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Properties

Label 342.2.p.a
Level $342$
Weight $2$
Character orbit 342.p
Analytic conductor $2.731$
Analytic rank $0$
Dimension $20$
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] = [342,2,Mod(113,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.p (of order \(6\), degree \(2\), 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + x^{18} + 3 x^{17} - 15 x^{16} + 33 x^{15} - 42 x^{14} + 72 x^{12} - 243 x^{11} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + x^{18} + 3 x^{17} - 15 x^{16} + 33 x^{15} - 42 x^{14} + 72 x^{12} - 243 x^{11} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 968 \nu^{19} + 8021 \nu^{18} - 24050 \nu^{17} + 10350 \nu^{16} + 1614 \nu^{15} + \cdots - 183504609 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 208 \nu^{19} - 625 \nu^{18} + 1093 \nu^{17} - 1746 \nu^{16} - 1365 \nu^{15} + 10284 \nu^{14} + \cdots + 2972133 ) / 1594323 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2099 \nu^{19} + 5633 \nu^{18} - 14912 \nu^{17} + 19935 \nu^{16} - 3129 \nu^{15} + \cdots - 115066818 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2426 \nu^{19} - 9479 \nu^{18} + 12386 \nu^{17} - 12537 \nu^{16} - 16923 \nu^{15} + \cdots + 83062260 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 836 \nu^{19} - 2984 \nu^{18} + 5594 \nu^{17} - 2709 \nu^{16} - 6762 \nu^{15} + 55353 \nu^{14} + \cdots + 45211851 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2672 \nu^{19} - 5711 \nu^{18} + 3704 \nu^{17} - 16803 \nu^{16} - 768 \nu^{15} + 146802 \nu^{14} + \cdots + 68595255 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1178 \nu^{19} + 4271 \nu^{18} - 8744 \nu^{17} + 11538 \nu^{16} - 15 \nu^{15} - 56028 \nu^{14} + \cdots - 41314617 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3847 \nu^{19} + 1600 \nu^{18} - 4606 \nu^{17} + 19701 \nu^{16} + 7431 \nu^{15} + \cdots - 71764218 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4388 \nu^{19} - 10685 \nu^{18} + 21287 \nu^{17} - 31572 \nu^{16} - 6015 \nu^{15} + \cdots + 134966331 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4658 \nu^{19} + 12737 \nu^{18} - 20666 \nu^{17} + 24039 \nu^{16} + 34851 \nu^{15} + \cdots - 95639697 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1603 \nu^{19} + 4180 \nu^{18} - 8698 \nu^{17} + 10818 \nu^{16} - 1281 \nu^{15} + \cdots - 54876204 ) / 4782969 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 5062 \nu^{19} + 10834 \nu^{18} - 24775 \nu^{17} + 28449 \nu^{16} + 45339 \nu^{15} + \cdots - 196121412 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 5399 \nu^{19} + 20750 \nu^{18} - 39641 \nu^{17} + 49851 \nu^{16} - 7170 \nu^{15} + \cdots - 205175592 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 5846 \nu^{19} + 9956 \nu^{18} - 20558 \nu^{17} + 25011 \nu^{16} + 21324 \nu^{15} + \cdots - 120617424 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 6575 \nu^{19} - 12872 \nu^{18} + 23474 \nu^{17} - 25011 \nu^{16} - 38820 \nu^{15} + \cdots + 91919610 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 7717 \nu^{19} - 21535 \nu^{18} + 49651 \nu^{17} - 50967 \nu^{16} - 20526 \nu^{15} + \cdots + 292489380 ) / 14348907 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 1031 \nu^{19} + 2522 \nu^{18} - 5024 \nu^{17} + 5895 \nu^{16} + 5718 \nu^{15} - 34923 \nu^{14} + \cdots - 23186574 ) / 1594323 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} + 2\beta_{5} + \beta_{4} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + 2 \beta_{17} + \beta_{16} + \beta_{15} + \beta_{12} + 2 \beta_{11} - \beta_{10} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{19} - 2 \beta_{18} + 2 \beta_{17} + 2 \beta_{16} - \beta_{15} - 3 \beta_{14} - 3 \beta_{13} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{19} + 5 \beta_{18} + \beta_{17} + \beta_{16} + \beta_{15} - 3 \beta_{14} + 4 \beta_{12} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{19} + 10 \beta_{18} - 7 \beta_{17} - 10 \beta_{16} + 8 \beta_{15} + 3 \beta_{13} + 5 \beta_{12} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7 \beta_{19} - \beta_{18} + 13 \beta_{17} - 5 \beta_{16} - 5 \beta_{15} + 15 \beta_{13} + \beta_{12} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14 \beta_{19} + 7 \beta_{18} + 5 \beta_{17} + 17 \beta_{16} - 28 \beta_{15} - 9 \beta_{14} + 3 \beta_{13} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26 \beta_{19} + 23 \beta_{18} - 17 \beta_{17} - 2 \beta_{16} + 25 \beta_{15} + 27 \beta_{14} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 56 \beta_{19} - 8 \beta_{18} + 29 \beta_{17} + 32 \beta_{16} + 32 \beta_{15} + 36 \beta_{14} + \cdots - 109 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 22 \beta_{19} - 7 \beta_{18} + 166 \beta_{17} + 154 \beta_{16} - 8 \beta_{15} + 45 \beta_{14} + \cdots + 259 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 226 \beta_{19} + 67 \beta_{18} - 280 \beta_{17} - 16 \beta_{16} + 38 \beta_{15} + 9 \beta_{14} + \cdots + 59 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 263 \beta_{19} + 242 \beta_{18} - 290 \beta_{17} - 356 \beta_{16} + 292 \beta_{15} + 288 \beta_{14} + \cdots + 307 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 527 \beta_{19} + 484 \beta_{18} + 95 \beta_{17} - 172 \beta_{16} + 314 \beta_{15} + 711 \beta_{14} + \cdots - 7 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 296 \beta_{19} - 517 \beta_{18} + 352 \beta_{17} - 533 \beta_{16} + 34 \beta_{15} + 855 \beta_{14} + \cdots + 661 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 380 \beta_{19} - 1196 \beta_{18} - 376 \beta_{17} + 473 \beta_{16} - 13 \beta_{15} - 153 \beta_{14} + \cdots + 1322 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 274 \beta_{19} + 362 \beta_{18} + 1192 \beta_{17} - 269 \beta_{16} + 5158 \beta_{15} + 585 \beta_{14} + \cdots - 5375 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1601 \beta_{19} + 643 \beta_{18} + 1493 \beta_{17} - 3292 \beta_{16} + 4160 \beta_{15} - 450 \beta_{14} + \cdots - 10993 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(1 + \beta_{5}\) \(-1\)

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
1.06721 + 1.36421i
0.0940817 + 1.72949i
−0.424393 + 1.67925i
1.70846 − 0.284919i
1.57590 − 0.718716i
1.01548 − 1.40314i
−1.73094 + 0.0619519i
−1.72527 − 0.153167i
−0.0754025 − 1.73041i
−1.00512 − 1.41058i
1.06721 − 1.36421i
0.0940817 − 1.72949i
−0.424393 − 1.67925i
1.70846 + 0.284919i
1.57590 + 0.718716i
1.01548 + 1.40314i
−1.73094 − 0.0619519i
−1.72527 + 0.153167i
−0.0754025 + 1.73041i
−1.00512 + 1.41058i
−0.500000 − 0.866025i −1.71504 + 0.242123i −0.500000 + 0.866025i 0.446642 + 0.257869i 1.06721 + 1.36421i −0.534840 − 0.926369i 1.00000 2.88275 − 0.830504i − 0.515738i
113.2 −0.500000 − 0.866025i −1.54483 − 0.783270i −0.500000 + 0.866025i −2.13944 − 1.23520i 0.0940817 + 1.72949i 2.17969 + 3.77534i 1.00000 1.77298 + 2.42003i 2.47041i
113.3 −0.500000 − 0.866025i −1.24208 − 1.20716i −0.500000 + 0.866025i 2.76481 + 1.59626i −0.424393 + 1.67925i −0.965307 − 1.67196i 1.00000 0.0855221 + 2.99878i − 3.19253i
113.4 −0.500000 − 0.866025i −0.607481 + 1.62203i −0.500000 + 0.866025i 3.41314 + 1.97058i 1.70846 − 0.284919i 1.07232 + 1.85732i 1.00000 −2.26193 − 1.97070i − 3.94116i
113.5 −0.500000 − 0.866025i −0.165521 + 1.72412i −0.500000 + 0.866025i −0.379645 − 0.219188i 1.57590 − 0.718716i −2.44148 − 4.22876i 1.00000 −2.94521 − 0.570758i 0.438376i
113.6 −0.500000 − 0.866025i 0.707416 + 1.58100i −0.500000 + 0.866025i −2.77331 − 1.60117i 1.01548 − 1.40314i 0.760344 + 1.31695i 1.00000 −1.99912 + 2.23685i 3.20235i
113.7 −0.500000 − 0.866025i 0.811819 − 1.53002i −0.500000 + 0.866025i 1.18812 + 0.685961i −1.73094 + 0.0619519i 2.26558 + 3.92410i 1.00000 −1.68190 − 2.48419i − 1.37192i
113.8 −0.500000 − 0.866025i 0.995279 − 1.41754i −0.500000 + 0.866025i −1.43477 − 0.828367i −1.72527 − 0.153167i −1.29562 − 2.24408i 1.00000 −1.01884 − 2.82170i 1.65673i
113.9 −0.500000 − 0.866025i 1.53628 + 0.799904i −0.500000 + 0.866025i 1.58264 + 0.913740i −0.0754025 − 1.73041i 0.101997 + 0.176663i 1.00000 1.72031 + 2.45775i − 1.82748i
113.10 −0.500000 − 0.866025i 1.72416 − 0.165164i −0.500000 + 0.866025i −2.66819 − 1.54048i −1.00512 − 1.41058i −0.142695 − 0.247156i 1.00000 2.94544 − 0.569538i 3.08096i
227.1 −0.500000 + 0.866025i −1.71504 − 0.242123i −0.500000 − 0.866025i 0.446642 − 0.257869i 1.06721 − 1.36421i −0.534840 + 0.926369i 1.00000 2.88275 + 0.830504i 0.515738i
227.2 −0.500000 + 0.866025i −1.54483 + 0.783270i −0.500000 − 0.866025i −2.13944 + 1.23520i 0.0940817 − 1.72949i 2.17969 − 3.77534i 1.00000 1.77298 − 2.42003i − 2.47041i
227.3 −0.500000 + 0.866025i −1.24208 + 1.20716i −0.500000 − 0.866025i 2.76481 − 1.59626i −0.424393 − 1.67925i −0.965307 + 1.67196i 1.00000 0.0855221 − 2.99878i 3.19253i
227.4 −0.500000 + 0.866025i −0.607481 − 1.62203i −0.500000 − 0.866025i 3.41314 − 1.97058i 1.70846 + 0.284919i 1.07232 − 1.85732i 1.00000 −2.26193 + 1.97070i 3.94116i
227.5 −0.500000 + 0.866025i −0.165521 − 1.72412i −0.500000 − 0.866025i −0.379645 + 0.219188i 1.57590 + 0.718716i −2.44148 + 4.22876i 1.00000 −2.94521 + 0.570758i − 0.438376i
227.6 −0.500000 + 0.866025i 0.707416 − 1.58100i −0.500000 − 0.866025i −2.77331 + 1.60117i 1.01548 + 1.40314i 0.760344 − 1.31695i 1.00000 −1.99912 − 2.23685i − 3.20235i
227.7 −0.500000 + 0.866025i 0.811819 + 1.53002i −0.500000 − 0.866025i 1.18812 − 0.685961i −1.73094 − 0.0619519i 2.26558 − 3.92410i 1.00000 −1.68190 + 2.48419i 1.37192i
227.8 −0.500000 + 0.866025i 0.995279 + 1.41754i −0.500000 − 0.866025i −1.43477 + 0.828367i −1.72527 + 0.153167i −1.29562 + 2.24408i 1.00000 −1.01884 + 2.82170i − 1.65673i
227.9 −0.500000 + 0.866025i 1.53628 − 0.799904i −0.500000 − 0.866025i 1.58264 − 0.913740i −0.0754025 + 1.73041i 0.101997 − 0.176663i 1.00000 1.72031 − 2.45775i 1.82748i
227.10 −0.500000 + 0.866025i 1.72416 + 0.165164i −0.500000 − 0.866025i −2.66819 + 1.54048i −1.00512 + 1.41058i −0.142695 + 0.247156i 1.00000 2.94544 + 0.569538i − 3.08096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.p.a ✓ 20
3.b odd 2 1 1026.2.p.b 20
9.c even 3 1 1026.2.p.a 20
9.c even 3 1 3078.2.b.c 20
9.d odd 6 1 342.2.p.b yes 20
9.d odd 6 1 3078.2.b.a 20
19.b odd 2 1 342.2.p.b yes 20
57.d even 2 1 1026.2.p.a 20
171.l even 6 1 inner 342.2.p.a ✓ 20
171.l even 6 1 3078.2.b.c 20
171.o odd 6 1 1026.2.p.b 20
171.o odd 6 1 3078.2.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.p.a ✓ 20 1.a even 1 1 trivial
342.2.p.a ✓ 20 171.l even 6 1 inner
342.2.p.b yes 20 9.d odd 6 1
342.2.p.b yes 20 19.b odd 2 1
1026.2.p.a 20 9.c even 3 1
1026.2.p.a 20 57.d even 2 1
1026.2.p.b 20 3.b odd 2 1
1026.2.p.b 20 171.o odd 6 1
3078.2.b.a 20 9.d odd 6 1
3078.2.b.a 20 171.o odd 6 1
3078.2.b.c 20 9.c even 3 1
3078.2.b.c 20 171.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{20} - 63 T_{13}^{18} + 2832 T_{13}^{16} - 3357 T_{13}^{15} - 58500 T_{13}^{14} + 111942 T_{13}^{13} + \cdots + 57972996 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} - T^{19} + \cdots + 59049 \) Copy content Toggle raw display
$5$ \( T^{20} - 30 T^{18} + \cdots + 82944 \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 9604 \) Copy content Toggle raw display
$11$ \( T^{20} + 3 T^{19} + \cdots + 254016 \) Copy content Toggle raw display
$13$ \( T^{20} - 63 T^{18} + \cdots + 57972996 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 27575591481 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 256064004 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 13947137604 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 152288236422144 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 529184016 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 6085773228096 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 637987982963776 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 2499200544996 \) Copy content Toggle raw display
$53$ \( (T^{10} - 219 T^{8} + \cdots + 2207412)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 423797094009 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 42955545186304 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 37008140625 \) Copy content Toggle raw display
$71$ \( (T^{10} - 12 T^{9} + \cdots + 298878336)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} - 17 T^{9} + \cdots - 584699069)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 263390662656 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{10} - 18 T^{9} + \cdots + 73284912)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
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