Properties

Label 845.2.t.d
Level $845$
Weight $2$
Character orbit 845.t
Analytic conductor $6.747$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(188,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.188");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.t (of order \(12\), 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.2520509501904273801216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{13} - 14 x^{12} + 8 x^{11} + 8 x^{10} + 26 x^{9} + 179 x^{8} + 104 x^{7} + 40 x^{6} + \cdots + 1 \) 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 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{8} + \cdots + \beta_{5}) q^{2}+ \cdots + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{7}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{8} + \cdots + \beta_{5}) q^{2}+ \cdots + (\beta_{10} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 6 q^{3} + 8 q^{4} + 4 q^{5} - 6 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 6 q^{3} + 8 q^{4} + 4 q^{5} - 6 q^{6} - 6 q^{10} + 6 q^{11} - 4 q^{12} + 2 q^{15} + 8 q^{16} - 16 q^{17} - 40 q^{18} - 14 q^{19} - 2 q^{20} + 24 q^{21} - 10 q^{22} + 14 q^{23} - 2 q^{24} - 24 q^{25} + 24 q^{27} - 8 q^{28} + 14 q^{30} - 4 q^{31} + 24 q^{35} - 44 q^{37} - 4 q^{38} + 44 q^{40} + 16 q^{41} - 24 q^{42} + 6 q^{43} + 20 q^{44} + 22 q^{45} + 2 q^{46} - 32 q^{47} + 14 q^{48} - 24 q^{49} + 44 q^{50} - 48 q^{53} + 20 q^{54} - 10 q^{55} + 24 q^{58} - 22 q^{59} + 20 q^{60} - 20 q^{61} + 30 q^{62} + 96 q^{64} - 72 q^{66} - 4 q^{68} - 4 q^{69} + 136 q^{70} - 10 q^{71} - 16 q^{72} - 30 q^{75} + 6 q^{76} - 48 q^{77} - 26 q^{80} + 20 q^{81} - 20 q^{82} - 96 q^{83} - 16 q^{84} + 32 q^{85} + 92 q^{86} - 16 q^{87} + 10 q^{88} + 28 q^{89} - 28 q^{90} + 100 q^{92} - 40 q^{93} - 2 q^{95} - 60 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{13} - 14 x^{12} + 8 x^{11} + 8 x^{10} + 26 x^{9} + 179 x^{8} + 104 x^{7} + 40 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 19751232 \nu^{15} + 17121420 \nu^{14} - 26059230 \nu^{13} + 83661521 \nu^{12} + \cdots + 346885583 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 21862109 \nu^{15} - 82228832 \nu^{14} + 54829334 \nu^{13} - 103879948 \nu^{12} + \cdots - 1299442096 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43223872 \nu^{15} - 37505470 \nu^{14} + 57560755 \nu^{13} - 179855583 \nu^{12} + \cdots - 1150180157 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 65367994 \nu^{15} + 99782383 \nu^{14} - 17186648 \nu^{13} + 236152604 \nu^{12} + \cdots - 75382536 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 73422053 \nu^{15} + 6270716 \nu^{14} - 68196002 \nu^{13} + 311340841 \nu^{12} + \cdots + 169167894 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 247375 \nu^{15} - 413104 \nu^{14} - 54152 \nu^{13} + 961664 \nu^{12} + 5139712 \nu^{11} + \cdots + 823277 ) / 2353901 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 91701184 \nu^{15} + 14401668 \nu^{14} + 25144234 \nu^{13} + 369822303 \nu^{12} + \cdots - 59997442 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 99782383 \nu^{15} - 17186648 \nu^{14} - 25319372 \nu^{13} - 403585088 \nu^{12} - 1327799731 \nu^{11} + \cdots + 65367994 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 149261939 \nu^{15} + 21361763 \nu^{14} + 44723362 \nu^{13} + 599779177 \nu^{12} + \cdots - 97502912 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 6655443 \nu^{15} - 8803488 \nu^{14} - 2325954 \nu^{13} + 26513692 \nu^{12} + 127187548 \nu^{11} + \cdots - 23873676 ) / 28652657 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 221411843 \nu^{15} + 15075771 \nu^{14} + 90426784 \nu^{13} + 861332824 \nu^{12} + \cdots + 59813086 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 233772946 \nu^{15} + 15658393 \nu^{14} - 76858280 \nu^{13} + 981839919 \nu^{12} + \cdots + 427938242 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8520800 \nu^{15} + 1273720 \nu^{14} + 282688 \nu^{13} - 33030966 \nu^{12} - 124560688 \nu^{11} + \cdots - 34034380 ) / 28652657 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 253678901 \nu^{15} + 79125423 \nu^{14} + 115310926 \nu^{13} - 990536316 \nu^{12} + \cdots - 1768355242 ) / 830927053 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 34034380 \nu^{15} + 8520800 \nu^{14} + 1273720 \nu^{13} - 135854832 \nu^{12} - 509512286 \nu^{11} + \cdots - 24910838 ) / 28652657 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{12} - \beta_{11} + \beta_{9} - 3\beta_{8} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{10} + 3\beta_{6} + 5\beta_{5} + 5\beta_{2} + 3\beta _1 + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 6\beta_{13} + \beta_{12} + 5\beta_{11} + 5\beta_{6} + 5\beta_{5} + 4\beta_{4} - 4\beta_{3} + 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{15} - 12 \beta_{14} + 3 \beta_{13} + 9 \beta_{12} + 11 \beta_{11} + 2 \beta_{9} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{14} - 7\beta_{10} - 20\beta_{6} + 8\beta_{5} - 16\beta_{4} + 23\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 43 \beta_{15} - 29 \beta_{13} + 89 \beta_{12} + 105 \beta_{11} - 43 \beta_{10} - 16 \beta_{9} + \cdots + 60 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 38 \beta_{15} - 66 \beta_{14} + 71 \beta_{13} + 46 \beta_{12} + 104 \beta_{11} + 38 \beta_{6} + \cdots - 38 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 300 \beta_{14} - 175 \beta_{10} - 201 \beta_{6} + 83 \beta_{5} - 300 \beta_{4} + 92 \beta_{3} + \cdots - 284 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 279 \beta_{13} + 279 \beta_{12} + 279 \beta_{11} - 188 \beta_{10} - 233 \beta_{9} + \cdots + 467 \beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 733 \beta_{15} - 357 \beta_{13} + 1667 \beta_{12} + 2135 \beta_{11} - 468 \beta_{9} + 1778 \beta_{8} + \cdots - 733 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -1200\beta_{14} - 1200\beta_{6} - 1200\beta_{5} - 1200\beta_{4} + 966\beta_{3} - 1200\beta_{2} - 889\beta _1 - 2167 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3133 \beta_{15} - 5069 \beta_{13} - 887 \beta_{12} - 3133 \beta_{11} - 3133 \beta_{10} - 2246 \beta_{9} + \cdots - 5956 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 5222\beta_{14} - 5222\beta_{13} + 5222\beta_{12} + 5222\beta_{11} - 4099\beta_{9} + 13577\beta_{8} - 4101\beta_{7} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 13577\beta_{10} - 5375\beta_{6} - 32223\beta_{5} + 10448\beta_{3} - 42671\beta_{2} - 13577\beta _1 - 26848 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(\beta_{8}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
188.1
−0.766482 + 0.205378i
2.03666 0.545721i
0.345207 0.0924980i
−1.61538 + 0.432841i
0.432841 1.61538i
−0.0924980 + 0.345207i
−0.545721 + 2.03666i
0.205378 0.766482i
−0.766482 0.205378i
2.03666 + 0.545721i
0.345207 + 0.0924980i
−1.61538 0.432841i
0.432841 + 1.61538i
−0.0924980 0.345207i
−0.545721 2.03666i
0.205378 + 0.766482i
−1.75831 1.01516i 1.81681 0.486813i 1.06110 + 1.83789i −1.70032 1.45220i −3.68871 0.988387i −0.809223 1.40161i 0.248119i 0.465739 0.268895i 1.51548 + 4.27953i
188.2 0.116595 + 0.0673159i 2.94457 0.788996i −0.990937 1.71635i 1.29021 + 1.82630i 0.396433 + 0.106224i 0.954850 + 1.65385i 0.536087i 5.44992 3.14651i 0.0274926 + 0.299788i
188.3 1.36782 + 0.789712i −0.991530 + 0.265680i 0.247291 + 0.428320i −0.146426 + 2.23127i −1.56605 0.419621i −2.12499 3.68058i 2.37769i −1.68553 + 0.973141i −1.96234 + 2.93634i
188.4 2.00594 + 1.15813i 0.328222 0.0879467i 1.68254 + 2.91425i 1.55654 1.60536i 0.760248 + 0.203708i 1.97936 + 3.42835i 3.16190i −2.49808 + 1.44227i 4.98155 1.41758i
418.1 −2.00594 + 1.15813i −0.0879467 + 0.328222i 1.68254 2.91425i 1.55654 1.60536i −0.203708 0.760248i 1.97936 3.42835i 3.16190i 2.49808 + 1.44227i −1.26311 + 5.02294i
418.2 −1.36782 + 0.789712i 0.265680 0.991530i 0.247291 0.428320i −0.146426 + 2.23127i 0.419621 + 1.56605i −2.12499 + 3.68058i 2.37769i 1.68553 + 0.973141i −1.56178 3.16761i
418.3 −0.116595 + 0.0673159i −0.788996 + 2.94457i −0.990937 + 1.71635i 1.29021 + 1.82630i −0.106224 0.396433i 0.954850 1.65385i 0.536087i −5.44992 3.14651i −0.273370 0.126085i
418.4 1.75831 1.01516i −0.486813 + 1.81681i 1.06110 1.83789i −1.70032 1.45220i 0.988387 + 3.68871i −0.809223 + 1.40161i 0.248119i −0.465739 0.268895i −4.46392 0.827324i
427.1 −1.75831 + 1.01516i 1.81681 + 0.486813i 1.06110 1.83789i −1.70032 + 1.45220i −3.68871 + 0.988387i −0.809223 + 1.40161i 0.248119i 0.465739 + 0.268895i 1.51548 4.27953i
427.2 0.116595 0.0673159i 2.94457 + 0.788996i −0.990937 + 1.71635i 1.29021 1.82630i 0.396433 0.106224i 0.954850 1.65385i 0.536087i 5.44992 + 3.14651i 0.0274926 0.299788i
427.3 1.36782 0.789712i −0.991530 0.265680i 0.247291 0.428320i −0.146426 2.23127i −1.56605 + 0.419621i −2.12499 + 3.68058i 2.37769i −1.68553 0.973141i −1.96234 2.93634i
427.4 2.00594 1.15813i 0.328222 + 0.0879467i 1.68254 2.91425i 1.55654 + 1.60536i 0.760248 0.203708i 1.97936 3.42835i 3.16190i −2.49808 1.44227i 4.98155 + 1.41758i
657.1 −2.00594 1.15813i −0.0879467 0.328222i 1.68254 + 2.91425i 1.55654 + 1.60536i −0.203708 + 0.760248i 1.97936 + 3.42835i 3.16190i 2.49808 1.44227i −1.26311 5.02294i
657.2 −1.36782 0.789712i 0.265680 + 0.991530i 0.247291 + 0.428320i −0.146426 2.23127i 0.419621 1.56605i −2.12499 3.68058i 2.37769i 1.68553 0.973141i −1.56178 + 3.16761i
657.3 −0.116595 0.0673159i −0.788996 2.94457i −0.990937 1.71635i 1.29021 1.82630i −0.106224 + 0.396433i 0.954850 + 1.65385i 0.536087i −5.44992 + 3.14651i −0.273370 + 0.126085i
657.4 1.75831 + 1.01516i −0.486813 1.81681i 1.06110 + 1.83789i −1.70032 + 1.45220i 0.988387 3.68871i −0.809223 1.40161i 0.248119i −0.465739 + 0.268895i −4.46392 + 0.827324i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 188.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
65.f even 4 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.t.d 16
5.c odd 4 1 845.2.o.c 16
13.b even 2 1 845.2.t.c 16
13.c even 3 1 845.2.f.b 8
13.c even 3 1 inner 845.2.t.d 16
13.d odd 4 1 845.2.o.c 16
13.d odd 4 1 845.2.o.d 16
13.e even 6 1 65.2.f.b 8
13.e even 6 1 845.2.t.c 16
13.f odd 12 1 65.2.k.b yes 8
13.f odd 12 1 845.2.k.b 8
13.f odd 12 1 845.2.o.c 16
13.f odd 12 1 845.2.o.d 16
39.h odd 6 1 585.2.n.e 8
39.k even 12 1 585.2.w.e 8
52.i odd 6 1 1040.2.cd.n 8
52.l even 12 1 1040.2.bg.n 8
65.f even 4 1 inner 845.2.t.d 16
65.h odd 4 1 845.2.o.d 16
65.k even 4 1 845.2.t.c 16
65.l even 6 1 325.2.f.b 8
65.o even 12 1 65.2.f.b 8
65.o even 12 1 845.2.t.c 16
65.q odd 12 1 845.2.k.b 8
65.q odd 12 1 845.2.o.c 16
65.r odd 12 1 65.2.k.b yes 8
65.r odd 12 1 325.2.k.b 8
65.r odd 12 1 845.2.o.d 16
65.s odd 12 1 325.2.k.b 8
65.t even 12 1 325.2.f.b 8
65.t even 12 1 845.2.f.b 8
65.t even 12 1 inner 845.2.t.d 16
195.bf even 12 1 585.2.w.e 8
195.bn odd 12 1 585.2.n.e 8
260.be odd 12 1 1040.2.cd.n 8
260.bg even 12 1 1040.2.bg.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.f.b 8 13.e even 6 1
65.2.f.b 8 65.o even 12 1
65.2.k.b yes 8 13.f odd 12 1
65.2.k.b yes 8 65.r odd 12 1
325.2.f.b 8 65.l even 6 1
325.2.f.b 8 65.t even 12 1
325.2.k.b 8 65.r odd 12 1
325.2.k.b 8 65.s odd 12 1
585.2.n.e 8 39.h odd 6 1
585.2.n.e 8 195.bn odd 12 1
585.2.w.e 8 39.k even 12 1
585.2.w.e 8 195.bf even 12 1
845.2.f.b 8 13.c even 3 1
845.2.f.b 8 65.t even 12 1
845.2.k.b 8 13.f odd 12 1
845.2.k.b 8 65.q odd 12 1
845.2.o.c 16 5.c odd 4 1
845.2.o.c 16 13.d odd 4 1
845.2.o.c 16 13.f odd 12 1
845.2.o.c 16 65.q odd 12 1
845.2.o.d 16 13.d odd 4 1
845.2.o.d 16 13.f odd 12 1
845.2.o.d 16 65.h odd 4 1
845.2.o.d 16 65.r odd 12 1
845.2.t.c 16 13.b even 2 1
845.2.t.c 16 13.e even 6 1
845.2.t.c 16 65.k even 4 1
845.2.t.c 16 65.o even 12 1
845.2.t.d 16 1.a even 1 1 trivial
845.2.t.d 16 13.c even 3 1 inner
845.2.t.d 16 65.f even 4 1 inner
845.2.t.d 16 65.t even 12 1 inner
1040.2.bg.n 8 52.l even 12 1
1040.2.bg.n 8 260.bg even 12 1
1040.2.cd.n 8 52.i odd 6 1
1040.2.cd.n 8 260.be odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\):

\( T_{2}^{16} - 12T_{2}^{14} + 98T_{2}^{12} - 440T_{2}^{10} + 1443T_{2}^{8} - 2552T_{2}^{6} + 3090T_{2}^{4} - 56T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} - 6 T_{3}^{15} + 18 T_{3}^{14} - 68 T_{3}^{13} + 196 T_{3}^{12} - 260 T_{3}^{11} + 344 T_{3}^{10} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 20T_{7}^{6} - 8T_{7}^{5} + 348T_{7}^{4} - 80T_{7}^{3} + 1056T_{7}^{2} + 208T_{7} + 2704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 12 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{8} - 2 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 20 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 181063936 \) Copy content Toggle raw display
$19$ \( T^{16} + 14 T^{15} + \cdots + 10000 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1664966416 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 100000000 \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + \cdots + 16900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 22 T^{7} + \cdots + 336400)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 181063936 \) Copy content Toggle raw display
$43$ \( T^{16} - 6 T^{15} + \cdots + 78074896 \) Copy content Toggle raw display
$47$ \( (T^{4} + 8 T^{3} + \cdots - 164)^{4} \) Copy content Toggle raw display
$53$ \( (T^{8} + 24 T^{7} + \cdots + 19600)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 14331920656 \) Copy content Toggle raw display
$61$ \( (T^{8} + 10 T^{7} + \cdots + 13162384)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 479785216 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 1496306311696 \) Copy content Toggle raw display
$73$ \( (T^{8} + 116 T^{6} + \cdots + 547600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 292 T^{6} + \cdots + 13719616)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots - 7372)^{4} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 3224179360000 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 188227863187456 \) Copy content Toggle raw display
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