Properties

Label 2400.2.m.c
Level $2400$
Weight $2$
Character orbit 2400.m
Analytic conductor $19.164$
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] = [2400,2,Mod(1199,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("2400.1199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.m (of order \(2\), degree \(1\), 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: \(19.1640964851\)
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: \( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{19} \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{9} q^{7} + \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{9} q^{7} + \beta_{7} q^{9} + \beta_{10} q^{11} + ( - \beta_{13} + \beta_{4} + \beta_{3}) q^{13} + ( - \beta_{9} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{7} - \beta_{5} - \beta_{2} - 1) q^{19} + ( - \beta_{14} + \beta_{10} + \beta_{5} + \beta_{2}) q^{21} + (\beta_{12} + \beta_{6} - \beta_{3}) q^{23} + ( - \beta_{13} + \beta_{6} + \beta_{3} - \beta_1) q^{27} + ( - \beta_{14} - \beta_{11} + \beta_{7} - \beta_{5}) q^{29} + (\beta_{15} - \beta_{10} - \beta_{7} - \beta_{5}) q^{31} + ( - \beta_{9} - \beta_{8} - \beta_{6}) q^{33} + (\beta_{13} - 2 \beta_{8} + \beta_{4} + \beta_{3}) q^{37} + (\beta_{15} - \beta_{14} - \beta_{10} + \beta_{7} - \beta_{5} + 2) q^{39} + (\beta_{7} + \beta_{5}) q^{41} + ( - \beta_{4} + \beta_{3}) q^{43} + (\beta_{12} - \beta_{6} - \beta_{3} + 2 \beta_1) q^{47} + (\beta_{7} - \beta_{5} - 2 \beta_{2} + 1) q^{49} + (\beta_{14} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{2} - 3) q^{51} + (2 \beta_{4} - 2 \beta_{3} + \beta_1) q^{53} + ( - \beta_{13} + \beta_{12} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} + 2 \beta_{3}) q^{57} + ( - 2 \beta_{15} + \beta_{14} - \beta_{11} + \beta_{10}) q^{59} + (\beta_{14} - \beta_{11} + \beta_{7} + \beta_{5}) q^{61} + ( - \beta_{12} - 2 \beta_{8} - \beta_{6} - \beta_{3} + 2 \beta_1) q^{63} + (2 \beta_{12} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{67} + (2 \beta_{15} + \beta_{11} - \beta_{10} + \beta_{7} + 2 \beta_{5} + \beta_{2} - 2) q^{69} + (\beta_{14} + \beta_{11} - 2 \beta_{2} + 2) q^{71} + (2 \beta_{6} - \beta_1) q^{73} + (2 \beta_{12} + 2 \beta_{4} - 4 \beta_{3}) q^{77} + (\beta_{15} - \beta_{14} + \beta_{11} - \beta_{10} + \beta_{7} + \beta_{5}) q^{79} + (2 \beta_{15} - 2 \beta_{14} - 2 \beta_{10} + \beta_{7} + \beta_{5} + 1) q^{81} + ( - 2 \beta_{9} - \beta_{4} - \beta_{3}) q^{83} + ( - 2 \beta_{13} + \beta_{12} + \beta_{9} + \beta_{6} + 3 \beta_{3} + 2 \beta_1) q^{87} + ( - 4 \beta_{15} + 2 \beta_{14} - 2 \beta_{11} - 2 \beta_{7} - 2 \beta_{5}) q^{89} + ( - \beta_{14} - \beta_{11} + 2 \beta_{2} - 2) q^{91} + (2 \beta_{13} + \beta_{12} + \beta_{9} - \beta_{4} + 2 \beta_{3}) q^{93} + (2 \beta_{12} + 2 \beta_{6} - 2 \beta_{3} - \beta_1) q^{97} + ( - 2 \beta_{15} + 2 \beta_{14} - 2 \beta_{11} - \beta_{10} - 2 \beta_{7} - 4 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{19} - 8 q^{21} + 32 q^{39} + 32 q^{49} - 40 q^{51} - 40 q^{69} + 48 q^{71} + 16 q^{81} - 48 q^{91} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 3\nu^{13} + 10\nu^{11} + 8\nu^{9} + 8\nu^{7} + 64\nu^{5} + 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + \nu^{8} + 8\nu^{4} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 3\nu^{13} - 2\nu^{11} + 8\nu^{9} - 4\nu^{7} - 8\nu^{5} + 48\nu^{3} - 32\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - 3\nu^{13} - 10\nu^{11} - 20\nu^{9} - 44\nu^{7} - 40\nu^{5} - 144\nu^{3} - 160\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} + 9\nu^{12} + 16\nu^{10} + 32\nu^{8} + 128\nu^{6} + 112\nu^{4} + 384\nu^{2} + 448 ) / 192 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 3\nu^{13} + 4\nu^{11} + 8\nu^{9} + 8\nu^{7} + 112\nu^{5} + 448\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{14} - 3\nu^{12} + 4\nu^{10} - 16\nu^{8} - 16\nu^{6} + 112\nu^{4} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{9} + \nu^{7} - 2\nu^{5} - 4\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + \nu^{11} + 4\nu^{9} + 12\nu^{7} + 16\nu^{5} + 16\nu^{3} + 32\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{14} + 3\nu^{12} - 4\nu^{10} - 8\nu^{8} - 8\nu^{6} - 16\nu^{4} - 96\nu^{2} - 64 ) / 96 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} + 2\nu^{10} - 20\nu^{8} - 32\nu^{6} - 64\nu^{4} - 96\nu^{2} - 64 ) / 96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + \nu^{13} - 2\nu^{11} - 8\nu^{9} - 4\nu^{7} - 8\nu^{5} + 48\nu^{3} - 32\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{15} + \nu^{13} + 8\nu^{7} + 16\nu^{5} - 64\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{14} - \nu^{10} - 2\nu^{8} - 8\nu^{6} + 8\nu^{4} + 48\nu^{2} + 32 ) / 48 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{14} - \nu^{12} - 4\nu^{10} - 4\nu^{8} - 8\nu^{6} - 16\nu^{4} - 32\nu^{2} ) / 32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{8} + \beta_{6} + 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + 2\beta_{14} - \beta_{10} - \beta_{2} - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{13} + 2\beta_{12} - 2\beta_{9} - \beta_{8} + \beta_{6} - 4\beta_{4} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} - 2\beta_{11} + \beta_{10} + 4\beta_{7} + \beta_{2} - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{13} + 2\beta_{12} + 2\beta_{9} - 3\beta_{8} + 3\beta_{6} + 4\beta_{4} - 4\beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 7\beta_{15} - 8\beta_{14} + 2\beta_{11} - \beta_{10} + 4\beta_{7} + 8\beta_{5} - \beta_{2} - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5\beta_{13} - 2\beta_{12} + 6\beta_{9} + 3\beta_{8} - 3\beta_{6} - 4\beta_{4} - 12\beta_{3} - 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{15} - 10\beta_{11} - 7\beta_{10} - 20\beta_{7} - 8\beta_{5} + 9\beta_{2} + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -13\beta_{13} - 14\beta_{12} + 10\beta_{9} - 11\beta_{8} - 5\beta_{6} + 4\beta_{4} + 12\beta_{3} - 6\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -9\beta_{15} + 26\beta_{11} - \beta_{10} - 12\beta_{7} + 8\beta_{5} + 15\beta_{2} - 13 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -11\beta_{13} - 2\beta_{12} - 26\beta_{9} + 3\beta_{8} - 19\beta_{6} - 36\beta_{4} - 44\beta_{3} + 54\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -47\beta_{15} + 32\beta_{14} - 10\beta_{11} + 73\beta_{10} - 20\beta_{7} - 8\beta_{5} - 7\beta_{2} + 53 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -29\beta_{13} + 18\beta_{12} + 42\beta_{9} + 37\beta_{8} - 21\beta_{6} + 68\beta_{4} + 140\beta_{3} + 90\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -89\beta_{15} - 32\beta_{14} - 38\beta_{11} - 17\beta_{10} + 52\beta_{7} - 56\beta_{5} - 65\beta_{2} + 163 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 133\beta_{13} - 34\beta_{12} - 122\beta_{9} + 51\beta_{8} + 61\beta_{6} - 100\beta_{4} + 148\beta_{3} - 42\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(1951\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1
−0.199044 1.40014i
−0.199044 + 1.40014i
−1.29041 0.578647i
−1.29041 + 0.578647i
1.15595 + 0.814732i
1.15595 0.814732i
−0.842022 + 1.13622i
−0.842022 1.13622i
0.842022 + 1.13622i
0.842022 1.13622i
−1.15595 + 0.814732i
−1.15595 0.814732i
1.29041 0.578647i
1.29041 + 0.578647i
0.199044 1.40014i
0.199044 + 1.40014i
0 −1.65195 0.520627i 0 0 0 −1.92736 0 2.45790 + 1.72010i 0
1199.2 0 −1.65195 + 0.520627i 0 0 0 −1.92736 0 2.45790 1.72010i 0
1199.3 0 −1.56044 0.751690i 0 0 0 4.28591 0 1.86993 + 2.34593i 0
1199.4 0 −1.56044 + 0.751690i 0 0 0 4.28591 0 1.86993 2.34593i 0
1199.5 0 −0.887900 1.48716i 0 0 0 −0.797253 0 −1.42327 + 2.64089i 0
1199.6 0 −0.887900 + 1.48716i 0 0 0 −0.797253 0 −1.42327 2.64089i 0
1199.7 0 −0.218455 1.71822i 0 0 0 −3.64426 0 −2.90455 + 0.750707i 0
1199.8 0 −0.218455 + 1.71822i 0 0 0 −3.64426 0 −2.90455 0.750707i 0
1199.9 0 0.218455 1.71822i 0 0 0 3.64426 0 −2.90455 0.750707i 0
1199.10 0 0.218455 + 1.71822i 0 0 0 3.64426 0 −2.90455 + 0.750707i 0
1199.11 0 0.887900 1.48716i 0 0 0 0.797253 0 −1.42327 2.64089i 0
1199.12 0 0.887900 + 1.48716i 0 0 0 0.797253 0 −1.42327 + 2.64089i 0
1199.13 0 1.56044 0.751690i 0 0 0 −4.28591 0 1.86993 2.34593i 0
1199.14 0 1.56044 + 0.751690i 0 0 0 −4.28591 0 1.86993 + 2.34593i 0
1199.15 0 1.65195 0.520627i 0 0 0 1.92736 0 2.45790 1.72010i 0
1199.16 0 1.65195 + 0.520627i 0 0 0 1.92736 0 2.45790 + 1.72010i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1199.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
24.f even 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.m.c 16
3.b odd 2 1 2400.2.m.d 16
4.b odd 2 1 600.2.m.d 16
5.b even 2 1 inner 2400.2.m.c 16
5.c odd 4 1 480.2.b.a 8
5.c odd 4 1 2400.2.b.e 8
8.b even 2 1 600.2.m.c 16
8.d odd 2 1 2400.2.m.d 16
12.b even 2 1 600.2.m.c 16
15.d odd 2 1 2400.2.m.d 16
15.e even 4 1 480.2.b.b 8
15.e even 4 1 2400.2.b.f 8
20.d odd 2 1 600.2.m.d 16
20.e even 4 1 120.2.b.b yes 8
20.e even 4 1 600.2.b.e 8
24.f even 2 1 inner 2400.2.m.c 16
24.h odd 2 1 600.2.m.d 16
40.e odd 2 1 2400.2.m.d 16
40.f even 2 1 600.2.m.c 16
40.i odd 4 1 120.2.b.a 8
40.i odd 4 1 600.2.b.f 8
40.k even 4 1 480.2.b.b 8
40.k even 4 1 2400.2.b.f 8
60.h even 2 1 600.2.m.c 16
60.l odd 4 1 120.2.b.a 8
60.l odd 4 1 600.2.b.f 8
120.i odd 2 1 600.2.m.d 16
120.m even 2 1 inner 2400.2.m.c 16
120.q odd 4 1 480.2.b.a 8
120.q odd 4 1 2400.2.b.e 8
120.w even 4 1 120.2.b.b yes 8
120.w even 4 1 600.2.b.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.b.a 8 40.i odd 4 1
120.2.b.a 8 60.l odd 4 1
120.2.b.b yes 8 20.e even 4 1
120.2.b.b yes 8 120.w even 4 1
480.2.b.a 8 5.c odd 4 1
480.2.b.a 8 120.q odd 4 1
480.2.b.b 8 15.e even 4 1
480.2.b.b 8 40.k even 4 1
600.2.b.e 8 20.e even 4 1
600.2.b.e 8 120.w even 4 1
600.2.b.f 8 40.i odd 4 1
600.2.b.f 8 60.l odd 4 1
600.2.m.c 16 8.b even 2 1
600.2.m.c 16 12.b even 2 1
600.2.m.c 16 40.f even 2 1
600.2.m.c 16 60.h even 2 1
600.2.m.d 16 4.b odd 2 1
600.2.m.d 16 20.d odd 2 1
600.2.m.d 16 24.h odd 2 1
600.2.m.d 16 120.i odd 2 1
2400.2.b.e 8 5.c odd 4 1
2400.2.b.e 8 120.q odd 4 1
2400.2.b.f 8 15.e even 4 1
2400.2.b.f 8 40.k even 4 1
2400.2.m.c 16 1.a even 1 1 trivial
2400.2.m.c 16 5.b even 2 1 inner
2400.2.m.c 16 24.f even 2 1 inner
2400.2.m.c 16 120.m even 2 1 inner
2400.2.m.d 16 3.b odd 2 1
2400.2.m.d 16 8.d odd 2 1
2400.2.m.d 16 15.d odd 2 1
2400.2.m.d 16 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{8} - 36T_{7}^{6} + 384T_{7}^{4} - 1136T_{7}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{8} + 48T_{11}^{6} + 672T_{11}^{4} + 2560T_{11}^{2} + 256 \) Copy content Toggle raw display
\( T_{29}^{4} - 64T_{29}^{2} + 112T_{29} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{12} + 16 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 36 T^{6} + 384 T^{4} - 1136 T^{2} + \cdots + 576)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 48 T^{6} + 672 T^{4} + 2560 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 52 T^{6} + 880 T^{4} - 5312 T^{2} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 52 T^{6} + 816 T^{4} - 4032 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} - 36 T^{2} - 72 T - 32)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 92 T^{6} + 2304 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 64 T^{2} + 112 T - 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 140 T^{6} + 4544 T^{4} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 228 T^{6} + 17072 T^{4} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 64 T^{6} + 1344 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 24 T^{6} + 176 T^{4} + 400 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 236 T^{6} + 18048 T^{4} + \cdots + 4875264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 112 T^{6} + 2912 T^{4} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 160 T^{6} + 672 T^{4} + 768 T^{2} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 208 T^{6} + 11392 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 360 T^{6} + 34992 T^{4} + \cdots + 8386816)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 12 T^{3} - 64 T^{2} + 992 T - 2304)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 240 T^{6} + 16224 T^{4} + \cdots + 430336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 108 T^{6} + 2048 T^{4} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 152 T^{6} + 7952 T^{4} + \cdots + 1032256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 384 T^{6} + 45056 T^{4} + \cdots + 1048576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 368 T^{6} + 23136 T^{4} + \cdots + 614656)^{2} \) Copy content Toggle raw display
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