Properties

Label 475.2.l.a
Level $475$
Weight $2$
Character orbit 475.l
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.939693 + 0.342020i
−0.173648 + 0.984808i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
1.93969 1.62760i −0.613341 0.223238i 0.766044 4.34445i 0 −1.55303 + 0.565258i 0.766044 1.32683i −3.05303 5.28801i −1.97178 1.65452i 0
176.1 0.826352 0.300767i −0.0923963 + 0.524005i −0.939693 + 0.788496i 0 0.0812519 + 0.460802i −0.939693 1.62760i −1.41875 + 2.45734i 2.55303 + 0.929228i 0
226.1 0.233956 1.32683i 2.20574 + 1.85083i 0.173648 + 0.0632028i 0 2.97178 2.49362i 0.173648 + 0.300767i 1.47178 2.54920i 0.918748 + 5.21048i 0
251.1 0.826352 + 0.300767i −0.0923963 0.524005i −0.939693 0.788496i 0 0.0812519 0.460802i −0.939693 + 1.62760i −1.41875 2.45734i 2.55303 0.929228i 0
301.1 1.93969 + 1.62760i −0.613341 + 0.223238i 0.766044 + 4.34445i 0 −1.55303 0.565258i 0.766044 + 1.32683i −3.05303 + 5.28801i −1.97178 + 1.65452i 0
351.1 0.233956 + 1.32683i 2.20574 1.85083i 0.173648 0.0632028i 0 2.97178 + 2.49362i 0.173648 0.300767i 1.47178 + 2.54920i 0.918748 5.21048i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.a 6
5.b even 2 1 19.2.e.a 6
5.c odd 4 2 475.2.u.a 12
15.d odd 2 1 171.2.u.c 6
19.e even 9 1 inner 475.2.l.a 6
19.e even 9 1 9025.2.a.bd 3
19.f odd 18 1 9025.2.a.x 3
20.d odd 2 1 304.2.u.b 6
35.c odd 2 1 931.2.w.a 6
35.i odd 6 1 931.2.v.a 6
35.i odd 6 1 931.2.x.b 6
35.j even 6 1 931.2.v.b 6
35.j even 6 1 931.2.x.a 6
95.d odd 2 1 361.2.e.h 6
95.h odd 6 1 361.2.e.a 6
95.h odd 6 1 361.2.e.b 6
95.i even 6 1 361.2.e.f 6
95.i even 6 1 361.2.e.g 6
95.o odd 18 1 361.2.a.h 3
95.o odd 18 2 361.2.c.h 6
95.o odd 18 1 361.2.e.a 6
95.o odd 18 1 361.2.e.b 6
95.o odd 18 1 361.2.e.h 6
95.p even 18 1 19.2.e.a 6
95.p even 18 1 361.2.a.g 3
95.p even 18 2 361.2.c.i 6
95.p even 18 1 361.2.e.f 6
95.p even 18 1 361.2.e.g 6
95.q odd 36 2 475.2.u.a 12
285.bd odd 18 1 171.2.u.c 6
285.bd odd 18 1 3249.2.a.z 3
285.bf even 18 1 3249.2.a.s 3
380.ba odd 18 1 304.2.u.b 6
380.ba odd 18 1 5776.2.a.br 3
380.bb even 18 1 5776.2.a.bi 3
665.cv odd 18 1 931.2.w.a 6
665.cw odd 18 1 931.2.x.b 6
665.db even 18 1 931.2.x.a 6
665.dc even 18 1 931.2.v.b 6
665.df odd 18 1 931.2.v.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 5.b even 2 1
19.2.e.a 6 95.p even 18 1
171.2.u.c 6 15.d odd 2 1
171.2.u.c 6 285.bd odd 18 1
304.2.u.b 6 20.d odd 2 1
304.2.u.b 6 380.ba odd 18 1
361.2.a.g 3 95.p even 18 1
361.2.a.h 3 95.o odd 18 1
361.2.c.h 6 95.o odd 18 2
361.2.c.i 6 95.p even 18 2
361.2.e.a 6 95.h odd 6 1
361.2.e.a 6 95.o odd 18 1
361.2.e.b 6 95.h odd 6 1
361.2.e.b 6 95.o odd 18 1
361.2.e.f 6 95.i even 6 1
361.2.e.f 6 95.p even 18 1
361.2.e.g 6 95.i even 6 1
361.2.e.g 6 95.p even 18 1
361.2.e.h 6 95.d odd 2 1
361.2.e.h 6 95.o odd 18 1
475.2.l.a 6 1.a even 1 1 trivial
475.2.l.a 6 19.e even 9 1 inner
475.2.u.a 12 5.c odd 4 2
475.2.u.a 12 95.q odd 36 2
931.2.v.a 6 35.i odd 6 1
931.2.v.a 6 665.df odd 18 1
931.2.v.b 6 35.j even 6 1
931.2.v.b 6 665.dc even 18 1
931.2.w.a 6 35.c odd 2 1
931.2.w.a 6 665.cv odd 18 1
931.2.x.a 6 35.j even 6 1
931.2.x.a 6 665.db even 18 1
931.2.x.b 6 35.i odd 6 1
931.2.x.b 6 665.cw odd 18 1
3249.2.a.s 3 285.bf even 18 1
3249.2.a.z 3 285.bd odd 18 1
5776.2.a.bi 3 380.bb even 18 1
5776.2.a.br 3 380.ba odd 18 1
9025.2.a.x 3 19.f odd 18 1
9025.2.a.bd 3 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 6 T_{2}^{5} + 18 T_{2}^{4} - 30 T_{2}^{3} + 36 T_{2}^{2} - 27 T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 27 T + 36 T^{2} - 30 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} \)
$3$ \( 1 + 3 T + 6 T^{2} + 8 T^{3} + 3 T^{4} - 3 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6} \)
$11$ \( 81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6} \)
$13$ \( 1369 - 222 T - 114 T^{2} - 26 T^{3} + 24 T^{4} - 3 T^{5} + T^{6} \)
$17$ \( 9 + 36 T^{2} - 30 T^{3} + 3 T^{5} + T^{6} \)
$19$ \( 6859 + 4332 T + 1482 T^{2} + 385 T^{3} + 78 T^{4} + 12 T^{5} + T^{6} \)
$23$ \( 576 + 864 T + 576 T^{2} + 192 T^{3} + 36 T^{4} + 6 T^{5} + T^{6} \)
$29$ \( 12321 - 1998 T - 477 T^{2} - 57 T^{3} + 36 T^{4} + 3 T^{5} + T^{6} \)
$31$ \( 2809 + 318 T + 513 T^{2} - 160 T^{3} + 75 T^{4} - 9 T^{5} + T^{6} \)
$37$ \( ( 17 - 21 T + T^{3} )^{2} \)
$41$ \( 12321 - 8991 T + 3411 T^{2} - 672 T^{3} + 162 T^{4} - 21 T^{5} + T^{6} \)
$43$ \( 26569 - 5379 T - 663 T^{2} - 8 T^{3} + 60 T^{4} - 3 T^{5} + T^{6} \)
$47$ \( 9 - 18 T^{2} - 24 T^{3} + 54 T^{4} - 3 T^{5} + T^{6} \)
$53$ \( 2601 + 1377 T + 387 T^{2} + 84 T^{3} - 3 T^{5} + T^{6} \)
$59$ \( 71289 + 19224 T + 3006 T^{2} - 159 T^{3} + 18 T^{4} - 12 T^{5} + T^{6} \)
$61$ \( 32761 - 2172 T + 984 T^{2} - 37 T^{3} + 24 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( 179776 - 86496 T + 20928 T^{2} - 2528 T^{3} + 348 T^{4} - 30 T^{5} + T^{6} \)
$71$ \( 788544 - 31968 T + 8352 T^{2} - 1536 T^{3} - 36 T^{4} + 6 T^{5} + T^{6} \)
$73$ \( 4096 + 3072 T + 768 T^{2} - 512 T^{3} + 96 T^{4} - 12 T^{5} + T^{6} \)
$79$ \( 654481 + 242700 T + 51663 T^{2} + 7487 T^{3} + 708 T^{4} + 39 T^{5} + T^{6} \)
$83$ \( 210681 - 86751 T + 35721 T^{2} - 918 T^{3} + 189 T^{4} + T^{6} \)
$89$ \( 3249 - 1539 T + 522 T^{2} - 300 T^{3} + 54 T^{4} + 12 T^{5} + T^{6} \)
$97$ \( 16129 + 20574 T + 9522 T^{2} + 1855 T^{3} + 234 T^{4} + 18 T^{5} + T^{6} \)
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