Properties

Label 273.2.i.d
Level $273$
Weight $2$
Character orbit 273.i
Analytic conductor $2.180$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4868829729.1
Defining polynomial: \(x^{8} - 2 x^{7} - x^{6} + 5 x^{5} - 8 x^{4} + 15 x^{3} - 9 x^{2} - 54 x + 81\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - x^{6} + 5 x^{5} - 8 x^{4} + 15 x^{3} - 9 x^{2} - 54 x + 81\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 4 \nu^{5} + 3 \nu^{4} - 7 \nu^{3} + 26 \nu^{2} + 3 \nu - 18 \)\()/45\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + \nu^{5} - 5 \nu^{4} + 8 \nu^{3} - 15 \nu^{2} + 9 \nu + 54 \)\()/27\)
\(\beta_{4}\)\(=\)\((\)\( 11 \nu^{7} - 7 \nu^{6} - 14 \nu^{5} + 13 \nu^{4} - 13 \nu^{3} + 126 \nu^{2} - 9 \nu - 567 \)\()/270\)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{7} - 11 \nu^{6} + 32 \nu^{5} - 13 \nu^{4} + 49 \nu^{3} - 126 \nu^{2} - 279 \nu + 621 \)\()/270\)
\(\beta_{6}\)\(=\)\((\)\( 5 \nu^{7} - \nu^{6} - 14 \nu^{5} + 25 \nu^{4} - 13 \nu^{3} + 30 \nu^{2} + 99 \nu - 297 \)\()/90\)
\(\beta_{7}\)\(=\)\((\)\( -16 \nu^{7} + 8 \nu^{6} + 28 \nu^{5} - 38 \nu^{4} + 71 \nu^{3} - 66 \nu^{2} - 90 \nu + 729 \)\()/135\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{7} + 5 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} + 2 \beta_{2} - 3 \beta_{1} + 1\)
\(\nu^{5}\)\(=\)\(-2 \beta_{7} + 5 \beta_{5} - \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + 2 \beta_{1} - 8\)
\(\nu^{6}\)\(=\)\(4 \beta_{6} - 6 \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 7 \beta_{2} - 14 \beta_{1} - 3\)
\(\nu^{7}\)\(=\)\(-14 \beta_{7} - \beta_{6} + 14 \beta_{5} + 15 \beta_{4} + 24 \beta_{3} + 11 \beta_{2} - 2 \beta_{1} + 28\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−1.70103 0.326320i
1.25184 1.19703i
1.72192 0.187090i
−0.272725 + 1.71044i
−1.70103 + 0.326320i
1.25184 + 1.19703i
1.72192 + 0.187090i
−0.272725 1.71044i
−1.13312 + 1.96262i −0.500000 0.866025i −1.56792 2.71571i −1.63312 + 2.82864i 2.26624 −0.133118 2.64240i 2.57406 −0.500000 + 0.866025i −3.70103 6.41038i
79.2 −0.410741 + 0.711425i −0.500000 0.866025i 0.662583 + 1.14763i −0.910741 + 1.57745i 0.821482 0.589259 + 2.57930i −2.73157 −0.500000 + 0.866025i −0.748158 1.29585i
79.3 0.698934 1.21059i −0.500000 0.866025i 0.0229829 + 0.0398076i 0.198934 0.344564i −1.39787 1.69893 2.02821i 2.85999 −0.500000 + 0.866025i −0.278083 0.481654i
79.4 1.34493 2.32948i −0.500000 0.866025i −2.61765 4.53390i 0.844926 1.46345i −2.68985 2.34493 + 1.22528i −8.70248 −0.500000 + 0.866025i −2.27273 3.93648i
235.1 −1.13312 1.96262i −0.500000 + 0.866025i −1.56792 + 2.71571i −1.63312 2.82864i 2.26624 −0.133118 + 2.64240i 2.57406 −0.500000 0.866025i −3.70103 + 6.41038i
235.2 −0.410741 0.711425i −0.500000 + 0.866025i 0.662583 1.14763i −0.910741 1.57745i 0.821482 0.589259 2.57930i −2.73157 −0.500000 0.866025i −0.748158 + 1.29585i
235.3 0.698934 + 1.21059i −0.500000 + 0.866025i 0.0229829 0.0398076i 0.198934 + 0.344564i −1.39787 1.69893 + 2.02821i 2.85999 −0.500000 0.866025i −0.278083 + 0.481654i
235.4 1.34493 + 2.32948i −0.500000 + 0.866025i −2.61765 + 4.53390i 0.844926 + 1.46345i −2.68985 2.34493 1.22528i −8.70248 −0.500000 0.866025i −2.27273 + 3.93648i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.i.d 8
3.b odd 2 1 819.2.j.f 8
7.c even 3 1 inner 273.2.i.d 8
7.c even 3 1 1911.2.a.r 4
7.d odd 6 1 1911.2.a.q 4
21.g even 6 1 5733.2.a.bk 4
21.h odd 6 1 819.2.j.f 8
21.h odd 6 1 5733.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.d 8 1.a even 1 1 trivial
273.2.i.d 8 7.c even 3 1 inner
819.2.j.f 8 3.b odd 2 1
819.2.j.f 8 21.h odd 6 1
1911.2.a.q 4 7.d odd 6 1
1911.2.a.r 4 7.c even 3 1
5733.2.a.bj 4 21.h odd 6 1
5733.2.a.bk 4 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 49 + 28 T + 65 T^{2} - 14 T^{3} + 46 T^{4} - T^{5} + 8 T^{6} - T^{7} + T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 16 - 36 T + 97 T^{2} + 12 T^{3} + 39 T^{4} + 6 T^{5} + 13 T^{6} + 3 T^{7} + T^{8} \)
$7$ \( 2401 - 3087 T + 2499 T^{2} - 1407 T^{3} + 611 T^{4} - 201 T^{5} + 51 T^{6} - 9 T^{7} + T^{8} \)
$11$ \( 7921 + 8455 T + 7156 T^{2} + 2707 T^{3} + 910 T^{4} + 106 T^{5} + 37 T^{6} + 4 T^{7} + T^{8} \)
$13$ \( ( 1 + T )^{8} \)
$17$ \( 3025 - 2915 T + 4184 T^{2} + 1105 T^{3} + 676 T^{4} + 56 T^{5} + 29 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( 73441 - 47154 T + 41116 T^{2} + 3708 T^{3} + 2373 T^{4} + 108 T^{5} + 76 T^{6} + 6 T^{7} + T^{8} \)
$23$ \( 62500 + 7250 T + 13591 T^{2} + 521 T^{3} + 2467 T^{4} + 146 T^{5} + 67 T^{6} - 4 T^{7} + T^{8} \)
$29$ \( ( 181 - 176 T + 15 T^{2} + 13 T^{3} + T^{4} )^{2} \)
$31$ \( 33856 + 5704 T + 8873 T^{2} - 2069 T^{3} + 1603 T^{4} - 148 T^{5} + 47 T^{6} + 2 T^{7} + T^{8} \)
$37$ \( 238144 - 230824 T + 172001 T^{2} - 55018 T^{3} + 14089 T^{4} - 416 T^{5} + 131 T^{6} - 5 T^{7} + T^{8} \)
$41$ \( ( -320 + 173 T - 7 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$43$ \( ( -9050 + 2227 T - 73 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$47$ \( 760384 + 219744 T + 96640 T^{2} + 16584 T^{3} + 6096 T^{4} + 1074 T^{5} + 187 T^{6} + 15 T^{7} + T^{8} \)
$53$ \( 3222025 + 669535 T + 370684 T^{2} - 40937 T^{3} + 15592 T^{4} - 488 T^{5} + 133 T^{6} - 2 T^{7} + T^{8} \)
$59$ \( 366025 - 65945 T + 83876 T^{2} + 37171 T^{3} + 12586 T^{4} + 2162 T^{5} + 281 T^{6} + 20 T^{7} + T^{8} \)
$61$ \( 483025 + 74365 T + 77474 T^{2} + 17635 T^{3} + 11860 T^{4} + 2114 T^{5} + 305 T^{6} + 20 T^{7} + T^{8} \)
$67$ \( 25 + 5 T + 116 T^{2} + 77 T^{3} + 544 T^{4} + 232 T^{5} + 77 T^{6} + 10 T^{7} + T^{8} \)
$71$ \( ( 919 + 187 T - 105 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$73$ \( 2500 - 4050 T + 5011 T^{2} - 2511 T^{3} + 1011 T^{4} - 162 T^{5} + 31 T^{6} + T^{8} \)
$79$ \( 556516 + 311082 T + 137335 T^{2} + 29385 T^{3} + 5649 T^{4} + 540 T^{5} + 85 T^{6} + 6 T^{7} + T^{8} \)
$83$ \( ( 16984 + 2317 T - 211 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$89$ \( 246866944 + 106008864 T + 31066969 T^{2} + 4604616 T^{3} + 486591 T^{4} + 33426 T^{5} + 1681 T^{6} + 51 T^{7} + T^{8} \)
$97$ \( ( -1802 + 805 T - 52 T^{2} - 13 T^{3} + T^{4} )^{2} \)
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