Properties

Label 900.2.o.a
Level $900$
Weight $2$
Character orbit 900.o
Analytic conductor $7.187$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(299,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.299");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.o (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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.7465802011608416256.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 36)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{15} - \beta_{14} - \beta_{5}) q^{3} + ( - \beta_{12} + 2 \beta_{11} + \beta_{8} + \beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} - \beta_{7}) q^{6} + ( - \beta_{14} - \beta_{13} - \beta_{9} - \beta_{5}) q^{7} + ( - \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{9} + 2 \beta_{6} + \beta_{2}) q^{8} + ( - 2 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{15} - \beta_{14} - \beta_{5}) q^{3} + ( - \beta_{12} + 2 \beta_{11} + \beta_{8} + \beta_{7} - \beta_{4} + 2 \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} - \beta_{7}) q^{6} + ( - \beta_{14} - \beta_{13} - \beta_{9} - \beta_{5}) q^{7} + ( - \beta_{15} - \beta_{14} + \beta_{13} + 2 \beta_{9} + 2 \beta_{6} + \beta_{2}) q^{8} + ( - 2 \beta_{12} + \beta_{11} - \beta_{10} + \beta_{4} + \beta_{3}) q^{9} + (\beta_{12} - \beta_{11} - \beta_{8} - 2 \beta_{7} + \beta_{4} - 2 \beta_{3}) q^{11} + (\beta_{15} - \beta_{14} - \beta_{13} + 2 \beta_{6} + \beta_{2} - 2 \beta_1) q^{12} + (\beta_{14} - \beta_{13} - \beta_{6} + 2 \beta_{5} - 2 \beta_1) q^{13} + (\beta_{12} + \beta_{8} - \beta_{7} - 1) q^{14} + ( - 3 \beta_{12} + \beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{3} - 2) q^{16} + (\beta_{14} - \beta_{13} + \beta_{5} - \beta_1) q^{17} + (\beta_{15} - 2 \beta_{13} + \beta_{9} + \beta_{6} + 2 \beta_{2} - \beta_1) q^{18} + (\beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{4} - \beta_{3} - 1) q^{19} + ( - 2 \beta_{12} + \beta_{11} + 2 \beta_{10} + \beta_{4} + \beta_{3}) q^{21} + (2 \beta_{15} + \beta_{14} - 2 \beta_{13} - 2 \beta_{9} - 2 \beta_1) q^{22} + ( - \beta_{15} + \beta_{13} - \beta_{9} - \beta_{5} + \beta_1) q^{23} + ( - \beta_{12} - \beta_{11} - 4 \beta_{10} + \beta_{8} - \beta_{3} - 4) q^{24} + (2 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - 2 \beta_{8} - \beta_{7} + \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{26}+ \cdots + (4 \beta_{12} - \beta_{11} + 2 \beta_{10} + \beta_{8} - 4 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{4} - 6 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{4} - 6 q^{6} + 12 q^{9} - 24 q^{14} - 2 q^{16} - 12 q^{21} - 6 q^{24} - 12 q^{29} - 14 q^{34} - 66 q^{36} + 48 q^{41} + 24 q^{46} + 20 q^{49} - 78 q^{54} + 36 q^{56} - 4 q^{61} - 52 q^{64} - 48 q^{66} + 60 q^{69} + 60 q^{74} - 6 q^{76} - 60 q^{81} - 60 q^{84} + 42 q^{86} + 36 q^{94} + 24 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{14} + x^{12} + 8x^{10} - 20x^{8} + 32x^{6} + 16x^{4} - 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 7\nu^{13} + 9\nu^{11} - 16\nu^{9} + 44\nu^{7} + 144\nu^{5} - 80\nu^{3} + 640\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 7\nu^{12} + 15\nu^{10} - 8\nu^{8} - 20\nu^{6} + 240\nu^{4} - 16\nu^{2} - 256 ) / 576 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 7\nu^{12} + 9\nu^{10} - 16\nu^{8} + 44\nu^{6} + 144\nu^{4} - 80\nu^{2} + 640 ) / 576 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{13} + \nu^{11} - \nu^{9} + 8\nu^{7} + 4\nu^{5} - 16\nu^{3} + 48\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{15} - 17\nu^{13} + 9\nu^{11} - 16\nu^{9} - 76\nu^{7} + 144\nu^{5} - 80\nu^{3} - 320\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{14} - \nu^{12} + 9\nu^{10} - 26\nu^{8} + 52\nu^{6} - 304\nu^{2} + 512 ) / 288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{14} + \nu^{12} - 9\nu^{10} + 26\nu^{8} - 52\nu^{6} + 16\nu^{2} - 224 ) / 288 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{15} + \nu^{13} - 9\nu^{11} - 20\nu^{7} - 80\nu^{3} ) / 384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{14} - \nu^{12} + \nu^{10} - 8\nu^{8} - 4\nu^{6} + 16\nu^{4} - 48\nu^{2} ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} + \nu^{12} - 3\nu^{10} + 8\nu^{8} + 14\nu^{6} - 12\nu^{4} + 88\nu^{2} - 32 ) / 144 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{14} + 2\nu^{12} - 6\nu^{10} + \nu^{8} + 28\nu^{6} - 24\nu^{4} + 56\nu^{2} - 64 ) / 144 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -\nu^{15} + \nu^{13} - \nu^{11} + 8\nu^{9} + 4\nu^{7} - 16\nu^{5} + 48\nu^{3} - 192\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{15} - \nu^{13} + \nu^{11} + 8\nu^{9} - 20\nu^{7} + 32\nu^{5} + 16\nu^{3} - 64\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -3\nu^{15} + 7\nu^{13} - 23\nu^{11} + 12\nu^{9} + 28\nu^{7} - 128\nu^{5} + 240\nu^{3} - 192\nu ) / 384 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - \beta_{7} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - 2\beta_{9} + 2\beta_{6} - \beta_{5} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{10} + \beta_{8} + \beta_{4} + 2\beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{9} + 3\beta_{6} + \beta_{5} + 6\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + 4\beta_{11} - \beta_{10} + 3\beta_{7} - \beta_{4} + 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - 3\beta_{6} + 8\beta_{5} + 3\beta_{2} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{12} + 8\beta_{11} - 14\beta_{10} + 7\beta_{8} + 7\beta_{7} - 2\beta_{4} + 4\beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3\beta_{15} + 8\beta_{14} + 10\beta_{13} + 6\beta_{9} - 14\beta_{6} + 11\beta_{5} - 7\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -21\beta_{12} + 4\beta_{11} - 29\beta_{10} - 13\beta_{8} + 7\beta_{4} - 6\beta_{3} - 29 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -17\beta_{15} - 9\beta_{14} + 19\beta_{13} - 17\beta_{9} - 3\beta_{6} - 17\beta_{5} - 6\beta_{2} + 17\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -5\beta_{12} - 20\beta_{11} - 19\beta_{10} - 15\beta_{7} + 53\beta_{4} - 34\beta_{3} - 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -10\beta_{15} - 5\beta_{14} + 5\beta_{13} + 5\beta_{9} - 33\beta_{6} - 40\beta_{5} + 33\beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -28\beta_{12} + 56\beta_{11} + 70\beta_{10} + 5\beta_{8} + 5\beta_{7} + 10\beta_{4} - 20\beta_{3} - 25 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23\beta_{15} + 56\beta_{14} - 146\beta_{13} - 46\beta_{9} - 106\beta_{6} + 33\beta_{5} - 53\beta_{2} - 123\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-\beta_{10}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
299.1
0.977642 + 1.02187i
−0.0786378 + 1.41203i
1.37379 + 0.335728i
−1.18353 + 0.774115i
1.18353 0.774115i
−1.37379 0.335728i
0.0786378 1.41203i
−0.977642 1.02187i
0.977642 1.02187i
−0.0786378 1.41203i
1.37379 0.335728i
−1.18353 0.774115i
1.18353 + 0.774115i
−1.37379 + 0.335728i
0.0786378 + 1.41203i
−0.977642 + 1.02187i
−1.37379 0.335728i −1.07561 1.35760i 1.77457 + 0.922437i 0 1.02187 + 2.22616i 0.637910 1.10489i −2.12819 1.86301i −0.686141 + 2.92048i 0
299.2 −1.18353 + 0.774115i 1.61030 0.637910i 0.801492 1.83238i 0 −1.41203 + 2.00155i 1.35760 2.35143i 0.469882 + 2.78912i 2.18614 2.05446i 0
299.3 −0.977642 1.02187i −1.07561 1.35760i −0.0884324 + 1.99804i 0 −0.335728 + 2.42637i 0.637910 1.10489i 2.12819 1.86301i −0.686141 + 2.92048i 0
299.4 −0.0786378 + 1.41203i −1.61030 + 0.637910i −1.98763 0.222077i 0 −0.774115 2.32395i −1.35760 + 2.35143i 0.469882 2.78912i 2.18614 2.05446i 0
299.5 0.0786378 1.41203i 1.61030 0.637910i −1.98763 0.222077i 0 −0.774115 2.32395i 1.35760 2.35143i −0.469882 + 2.78912i 2.18614 2.05446i 0
299.6 0.977642 + 1.02187i 1.07561 + 1.35760i −0.0884324 + 1.99804i 0 −0.335728 + 2.42637i −0.637910 + 1.10489i −2.12819 + 1.86301i −0.686141 + 2.92048i 0
299.7 1.18353 0.774115i −1.61030 + 0.637910i 0.801492 1.83238i 0 −1.41203 + 2.00155i −1.35760 + 2.35143i −0.469882 2.78912i 2.18614 2.05446i 0
299.8 1.37379 + 0.335728i 1.07561 + 1.35760i 1.77457 + 0.922437i 0 1.02187 + 2.22616i −0.637910 + 1.10489i 2.12819 + 1.86301i −0.686141 + 2.92048i 0
599.1 −1.37379 + 0.335728i −1.07561 + 1.35760i 1.77457 0.922437i 0 1.02187 2.22616i 0.637910 + 1.10489i −2.12819 + 1.86301i −0.686141 2.92048i 0
599.2 −1.18353 0.774115i 1.61030 + 0.637910i 0.801492 + 1.83238i 0 −1.41203 2.00155i 1.35760 + 2.35143i 0.469882 2.78912i 2.18614 + 2.05446i 0
599.3 −0.977642 + 1.02187i −1.07561 + 1.35760i −0.0884324 1.99804i 0 −0.335728 2.42637i 0.637910 + 1.10489i 2.12819 + 1.86301i −0.686141 2.92048i 0
599.4 −0.0786378 1.41203i −1.61030 0.637910i −1.98763 + 0.222077i 0 −0.774115 + 2.32395i −1.35760 2.35143i 0.469882 + 2.78912i 2.18614 + 2.05446i 0
599.5 0.0786378 + 1.41203i 1.61030 + 0.637910i −1.98763 + 0.222077i 0 −0.774115 + 2.32395i 1.35760 + 2.35143i −0.469882 2.78912i 2.18614 + 2.05446i 0
599.6 0.977642 1.02187i 1.07561 1.35760i −0.0884324 1.99804i 0 −0.335728 2.42637i −0.637910 1.10489i −2.12819 1.86301i −0.686141 2.92048i 0
599.7 1.18353 + 0.774115i −1.61030 0.637910i 0.801492 + 1.83238i 0 −1.41203 2.00155i −1.35760 2.35143i −0.469882 + 2.78912i 2.18614 + 2.05446i 0
599.8 1.37379 0.335728i 1.07561 1.35760i 1.77457 0.922437i 0 1.02187 2.22616i −0.637910 1.10489i 2.12819 1.86301i −0.686141 2.92048i 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
4.b odd 2 1 inner
5.b even 2 1 inner
9.d odd 6 1 inner
20.d odd 2 1 inner
36.h even 6 1 inner
45.h odd 6 1 inner
180.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.o.a 16
4.b odd 2 1 inner 900.2.o.a 16
5.b even 2 1 inner 900.2.o.a 16
5.c odd 4 1 36.2.h.a 8
5.c odd 4 1 900.2.r.c 8
9.d odd 6 1 inner 900.2.o.a 16
15.e even 4 1 108.2.h.a 8
20.d odd 2 1 inner 900.2.o.a 16
20.e even 4 1 36.2.h.a 8
20.e even 4 1 900.2.r.c 8
36.h even 6 1 inner 900.2.o.a 16
40.i odd 4 1 576.2.s.f 8
40.k even 4 1 576.2.s.f 8
45.h odd 6 1 inner 900.2.o.a 16
45.k odd 12 1 108.2.h.a 8
45.k odd 12 1 324.2.b.b 8
45.l even 12 1 36.2.h.a 8
45.l even 12 1 324.2.b.b 8
45.l even 12 1 900.2.r.c 8
60.l odd 4 1 108.2.h.a 8
120.q odd 4 1 1728.2.s.f 8
120.w even 4 1 1728.2.s.f 8
180.n even 6 1 inner 900.2.o.a 16
180.v odd 12 1 36.2.h.a 8
180.v odd 12 1 324.2.b.b 8
180.v odd 12 1 900.2.r.c 8
180.x even 12 1 108.2.h.a 8
180.x even 12 1 324.2.b.b 8
360.bo even 12 1 1728.2.s.f 8
360.bo even 12 1 5184.2.c.j 8
360.br even 12 1 576.2.s.f 8
360.br even 12 1 5184.2.c.j 8
360.bt odd 12 1 576.2.s.f 8
360.bt odd 12 1 5184.2.c.j 8
360.bu odd 12 1 1728.2.s.f 8
360.bu odd 12 1 5184.2.c.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a 8 5.c odd 4 1
36.2.h.a 8 20.e even 4 1
36.2.h.a 8 45.l even 12 1
36.2.h.a 8 180.v odd 12 1
108.2.h.a 8 15.e even 4 1
108.2.h.a 8 45.k odd 12 1
108.2.h.a 8 60.l odd 4 1
108.2.h.a 8 180.x even 12 1
324.2.b.b 8 45.k odd 12 1
324.2.b.b 8 45.l even 12 1
324.2.b.b 8 180.v odd 12 1
324.2.b.b 8 180.x even 12 1
576.2.s.f 8 40.i odd 4 1
576.2.s.f 8 40.k even 4 1
576.2.s.f 8 360.br even 12 1
576.2.s.f 8 360.bt odd 12 1
900.2.o.a 16 1.a even 1 1 trivial
900.2.o.a 16 4.b odd 2 1 inner
900.2.o.a 16 5.b even 2 1 inner
900.2.o.a 16 9.d odd 6 1 inner
900.2.o.a 16 20.d odd 2 1 inner
900.2.o.a 16 36.h even 6 1 inner
900.2.o.a 16 45.h odd 6 1 inner
900.2.o.a 16 180.n even 6 1 inner
900.2.r.c 8 5.c odd 4 1
900.2.r.c 8 20.e even 4 1
900.2.r.c 8 45.l even 12 1
900.2.r.c 8 180.v odd 12 1
1728.2.s.f 8 120.q odd 4 1
1728.2.s.f 8 120.w even 4 1
1728.2.s.f 8 360.bo even 12 1
1728.2.s.f 8 360.bu odd 12 1
5184.2.c.j 8 360.bo even 12 1
5184.2.c.j 8 360.br even 12 1
5184.2.c.j 8 360.bt odd 12 1
5184.2.c.j 8 360.bu odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\):

\( T_{7}^{8} + 9T_{7}^{6} + 69T_{7}^{4} + 108T_{7}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{8} - 17T_{13}^{6} + 225T_{13}^{4} - 1088T_{13}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{14} + T^{12} + 8 T^{10} - 20 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} - 3 T^{6} + 12 T^{4} - 27 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 9 T^{6} + 69 T^{4} + 108 T^{2} + \cdots + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 12 T^{6} + 141 T^{4} + 36 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 17 T^{6} + 225 T^{4} - 1088 T^{2} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 7 T^{2} + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 108)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 15 T^{6} + 177 T^{4} - 720 T^{2} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 3 T^{3} + T^{2} - 6 T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 69 T^{6} + 4569 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 68 T^{2} + 1024)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 49 T^{2} - 12 T + 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 108 T^{6} + 8781 T^{4} + \cdots + 8311689)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 135 T^{6} + 18033 T^{4} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 76 T^{2} + 256)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 180 T^{6} + 28293 T^{4} + \cdots + 16867449)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 108 T^{6} + 8781 T^{4} + \cdots + 8311689)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 432)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 17 T^{2} + 64)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 201 T^{6} + 30309 T^{4} + \cdots + 101848464)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 111 T^{6} + 9249 T^{4} + \cdots + 9437184)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 172 T^{2} + 4096)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} - 266 T^{6} + 53595 T^{4} + \cdots + 294499921)^{2} \) Copy content Toggle raw display
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