Properties

Label 322.2.e.b
Level $322$
Weight $2$
Character orbit 322.e
Analytic conductor $2.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 322 = 2 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 322.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.57118294509\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1767277521.3
Defining polynomial: \(x^{8} - 2 x^{7} + x^{6} - 10 x^{5} + 38 x^{4} - 40 x^{3} + 64 x^{2} - 38 x + 7\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{3} + ( -1 + \beta_{5} ) q^{4} + ( \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{6} ) q^{7} + q^{8} + ( -\beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{2} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{3} + ( -1 + \beta_{5} ) q^{4} + ( \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{5} + ( -1 - \beta_{2} ) q^{6} + ( \beta_{3} - \beta_{6} ) q^{7} + q^{8} + ( -\beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{9} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{10} + ( \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{5} - \beta_{6} ) q^{12} + ( -2 + \beta_{2} + \beta_{3} ) q^{13} + ( \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{14} + ( -1 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{15} -\beta_{5} q^{16} + ( -3 \beta_{1} - \beta_{4} - 3 \beta_{7} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{18} + ( -\beta_{3} + \beta_{4} + 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{19} + ( -1 - \beta_{1} - \beta_{3} ) q^{20} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{21} + ( -\beta_{1} - \beta_{2} ) q^{22} -\beta_{5} q^{23} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} ) q^{24} + ( -2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{25} + ( -\beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{26} + ( -3 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{27} + ( -\beta_{2} - \beta_{4} ) q^{28} + ( -2 \beta_{2} - \beta_{3} ) q^{29} + ( -3 \beta_{3} + 3 \beta_{4} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{30} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( -1 + \beta_{5} ) q^{32} + ( -\beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{33} + ( 3 \beta_{1} + \beta_{3} ) q^{34} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{35} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{36} + ( -\beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{37} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{38} + ( 2 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{39} + ( \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{40} + ( -4 + \beta_{2} - \beta_{3} ) q^{41} + ( 1 - \beta_{1} - 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} ) q^{42} + ( -5 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{43} + ( -\beta_{6} - \beta_{7} ) q^{44} + ( 5 \beta_{1} + 6 \beta_{2} + 5 \beta_{4} + 6 \beta_{6} + 5 \beta_{7} ) q^{45} + ( -1 + \beta_{5} ) q^{46} + ( \beta_{3} - \beta_{4} + 5 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{47} + ( -1 - \beta_{2} ) q^{48} + ( 2 - 3 \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{49} + ( 2 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( 5 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} ) q^{51} + ( 2 - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{52} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{53} + ( \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{54} + ( 1 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} ) q^{55} + ( \beta_{3} - \beta_{6} ) q^{56} + ( -1 - 4 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{57} + ( \beta_{3} - \beta_{4} - 2 \beta_{6} ) q^{58} + ( 4 + \beta_{1} - \beta_{2} - 4 \beta_{4} - 4 \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( 1 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{60} + ( \beta_{3} - \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{61} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{62} + ( 5 + 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + 4 \beta_{6} + 5 \beta_{7} ) q^{63} + q^{64} + ( -\beta_{3} + \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{65} + ( -2 - 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{66} + ( 4 + \beta_{2} - 2 \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{67} + ( -\beta_{3} + \beta_{4} + 3 \beta_{7} ) q^{68} + ( -1 - \beta_{2} ) q^{69} + ( 1 + 4 \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{70} + ( -\beta_{2} - 5 \beta_{3} ) q^{71} + ( -\beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{72} + ( \beta_{1} + 3 \beta_{2} + 4 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{73} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{74} + ( 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - 7 \beta_{6} - 3 \beta_{7} ) q^{75} + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{76} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + \beta_{3} + 3 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{77} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{78} + ( -\beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{79} + ( 1 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{80} + ( -2 - 3 \beta_{1} - 5 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} - 5 \beta_{6} - 3 \beta_{7} ) q^{81} + ( \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{82} + ( -4 \beta_{2} - 5 \beta_{3} ) q^{83} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{84} + ( -12 - 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{85} + ( -2 \beta_{3} + 2 \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{86} + ( -5 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{87} + ( \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{88} + ( -\beta_{3} + \beta_{4} + \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{89} + ( -5 \beta_{1} - 6 \beta_{2} - 5 \beta_{3} ) q^{90} + ( 4 - \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{91} + q^{92} + ( 5 \beta_{3} - 5 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{93} + ( 5 + 2 \beta_{1} - \beta_{2} + \beta_{4} - 5 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{94} + ( -1 + 7 \beta_{2} + \beta_{5} + 7 \beta_{6} ) q^{95} + ( \beta_{5} - \beta_{6} ) q^{96} + ( 9 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{97} + ( 1 + 3 \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{98} + ( -8 - 3 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} + 5q^{3} - 4q^{4} + 5q^{5} - 10q^{6} + q^{7} + 8q^{8} - 7q^{9} + O(q^{10}) \) \( 8q - 4q^{2} + 5q^{3} - 4q^{4} + 5q^{5} - 10q^{6} + q^{7} + 8q^{8} - 7q^{9} + 5q^{10} + 2q^{11} + 5q^{12} - 14q^{13} + q^{14} + 2q^{15} - 4q^{16} - 3q^{17} - 7q^{18} + 9q^{19} - 10q^{20} + 25q^{21} - 4q^{22} - 4q^{23} + 5q^{24} - 11q^{25} + 7q^{26} - 34q^{27} - 2q^{28} - 4q^{29} - q^{30} + 14q^{31} - 4q^{32} - 13q^{33} + 6q^{34} + 16q^{35} + 14q^{36} + 9q^{38} + q^{39} + 5q^{40} - 30q^{41} - 8q^{42} - 36q^{43} + 2q^{44} + 11q^{45} - 4q^{46} + 21q^{47} - 10q^{48} + 17q^{49} + 22q^{50} - 6q^{51} + 7q^{52} + 3q^{53} + 17q^{54} + 20q^{55} + q^{56} - 18q^{57} + 2q^{58} + 16q^{59} - q^{60} + q^{61} - 28q^{62} + 37q^{63} + 8q^{64} - 2q^{65} - 13q^{66} + 17q^{67} - 3q^{68} - 10q^{69} + 4q^{70} - 2q^{71} - 7q^{72} + 4q^{73} + 22q^{75} - 18q^{76} + 13q^{77} - 2q^{78} - 5q^{79} + 5q^{80} - 16q^{81} + 15q^{82} - 8q^{83} - 17q^{84} - 108q^{85} + 18q^{86} - 25q^{87} + 2q^{88} + 9q^{89} - 22q^{90} + 38q^{91} + 8q^{92} - 13q^{93} + 21q^{94} + 3q^{95} + 5q^{96} + 80q^{97} + 2q^{98} - 82q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + x^{6} - 10 x^{5} + 38 x^{4} - 40 x^{3} + 64 x^{2} - 38 x + 7\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -14 \nu^{7} + 688 \nu^{6} - 619 \nu^{5} - 193 \nu^{4} - 6480 \nu^{3} + 17846 \nu^{2} - 10595 \nu + 23150 \)\()/8102\)
\(\beta_{2}\)\(=\)\((\)\( 74 \nu^{7} - 743 \nu^{6} + 957 \nu^{5} - 716 \nu^{4} + 8788 \nu^{3} - 23147 \nu^{2} + 13756 \nu - 21668 \)\()/8102\)
\(\beta_{3}\)\(=\)\((\)\( -139 \nu^{7} + 465 \nu^{6} - 648 \nu^{5} + 688 \nu^{4} - 5887 \nu^{3} + 15724 \nu^{2} - 9416 \nu - 2218 \)\()/8102\)
\(\beta_{4}\)\(=\)\((\)\( -1269 \nu^{7} + 2176 \nu^{6} - 262 \nu^{5} + 11731 \nu^{4} - 43953 \nu^{3} + 36565 \nu^{2} - 52069 \nu + 14432 \)\()/8102\)
\(\beta_{5}\)\(=\)\((\)\( -962 \nu^{7} + 1557 \nu^{6} - 288 \nu^{5} + 9308 \nu^{4} - 33224 \nu^{3} + 25443 \nu^{2} - 49196 \nu + 18369 \)\()/4051\)
\(\beta_{6}\)\(=\)\((\)\( -4270 \nu^{7} + 7290 \nu^{6} - 2449 \nu^{5} + 42410 \nu^{4} - 149399 \nu^{3} + 128118 \nu^{2} - 241837 \nu + 88979 \)\()/8102\)
\(\beta_{7}\)\(=\)\((\)\( 4805 \nu^{7} - 8118 \nu^{6} + 2087 \nu^{5} - 47477 \nu^{4} + 167278 \nu^{3} - 139939 \nu^{2} + 265634 \nu - 98045 \)\()/8102\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - \beta_{1} + 2\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-5 \beta_{7} - 5 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - 4 \beta_{1} + 2\)\()/3\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{6} - 6 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_{1} + 6\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{7} + 5 \beta_{6} - 38 \beta_{5} + 16 \beta_{4} - 23 \beta_{3} - 17 \beta_{2} - 17 \beta_{1} + 1\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-70 \beta_{7} - 70 \beta_{6} - 11 \beta_{5} - 14 \beta_{4} + 4 \beta_{3} - 14 \beta_{2} - 8 \beta_{1} - 17\)\()/3\)
\(\nu^{6}\)\(=\)\(17 \beta_{7} + 37 \beta_{6} - 60 \beta_{5} + 35 \beta_{4} - 16 \beta_{3} + 50 \beta_{2} + 46 \beta_{1} + 7\)
\(\nu^{7}\)\(=\)\((\)\(-44 \beta_{7} - 38 \beta_{6} - 22 \beta_{5} - 7 \beta_{4} - 301 \beta_{3} - 283 \beta_{2} - 97 \beta_{1} - 565\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/322\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
0.373419 + 0.0835272i
2.11692 + 0.978886i
0.0512865 1.21608i
−1.54162 + 1.88572i
0.373419 0.0835272i
2.11692 0.978886i
0.0512865 + 1.21608i
−1.54162 1.88572i
−0.500000 + 0.866025i −0.685837 1.18790i −0.500000 0.866025i 1.57146 2.72186i 1.37167 −1.66774 + 2.05393i 1.00000 0.559256 0.968659i 1.57146 + 2.72186i
93.2 −0.500000 + 0.866025i 0.313577 + 0.543132i −0.500000 0.866025i 0.475705 0.823946i −0.627155 2.62597 + 0.322894i 1.00000 1.30334 2.25745i 0.475705 + 0.823946i
93.3 −0.500000 + 0.866025i 1.26826 + 2.19669i −0.500000 0.866025i −1.34706 + 2.33318i −2.53652 −2.28677 1.33067i 1.00000 −1.71698 + 2.97389i −1.34706 2.33318i
93.4 −0.500000 + 0.866025i 1.60400 + 2.77821i −0.500000 0.866025i 1.79990 3.11751i −3.20800 1.82854 1.91218i 1.00000 −3.64562 + 6.31440i 1.79990 + 3.11751i
277.1 −0.500000 0.866025i −0.685837 + 1.18790i −0.500000 + 0.866025i 1.57146 + 2.72186i 1.37167 −1.66774 2.05393i 1.00000 0.559256 + 0.968659i 1.57146 2.72186i
277.2 −0.500000 0.866025i 0.313577 0.543132i −0.500000 + 0.866025i 0.475705 + 0.823946i −0.627155 2.62597 0.322894i 1.00000 1.30334 + 2.25745i 0.475705 0.823946i
277.3 −0.500000 0.866025i 1.26826 2.19669i −0.500000 + 0.866025i −1.34706 2.33318i −2.53652 −2.28677 + 1.33067i 1.00000 −1.71698 2.97389i −1.34706 + 2.33318i
277.4 −0.500000 0.866025i 1.60400 2.77821i −0.500000 + 0.866025i 1.79990 + 3.11751i −3.20800 1.82854 + 1.91218i 1.00000 −3.64562 6.31440i 1.79990 3.11751i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 277.4
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 322.2.e.b 8
7.c even 3 1 inner 322.2.e.b 8
7.c even 3 1 2254.2.a.w 4
7.d odd 6 1 2254.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.e.b 8 1.a even 1 1 trivial
322.2.e.b 8 7.c even 3 1 inner
2254.2.a.w 4 7.c even 3 1
2254.2.a.ba 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(322, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( 49 - 77 T + 142 T^{2} - 37 T^{3} + 71 T^{4} - 37 T^{5} + 22 T^{6} - 5 T^{7} + T^{8} \)
$5$ \( 841 - 1073 T + 1282 T^{2} - 401 T^{3} + 223 T^{4} - 59 T^{5} + 28 T^{6} - 5 T^{7} + T^{8} \)
$7$ \( 2401 - 343 T - 392 T^{2} - 35 T^{3} + 83 T^{4} - 5 T^{5} - 8 T^{6} - T^{7} + T^{8} \)
$11$ \( 9 - 45 T + 189 T^{2} - 192 T^{3} + 177 T^{4} - 6 T^{5} + 16 T^{6} - 2 T^{7} + T^{8} \)
$13$ \( ( 7 - 20 T + 6 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$17$ \( 1814409 - 148170 T + 109084 T^{2} - 162 T^{3} + 4167 T^{4} + 4 T^{5} + 81 T^{6} + 3 T^{7} + T^{8} \)
$19$ \( 95481 - 44805 T + 21952 T^{2} - 5127 T^{3} + 1623 T^{4} - 317 T^{5} + 78 T^{6} - 9 T^{7} + T^{8} \)
$23$ \( ( 1 + T + T^{2} )^{4} \)
$29$ \( ( -1 - 56 T - 27 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$31$ \( 11449 - 3317 T + 5455 T^{2} - 1694 T^{3} + 2305 T^{4} - 650 T^{5} + 154 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( 729 - 135 T + 835 T^{2} + 150 T^{3} + 873 T^{4} + 10 T^{5} + 30 T^{6} + T^{8} \)
$41$ \( ( -63 + 62 T + 66 T^{2} + 15 T^{3} + T^{4} )^{2} \)
$43$ \( ( -81 + 99 T + 90 T^{2} + 18 T^{3} + T^{4} )^{2} \)
$47$ \( 35721 + 18711 T + 31914 T^{2} - 19521 T^{3} + 11799 T^{4} - 2259 T^{5} + 324 T^{6} - 21 T^{7} + T^{8} \)
$53$ \( 13689 + 25623 T + 63756 T^{2} - 28863 T^{3} + 18765 T^{4} - 33 T^{5} + 144 T^{6} - 3 T^{7} + T^{8} \)
$59$ \( 1320201 - 1795887 T + 2353347 T^{2} - 158682 T^{3} + 32241 T^{4} - 1878 T^{5} + 334 T^{6} - 16 T^{7} + T^{8} \)
$61$ \( 8346321 - 936036 T + 676998 T^{2} + 69930 T^{3} + 35991 T^{4} + 846 T^{5} + 199 T^{6} - T^{7} + T^{8} \)
$67$ \( 82369 - 14924 T + 19924 T^{2} - 6638 T^{3} + 4771 T^{4} - 1124 T^{5} + 229 T^{6} - 17 T^{7} + T^{8} \)
$71$ \( ( 5243 - 196 T - 216 T^{2} + T^{3} + T^{4} )^{2} \)
$73$ \( 19245769 + 995849 T + 656935 T^{2} + 3770 T^{3} + 15565 T^{4} + 98 T^{5} + 154 T^{6} - 4 T^{7} + T^{8} \)
$79$ \( 145161 - 25146 T + 20358 T^{2} - 1038 T^{3} + 1713 T^{4} - 78 T^{5} + 67 T^{6} + 5 T^{7} + T^{8} \)
$83$ \( ( 17003 - 574 T - 261 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$89$ \( 12131289 + 1849473 T + 626778 T^{2} + 10125 T^{3} + 11097 T^{4} - 171 T^{5} + 180 T^{6} - 9 T^{7} + T^{8} \)
$97$ \( ( -637 - 2126 T + 519 T^{2} - 40 T^{3} + T^{4} )^{2} \)
show more
show less