c even 3 1 441.2.g.b 6
7.d odd 6 1 63.2.f.a 6
7.d odd 6 1 441.2.g.c 6
9.c even 3 1 441.2.g.b 6
9.d odd 6 1 1323.2.g.e 6
21.c even 2 1 1323.2.h.c 6
21.g even 6 1 189.2.f.b 6
21.g even 6 1 1323.2.g.d 6
21.h odd 6 1 1323.2.f.d 6
21.h odd 6 1 1323.2.g.e 6
28.f even 6 1 1008.2.r.h 6
63.g even 3 1 441.2.f.c 6
63.h even 3 1 inner 441.2.h.e 6
63.h even 3 1 3969.2.a.q 3
63.i even 6 1 567.2.a.c 3
63.i even 6 1 1323.2.h.c 6
63.j odd 6 1 1323.2.h.b 6
63.j odd 6 1 3969.2.a.l 3
63.k odd 6 1 63.2.f.a 6
63.l odd 6 1 441.2.g.c 6
63.n odd 6 1 1323.2.f.d 6
63.o even 6 1 1323.2.g.d 6
63.s even 6 1 189.2.f.b 6
63.t odd 6 1 441.2.h.d 6
63.t odd 6 1 567.2.a.h 3
84.j odd 6 1 3024.2.r.k 6
252.n even 6 1 1008.2.r.h 6
252.r odd 6 1 9072.2.a.bs 3
252.bj even 6 1 9072.2.a.ca 3
252.bn odd 6 1 3024.2.r.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 7.d odd 6 1
63.2.f.a 6 63.k odd 6 1
189.2.f.b 6 21.g even 6 1
189.2.f.b 6 63.s even 6 1
441.2.f.c 6 7.c even 3 1
441.2.f.c 6 63.g even 3 1
441.2.g.b 6 7.c even 3 1
441.2.g.b 6 9.c even 3 1
441.2.g.c 6 7.d odd 6 1
441.2.g.c 6 63.l odd 6 1
441.2.h.d 6 7.b odd 2 1
441.2.h.d 6 63.t odd 6 1
441.2.h.e 6 1.a even 1 1 trivial
441.2.h.e 6 63.h even 3 1 inner
567.2.a.c 3 63.i even 6 1
567.2.a.h 3 63.t odd 6 1
1008.2.r.h 6 28.f even 6 1
1008.2.r.h 6 252.n even 6 1
1323.2.f.d 6 21.h odd 6 1
1323.2.f.d 6 63.n odd 6 1
1323.2.g.d 6 21.g even 6 1
1323.2.g.d 6 63.o even 6 1
1323.2.g.e 6 9.d odd 6 1
1323.2.g.e 6 21.h odd 6 1
1323.2.h.b 6 3.b odd 2 1
1323.2.h.b 6 63.j odd 6 1
1323.2.h.c 6 21.c even 2 1
1323.2.h.c 6 63.i even 6 1
3024.2.r.k 6 84.j odd 6 1
3024.2.r.k 6 252.bn odd 6 1
3969.2.a.l 3 63.j odd 6 1
3969.2.a.q 3 63.h even 3 1
9072.2.a.bs 3 252.r odd 6 1
9072.2.a.ca 3 252.bj even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{3} - 3 T_{2}^{2} + 3 \)
\( T_{5}^{6} - 3 T_{5}^{5} + 9 T_{5}^{4} - 6 T_{5}^{3} + 9 T_{5}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 3 - 3 T^{2} + T^{3} )^{2} \)
$3$ \( 27 + 9 T^{3} + T^{6} \)
$5$ \( 9 + 9 T^{2} - 6 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( 9 + 27 T + 63 T^{2} + 48 T^{3} + 27 T^{4} + 6 T^{5} + T^{6} \)
$13$ \( 11449 + 3531 T + 1410 T^{2} + 115 T^{3} + 42 T^{4} + 3 T^{5} + T^{6} \)
$17$ \( 9 - 27 T + 63 T^{2} - 48 T^{3} + 27 T^{4} - 6 T^{5} + T^{6} \)
$19$ \( 289 + 102 T + 87 T^{2} + 16 T^{3} + 15 T^{4} + 3 T^{5} + T^{6} \)
$23$ \( 9 - 81 T + 765 T^{2} + 330 T^{3} + 117 T^{4} + 12 T^{5} + T^{6} \)
$29$ \( 110889 + 11988 T + 4293 T^{2} + 342 T^{3} + 117 T^{4} + 9 T^{5} + T^{6} \)
$31$ \( ( 323 - 78 T - 3 T^{2} + T^{3} )^{2} \)
$37$ \( 104329 - 25194 T + 7053 T^{2} - 412 T^{3} + 87 T^{4} - 3 T^{5} + T^{6} \)
$41$ \( 81 + 81 T + 81 T^{2} + 18 T^{3} + 9 T^{4} + T^{6} \)
$43$ \( 1 + 6 T + 33 T^{2} + 20 T^{3} + 15 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( ( 51 - 54 T + 3 T^{2} + T^{3} )^{2} \)
$53$ \( 9 - 27 T + 63 T^{2} - 60 T^{3} + 45 T^{4} + 6 T^{5} + T^{6} \)
$59$ \( ( -51 - 72 T - 3 T^{2} + T^{3} )^{2} \)
$61$ \( ( -19 - 15 T + 6 T^{2} + T^{3} )^{2} \)
$67$ \( ( -17 + 21 T + 12 T^{2} + T^{3} )^{2} \)
$71$ \( ( -27 - 54 T - 9 T^{2} + T^{3} )^{2} \)
$73$ \( 72361 - 22596 T + 12705 T^{2} + 2302 T^{3} + 357 T^{4} + 21 T^{5} + T^{6} \)
$79$ \( ( 181 + 120 T + 21 T^{2} + T^{3} )^{2} \)
$83$ \( 81 + 405 T + 1863 T^{2} + 792 T^{3} + 279 T^{4} + 18 T^{5} + T^{6} \)
$89$ \( 660969 - 51219 T + 13725 T^{2} - 870 T^{3} + 207 T^{4} - 12 T^{5} + T^{6} \)
$97$ \( 104329 + 54264 T + 29193 T^{2} + 142 T^{3} + 177 T^{4} + 3 T^{5} + T^{6} \)
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