Properties

Label 6422.2.a.bl
Level $6422$
Weight $2$
Character orbit 6422.a
Self dual yes
Analytic conductor $51.280$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.2799281781\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - x^{8} - 12 x^{7} + 14 x^{6} + 35 x^{5} - 35 x^{4} - 28 x^{3} + 10 x^{2} + 4 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 12 x^{7} + 14 x^{6} + 35 x^{5} - 35 x^{4} - 28 x^{3} + 10 x^{2} + 4 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -2 \nu^{7} - \nu^{6} + 21 \nu^{5} + 3 \nu^{4} - 50 \nu^{3} - 5 \nu^{2} + 14 \nu \)
\(\beta_{3}\)\(=\)\( \nu^{8} + \nu^{7} - 11 \nu^{6} - 7 \nu^{5} + 32 \nu^{4} + 15 \nu^{3} - 23 \nu^{2} - 4 \nu + 3 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 20 \nu^{5} - 13 \nu^{4} + 44 \nu^{3} + 28 \nu^{2} - 3 \nu - 4 \)
\(\beta_{5}\)\(=\)\( \nu^{8} + 2 \nu^{7} - 10 \nu^{6} - 17 \nu^{5} + 26 \nu^{4} + 38 \nu^{3} - 12 \nu^{2} - 9 \nu + 3 \)
\(\beta_{6}\)\(=\)\( -\nu^{8} + \nu^{7} + 10 \nu^{6} - 15 \nu^{5} - 14 \nu^{4} + 38 \nu^{3} - 21 \nu^{2} - 14 \nu + 5 \)
\(\beta_{7}\)\(=\)\( 3 \nu^{7} + 2 \nu^{6} - 31 \nu^{5} - 9 \nu^{4} + 72 \nu^{3} + 15 \nu^{2} - 14 \nu + 1 \)
\(\beta_{8}\)\(=\)\( 2 \nu^{8} - \nu^{7} - 22 \nu^{6} + 18 \nu^{5} + 53 \nu^{4} - 44 \nu^{3} - 18 \nu^{2} + 8 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 5 \beta_{1} - 2\)
\(\nu^{4}\)\(=\)\(8 \beta_{8} + 2 \beta_{7} + 8 \beta_{6} - 8 \beta_{5} + 7 \beta_{4} + 2 \beta_{2} - 3 \beta_{1} + 18\)
\(\nu^{5}\)\(=\)\(-14 \beta_{8} - 11 \beta_{7} - 13 \beta_{6} + 23 \beta_{5} - 11 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} + 30 \beta_{1} - 27\)
\(\nu^{6}\)\(=\)\(65 \beta_{8} + 25 \beta_{7} + 64 \beta_{6} - 68 \beta_{5} + 53 \beta_{4} + 2 \beta_{3} + 23 \beta_{2} - 41 \beta_{1} + 130\)
\(\nu^{7}\)\(=\)\(-145 \beta_{8} - 100 \beta_{7} - 134 \beta_{6} + 216 \beta_{5} - 109 \beta_{4} - 60 \beta_{3} - 68 \beta_{2} + 213 \beta_{1} - 279\)
\(\nu^{8}\)\(=\)\(544 \beta_{8} + 249 \beta_{7} + 529 \beta_{6} - 600 \beta_{5} + 429 \beta_{4} + 42 \beta_{3} + 216 \beta_{2} - 429 \beta_{1} + 1040\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.588380
0.380781
−1.66806
1.89407
1.91149
−2.97314
0.219034
−0.490336
2.31454
1.00000 −2.01699 1.00000 2.32439 −2.01699 1.49240 1.00000 1.06824 2.32439
1.2 1.00000 −1.66099 1.00000 −1.69394 −1.66099 0.961642 1.00000 −0.241101 −1.69394
1.3 1.00000 −1.57968 1.00000 1.25818 −1.57968 −0.329623 1.00000 −0.504623 1.25818
1.4 1.00000 −0.953982 1.00000 −3.75463 −0.953982 −3.81192 1.00000 −2.08992 −3.75463
1.5 1.00000 0.0255071 1.00000 1.15077 0.0255071 2.71462 1.00000 −2.99935 1.15077
1.6 1.00000 1.04618 1.00000 0.929245 1.04618 −2.89421 1.00000 −1.90551 0.929245
1.7 1.00000 1.28668 1.00000 4.29839 1.28668 −4.70931 1.00000 −1.34446 4.29839
1.8 1.00000 2.08053 1.00000 −0.703813 2.08053 −4.73726 1.00000 1.32860 −0.703813
1.9 1.00000 2.77275 1.00000 −2.80860 2.77275 −1.68633 1.00000 4.68813 −2.80860
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(13\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6422.2.a.bl yes 9
13.b even 2 1 6422.2.a.bj 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6422.2.a.bj 9 13.b even 2 1
6422.2.a.bl yes 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6422))\):

\(T_{3}^{9} - \cdots\)
\(T_{5}^{9} - \cdots\)
\(T_{7}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{9} \)
$3$ \( -1 + 39 T + 10 T^{2} - 77 T^{3} - 14 T^{4} + 49 T^{5} + 7 T^{6} - 12 T^{7} - T^{8} + T^{9} \)
$5$ \( -169 + 130 T + 402 T^{2} - 371 T^{3} - 168 T^{4} + 182 T^{5} + 21 T^{6} - 26 T^{7} - T^{8} + T^{9} \)
$7$ \( -533 - 1375 T + 1291 T^{2} + 1631 T^{3} - 357 T^{4} - 574 T^{5} - 70 T^{6} + 44 T^{7} + 13 T^{8} + T^{9} \)
$11$ \( 73177 + 21554 T - 29024 T^{2} - 8603 T^{3} + 3780 T^{4} + 1141 T^{5} - 189 T^{6} - 59 T^{7} + 3 T^{8} + T^{9} \)
$13$ \( T^{9} \)
$17$ \( 2059 - 773 T - 3034 T^{2} + 819 T^{3} + 1253 T^{4} - 112 T^{5} - 196 T^{6} - 12 T^{7} + 8 T^{8} + T^{9} \)
$19$ \( ( 1 + T )^{9} \)
$23$ \( 42391 - 278256 T - 115397 T^{2} + 45710 T^{3} + 24101 T^{4} + 294 T^{5} - 938 T^{6} - 73 T^{7} + 10 T^{8} + T^{9} \)
$29$ \( 312997 - 505207 T + 5827 T^{2} + 118944 T^{3} + 10507 T^{4} - 7049 T^{5} - 1190 T^{6} + 51 T^{7} + 20 T^{8} + T^{9} \)
$31$ \( 1274533 + 644851 T - 526340 T^{2} - 140063 T^{3} + 43351 T^{4} + 11102 T^{5} - 525 T^{6} - 194 T^{7} + T^{8} + T^{9} \)
$37$ \( -11479 - 19925 T + 9328 T^{2} + 26425 T^{3} + 7420 T^{4} - 3136 T^{5} - 1057 T^{6} - 19 T^{7} + 15 T^{8} + T^{9} \)
$41$ \( 2099 + 11216 T + 20705 T^{2} + 14658 T^{3} + 1365 T^{4} - 2226 T^{5} - 462 T^{6} + 64 T^{7} + 19 T^{8} + T^{9} \)
$43$ \( 105911 + 356666 T - 292704 T^{2} - 28224 T^{3} + 33187 T^{4} + 1064 T^{5} - 1260 T^{6} - 48 T^{7} + 16 T^{8} + T^{9} \)
$47$ \( 46789 + 133323 T + 134388 T^{2} + 50925 T^{3} - 938 T^{4} - 4858 T^{5} - 791 T^{6} + 46 T^{7} + 18 T^{8} + T^{9} \)
$53$ \( 642797 - 705816 T - 43125 T^{2} + 168938 T^{3} - 24731 T^{4} - 8316 T^{5} + 2065 T^{6} - 45 T^{7} - 17 T^{8} + T^{9} \)
$59$ \( 3093047 + 1658336 T - 2594020 T^{2} + 245147 T^{3} + 162785 T^{4} - 8106 T^{5} - 3549 T^{6} - 24 T^{7} + 24 T^{8} + T^{9} \)
$61$ \( -23491 - 86229 T - 110176 T^{2} - 53543 T^{3} - 1841 T^{4} + 5026 T^{5} + 539 T^{6} - 159 T^{7} - 6 T^{8} + T^{9} \)
$67$ \( -772001 - 272940 T + 544093 T^{2} + 224392 T^{3} - 19516 T^{4} - 17486 T^{5} - 1540 T^{6} + 177 T^{7} + 29 T^{8} + T^{9} \)
$71$ \( 4947181 - 8556007 T - 92980 T^{2} + 1509501 T^{3} + 233051 T^{4} - 20713 T^{5} - 5243 T^{6} - 97 T^{7} + 23 T^{8} + T^{9} \)
$73$ \( -63211 - 2288 T + 110430 T^{2} - 1211 T^{3} - 29211 T^{4} - 2884 T^{5} + 2058 T^{6} + 487 T^{7} + 38 T^{8} + T^{9} \)
$79$ \( -7060187 - 3175063 T + 3530481 T^{2} + 1871821 T^{3} + 217938 T^{4} - 18277 T^{5} - 4613 T^{6} - 110 T^{7} + 20 T^{8} + T^{9} \)
$83$ \( -34963811 - 18258768 T - 655484 T^{2} + 1096312 T^{3} + 167937 T^{4} - 11480 T^{5} - 3549 T^{6} - 75 T^{7} + 20 T^{8} + T^{9} \)
$89$ \( -187720771 + 28686707 T + 11389987 T^{2} - 1722455 T^{3} - 244426 T^{4} + 36414 T^{5} + 2205 T^{6} - 322 T^{7} - 7 T^{8} + T^{9} \)
$97$ \( 20086087 - 7455903 T - 1728741 T^{2} + 625891 T^{3} + 82705 T^{4} - 17654 T^{5} - 2429 T^{6} + 126 T^{7} + 28 T^{8} + T^{9} \)
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