Properties

Label 38.2.e.a
Level $38$
Weight $2$
Character orbit 38.e
Analytic conductor $0.303$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.303431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(\zeta_{18}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.939693 0.342020i
−0.173648 0.984808i
−0.173648 + 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.766044 0.642788i
−0.939693 + 0.342020i −0.326352 + 1.85083i 0.766044 0.642788i 1.53209 + 1.28558i −0.326352 1.85083i −2.53209 4.38571i −0.500000 + 0.866025i −0.500000 0.181985i −1.87939 0.684040i
9.1 0.173648 + 0.984808i 0.266044 0.223238i −0.939693 + 0.342020i −1.87939 0.684040i 0.266044 + 0.223238i 0.879385 1.52314i −0.500000 0.866025i −0.500000 + 2.83564i 0.347296 1.96962i
17.1 0.173648 0.984808i 0.266044 + 0.223238i −0.939693 0.342020i −1.87939 + 0.684040i 0.266044 0.223238i 0.879385 + 1.52314i −0.500000 + 0.866025i −0.500000 2.83564i 0.347296 + 1.96962i
23.1 −0.939693 0.342020i −0.326352 1.85083i 0.766044 + 0.642788i 1.53209 1.28558i −0.326352 + 1.85083i −2.53209 + 4.38571i −0.500000 0.866025i −0.500000 + 0.181985i −1.87939 + 0.684040i
25.1 0.766044 0.642788i −1.43969 0.524005i 0.173648 0.984808i 0.347296 + 1.96962i −1.43969 + 0.524005i −1.34730 + 2.33359i −0.500000 0.866025i −0.500000 0.419550i 1.53209 + 1.28558i
35.1 0.766044 + 0.642788i −1.43969 + 0.524005i 0.173648 + 0.984808i 0.347296 1.96962i −1.43969 0.524005i −1.34730 2.33359i −0.500000 + 0.866025i −0.500000 + 0.419550i 1.53209 1.28558i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.1
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 38.2.e.a 6
3.b odd 2 1 342.2.u.c 6
4.b odd 2 1 304.2.u.c 6
5.b even 2 1 950.2.l.d 6
5.c odd 4 2 950.2.u.b 12
19.b odd 2 1 722.2.e.k 6
19.c even 3 1 722.2.e.b 6
19.c even 3 1 722.2.e.m 6
19.d odd 6 1 722.2.e.a 6
19.d odd 6 1 722.2.e.l 6
19.e even 9 1 inner 38.2.e.a 6
19.e even 9 1 722.2.a.l 3
19.e even 9 2 722.2.c.k 6
19.e even 9 1 722.2.e.b 6
19.e even 9 1 722.2.e.m 6
19.f odd 18 1 722.2.a.k 3
19.f odd 18 2 722.2.c.l 6
19.f odd 18 1 722.2.e.a 6
19.f odd 18 1 722.2.e.k 6
19.f odd 18 1 722.2.e.l 6
57.j even 18 1 6498.2.a.bq 3
57.l odd 18 1 342.2.u.c 6
57.l odd 18 1 6498.2.a.bl 3
76.k even 18 1 5776.2.a.bo 3
76.l odd 18 1 304.2.u.c 6
76.l odd 18 1 5776.2.a.bn 3
95.p even 18 1 950.2.l.d 6
95.q odd 36 2 950.2.u.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.2.e.a 6 1.a even 1 1 trivial
38.2.e.a 6 19.e even 9 1 inner
304.2.u.c 6 4.b odd 2 1
304.2.u.c 6 76.l odd 18 1
342.2.u.c 6 3.b odd 2 1
342.2.u.c 6 57.l odd 18 1
722.2.a.k 3 19.f odd 18 1
722.2.a.l 3 19.e even 9 1
722.2.c.k 6 19.e even 9 2
722.2.c.l 6 19.f odd 18 2
722.2.e.a 6 19.d odd 6 1
722.2.e.a 6 19.f odd 18 1
722.2.e.b 6 19.c even 3 1
722.2.e.b 6 19.e even 9 1
722.2.e.k 6 19.b odd 2 1
722.2.e.k 6 19.f odd 18 1
722.2.e.l 6 19.d odd 6 1
722.2.e.l 6 19.f odd 18 1
722.2.e.m 6 19.c even 3 1
722.2.e.m 6 19.e even 9 1
950.2.l.d 6 5.b even 2 1
950.2.l.d 6 95.p even 18 1
950.2.u.b 12 5.c odd 4 2
950.2.u.b 12 95.q odd 36 2
5776.2.a.bn 3 76.l odd 18 1
5776.2.a.bo 3 76.k even 18 1
6498.2.a.bl 3 57.l odd 18 1
6498.2.a.bq 3 57.j even 18 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( 1 + 3 T + 6 T^{2} + 8 T^{3} + 21 T^{4} + 51 T^{5} + 109 T^{6} + 153 T^{7} + 189 T^{8} + 216 T^{9} + 486 T^{10} + 729 T^{11} + 729 T^{12} \)
$5$ \( 1 - 22 T^{3} + 359 T^{6} - 2750 T^{9} + 15625 T^{12} \)
$7$ \( 1 + 6 T + 15 T^{2} + 6 T^{3} - 66 T^{4} - 210 T^{5} - 565 T^{6} - 1470 T^{7} - 3234 T^{8} + 2058 T^{9} + 36015 T^{10} + 100842 T^{11} + 117649 T^{12} \)
$11$ \( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 462 T^{7} + 26862 T^{8} - 13310 T^{9} + 966306 T^{11} + 1771561 T^{12} \)
$13$ \( 1 - 12 T + 48 T^{2} + 14 T^{3} - 684 T^{4} + 1854 T^{5} - 2757 T^{6} + 24102 T^{7} - 115596 T^{8} + 30758 T^{9} + 1370928 T^{10} - 4455516 T^{11} + 4826809 T^{12} \)
$17$ \( 1 + 12 T + 108 T^{2} + 702 T^{3} + 3744 T^{4} + 17904 T^{5} + 75187 T^{6} + 304368 T^{7} + 1082016 T^{8} + 3448926 T^{9} + 9020268 T^{10} + 17038284 T^{11} + 24137569 T^{12} \)
$19$ \( 1 - 18 T + 162 T^{2} - 883 T^{3} + 3078 T^{4} - 6498 T^{5} + 6859 T^{6} \)
$23$ \( 1 + 12 T + 60 T^{2} + 152 T^{3} - 84 T^{4} - 4734 T^{5} - 34459 T^{6} - 108882 T^{7} - 44436 T^{8} + 1849384 T^{9} + 16790460 T^{10} + 77236116 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 18 T + 144 T^{2} + 580 T^{3} + 1440 T^{4} + 7542 T^{5} + 56483 T^{6} + 218718 T^{7} + 1211040 T^{8} + 14145620 T^{9} + 101848464 T^{10} + 369200682 T^{11} + 594823321 T^{12} \)
$31$ \( 1 - 6 T - 33 T^{2} + 346 T^{3} + 342 T^{4} - 6318 T^{5} + 21795 T^{6} - 195858 T^{7} + 328662 T^{8} + 10307686 T^{9} - 30476193 T^{10} - 171774906 T^{11} + 887503681 T^{12} \)
$37$ \( ( 1 + 6 T + 87 T^{2} + 308 T^{3} + 3219 T^{4} + 8214 T^{5} + 50653 T^{6} )^{2} \)
$41$ \( 1 - 3 T + 6 T^{2} - 8 T^{3} - 111 T^{4} + 9711 T^{5} - 83311 T^{6} + 398151 T^{7} - 186591 T^{8} - 551368 T^{9} + 16954566 T^{10} - 347568603 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 6 T + 42 T^{2} - 116 T^{3} + 1044 T^{4} - 6768 T^{5} + 34173 T^{6} - 291024 T^{7} + 1930356 T^{8} - 9222812 T^{9} + 143589642 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 - 30 T + 372 T^{2} - 2086 T^{3} - 2544 T^{4} + 153360 T^{5} - 1460047 T^{6} + 7207920 T^{7} - 5619696 T^{8} - 216574778 T^{9} + 1815241332 T^{10} - 6880350210 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 - 24 T + 276 T^{2} - 2152 T^{3} + 18132 T^{4} - 179262 T^{5} + 1523855 T^{6} - 9500886 T^{7} + 50932788 T^{8} - 320383304 T^{9} + 2177772756 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 + 3 T + 54 T^{2} + 378 T^{3} + 5823 T^{4} + 40179 T^{5} + 259255 T^{6} + 2370561 T^{7} + 20269863 T^{8} + 77633262 T^{9} + 654337494 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \)
$61$ \( 1 - 6 T - 12 T^{2} - 586 T^{3} - 1008 T^{4} + 15732 T^{5} + 419787 T^{6} + 959652 T^{7} - 3750768 T^{8} - 133010866 T^{9} - 166150092 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 + 9 T + 162 T^{2} + 1854 T^{3} + 22437 T^{4} + 180369 T^{5} + 1838609 T^{6} + 12084723 T^{7} + 100719693 T^{8} + 557614602 T^{9} + 3264481602 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 18 T + 216 T^{2} + 2516 T^{3} + 26244 T^{4} + 221256 T^{5} + 1869317 T^{6} + 15709176 T^{7} + 132296004 T^{8} + 900504076 T^{9} + 5488923096 T^{10} + 32476128318 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 30 T + 279 T^{2} - 231 T^{3} - 13599 T^{4} + 110865 T^{5} + 2572046 T^{6} + 8093145 T^{7} - 72469071 T^{8} - 89862927 T^{9} + 7923109239 T^{10} + 62192147790 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 6 T - 48 T^{2} + 1430 T^{3} - 8820 T^{4} - 56934 T^{5} + 1359837 T^{6} - 4497786 T^{7} - 55045620 T^{8} + 705045770 T^{9} - 1869603888 T^{10} - 18462338394 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 2490996 T^{7} + 173520132 T^{8} - 319057146 T^{9} - 8827247706 T^{10} + 23634243858 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 36 T^{2} - 486 T^{3} - 1548 T^{4} - 34956 T^{5} + 586603 T^{6} - 3111084 T^{7} - 12261708 T^{8} - 342614934 T^{9} + 2258720676 T^{10} + 496981290961 T^{12} \)
$97$ \( 1 - 3 T + 6 T^{2} - 8 T^{3} + 585 T^{4} - 29097 T^{5} - 744183 T^{6} - 2822409 T^{7} + 5504265 T^{8} - 7301384 T^{9} + 531175686 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \)
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