Properties

Label 315.4.d.d
Level $315$
Weight $4$
Character orbit 315.d
Analytic conductor $18.586$
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] = [315,4,Mod(64,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.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: \(18.5856016518\)
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: \( x^{20} + 144 x^{18} + 8518 x^{16} + 269932 x^{14} + 5002289 x^{12} + 55478700 x^{10} + 361614704 x^{8} + 1303665216 x^{6} + 2335301632 x^{4} + \cdots + 603979776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 5^{2}\cdot 7^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_1 - 5) q^{4} + \beta_{10} q^{5} + \beta_{9} q^{7} + (\beta_{11} + 6 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + (\beta_1 - 5) q^{4} + \beta_{10} q^{5} + \beta_{9} q^{7} + (\beta_{11} + 6 \beta_{4}) q^{8} + (\beta_{9} - \beta_{6} + 6) q^{10} + (\beta_{3} + \beta_{2}) q^{11} + ( - \beta_{9} - \beta_{5}) q^{13} - \beta_{2} q^{14} + (\beta_{8} + \beta_{7} - 5 \beta_1 + 29) q^{16} + (\beta_{19} + \beta_{11} + \beta_{10} - \beta_{4}) q^{17} + (\beta_{7} - \beta_{6} - 4 \beta_1 - 5) q^{19} + (\beta_{19} - \beta_{13} - \beta_{11} - 5 \beta_{10} - 2 \beta_{4} - \beta_{3} - \beta_{2}) q^{20} + (2 \beta_{18} + \beta_{14} + 9 \beta_{9} - \beta_{6} - \beta_{5}) q^{22} + (\beta_{17} + \beta_{15} + \beta_{12} - \beta_{11} + 3 \beta_{10} - 3 \beta_{4}) q^{23} + (\beta_{14} - \beta_{9} + \beta_{7} - \beta_{5} - 2 \beta_1 + 20) q^{25} + ( - 2 \beta_{19} + 2 \beta_{17} - 3 \beta_{15} + 2 \beta_{13} - 2 \beta_{12} + 5 \beta_{10}) q^{26} + ( - \beta_{18} - 5 \beta_{9}) q^{28} + ( - \beta_{19} + \beta_{17} - \beta_{15} - \beta_{12} + 2 \beta_{10} - \beta_{3} - 5 \beta_{2}) q^{29} + (\beta_{18} - \beta_{16} + \beta_{14} - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} + 3 \beta_1 - 2) q^{31} + (2 \beta_{17} - 2 \beta_{15} + 2 \beta_{12} - 10 \beta_{11} - 6 \beta_{10} - 31 \beta_{4}) q^{32} + (\beta_{8} + \beta_{7} - 6 \beta_1 - 14) q^{34} + (\beta_{19} + \beta_{12} - \beta_{11} - \beta_{10} - 3 \beta_{4} + \beta_{2}) q^{35} + (2 \beta_{18} - \beta_{14} + 11 \beta_{9} + \beta_{6} - 3 \beta_{5}) q^{37} + (2 \beta_{17} - 4 \beta_{15} + 2 \beta_{12} - 11 \beta_{11} - 12 \beta_{10} - 27 \beta_{4}) q^{38} + ( - 4 \beta_{18} - 2 \beta_{14} - 13 \beta_{9} - \beta_{8} - \beta_{7} + 3 \beta_{6} + \cdots + 2) q^{40}+ \cdots + 49 \beta_{4} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 108 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 108 q^{4} + 112 q^{10} + 620 q^{16} - 72 q^{19} + 428 q^{25} - 48 q^{31} - 232 q^{34} - 16 q^{40} - 368 q^{46} - 980 q^{49} + 1904 q^{55} + 2048 q^{61} - 3180 q^{64} - 756 q^{70} - 8368 q^{76} + 1552 q^{79} - 2616 q^{85} + 1456 q^{91} + 8056 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 144 x^{18} + 8518 x^{16} + 269932 x^{14} + 5002289 x^{12} + 55478700 x^{10} + 361614704 x^{8} + 1303665216 x^{6} + 2335301632 x^{4} + \cdots + 603979776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 465021503 \nu^{18} - 77684214024 \nu^{16} - 5383318221626 \nu^{14} - 200420748745508 \nu^{12} + \cdots - 56\!\cdots\!96 ) / 11\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3255150521 \nu^{18} + 543789498168 \nu^{16} + 37683227551382 \nu^{14} + \cdots + 50\!\cdots\!16 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 717392106679 \nu^{18} - 103623757921560 \nu^{16} + \cdots - 42\!\cdots\!24 ) / 15\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 221279727269 \nu^{19} + 31774996598160 \nu^{17} + \cdots + 10\!\cdots\!96 \nu ) / 43\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4351528823527 \nu^{19} + 637080822795696 \nu^{17} + \cdots + 44\!\cdots\!72 \nu ) / 30\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4199576014411 \nu^{19} - 5465745867120 \nu^{18} + 590865472700928 \nu^{17} - 750679005920256 \nu^{16} + \cdots - 12\!\cdots\!16 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4199576014411 \nu^{19} + 74855304483120 \nu^{18} + 590865472700928 \nu^{17} + \cdots + 23\!\cdots\!44 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4199576014411 \nu^{19} - 112932858139920 \nu^{18} - 590865472700928 \nu^{17} + \cdots - 43\!\cdots\!84 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1548958090883 \nu^{19} - 222424976187120 \nu^{17} + \cdots - 68\!\cdots\!08 \nu ) / 43\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35670797971781 \nu^{19} - 12416725219872 \nu^{18} + \cdots - 99\!\cdots\!96 ) / 75\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1459673962307 \nu^{19} - 207509607094512 \nu^{17} + \cdots - 57\!\cdots\!60 \nu ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 204494539235861 \nu^{19} + 690720097783488 \nu^{18} + \cdots + 34\!\cdots\!64 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 37912440093937 \nu^{19} + 201405300010944 \nu^{18} + \cdots + 11\!\cdots\!12 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 24714338304557 \nu^{19} - 4684925028960 \nu^{18} + \cdots - 10\!\cdots\!28 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 114858141342889 \nu^{19} + 18625087829808 \nu^{18} + \cdots + 14\!\cdots\!44 ) / 37\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 162695106701881 \nu^{19} - 192189596923520 \nu^{18} + \cdots - 71\!\cdots\!16 ) / 50\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 505350503918939 \nu^{19} - 690720097783488 \nu^{18} + \cdots - 34\!\cdots\!64 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 498958560849 \nu^{19} + 71241448072144 \nu^{17} + \cdots + 19\!\cdots\!60 \nu ) / 12\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 150495286990293 \nu^{19} + 4138908406624 \nu^{18} + \cdots + 33\!\cdots\!32 ) / 25\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + 7\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 7\beta _1 - 98 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{18} - 7\beta_{11} - 40\beta_{9} - 175\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4 \beta_{19} - 4 \beta_{17} + 8 \beta_{15} - 8 \beta_{13} + 4 \beta_{12} - 12 \beta_{10} + 7 \beta_{8} + 7 \beta_{7} - 12 \beta_{3} - 84 \beta_{2} - 245 \beta _1 + 2492 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 150 \beta_{18} - 14 \beta_{17} - 5 \beta_{16} + 14 \beta_{15} + 20 \beta_{14} - 14 \beta_{12} + 364 \beta_{11} + 42 \beta_{10} + 1516 \beta_{9} - 5 \beta_{7} - 45 \beta_{6} + 25 \beta_{5} + 5376 \beta_{4} + 5 \beta _1 + 15 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 332 \beta_{19} + 224 \beta_{17} - 592 \beta_{15} + 520 \beta_{13} - 224 \beta_{12} + 1356 \beta_{10} - 399 \beta_{8} - 455 \beta_{7} + 56 \beta_{6} - 216 \beta_{4} + 696 \beta_{3} + 3160 \beta_{2} + 8925 \beta _1 - 77644 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6482 \beta_{18} + 994 \beta_{17} + 441 \beta_{16} - 1106 \beta_{15} - 1372 \beta_{14} + 994 \beta_{12} - 16408 \beta_{11} - 3318 \beta_{10} - 57436 \beta_{9} + 441 \beta_{7} + 3577 \beta_{6} - 1421 \beta_{5} + \cdots - 1323 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 18128 \beta_{19} - 112 \beta_{18} - 9664 \beta_{17} + 112 \beta_{16} + 30912 \beta_{15} - 112 \beta_{14} - 25568 \beta_{13} + 9664 \beta_{12} - 82896 \beta_{10} + 112 \beta_{9} + 18711 \beta_{8} + 23191 \beta_{7} + \cdots + 2698500 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 224 \beta_{19} + 271386 \beta_{18} - 52542 \beta_{17} - 25185 \beta_{16} + 64078 \beta_{15} + 71844 \beta_{14} - 52542 \beta_{12} + 713720 \beta_{11} + 192458 \beta_{10} + 2217676 \beta_{9} + \cdots + 75555 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 853364 \beta_{19} + 12544 \beta_{18} + 387008 \beta_{17} - 12544 \beta_{16} - 1426064 \beta_{15} + 12544 \beta_{14} + 1145400 \beta_{13} - 387008 \beta_{12} + 4144852 \beta_{10} + \cdots - 100170588 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 27328 \beta_{19} - 11276210 \beta_{18} + 2481850 \beta_{17} + 1227897 \beta_{16} - 3263722 \beta_{15} - 3399308 \beta_{14} + 2481850 \beta_{12} - 30561832 \beta_{11} + \cdots - 3683691 ) / 7 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 37677528 \beta_{19} - 882896 \beta_{18} - 15189504 \beta_{17} + 882896 \beta_{16} + 62348128 \beta_{15} - 882896 \beta_{14} - 49341200 \beta_{13} + 15189504 \beta_{12} + \cdots + 3874329620 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1995168 \beta_{19} + 467849642 \beta_{18} - 110984118 \beta_{17} - 55700593 \beta_{16} + 154800646 \beta_{15} + 152895444 \beta_{14} - 110984118 \beta_{12} + 1296152984 \beta_{11} + \cdots + 167101779 ) / 7 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1613956284 \beta_{19} + 50425312 \beta_{18} + 596231616 \beta_{17} - 50425312 \beta_{16} - 2657566128 \beta_{15} + 50425312 \beta_{14} + 2087219688 \beta_{13} + \cdots - 153784357804 ) / 7 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 116063360 \beta_{19} - 19413185602 \beta_{18} + 4818281650 \beta_{17} + 2435298313 \beta_{16} - 7025613602 \beta_{15} - 6682189948 \beta_{14} + 4818281650 \beta_{12} + \cdots - 7305894939 ) / 7 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 68110340384 \beta_{19} - 2566954544 \beta_{18} - 23583958016 \beta_{17} + 2566954544 \beta_{16} + 111867584640 \beta_{15} - 2566954544 \beta_{14} + \cdots + 6205216413732 ) / 7 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 5969217632 \beta_{19} + 805865205370 \beta_{18} - 205666851502 \beta_{17} - 104342791233 \beta_{16} + 309846751550 \beta_{15} + 286980690564 \beta_{14} + \cdots + 313028373699 ) / 7 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 2852683485700 \beta_{19} + 121905022400 \beta_{18} + 941952442816 \beta_{17} - 121905022400 \beta_{16} - 4679265488080 \beta_{15} + \cdots - 253003481109372 ) / 7 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 285229507136 \beta_{19} - 33464516148818 \beta_{18} + 8690900154538 \beta_{17} + 4418848167705 \beta_{16} - 13406589714266 \beta_{15} + \cdots - 13256544503115 ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
6.44617i
4.44617i
3.31226i
5.31226i
2.73445i
4.73445i
3.18874i
1.18874i
0.916651i
1.08335i
0.916651i
1.08335i
3.18874i
1.18874i
2.73445i
4.73445i
3.31226i
5.31226i
6.44617i
4.44617i
5.44617i 0 −21.6607 −10.6013 + 3.55149i 0 7.00000i 74.3987i 0 19.3420 + 57.7363i
64.2 5.44617i 0 −21.6607 10.6013 + 3.55149i 0 7.00000i 74.3987i 0 19.3420 57.7363i
64.3 4.31226i 0 −10.5956 −7.55443 8.24200i 0 7.00000i 11.1930i 0 −35.5417 + 32.5767i
64.4 4.31226i 0 −10.5956 7.55443 8.24200i 0 7.00000i 11.1930i 0 −35.5417 32.5767i
64.5 3.73445i 0 −5.94609 −5.59787 + 9.67801i 0 7.00000i 7.67022i 0 36.1420 + 20.9049i
64.6 3.73445i 0 −5.94609 5.59787 + 9.67801i 0 7.00000i 7.67022i 0 36.1420 20.9049i
64.7 2.18874i 0 3.20940 −10.6651 + 3.35486i 0 7.00000i 24.5345i 0 7.34293 + 23.3432i
64.8 2.18874i 0 3.20940 10.6651 + 3.35486i 0 7.00000i 24.5345i 0 7.34293 23.3432i
64.9 0.0833494i 0 7.99305 −7.17375 + 8.57539i 0 7.00000i 1.33301i 0 0.714753 + 0.597928i
64.10 0.0833494i 0 7.99305 7.17375 + 8.57539i 0 7.00000i 1.33301i 0 0.714753 0.597928i
64.11 0.0833494i 0 7.99305 −7.17375 8.57539i 0 7.00000i 1.33301i 0 0.714753 0.597928i
64.12 0.0833494i 0 7.99305 7.17375 8.57539i 0 7.00000i 1.33301i 0 0.714753 + 0.597928i
64.13 2.18874i 0 3.20940 −10.6651 3.35486i 0 7.00000i 24.5345i 0 7.34293 23.3432i
64.14 2.18874i 0 3.20940 10.6651 3.35486i 0 7.00000i 24.5345i 0 7.34293 + 23.3432i
64.15 3.73445i 0 −5.94609 −5.59787 9.67801i 0 7.00000i 7.67022i 0 36.1420 20.9049i
64.16 3.73445i 0 −5.94609 5.59787 9.67801i 0 7.00000i 7.67022i 0 36.1420 + 20.9049i
64.17 4.31226i 0 −10.5956 −7.55443 + 8.24200i 0 7.00000i 11.1930i 0 −35.5417 32.5767i
64.18 4.31226i 0 −10.5956 7.55443 + 8.24200i 0 7.00000i 11.1930i 0 −35.5417 + 32.5767i
64.19 5.44617i 0 −21.6607 −10.6013 3.55149i 0 7.00000i 74.3987i 0 19.3420 57.7363i
64.20 5.44617i 0 −21.6607 10.6013 3.55149i 0 7.00000i 74.3987i 0 19.3420 + 57.7363i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 64.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.d.d 20
3.b odd 2 1 inner 315.4.d.d 20
5.b even 2 1 inner 315.4.d.d 20
5.c odd 4 1 1575.4.a.bt 10
5.c odd 4 1 1575.4.a.bu 10
15.d odd 2 1 inner 315.4.d.d 20
15.e even 4 1 1575.4.a.bt 10
15.e even 4 1 1575.4.a.bu 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.d.d 20 1.a even 1 1 trivial
315.4.d.d 20 3.b odd 2 1 inner
315.4.d.d 20 5.b even 2 1 inner
315.4.d.d 20 15.d odd 2 1 inner
1575.4.a.bt 10 5.c odd 4 1
1575.4.a.bt 10 15.e even 4 1
1575.4.a.bu 10 5.c odd 4 1
1575.4.a.bu 10 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 67T_{2}^{8} + 1523T_{2}^{6} + 13569T_{2}^{4} + 36944T_{2}^{2} + 256 \) acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} + 67 T^{8} + 1523 T^{6} + 13569 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 214 T^{18} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{10} \) Copy content Toggle raw display
$11$ \( (T^{10} - 8988 T^{8} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + 15472 T^{8} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + 12880 T^{8} + \cdots + 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 18 T^{4} - 20572 T^{3} + \cdots + 1947671040)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + 73124 T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} - 93332 T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 12 T^{4} - 102304 T^{3} + \cdots + 76972365952)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + 292996 T^{8} + \cdots + 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} - 222504 T^{8} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + 503028 T^{8} + \cdots + 34\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + 320468 T^{8} + \cdots + 24\!\cdots\!24)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + 891664 T^{8} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} - 1281632 T^{8} + \cdots - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 512 T^{4} + \cdots - 1370493512000)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + 1725796 T^{8} + \cdots + 40\!\cdots\!64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} - 2602184 T^{8} + \cdots - 90\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + 2167376 T^{8} + \cdots + 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 388 T^{4} + \cdots - 8769624080384)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + 774896 T^{8} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} - 4067576 T^{8} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + 4967664 T^{8} + \cdots + 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
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