Properties

Label 45.4.e.b
Level $45$
Weight $4$
Character orbit 45.e
Analytic conductor $2.655$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.65508595026\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.15759792.1
Defining polynomial: \(x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} + \beta_{5} ) q^{2} + ( 1 + \beta_{2} - \beta_{5} ) q^{3} + ( 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{4} + 5 \beta_{3} q^{5} + ( 17 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{6} + ( 16 - \beta_{1} - 6 \beta_{2} + 16 \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{7} + ( -9 + 3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -2 + 4 \beta_{1} + 7 \beta_{2} - 22 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -\beta_{1} + \beta_{5} ) q^{2} + ( 1 + \beta_{2} - \beta_{5} ) q^{3} + ( 4 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{4} + 5 \beta_{3} q^{5} + ( 17 - 2 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - \beta_{4} ) q^{6} + ( 16 - \beta_{1} - 6 \beta_{2} + 16 \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{7} + ( -9 + 3 \beta_{1} + 3 \beta_{2} ) q^{8} + ( -2 + 4 \beta_{1} + 7 \beta_{2} - 22 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} ) q^{9} + 5 \beta_{1} q^{10} + ( -3 + 14 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - 14 \beta_{5} ) q^{11} + ( 9 - 11 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} - 10 \beta_{4} + 17 \beta_{5} ) q^{12} + ( 17 \beta_{3} + 3 \beta_{4} - 14 \beta_{5} ) q^{13} + ( -18 \beta_{3} + 3 \beta_{4} + 24 \beta_{5} ) q^{14} + ( -5 \beta_{1} + 5 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{15} + ( 11 + 2 \beta_{1} - 18 \beta_{2} + 11 \beta_{3} - 18 \beta_{4} - 2 \beta_{5} ) q^{16} + ( -57 - 2 \beta_{1} + 3 \beta_{2} ) q^{17} + ( -14 - 14 \beta_{1} + 4 \beta_{2} - 13 \beta_{3} + 5 \beta_{4} - 17 \beta_{5} ) q^{18} + ( -55 - 10 \beta_{1} - 9 \beta_{2} ) q^{19} + ( -20 + 10 \beta_{1} + 15 \beta_{2} - 20 \beta_{3} + 15 \beta_{4} - 10 \beta_{5} ) q^{20} + ( -51 - 36 \beta_{1} - 33 \beta_{3} - 3 \beta_{4} + 12 \beta_{5} ) q^{21} + ( -123 \beta_{3} + 33 \beta_{4} - 40 \beta_{5} ) q^{22} + ( 48 \beta_{3} + 36 \beta_{4} - 9 \beta_{5} ) q^{23} + ( 3 + 6 \beta_{1} + 3 \beta_{2} + 21 \beta_{3} - 6 \beta_{4} + 18 \beta_{5} ) q^{24} + ( -25 - 25 \beta_{3} ) q^{25} + ( 153 - 14 \beta_{1} - 39 \beta_{2} ) q^{26} + ( 42 + 45 \beta_{1} + 24 \beta_{2} + 27 \beta_{3} - 9 \beta_{4} - 24 \beta_{5} ) q^{27} + ( -175 + 19 \beta_{1} + 27 \beta_{2} ) q^{28} + ( 117 - 14 \beta_{1} - 30 \beta_{2} + 117 \beta_{3} - 30 \beta_{4} + 14 \beta_{5} ) q^{29} + ( -25 + 5 \beta_{2} + 60 \beta_{3} - 15 \beta_{4} + 10 \beta_{5} ) q^{30} + ( -127 \beta_{3} - 33 \beta_{4} + 62 \beta_{5} ) q^{31} + ( -42 \beta_{3} + 49 \beta_{5} ) q^{32} + ( -115 + 58 \beta_{1} + 71 \beta_{2} + 8 \beta_{3} + 38 \beta_{4} - 18 \beta_{5} ) q^{33} + ( 39 + 56 \beta_{1} - 9 \beta_{2} + 39 \beta_{3} - 9 \beta_{4} - 56 \beta_{5} ) q^{34} + ( -80 + 5 \beta_{1} + 30 \beta_{2} ) q^{35} + ( 191 - 46 \beta_{1} - 52 \beta_{2} + 28 \beta_{3} - 62 \beta_{4} + 26 \beta_{5} ) q^{36} + ( 143 - 24 \beta_{1} - 51 \beta_{2} ) q^{37} + ( 75 + 26 \beta_{1} - 21 \beta_{2} + 75 \beta_{3} - 21 \beta_{4} - 26 \beta_{5} ) q^{38} + ( -153 + 14 \beta_{1} + 39 \beta_{2} - 179 \beta_{3} + 79 \beta_{4} - 20 \beta_{5} ) q^{39} + ( -45 \beta_{3} + 15 \beta_{4} - 15 \beta_{5} ) q^{40} + ( 72 \beta_{3} - 69 \beta_{4} + 40 \beta_{5} ) q^{41} + ( 303 - 27 \beta_{1} - 75 \beta_{2} + 432 \beta_{3} - 108 \beta_{4} + 21 \beta_{5} ) q^{42} + ( 169 - 8 \beta_{1} + 87 \beta_{2} + 169 \beta_{3} + 87 \beta_{4} + 8 \beta_{5} ) q^{43} + ( 291 - 124 \beta_{1} - 15 \beta_{2} ) q^{44} + ( 110 - 10 \beta_{1} - 25 \beta_{2} + 100 \beta_{3} + 10 \beta_{4} - 10 \beta_{5} ) q^{45} + ( -72 - 6 \beta_{1} + 9 \beta_{2} ) q^{46} + ( -207 + 59 \beta_{1} + 15 \beta_{2} - 207 \beta_{3} + 15 \beta_{4} - 59 \beta_{5} ) q^{47} + ( -275 - 61 \beta_{1} - 17 \beta_{2} - 161 \beta_{3} - 41 \beta_{4} + 36 \beta_{5} ) q^{48} + ( 189 \beta_{3} - 111 \beta_{4} - 26 \beta_{5} ) q^{49} -25 \beta_{5} q^{50} + ( -20 + 6 \beta_{1} - 50 \beta_{2} - 39 \beta_{3} + 9 \beta_{4} + 56 \beta_{5} ) q^{51} + ( 109 - 108 \beta_{1} + 21 \beta_{2} + 109 \beta_{3} + 21 \beta_{4} + 108 \beta_{5} ) q^{52} + ( -228 + 70 \beta_{1} + 24 \beta_{2} ) q^{53} + ( -87 + 60 \beta_{1} + 30 \beta_{2} - 420 \beta_{3} + 111 \beta_{4} - 72 \beta_{5} ) q^{54} + ( 15 - 70 \beta_{1} - 45 \beta_{2} ) q^{55} + ( -237 + 48 \beta_{1} + 54 \beta_{2} - 237 \beta_{3} + 54 \beta_{4} - 48 \beta_{5} ) q^{56} + ( -86 - 18 \beta_{1} - 92 \beta_{2} - 75 \beta_{3} + 21 \beta_{4} + 26 \beta_{5} ) q^{57} + ( 18 \beta_{3} - 12 \beta_{4} + 175 \beta_{5} ) q^{58} + ( -36 \beta_{3} - 108 \beta_{4} - 82 \beta_{5} ) q^{59} + ( 20 + 85 \beta_{1} + 50 \beta_{2} + 65 \beta_{3} + 35 \beta_{4} - 30 \beta_{5} ) q^{60} + ( 448 - 20 \beta_{1} + 75 \beta_{2} + 448 \beta_{3} + 75 \beta_{4} + 20 \beta_{5} ) q^{61} + ( -579 + 30 \beta_{1} + 153 \beta_{2} ) q^{62} + ( 258 + 6 \beta_{1} - 120 \beta_{2} - 303 \beta_{3} + 21 \beta_{4} - 21 \beta_{5} ) q^{63} + ( -500 + 72 \beta_{1} + 3 \beta_{2} ) q^{64} + ( -85 - 70 \beta_{1} - 15 \beta_{2} - 85 \beta_{3} - 15 \beta_{4} + 70 \beta_{5} ) q^{65} + ( -315 + 236 \beta_{1} + 87 \beta_{2} - 341 \beta_{3} + 103 \beta_{4} - 302 \beta_{5} ) q^{66} + ( -637 \beta_{3} + 21 \beta_{4} + 41 \beta_{5} ) q^{67} + ( -261 \beta_{3} + 153 \beta_{4} - 80 \beta_{5} ) q^{68} + ( 72 + 6 \beta_{1} - 9 \beta_{2} + 399 \beta_{3} + 156 \beta_{4} - 78 \beta_{5} ) q^{69} + ( 90 + 120 \beta_{1} - 15 \beta_{2} + 90 \beta_{3} - 15 \beta_{4} - 120 \beta_{5} ) q^{70} + ( 81 - 76 \beta_{1} - 249 \beta_{2} ) q^{71} + ( 186 - 57 \beta_{1} - 30 \beta_{2} + 300 \beta_{3} - 78 \beta_{4} + 87 \beta_{5} ) q^{72} + ( -172 + 44 \beta_{1} + 192 \beta_{2} ) q^{73} + ( 33 - 242 \beta_{1} - 21 \beta_{2} + 33 \beta_{3} - 21 \beta_{4} + 242 \beta_{5} ) q^{74} + ( -25 + 25 \beta_{1} - 25 \beta_{2} - 25 \beta_{3} - 25 \beta_{4} ) q^{75} + ( 23 \beta_{3} + 171 \beta_{4} - 36 \beta_{5} ) q^{76} + ( -237 \beta_{3} + 3 \beta_{4} - 240 \beta_{5} ) q^{77} + ( -128 - 78 \beta_{1} + 22 \beta_{2} + 27 \beta_{3} + 3 \beta_{4} - 220 \beta_{5} ) q^{78} + ( 154 - 112 \beta_{1} - 192 \beta_{2} + 154 \beta_{3} - 192 \beta_{4} + 112 \beta_{5} ) q^{79} + ( -55 - 10 \beta_{1} + 90 \beta_{2} ) q^{80} + ( -300 + 60 \beta_{1} + 240 \beta_{2} - 87 \beta_{3} - 6 \beta_{4} + 78 \beta_{5} ) q^{81} + ( -135 + 221 \beta_{1} + 51 \beta_{2} ) q^{82} + ( 360 - 105 \beta_{1} - 354 \beta_{2} + 360 \beta_{3} - 354 \beta_{4} + 105 \beta_{5} ) q^{83} + ( -27 + 54 \beta_{1} - 75 \beta_{2} + 93 \beta_{3} - 30 \beta_{4} + 240 \beta_{5} ) q^{84} + ( -285 \beta_{3} + 15 \beta_{4} + 10 \beta_{5} ) q^{85} + ( 531 \beta_{3} - 111 \beta_{4} + 98 \beta_{5} ) q^{86} + ( -65 - 235 \beta_{1} + \beta_{2} - 83 \beta_{3} + 13 \beta_{4} + 60 \beta_{5} ) q^{87} + ( 429 - 234 \beta_{1} - 93 \beta_{2} + 429 \beta_{3} - 93 \beta_{4} + 234 \beta_{5} ) q^{88} + ( 549 + 168 \beta_{1} + 240 \beta_{2} ) q^{89} + ( 65 - 85 \beta_{1} - 25 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} + 155 \beta_{5} ) q^{90} + ( -377 - 286 \beta_{1} + 135 \beta_{2} ) q^{91} + ( 501 + 141 \beta_{1} + 261 \beta_{2} + 501 \beta_{3} + 261 \beta_{4} - 141 \beta_{5} ) q^{92} + ( 579 - 30 \beta_{1} - 153 \beta_{2} + 465 \beta_{3} - 441 \beta_{4} + 96 \beta_{5} ) q^{93} + ( -633 \beta_{3} + 162 \beta_{4} - 340 \beta_{5} ) q^{94} + ( -275 \beta_{3} - 45 \beta_{4} + 50 \beta_{5} ) q^{95} + ( 588 - 56 \beta_{1} - 147 \beta_{2} + 791 \beta_{3} - 238 \beta_{4} + 56 \beta_{5} ) q^{96} + ( 100 + 78 \beta_{1} + 60 \beta_{2} + 100 \beta_{3} + 60 \beta_{4} - 78 \beta_{5} ) q^{97} + ( 867 + 248 \beta_{1} - 189 \beta_{2} ) q^{98} + ( 208 + 208 \beta_{1} + 172 \beta_{2} + 575 \beta_{3} + 53 \beta_{4} + 160 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} + 9 q^{3} - 11 q^{4} - 15 q^{5} + 84 q^{6} + 43 q^{7} - 54 q^{8} + 57 q^{9} + O(q^{10}) \) \( 6 q + q^{2} + 9 q^{3} - 11 q^{4} - 15 q^{5} + 84 q^{6} + 43 q^{7} - 54 q^{8} + 57 q^{9} - 10 q^{10} - 14 q^{11} + 75 q^{12} - 40 q^{13} + 27 q^{14} - 15 q^{15} + 13 q^{16} - 332 q^{17} + 3 q^{18} - 328 q^{19} - 55 q^{20} - 144 q^{21} + 376 q^{22} - 171 q^{23} - 63 q^{24} - 75 q^{25} + 868 q^{26} + 162 q^{27} - 1034 q^{28} + 335 q^{29} - 315 q^{30} + 352 q^{31} + 77 q^{32} - 708 q^{33} + 52 q^{34} - 430 q^{35} + 1086 q^{36} + 804 q^{37} + 178 q^{38} - 390 q^{39} + 135 q^{40} - 187 q^{41} + 513 q^{42} + 602 q^{43} + 1964 q^{44} + 330 q^{45} - 402 q^{46} - 665 q^{47} - 1074 q^{48} - 430 q^{49} + 25 q^{50} - 180 q^{51} + 456 q^{52} - 1460 q^{53} + 639 q^{54} + 140 q^{55} - 705 q^{56} - 486 q^{57} - 217 q^{58} + 298 q^{59} - 150 q^{60} + 1439 q^{61} - 3228 q^{62} + 2205 q^{63} - 3138 q^{64} - 200 q^{65} - 966 q^{66} + 1849 q^{67} + 710 q^{68} - 873 q^{69} + 135 q^{70} + 140 q^{71} + 261 q^{72} - 736 q^{73} + 320 q^{74} - 150 q^{75} - 204 q^{76} + 948 q^{77} - 432 q^{78} + 382 q^{79} - 130 q^{80} - 1251 q^{81} - 1150 q^{82} + 831 q^{83} - 909 q^{84} + 830 q^{85} - 1580 q^{86} + 258 q^{87} + 1428 q^{88} + 3438 q^{89} + 375 q^{90} - 1420 q^{91} + 1623 q^{92} + 2178 q^{93} + 2077 q^{94} + 820 q^{95} + 1155 q^{96} + 282 q^{97} + 4328 q^{98} - 762 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} + 16 x^{4} - 27 x^{3} + 52 x^{2} - 39 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu + 4 \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{3} - 11 \nu^{2} + 10 \nu - 12 \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{5} - 5 \nu^{4} + 30 \nu^{3} - 40 \nu^{2} + 88 \nu - 39 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( -7 \nu^{5} + 18 \nu^{4} - 103 \nu^{3} + 141 \nu^{2} - 289 \nu + 126 \)\()/3\)
\(\beta_{5}\)\(=\)\((\)\( -8 \nu^{5} + 20 \nu^{4} - 117 \nu^{3} + 157 \nu^{2} - 334 \nu + 147 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + 4 \beta_{1} - 11\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(13 \beta_{5} - 10 \beta_{4} + 17 \beta_{3} - 5 \beta_{2} - 2 \beta_{1} - 8\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(28 \beta_{5} - 22 \beta_{4} + 35 \beta_{3} - 20 \beta_{2} - 38 \beta_{1} + 79\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(-77 \beta_{5} + 47 \beta_{4} - 139 \beta_{3} + \beta_{2} - 29 \beta_{1} + 112\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1
0.500000 0.0378788i
0.500000 1.98116i
0.500000 + 2.88506i
0.500000 + 0.0378788i
0.500000 + 1.98116i
0.500000 2.88506i
−1.87428 + 3.24635i −4.05724 3.24635i −3.02587 5.24096i −2.50000 4.33013i 18.1432 7.08665i 15.6746 27.1492i −7.30318 5.92239 + 26.3425i 18.7428
16.2 0.0874923 0.151541i 5.19394 + 0.151541i 3.98469 + 6.90169i −2.50000 4.33013i 0.477395 0.773837i −4.23186 + 7.32979i 2.79440 26.9541 + 1.57419i −0.874923
16.3 2.28679 3.96084i 3.36330 + 3.96084i −6.45882 11.1870i −2.50000 4.33013i 23.3794 4.26387i 10.0573 17.4197i −22.4912 −4.37646 + 26.6429i −22.8679
31.1 −1.87428 3.24635i −4.05724 + 3.24635i −3.02587 + 5.24096i −2.50000 + 4.33013i 18.1432 + 7.08665i 15.6746 + 27.1492i −7.30318 5.92239 26.3425i 18.7428
31.2 0.0874923 + 0.151541i 5.19394 0.151541i 3.98469 6.90169i −2.50000 + 4.33013i 0.477395 + 0.773837i −4.23186 7.32979i 2.79440 26.9541 1.57419i −0.874923
31.3 2.28679 + 3.96084i 3.36330 3.96084i −6.45882 + 11.1870i −2.50000 + 4.33013i 23.3794 + 4.26387i 10.0573 + 17.4197i −22.4912 −4.37646 26.6429i −22.8679
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.4.e.b 6
3.b odd 2 1 135.4.e.b 6
5.b even 2 1 225.4.e.c 6
5.c odd 4 2 225.4.k.c 12
9.c even 3 1 inner 45.4.e.b 6
9.c even 3 1 405.4.a.h 3
9.d odd 6 1 135.4.e.b 6
9.d odd 6 1 405.4.a.j 3
45.h odd 6 1 2025.4.a.q 3
45.j even 6 1 225.4.e.c 6
45.j even 6 1 2025.4.a.s 3
45.k odd 12 2 225.4.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.e.b 6 1.a even 1 1 trivial
45.4.e.b 6 9.c even 3 1 inner
135.4.e.b 6 3.b odd 2 1
135.4.e.b 6 9.d odd 6 1
225.4.e.c 6 5.b even 2 1
225.4.e.c 6 45.j even 6 1
225.4.k.c 12 5.c odd 4 2
225.4.k.c 12 45.k odd 12 2
405.4.a.h 3 9.c even 3 1
405.4.a.j 3 9.d odd 6 1
2025.4.a.q 3 45.h odd 6 1
2025.4.a.s 3 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - T_{2}^{5} + 18 T_{2}^{4} + 11 T_{2}^{3} + 292 T_{2}^{2} - 51 T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 - 51 T + 292 T^{2} + 11 T^{3} + 18 T^{4} - T^{5} + T^{6} \)
$3$ \( 19683 - 6561 T + 324 T^{2} + 81 T^{3} + 12 T^{4} - 9 T^{5} + T^{6} \)
$5$ \( ( 25 + 5 T + T^{2} )^{3} \)
$7$ \( 28483569 + 1040715 T + 267516 T^{2} - 19059 T^{3} + 1654 T^{4} - 43 T^{5} + T^{6} \)
$11$ \( 1896428304 - 122631168 T + 7320184 T^{2} - 126520 T^{3} + 3012 T^{4} + 14 T^{5} + T^{6} \)
$13$ \( 5679732496 + 184792528 T + 9026864 T^{2} + 52648 T^{3} + 4052 T^{4} + 40 T^{5} + T^{6} \)
$17$ \( ( 156324 + 8920 T + 166 T^{2} + T^{3} )^{2} \)
$19$ \( ( 57316 + 7292 T + 164 T^{2} + T^{3} )^{2} \)
$23$ \( 3746541681 + 295823097 T + 33824628 T^{2} - 704025 T^{3} + 34074 T^{4} + 171 T^{5} + T^{6} \)
$29$ \( 11463342489 + 2926248177 T + 782851006 T^{2} - 9370019 T^{3} + 84894 T^{4} - 335 T^{5} + T^{6} \)
$31$ \( 97238137683600 - 137382616080 T + 3665151504 T^{2} - 14817816 T^{3} + 137836 T^{4} - 352 T^{5} + T^{6} \)
$37$ \( ( 3335284 + 24708 T - 402 T^{2} + T^{3} )^{2} \)
$41$ \( 52247959475625 + 322359380175 T + 3340579834 T^{2} + 6116911 T^{3} + 79566 T^{4} + 187 T^{5} + T^{6} \)
$43$ \( 238533321586576 + 153765680944 T + 9396725384 T^{2} - 36882560 T^{3} + 352448 T^{4} - 602 T^{5} + T^{6} \)
$47$ \( 6187571675289 + 237009867723 T + 7424292766 T^{2} + 58386899 T^{3} + 346944 T^{4} + 665 T^{5} + T^{6} \)
$53$ \( ( 3250536 + 106300 T + 730 T^{2} + T^{3} )^{2} \)
$59$ \( 16224618881802816 - 45173097261024 T + 163730383744 T^{2} - 149067880 T^{3} + 443448 T^{4} - 298 T^{5} + T^{6} \)
$61$ \( 3092861048569009 - 32773423076579 T + 267254918066 T^{2} - 736785779 T^{3} + 1481414 T^{4} - 1439 T^{5} + T^{6} \)
$67$ \( 43587484566793281 - 228333683246643 T + 810102262338 T^{2} - 1604656455 T^{3} + 2325124 T^{4} - 1849 T^{5} + T^{6} \)
$71$ \( ( 223775052 - 685460 T - 70 T^{2} + T^{3} )^{2} \)
$73$ \( ( -134927744 - 372928 T + 368 T^{2} + T^{3} )^{2} \)
$79$ \( 19133137244437056 - 53658650075616 T + 203324256864 T^{2} - 128458200 T^{3} + 533848 T^{4} - 382 T^{5} + T^{6} \)
$83$ \( 913637410143880041 - 1109708868397833 T + 2142164521980 T^{2} - 946919079 T^{3} + 1851534 T^{4} - 831 T^{5} + T^{6} \)
$89$ \( ( 125506395 + 238491 T - 1719 T^{2} + T^{3} )^{2} \)
$97$ \( 285551326213696 - 1143471728352 T + 9344268672 T^{2} - 14714152 T^{3} + 147192 T^{4} - 282 T^{5} + T^{6} \)
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