Properties

Label 240.2.s.b
Level $240$
Weight $2$
Character orbit 240.s
Analytic conductor $1.916$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.91640964851\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{7} - 7 \nu^{6} + 24 \nu^{5} - 42 \nu^{4} + 59 \nu^{3} - 48 \nu^{2} + 24 \nu - 5 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 43 \nu^{4} - 61 \nu^{3} + 54 \nu^{2} - 29 \nu + 8 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11 \)
\(\beta_{4}\)\(=\)\( -3 \nu^{7} + 11 \nu^{6} - 38 \nu^{5} + 70 \nu^{4} - 102 \nu^{3} + 91 \nu^{2} - 53 \nu + 13 \)
\(\beta_{5}\)\(=\)\( 5 \nu^{7} - 17 \nu^{6} + 60 \nu^{5} - 105 \nu^{4} + 155 \nu^{3} - 133 \nu^{2} + 77 \nu - 19 \)
\(\beta_{6}\)\(=\)\( -5 \nu^{7} + 18 \nu^{6} - 63 \nu^{5} + 115 \nu^{4} - 170 \nu^{3} + 152 \nu^{2} - 89 \nu + 23 \)
\(\beta_{7}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 3 \beta_{6} + \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} - 4\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} - \beta_{6} + 7 \beta_{5} - \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 3 \beta_{1} - 2\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(11 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} + 11 \beta_{4} - \beta_{3} - 5 \beta_{2} + 9 \beta_{1} + 14\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-13 \beta_{7} - 11 \beta_{6} - 29 \beta_{5} + 17 \beta_{4} - 23 \beta_{3} + 13 \beta_{2} - 3 \beta_{1} + 18\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-67 \beta_{7} + 59 \beta_{6} - 41 \beta_{5} - 29 \beta_{4} - 15 \beta_{3} + 37 \beta_{2} - 47 \beta_{1} - 48\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-7 \beta_{7} + 113 \beta_{6} + 97 \beta_{5} - 105 \beta_{4} + 91 \beta_{3} - 31 \beta_{2} - 39 \beta_{1} - 122\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 2.10607i
0.500000 + 1.44392i
0.500000 + 0.691860i
0.500000 0.0297061i
0.500000 + 2.10607i
0.500000 1.44392i
0.500000 0.691860i
0.500000 + 0.0297061i
−1.34277 + 0.443806i 0.707107 + 0.707107i 1.60607 1.19186i −0.707107 + 0.707107i −1.26330 0.635665i 1.41421i −1.62764 + 2.31318i 1.00000i 0.635665 1.26330i
61.2 −0.167452 + 1.40426i −0.707107 0.707107i −1.94392 0.470294i 0.707107 0.707107i 1.11137 0.874559i 1.41421i 0.985930 2.65103i 1.00000i 0.874559 + 1.11137i
61.3 0.635665 + 1.26330i 0.707107 + 0.707107i −1.19186 + 1.60607i −0.707107 + 0.707107i −0.443806 + 1.34277i 1.41421i −2.78658 0.484753i 1.00000i −1.34277 0.443806i
61.4 0.874559 1.11137i −0.707107 0.707107i −0.470294 1.94392i 0.707107 0.707107i −1.40426 + 0.167452i 1.41421i −2.57172 1.17740i 1.00000i −0.167452 1.40426i
181.1 −1.34277 0.443806i 0.707107 0.707107i 1.60607 + 1.19186i −0.707107 0.707107i −1.26330 + 0.635665i 1.41421i −1.62764 2.31318i 1.00000i 0.635665 + 1.26330i
181.2 −0.167452 1.40426i −0.707107 + 0.707107i −1.94392 + 0.470294i 0.707107 + 0.707107i 1.11137 + 0.874559i 1.41421i 0.985930 + 2.65103i 1.00000i 0.874559 1.11137i
181.3 0.635665 1.26330i 0.707107 0.707107i −1.19186 1.60607i −0.707107 0.707107i −0.443806 1.34277i 1.41421i −2.78658 + 0.484753i 1.00000i −1.34277 + 0.443806i
181.4 0.874559 + 1.11137i −0.707107 + 0.707107i −0.470294 + 1.94392i 0.707107 + 0.707107i −1.40426 0.167452i 1.41421i −2.57172 + 1.17740i 1.00000i −0.167452 + 1.40426i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.2.s.b 8
3.b odd 2 1 720.2.t.b 8
4.b odd 2 1 960.2.s.b 8
8.b even 2 1 1920.2.s.c 8
8.d odd 2 1 1920.2.s.d 8
12.b even 2 1 2880.2.t.b 8
16.e even 4 1 inner 240.2.s.b 8
16.e even 4 1 1920.2.s.c 8
16.f odd 4 1 960.2.s.b 8
16.f odd 4 1 1920.2.s.d 8
48.i odd 4 1 720.2.t.b 8
48.k even 4 1 2880.2.t.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.s.b 8 1.a even 1 1 trivial
240.2.s.b 8 16.e even 4 1 inner
720.2.t.b 8 3.b odd 2 1
720.2.t.b 8 48.i odd 4 1
960.2.s.b 8 4.b odd 2 1
960.2.s.b 8 16.f odd 4 1
1920.2.s.c 8 8.b even 2 1
1920.2.s.c 8 16.e even 4 1
1920.2.s.d 8 8.d odd 2 1
1920.2.s.d 8 16.f odd 4 1
2880.2.t.b 8 12.b even 2 1
2880.2.t.b 8 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(240, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 8 T^{2} + 8 T^{3} + 2 T^{4} + 4 T^{5} + 2 T^{6} + T^{8} \)
$3$ \( ( 1 + T^{4} )^{2} \)
$5$ \( ( 1 + T^{4} )^{2} \)
$7$ \( ( 2 + T^{2} )^{4} \)
$11$ \( 64 + 128 T + 128 T^{2} + 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 16 + 64 T + 128 T^{2} + 32 T^{3} - 8 T^{4} - 16 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$17$ \( ( 8 - 12 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$19$ \( 256 + 512 T + 512 T^{2} - 384 T^{3} + 224 T^{4} + 96 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$23$ \( 200704 + 73728 T^{2} + 5248 T^{4} + 128 T^{6} + T^{8} \)
$29$ \( 135424 + 11776 T + 512 T^{2} - 11392 T^{3} + 7136 T^{4} - 1888 T^{5} + 288 T^{6} - 24 T^{7} + T^{8} \)
$31$ \( ( -28 + 88 T - 40 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$37$ \( 414736 - 154560 T + 28800 T^{2} + 6368 T^{3} + 1016 T^{4} - 144 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$41$ \( 17707264 + 1276672 T^{2} + 31072 T^{4} + 304 T^{6} + T^{8} \)
$43$ \( 12544 + 28672 T + 32768 T^{2} + 18432 T^{3} + 5408 T^{4} + 256 T^{5} + T^{8} \)
$47$ \( ( -224 - 416 T - 152 T^{2} + T^{4} )^{2} \)
$53$ \( 73984 + 33312 T^{4} + T^{8} \)
$59$ \( 7529536 - 6453888 T + 2765952 T^{2} - 482944 T^{3} + 43904 T^{4} - 784 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$61$ \( 4343056 - 1667200 T + 320000 T^{2} + 27456 T^{3} + 1608 T^{4} - 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$67$ \( 1183744 + 4224 T^{4} + T^{8} \)
$71$ \( 5161984 + 699904 T^{2} + 23168 T^{4} + 272 T^{6} + T^{8} \)
$73$ \( 50176 + 27136 T^{2} + 4224 T^{4} + 176 T^{6} + T^{8} \)
$79$ \( ( 164 + 264 T + 120 T^{2} + 20 T^{3} + T^{4} )^{2} \)
$83$ \( 2166784 - 1130496 T + 294912 T^{2} + 14336 T^{3} + 9344 T^{4} - 3328 T^{5} + 512 T^{6} - 32 T^{7} + T^{8} \)
$89$ \( 118026496 + 6544640 T^{2} + 102752 T^{4} + 592 T^{6} + T^{8} \)
$97$ \( ( -736 - 32 T + 120 T^{2} + 24 T^{3} + T^{4} )^{2} \)
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