Properties

Label 136.2.n.c
Level $136$
Weight $2$
Character orbit 136.n
Analytic conductor $1.086$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 136.n (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.08596546749\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 28 x^{10} + 258 x^{8} + 880 x^{6} + 1033 x^{4} + 132 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{8} q^{5} + ( 1 - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{7} + ( \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + \beta_{8} q^{5} + ( 1 - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} ) q^{7} + ( \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{9} + ( -1 + \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{11} ) q^{11} + ( -2 \beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{9} - \beta_{11} ) q^{15} + ( \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{9} - \beta_{11} ) q^{19} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} + \beta_{11} ) q^{21} + ( -1 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} ) q^{23} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{25} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{27} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{29} + ( -3 - \beta_{2} - \beta_{3} - \beta_{4} - 3 \beta_{7} - \beta_{11} ) q^{31} + ( -2 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{33} + ( -2 + \beta_{3} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{10} + \beta_{11} ) q^{35} + ( -2 \beta_{1} + 2 \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{37} + ( 4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{8} ) q^{39} + ( -3 \beta_{1} - 2 \beta_{2} + 2 \beta_{5} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{11} ) q^{41} + ( -\beta_{2} - \beta_{3} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{43} + ( -5 + \beta_{1} + \beta_{2} + \beta_{5} - 5 \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{45} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{47} + ( 5 + 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 5 \beta_{6} - \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{49} + ( -4 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{51} + ( -\beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{9} + \beta_{11} ) q^{53} + ( 4 \beta_{1} + \beta_{3} - \beta_{5} + 5 \beta_{6} - 5 \beta_{7} + \beta_{10} - \beta_{11} ) q^{55} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{11} ) q^{57} + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{59} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{61} + ( 8 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 8 \beta_{7} + \beta_{8} ) q^{63} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{11} ) q^{65} + ( -4 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} + \beta_{8} + \beta_{9} ) q^{67} + ( 4 + \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{10} - \beta_{11} ) q^{69} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{6} + 3 \beta_{7} + \beta_{11} ) q^{71} + ( 2 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} ) q^{73} + ( 5 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - 5 \beta_{6} + 3 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{11} ) q^{75} + ( -3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + 10 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{11} ) q^{77} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{79} + ( -5 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 4 \beta_{6} - 4 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{81} + ( 4 + 4 \beta_{1} - \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{83} + ( 5 + \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + 3 \beta_{6} + 3 \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{85} + ( -5 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{9} + 3 \beta_{11} ) q^{87} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} + 4 \beta_{6} - 4 \beta_{7} + 3 \beta_{10} - 3 \beta_{11} ) q^{89} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + 4 \beta_{10} - 2 \beta_{11} ) q^{91} + ( -5 + 5 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{93} + ( 3 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - \beta_{8} + \beta_{11} ) q^{95} + ( -2 - 3 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{97} + ( -5 + 7 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 7 \beta_{6} - 5 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} - 8 q^{9} + O(q^{10}) \) \( 12 q + 4 q^{5} - 8 q^{9} - 12 q^{11} + 20 q^{15} + 4 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{25} + 24 q^{27} + 8 q^{29} - 32 q^{31} - 24 q^{33} - 32 q^{35} + 4 q^{37} - 8 q^{39} + 16 q^{41} + 8 q^{43} - 64 q^{45} + 44 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} + 16 q^{59} + 44 q^{61} + 100 q^{63} - 20 q^{65} - 40 q^{67} + 56 q^{69} + 32 q^{71} + 8 q^{73} + 92 q^{75} - 12 q^{77} - 8 q^{79} + 40 q^{83} + 40 q^{85} - 84 q^{87} - 40 q^{91} - 76 q^{93} + 28 q^{95} - 16 q^{97} - 68 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 28 x^{10} + 258 x^{8} + 880 x^{6} + 1033 x^{4} + 132 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -9 \nu^{11} - 250 \nu^{9} - 2274 \nu^{7} - 7596 \nu^{5} - 8901 \nu^{3} - 1658 \nu \)\()/272\)
\(\beta_{2}\)\(=\)\((\)\( -39 \nu^{11} + 17 \nu^{10} - 1089 \nu^{9} + 459 \nu^{8} - 9973 \nu^{7} + 3927 \nu^{6} - 33443 \nu^{5} + 11305 \nu^{4} - 37228 \nu^{3} + 10064 \nu^{2} - 3524 \nu + 612 \)\()/1088\)
\(\beta_{3}\)\(=\)\((\)\( 39 \nu^{11} - 25 \nu^{10} + 1089 \nu^{9} - 651 \nu^{8} + 9973 \nu^{7} - 5223 \nu^{6} + 33443 \nu^{5} - 12889 \nu^{4} + 37228 \nu^{3} - 8184 \nu^{2} + 2436 \nu - 756 \)\()/1088\)
\(\beta_{4}\)\(=\)\((\)\( 39 \nu^{11} + 17 \nu^{10} + 1089 \nu^{9} + 459 \nu^{8} + 9973 \nu^{7} + 3927 \nu^{6} + 33443 \nu^{5} + 11305 \nu^{4} + 37228 \nu^{3} + 10064 \nu^{2} + 3524 \nu + 612 \)\()/1088\)
\(\beta_{5}\)\(=\)\((\)\( -39 \nu^{11} - 25 \nu^{10} - 1089 \nu^{9} - 651 \nu^{8} - 9973 \nu^{7} - 5223 \nu^{6} - 33443 \nu^{5} - 12889 \nu^{4} - 37228 \nu^{3} - 8184 \nu^{2} - 2436 \nu - 756 \)\()/1088\)
\(\beta_{6}\)\(=\)\((\)\( 153 \nu^{11} - 39 \nu^{10} + 4267 \nu^{9} - 1089 \nu^{8} + 39015 \nu^{7} - 9973 \nu^{6} + 130713 \nu^{5} - 33443 \nu^{4} + 146744 \nu^{3} - 37228 \nu^{2} + 10132 \nu - 2436 \)\()/1088\)
\(\beta_{7}\)\(=\)\((\)\( -153 \nu^{11} - 39 \nu^{10} - 4267 \nu^{9} - 1089 \nu^{8} - 39015 \nu^{7} - 9973 \nu^{6} - 130713 \nu^{5} - 33443 \nu^{4} - 146744 \nu^{3} - 37228 \nu^{2} - 10132 \nu - 2436 \)\()/1088\)
\(\beta_{8}\)\(=\)\((\)\( 119 \nu^{11} - 27 \nu^{10} + 3315 \nu^{9} - 767 \nu^{8} + 30243 \nu^{7} - 7247 \nu^{6} + 100725 \nu^{5} - 25865 \nu^{4} + 110670 \nu^{3} - 31582 \nu^{2} + 3808 \nu - 2016 \)\()/544\)
\(\beta_{9}\)\(=\)\((\)\( -119 \nu^{11} - 27 \nu^{10} - 3315 \nu^{9} - 767 \nu^{8} - 30243 \nu^{7} - 7247 \nu^{6} - 100725 \nu^{5} - 25865 \nu^{4} - 110670 \nu^{3} - 31582 \nu^{2} - 3808 \nu - 2016 \)\()/544\)
\(\beta_{10}\)\(=\)\((\)\( 123 \nu^{11} - 41 \nu^{10} + 3445 \nu^{9} - 1137 \nu^{8} + 31741 \nu^{7} - 10297 \nu^{6} + 107943 \nu^{5} - 33839 \nu^{4} + 123568 \nu^{3} - 36486 \nu^{2} + 8300 \nu - 1384 \)\()/544\)
\(\beta_{11}\)\(=\)\((\)\( -123 \nu^{11} - 41 \nu^{10} - 3445 \nu^{9} - 1137 \nu^{8} - 31741 \nu^{7} - 10297 \nu^{6} - 107943 \nu^{5} - 33839 \nu^{4} - 123568 \nu^{3} - 36486 \nu^{2} - 8300 \nu - 1384 \)\()/544\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{11} + 2 \beta_{10} - 4 \beta_{7} - 4 \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 8\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} - 2 \beta_{6} - 13 \beta_{5} - 7 \beta_{4} + 13 \beta_{3} + 7 \beta_{2} - 4 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-20 \beta_{11} - 20 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} + 52 \beta_{7} + 52 \beta_{6} + 9 \beta_{5} + 17 \beta_{4} + 9 \beta_{3} + 17 \beta_{2} + 80\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{11} - 2 \beta_{10} + 6 \beta_{9} - 6 \beta_{8} - 42 \beta_{7} + 42 \beta_{6} + 155 \beta_{5} + 69 \beta_{4} - 155 \beta_{3} - 69 \beta_{2} + 64 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(218 \beta_{11} + 218 \beta_{10} + 84 \beta_{9} + 84 \beta_{8} - 612 \beta_{7} - 612 \beta_{6} - 107 \beta_{5} - 243 \beta_{4} - 107 \beta_{3} - 243 \beta_{2} - 884\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-52 \beta_{11} + 52 \beta_{10} - 132 \beta_{9} + 132 \beta_{8} + 670 \beta_{7} - 670 \beta_{6} - 1817 \beta_{5} - 767 \beta_{4} + 1817 \beta_{3} + 767 \beta_{2} - 964 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-2452 \beta_{11} - 2452 \beta_{10} - 998 \beta_{9} - 998 \beta_{8} + 7084 \beta_{7} + 7084 \beta_{6} + 1381 \beta_{5} + 3285 \beta_{4} + 1381 \beta_{3} + 3285 \beta_{2} + 10072\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(906 \beta_{11} - 906 \beta_{10} + 2214 \beta_{9} - 2214 \beta_{8} - 9690 \beta_{7} + 9690 \beta_{6} + 21307 \beta_{5} + 8869 \beta_{4} - 21307 \beta_{3} - 8869 \beta_{2} + 13760 \beta_{1}\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(27962 \beta_{11} + 27962 \beta_{10} + 11532 \beta_{9} + 11532 \beta_{8} - 82108 \beta_{7} - 82108 \beta_{6} - 17963 \beta_{5} - 43211 \beta_{4} - 17963 \beta_{3} - 43211 \beta_{2} - 116276\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-13716 \beta_{11} + 13716 \beta_{10} - 33212 \beta_{9} + 33212 \beta_{8} + 133350 \beta_{7} - 133350 \beta_{6} - 250913 \beta_{5} - 104063 \beta_{4} + 250913 \beta_{3} + 104063 \beta_{2} - 188772 \beta_{1}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
3.18938i
0.306239i
3.49562i
1.75800i
0.216105i
1.54190i
1.75800i
0.216105i
1.54190i
3.18938i
0.306239i
3.49562i
0 −2.25523 0.934148i 0 −1.35305 + 3.26655i 0 0.666590 + 1.60929i 0 2.09212 + 2.09212i 0
9.2 0 −0.216544 0.0896953i 0 1.33137 3.21420i 0 0.934059 + 2.25502i 0 −2.08247 2.08247i 0
9.3 0 2.47178 + 1.02384i 0 −0.392531 + 0.947653i 0 −0.893542 2.15720i 0 2.94010 + 2.94010i 0
25.1 0 −1.24310 + 3.00110i 0 0.194339 + 0.0804980i 0 −1.76317 + 0.730328i 0 −5.33998 5.33998i 0
25.2 0 0.152809 0.368914i 0 −1.09698 0.454383i 0 4.72436 1.95689i 0 2.00857 + 2.00857i 0
25.3 0 1.09029 2.63218i 0 3.31685 + 1.37389i 0 −3.66830 + 1.51946i 0 −3.61834 3.61834i 0
49.1 0 −1.24310 3.00110i 0 0.194339 0.0804980i 0 −1.76317 0.730328i 0 −5.33998 + 5.33998i 0
49.2 0 0.152809 + 0.368914i 0 −1.09698 + 0.454383i 0 4.72436 + 1.95689i 0 2.00857 2.00857i 0
49.3 0 1.09029 + 2.63218i 0 3.31685 1.37389i 0 −3.66830 1.51946i 0 −3.61834 + 3.61834i 0
121.1 0 −2.25523 + 0.934148i 0 −1.35305 3.26655i 0 0.666590 1.60929i 0 2.09212 2.09212i 0
121.2 0 −0.216544 + 0.0896953i 0 1.33137 + 3.21420i 0 0.934059 2.25502i 0 −2.08247 + 2.08247i 0
121.3 0 2.47178 1.02384i 0 −0.392531 0.947653i 0 −0.893542 + 2.15720i 0 2.94010 2.94010i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 136.2.n.c 12
3.b odd 2 1 1224.2.bq.c 12
4.b odd 2 1 272.2.v.f 12
17.d even 8 1 inner 136.2.n.c 12
17.e odd 16 2 2312.2.a.w 12
17.e odd 16 2 2312.2.b.n 12
51.g odd 8 1 1224.2.bq.c 12
68.g odd 8 1 272.2.v.f 12
68.i even 16 2 4624.2.a.bt 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.c 12 1.a even 1 1 trivial
136.2.n.c 12 17.d even 8 1 inner
272.2.v.f 12 4.b odd 2 1
272.2.v.f 12 68.g odd 8 1
1224.2.bq.c 12 3.b odd 2 1
1224.2.bq.c 12 51.g odd 8 1
2312.2.a.w 12 17.e odd 16 2
2312.2.b.n 12 17.e odd 16 2
4624.2.a.bt 12 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{12} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(136, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( 32 + 192 T + 304 T^{2} + 464 T^{3} + 3652 T^{4} + 96 T^{5} - 224 T^{6} + 40 T^{7} + 8 T^{8} - 8 T^{9} + 4 T^{10} + T^{12} \)
$5$ \( 128 - 896 T + 1120 T^{2} + 3232 T^{3} + 4228 T^{4} + 2624 T^{5} + 288 T^{6} - 328 T^{7} + 160 T^{8} - 56 T^{9} + 16 T^{10} - 4 T^{11} + T^{12} \)
$7$ \( 147968 + 95744 T + 62848 T^{2} + 49408 T^{3} + 25488 T^{4} + 9024 T^{5} + 3352 T^{6} + 632 T^{7} + 242 T^{8} - 12 T^{9} - 22 T^{10} + T^{12} \)
$11$ \( 16928 + 29440 T + 118608 T^{2} + 6640 T^{3} + 107044 T^{4} + 68416 T^{5} + 10088 T^{6} - 2592 T^{7} - 960 T^{8} - 32 T^{9} + 48 T^{10} + 12 T^{11} + T^{12} \)
$13$ \( 262144 + 483328 T^{2} + 265280 T^{4} + 44448 T^{6} + 3060 T^{8} + 92 T^{10} + T^{12} \)
$17$ \( 24137569 - 5679428 T - 3340840 T^{2} + 884340 T^{3} + 161551 T^{4} - 32912 T^{5} - 5136 T^{6} - 1936 T^{7} + 559 T^{8} + 180 T^{9} - 40 T^{10} - 4 T^{11} + T^{12} \)
$19$ \( 1024 + 32768 T + 524288 T^{2} + 250368 T^{3} + 59776 T^{4} + 128 T^{5} + 6272 T^{6} + 2912 T^{7} + 676 T^{8} + 8 T^{9} + 8 T^{10} + 4 T^{11} + T^{12} \)
$23$ \( 43655168 - 29601792 T + 21729792 T^{2} - 2947072 T^{3} - 277488 T^{4} + 227040 T^{5} - 24200 T^{6} - 5416 T^{7} + 1970 T^{8} + 68 T^{9} - 6 T^{10} + 8 T^{11} + T^{12} \)
$29$ \( 118210688 - 25462656 T + 2452512 T^{2} - 2766336 T^{3} + 571140 T^{4} + 70160 T^{5} + 17480 T^{6} + 96 T^{7} + 8 T^{8} - 8 T^{9} + 20 T^{10} - 8 T^{11} + T^{12} \)
$31$ \( 103219712 + 5747200 T + 18451328 T^{2} + 1519616 T^{3} + 1676816 T^{4} - 39200 T^{5} - 39864 T^{6} - 15752 T^{7} + 1154 T^{8} + 1868 T^{9} + 386 T^{10} + 32 T^{11} + T^{12} \)
$37$ \( 512 - 15360 T + 147520 T^{2} - 267872 T^{3} + 200260 T^{4} - 72992 T^{5} + 24832 T^{6} - 8616 T^{7} + 2048 T^{8} - 328 T^{9} + 56 T^{10} - 4 T^{11} + T^{12} \)
$41$ \( 591542408 - 497916496 T + 150106584 T^{2} - 49805920 T^{3} + 28852132 T^{4} - 3730464 T^{5} + 25104 T^{6} + 61328 T^{7} - 6558 T^{8} - 404 T^{9} + 174 T^{10} - 16 T^{11} + T^{12} \)
$43$ \( 148254976 + 14806016 T + 739328 T^{2} + 1186816 T^{3} + 2328464 T^{4} + 433984 T^{5} + 36480 T^{6} - 6560 T^{7} + 2264 T^{8} + 336 T^{9} + 32 T^{10} - 8 T^{11} + T^{12} \)
$47$ \( 440664064 + 288784384 T^{2} + 46576640 T^{4} + 2438912 T^{6} + 46032 T^{8} + 360 T^{10} + T^{12} \)
$53$ \( 25080064 - 43749888 T + 38158848 T^{2} - 19102336 T^{3} + 5908880 T^{4} - 1028224 T^{5} + 78080 T^{6} + 896 T^{7} + 1880 T^{8} - 512 T^{9} + 32 T^{10} + 8 T^{11} + T^{12} \)
$59$ \( 9339136 - 32564736 T + 56775168 T^{2} - 47992960 T^{3} + 24599952 T^{4} - 8008640 T^{5} + 1691264 T^{6} - 216288 T^{7} + 15192 T^{8} - 592 T^{9} + 128 T^{10} - 16 T^{11} + T^{12} \)
$61$ \( 118949888 + 455810048 T + 511760768 T^{2} + 9096640 T^{3} + 32975108 T^{4} - 7793088 T^{5} + 1608928 T^{6} - 338280 T^{7} + 61056 T^{8} - 8632 T^{9} + 824 T^{10} - 44 T^{11} + T^{12} \)
$67$ \( ( 28928 + 1664 T - 4784 T^{2} - 880 T^{3} + 58 T^{4} + 20 T^{5} + T^{6} )^{2} \)
$71$ \( 10913792 - 24817664 T + 24433920 T^{2} - 14803584 T^{3} + 7623056 T^{4} - 3716800 T^{5} + 1384040 T^{6} - 336872 T^{7} + 57090 T^{8} - 6868 T^{9} + 578 T^{10} - 32 T^{11} + T^{12} \)
$73$ \( 545424392 - 757133872 T + 196591256 T^{2} + 86347968 T^{3} + 43060708 T^{4} + 5009888 T^{5} - 429104 T^{6} - 77360 T^{7} + 7074 T^{8} - 460 T^{9} + 158 T^{10} - 8 T^{11} + T^{12} \)
$79$ \( 1130596352 + 101951488 T + 374777600 T^{2} - 293761152 T^{3} + 32598160 T^{4} + 1103008 T^{5} + 5894168 T^{6} + 385432 T^{7} - 4654 T^{8} - 2524 T^{9} - 118 T^{10} + 8 T^{11} + T^{12} \)
$83$ \( 256 - 11776 T + 270848 T^{2} - 774528 T^{3} + 1150352 T^{4} - 999296 T^{5} + 562432 T^{6} - 208640 T^{7} + 51672 T^{8} - 8192 T^{9} + 800 T^{10} - 40 T^{11} + T^{12} \)
$89$ \( 12558340096 + 3825710592 T^{2} + 325147152 T^{4} + 9151008 T^{6} + 107672 T^{8} + 552 T^{10} + T^{12} \)
$97$ \( 30451208 - 905264 T + 42001240 T^{2} + 57668896 T^{3} + 27214564 T^{4} + 5368608 T^{5} + 235472 T^{6} - 77296 T^{7} - 13022 T^{8} + 20 T^{9} + 334 T^{10} + 16 T^{11} + T^{12} \)
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