Properties

Label 1078.2.c.c
Level $1078$
Weight $2$
Character orbit 1078.c
Analytic conductor $8.608$
Analytic rank $0$
Dimension $16$
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] = [1078,2,Mod(1077,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1078.1077");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.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: \(8.60787333789\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 32x^{14} + 512x^{12} - 2272x^{10} - 1087x^{8} + 72448x^{6} + 819200x^{4} + 1310720x^{2} + 1048576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + (\beta_{11} - \beta_{9}) q^{5} - \beta_{12} q^{6} + \beta_{3} q^{8} + ( - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{5} q^{3} - q^{4} + (\beta_{11} - \beta_{9}) q^{5} - \beta_{12} q^{6} + \beta_{3} q^{8} + ( - \beta_1 + 1) q^{9} + (\beta_{15} - \beta_{13}) q^{10} + ( - \beta_{8} + \beta_{4} + 1) q^{11} + \beta_{5} q^{12} + (\beta_{15} - \beta_{14} - \beta_{12} + \beta_{10}) q^{13} + (\beta_{8} + \beta_{7} - \beta_1) q^{15} + q^{16} + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{10}) q^{17} + ( - \beta_{4} - \beta_{3}) q^{18} + ( - 3 \beta_{15} + \beta_{13} + \beta_{12}) q^{19} + ( - \beta_{11} + \beta_{9}) q^{20} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{22} + ( - \beta_{6} - \beta_{2} - \beta_1 - 1) q^{23} + \beta_{12} q^{24} + ( - \beta_{6} - \beta_{2} + \beta_1 - 4) q^{25} + ( - \beta_{14} - \beta_{10} + \beta_{9} + \beta_{5}) q^{26} + (\beta_{9} - 3 \beta_{5}) q^{27} + (\beta_{8} - \beta_{7} + \beta_{6} - 4 \beta_{3} - \beta_{2} + 1) q^{29} + (\beta_{6} - \beta_{4} - \beta_{2} + 1) q^{30} + ( - \beta_{14} + \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{5}) q^{31} - \beta_{3} q^{32} + (\beta_{15} + \beta_{14} - \beta_{12} + \beta_{11} - \beta_{5}) q^{33} + ( - \beta_{14} - \beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{5}) q^{34} + (\beta_1 - 1) q^{36} + ( - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{2} + 2 \beta_1 - 5) q^{37} + (\beta_{11} - 3 \beta_{9} - \beta_{5}) q^{38} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{39} + ( - \beta_{15} + \beta_{13}) q^{40} + ( - \beta_{13} + 2 \beta_{12}) q^{41} + (\beta_{6} + 2 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{43} + (\beta_{8} - \beta_{4} - 1) q^{44} + ( - \beta_{14} + \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{5}) q^{45} + ( - \beta_{8} + \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{46} + (\beta_{11} + \beta_{9}) q^{47} - \beta_{5} q^{48} + ( - \beta_{8} + \beta_{7} + \beta_{4} + 5 \beta_{3}) q^{50} + ( - \beta_{8} + \beta_{7} + 2 \beta_{6} + 4 \beta_{3} - 2 \beta_{2} + 2) q^{51} + ( - \beta_{15} + \beta_{14} + \beta_{12} - \beta_{10}) q^{52} + (\beta_{8} + \beta_{7} - \beta_1 + 2) q^{53} + ( - \beta_{15} - 3 \beta_{12}) q^{54} + ( - 5 \beta_{15} - \beta_{14} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} - 4 \beta_{5}) q^{55} + ( - \beta_{6} + 4 \beta_{4} - 2 \beta_{3} + \beta_{2} - 1) q^{57} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} - 3) q^{58} + ( - \beta_{14} - \beta_{10} + 3 \beta_{5}) q^{59} + ( - \beta_{8} - \beta_{7} + \beta_1) q^{60} + ( - \beta_{15} + \beta_{14} - 2 \beta_{13} + 5 \beta_{12} - \beta_{10}) q^{61} + (2 \beta_{15} + \beta_{14} - \beta_{13} - \beta_{12} - \beta_{10}) q^{62} - q^{64} + ( - \beta_{8} + \beta_{7} + \beta_{6} - 8 \beta_{4} + 2 \beta_{3} - \beta_{2} + 1) q^{65} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9} + \beta_{5}) q^{66} + ( - \beta_{6} - \beta_{2} + 3 \beta_1 + 1) q^{67} + (2 \beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - \beta_{10}) q^{68} + ( - \beta_{14} - \beta_{10} + \beta_{9} + 3 \beta_{5}) q^{69} + (\beta_{6} + \beta_{2} - 5 \beta_1 - 3) q^{71} + (\beta_{4} + \beta_{3}) q^{72} + ( - 2 \beta_{15} + 2 \beta_{14} + \beta_{13} - 2 \beta_{10}) q^{73} + (\beta_{8} - \beta_{7} - \beta_{6} + 2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 1) q^{74} + ( - \beta_{14} - \beta_{10} - \beta_{9} + 4 \beta_{5}) q^{75} + (3 \beta_{15} - \beta_{13} - \beta_{12}) q^{76} + ( - \beta_{8} - \beta_{7} + \beta_{6} + \beta_{2} + 2 \beta_1 + 1) q^{78} + 6 \beta_{4} q^{79} + (\beta_{11} - \beta_{9}) q^{80} + ( - 5 \beta_1 - 3) q^{81} + ( - \beta_{11} - 2 \beta_{5}) q^{82} + ( - \beta_{15} - 2 \beta_{14} + \beta_{13} - \beta_{12} + 2 \beta_{10}) q^{83} + (2 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 12 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{85}+ \cdots + ( - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{9} + 16 q^{11} + 16 q^{16} - 8 q^{22} - 16 q^{23} - 64 q^{25} - 16 q^{36} - 80 q^{37} - 16 q^{44} + 32 q^{53} - 48 q^{58} - 16 q^{64} + 16 q^{67} - 48 q^{71} + 16 q^{78} - 48 q^{81} + 32 q^{86} + 8 q^{88} + 16 q^{92} - 48 q^{93} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 32x^{14} + 512x^{12} - 2272x^{10} - 1087x^{8} + 72448x^{6} + 819200x^{4} + 1310720x^{2} + 1048576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 32173 \nu^{14} + 668960 \nu^{12} - 6919680 \nu^{10} - 46669984 \nu^{8} - 133787245 \nu^{6} + 474538112 \nu^{4} + 1011889152 \nu^{2} + \cdots - 481919500288 ) / 341451177984 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21649515 \nu^{14} - 500615872 \nu^{12} + 4404178432 \nu^{10} + 73242356320 \nu^{8} - 905783138901 \nu^{6} + \cdots + 115772308922368 ) / 28937987334144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 240905 \nu^{14} - 7913160 \nu^{12} + 129966848 \nu^{10} - 662732768 \nu^{8} + 491616713 \nu^{6} + 13417758552 \nu^{4} + \cdots + 167526400000 ) / 149937758208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 42729 \nu^{14} - 1400096 \nu^{12} + 23098880 \nu^{10} - 119264224 \nu^{8} + 119589033 \nu^{6} + 2634257152 \nu^{4} + 35772670976 \nu^{2} + \cdots + 30704009216 ) / 19295207424 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 116266153 \nu^{15} - 3358638720 \nu^{13} + 46096726528 \nu^{11} - 28654574560 \nu^{9} - 1703395919255 \nu^{7} + \cdots + 298669982056448 \nu ) / 308671864897536 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26548369 \nu^{14} + 933827904 \nu^{12} - 16598608384 \nu^{10} + 111831131872 \nu^{8} - 267579354193 \nu^{6} + \cdots + 8620523487232 ) / 9645995778048 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1040915 \nu^{14} - 33066624 \nu^{12} + 519454208 \nu^{10} - 2062231712 \nu^{8} - 5141852845 \nu^{6} + 101561457888 \nu^{4} + \cdots + 849802264576 ) / 256088383488 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 135220701 \nu^{14} + 4561306816 \nu^{12} - 76278327808 \nu^{10} + 410096159072 \nu^{8} - 107080965789 \nu^{6} + \cdots - 49543323222016 ) / 28937987334144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 263077499 \nu^{15} + 7568932160 \nu^{13} - 104813186560 \nu^{11} + 66556609440 \nu^{9} + 3864561461317 \nu^{7} + \cdots - 723250826477568 \nu ) / 308671864897536 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15339039 \nu^{15} - 510774904 \nu^{13} + 8514025600 \nu^{11} - 45705450272 \nu^{9} + 42003795807 \nu^{7} + 997736067176 \nu^{5} + \cdots - 13167578624000 \nu ) / 14468993667072 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 133618959 \nu^{15} - 4312976672 \nu^{13} + 70457976320 \nu^{11} - 349822477600 \nu^{9} + 381384242511 \nu^{7} + \cdots + 289533980278784 \nu ) / 115751949336576 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 10897203 \nu^{15} + 359824640 \nu^{13} - 5954700800 \nu^{11} + 30984053920 \nu^{9} - 17555552883 \nu^{7} + \cdots - 4233948004352 \nu ) / 8194828271616 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 77172743 \nu^{15} - 2548789688 \nu^{13} + 42111652864 \nu^{11} - 219318324768 \nu^{9} + 178889617223 \nu^{7} + \cdots + 34935425286144 \nu ) / 28937987334144 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 319597619 \nu^{15} + 10421936192 \nu^{13} - 170792382976 \nu^{11} + 861452174496 \nu^{9} - 760912344947 \nu^{7} + \cdots - 303112411742208 \nu ) / 115751949336576 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 25933907 \nu^{15} + 863194304 \nu^{13} - 14336292352 \nu^{11} + 75544371360 \nu^{9} - 37801257875 \nu^{7} + \cdots - 10214744358912 \nu ) / 8194828271616 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{6} + 8\beta_{4} - 8\beta_{3} - \beta_{2} + 8\beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{15} + 17\beta_{14} + 9\beta_{13} - 4\beta_{12} + 17\beta_{11} + 9\beta_{10} + 4\beta_{9} + 12\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -17\beta_{8} + 17\beta_{7} + 33\beta_{6} + 136\beta_{4} - 208\beta_{3} - 33\beta_{2} + 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 68 \beta_{15} + 121 \beta_{14} - 121 \beta_{13} + 132 \beta_{12} + 273 \beta_{11} + 273 \beta_{10} + 132 \beta_{9} + 332 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -737\beta_{8} - 321\beta_{7} + 737\beta_{6} + 1576\beta_{4} - 2184\beta_{3} - 321\beta_{2} - 1576\beta _1 - 1863 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2948 \beta_{15} - 1897 \beta_{14} - 4497 \beta_{13} + 7180 \beta_{12} + 1897 \beta_{11} + 4497 \beta_{10} + 1284 \beta_{9} + 2948 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -14625\beta_{8} - 14625\beta_{7} + 6129\beta_{6} + 6129\beta_{2} - 35976\beta _1 - 57281 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 58500 \beta_{15} - 76177 \beta_{14} - 76177 \beta_{13} + 141516 \beta_{12} - 31705 \beta_{11} + 31705 \beta_{10} - 24516 \beta_{9} - 58500 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 114721 \beta_{8} - 276193 \beta_{7} - 114721 \beta_{6} - 431528 \beta_{4} + 609416 \beta_{3} + 276193 \beta_{2} - 431528 \beta _1 - 885609 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 458884 \beta_{15} - 1317137 \beta_{14} - 546249 \beta_{13} + 1104772 \beta_{12} - 1317137 \beta_{11} - 546249 \beta_{10} - 1104772 \beta_{9} - 2668428 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2109905 \beta_{8} - 2109905 \beta_{7} - 5090337 \beta_{6} - 10537096 \beta_{4} + 14907088 \beta_{3} + 5090337 \beta_{2} - 5090337 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 8439620 \beta_{15} - 9563449 \beta_{14} + 9563449 \beta_{13} - 20361348 \beta_{12} - 23080977 \beta_{11} - 23080977 \beta_{10} - 20361348 \beta_{9} - 49162316 \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 92604641 \beta_{8} + 38364417 \beta_{7} - 92604641 \beta_{6} - 130577704 \beta_{4} + 184647816 \beta_{3} + 38364417 \beta_{2} + 130577704 \beta _1 + 146283399 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 370418564 \beta_{15} + 168942121 \beta_{14} + 407830161 \beta_{13} - 894294796 \beta_{12} - 168942121 \beta_{11} - 407830161 \beta_{10} - 153457668 \beta_{9} + \cdots - 370418564 \beta_{5} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(981\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1077.1
2.98338 1.23576i
−3.90726 + 1.61844i
−0.807586 1.94969i
0.424903 + 1.02581i
−0.424903 1.02581i
0.807586 + 1.94969i
3.90726 1.61844i
−2.98338 + 1.23576i
−2.98338 1.23576i
3.90726 + 1.61844i
0.807586 1.94969i
−0.424903 + 1.02581i
0.424903 1.02581i
−0.807586 + 1.94969i
−3.90726 1.61844i
2.98338 + 1.23576i
1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
1077.2 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.3 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.4 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.5 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.6 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.7 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.8 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.9 1.00000i 1.84776i −1.00000 3.23688i 1.84776 0 1.00000i −0.414214 3.23688
1077.10 1.00000i 1.84776i −1.00000 2.47151i 1.84776 0 1.00000i −0.414214 −2.47151
1077.11 1.00000i 0.765367i −1.00000 2.05161i 0.765367 0 1.00000i 2.41421 2.05161
1077.12 1.00000i 0.765367i −1.00000 3.89937i 0.765367 0 1.00000i 2.41421 −3.89937
1077.13 1.00000i 0.765367i −1.00000 3.89937i −0.765367 0 1.00000i 2.41421 3.89937
1077.14 1.00000i 0.765367i −1.00000 2.05161i −0.765367 0 1.00000i 2.41421 −2.05161
1077.15 1.00000i 1.84776i −1.00000 2.47151i −1.84776 0 1.00000i −0.414214 2.47151
1077.16 1.00000i 1.84776i −1.00000 3.23688i −1.84776 0 1.00000i −0.414214 −3.23688
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1077.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.b odd 2 1 inner
77.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1078.2.c.c 16
7.b odd 2 1 inner 1078.2.c.c 16
7.c even 3 2 1078.2.i.d 32
7.d odd 6 2 1078.2.i.d 32
11.b odd 2 1 inner 1078.2.c.c 16
77.b even 2 1 inner 1078.2.c.c 16
77.h odd 6 2 1078.2.i.d 32
77.i even 6 2 1078.2.i.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.c.c 16 1.a even 1 1 trivial
1078.2.c.c 16 7.b odd 2 1 inner
1078.2.c.c 16 11.b odd 2 1 inner
1078.2.c.c 16 77.b even 2 1 inner
1078.2.i.d 32 7.c even 3 2
1078.2.i.d 32 7.d odd 6 2
1078.2.i.d 32 77.h odd 6 2
1078.2.i.d 32 77.i even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1078, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} + 36 T^{6} + 450 T^{4} + 2304 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 8 T^{7} + 34 T^{6} - 128 T^{5} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 88 T^{6} + 2216 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 148 T^{6} + 7490 T^{4} + \cdots + 1016064)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 92 T^{6} + 2186 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 4 T^{3} - 38 T^{2} - 192 T - 144)^{4} \) Copy content Toggle raw display
$29$ \( (T^{8} + 184 T^{6} + 10768 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 132 T^{6} + 5634 T^{4} + \cdots + 322624)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + 68 T^{2} - 576 T - 2592)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} - 68 T^{6} + 1058 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 132 T^{6} + 5620 T^{4} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 52 T^{6} + 674 T^{4} + 1568 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8 T^{3} - 10 T^{2} + 136 T - 56)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 120 T^{6} + 3444 T^{4} + \cdots + 21316)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 424 T^{6} + 65672 T^{4} + \cdots + 107495424)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} - 54 T^{2} + 296 T - 376)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} - 62 T^{2} - 912 T - 1568)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 356 T^{6} + 42242 T^{4} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 72)^{8} \) Copy content Toggle raw display
$83$ \( (T^{8} - 316 T^{6} + 31658 T^{4} + \cdots + 12194064)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 232 T^{6} + 12500 T^{4} + \cdots + 454276)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 1250)^{4} \) Copy content Toggle raw display
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