Properties

Label 325.2.e.c
Level $325$
Weight $2$
Character orbit 325.e
Analytic conductor $2.595$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 8 x^{8} - 2 x^{7} + 52 x^{6} - 5 x^{5} + 97 x^{4} + 60 x^{3} + 141 x^{2} + 36 x + 9\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{4} + \beta_{6} + \beta_{8} ) q^{3} + ( -\beta_{6} + \beta_{9} ) q^{4} + ( \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( \beta_{1} - \beta_{5} - \beta_{7} ) q^{7} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( \beta_{1} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -1 - \beta_{4} + \beta_{6} + \beta_{8} ) q^{3} + ( -\beta_{6} + \beta_{9} ) q^{4} + ( \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{6} + ( \beta_{1} - \beta_{5} - \beta_{7} ) q^{7} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{8} + ( \beta_{1} - \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{9} + ( -1 + \beta_{6} + \beta_{7} ) q^{11} + ( \beta_{2} - \beta_{5} ) q^{12} + ( 1 + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{13} + ( -1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{14} + ( 3 \beta_{1} + \beta_{4} - \beta_{7} - \beta_{8} ) q^{16} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{17} + ( 1 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{18} + ( \beta_{6} + \beta_{9} ) q^{19} + ( -1 - 2 \beta_{2} - \beta_{3} + \beta_{4} ) q^{21} + ( -\beta_{2} + \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{22} + ( -3 - \beta_{1} - 2 \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{23} + ( 4 + \beta_{1} + 2 \beta_{4} - 4 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{24} + ( \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{26} + ( 6 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{27} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{28} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{29} + ( -1 - \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{31} + ( -6 \beta_{6} + \beta_{9} ) q^{32} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{33} + ( 5 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{34} + ( 3 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{36} + ( -3 + 2 \beta_{4} + 3 \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{37} + ( -\beta_{2} + \beta_{3} - \beta_{4} ) q^{38} + ( -4 - \beta_{1} + 2 \beta_{2} + \beta_{3} - 3 \beta_{4} + 4 \beta_{6} + 2 \beta_{8} ) q^{39} + ( -2 - 3 \beta_{1} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{41} + ( -7 - 4 \beta_{1} - \beta_{3} - \beta_{4} + 7 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{42} + ( 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{6} + 2 \beta_{9} ) q^{43} + ( -2 - 2 \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{44} + ( -6 \beta_{1} + 7 \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{46} + ( -2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{48} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{49} + ( 1 - 3 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{51} + ( 3 - \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{52} + ( 1 - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{53} + ( -6 + \beta_{3} - 5 \beta_{4} + 6 \beta_{6} + 5 \beta_{8} + \beta_{9} ) q^{54} + ( 5 \beta_{1} - 5 \beta_{2} + 3 \beta_{6} - 2 \beta_{9} ) q^{56} + ( -2 + \beta_{2} - 2 \beta_{4} - \beta_{5} ) q^{57} + ( -\beta_{1} + 2 \beta_{2} - \beta_{5} - 7 \beta_{6} - \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{58} + ( -2 \beta_{1} + \beta_{2} + \beta_{5} + 4 \beta_{6} + \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{59} + ( \beta_{1} - \beta_{2} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{61} + ( -1 + 3 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{62} + ( -6 + 2 \beta_{1} - \beta_{4} + 6 \beta_{6} + \beta_{7} + \beta_{8} ) q^{63} + ( -4 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{64} + ( 10 + 3 \beta_{2} + \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{66} + ( 1 - \beta_{1} + \beta_{3} - 5 \beta_{4} - \beta_{6} + 5 \beta_{8} + \beta_{9} ) q^{67} + ( -3 + 7 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{6} - 4 \beta_{8} - 2 \beta_{9} ) q^{68} + ( \beta_{1} - 3 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} ) q^{69} + ( -2 \beta_{1} + 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} - \beta_{8} - 3 \beta_{9} ) q^{71} + ( 3 \beta_{1} - 5 \beta_{2} + 2 \beta_{5} + 11 \beta_{6} + 2 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{72} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{73} + ( -4 \beta_{1} + \beta_{2} + 3 \beta_{5} + 3 \beta_{7} - 2 \beta_{8} ) q^{74} + ( -4 + 3 \beta_{1} + \beta_{4} + 4 \beta_{6} - \beta_{7} - \beta_{8} ) q^{76} + ( 7 + 3 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{77} + ( 7 + \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 6 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{78} + ( -2 - 4 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{79} + ( -9 + 5 \beta_{1} + 2 \beta_{3} - 5 \beta_{4} + 9 \beta_{6} + 2 \beta_{7} + 5 \beta_{8} + 2 \beta_{9} ) q^{81} + ( -3 \beta_{1} + 2 \beta_{2} + \beta_{5} + 4 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{82} + ( -4 - \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{83} + ( -3 \beta_{1} + 3 \beta_{2} + 7 \beta_{6} + \beta_{8} - \beta_{9} ) q^{84} + ( -9 - 6 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{86} + ( \beta_{1} + \beta_{2} - 2 \beta_{5} - 5 \beta_{6} - 2 \beta_{7} ) q^{87} + ( 1 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{88} + ( 1 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{89} + ( 4 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{5} - 4 \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{91} + ( 12 + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{92} + ( 9 + 2 \beta_{1} + 4 \beta_{4} - 9 \beta_{6} - 2 \beta_{7} - 4 \beta_{8} ) q^{93} + ( 6 + \beta_{3} - 6 \beta_{6} + \beta_{9} ) q^{94} + ( 5 + \beta_{2} + 5 \beta_{4} - \beta_{5} ) q^{96} + ( 3 \beta_{1} - 4 \beta_{2} + \beta_{5} - 7 \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{97} + ( -5 \beta_{1} + 3 \beta_{2} + 2 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{98} + ( 9 - 3 \beta_{2} - \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 6 q^{4} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 4 q^{9} + O(q^{10}) \) \( 10 q - 3 q^{3} - 6 q^{4} - 3 q^{6} + 2 q^{7} + 6 q^{8} - 4 q^{9} - 3 q^{11} + 4 q^{12} + 10 q^{13} - 16 q^{14} - 4 q^{16} - 4 q^{17} - 4 q^{18} + 4 q^{19} - 16 q^{21} - 8 q^{22} - 15 q^{23} + 14 q^{24} - 9 q^{26} + 42 q^{27} + 17 q^{28} + q^{29} - 31 q^{32} + 2 q^{33} + 54 q^{34} + 13 q^{36} - 17 q^{37} + 6 q^{38} - 10 q^{39} - 6 q^{41} - 32 q^{42} - 12 q^{43} - 16 q^{44} + 7 q^{46} - 24 q^{47} + 2 q^{48} - 7 q^{49} + 23 q^{52} + 16 q^{53} - 19 q^{54} + 17 q^{56} - 8 q^{57} - 38 q^{58} + 12 q^{59} - 5 q^{61} - 13 q^{62} - 26 q^{63} - 10 q^{64} + 86 q^{66} + 16 q^{67} - 25 q^{68} - 20 q^{69} - 19 q^{71} + 45 q^{72} - 16 q^{73} - 2 q^{74} - 24 q^{76} + 64 q^{77} + 42 q^{78} - 28 q^{79} - 29 q^{81} + 23 q^{82} - 14 q^{83} + 34 q^{84} - 84 q^{86} - 21 q^{87} - 2 q^{88} + 10 q^{89} + 17 q^{91} + 142 q^{92} + 33 q^{93} + 31 q^{94} + 34 q^{96} - 37 q^{97} + 21 q^{98} + 56 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 8 x^{8} - 2 x^{7} + 52 x^{6} - 5 x^{5} + 97 x^{4} + 60 x^{3} + 141 x^{2} + 36 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 1304 \nu^{9} + 1776 \nu^{8} - 1628 \nu^{7} + 8447 \nu^{6} - 14800 \nu^{5} + 93980 \nu^{4} - 511687 \nu^{3} + 165612 \nu^{2} + 42624 \nu - 242496 \)\()/983355\)
\(\beta_{3}\)\(=\)\((\)\( 592 \nu^{9} - 4020 \nu^{8} + 3685 \nu^{7} - 27536 \nu^{6} + 33500 \nu^{5} - 212725 \nu^{4} + 29124 \nu^{3} - 374865 \nu^{2} - 96480 \nu - 987267 \)\()/327785\)
\(\beta_{4}\)\(=\)\((\)\( -4744 \nu^{9} - 20940 \nu^{8} + 19195 \nu^{7} - 124843 \nu^{6} + 174500 \nu^{5} - 1108075 \nu^{4} + 1662452 \nu^{3} - 1952655 \nu^{2} - 502560 \nu - 1749321 \)\()/983355\)
\(\beta_{5}\)\(=\)\((\)\( 3213 \nu^{9} - 12516 \nu^{8} + 11473 \nu^{7} - 129958 \nu^{6} + 104300 \nu^{5} - 662305 \nu^{4} + 22966 \nu^{3} - 1167117 \nu^{2} - 300384 \nu - 553661 \)\()/327785\)
\(\beta_{6}\)\(=\)\((\)\( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + 2707548 \nu^{3} + 1104953 \nu^{2} + 3964716 \nu + 1012608 \)\()/983355\)
\(\beta_{7}\)\(=\)\((\)\( -47884 \nu^{9} + 55843 \nu^{8} - 351659 \nu^{7} + 476704 \nu^{6} - 2541330 \nu^{5} + 2272140 \nu^{4} - 4375563 \nu^{3} - 938609 \nu^{2} - 4559898 \nu + 13443 \)\()/983355\)
\(\beta_{8}\)\(=\)\((\)\( -75292 \nu^{9} - 9639 \nu^{8} - 564788 \nu^{7} + 116165 \nu^{6} - 3525310 \nu^{5} + 63560 \nu^{4} - 5316409 \nu^{3} - 4586418 \nu^{2} - 7114821 \nu - 1809360 \)\()/983355\)
\(\beta_{9}\)\(=\)\((\)\( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + 2707548 \nu^{3} + 1432738 \nu^{2} + 3964716 \nu + 1012608 \)\()/327785\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} - 3 \beta_{6}\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} - 5 \beta_{2}\)
\(\nu^{4}\)\(=\)\(-6 \beta_{9} - \beta_{8} - \beta_{7} + 14 \beta_{6} + \beta_{4} - 6 \beta_{3} + 3 \beta_{1} - 14\)
\(\nu^{5}\)\(=\)\(9 \beta_{9} + 8 \beta_{8} - 6 \beta_{6} + 28 \beta_{2} - 28 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-8 \beta_{5} - 9 \beta_{4} + 37 \beta_{3} - 24 \beta_{2} + 76\)
\(\nu^{7}\)\(=\)\(-69 \beta_{9} - 53 \beta_{8} - \beta_{7} + 71 \beta_{6} + 53 \beta_{4} - 69 \beta_{3} + 167 \beta_{1} - 71\)
\(\nu^{8}\)\(=\)\(236 \beta_{9} + 71 \beta_{8} + 52 \beta_{7} - 446 \beta_{6} + 52 \beta_{5} + 211 \beta_{2} - 263 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-19 \beta_{5} - 340 \beta_{4} + 499 \beta_{3} - 1022 \beta_{2} + 614\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−1.30241 + 2.25583i
−0.547998 + 0.949161i
−0.134432 + 0.232843i
0.904178 1.56608i
1.08066 1.87176i
−1.30241 2.25583i
−0.547998 0.949161i
−0.134432 0.232843i
0.904178 + 1.56608i
1.08066 + 1.87176i
−1.30241 + 2.25583i −0.0676602 + 0.117191i −2.39253 4.14398i 0 −0.176242 0.305261i 1.62616 + 2.81660i 7.25455 1.49084 + 2.58222i 0
126.2 −0.547998 + 0.949161i −1.68234 + 2.91389i 0.399395 + 0.691773i 0 −1.84383 3.19362i −0.795836 1.37843i −3.06747 −4.16051 7.20621i 0
126.3 −0.134432 + 0.232843i 0.301414 0.522064i 0.963856 + 1.66945i 0 0.0810394 + 0.140364i 0.715471 + 1.23923i −1.05602 1.31830 + 2.28336i 0
126.4 0.904178 1.56608i 0.929015 1.60910i −0.635076 1.09998i 0 −1.67999 2.90983i −2.08417 3.60988i 1.31983 −0.226138 0.391682i 0
126.5 1.08066 1.87176i −0.980433 + 1.69816i −1.33565 2.31341i 0 2.11903 + 3.67026i 1.53837 + 2.66453i −1.45089 −0.422497 0.731787i 0
276.1 −1.30241 2.25583i −0.0676602 0.117191i −2.39253 + 4.14398i 0 −0.176242 + 0.305261i 1.62616 2.81660i 7.25455 1.49084 2.58222i 0
276.2 −0.547998 0.949161i −1.68234 2.91389i 0.399395 0.691773i 0 −1.84383 + 3.19362i −0.795836 + 1.37843i −3.06747 −4.16051 + 7.20621i 0
276.3 −0.134432 0.232843i 0.301414 + 0.522064i 0.963856 1.66945i 0 0.0810394 0.140364i 0.715471 1.23923i −1.05602 1.31830 2.28336i 0
276.4 0.904178 + 1.56608i 0.929015 + 1.60910i −0.635076 + 1.09998i 0 −1.67999 + 2.90983i −2.08417 + 3.60988i 1.31983 −0.226138 + 0.391682i 0
276.5 1.08066 + 1.87176i −0.980433 1.69816i −1.33565 + 2.31341i 0 2.11903 3.67026i 1.53837 2.66453i −1.45089 −0.422497 + 0.731787i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 276.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.c 10
5.b even 2 1 325.2.e.d yes 10
5.c odd 4 2 325.2.o.c 20
13.c even 3 1 inner 325.2.e.c 10
13.c even 3 1 4225.2.a.bp 5
13.e even 6 1 4225.2.a.bo 5
65.l even 6 1 4225.2.a.bm 5
65.n even 6 1 325.2.e.d yes 10
65.n even 6 1 4225.2.a.bn 5
65.q odd 12 2 325.2.o.c 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.e.c 10 1.a even 1 1 trivial
325.2.e.c 10 13.c even 3 1 inner
325.2.e.d yes 10 5.b even 2 1
325.2.e.d yes 10 65.n even 6 1
325.2.o.c 20 5.c odd 4 2
325.2.o.c 20 65.q odd 12 2
4225.2.a.bm 5 65.l even 6 1
4225.2.a.bn 5 65.n even 6 1
4225.2.a.bo 5 13.e even 6 1
4225.2.a.bp 5 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 36 T + 141 T^{2} + 60 T^{3} + 97 T^{4} - 5 T^{5} + 52 T^{6} - 2 T^{7} + 8 T^{8} + T^{10} \)
$3$ \( 1 + 6 T + 47 T^{2} - 56 T^{3} + 148 T^{4} + 20 T^{5} + 52 T^{6} + 7 T^{7} + 14 T^{8} + 3 T^{9} + T^{10} \)
$5$ \( T^{10} \)
$7$ \( 9025 - 4180 T + 6116 T^{2} - 1674 T^{3} + 2582 T^{4} - 755 T^{5} + 405 T^{6} - 50 T^{7} + 23 T^{8} - 2 T^{9} + T^{10} \)
$11$ \( 9 - 27 T + 249 T^{2} + 606 T^{3} + 2974 T^{4} + 1009 T^{5} + 466 T^{6} + 61 T^{7} + 26 T^{8} + 3 T^{9} + T^{10} \)
$13$ \( 371293 - 285610 T + 131820 T^{2} - 44447 T^{3} + 13988 T^{4} - 3957 T^{5} + 1076 T^{6} - 263 T^{7} + 60 T^{8} - 10 T^{9} + T^{10} \)
$17$ \( 239121 + 111492 T + 98439 T^{2} + 13548 T^{3} + 15277 T^{4} + 2085 T^{5} + 1448 T^{6} + 46 T^{7} + 52 T^{8} + 4 T^{9} + T^{10} \)
$19$ \( 225 - 30 T + 499 T^{2} - 174 T^{3} + 1045 T^{4} - 263 T^{5} + 194 T^{6} - 34 T^{7} + 24 T^{8} - 4 T^{9} + T^{10} \)
$23$ \( 281961 + 543213 T + 869706 T^{2} + 370395 T^{3} + 147498 T^{4} + 20835 T^{5} + 6802 T^{6} + 1086 T^{7} + 197 T^{8} + 15 T^{9} + T^{10} \)
$29$ \( 881721 - 456354 T + 376107 T^{2} - 49656 T^{3} + 52852 T^{4} - 9652 T^{5} + 3888 T^{6} - 233 T^{7} + 66 T^{8} - T^{9} + T^{10} \)
$31$ \( ( 225 + 495 T - 29 T^{2} - 57 T^{3} + T^{5} )^{2} \)
$37$ \( 826281 + 1104435 T + 1256247 T^{2} + 390384 T^{3} + 138412 T^{4} + 27575 T^{5} + 8138 T^{6} + 1385 T^{7} + 236 T^{8} + 17 T^{9} + T^{10} \)
$41$ \( 50625 - 86400 T + 237231 T^{2} + 186516 T^{3} + 129435 T^{4} + 34359 T^{5} + 8254 T^{6} + 354 T^{7} + 110 T^{8} + 6 T^{9} + T^{10} \)
$43$ \( 14190289 + 5115586 T + 4010189 T^{2} - 374014 T^{3} + 358753 T^{4} + 2225 T^{5} + 8458 T^{6} + 502 T^{7} + 198 T^{8} + 12 T^{9} + T^{10} \)
$47$ \( ( 2025 + 90 T - 311 T^{2} - 20 T^{3} + 12 T^{4} + T^{5} )^{2} \)
$53$ \( ( 6075 + 3645 T + 279 T^{2} - 117 T^{3} - 8 T^{4} + T^{5} )^{2} \)
$59$ \( 4782969 + 2657205 T + 4133430 T^{2} - 1856763 T^{3} + 1344276 T^{4} - 137052 T^{5} + 23364 T^{6} - 1386 T^{7} + 231 T^{8} - 12 T^{9} + T^{10} \)
$61$ \( 403225 + 172085 T + 194091 T^{2} + 8200 T^{3} + 45662 T^{4} + 6855 T^{5} + 2888 T^{6} + 145 T^{7} + 72 T^{8} + 5 T^{9} + T^{10} \)
$67$ \( 2567550241 + 10742252 T + 127229154 T^{2} - 13807922 T^{3} + 5461592 T^{4} - 386265 T^{5} + 57533 T^{6} - 2924 T^{7} + 387 T^{8} - 16 T^{9} + T^{10} \)
$71$ \( 101787921 - 36259866 T + 26869923 T^{2} + 5111748 T^{3} + 1695840 T^{4} + 156342 T^{5} + 29920 T^{6} + 2633 T^{7} + 368 T^{8} + 19 T^{9} + T^{10} \)
$73$ \( ( 10125 + 900 T - 990 T^{2} - 119 T^{3} + 8 T^{4} + T^{5} )^{2} \)
$79$ \( ( 16875 + 2250 T - 1350 T^{2} - 95 T^{3} + 14 T^{4} + T^{5} )^{2} \)
$83$ \( ( -53775 + 19110 T - 461 T^{2} - 257 T^{3} + 7 T^{4} + T^{5} )^{2} \)
$89$ \( 2537649 + 200718 T + 770958 T^{2} - 183978 T^{3} + 203832 T^{4} - 22599 T^{5} + 6387 T^{6} - 558 T^{7} + 139 T^{8} - 10 T^{9} + T^{10} \)
$97$ \( 1683378841 + 867147915 T + 419732172 T^{2} + 44083039 T^{3} + 9727402 T^{4} + 1281185 T^{5} + 180868 T^{6} + 14930 T^{7} + 1001 T^{8} + 37 T^{9} + T^{10} \)
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