Properties

Label 1148.2.d.a
Level $1148$
Weight $2$
Character orbit 1148.d
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1148,2,Mod(1065,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.1065");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), 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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{3} + \beta_{2} q^{5} - \beta_{9} q^{7} + ( - \beta_{11} - \beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{3} + \beta_{2} q^{5} - \beta_{9} q^{7} + ( - \beta_{11} - \beta_{7} - 1) q^{9} + \beta_{19} q^{11} + \beta_{15} q^{13} + (\beta_{19} + \beta_{18} - \beta_{17} - \beta_{14} - \beta_{10} - \beta_{9}) q^{15} + (\beta_{18} - \beta_{16} - \beta_{12}) q^{17} + (\beta_{19} + \beta_{16} - \beta_{14} + \beta_{13} + \beta_{12} - \beta_{9}) q^{19} - \beta_{4} q^{21} + ( - \beta_{11} + \beta_{8} - \beta_1) q^{23} + ( - \beta_{7} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{25} + (\beta_{18} - \beta_{17} - \beta_{15} - \beta_{12} - 2 \beta_{9}) q^{27} + (\beta_{19} - \beta_{17} - 2 \beta_{14} + \beta_{13} - \beta_{9}) q^{29} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{31} + ( - \beta_{11} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{33} + \beta_{14} q^{35} + ( - \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{37} + ( - \beta_{11} + \beta_{8} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{39} + ( - \beta_{17} - \beta_{16} - \beta_{15} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - 2) q^{41} + ( - \beta_{11} - \beta_{4} - \beta_{3} + \beta_1) q^{43} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{45} + ( - \beta_{19} + \beta_{17} + \beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{10} + \beta_{9}) q^{47} - q^{49} + (\beta_{11} + 2 \beta_{8} + \beta_{7} - 2 \beta_{2} + 3) q^{51} + ( - \beta_{19} - \beta_{18} - \beta_{16} + \beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 2 \beta_{10} + \beta_{9}) q^{53} + (\beta_{18} + \beta_{17} - 2 \beta_{9}) q^{55} + ( - \beta_{11} + \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{57} + ( - \beta_{11} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 1) q^{59} + ( - \beta_{11} - \beta_{8} - \beta_{5} - \beta_1) q^{61} + (\beta_{15} - \beta_{13} + \beta_{9}) q^{63} + ( - \beta_{18} + \beta_{16} - \beta_{15} + \beta_{13} + \beta_{12} - 2 \beta_{9}) q^{65} + ( - \beta_{19} - \beta_{17} - 2 \beta_{13} - \beta_{12} - 2 \beta_{9}) q^{67} + ( - \beta_{17} + \beta_{16} - \beta_{12}) q^{69} + ( - \beta_{19} - 3 \beta_{17} - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9}) q^{71} + (\beta_{11} + \beta_{8} + \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{73} + (3 \beta_{19} + 2 \beta_{18} - \beta_{17} + 2 \beta_{16} - 5 \beta_{14} + \beta_{13} - \beta_{12} + \cdots - 3 \beta_{9}) q^{75}+ \cdots + (\beta_{18} - 2 \beta_{17} - \beta_{16} - 5 \beta_{15} + \beta_{13} - \beta_{12} - 6 \beta_{9}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 4 q^{5} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 4 q^{5} - 20 q^{9} + 4 q^{21} + 8 q^{31} + 20 q^{37} + 4 q^{39} - 16 q^{41} + 20 q^{43} - 4 q^{45} - 20 q^{49} + 52 q^{51} - 36 q^{57} + 20 q^{59} - 4 q^{61} - 12 q^{73} + 8 q^{77} + 20 q^{81} - 48 q^{83} + 44 q^{87} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 31787625 \nu^{18} - 1234487653 \nu^{16} - 16663895421 \nu^{14} - 107644253652 \nu^{12} - 377905031721 \nu^{10} - 752256795415 \nu^{8} + \cdots - 2443071016 ) / 1308220948 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 78534511 \nu^{18} + 3023650107 \nu^{16} + 40148207403 \nu^{14} + 252303406944 \nu^{12} + 850222236003 \nu^{10} + 1608764602793 \nu^{8} + \cdots + 25131533044 ) / 1308220948 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 65716921 \nu^{18} + 2470157524 \nu^{16} + 31428482111 \nu^{14} + 185642737294 \nu^{12} + 580415674738 \nu^{10} + 1021258449659 \nu^{8} + \cdots + 21827912202 ) / 654110474 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 132705459 \nu^{18} - 5017740939 \nu^{16} - 64447009489 \nu^{14} - 384270456442 \nu^{12} - 1199136434943 \nu^{10} - 2027852552593 \nu^{8} + \cdots - 1710402836 ) / 1308220948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 157161803 \nu^{18} - 5956564083 \nu^{16} - 76935253097 \nu^{14} - 464683022950 \nu^{12} - 1491603036323 \nu^{10} + \cdots - 32192745056 ) / 1308220948 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 81400012 \nu^{18} - 3072480759 \nu^{16} - 39346942920 \nu^{14} - 233733680863 \nu^{12} - 726779307974 \nu^{10} - 1225255445524 \nu^{8} + \cdots + 2586015382 ) / 654110474 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 254818261 \nu^{18} + 9591778621 \nu^{16} + 122242189893 \nu^{14} + 721325370166 \nu^{12} + 2227511191483 \nu^{10} + 3743842788823 \nu^{8} + \cdots - 469951060 ) / 1308220948 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 147573535 \nu^{18} - 5598394419 \nu^{16} - 72436339599 \nu^{14} - 438766175071 \nu^{12} - 1414547618202 \nu^{10} - 2551719292225 \nu^{8} + \cdots - 25607577556 ) / 654110474 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5623603 \nu^{19} + 216227373 \nu^{17} + 2866746153 \nu^{15} + 18022264568 \nu^{13} + 61108447099 \nu^{11} + 117555643485 \nu^{9} + \cdots + 1935913456 \nu ) / 73358184 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 608503607 \nu^{19} + 22567526841 \nu^{17} + 279469541481 \nu^{15} + 1569995546788 \nu^{13} + 4472860961675 \nu^{11} + \cdots - 155209856212 \nu ) / 7849325688 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 313679163 \nu^{18} - 11920654724 \nu^{16} - 154696614366 \nu^{14} - 940624815713 \nu^{12} - 3043090741214 \nu^{10} + \cdots - 47870220358 ) / 654110474 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 232671212 \nu^{19} - 8702946798 \nu^{17} - 109604242404 \nu^{15} - 634457380336 \nu^{13} - 1904709690599 \nu^{11} + \cdots - 9014188499 \nu ) / 1962331422 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 588668419 \nu^{19} + 22208188653 \nu^{17} + 284385522525 \nu^{15} + 1694539204220 \nu^{13} + 5340504255169 \nu^{11} + \cdots + 182489439166 \nu ) / 3924662844 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 302557282 \nu^{19} + 11425963938 \nu^{17} + 146552844963 \nu^{15} + 874594463891 \nu^{13} + 2752866200317 \nu^{11} + 4778882242827 \nu^{9} + \cdots + 9058558051 \nu ) / 1962331422 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 33910453 \nu^{19} + 1287273639 \nu^{17} + 16680088383 \nu^{15} + 101368174544 \nu^{13} + 329355900889 \nu^{11} + 605696469483 \nu^{9} + \cdots + 11249472208 \nu ) / 191446968 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 751498207 \nu^{19} - 28683213891 \nu^{17} - 375155900355 \nu^{15} - 2308300695836 \nu^{13} - 7588073741251 \nu^{11} + \cdots - 152958371824 \nu ) / 3924662844 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 1850657971 \nu^{19} - 70481463165 \nu^{17} - 918223917393 \nu^{15} - 5616400889096 \nu^{13} - 18316670673547 \nu^{11} + \cdots - 465025308544 \nu ) / 7849325688 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2134858519 \nu^{19} - 81836573685 \nu^{17} - 1079672604993 \nu^{15} - 6749307948548 \nu^{13} - 22810245655159 \nu^{11} + \cdots - 905372335252 \nu ) / 7849325688 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1147322905 \nu^{19} + 43601198779 \nu^{17} + 565803981419 \nu^{15} + 3439887938000 \nu^{13} + 11123703697949 \nu^{11} + \cdots + 125108902800 \nu ) / 2616441896 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{10} + \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{14} + \beta_{13} - 5 \beta_{12} - 15 \beta_{10} - 18 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{11} - 14\beta_{7} - 8\beta_{6} + 12\beta_{5} - 28\beta_{4} + 20\beta_{3} - 15\beta_{2} - 47\beta _1 + 111 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 95 \beta_{19} + 29 \beta_{18} - 11 \beta_{17} - 22 \beta_{16} + 32 \beta_{15} + 15 \beta_{14} + 9 \beta_{13} + 119 \beta_{12} + 254 \beta_{10} + 363 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 102 \beta_{11} - 20 \beta_{8} + 193 \beta_{7} + 63 \beta_{6} - 189 \beta_{5} + 559 \beta_{4} - 389 \beta_{3} + 224 \beta_{2} + 969 \beta _1 - 1860 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1772 \beta_{19} - 650 \beta_{18} + 78 \beta_{17} + 276 \beta_{16} - 558 \beta_{15} - 573 \beta_{14} - 418 \beta_{13} - 2394 \beta_{12} - 4629 \beta_{10} - 7275 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2146 \beta_{11} + 616 \beta_{8} - 3045 \beta_{7} - 573 \beta_{6} + 3341 \beta_{5} - 10743 \beta_{4} + 7543 \beta_{3} - 3701 \beta_{2} - 19172 \beta _1 + 34091 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 33621 \beta_{19} + 13255 \beta_{18} - 213 \beta_{17} - 4234 \beta_{16} + 10268 \beta_{15} + 12974 \beta_{14} + 9957 \beta_{13} + 46559 \beta_{12} + 87061 \beta_{10} + 142784 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 42642 \beta_{11} - 13978 \beta_{8} + 53164 \beta_{7} + 6392 \beta_{6} - 61972 \beta_{5} + 205870 \beta_{4} - 145630 \beta_{3} + 66091 \beta_{2} + 372887 \beta _1 - 643495 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 642815 \beta_{19} - 260737 \beta_{18} - 6873 \beta_{17} + 73534 \beta_{16} - 193624 \beta_{15} - 264057 \beta_{14} - 206351 \beta_{13} - 897873 \beta_{12} - 1658276 \beta_{10} - 2769101 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 830018 \beta_{11} + 286758 \beta_{8} - 979163 \beta_{7} - 88163 \beta_{6} + 1173239 \beta_{5} - 3948889 \beta_{4} + 2804025 \beta_{3} - 1229500 \beta_{2} - 7200091 \beta _1 + 12276920 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 12325824 \beta_{19} + 5056778 \beta_{18} + 221408 \beta_{17} - 1354084 \beta_{16} + 3690506 \beta_{15} + 5187285 \beta_{14} + 4082134 \beta_{13} + 17271626 \beta_{12} + \cdots + 53408601 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 16028518 \beta_{11} - 5657626 \beta_{8} + 18478287 \beta_{7} + 1428069 \beta_{6} - 22402389 \beta_{5} + 75804203 \beta_{4} - 53918615 \beta_{3} + 23301041 \beta_{2} + \cdots - 235189781 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 236604049 \beta_{19} - 97516461 \beta_{18} - 4967495 \beta_{17} + 25563982 \beta_{16} - 70663378 \beta_{15} - 100542516 \beta_{14} - 79343629 \beta_{13} - 331954503 \beta_{12} + \cdots - 1027679528 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 308556528 \beta_{11} + 109875106 \beta_{8} - 352359700 \beta_{7} - 25372692 \beta_{6} + 429279674 \beta_{5} - 1455728826 \beta_{4} + 1036192852 \beta_{3} + \cdots + 4512893735 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 4543748283 \beta_{19} + 1876213613 \beta_{18} + 101080219 \beta_{17} - 487607498 \beta_{16} + 1355597420 \beta_{15} + 1938402667 \beta_{14} + 1531436353 \beta_{13} + \cdots + 19754909633 \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 5932354398 \beta_{11} - 2120124070 \beta_{8} + 6748252039 \beta_{7} + 471320859 \beta_{6} - 8237938373 \beta_{5} + 27960355095 \beta_{4} + \cdots - 86651605338 ) / 2 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 87273104296 \beta_{19} - 36064473216 \beta_{18} - 1986303818 \beta_{17} + 9339696968 \beta_{16} - 26026104824 \beta_{15} - 37290917315 \beta_{14} + \cdots - 379590218363 \beta_{9} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1065.1
0.544821i
1.52399i
1.02366i
1.40239i
1.20144i
4.38287i
1.97172i
0.379831i
0.911088i
2.80197i
2.80197i
0.911088i
0.379831i
1.97172i
4.38287i
1.20144i
1.40239i
1.02366i
1.52399i
0.544821i
0 3.32626i 0 −0.885775 0 1.00000i 0 −8.06402 0
1065.2 0 2.64112i 0 0.508610 0 1.00000i 0 −3.97553 0
1065.3 0 2.57093i 0 −2.38500 0 1.00000i 0 −3.60969 0
1065.4 0 2.39349i 0 3.99395 0 1.00000i 0 −2.72882 0
1065.5 0 2.23543i 0 0.468853 0 1.00000i 0 −1.99714 0
1065.6 0 1.40464i 0 −1.35828 0 1.00000i 0 1.02697 0
1065.7 0 1.24486i 0 3.14516 0 1.00000i 0 1.45033 0
1065.8 0 1.03684i 0 2.44552 0 1.00000i 0 1.92496 0
1065.9 0 0.162086i 0 −1.05958 0 1.00000i 0 2.97373 0
1065.10 0 0.0281596i 0 −2.87346 0 1.00000i 0 2.99921 0
1065.11 0 0.0281596i 0 −2.87346 0 1.00000i 0 2.99921 0
1065.12 0 0.162086i 0 −1.05958 0 1.00000i 0 2.97373 0
1065.13 0 1.03684i 0 2.44552 0 1.00000i 0 1.92496 0
1065.14 0 1.24486i 0 3.14516 0 1.00000i 0 1.45033 0
1065.15 0 1.40464i 0 −1.35828 0 1.00000i 0 1.02697 0
1065.16 0 2.23543i 0 0.468853 0 1.00000i 0 −1.99714 0
1065.17 0 2.39349i 0 3.99395 0 1.00000i 0 −2.72882 0
1065.18 0 2.57093i 0 −2.38500 0 1.00000i 0 −3.60969 0
1065.19 0 2.64112i 0 0.508610 0 1.00000i 0 −3.97553 0
1065.20 0 3.32626i 0 −0.885775 0 1.00000i 0 −8.06402 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1065.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1148.2.d.a 20
41.b even 2 1 inner 1148.2.d.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.2.d.a 20 1.a even 1 1 trivial
1148.2.d.a 20 41.b even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1148, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 40 T^{18} + 660 T^{16} + 5836 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{10} - 2 T^{9} - 23 T^{8} + 26 T^{7} + \cdots - 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$11$ \( T^{20} + 128 T^{18} + 6470 T^{16} + \cdots + 2985984 \) Copy content Toggle raw display
$13$ \( T^{20} + 110 T^{18} + 4943 T^{16} + \cdots + 4609609 \) Copy content Toggle raw display
$17$ \( T^{20} + 194 T^{18} + \cdots + 16643322081 \) Copy content Toggle raw display
$19$ \( T^{20} + 160 T^{18} + 9800 T^{16} + \cdots + 273529 \) Copy content Toggle raw display
$23$ \( (T^{10} - 116 T^{8} - 232 T^{7} + \cdots - 27743)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + 286 T^{18} + \cdots + 1220370927616 \) Copy content Toggle raw display
$31$ \( (T^{10} - 4 T^{9} - 91 T^{8} + 244 T^{7} + \cdots + 24000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} - 10 T^{9} - 211 T^{8} + \cdots + 1664144)^{2} \) Copy content Toggle raw display
$41$ \( T^{20} + 16 T^{19} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{10} - 10 T^{9} - 133 T^{8} + \cdots + 261201)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 190650259603456 \) Copy content Toggle raw display
$53$ \( T^{20} + 614 T^{18} + \cdots + 1684097961984 \) Copy content Toggle raw display
$59$ \( (T^{10} - 10 T^{9} - 195 T^{8} + \cdots - 4392448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 2 T^{9} - 214 T^{8} + \cdots + 1741312)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 494 T^{18} + \cdots + 11529990701056 \) Copy content Toggle raw display
$71$ \( T^{20} + 1080 T^{18} + \cdots + 62\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( (T^{10} + 6 T^{9} - 298 T^{8} + \cdots - 5926912)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + 600 T^{18} + \cdots + 89369947930624 \) Copy content Toggle raw display
$83$ \( (T^{10} + 24 T^{9} - 190 T^{8} + \cdots + 897075264)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + 1044 T^{18} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( T^{20} + 984 T^{18} + \cdots + 69\!\cdots\!01 \) Copy content Toggle raw display
show more
show less