Properties

Label 976.2.bd.b
Level $976$
Weight $2$
Character orbit 976.bd
Analytic conductor $7.793$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [976,2,Mod(113,976)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(976, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("976.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 976 = 2^{4} \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 976.bd (of order \(10\), degree \(4\), 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: \(7.79339923728\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 17x^{14} + 111x^{12} + 361x^{10} + 624x^{8} + 558x^{6} + 229x^{4} + 34x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 61)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{15} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 1) q^{3}+ \cdots + (3 \beta_{12} + \beta_{10} + 2 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{15} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + \beta_{8} + \cdots + 1) q^{3}+ \cdots + ( - 5 \beta_{15} - 3 \beta_{14} + 9 \beta_{13} + 8 \beta_{12} - 11 \beta_{11} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{3} - 10 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{3} - 10 q^{7} + q^{9} - 12 q^{13} + 13 q^{15} - 3 q^{19} + 15 q^{23} - 2 q^{25} + 4 q^{27} + 15 q^{31} + 25 q^{33} - 10 q^{35} - 5 q^{37} + 3 q^{39} + 12 q^{41} + 25 q^{43} + 36 q^{45} - 6 q^{47} - 30 q^{49} - 50 q^{51} - 20 q^{53} - 20 q^{55} - 11 q^{57} - 5 q^{59} - 53 q^{61} + 5 q^{63} + 20 q^{65} + 55 q^{67} - 15 q^{69} + 50 q^{71} - 11 q^{73} + 88 q^{75} + 63 q^{77} - 40 q^{79} - 19 q^{81} - 31 q^{83} + 55 q^{85} - 25 q^{87} + 60 q^{89} + 15 q^{91} - 48 q^{95} + 45 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 17x^{14} + 111x^{12} + 361x^{10} + 624x^{8} + 558x^{6} + 229x^{4} + 34x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15 \nu^{15} - 9 \nu^{14} + 226 \nu^{13} - 162 \nu^{12} + 1231 \nu^{11} - 1117 \nu^{10} + 3082 \nu^{9} - 3750 \nu^{8} + 3686 \nu^{7} - 6374 \nu^{6} + 2112 \nu^{5} - 5104 \nu^{4} + 839 \nu^{3} + \cdots - 123 ) / 88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9 \nu^{15} - 3 \nu^{14} + 162 \nu^{13} - 54 \nu^{12} + 1117 \nu^{11} - 387 \nu^{10} + 3750 \nu^{9} - 1426 \nu^{8} + 6374 \nu^{7} - 2770 \nu^{6} + 5104 \nu^{5} - 2508 \nu^{4} + 1533 \nu^{3} - 731 \nu^{2} + \cdots - 41 ) / 88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9 \nu^{15} - 29 \nu^{14} + 162 \nu^{13} - 434 \nu^{12} + 1117 \nu^{11} - 2333 \nu^{10} + 3750 \nu^{9} - 5674 \nu^{8} + 6374 \nu^{7} - 6258 \nu^{6} + 5104 \nu^{5} - 2596 \nu^{4} + 1533 \nu^{3} + \cdots - 15 ) / 88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 9 \nu^{15} - 3 \nu^{14} - 162 \nu^{13} - 54 \nu^{12} - 1117 \nu^{11} - 387 \nu^{10} - 3750 \nu^{9} - 1426 \nu^{8} - 6374 \nu^{7} - 2770 \nu^{6} - 5104 \nu^{5} - 2508 \nu^{4} - 1533 \nu^{3} + \cdots - 41 ) / 88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9 \nu^{15} + 29 \nu^{14} + 162 \nu^{13} + 434 \nu^{12} + 1117 \nu^{11} + 2333 \nu^{10} + 3750 \nu^{9} + 5674 \nu^{8} + 6374 \nu^{7} + 6258 \nu^{6} + 5104 \nu^{5} + 2596 \nu^{4} + 1533 \nu^{3} + \cdots + 15 ) / 88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41 \nu^{15} - 9 \nu^{14} - 650 \nu^{13} - 118 \nu^{12} - 3837 \nu^{11} - 501 \nu^{10} - 10850 \nu^{9} - 758 \nu^{8} - 15402 \nu^{7} - 82 \nu^{6} - 10164 \nu^{5} + 484 \nu^{4} - 2217 \nu^{3} + \cdots - 123 ) / 88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18 \nu^{15} + 21 \nu^{14} - 302 \nu^{13} + 334 \nu^{12} - 1926 \nu^{11} + 1983 \nu^{10} - 6004 \nu^{9} + 5670 \nu^{8} - 9602 \nu^{7} + 8258 \nu^{6} - 7392 \nu^{5} + 5896 \nu^{4} - 2186 \nu^{3} + \cdots + 133 ) / 44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 25 \nu^{15} + 5 \nu^{14} - 406 \nu^{13} + 68 \nu^{12} - 2477 \nu^{11} + 293 \nu^{10} - 7300 \nu^{9} + 338 \nu^{8} - 10888 \nu^{7} - 634 \nu^{6} - 7634 \nu^{5} - 1606 \nu^{4} - 1897 \nu^{3} + \cdots - 49 ) / 44 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{15} - 13 \nu^{14} + 406 \nu^{13} - 212 \nu^{12} + 2477 \nu^{11} - 1303 \nu^{10} + 7300 \nu^{9} - 3884 \nu^{8} + 10888 \nu^{7} - 5858 \nu^{6} + 7634 \nu^{5} - 4048 \nu^{4} + 1875 \nu^{3} + \cdots - 31 ) / 44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 41 \nu^{15} + 9 \nu^{14} - 694 \nu^{13} + 162 \nu^{12} - 4497 \nu^{11} + 1117 \nu^{10} - 14414 \nu^{9} + 3750 \nu^{8} - 24158 \nu^{7} + 6374 \nu^{6} - 20108 \nu^{5} + 5104 \nu^{4} + \cdots + 79 ) / 88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 25 \nu^{15} + 5 \nu^{14} + 406 \nu^{13} + 68 \nu^{12} + 2477 \nu^{11} + 293 \nu^{10} + 7300 \nu^{9} + 338 \nu^{8} + 10888 \nu^{7} - 634 \nu^{6} + 7634 \nu^{5} - 1606 \nu^{4} + 1897 \nu^{3} - 835 \nu^{2} + \cdots - 49 ) / 44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43 \nu^{15} - 15 \nu^{14} + 730 \nu^{13} - 226 \nu^{12} + 4755 \nu^{11} - 1231 \nu^{10} + 15394 \nu^{9} - 3082 \nu^{8} + 26342 \nu^{7} - 3686 \nu^{6} + 22924 \nu^{5} - 2112 \nu^{4} + 8659 \nu^{3} + \cdots - 117 ) / 88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 41 \nu^{15} + 9 \nu^{14} + 694 \nu^{13} + 162 \nu^{12} + 4497 \nu^{11} + 1117 \nu^{10} + 14414 \nu^{9} + 3750 \nu^{8} + 24158 \nu^{7} + 6374 \nu^{6} + 20108 \nu^{5} + 5104 \nu^{4} + 6881 \nu^{3} + \cdots + 79 ) / 88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 43 \nu^{15} - 15 \nu^{14} - 730 \nu^{13} - 226 \nu^{12} - 4755 \nu^{11} - 1231 \nu^{10} - 15394 \nu^{9} - 3082 \nu^{8} - 26342 \nu^{7} - 3686 \nu^{6} - 22924 \nu^{5} - 2112 \nu^{4} + \cdots - 117 ) / 88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 18 \nu^{15} - 21 \nu^{14} - 302 \nu^{13} - 334 \nu^{12} - 1926 \nu^{11} - 1983 \nu^{10} - 6004 \nu^{9} - 5670 \nu^{8} - 9602 \nu^{7} - 8258 \nu^{6} - 7392 \nu^{5} - 5896 \nu^{4} - 2186 \nu^{3} + \cdots - 133 ) / 44 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{14} - \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} - 5 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} + 5 \beta_{9} - \beta_{7} + 5 \beta_{6} + \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 7 \beta_{14} + 9 \beta_{13} + 5 \beta_{12} - 6 \beta_{11} + 5 \beta_{10} + 6 \beta_{9} + 6 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 26 \beta_{5} + 18 \beta_{4} + 20 \beta_{3} - 18 \beta_{2} + 2 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} + 17 \beta_{14} - 10 \beta_{13} + 17 \beta_{12} + \beta_{11} - 10 \beta_{10} - 27 \beta_{9} + \beta_{8} + 10 \beta_{7} - 27 \beta_{6} - 12 \beta_{5} + 26 \beta_{4} + 19 \beta_{3} + 46 \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22 \beta_{15} - 45 \beta_{14} - 63 \beta_{13} - 27 \beta_{12} + 34 \beta_{11} - 25 \beta_{10} - 36 \beta_{9} - 34 \beta_{8} - 22 \beta_{7} + 36 \beta_{6} - 149 \beta_{5} - 93 \beta_{4} - 113 \beta_{3} + 93 \beta_{2} - 16 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 76 \beta_{15} - 119 \beta_{14} + 76 \beta_{13} - 119 \beta_{12} - 13 \beta_{11} + 76 \beta_{10} + 158 \beta_{9} - 13 \beta_{8} - 76 \beta_{7} + 158 \beta_{6} + 99 \beta_{5} - 144 \beta_{4} - 105 \beta_{3} - 296 \beta_{2} + 70 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 174 \beta_{15} + 286 \beta_{14} + 414 \beta_{13} + 162 \beta_{12} - 201 \beta_{11} + 140 \beta_{10} + 224 \beta_{9} + 201 \beta_{8} + 174 \beta_{7} - 224 \beta_{6} + 906 \beta_{5} + 530 \beta_{4} + 682 \beta_{3} - 530 \beta_{2} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 527 \beta_{15} + 790 \beta_{14} - 528 \beta_{13} + 790 \beta_{12} + 112 \beta_{11} - 528 \beta_{10} - 968 \beta_{9} + 112 \beta_{8} + 527 \beta_{7} - 968 \beta_{6} - 715 \beta_{5} + 841 \beta_{4} + 629 \beta_{3} + 1895 \beta_{2} + \cdots - 419 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1227 \beta_{15} - 1819 \beta_{14} - 2670 \beta_{13} - 1019 \beta_{12} + 1231 \beta_{11} - 846 \beta_{10} - 1419 \beta_{9} - 1231 \beta_{8} - 1227 \beta_{7} + 1419 \beta_{6} - 5669 \beta_{5} - 3195 \beta_{4} + \cdots - 339 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3513 \beta_{15} - 5144 \beta_{14} + 3529 \beta_{13} - 5144 \beta_{12} - 827 \beta_{11} + 3529 \beta_{10} + 6069 \beta_{9} - 827 \beta_{8} - 3513 \beta_{7} + 6069 \beta_{6} + 4867 \beta_{5} - 5099 \beta_{4} + \cdots + 2619 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8237 \beta_{15} + 11597 \beta_{14} + 17125 \beta_{13} + 6511 \beta_{12} - 7700 \beta_{11} + 5299 \beta_{10} + 9054 \beta_{9} + 7700 \beta_{8} + 8237 \beta_{7} - 9054 \beta_{6} + 35936 \beta_{5} + 19840 \beta_{4} + \cdots + 2158 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 22979 \beta_{15} + 33222 \beta_{14} - 23138 \beta_{13} + 33222 \beta_{12} + 5694 \beta_{11} - 23138 \beta_{10} - 38479 \beta_{9} + 5694 \beta_{8} + 22979 \beta_{7} - 38479 \beta_{6} - 32185 \beta_{5} + \cdots - 16639 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54035 \beta_{15} - 74084 \beta_{14} - 109689 \beta_{13} - 41772 \beta_{12} + 48722 \beta_{11} - 33677 \beta_{10} - 57928 \beta_{9} - 48722 \beta_{8} - 54035 \beta_{7} + 57928 \beta_{6} + \cdots - 13755 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/976\mathbb{Z}\right)^\times\).

\(n\) \(245\) \(367\) \(673\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
1.85647i
0.475317i
1.25523i
1.46081i
0.776536i
1.60228i
2.53165i
0.196205i
0.776536i
1.60228i
2.53165i
0.196205i
1.85647i
0.475317i
1.25523i
1.46081i
0 −2.38911 1.73579i 0 −1.04459 3.21492i 0 0.380995 + 0.524395i 0 1.76783 + 5.44082i 0
113.2 0 −0.261794 0.190204i 0 0.00765597 + 0.0235626i 0 2.52527 + 3.47573i 0 −0.894693 2.75358i 0
113.3 0 0.446713 + 0.324556i 0 0.620635 + 1.91012i 0 −1.40505 1.93389i 0 −0.832835 2.56320i 0
113.4 0 1.89518 + 1.37693i 0 −0.701732 2.15971i 0 −2.88318 3.96835i 0 0.768714 + 2.36586i 0
369.1 0 −0.554995 + 1.70810i 0 −0.515969 0.374873i 0 −1.31473 + 0.427183i 0 −0.182527 0.132614i 0
369.2 0 0.221623 0.682087i 0 −1.72728 1.25494i 0 1.05248 0.341973i 0 2.01093 + 1.46102i 0
369.3 0 0.277331 0.853536i 0 0.429180 + 0.311818i 0 −3.57843 + 1.16270i 0 1.77544 + 1.28993i 0
369.4 0 0.865057 2.66237i 0 2.93210 + 2.13030i 0 0.222647 0.0723423i 0 −3.91286 2.84286i 0
529.1 0 −0.554995 1.70810i 0 −0.515969 + 0.374873i 0 −1.31473 0.427183i 0 −0.182527 + 0.132614i 0
529.2 0 0.221623 + 0.682087i 0 −1.72728 + 1.25494i 0 1.05248 + 0.341973i 0 2.01093 1.46102i 0
529.3 0 0.277331 + 0.853536i 0 0.429180 0.311818i 0 −3.57843 1.16270i 0 1.77544 1.28993i 0
529.4 0 0.865057 + 2.66237i 0 2.93210 2.13030i 0 0.222647 + 0.0723423i 0 −3.91286 + 2.84286i 0
881.1 0 −2.38911 + 1.73579i 0 −1.04459 + 3.21492i 0 0.380995 0.524395i 0 1.76783 5.44082i 0
881.2 0 −0.261794 + 0.190204i 0 0.00765597 0.0235626i 0 2.52527 3.47573i 0 −0.894693 + 2.75358i 0
881.3 0 0.446713 0.324556i 0 0.620635 1.91012i 0 −1.40505 + 1.93389i 0 −0.832835 + 2.56320i 0
881.4 0 1.89518 1.37693i 0 −0.701732 + 2.15971i 0 −2.88318 + 3.96835i 0 0.768714 2.36586i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 976.2.bd.b 16
4.b odd 2 1 61.2.g.a 16
12.b even 2 1 549.2.y.b 16
61.g even 10 1 inner 976.2.bd.b 16
244.m odd 10 1 61.2.g.a 16
244.r even 20 2 3721.2.a.k 16
732.y even 10 1 549.2.y.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.g.a 16 4.b odd 2 1
61.2.g.a 16 244.m odd 10 1
549.2.y.b 16 12.b even 2 1
549.2.y.b 16 732.y even 10 1
976.2.bd.b 16 1.a even 1 1 trivial
976.2.bd.b 16 61.g even 10 1 inner
3721.2.a.k 16 244.r even 20 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - T_{3}^{15} + 6 T_{3}^{14} + T_{3}^{13} + 43 T_{3}^{12} - 131 T_{3}^{11} + 504 T_{3}^{10} - 652 T_{3}^{9} + 1948 T_{3}^{8} - 2091 T_{3}^{7} + 2466 T_{3}^{6} - 1352 T_{3}^{5} + 633 T_{3}^{4} - 52 T_{3}^{3} - 16 T_{3}^{2} + 8 T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(976, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + 6 T^{14} + T^{13} + 43 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} + 11 T^{14} - 15 T^{13} + 112 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{16} + 10 T^{15} + 51 T^{14} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{16} + 87 T^{14} + 2756 T^{12} + \cdots + 1936 \) Copy content Toggle raw display
$13$ \( (T^{8} + 6 T^{7} - 38 T^{6} - 242 T^{5} + \cdots + 10261)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 25 T^{14} - 210 T^{13} + \cdots + 4879681 \) Copy content Toggle raw display
$19$ \( T^{16} + 3 T^{15} + 36 T^{14} + \cdots + 7311616 \) Copy content Toggle raw display
$23$ \( T^{16} - 15 T^{15} + 115 T^{14} + \cdots + 30976 \) Copy content Toggle raw display
$29$ \( T^{16} + 185 T^{14} + 11273 T^{12} + \cdots + 383161 \) Copy content Toggle raw display
$31$ \( T^{16} - 15 T^{15} + 105 T^{14} + \cdots + 1008016 \) Copy content Toggle raw display
$37$ \( T^{16} + 5 T^{15} + \cdots + 46787420416 \) Copy content Toggle raw display
$41$ \( T^{16} - 12 T^{15} + 159 T^{14} + \cdots + 1437601 \) Copy content Toggle raw display
$43$ \( T^{16} - 25 T^{15} + \cdots + 591267856 \) Copy content Toggle raw display
$47$ \( (T^{8} + 3 T^{7} - 124 T^{6} - 716 T^{5} + \cdots - 3524)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 20 T^{15} + \cdots + 162537001 \) Copy content Toggle raw display
$59$ \( T^{16} + 5 T^{15} + \cdots + 138529862416 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 191707312997281 \) Copy content Toggle raw display
$67$ \( T^{16} - 55 T^{15} + \cdots + 395837272336 \) Copy content Toggle raw display
$71$ \( T^{16} - 50 T^{15} + \cdots + 121101216016 \) Copy content Toggle raw display
$73$ \( T^{16} + 11 T^{15} + \cdots + 687101735056 \) Copy content Toggle raw display
$79$ \( T^{16} + 40 T^{15} + \cdots + 2515192996096 \) Copy content Toggle raw display
$83$ \( T^{16} + 31 T^{15} + \cdots + 5678526736 \) Copy content Toggle raw display
$89$ \( T^{16} - 60 T^{15} + \cdots + 10\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{16} - 45 T^{15} + \cdots + 394856641 \) Copy content Toggle raw display
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