Properties

Label 525.2.q.f
Level $525$
Weight $2$
Character orbit 525.q
Analytic conductor $4.192$
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] = [525,2,Mod(299,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.q (of order \(6\), degree \(2\), not 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: \(4.19214610612\)
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: \( x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{15} - \beta_{13} + \beta_{12} - \beta_{8} + \beta_{7} - \beta_1) q^{3} + (\beta_{9} + \beta_{5}) q^{4} + (\beta_{11} - \beta_{9} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{6} + (\beta_{15} + \beta_{10} + 2 \beta_{7} - \beta_1) q^{7} + (\beta_{13} - \beta_{12} - \beta_{10} - \beta_{7} + \beta_1) q^{8} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{15} - \beta_{13} + \beta_{12} - \beta_{8} + \beta_{7} - \beta_1) q^{3} + (\beta_{9} + \beta_{5}) q^{4} + (\beta_{11} - \beta_{9} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{6} + (\beta_{15} + \beta_{10} + 2 \beta_{7} - \beta_1) q^{7} + (\beta_{13} - \beta_{12} - \beta_{10} - \beta_{7} + \beta_1) q^{8} + ( - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{9} + (\beta_{9} - \beta_{5} + \beta_{3}) q^{11} + (\beta_{14} - 3 \beta_{12} + \beta_{10} + 2 \beta_{8} - \beta_{7} - \beta_1) q^{12} + (\beta_{15} + \beta_{14}) q^{13} + (2 \beta_{11} - \beta_{9} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{2} + 1) q^{14} + (\beta_{11} + \beta_{9} + \beta_{6} - 2 \beta_{3} + \beta_{2} + 1) q^{16} + (\beta_{15} + \beta_{14} - \beta_{13} - 3 \beta_{12}) q^{17} + ( - 2 \beta_{14} - \beta_{13} - \beta_{12} - \beta_{10} + 2 \beta_{8} - \beta_1) q^{18} + (\beta_{11} + \beta_{5} - \beta_{4} - 1) q^{19} + ( - \beta_{9} - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 1) q^{21} + ( - \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{10} - 3 \beta_{8} + \beta_{7} + \cdots + \beta_1) q^{22}+ \cdots + ( - 3 \beta_{11} + \beta_{9} - 11 \beta_{6} - 2 \beta_{5} - \beta_{4} + 3 \beta_{3} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{4} + 10 q^{6} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{4} + 10 q^{6} + 10 q^{9} + 24 q^{14} + 2 q^{16} - 18 q^{19} + 38 q^{21} - 32 q^{24} - 12 q^{26} - 42 q^{31} + 18 q^{36} + 6 q^{39} - 60 q^{41} - 14 q^{46} + 8 q^{49} - 12 q^{51} - 34 q^{54} - 42 q^{56} + 24 q^{59} + 30 q^{61} - 76 q^{64} + 44 q^{66} + 26 q^{69} - 108 q^{74} + 58 q^{79} - 82 q^{81} + 6 q^{84} + 18 q^{86} + 6 q^{89} - 6 q^{91} + 48 q^{94} - 6 q^{96} + 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 11x^{14} + 85x^{12} + 332x^{10} + 940x^{8} + 1064x^{6} + 880x^{4} + 128x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{14} - 127\nu^{12} - 1059\nu^{10} - 4226\nu^{8} - 14630\nu^{6} - 20976\nu^{4} - 28864\nu^{2} + 3416 ) / 17208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8072 \nu^{14} + 83435 \nu^{12} + 619005 \nu^{10} + 2199943 \nu^{8} + 5759872 \nu^{6} + 4542924 \nu^{4} + 4197620 \nu^{2} + 1131680 ) / 2357496 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1699 \nu^{14} - 18896 \nu^{12} - 149820 \nu^{10} - 612405 \nu^{8} - 1879598 \nu^{6} - 2699154 \nu^{4} - 3252680 \nu^{2} - 749364 ) / 392916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9357 \nu^{14} - 91521 \nu^{12} - 672151 \nu^{10} - 2181786 \nu^{8} - 5398232 \nu^{6} - 1173004 \nu^{4} - 1016664 \nu^{2} + 2648608 ) / 1571664 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15229 \nu^{14} + 164807 \nu^{12} + 1267309 \nu^{10} + 4851724 \nu^{8} + 13616468 \nu^{6} + 14343892 \nu^{4} + 12681936 \nu^{2} + 272784 ) / 1571664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15229 \nu^{15} + 164807 \nu^{13} + 1267309 \nu^{11} + 4851724 \nu^{9} + 13616468 \nu^{7} + 14343892 \nu^{5} + 12681936 \nu^{3} + 1844448 \nu ) / 1571664 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 427 \nu^{15} + 4723 \nu^{13} + 36549 \nu^{11} + 143882 \nu^{9} + 409832 \nu^{7} + 483588 \nu^{5} + 417712 \nu^{3} + 112384 \nu ) / 34416 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21101 \nu^{14} - 238093 \nu^{12} - 1862467 \nu^{10} - 7521662 \nu^{8} - 21834704 \nu^{6} - 27514780 \nu^{4} - 22775544 \nu^{2} - 3194176 ) / 1571664 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30473 \nu^{15} - 308177 \nu^{13} - 2306547 \nu^{11} - 7971934 \nu^{9} - 20873260 \nu^{7} - 11665404 \nu^{5} - 10957532 \nu^{3} + 8495368 \nu ) / 2357496 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13411 \nu^{14} - 150080 \nu^{12} - 1163096 \nu^{10} - 4612487 \nu^{8} - 13036508 \nu^{6} - 15171236 \nu^{4} - 11170948 \nu^{2} - 1684584 ) / 785832 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 82961 \nu^{15} - 892865 \nu^{13} - 6850017 \nu^{11} - 26014876 \nu^{9} - 72425530 \nu^{7} - 72742140 \nu^{5} - 58172408 \nu^{3} + 5684968 \nu ) / 4714992 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 14088 \nu^{15} + 160481 \nu^{13} + 1257754 \nu^{11} + 5136186 \nu^{9} + 15012495 \nu^{7} + 19847572 \nu^{5} + 17571096 \nu^{3} + 4461244 \nu ) / 785832 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29781 \nu^{15} + 319213 \nu^{13} + 2439960 \nu^{11} + 9179749 \nu^{9} + 25256949 \nu^{7} + 24011448 \nu^{5} + 18177892 \nu^{3} - 1445260 \nu ) / 785832 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 40479 \nu^{15} + 445749 \nu^{13} + 3442334 \nu^{11} + 13442445 \nu^{9} + 37944739 \nu^{7} + 42566624 \nu^{5} + 33653328 \nu^{3} + 2711404 \nu ) / 785832 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{13} + \beta_{12} + \beta_{10} + 5\beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{11} + \beta_{9} - 13\beta_{6} - 6\beta_{5} - 6\beta_{4} - 2\beta_{3} + \beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{15} + 7\beta_{14} + 7\beta_{13} - 5\beta_{12} + 10\beta_{8} - 25\beta_{7} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 32\beta_{11} - 32\beta_{9} + 25\beta_{6} + 7\beta_{5} + 25\beta_{4} + 9\beta_{3} + 9\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 43\beta_{15} - 41\beta_{14} - 2\beta_{13} + 22\beta_{12} - 43\beta_{10} - 64\beta_{8} + 125\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -41\beta_{11} + 127\beta_{9} + 168\beta_{6} + 127\beta_{5} + 41\beta_{4} + 63\beta_{3} - 126\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 22 \beta_{15} - 22 \beta_{14} - 231 \beta_{13} - 65 \beta_{12} + 231 \beta_{10} + 148 \beta_{8} + 631 \beta_{7} - 631 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -653\beta_{11} + 231\beta_{9} - 1453\beta_{6} - 884\beta_{5} - 884\beta_{4} - 802\beta_{3} + 401\beta_{2} - 569 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1285\beta_{15} + 1455\beta_{14} + 1455\beta_{13} - 483\beta_{12} + 170\beta_{10} + 796\beta_{8} - 3221\beta_{7} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4676 \beta_{11} - 4676 \beta_{9} + 3391 \beta_{6} + 1285 \beta_{5} + 3391 \beta_{4} + 2427 \beta_{3} + 2427 \beta_{2} + 2587 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8245\beta_{15} - 7103\beta_{14} - 1142\beta_{13} + 7138\beta_{12} - 8245\beta_{10} - 9352\beta_{8} + 16615\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7103 \beta_{11} + 17757 \beta_{9} + 19024 \beta_{6} + 17757 \beta_{5} + 7103 \beta_{4} + 14241 \beta_{3} - 28482 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 7138 \beta_{15} - 7138 \beta_{14} - 39101 \beta_{13} - 32139 \beta_{12} + 39101 \beta_{10} + 35620 \beta_{8} + 86501 \beta_{7} - 86501 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(1 + \beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
1.16543 2.01859i
1.03144 1.78651i
0.539169 0.933868i
0.192865 0.334053i
−0.192865 + 0.334053i
−0.539169 + 0.933868i
−1.03144 + 1.78651i
−1.16543 + 2.01859i
1.16543 + 2.01859i
1.03144 + 1.78651i
0.539169 + 0.933868i
0.192865 + 0.334053i
−0.192865 0.334053i
−0.539169 0.933868i
−1.03144 1.78651i
−1.16543 2.01859i
−1.16543 + 2.01859i −1.23297 1.21646i −1.71646 2.97300i 0 3.89248 1.07116i −2.39840 1.11699i 3.33995 0.0404447 + 2.99973i 0
299.2 −1.03144 + 1.78651i −1.61429 + 0.627739i −1.12774 1.95330i 0 0.543588 3.53142i −2.64573 0.00953166i 0.527019 2.21189 2.02671i 0
299.3 −0.539169 + 0.933868i 1.46840 + 0.918594i 0.418594 + 0.725026i 0 −1.64956 + 0.876010i 0.929227 + 2.47720i −3.05945 1.31237 + 2.69772i 0
299.4 −0.192865 + 0.334053i −0.983691 + 1.42561i 0.925606 + 1.60320i 0 −0.286507 0.603555i −1.17656 2.36975i −1.48553 −1.06470 2.80471i 0
299.5 0.192865 0.334053i 0.983691 1.42561i 0.925606 + 1.60320i 0 −0.286507 0.603555i 1.17656 + 2.36975i 1.48553 −1.06470 2.80471i 0
299.6 0.539169 0.933868i −1.46840 0.918594i 0.418594 + 0.725026i 0 −1.64956 + 0.876010i −0.929227 2.47720i 3.05945 1.31237 + 2.69772i 0
299.7 1.03144 1.78651i 1.61429 0.627739i −1.12774 1.95330i 0 0.543588 3.53142i 2.64573 + 0.00953166i −0.527019 2.21189 2.02671i 0
299.8 1.16543 2.01859i 1.23297 + 1.21646i −1.71646 2.97300i 0 3.89248 1.07116i 2.39840 + 1.11699i −3.33995 0.0404447 + 2.99973i 0
374.1 −1.16543 2.01859i −1.23297 + 1.21646i −1.71646 + 2.97300i 0 3.89248 + 1.07116i −2.39840 + 1.11699i 3.33995 0.0404447 2.99973i 0
374.2 −1.03144 1.78651i −1.61429 0.627739i −1.12774 + 1.95330i 0 0.543588 + 3.53142i −2.64573 + 0.00953166i 0.527019 2.21189 + 2.02671i 0
374.3 −0.539169 0.933868i 1.46840 0.918594i 0.418594 0.725026i 0 −1.64956 0.876010i 0.929227 2.47720i −3.05945 1.31237 2.69772i 0
374.4 −0.192865 0.334053i −0.983691 1.42561i 0.925606 1.60320i 0 −0.286507 + 0.603555i −1.17656 + 2.36975i −1.48553 −1.06470 + 2.80471i 0
374.5 0.192865 + 0.334053i 0.983691 + 1.42561i 0.925606 1.60320i 0 −0.286507 + 0.603555i 1.17656 2.36975i 1.48553 −1.06470 + 2.80471i 0
374.6 0.539169 + 0.933868i −1.46840 + 0.918594i 0.418594 0.725026i 0 −1.64956 0.876010i −0.929227 + 2.47720i 3.05945 1.31237 2.69772i 0
374.7 1.03144 + 1.78651i 1.61429 + 0.627739i −1.12774 + 1.95330i 0 0.543588 + 3.53142i 2.64573 0.00953166i −0.527019 2.21189 + 2.02671i 0
374.8 1.16543 + 2.01859i 1.23297 1.21646i −1.71646 + 2.97300i 0 3.89248 + 1.07116i 2.39840 1.11699i −3.33995 0.0404447 2.99973i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 299.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
21.g even 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.q.f 16
3.b odd 2 1 525.2.q.e 16
5.b even 2 1 inner 525.2.q.f 16
5.c odd 4 1 105.2.s.c 8
5.c odd 4 1 525.2.t.g 8
7.d odd 6 1 525.2.q.e 16
15.d odd 2 1 525.2.q.e 16
15.e even 4 1 105.2.s.d yes 8
15.e even 4 1 525.2.t.f 8
21.g even 6 1 inner 525.2.q.f 16
35.f even 4 1 735.2.s.k 8
35.i odd 6 1 525.2.q.e 16
35.k even 12 1 105.2.s.d yes 8
35.k even 12 1 525.2.t.f 8
35.k even 12 1 735.2.b.c 8
35.l odd 12 1 735.2.b.d 8
35.l odd 12 1 735.2.s.l 8
105.k odd 4 1 735.2.s.l 8
105.p even 6 1 inner 525.2.q.f 16
105.w odd 12 1 105.2.s.c 8
105.w odd 12 1 525.2.t.g 8
105.w odd 12 1 735.2.b.d 8
105.x even 12 1 735.2.b.c 8
105.x even 12 1 735.2.s.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c 8 5.c odd 4 1
105.2.s.c 8 105.w odd 12 1
105.2.s.d yes 8 15.e even 4 1
105.2.s.d yes 8 35.k even 12 1
525.2.q.e 16 3.b odd 2 1
525.2.q.e 16 7.d odd 6 1
525.2.q.e 16 15.d odd 2 1
525.2.q.e 16 35.i odd 6 1
525.2.q.f 16 1.a even 1 1 trivial
525.2.q.f 16 5.b even 2 1 inner
525.2.q.f 16 21.g even 6 1 inner
525.2.q.f 16 105.p even 6 1 inner
525.2.t.f 8 15.e even 4 1
525.2.t.f 8 35.k even 12 1
525.2.t.g 8 5.c odd 4 1
525.2.t.g 8 105.w odd 12 1
735.2.b.c 8 35.k even 12 1
735.2.b.c 8 105.x even 12 1
735.2.b.d 8 35.l odd 12 1
735.2.b.d 8 105.w odd 12 1
735.2.s.k 8 35.f even 4 1
735.2.s.k 8 105.x even 12 1
735.2.s.l 8 35.l odd 12 1
735.2.s.l 8 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{16} + 11T_{2}^{14} + 85T_{2}^{12} + 332T_{2}^{10} + 940T_{2}^{8} + 1064T_{2}^{6} + 880T_{2}^{4} + 128T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 28T_{11}^{6} + 636T_{11}^{4} - 168T_{11}^{3} - 4132T_{11}^{2} + 888T_{11} + 21904 \) Copy content Toggle raw display
\( T_{13}^{8} - 21T_{13}^{6} + 123T_{13}^{4} - 135T_{13}^{2} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 11 T^{14} + 85 T^{12} + 332 T^{10} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{16} - 5 T^{14} + 33 T^{12} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{14} - 26 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{6} + 636 T^{4} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 21 T^{6} + 123 T^{4} - 135 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 60 T^{14} + 2820 T^{12} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + 24 T^{6} - 27 T^{5} + \cdots + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 131 T^{14} + \cdots + 206366684176 \) Copy content Toggle raw display
$29$ \( (T^{8} + 179 T^{6} + 10848 T^{4} + \cdots + 879844)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 21 T^{7} + 138 T^{6} + \cdots + 695556)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 199 T^{14} + \cdots + 236421376 \) Copy content Toggle raw display
$41$ \( (T^{4} + 15 T^{3} + 54 T^{2} - 54 T - 378)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 40 T^{6} + 306 T^{4} + 76 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 108 T^{14} + \cdots + 21743271936 \) Copy content Toggle raw display
$53$ \( T^{16} + 320 T^{14} + \cdots + 17592186044416 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + 156 T^{6} - 252 T^{5} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 15 T^{7} - 21 T^{6} + 1440 T^{5} + \cdots + 76176)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 52 T^{14} + \cdots + 104060401 \) Copy content Toggle raw display
$71$ \( (T^{8} + 104 T^{6} + 2688 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 249 T^{14} + 53634 T^{12} + \cdots + 1296 \) Copy content Toggle raw display
$79$ \( (T^{8} - 29 T^{7} + 610 T^{6} + \cdots + 12166144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 129 T^{6} + 1500 T^{4} + \cdots + 1764)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 3 T^{7} + 303 T^{6} + \cdots + 2643876)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 408 T^{6} + 55680 T^{4} + \cdots + 49449024)^{2} \) Copy content Toggle raw display
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