Properties

Label 275.2.h.e
Level $275$
Weight $2$
Character orbit 275.h
Analytic conductor $2.196$
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] = [275,2,Mod(26,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.26");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.h (of order \(5\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} - 256 x^{7} + 472 x^{6} - 48 x^{5} + 432 x^{4} + 88 x^{3} + 128 x^{2} + 64 x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{9} - \beta_{5} - \beta_{2}) q^{2} + \beta_{14} q^{3} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{15} - \beta_{12} - \beta_{10}) q^{6} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + (\beta_{15} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} - 1) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} - \beta_{5} - \beta_{2}) q^{2} + \beta_{14} q^{3} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} - \beta_1 - 1) q^{4} + ( - \beta_{15} - \beta_{12} - \beta_{10}) q^{6} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - 1) q^{7} + (\beta_{15} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} - 1) q^{8} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{9} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 1) q^{11}+ \cdots + (\beta_{15} + \beta_{14} + 2 \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - \beta_{10} - 3 \beta_{9} + \cdots - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{3} - 2 q^{4} - 3 q^{6} - 4 q^{7} - 16 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{3} - 2 q^{4} - 3 q^{6} - 4 q^{7} - 16 q^{8} + 8 q^{9} - 5 q^{11} + 6 q^{12} - 7 q^{13} + 3 q^{14} - 4 q^{16} - 12 q^{17} - 16 q^{18} - 13 q^{19} + 10 q^{21} - 28 q^{22} + 4 q^{23} - 43 q^{24} - 34 q^{26} + 11 q^{27} - 47 q^{28} - 11 q^{29} - 10 q^{31} + 58 q^{32} + 34 q^{33} + 20 q^{34} + 3 q^{36} + 4 q^{37} + 36 q^{38} + 3 q^{39} + 25 q^{41} - 13 q^{42} + 28 q^{43} - q^{44} + 40 q^{46} - 8 q^{47} + 106 q^{48} + 16 q^{49} + 35 q^{51} - 39 q^{52} + 22 q^{53} + 60 q^{54} - 20 q^{56} - 29 q^{57} - 6 q^{58} + 14 q^{59} + 16 q^{61} + 10 q^{62} - 73 q^{63} + 40 q^{64} - 55 q^{66} + 14 q^{67} - 83 q^{68} + 35 q^{69} - 46 q^{71} - 28 q^{72} - 7 q^{73} - 7 q^{74} - 62 q^{76} + 51 q^{77} - 34 q^{78} - 39 q^{79} - 43 q^{81} + 51 q^{82} - 28 q^{83} - 54 q^{84} + 2 q^{86} + 50 q^{87} + 76 q^{88} - 22 q^{89} - 34 q^{91} - 4 q^{92} - 3 q^{93} - 40 q^{94} + 108 q^{96} + 39 q^{97} + 52 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 10 x^{14} - 13 x^{13} + 53 x^{12} - 12 x^{11} + 136 x^{10} + 8 x^{9} + 300 x^{8} - 256 x^{7} + 472 x^{6} - 48 x^{5} + 432 x^{4} + 88 x^{3} + 128 x^{2} + 64 x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26449018002 \nu^{15} - 236888516211 \nu^{14} + 788099178184 \nu^{13} - 1526842653922 \nu^{12} + 2602718491889 \nu^{11} + \cdots + 6985303901728 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71383474437 \nu^{15} - 289509004196 \nu^{14} + 724686737314 \nu^{13} - 880677312057 \nu^{12} + 3525945524039 \nu^{11} + \cdots - 26659476408112 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 104750151572 \nu^{15} + 673159972921 \nu^{14} - 2076349151074 \nu^{13} + 3882890610742 \nu^{12} - 8663636557279 \nu^{11} + \cdots + 2568824593792 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 64382245392 \nu^{15} - 136843233291 \nu^{14} + 153040707434 \nu^{13} + 333330036813 \nu^{12} + 2058624775804 \nu^{11} + \cdots + 3970336916928 ) / 12763543546500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 160551537112 \nu^{15} + 537455996876 \nu^{14} - 932355398199 \nu^{13} + 10820831382 \nu^{12} - 4626340856194 \nu^{11} + \cdots - 10162625679608 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 335740577504 \nu^{15} - 1576476952372 \nu^{14} + 4304252214543 \nu^{13} - 6747058073494 \nu^{12} + 21010481144828 \nu^{11} + \cdots + 2834825497856 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 169847619014 \nu^{15} - 638541728247 \nu^{14} + 1347321043353 \nu^{13} - 1028723196929 \nu^{12} + 6551150466593 \nu^{11} + \cdots - 8155768916224 ) / 12763543546500 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 436581493858 \nu^{15} + 1772774993434 \nu^{14} - 4602703454791 \nu^{13} + 6463658598338 \nu^{12} - 24665661828396 \nu^{11} + \cdots - 32620021539072 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 336455799554 \nu^{15} - 1405015474407 \nu^{14} + 3422183399388 \nu^{13} - 4226220696494 \nu^{12} + 16839370278338 \nu^{11} + \cdots - 6722455741504 ) / 12763543546500 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 784893186521 \nu^{15} + 3081039362473 \nu^{14} - 7454798472622 \nu^{13} + 8976897755181 \nu^{12} - 39074125561792 \nu^{11} + \cdots - 17909140090624 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 825910307819 \nu^{15} + 3905169355947 \nu^{14} - 10779625022808 \nu^{13} + 16856514063109 \nu^{12} + \cdots - 50705453705536 ) / 25527087093000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 420153483844 \nu^{15} - 1344158135822 \nu^{14} + 2796519364033 \nu^{13} - 2039811890584 \nu^{12} + 18041913947238 \nu^{11} + \cdots + 30173885523636 ) / 12763543546500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 509735557264 \nu^{15} - 1869094610042 \nu^{14} + 4458813844393 \nu^{13} - 5279241201079 \nu^{12} + 25987261338063 \nu^{11} + \cdots + 14317509681756 ) / 12763543546500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1298485402504 \nu^{15} - 6312051157617 \nu^{14} + 17598840879833 \nu^{13} - 28484833417244 \nu^{12} + 84191619087023 \nu^{11} + \cdots + 5900772630536 ) / 25527087093000 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{5} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} - \beta_{13} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + \beta_{6} + 8 \beta_{5} - 8 \beta_{4} + 3 \beta_{2} - 5 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} + \beta_{14} - 3 \beta_{13} + 2 \beta_{12} - 9 \beta_{11} + 3 \beta_{10} + 11 \beta_{9} - 9 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + 23 \beta_{5} - 13 \beta_{4} - 3 \beta_{3} + 23 \beta_{2} - 13 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{14} - 12 \beta_{13} + 9 \beta_{12} - 14 \beta_{11} + 14 \beta_{9} + 14 \beta_{6} + 42 \beta_{5} - 25 \beta_{3} + 79 \beta_{2} - 42 \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{15} + 25 \beta_{14} - 25 \beta_{13} + 14 \beta_{12} - 45 \beta_{10} + 88 \beta_{8} + 45 \beta_{7} + 49 \beta_{6} + 149 \beta_{4} - 88 \beta_{3} + 149 \beta_{2} - 101 \beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88 \beta_{15} + 45 \beta_{14} + 163 \beta_{11} - 275 \beta_{10} - 165 \beta_{9} + 438 \beta_{8} + 279 \beta_{7} - 489 \beta_{5} + 831 \beta_{4} - 163 \beta_{3} - 165 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 275 \beta_{15} + 275 \beta_{13} - 163 \beta_{12} + 919 \beta_{11} - 919 \beta_{10} - 949 \beta_{9} + 1453 \beta_{8} + 949 \beta_{7} - 409 \beta_{6} - 2705 \beta_{5} + 2705 \beta_{4} - 1648 \beta_{2} + 1057 \beta _1 - 409 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 534 \beta_{15} - 534 \beta_{14} + 1453 \beta_{13} - 919 \beta_{12} + 2980 \beta_{11} - 1811 \beta_{10} - 3038 \beta_{9} + 2980 \beta_{8} + 1843 \beta_{7} - 1843 \beta_{6} - 8926 \beta_{5} + 5427 \beta_{4} + \cdots + 5427 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2980 \beta_{14} + 4791 \beta_{13} - 2980 \beta_{12} + 5961 \beta_{11} - 6063 \beta_{9} - 6063 \beta_{6} - 18007 \beta_{5} + 9845 \beta_{3} - 29297 \beta_{2} + 18007 \beta _1 + 4018 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5961 \beta_{15} - 9845 \beta_{14} + 9845 \beta_{13} - 5961 \beta_{12} + 19818 \beta_{10} - 32277 \beta_{8} - 20208 \beta_{7} - 12735 \beta_{6} - 59328 \beta_{4} + 32277 \beta_{3} - 59328 \beta_{2} + \cdots + 20208 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 32277 \beta_{15} - 19818 \beta_{14} - 65289 \beta_{11} + 106372 \beta_{10} + 66561 \beta_{9} - 171661 \beta_{8} - 108682 \beta_{7} + 195881 \beta_{5} - 317602 \beta_{4} + 65289 \beta_{3} + \cdots + 66561 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 106372 \beta_{15} - 106372 \beta_{13} + 65289 \beta_{12} - 349879 \beta_{11} + 349879 \beta_{10} + 357171 \beta_{9} - 565578 \beta_{8} - 357171 \beta_{7} + 137070 \beta_{6} + \cdots + 137070 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 215699 \beta_{15} + 215699 \beta_{14} - 565578 \beta_{13} + 349879 \beta_{12} - 1152511 \beta_{11} + 710622 \beta_{10} + 1176821 \beta_{9} - 1152511 \beta_{8} - 725088 \beta_{7} + \cdots - 2127272 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1152511 \beta_{14} - 1863133 \beta_{13} + 1152511 \beta_{12} - 2342971 \beta_{11} + 2391411 \beta_{9} + 2391411 \beta_{6} + 7007643 \beta_{5} - 3794540 \beta_{3} + 11345926 \beta_{2} + \cdots - 1482267 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-1 - \beta_{6} + \beta_{7} - \beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−1.26407 + 0.918397i
−0.260198 + 0.189045i
0.977523 0.710212i
2.66477 1.93607i
−0.605307 + 1.86294i
−0.265939 + 0.818476i
0.207669 0.639141i
0.545543 1.67901i
−1.26407 0.918397i
−0.260198 0.189045i
0.977523 + 0.710212i
2.66477 + 1.93607i
−0.605307 1.86294i
−0.265939 0.818476i
0.207669 + 0.639141i
0.545543 + 1.67901i
−2.07308 1.50618i 0.553942 1.70486i 1.41105 + 4.34277i 0 −3.71620 + 2.69998i 1.17454 + 3.61485i 2.03208 6.25410i −0.172643 0.125433i 0
26.2 −1.06921 0.776830i −0.626199 + 1.92724i −0.0782786 0.240917i 0 2.16668 1.57419i 0.107618 + 0.331213i −0.920262 + 2.83228i −0.895086 0.650318i 0
26.3 0.168506 + 0.122427i 0.585361 1.80155i −0.604628 1.86085i 0 0.319195 0.231909i −0.398265 1.22573i 0.254662 0.783769i −0.475901 0.345762i 0
26.4 1.85576 + 1.34829i −0.0131041 + 0.0403304i 1.00792 + 3.10207i 0 −0.0786950 + 0.0571753i 0.352180 + 1.08390i −0.894346 + 2.75251i 2.42560 + 1.76230i 0
126.1 −0.296290 0.911888i −0.727583 0.528620i 0.874283 0.635204i 0 −0.266466 + 0.820099i 0.616184 0.447684i −2.38967 1.73620i −0.677113 2.08394i 0
126.2 0.0430779 + 0.132580i 2.41558 + 1.75502i 1.60231 1.16415i 0 −0.128623 + 0.395861i −2.88786 + 2.09815i 0.448926 + 0.326164i 1.82787 + 5.62561i 0
126.3 0.516686 + 1.59020i −2.51599 1.82798i −0.643729 + 0.467696i 0 1.60686 4.94542i −2.49985 + 1.81624i 1.62907 + 1.18359i 2.06168 + 6.34519i 0
126.4 0.854560 + 2.63006i 1.32800 + 0.964848i −4.56893 + 3.31953i 0 −1.40276 + 4.31724i 1.53545 1.11557i −8.16046 5.92892i −0.0944009 0.290536i 0
201.1 −2.07308 + 1.50618i 0.553942 + 1.70486i 1.41105 4.34277i 0 −3.71620 2.69998i 1.17454 3.61485i 2.03208 + 6.25410i −0.172643 + 0.125433i 0
201.2 −1.06921 + 0.776830i −0.626199 1.92724i −0.0782786 + 0.240917i 0 2.16668 + 1.57419i 0.107618 0.331213i −0.920262 2.83228i −0.895086 + 0.650318i 0
201.3 0.168506 0.122427i 0.585361 + 1.80155i −0.604628 + 1.86085i 0 0.319195 + 0.231909i −0.398265 + 1.22573i 0.254662 + 0.783769i −0.475901 + 0.345762i 0
201.4 1.85576 1.34829i −0.0131041 0.0403304i 1.00792 3.10207i 0 −0.0786950 0.0571753i 0.352180 1.08390i −0.894346 2.75251i 2.42560 1.76230i 0
251.1 −0.296290 + 0.911888i −0.727583 + 0.528620i 0.874283 + 0.635204i 0 −0.266466 0.820099i 0.616184 + 0.447684i −2.38967 + 1.73620i −0.677113 + 2.08394i 0
251.2 0.0430779 0.132580i 2.41558 1.75502i 1.60231 + 1.16415i 0 −0.128623 0.395861i −2.88786 2.09815i 0.448926 0.326164i 1.82787 5.62561i 0
251.3 0.516686 1.59020i −2.51599 + 1.82798i −0.643729 0.467696i 0 1.60686 + 4.94542i −2.49985 1.81624i 1.62907 1.18359i 2.06168 6.34519i 0
251.4 0.854560 2.63006i 1.32800 0.964848i −4.56893 3.31953i 0 −1.40276 4.31724i 1.53545 + 1.11557i −8.16046 + 5.92892i −0.0944009 + 0.290536i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.h.e yes 16
5.b even 2 1 275.2.h.c 16
5.c odd 4 2 275.2.z.c 32
11.c even 5 1 inner 275.2.h.e yes 16
11.c even 5 1 3025.2.a.bn 8
11.d odd 10 1 3025.2.a.bj 8
55.h odd 10 1 3025.2.a.bm 8
55.j even 10 1 275.2.h.c 16
55.j even 10 1 3025.2.a.bi 8
55.k odd 20 2 275.2.z.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.h.c 16 5.b even 2 1
275.2.h.c 16 55.j even 10 1
275.2.h.e yes 16 1.a even 1 1 trivial
275.2.h.e yes 16 11.c even 5 1 inner
275.2.z.c 32 5.c odd 4 2
275.2.z.c 32 55.k odd 20 2
3025.2.a.bi 8 55.j even 10 1
3025.2.a.bj 8 11.d odd 10 1
3025.2.a.bm 8 55.h odd 10 1
3025.2.a.bn 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 5 T_{2}^{14} + 12 T_{2}^{13} + 32 T_{2}^{12} - 30 T_{2}^{11} + 259 T_{2}^{10} + 193 T_{2}^{9} + 584 T_{2}^{8} + 833 T_{2}^{7} + 1629 T_{2}^{6} + 659 T_{2}^{5} + 759 T_{2}^{4} - 391 T_{2}^{3} + 96 T_{2}^{2} - 11 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 5 T^{14} + 12 T^{13} + 32 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} - 2 T^{15} + 4 T^{14} - 3 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{15} + 14 T^{14} + 50 T^{13} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( T^{16} + 5 T^{15} + 17 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 7 T^{15} + 28 T^{14} + \cdots + 12075625 \) Copy content Toggle raw display
$17$ \( T^{16} + 12 T^{15} + 75 T^{14} + \cdots + 192721 \) Copy content Toggle raw display
$19$ \( T^{16} + 13 T^{15} + \cdots + 109830400 \) Copy content Toggle raw display
$23$ \( (T^{8} - 2 T^{7} - 62 T^{6} + 252 T^{5} + \cdots - 121)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 11 T^{15} + \cdots + 1259895025 \) Copy content Toggle raw display
$31$ \( T^{16} + 10 T^{15} + \cdots + 1126877761 \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + 96 T^{14} + \cdots + 19811401 \) Copy content Toggle raw display
$41$ \( T^{16} - 25 T^{15} + \cdots + 195873515776 \) Copy content Toggle raw display
$43$ \( (T^{8} - 14 T^{7} - 28 T^{6} + 732 T^{5} + \cdots - 101)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 8 T^{15} + \cdots + 211111761961 \) Copy content Toggle raw display
$53$ \( T^{16} - 22 T^{15} + \cdots + 2390818816 \) Copy content Toggle raw display
$59$ \( T^{16} - 14 T^{15} + \cdots + 6178746025 \) Copy content Toggle raw display
$61$ \( T^{16} - 16 T^{15} + \cdots + 11020667028121 \) Copy content Toggle raw display
$67$ \( (T^{8} - 7 T^{7} - 323 T^{6} + \cdots + 23994961)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 46 T^{15} + \cdots + 923250017881 \) Copy content Toggle raw display
$73$ \( T^{16} + 7 T^{15} + \cdots + 217828491841 \) Copy content Toggle raw display
$79$ \( T^{16} + 39 T^{15} + \cdots + 275304843025 \) Copy content Toggle raw display
$83$ \( T^{16} + 28 T^{15} + 467 T^{14} + \cdots + 69372241 \) Copy content Toggle raw display
$89$ \( (T^{8} + 11 T^{7} - 258 T^{6} + \cdots - 278125)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 39 T^{15} + \cdots + 7330099456 \) Copy content Toggle raw display
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