Properties

Label 45.3.h.a
Level $45$
Weight $3$
Character orbit 45.h
Analytic conductor $1.226$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,3,Mod(14,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 45.h (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.22616118962\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{5}) q^{3} + (2 \beta_{4} - \beta_{2} - 2) q^{4} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{5} + ( - \beta_{16} + \beta_{11} + \beta_{9} + \cdots + 1) q^{6}+ \cdots + ( - \beta_{19} + \beta_{16} - \beta_{15} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{6} + \beta_{5}) q^{3} + (2 \beta_{4} - \beta_{2} - 2) q^{4} + (\beta_{10} + \beta_{8} - \beta_{4}) q^{5} + ( - \beta_{16} + \beta_{11} + \beta_{9} + \cdots + 1) q^{6}+ \cdots + ( - 4 \beta_{19} + \beta_{18} + \cdots - 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 18 q^{4} - 12 q^{5} + 12 q^{6} - 18 q^{9} + 4 q^{10} - 24 q^{11} + 30 q^{14} + 24 q^{15} - 26 q^{16} - 8 q^{19} + 144 q^{20} - 96 q^{21} - 102 q^{24} + 2 q^{25} - 114 q^{29} - 48 q^{30} + 28 q^{31} - 4 q^{34} + 432 q^{36} + 240 q^{39} - 34 q^{40} + 102 q^{41} - 162 q^{45} + 116 q^{46} - 40 q^{49} - 408 q^{50} - 156 q^{51} - 270 q^{54} + 36 q^{55} - 618 q^{56} + 120 q^{59} + 330 q^{60} - 50 q^{61} + 140 q^{64} + 492 q^{65} - 768 q^{66} + 162 q^{69} - 54 q^{70} + 504 q^{74} + 276 q^{75} - 96 q^{76} - 128 q^{79} + 846 q^{81} + 450 q^{84} - 74 q^{85} + 1488 q^{86} - 990 q^{90} - 288 q^{91} + 218 q^{94} - 762 q^{95} - 474 q^{96} - 468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29 \nu^{19} + 609 \nu^{17} - 286 \nu^{15} - 8916 \nu^{13} + 19627 \nu^{11} + 25071 \nu^{9} + \cdots - 2407887 \nu ) / 393660 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23 \nu^{18} + 327 \nu^{16} - 1058 \nu^{14} - 3333 \nu^{12} + 13496 \nu^{10} - 8097 \nu^{8} + \cdots - 1154736 ) / 65610 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{19} + 3 \nu^{17} + 19 \nu^{15} - 66 \nu^{13} - 109 \nu^{11} + 813 \nu^{9} - 981 \nu^{7} + \cdots + 19683 \nu ) / 6561 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6 \nu^{18} + 4 \nu^{16} - 126 \nu^{14} + 59 \nu^{12} + 837 \nu^{10} - 2804 \nu^{8} - 1497 \nu^{6} + \cdots - 89667 ) / 7290 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6 \nu^{19} + 4 \nu^{17} - 126 \nu^{15} + 59 \nu^{13} + 837 \nu^{11} - 2804 \nu^{9} - 1497 \nu^{7} + \cdots - 82377 \nu ) / 7290 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{19} - 4 \nu^{17} + 126 \nu^{15} - 59 \nu^{13} - 837 \nu^{11} + 2804 \nu^{9} + \cdots + 104247 \nu ) / 7290 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 338 \nu^{19} - 213 \nu^{17} - 7448 \nu^{15} + 7332 \nu^{13} + 48236 \nu^{11} - 166587 \nu^{9} + \cdots - 6436341 \nu ) / 393660 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 270 \nu^{19} + 281 \nu^{18} + 105 \nu^{17} + 219 \nu^{16} + 5760 \nu^{15} - 4556 \nu^{14} + \cdots - 2998377 ) / 262440 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 163 \nu^{18} + 93 \nu^{16} + 2908 \nu^{14} - 3477 \nu^{12} - 13366 \nu^{10} + 68457 \nu^{8} + \cdots + 1614006 ) / 65610 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 235 \nu^{19} + 219 \nu^{18} + 435 \nu^{17} + 234 \nu^{16} + 2764 \nu^{15} - 6348 \nu^{14} + \cdots - 8936082 ) / 157464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 419 \nu^{18} - 249 \nu^{16} - 8474 \nu^{14} + 3156 \nu^{12} + 62303 \nu^{10} - 165651 \nu^{8} + \cdots - 10582893 ) / 131220 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 270 \nu^{19} + 281 \nu^{18} - 105 \nu^{17} + 219 \nu^{16} - 5760 \nu^{15} - 4556 \nu^{14} + \cdots - 2735937 ) / 262440 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 311 \nu^{19} + 365 \nu^{18} - 21 \nu^{17} + 390 \nu^{16} - 7376 \nu^{15} - 10580 \nu^{14} + \cdots - 14893470 ) / 262440 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 133 \nu^{19} - 87 \nu^{17} + 2203 \nu^{15} + 1428 \nu^{13} - 19276 \nu^{11} + 40332 \nu^{9} + \cdots + 2991816 \nu ) / 65610 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 235 \nu^{19} + 219 \nu^{18} - 435 \nu^{17} + 234 \nu^{16} - 2764 \nu^{15} - 6348 \nu^{14} + \cdots - 8936082 ) / 157464 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 677 \nu^{18} + 1482 \nu^{16} + 9812 \nu^{14} - 26718 \nu^{12} - 25139 \nu^{10} + 285468 \nu^{8} + \cdots + 4815774 ) / 131220 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 2177 \nu^{19} + 1290 \nu^{18} - 768 \nu^{17} + 1395 \nu^{16} + 48032 \nu^{15} - 31800 \nu^{14} + \cdots - 43204185 ) / 787320 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 509 \nu^{18} + 69 \nu^{16} + 7889 \nu^{14} - 546 \nu^{12} - 54428 \nu^{10} + 156966 \nu^{8} + \cdots + 8975448 ) / 65610 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 221 \nu^{18} + 123 \nu^{16} + 4280 \nu^{14} - 3354 \nu^{12} - 28301 \nu^{10} + 95055 \nu^{8} + \cdots + 4218723 ) / 26244 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{19} - \beta_{16} + \beta_{15} + \beta_{12} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{4} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{17} + \beta_{13} - \beta_{12} + 2\beta_{8} - \beta_{7} - \beta_{4} + \beta_{3} - 2\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{18} + 2\beta_{16} - 3\beta_{12} - \beta_{11} - 4\beta_{9} - 3\beta_{8} + 4\beta_{4} - 3\beta_{2} + 12 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3 \beta_{17} + 3 \beta_{15} + 4 \beta_{14} + 3 \beta_{12} - 12 \beta_{7} + 5 \beta_{6} + \cdots + 3 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{19} - \beta_{18} - 3 \beta_{16} + 9 \beta_{15} + 6 \beta_{12} - 9 \beta_{11} + 9 \beta_{10} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11 \beta_{17} + 12 \beta_{15} + 8 \beta_{14} - \beta_{13} - 8 \beta_{12} + 19 \beta_{8} + 16 \beta_{7} + \cdots + 19 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5 \beta_{19} - 7 \beta_{18} - 4 \beta_{16} + 8 \beta_{15} - 22 \beta_{12} - 11 \beta_{11} + 8 \beta_{10} + \cdots + 105 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15 \beta_{17} - 6 \beta_{15} + 17 \beta_{14} + 24 \beta_{13} + 21 \beta_{12} - 33 \beta_{10} - 36 \beta_{8} + \cdots - 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10 \beta_{19} - 4 \beta_{18} - 35 \beta_{16} + 29 \beta_{15} + 14 \beta_{12} - 54 \beta_{11} + \cdots + 467 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26 \beta_{17} - 51 \beta_{15} - 49 \beta_{14} + 106 \beta_{13} - 124 \beta_{12} - 81 \beta_{10} + \cdots + 98 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 140 \beta_{19} - 51 \beta_{18} - 159 \beta_{16} + 71 \beta_{15} - 163 \beta_{12} - \beta_{11} + \cdots - 105 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 144 \beta_{17} - 387 \beta_{15} - 59 \beta_{14} + 654 \beta_{13} - 99 \beta_{12} - 411 \beta_{10} + \cdots - 45 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 436 \beta_{19} - 309 \beta_{18} + 598 \beta_{16} - 268 \beta_{15} - 901 \beta_{12} - 621 \beta_{11} + \cdots + 2539 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 323 \beta_{17} + 126 \beta_{15} + 150 \beta_{14} + 791 \beta_{13} - 476 \beta_{12} - 594 \beta_{10} + \cdots + 799 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 747 \beta_{19} - 701 \beta_{18} - 452 \beta_{16} + 1188 \beta_{15} - 1122 \beta_{12} - 1997 \beta_{11} + \cdots - 11892 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 51 \beta_{17} - 516 \beta_{15} + 1103 \beta_{14} + 3402 \beta_{13} - 1515 \beta_{12} - 2835 \beta_{10} + \cdots + 1566 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 5781 \beta_{19} - 3536 \beta_{18} + 6024 \beta_{16} - 4410 \beta_{15} - 11697 \beta_{12} + \cdots + 30404 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 6545 \beta_{17} - 741 \beta_{15} - 17 \beta_{14} + 4861 \beta_{13} + 1493 \beta_{12} - 10665 \beta_{10} + \cdots - 8038 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1
1.44078 + 0.961330i
−0.346576 1.69702i
−1.72886 0.105167i
1.72212 0.185238i
−0.315300 + 1.70311i
0.315300 1.70311i
−1.72212 + 0.185238i
1.72886 + 0.105167i
0.346576 + 1.69702i
−1.44078 0.961330i
1.44078 0.961330i
−0.346576 + 1.69702i
−1.72886 + 0.105167i
1.72212 + 0.185238i
−0.315300 1.70311i
0.315300 + 1.70311i
−1.72212 0.185238i
1.72886 0.105167i
0.346576 1.69702i
−1.44078 + 0.961330i
−1.84364 3.19328i −1.66507 + 2.49550i −4.79800 + 8.31039i −4.90060 0.992036i 11.0386 + 0.716233i −2.22343 + 1.28370i 20.6340 −3.45506 8.31039i 5.86709 + 17.4779i
14.2 −1.42081 2.46092i 2.93933 0.600288i −2.03740 + 3.52889i 3.58357 3.48684i −5.65349 6.38055i −8.42581 + 4.86464i 0.212574 8.27931 3.52889i −13.6724 3.86473i
14.3 −1.14670 1.98614i 0.182154 2.99446i −0.629835 + 1.09091i −3.96090 + 3.05143i −6.15630 + 3.07197i 6.04559 3.49042i −6.28466 −8.93364 1.09091i 10.6025 + 4.36783i
14.4 −0.668935 1.15863i 0.320841 + 2.98279i 1.10505 1.91401i 4.23577 + 2.65674i 3.24133 2.36703i 7.10792 4.10376i −8.30831 −8.79412 + 1.91401i 0.244718 6.68487i
14.5 −0.264396 0.457947i −2.94987 0.546115i 1.86019 3.22194i −0.819902 4.93232i 0.529842 + 1.49528i 2.39593 1.38329i −4.08247 8.40352 + 3.22194i −2.04196 + 1.67955i
14.6 0.264396 + 0.457947i 2.94987 + 0.546115i 1.86019 3.22194i −4.68146 + 1.75610i 0.529842 + 1.49528i −2.39593 + 1.38329i 4.08247 8.40352 + 3.22194i −2.04196 1.67955i
14.7 0.668935 + 1.15863i −0.320841 2.98279i 1.10505 1.91401i 4.41869 + 2.33992i 3.24133 2.36703i −7.10792 + 4.10376i 8.30831 −8.79412 + 1.91401i 0.244718 + 6.68487i
14.8 1.14670 + 1.98614i −0.182154 + 2.99446i −0.629835 + 1.09091i 0.662169 4.95596i −6.15630 + 3.07197i −6.04559 + 3.49042i 6.28466 −8.93364 1.09091i 10.6025 4.36783i
14.9 1.42081 + 2.46092i −2.93933 + 0.600288i −2.03740 + 3.52889i −1.22790 + 4.84688i −5.65349 6.38055i 8.42581 4.86464i −0.212574 8.27931 3.52889i −13.6724 + 3.86473i
14.10 1.84364 + 3.19328i 1.66507 2.49550i −4.79800 + 8.31039i −3.30943 3.74802i 11.0386 + 0.716233i 2.22343 1.28370i −20.6340 −3.45506 8.31039i 5.86709 17.4779i
29.1 −1.84364 + 3.19328i −1.66507 2.49550i −4.79800 8.31039i −4.90060 + 0.992036i 11.0386 0.716233i −2.22343 1.28370i 20.6340 −3.45506 + 8.31039i 5.86709 17.4779i
29.2 −1.42081 + 2.46092i 2.93933 + 0.600288i −2.03740 3.52889i 3.58357 + 3.48684i −5.65349 + 6.38055i −8.42581 4.86464i 0.212574 8.27931 + 3.52889i −13.6724 + 3.86473i
29.3 −1.14670 + 1.98614i 0.182154 + 2.99446i −0.629835 1.09091i −3.96090 3.05143i −6.15630 3.07197i 6.04559 + 3.49042i −6.28466 −8.93364 + 1.09091i 10.6025 4.36783i
29.4 −0.668935 + 1.15863i 0.320841 2.98279i 1.10505 + 1.91401i 4.23577 2.65674i 3.24133 + 2.36703i 7.10792 + 4.10376i −8.30831 −8.79412 1.91401i 0.244718 + 6.68487i
29.5 −0.264396 + 0.457947i −2.94987 + 0.546115i 1.86019 + 3.22194i −0.819902 + 4.93232i 0.529842 1.49528i 2.39593 + 1.38329i −4.08247 8.40352 3.22194i −2.04196 1.67955i
29.6 0.264396 0.457947i 2.94987 0.546115i 1.86019 + 3.22194i −4.68146 1.75610i 0.529842 1.49528i −2.39593 1.38329i 4.08247 8.40352 3.22194i −2.04196 + 1.67955i
29.7 0.668935 1.15863i −0.320841 + 2.98279i 1.10505 + 1.91401i 4.41869 2.33992i 3.24133 + 2.36703i −7.10792 4.10376i 8.30831 −8.79412 1.91401i 0.244718 6.68487i
29.8 1.14670 1.98614i −0.182154 2.99446i −0.629835 1.09091i 0.662169 + 4.95596i −6.15630 3.07197i −6.04559 3.49042i 6.28466 −8.93364 + 1.09091i 10.6025 + 4.36783i
29.9 1.42081 2.46092i −2.93933 0.600288i −2.03740 3.52889i −1.22790 4.84688i −5.65349 + 6.38055i 8.42581 + 4.86464i −0.212574 8.27931 + 3.52889i −13.6724 3.86473i
29.10 1.84364 3.19328i 1.66507 + 2.49550i −4.79800 8.31039i −3.30943 + 3.74802i 11.0386 0.716233i 2.22343 + 1.28370i −20.6340 −3.45506 + 8.31039i 5.86709 + 17.4779i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.3.h.a 20
3.b odd 2 1 135.3.h.a 20
5.b even 2 1 inner 45.3.h.a 20
5.c odd 4 2 225.3.j.e 20
9.c even 3 1 135.3.h.a 20
9.c even 3 1 405.3.d.a 20
9.d odd 6 1 inner 45.3.h.a 20
9.d odd 6 1 405.3.d.a 20
15.d odd 2 1 135.3.h.a 20
15.e even 4 2 675.3.j.e 20
45.h odd 6 1 inner 45.3.h.a 20
45.h odd 6 1 405.3.d.a 20
45.j even 6 1 135.3.h.a 20
45.j even 6 1 405.3.d.a 20
45.k odd 12 2 675.3.j.e 20
45.l even 12 2 225.3.j.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.h.a 20 1.a even 1 1 trivial
45.3.h.a 20 5.b even 2 1 inner
45.3.h.a 20 9.d odd 6 1 inner
45.3.h.a 20 45.h odd 6 1 inner
135.3.h.a 20 3.b odd 2 1
135.3.h.a 20 9.c even 3 1
135.3.h.a 20 15.d odd 2 1
135.3.h.a 20 45.j even 6 1
225.3.j.e 20 5.c odd 4 2
225.3.j.e 20 45.l even 12 2
405.3.d.a 20 9.c even 3 1
405.3.d.a 20 9.d odd 6 1
405.3.d.a 20 45.h odd 6 1
405.3.d.a 20 45.j even 6 1
675.3.j.e 20 15.e even 4 2
675.3.j.e 20 45.k odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 29 T^{18} + \cdots + 83521 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 3486784401 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 95367431640625 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 245780494502544 \) Copy content Toggle raw display
$11$ \( (T^{10} + 12 T^{9} + \cdots + 21579372)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{10} - 704 T^{8} + \cdots - 1532096164)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 2 T^{4} + \cdots - 7946)^{4} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15352201216 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 19167316554672)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 116686747082896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 2777948395200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 957338655721083)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 70779040697088)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 457796433007684)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 42\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
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