Properties

Label 60.2.e.a
Level $60$
Weight $2$
Character orbit 60.e
Analytic conductor $0.479$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 60.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.479102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.342102016.5
Defining polynomial: \( x^{8} + x^{6} + 4x^{4} + 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{5} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{7} - \beta_{4}) q^{3} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{4} + \beta_1 q^{5} + (\beta_{6} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{6} + (\beta_{5} - \beta_{3}) q^{7} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + (\beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} + \beta_1) q^{9} + \beta_{7} q^{10} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{11} + (\beta_{6} + \beta_{4} + \beta_{2} + 2 \beta_1) q^{12} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{13} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{14} + (\beta_{3} - \beta_{2}) q^{15} + ( - 3 \beta_{7} - \beta_{6} - \beta_{4} - 2) q^{16} - 2 \beta_1 q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{18} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3}) q^{19} + ( - \beta_{6} + \beta_{4} + \beta_{2}) q^{20} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - 3 \beta_1 + 1) q^{21} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4}) q^{22} + (\beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} - 4 \beta_{2}) q^{23} + (\beta_{7} + \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_1) q^{24} - q^{25} + (2 \beta_{6} - 2 \beta_{4} + 4 \beta_1) q^{26} + (\beta_{7} - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{27} + (2 \beta_{7} + 4) q^{28} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{29} + (\beta_{7} + \beta_{6} - \beta_{5} - \beta_1 + 1) q^{30} + (2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3}) q^{31} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1) q^{32} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{33} - 2 \beta_{7} q^{34} + (\beta_{6} - \beta_{4}) q^{35} + ( - 3 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{36} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{37} + ( - 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2}) q^{38} + ( - 2 \beta_{7} - 3 \beta_{6} + 3 \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2}) q^{39} + ( - \beta_{7} - \beta_{6} - \beta_{4} + 2) q^{40} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{41} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1 - 2) q^{42} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3}) q^{43} + (2 \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - 4 \beta_1) q^{44} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{45} + (4 \beta_{7} + 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 2) q^{46} + ( - \beta_{6} + \beta_{4}) q^{47} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{48} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 1) q^{49} + \beta_{2} q^{50} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{51} + (4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4) q^{52} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1) q^{53} + ( - 2 \beta_{7} - 2 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{54} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3}) q^{55} + ( - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_{2}) q^{56} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{57} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} - 4) q^{58} + (3 \beta_{6} - 3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 6 \beta_{2}) q^{59} + ( - \beta_{7} - \beta_{5} - \beta_{3} - 2) q^{60} + (2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + ( - 2 \beta_{6} + 2 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{62} + (2 \beta_{7} + 3 \beta_{6} - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{63} + ( - \beta_{7} + 3 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} - 2 \beta_{3} - 2) q^{64} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{65} + (4 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 4) q^{66} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3}) q^{67} + (2 \beta_{6} - 2 \beta_{4} - 2 \beta_{2}) q^{68} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 1) q^{69} + (\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{70} + (2 \beta_{5} + 2 \beta_{3} - 4 \beta_{2}) q^{71} + (4 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 2 \beta_1 + 4) q^{72} + (4 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2) q^{73} + ( - 2 \beta_{6} + 2 \beta_{4} - 4 \beta_1) q^{74} + (\beta_{7} + \beta_{4}) q^{75} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} - 4) q^{76} + 4 \beta_1 q^{77} + (2 \beta_{7} - 2 \beta_{6} + 6 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 6) q^{78} + ( - 6 \beta_{7} - 3 \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_{3}) q^{79} + (\beta_{5} + \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{80} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 8 \beta_1 - 1) q^{81} + ( - 4 \beta_{7} + 2 \beta_{5} - 2 \beta_{3} + 4) q^{82} + ( - 5 \beta_{6} + 5 \beta_{4}) q^{83} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} - 2 \beta_1 + 2) q^{84} + 2 q^{85} + (3 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{86} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{3} - 4 \beta_{2}) q^{87} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{4} + 4) q^{88} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 4 \beta_1) q^{89} + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 + 2) q^{90} + ( - 4 \beta_{7} - 2 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3}) q^{91} + ( - 4 \beta_{5} - 4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{92} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 4) q^{93} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 2) q^{94} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2}) q^{95} + (\beta_{7} - 3 \beta_{6} + \beta_{4} + 6 \beta_{2} + 6) q^{96} - 6 q^{97} + (2 \beta_{6} - 2 \beta_{4} - 3 \beta_{2} + 4 \beta_1) q^{98} + (4 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + \beta_{4} + 3 \beta_{3} - 6 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 6 q^{6} - 4 q^{9} - 2 q^{10} + 4 q^{12} - 8 q^{13} - 14 q^{16} + 16 q^{18} + 4 q^{21} - 4 q^{22} - 2 q^{24} - 8 q^{25} + 28 q^{28} + 8 q^{30} - 16 q^{33} + 4 q^{34} + 18 q^{36} + 8 q^{37} + 14 q^{40} - 12 q^{42} - 4 q^{45} + 20 q^{46} - 36 q^{48} + 16 q^{49} - 32 q^{52} - 10 q^{54} + 24 q^{57} - 36 q^{58} - 14 q^{60} - 8 q^{61} - 2 q^{64} - 40 q^{66} + 12 q^{69} - 12 q^{70} + 24 q^{72} - 36 q^{76} + 40 q^{78} + 40 q^{82} + 16 q^{84} + 16 q^{85} + 44 q^{88} + 18 q^{90} + 32 q^{93} + 12 q^{94} + 42 q^{96} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{3} + 4\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - \nu^{6} + 3\nu^{5} - 3\nu^{4} + 2\nu^{3} - 10\nu^{2} + 16\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 3\nu^{5} + 3\nu^{4} - 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} + 10\nu^{2} + 16\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + \nu^{6} + 3\nu^{5} + 3\nu^{4} + 2\nu^{3} - 6\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + \nu^{4} + 2\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + 4\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} + 3\beta_{4} - \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{6} + \beta_{5} - 5\beta_{4} + \beta_{3} - 4\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12\beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} - 3\beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3\beta_{6} - \beta_{5} - 3\beta_{4} - \beta_{3} + 4\beta_{2} - 20\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−1.17915 + 0.780776i
−1.17915 0.780776i
−0.599676 + 1.28078i
−0.599676 1.28078i
0.599676 + 1.28078i
0.599676 1.28078i
1.17915 + 0.780776i
1.17915 0.780776i
−1.17915 0.780776i 1.51022 + 0.848071i 0.780776 + 1.84130i 1.00000i −1.11862 2.17915i 3.02045i 0.516994 2.78078i 1.56155 + 2.56155i 0.780776 1.17915i
11.2 −1.17915 + 0.780776i 1.51022 0.848071i 0.780776 1.84130i 1.00000i −1.11862 + 2.17915i 3.02045i 0.516994 + 2.78078i 1.56155 2.56155i 0.780776 + 1.17915i
11.3 −0.599676 1.28078i −0.468213 1.66757i −1.28078 + 1.53610i 1.00000i −1.85500 + 1.59968i 0.936426i 2.73546 + 0.719224i −2.56155 + 1.56155i −1.28078 + 0.599676i
11.4 −0.599676 + 1.28078i −0.468213 + 1.66757i −1.28078 1.53610i 1.00000i −1.85500 1.59968i 0.936426i 2.73546 0.719224i −2.56155 1.56155i −1.28078 0.599676i
11.5 0.599676 1.28078i 0.468213 + 1.66757i −1.28078 1.53610i 1.00000i 2.41656 + 0.400324i 0.936426i −2.73546 + 0.719224i −2.56155 + 1.56155i −1.28078 0.599676i
11.6 0.599676 + 1.28078i 0.468213 1.66757i −1.28078 + 1.53610i 1.00000i 2.41656 0.400324i 0.936426i −2.73546 0.719224i −2.56155 1.56155i −1.28078 + 0.599676i
11.7 1.17915 0.780776i −1.51022 0.848071i 0.780776 1.84130i 1.00000i −2.44293 + 0.179147i 3.02045i −0.516994 2.78078i 1.56155 + 2.56155i 0.780776 + 1.17915i
11.8 1.17915 + 0.780776i −1.51022 + 0.848071i 0.780776 + 1.84130i 1.00000i −2.44293 0.179147i 3.02045i −0.516994 + 2.78078i 1.56155 2.56155i 0.780776 1.17915i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.2.e.a 8
3.b odd 2 1 inner 60.2.e.a 8
4.b odd 2 1 inner 60.2.e.a 8
5.b even 2 1 300.2.e.c 8
5.c odd 4 1 300.2.h.a 8
5.c odd 4 1 300.2.h.b 8
8.b even 2 1 960.2.h.g 8
8.d odd 2 1 960.2.h.g 8
12.b even 2 1 inner 60.2.e.a 8
15.d odd 2 1 300.2.e.c 8
15.e even 4 1 300.2.h.a 8
15.e even 4 1 300.2.h.b 8
20.d odd 2 1 300.2.e.c 8
20.e even 4 1 300.2.h.a 8
20.e even 4 1 300.2.h.b 8
24.f even 2 1 960.2.h.g 8
24.h odd 2 1 960.2.h.g 8
60.h even 2 1 300.2.e.c 8
60.l odd 4 1 300.2.h.a 8
60.l odd 4 1 300.2.h.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.e.a 8 1.a even 1 1 trivial
60.2.e.a 8 3.b odd 2 1 inner
60.2.e.a 8 4.b odd 2 1 inner
60.2.e.a 8 12.b even 2 1 inner
300.2.e.c 8 5.b even 2 1
300.2.e.c 8 15.d odd 2 1
300.2.e.c 8 20.d odd 2 1
300.2.e.c 8 60.h even 2 1
300.2.h.a 8 5.c odd 4 1
300.2.h.a 8 15.e even 4 1
300.2.h.a 8 20.e even 4 1
300.2.h.a 8 60.l odd 4 1
300.2.h.b 8 5.c odd 4 1
300.2.h.b 8 15.e even 4 1
300.2.h.b 8 20.e even 4 1
300.2.h.b 8 60.l odd 4 1
960.2.h.g 8 8.b even 2 1
960.2.h.g 8 8.d odd 2 1
960.2.h.g 8 24.f even 2 1
960.2.h.g 8 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(60, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{6} + 4 T^{4} + 4 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{6} + 2 T^{4} + 18 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{2} + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 58 T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 36 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 28 T^{2} + 128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T - 16)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 52 T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 62 T^{2} + 128)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 10 T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 168 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 252 T^{2} + 10368)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T - 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 46 T^{2} + 512)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 56 T^{2} + 512)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 68)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 148 T^{2} + 5408)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 250 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 144 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T + 6)^{8} \) Copy content Toggle raw display
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