Properties

Label 37.4.e.a
Level $37$
Weight $4$
Character orbit 37.e
Analytic conductor $2.183$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [37,4,Mod(11,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.e (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: \(2.18307067021\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 82 x^{14} + 2679 x^{12} + 44392 x^{10} + 392767 x^{8} + 1779258 x^{6} + 3438825 x^{4} + 1208748 x^{2} + 82944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{6} q^{3} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{11} + \beta_{4} + 1) q^{5} + ( - \beta_{9} + 3 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{15} - \beta_{5} - 3 \beta_{4} + \beta_{3} + 3) q^{7} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{8} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 6 \beta_{4} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{6} q^{3} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2) q^{4} + ( - \beta_{11} + \beta_{4} + 1) q^{5} + ( - \beta_{9} + 3 \beta_{4} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{15} - \beta_{5} - 3 \beta_{4} + \beta_{3} + 3) q^{7} + ( - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} + 4 \beta_{4} + \beta_{2} + \beta_1 - 2) q^{8} + (\beta_{15} + \beta_{14} - 2 \beta_{13} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} - 6 \beta_{4} + \cdots + \beta_1) q^{9}+ \cdots + (29 \beta_{15} + 14 \beta_{14} - 28 \beta_{13} - 21 \beta_{12} + 42 \beta_{11} + \cdots + 438 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{2} - 3 q^{3} + 18 q^{4} + 18 q^{5} + 23 q^{7} - 53 q^{9} - 4 q^{10} + 36 q^{11} + 14 q^{12} - 9 q^{13} - 93 q^{15} + 90 q^{16} - 210 q^{17} - 144 q^{18} - 135 q^{19} - 18 q^{20} + 71 q^{21} + 18 q^{22} - 126 q^{24} - 72 q^{25} - 276 q^{26} + 1170 q^{27} + 256 q^{28} - 236 q^{30} - 552 q^{32} + 336 q^{33} + 274 q^{34} - 27 q^{35} + 180 q^{36} - 33 q^{37} + 1344 q^{38} - 909 q^{39} - 756 q^{40} + 642 q^{41} + 846 q^{42} - 6 q^{44} + 74 q^{46} - 468 q^{47} - 284 q^{48} + 187 q^{49} - 1932 q^{50} + 180 q^{52} - 249 q^{53} - 342 q^{54} + 162 q^{55} - 996 q^{56} - 141 q^{57} - 1496 q^{58} - 1455 q^{59} + 1188 q^{61} - 510 q^{62} - 3472 q^{63} + 3476 q^{64} + 579 q^{65} - 1033 q^{67} + 810 q^{69} + 2934 q^{70} + 2319 q^{71} + 5196 q^{72} - 1672 q^{73} - 1110 q^{74} + 4364 q^{75} - 3450 q^{76} - 2472 q^{77} + 2622 q^{78} + 1569 q^{79} - 1508 q^{81} + 975 q^{83} + 3064 q^{84} + 3128 q^{85} - 36 q^{86} - 5892 q^{87} + 522 q^{89} - 2908 q^{90} - 1773 q^{91} - 3462 q^{92} + 222 q^{93} - 1614 q^{94} - 4311 q^{95} + 378 q^{96} + 5748 q^{98} - 3606 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 82 x^{14} + 2679 x^{12} + 44392 x^{10} + 392767 x^{8} + 1779258 x^{6} + 3438825 x^{4} + 1208748 x^{2} + 82944 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{14} + 33 \nu^{12} + 2850 \nu^{10} + 38114 \nu^{8} - 57933 \nu^{6} - 2426979 \nu^{4} - 5725236 \nu^{2} + 2386944 \nu + 746496 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} - 33 \nu^{12} - 2850 \nu^{10} - 38114 \nu^{8} + 57933 \nu^{6} + 2426979 \nu^{4} + 5725236 \nu^{2} + 2386944 \nu - 746496 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{14} + 33 \nu^{12} + 2850 \nu^{10} + 38114 \nu^{8} - 57933 \nu^{6} - 2426979 \nu^{4} - 951348 \nu^{2} + 2386944 \nu + 48485376 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9 \nu^{15} + 739 \nu^{13} + 24078 \nu^{11} + 396678 \nu^{9} + 3496789 \nu^{7} + 16071255 \nu^{5} + 33376404 \nu^{3} + 16603968 \nu + 2386944 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 91 \nu^{15} - \nu^{14} + 7357 \nu^{13} + 33 \nu^{12} + 237930 \nu^{11} + 2850 \nu^{10} + 3928666 \nu^{9} + 38114 \nu^{8} + 35025823 \nu^{7} - 57933 \nu^{6} + 163139529 \nu^{5} + \cdots + 24615936 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17 \nu^{15} - 256 \nu^{14} + 2029 \nu^{13} - 19524 \nu^{12} + 94518 \nu^{11} - 575760 \nu^{10} + 2208314 \nu^{9} - 8200840 \nu^{8} + 27220525 \nu^{7} - 57273696 \nu^{6} + \cdots - 18948096 ) / 9547776 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 199 \nu^{14} - 16743 \nu^{12} - 570378 \nu^{10} - 10065646 \nu^{8} - 97272171 \nu^{6} - 490768347 \nu^{4} - 1018240500 \nu^{2} - 128332800 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 64 \nu^{14} + 4881 \nu^{12} + 143940 \nu^{10} + 2050210 \nu^{8} + 14318424 \nu^{6} + 43510917 \nu^{4} + 38107740 \nu^{2} + 4737024 ) / 1193472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 229 \nu^{15} - 17307 \nu^{13} - 503526 \nu^{11} - 7010806 \nu^{9} - 46783329 \nu^{7} - 125829903 \nu^{5} - 52301748 \nu^{3} + 26089920 \nu + 2386944 ) / 4773888 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 34 \nu^{15} + 91 \nu^{14} - 2578 \nu^{13} + 6987 \nu^{12} - 75372 \nu^{11} + 206850 \nu^{10} - 1058212 \nu^{9} + 2923894 \nu^{8} - 7182874 \nu^{7} + 19534959 \nu^{6} + \cdots + 1631232 ) / 1363968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 632 \nu^{15} - 101 \nu^{14} - 51664 \nu^{13} - 10653 \nu^{12} - 1679760 \nu^{11} - 439422 \nu^{10} - 27616640 \nu^{9} - 9082874 \nu^{8} - 241086904 \nu^{7} + \cdots - 254002176 ) / 9547776 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 632 \nu^{15} - 101 \nu^{14} + 51664 \nu^{13} - 10653 \nu^{12} + 1679760 \nu^{11} - 439422 \nu^{10} + 27616640 \nu^{9} - 9082874 \nu^{8} + 241086904 \nu^{7} + \cdots - 254002176 ) / 9547776 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 668 \nu^{15} - 297 \nu^{14} - 54620 \nu^{13} - 22833 \nu^{12} - 1776072 \nu^{11} - 701334 \nu^{10} - 29203352 \nu^{9} - 11048418 \nu^{8} - 255074060 \nu^{7} + \cdots - 308192256 ) / 9547776 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 668 \nu^{15} - 297 \nu^{14} + 54620 \nu^{13} - 22833 \nu^{12} + 1776072 \nu^{11} - 701334 \nu^{10} + 29203352 \nu^{9} - 11048418 \nu^{8} + 255074060 \nu^{7} + \cdots - 308192256 ) / 9547776 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1466 \nu^{15} + 199 \nu^{14} - 118418 \nu^{13} + 16743 \nu^{12} - 3790812 \nu^{11} + 570378 \nu^{10} - 61054052 \nu^{9} + 10065646 \nu^{8} - 518466770 \nu^{7} + \cdots + 128332800 ) / 9547776 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - 4\beta_{4} - 17\beta_{2} - 17\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2\beta_{7} - 23\beta_{3} + 23\beta _1 + 166 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{15} - 26 \beta_{14} + 26 \beta_{13} + 30 \beta_{12} - 30 \beta_{11} - 2 \beta_{9} + 2 \beta_{7} + 98 \beta_{4} + 321 \beta_{2} + 321 \beta _1 - 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26 \beta_{14} + 26 \beta_{13} + 34 \beta_{12} + 34 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} - 16 \beta_{8} - 64 \beta_{7} + 4 \beta_{4} + 493 \beta_{3} - 16 \beta_{2} - 477 \beta _1 - 3114 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 152 \beta_{15} + 571 \beta_{14} - 571 \beta_{13} - 723 \beta_{12} + 723 \beta_{11} + 92 \beta_{9} - 32 \beta_{8} - 76 \beta_{7} - 64 \beta_{6} + 32 \beta_{5} - 1912 \beta_{4} - 16 \beta_{3} - 6321 \beta_{2} - 6337 \beta _1 + 910 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 555 \beta_{14} - 555 \beta_{13} - 891 \beta_{12} - 891 \beta_{11} - 400 \beta_{10} + 200 \beta_{9} + 752 \beta_{8} + 1614 \beta_{7} - 200 \beta_{4} - 10371 \beta_{3} + 624 \beta_{2} + 9747 \beta _1 + 61550 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4300 \beta_{15} - 12036 \beta_{14} + 12036 \beta_{13} + 16272 \beta_{12} - 16272 \beta_{11} - 2966 \beta_{9} + 1632 \beta_{8} + 2150 \beta_{7} + 3264 \beta_{6} - 1504 \beta_{5} + 37550 \beta_{4} + \cdots - 17292 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11284 \beta_{14} + 11284 \beta_{13} + 21324 \beta_{12} + 21324 \beta_{11} + 13496 \beta_{10} - 6748 \beta_{9} - 24480 \beta_{8} - 37724 \beta_{7} + 6748 \beta_{4} + 217049 \beta_{3} - 16800 \beta_{2} + \cdots - 1245778 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 109024 \beta_{15} + 250901 \beta_{14} - 250901 \beta_{13} - 356469 \beta_{12} + 356469 \beta_{11} + 82448 \beta_{9} - 55616 \beta_{8} - 54512 \beta_{7} - 111232 \beta_{6} + 50368 \beta_{5} + \cdots + 346858 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 225717 \beta_{14} - 225717 \beta_{13} - 490677 \beta_{12} - 490677 \beta_{11} - 385152 \beta_{10} + 192576 \beta_{9} + 684064 \beta_{8} + 853354 \beta_{7} - 192576 \beta_{4} + \cdots + 25523478 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2621780 \beta_{15} - 5216782 \beta_{14} + 5216782 \beta_{13} + 7718370 \beta_{12} - 7718370 \beta_{11} - 2115146 \beta_{9} + 1596224 \beta_{8} + 1310890 \beta_{7} + 3192448 \beta_{6} + \cdots - 7358248 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4478190 \beta_{14} + 4478190 \beta_{13} + 11078742 \beta_{12} + 11078742 \beta_{11} + 10044520 \beta_{10} - 5022260 \beta_{9} - 17605232 \beta_{8} - 18982968 \beta_{7} + \cdots - 526752122 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 61211560 \beta_{15} + 108470415 \beta_{14} - 108470415 \beta_{13} - 166238007 \beta_{12} + 166238007 \beta_{11} + 51654980 \beta_{9} - 41774304 \beta_{8} + \cdots + 162850758 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
4.65090i
4.17312i
3.17259i
0.575927i
0.301955i
2.31004i
2.56981i
4.53054i
4.65090i
4.17312i
3.17259i
0.575927i
0.301955i
2.31004i
2.56981i
4.53054i
−4.02780 + 2.32545i 1.78194 3.08642i 6.81543 11.8047i 11.2971 + 6.52240i 16.5753i −10.9405 + 18.9495i 26.1886i 7.14934 + 12.3830i −60.6701
11.2 −3.61403 + 2.08656i −3.72941 + 6.45953i 4.70747 8.15358i −2.02566 1.16952i 31.1266i 3.33293 5.77281i 5.90472i −14.3170 24.7978i 9.76108
11.3 −2.74754 + 1.58629i 2.85325 4.94198i 1.03266 1.78861i −13.1878 7.61398i 18.1044i 14.3541 24.8621i 18.8283i −2.78209 4.81872i 48.3120
11.4 −0.498768 + 0.287964i −1.51417 + 2.62262i −3.83415 + 6.64095i −6.67348 3.85293i 1.74410i −10.3028 + 17.8450i 9.02381i 8.91459 + 15.4405i 4.43802
11.5 −0.261501 + 0.150978i 0.675960 1.17080i −3.95441 + 6.84924i 17.4040 + 10.0482i 0.408219i 10.5914 18.3449i 4.80375i 12.5862 + 21.7999i −6.06822
11.6 2.00056 1.15502i 3.80485 6.59020i −1.33185 + 2.30684i −1.68208 0.971152i 17.5787i −1.95249 + 3.38181i 24.6336i −15.4538 26.7667i −4.48680
11.7 2.22552 1.28490i −4.96296 + 8.59609i −0.698050 + 1.20906i 5.97182 + 3.44783i 25.5077i 8.67107 15.0187i 24.1461i −35.7619 61.9414i 17.7205
11.8 3.92356 2.26527i −0.409472 + 0.709227i 6.26291 10.8477i −2.10394 1.21471i 3.71026i −2.25379 + 3.90368i 20.5044i 13.1647 + 22.8019i −11.0066
27.1 −4.02780 2.32545i 1.78194 + 3.08642i 6.81543 + 11.8047i 11.2971 6.52240i 16.5753i −10.9405 18.9495i 26.1886i 7.14934 12.3830i −60.6701
27.2 −3.61403 2.08656i −3.72941 6.45953i 4.70747 + 8.15358i −2.02566 + 1.16952i 31.1266i 3.33293 + 5.77281i 5.90472i −14.3170 + 24.7978i 9.76108
27.3 −2.74754 1.58629i 2.85325 + 4.94198i 1.03266 + 1.78861i −13.1878 + 7.61398i 18.1044i 14.3541 + 24.8621i 18.8283i −2.78209 + 4.81872i 48.3120
27.4 −0.498768 0.287964i −1.51417 2.62262i −3.83415 6.64095i −6.67348 + 3.85293i 1.74410i −10.3028 17.8450i 9.02381i 8.91459 15.4405i 4.43802
27.5 −0.261501 0.150978i 0.675960 + 1.17080i −3.95441 6.84924i 17.4040 10.0482i 0.408219i 10.5914 + 18.3449i 4.80375i 12.5862 21.7999i −6.06822
27.6 2.00056 + 1.15502i 3.80485 + 6.59020i −1.33185 2.30684i −1.68208 + 0.971152i 17.5787i −1.95249 3.38181i 24.6336i −15.4538 + 26.7667i −4.48680
27.7 2.22552 + 1.28490i −4.96296 8.59609i −0.698050 1.20906i 5.97182 3.44783i 25.5077i 8.67107 + 15.0187i 24.1461i −35.7619 + 61.9414i 17.7205
27.8 3.92356 + 2.26527i −0.409472 0.709227i 6.26291 + 10.8477i −2.10394 + 1.21471i 3.71026i −2.25379 3.90368i 20.5044i 13.1647 22.8019i −11.0066
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.4.e.a 16
3.b odd 2 1 333.4.s.c 16
37.e even 6 1 inner 37.4.e.a 16
37.g odd 12 2 1369.4.a.g 16
111.h odd 6 1 333.4.s.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.4.e.a 16 1.a even 1 1 trivial
37.4.e.a 16 37.e even 6 1 inner
333.4.s.c 16 3.b odd 2 1
333.4.s.c 16 111.h odd 6 1
1369.4.a.g 16 37.g odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(37, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 6 T^{15} - 23 T^{14} + \cdots + 82944 \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 1475789056 \) Copy content Toggle raw display
$5$ \( T^{16} - 18 T^{15} + \cdots + 5481540530361 \) Copy content Toggle raw display
$7$ \( T^{16} - 23 T^{15} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{8} - 18 T^{7} + \cdots - 233130203136)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 9 T^{15} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{16} + 210 T^{15} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( T^{16} + 135 T^{15} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{16} + 121024 T^{14} + \cdots + 77\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + 228973 T^{14} + \cdots + 26\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{16} + 193236 T^{14} + \cdots + 91\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{16} + 33 T^{15} + \cdots + 43\!\cdots\!61 \) Copy content Toggle raw display
$41$ \( T^{16} - 642 T^{15} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{16} + 636216 T^{14} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{8} + 234 T^{7} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 249 T^{15} + \cdots + 29\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{16} + 1455 T^{15} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{16} - 1188 T^{15} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{16} + 1033 T^{15} + \cdots + 87\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{16} - 2319 T^{15} + \cdots + 35\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{8} + 836 T^{7} + \cdots + 72\!\cdots\!36)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} - 1569 T^{15} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{16} - 975 T^{15} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{16} - 522 T^{15} + \cdots + 46\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{16} + 5602893 T^{14} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
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