Properties

Label 96.6.c.a
Level $96$
Weight $6$
Character orbit 96.c
Analytic conductor $15.397$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [96,6,Mod(95,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.95");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 96.c (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: \(15.3968467020\)
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: \( x^{20} + 416x^{16} + 57056x^{12} + 3187216x^{8} + 63121536x^{4} + 49787136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{134}\cdot 3^{14} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{6} q^{5} + \beta_{2} q^{7} + ( - \beta_{7} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{6} q^{5} + \beta_{2} q^{7} + ( - \beta_{7} - 2) q^{9} + ( - \beta_{10} + 3 \beta_{3}) q^{11} + (\beta_1 - 12) q^{13} + (\beta_{12} + \beta_{9}) q^{15} + (\beta_{7} - 2 \beta_{6} - \beta_{5}) q^{17} + ( - \beta_{9} - \beta_{8} - 7 \beta_{3} + \beta_{2}) q^{19} + ( - \beta_{16} + \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_1 - 82) q^{21} + (\beta_{18} + \beta_{13} + \beta_{12} + \beta_{10} + 3 \beta_{9} - \beta_{4} - 20 \beta_{3} + \beta_{2}) q^{23} + ( - \beta_{16} + \beta_{15} + 3 \beta_{7} - \beta_{6} + \beta_1 - 763) q^{25} + (\beta_{18} + \beta_{14} - 2 \beta_{10} - \beta_{8} - \beta_{4} + 4 \beta_{3} - \beta_{2}) q^{27} + ( - \beta_{19} + \beta_{17} - \beta_{16} - 5 \beta_{7} - 9 \beta_{6} - 3 \beta_{5} + \beta_1 + 1) q^{29} + (\beta_{14} + 3 \beta_{12} + 3 \beta_{9} - 2 \beta_{8} - \beta_{4} + 12 \beta_{3} + 10 \beta_{2}) q^{31} + ( - \beta_{17} + \beta_{16} + \beta_{15} + \beta_{11} + 2 \beta_{7} + 6 \beta_{6} - 3 \beta_{5} + \cdots - 665) q^{33}+ \cdots + ( - 9 \beta_{18} - 5 \beta_{14} - 42 \beta_{13} + 16 \beta_{12} - 3 \beta_{10} + \cdots + 329 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 44 q^{9} - 232 q^{13} - 1640 q^{21} - 15228 q^{25} - 13264 q^{33} + 17208 q^{37} + 24224 q^{45} - 51780 q^{49} - 37512 q^{57} + 123416 q^{61} + 100192 q^{69} - 48120 q^{73} - 42508 q^{81} + 168960 q^{85} + 34840 q^{93} + 39368 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 416x^{16} + 57056x^{12} + 3187216x^{8} + 63121536x^{4} + 49787136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4143739 \nu^{16} + 1426187564 \nu^{12} + 133423868720 \nu^{8} + 3223423733296 \nu^{4} - 22985854105104 ) / 27240479928 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 723903389 \nu^{18} + 255418666612 \nu^{14} + 25836421256752 \nu^{10} + 875564384087312 \nu^{6} + 59\!\cdots\!52 \nu^{2} ) / 128139217581312 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1030803915 \nu^{19} - 421140118 \nu^{18} + 3077327106 \nu^{17} - 406843891548 \nu^{15} - 194508462680 \nu^{14} + \cdots + 43\!\cdots\!52 \nu ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1030803915 \nu^{19} + 5315924467 \nu^{18} - 3077327106 \nu^{17} + 406843891548 \nu^{15} + 2159424939980 \nu^{14} + \cdots - 43\!\cdots\!52 \nu ) / 896974523069184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7357094657 \nu^{19} + 182222209302 \nu^{17} - 423326681568 \nu^{16} + 2022007365268 \nu^{15} + 72197648157432 \nu^{13} + \cdots - 15\!\cdots\!88 ) / 53\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7957458937 \nu^{19} - 9775726086 \nu^{17} - 3291670217972 \nu^{15} - 4047882143928 \nu^{13} - 445789109896304 \nu^{11} + \cdots - 40\!\cdots\!04 \nu ) / 53\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3438551269 \nu^{19} + 7404560814 \nu^{17} - 141108893856 \nu^{16} + 1395208175204 \nu^{15} + 2841196874712 \nu^{13} + \cdots - 52\!\cdots\!96 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1030803915 \nu^{19} + 216731634280 \nu^{18} + 3077327106 \nu^{17} - 406843891548 \nu^{15} + 78360168014240 \nu^{14} + \cdots + 43\!\cdots\!52 \nu ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1030803915 \nu^{19} - 421140118 \nu^{18} - 3077327106 \nu^{17} + 406843891548 \nu^{15} - 194508462680 \nu^{14} + \cdots - 43\!\cdots\!52 \nu ) / 398655343586304 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 36482783039 \nu^{19} - 3790261062 \nu^{18} + 9140093382 \nu^{17} + 15416529773164 \nu^{15} - 1750576164120 \nu^{14} + \cdots - 23\!\cdots\!00 \nu ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2517787093 \nu^{19} + 196830877026 \nu^{17} - 52915835196 \nu^{16} - 681530892796 \nu^{15} + 67744773462408 \nu^{13} + \cdots - 19\!\cdots\!36 ) / 672730892301888 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14079898013 \nu^{19} + 115639438222 \nu^{18} - 43734664722 \nu^{17} - 6279081692356 \nu^{15} + 47686948001336 \nu^{14} + \cdots + 12\!\cdots\!84 \nu ) / 35\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15929354725 \nu^{19} + 9553162143 \nu^{18} + 431824308510 \nu^{17} - 2870272901732 \nu^{15} + 3968544145020 \nu^{14} + \cdots - 20\!\cdots\!32 \nu ) / 26\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13388817657 \nu^{19} + 181698674248 \nu^{18} + 88681950378 \nu^{17} + 6367293351924 \nu^{15} + 74230995252512 \nu^{14} + \cdots - 15\!\cdots\!36 \nu ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7408806935 \nu^{19} + 2342671674 \nu^{17} + 27477828000 \nu^{16} + 3076141758796 \nu^{15} + 694018548552 \nu^{13} + \cdots - 23\!\cdots\!12 ) / 768835305487872 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 10085118767 \nu^{19} + 10312830570 \nu^{17} - 58865100224 \nu^{16} + 4153504011820 \nu^{15} + 3830753761800 \nu^{13} + \cdots + 50\!\cdots\!72 ) / 597983015379456 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 49220074165 \nu^{19} - 98630455854 \nu^{17} - 2525968521216 \nu^{16} - 20034168320420 \nu^{15} - 38142244640472 \nu^{13} + \cdots - 56\!\cdots\!08 ) / 26\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 739043869309 \nu^{19} - 369513550770 \nu^{18} - 2826077750958 \nu^{17} + 290300114022980 \nu^{15} - 148641835419336 \nu^{14} + \cdots - 23\!\cdots\!56 \nu ) / 10\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 137159451379 \nu^{19} + 692629593474 \nu^{17} - 105622487040 \nu^{16} + 51690674461052 \nu^{15} + 243783443082984 \nu^{13} + \cdots - 25\!\cdots\!76 ) / 17\!\cdots\!68 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 8 \beta_{19} + 40 \beta_{17} + 8 \beta_{16} - 32 \beta_{14} + 6 \beta_{13} + 96 \beta_{12} - 5 \beta_{11} + 192 \beta_{10} - 119 \beta_{9} - 403 \beta_{7} - 901 \beta_{6} - 8 \beta_{5} - 3 \beta_{4} + 813 \beta_{3} + 32 \beta_{2} + 40 \beta _1 + 88 ) / 73728 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -16\beta_{14} - 48\beta_{12} - 477\beta_{9} + 285\beta_{4} - 2913\beta_{3} + 112\beta_{2} ) / 18432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 4 \beta_{19} + 12 \beta_{18} + 220 \beta_{17} - 4 \beta_{16} + 188 \beta_{14} - 114 \beta_{13} - 552 \beta_{12} + 31 \beta_{11} - 828 \beta_{10} - 433 \beta_{9} - 2179 \beta_{7} - 5761 \beta_{6} - 44 \beta_{5} + 51 \beta_{4} + \cdots + 436 ) / 36864 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 9 \beta_{19} - 46 \beta_{17} - 6 \beta_{16} + 51 \beta_{15} + 9 \beta_{11} - 269 \beta_{7} + 50 \beta_{6} + 9 \beta_{5} - 433 \beta _1 - 383195 ) / 4608 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 212 \beta_{19} + 180 \beta_{18} - 1260 \beta_{17} + 212 \beta_{16} + 1252 \beta_{14} - 930 \beta_{13} - 3576 \beta_{12} - 407 \beta_{11} - 3012 \beta_{10} - 6477 \beta_{9} + 12075 \beta_{7} + 43849 \beta_{6} + \cdots - 2308 ) / 18432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 143 \beta_{14} + 429 \beta_{12} + 5764 \beta_{9} + 153 \beta_{8} - 1866 \beta_{4} + 41961 \beta_{3} - 4376 \beta_{2} ) / 1152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1114 \beta_{19} - 750 \beta_{18} - 3878 \beta_{17} + 1114 \beta_{16} - 4310 \beta_{14} + 3588 \beta_{13} + 12180 \beta_{12} - 1825 \beta_{11} + 4230 \beta_{10} + 29309 \beta_{9} + 35771 \beta_{7} + 162447 \beta_{6} + \cdots - 6642 ) / 4608 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 115 \beta_{19} + 808 \beta_{17} + 680 \beta_{16} - 1255 \beta_{15} - 115 \beta_{11} + 3389 \beta_{7} - 476 \beta_{6} - 115 \beta_{5} + 7399 \beta _1 + 4524177 ) / 384 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 18838 \beta_{19} - 10782 \beta_{18} + 50506 \beta_{17} - 18838 \beta_{16} - 60422 \beta_{14} + 54348 \beta_{13} + 170484 \beta_{12} + 29389 \beta_{11} + 1974 \beta_{10} + 467137 \beta_{9} + \cdots + 82174 ) / 4608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 47456 \beta_{14} - 142368 \beta_{12} - 2207307 \beta_{9} - 94608 \beta_{8} + 541803 \beta_{4} - 16797495 \beta_{3} + 2356112 \beta_{2} ) / 2304 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 73469 \beta_{19} + 37059 \beta_{18} + 170891 \beta_{17} - 73469 \beta_{16} + 214159 \beta_{14} - 203313 \beta_{13} - 605418 \beta_{12} + 112907 \beta_{11} + 125361 \beta_{10} + \cdots + 268313 ) / 1152 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 20958 \beta_{19} - 251225 \beta_{17} - 308421 \beta_{16} + 413211 \beta_{15} + 20958 \beta_{11} - 895915 \beta_{7} + 110197 \beta_{6} + 20958 \beta_{5} - 2296664 \beta _1 - 1191202078 ) / 576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2209072 \beta_{19} + 1010880 \beta_{18} - 4742064 \beta_{17} + 2209072 \beta_{16} + 6116192 \beta_{14} - 6030006 \beta_{13} - 17337696 \beta_{12} - 3388003 \beta_{11} + \cdots - 7275056 ) / 2304 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1125332 \beta_{14} + 3375996 \beta_{12} + 54591109 \beta_{9} + 2794500 \beta_{8} - 12050301 \beta_{4} + 420649797 \beta_{3} - 67607336 \beta_{2} ) / 288 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 8151089 \beta_{19} - 3467661 \beta_{18} - 16707103 \beta_{17} + 8151089 \beta_{16} - 21934537 \beta_{14} + 22204011 \beta_{13} + 62335950 \beta_{12} - 12537734 \beta_{11} + \cdots - 25263117 ) / 576 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 422355 \beta_{19} + 8652368 \beta_{17} + 12315424 \beta_{16} - 14427199 \beta_{15} - 422355 \beta_{11} + 28471477 \beta_{7} - 3299892 \beta_{6} - 422355 \beta_{5} + \cdots + 38655411889 ) / 96 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 59580013 \beta_{19} - 24039423 \beta_{18} + 118872571 \beta_{17} - 59580013 \beta_{16} - 157792475 \beta_{14} + 162694401 \beta_{13} + 449338002 \beta_{12} + \cdots + 178165129 ) / 288 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 226510924 \beta_{14} - 679532772 \beta_{12} - 11095385079 \beta_{9} - 610186788 \beta_{8} + 2361984207 \beta_{4} - 85792491447 \beta_{3} + 14702610448 \beta_{2} ) / 288 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 1733248276 \beta_{19} + 673627620 \beta_{18} + 3403665868 \beta_{17} - 1733248276 \beta_{16} + 4549155188 \beta_{14} - 4750619190 \beta_{13} + \cdots + 5074083460 ) / 576 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
0.673373 0.673373i
0.673373 + 0.673373i
2.02378 + 2.02378i
2.02378 2.02378i
−2.68947 + 2.68947i
−2.68947 2.68947i
−2.18919 + 2.18919i
−2.18919 2.18919i
1.85069 + 1.85069i
1.85069 1.85069i
−1.85069 1.85069i
−1.85069 + 1.85069i
2.18919 2.18919i
2.18919 + 2.18919i
2.68947 2.68947i
2.68947 + 2.68947i
−2.02378 2.02378i
−2.02378 + 2.02378i
−0.673373 + 0.673373i
−0.673373 0.673373i
0 −15.2063 3.43054i 0 96.8365i 0 37.4486i 0 219.463 + 104.332i 0
95.2 0 −15.2063 + 3.43054i 0 96.8365i 0 37.4486i 0 219.463 104.332i 0
95.3 0 −14.6622 5.29334i 0 8.27630i 0 69.0235i 0 186.961 + 155.224i 0
95.4 0 −14.6622 + 5.29334i 0 8.27630i 0 69.0235i 0 186.961 155.224i 0
95.5 0 −7.80911 13.4914i 0 90.0279i 0 203.760i 0 −121.035 + 210.712i 0
95.6 0 −7.80911 + 13.4914i 0 90.0279i 0 203.760i 0 −121.035 210.712i 0
95.7 0 −7.49613 13.6678i 0 23.9222i 0 214.347i 0 −130.616 + 204.911i 0
95.8 0 −7.49613 + 13.6678i 0 23.9222i 0 214.347i 0 −130.616 204.911i 0
95.9 0 −6.21400 14.2964i 0 36.1786i 0 57.8847i 0 −165.772 + 177.675i 0
95.10 0 −6.21400 + 14.2964i 0 36.1786i 0 57.8847i 0 −165.772 177.675i 0
95.11 0 6.21400 14.2964i 0 36.1786i 0 57.8847i 0 −165.772 177.675i 0
95.12 0 6.21400 + 14.2964i 0 36.1786i 0 57.8847i 0 −165.772 + 177.675i 0
95.13 0 7.49613 13.6678i 0 23.9222i 0 214.347i 0 −130.616 204.911i 0
95.14 0 7.49613 + 13.6678i 0 23.9222i 0 214.347i 0 −130.616 + 204.911i 0
95.15 0 7.80911 13.4914i 0 90.0279i 0 203.760i 0 −121.035 210.712i 0
95.16 0 7.80911 + 13.4914i 0 90.0279i 0 203.760i 0 −121.035 + 210.712i 0
95.17 0 14.6622 5.29334i 0 8.27630i 0 69.0235i 0 186.961 155.224i 0
95.18 0 14.6622 + 5.29334i 0 8.27630i 0 69.0235i 0 186.961 + 155.224i 0
95.19 0 15.2063 3.43054i 0 96.8365i 0 37.4486i 0 219.463 104.332i 0
95.20 0 15.2063 + 3.43054i 0 96.8365i 0 37.4486i 0 219.463 + 104.332i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.6.c.a 20
3.b odd 2 1 inner 96.6.c.a 20
4.b odd 2 1 inner 96.6.c.a 20
8.b even 2 1 192.6.c.f 20
8.d odd 2 1 192.6.c.f 20
12.b even 2 1 inner 96.6.c.a 20
24.f even 2 1 192.6.c.f 20
24.h odd 2 1 192.6.c.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.6.c.a 20 1.a even 1 1 trivial
96.6.c.a 20 3.b odd 2 1 inner
96.6.c.a 20 4.b odd 2 1 inner
96.6.c.a 20 12.b even 2 1 inner
192.6.c.f 20 8.b even 2 1
192.6.c.f 20 8.d odd 2 1
192.6.c.f 20 24.f even 2 1
192.6.c.f 20 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} + 22 T^{18} + \cdots + 71\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( (T^{10} + 19432 T^{8} + \cdots + 38\!\cdots\!48)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + 96980 T^{8} + \cdots + 42\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} - 787048 T^{8} + \cdots - 57\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + 58 T^{4} + \cdots + 17600032211744)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + 5431296 T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + 14607924 T^{8} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} - 46609312 T^{8} + \cdots - 47\!\cdots\!52)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + 135046888 T^{8} + \cdots + 38\!\cdots\!48)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 117005396 T^{8} + \cdots + 82\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 4302 T^{4} + \cdots + 51\!\cdots\!48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + 640870816 T^{8} + \cdots + 22\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 519361556 T^{8} + \cdots + 31\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} - 1040993920 T^{8} + \cdots - 13\!\cdots\!12)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 2920253736 T^{8} + \cdots + 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 2533101160 T^{8} + \cdots - 13\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 30854 T^{4} + \cdots - 56\!\cdots\!04)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 5251781012 T^{8} + \cdots + 27\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 9994826656 T^{8} + \cdots - 46\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 12030 T^{4} + \cdots + 28\!\cdots\!92)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + 12484656404 T^{8} + \cdots + 40\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} - 19278936424 T^{8} + \cdots - 28\!\cdots\!12)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + 26495638176 T^{8} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} - 9842 T^{4} + \cdots - 53\!\cdots\!20)^{4} \) Copy content Toggle raw display
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