↑

Properties

Label 374.2.g.g
Level $374$
Weight $2$
Character orbit 374.g
Analytic conductor $2.986$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [374,2,Mod(69,374)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(374, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([8, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("374.69");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 374 = 2 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 374.g (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: \(2.98640503560\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{19} + 11 x^{18} + 3 x^{17} + 87 x^{16} + 144 x^{15} + 812 x^{14} + 2453 x^{13} + \cdots + 21025 \) 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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{9} - \beta_{7} + \beta_{3} + 1) q^{2} - \beta_{5} q^{3} - \beta_{7} q^{4} + ( - \beta_{18} - \beta_{14} + \cdots - \beta_{6}) q^{5}+ \cdots + (\beta_{19} - \beta_{16} - \beta_{10} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{9} - \beta_{7} + \beta_{3} + 1) q^{2} - \beta_{5} q^{3} - \beta_{7} q^{4} + ( - \beta_{18} - \beta_{14} + \cdots - \beta_{6}) q^{5}+ \cdots + (\beta_{19} - \beta_{18} - \beta_{16} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - q^{3} - 5 q^{4} + q^{6} + 6 q^{7} + 5 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 5 q^{2} - q^{3} - 5 q^{4} + q^{6} + 6 q^{7} + 5 q^{8} - 6 q^{9} - 3 q^{11} + 4 q^{12} + 10 q^{13} - 6 q^{14} + 10 q^{15} - 5 q^{16} + 5 q^{17} + 6 q^{18} + 7 q^{19} - 38 q^{21} - 17 q^{22} - 12 q^{23} + q^{24} + 3 q^{25} + 10 q^{26} + 47 q^{27} - 4 q^{28} + 5 q^{29} - 10 q^{30} - 7 q^{31} - 20 q^{32} - 56 q^{33} + 20 q^{34} + 19 q^{35} - 6 q^{36} - 6 q^{37} - 2 q^{38} + 3 q^{39} + 47 q^{41} - 7 q^{42} - 32 q^{43} - 3 q^{44} - 52 q^{45} + 12 q^{46} + 7 q^{47} - q^{48} + 41 q^{49} + 7 q^{50} + q^{51} - 10 q^{52} - 4 q^{53} - 22 q^{54} + 4 q^{56} + 57 q^{57} - 5 q^{58} + 7 q^{59} - 5 q^{60} - 4 q^{61} - 8 q^{62} + 37 q^{63} - 5 q^{64} - 104 q^{65} - 34 q^{66} - 26 q^{67} + 5 q^{68} + 45 q^{69} + 6 q^{70} + 36 q^{71} + 6 q^{72} + 22 q^{73} + 6 q^{74} - 25 q^{75} - 18 q^{76} - 13 q^{77} + 22 q^{78} + 82 q^{79} - 17 q^{81} + 8 q^{82} - 18 q^{83} + 12 q^{84} - 8 q^{86} - 58 q^{87} + 8 q^{88} - 16 q^{89} + 22 q^{90} + 43 q^{91} + 18 q^{92} + 41 q^{93} - 7 q^{94} - 46 q^{95} + q^{96} - 62 q^{97} - 36 q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - x^{19} + 11 x^{18} + 3 x^{17} + 87 x^{16} + 144 x^{15} + 812 x^{14} + 2453 x^{13} + \cdots + 21025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!36 \nu^{19} + \cdots - 39\!\cdots\!75 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!63 \nu^{19} + \cdots + 15\!\cdots\!75 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 98\!\cdots\!22 \nu^{19} + \cdots - 15\!\cdots\!75 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 52\!\cdots\!96 \nu^{19} + \cdots - 28\!\cdots\!50 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17\!\cdots\!88 \nu^{19} + \cdots + 18\!\cdots\!75 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!19 \nu^{19} + \cdots + 91\!\cdots\!85 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79\!\cdots\!82 \nu^{19} + \cdots + 15\!\cdots\!75 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 23\!\cdots\!58 \nu^{19} + \cdots + 90\!\cdots\!05 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!69 \nu^{19} + \cdots - 12\!\cdots\!00 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!51 \nu^{19} + \cdots + 91\!\cdots\!95 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!19 \nu^{19} + \cdots + 12\!\cdots\!75 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38\!\cdots\!90 \nu^{19} + \cdots - 82\!\cdots\!95 ) / 11\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 39\!\cdots\!77 \nu^{19} + \cdots + 10\!\cdots\!75 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46\!\cdots\!44 \nu^{19} + \cdots - 51\!\cdots\!10 ) / 10\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 18\!\cdots\!89 \nu^{19} + \cdots + 40\!\cdots\!55 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 22\!\cdots\!49 \nu^{19} + \cdots - 69\!\cdots\!70 ) / 32\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 29\!\cdots\!15 \nu^{19} + \cdots - 38\!\cdots\!25 ) / 37\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 34\!\cdots\!72 \nu^{19} + \cdots - 28\!\cdots\!85 ) / 32\!\cdots\!30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{18} + \beta_{17} - \beta_{16} + \beta_{14} + \beta_{12} - \beta_{10} + \beta_{6} + 4\beta_{3} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} - 2 \beta_{16} + 2 \beta_{15} - 2 \beta_{13} + \beta_{12} - 2 \beta_{11} - 4 \beta_{9} + \cdots + 7 \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{19} + 2 \beta_{18} - 11 \beta_{17} + 3 \beta_{15} - 10 \beta_{14} - 2 \beta_{11} + \cdots - 15 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 29 \beta_{19} + 10 \beta_{18} - 29 \beta_{17} + 28 \beta_{16} + 2 \beta_{15} - 19 \beta_{14} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 115 \beta_{19} - 36 \beta_{17} + 115 \beta_{16} - 36 \beta_{14} + 36 \beta_{13} - 133 \beta_{12} + \cdots - 17 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 97 \beta_{18} - 18 \beta_{17} + 18 \beta_{16} - 43 \beta_{15} - 97 \beta_{14} + 43 \beta_{13} + \cdots - 692 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1215 \beta_{19} - 505 \beta_{18} - 710 \beta_{16} - 582 \beta_{15} + 101 \beta_{13} - 505 \beta_{12} + \cdots - 2931 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4085 \beta_{19} - 3107 \beta_{18} + 295 \beta_{17} - 2722 \beta_{15} + 1005 \beta_{14} + \cdots + 2739 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6545 \beta_{19} - 10199 \beta_{18} + 6545 \beta_{17} + 6466 \beta_{16} - 5852 \beta_{15} + 6518 \beta_{14} + \cdots + 30354 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4651 \beta_{19} + 46722 \beta_{17} - 4651 \beta_{16} + 36550 \beta_{14} - 9261 \beta_{13} + 47694 \beta_{12} + \cdots + 91901 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 109004 \beta_{18} + 140903 \beta_{17} - 140903 \beta_{16} + 68632 \beta_{15} + 109004 \beta_{14} + \cdots + 94435 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 69663 \beta_{19} + 423593 \beta_{18} - 461962 \beta_{16} + 343087 \beta_{15} - 223204 \beta_{13} + \cdots - 77616 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 1004529 \beta_{19} + 982901 \beta_{18} - 1539262 \beta_{17} + 926626 \beta_{15} + \cdots - 3100538 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 6039154 \beta_{19} + 1549843 \beta_{18} - 6039154 \beta_{17} + 5045672 \beta_{16} + 1498885 \beta_{15} + \cdots + 416836 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 16928682 \beta_{19} - 12174026 \beta_{17} + 16928682 \beta_{16} - 11787887 \beta_{14} + \cdots - 16247865 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 18747426 \beta_{18} - 13606631 \beta_{17} + 13606631 \beta_{16} - 18356070 \beta_{15} + \cdots - 132590795 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 187169500 \beta_{19} - 140215791 \beta_{18} - 40866858 \beta_{16} - 125870017 \beta_{15} + \cdots - 402750775 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 779659132 \beta_{19} - 640905867 \beta_{18} + 180526241 \beta_{17} - 508891433 \beta_{15} + \cdots + 795940100 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/374\mathbb{Z}\right)^\times\).

\(n\) \(35\) \(309\)
\(\chi(n)\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.995789 − 3.06472i
0.254561 − 0.783459i
−0.298147 + 0.917601i
−0.450757 + 1.38729i
−0.810463 + 2.49435i
0.995789 + 3.06472i
0.254561 + 0.783459i
−0.298147 − 0.917601i
−0.450757 − 1.38729i
−0.810463 − 2.49435i
2.75223 − 1.99961i
1.43989 − 1.04614i
−0.864376 + 0.628006i
−0.961427 + 0.698518i
−1.55729 + 1.13144i
2.75223 + 1.99961i
1.43989 + 1.04614i
−0.864376 − 0.628006i
−0.961427 − 0.698518i
−1.55729 − 1.13144i
0.809017 − 0.587785i −0.995789 − 3.06472i 0.309017 − 0.951057i −0.255815 − 0.185860i −2.60701 − 1.89410i 0.428133 − 1.31766i −0.309017 − 0.951057i −5.97388 + 4.34028i −0.316204
69.2 0.809017 − 0.587785i −0.254561 − 0.783459i 0.309017 − 0.951057i 1.31175 + 0.953042i −0.666450 − 0.484204i 0.0179370 − 0.0552043i −0.309017 − 0.951057i 1.87804 − 1.36448i 1.62141
69.3 0.809017 − 0.587785i 0.298147 + 0.917601i 0.309017 − 0.951057i −3.23048 − 2.34708i 0.780558 + 0.567109i −0.0589366 + 0.181388i −0.309017 − 0.951057i 1.67395 − 1.21620i −3.99309
69.4 0.809017 − 0.587785i 0.450757 + 1.38729i 0.309017 − 0.951057i 2.59538 + 1.88566i 1.18010 + 0.857391i 0.966763 − 2.97539i −0.309017 − 0.951057i 0.705665 − 0.512696i 3.20807
69.5 0.809017 − 0.587785i 0.810463 + 2.49435i 0.309017 − 0.951057i −0.420836 − 0.305755i 2.12182 + 1.54159i −0.971930 + 2.99129i −0.309017 − 0.951057i −3.13788 + 2.27980i −0.520182
103.1 0.809017 + 0.587785i −0.995789 + 3.06472i 0.309017 + 0.951057i −0.255815 + 0.185860i −2.60701 + 1.89410i 0.428133 + 1.31766i −0.309017 + 0.951057i −5.97388 − 4.34028i −0.316204
103.2 0.809017 + 0.587785i −0.254561 + 0.783459i 0.309017 + 0.951057i 1.31175 − 0.953042i −0.666450 + 0.484204i 0.0179370 + 0.0552043i −0.309017 + 0.951057i 1.87804 + 1.36448i 1.62141
103.3 0.809017 + 0.587785i 0.298147 − 0.917601i 0.309017 + 0.951057i −3.23048 + 2.34708i 0.780558 − 0.567109i −0.0589366 − 0.181388i −0.309017 + 0.951057i 1.67395 + 1.21620i −3.99309
103.4 0.809017 + 0.587785i 0.450757 − 1.38729i 0.309017 + 0.951057i 2.59538 − 1.88566i 1.18010 − 0.857391i 0.966763 + 2.97539i −0.309017 + 0.951057i 0.705665 + 0.512696i 3.20807
103.5 0.809017 + 0.587785i 0.810463 − 2.49435i 0.309017 + 0.951057i −0.420836 + 0.305755i 2.12182 − 1.54159i −0.971930 − 2.99129i −0.309017 + 0.951057i −3.13788 − 2.27980i −0.520182
137.1 −0.309017 − 0.951057i −2.75223 − 1.99961i −0.809017 + 0.587785i −0.917596 + 2.82407i −1.05126 + 3.23544i 0.836626 − 0.607845i 0.809017 + 0.587785i 2.64926 + 8.15359i 2.96940
137.2 −0.309017 − 0.951057i −1.43989 − 1.04614i −0.809017 + 0.587785i 0.725811 − 2.23382i −0.549988 + 1.69269i 3.67291 − 2.66852i 0.809017 + 0.587785i 0.0518163 + 0.159474i −2.34877
137.3 −0.309017 − 0.951057i 0.864376 + 0.628006i −0.809017 + 0.587785i −0.294593 + 0.906664i 0.330162 − 1.01614i −3.77060 + 2.73950i 0.809017 + 0.587785i −0.574297 − 1.76750i 0.953323
137.4 −0.309017 − 0.951057i 0.961427 + 0.698518i −0.809017 + 0.587785i 0.872035 − 2.68385i 0.367233 − 1.13023i −1.63538 + 1.18817i 0.809017 + 0.587785i −0.490636 − 1.51002i −2.82196
137.5 −0.309017 − 0.951057i 1.55729 + 1.13144i −0.809017 + 0.587785i −0.385656 + 1.18693i 0.594833 − 1.83071i 3.51448 − 2.55342i 0.809017 + 0.587785i 0.217956 + 0.670798i 1.24801
273.1 −0.309017 + 0.951057i −2.75223 + 1.99961i −0.809017 − 0.587785i −0.917596 − 2.82407i −1.05126 − 3.23544i 0.836626 + 0.607845i 0.809017 − 0.587785i 2.64926 − 8.15359i 2.96940
273.2 −0.309017 + 0.951057i −1.43989 + 1.04614i −0.809017 − 0.587785i 0.725811 + 2.23382i −0.549988 − 1.69269i 3.67291 + 2.66852i 0.809017 − 0.587785i 0.0518163 − 0.159474i −2.34877
273.3 −0.309017 + 0.951057i 0.864376 − 0.628006i −0.809017 − 0.587785i −0.294593 − 0.906664i 0.330162 + 1.01614i −3.77060 − 2.73950i 0.809017 − 0.587785i −0.574297 + 1.76750i 0.953323
273.4 −0.309017 + 0.951057i 0.961427 − 0.698518i −0.809017 − 0.587785i 0.872035 + 2.68385i 0.367233 + 1.13023i −1.63538 − 1.18817i 0.809017 − 0.587785i −0.490636 + 1.51002i −2.82196
273.5 −0.309017 + 0.951057i 1.55729 − 1.13144i −0.809017 − 0.587785i −0.385656 − 1.18693i 0.594833 + 1.83071i 3.51448 + 2.55342i 0.809017 − 0.587785i 0.217956 − 0.670798i 1.24801
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.5
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 374.2.g.g ✓ 20
11.c even 5 1 inner 374.2.g.g ✓ 20
11.c even 5 1 4114.2.a.bk 10
11.d odd 10 1 4114.2.a.bl 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
374.2.g.g ✓ 20 1.a even 1 1 trivial
374.2.g.g ✓ 20 11.c even 5 1 inner
4114.2.a.bk 10 11.c even 5 1
4114.2.a.bl 10 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(374, [\chi])\):

\( T_{3}^{20} + T_{3}^{19} + 11 T_{3}^{18} - 3 T_{3}^{17} + 87 T_{3}^{16} - 144 T_{3}^{15} + 812 T_{3}^{14} + \cdots + 21025 \) Copy content Toggle raw display
\( T_{5}^{20} + 11 T_{5}^{18} - 2 T_{5}^{17} + 182 T_{5}^{16} - 294 T_{5}^{15} + 2812 T_{5}^{14} + \cdots + 6400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} + T^{19} + \cdots + 21025 \) Copy content Toggle raw display
$5$ \( T^{20} + 11 T^{18} + \cdots + 6400 \) Copy content Toggle raw display
$7$ \( T^{20} - 6 T^{19} + \cdots + 841 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 25937424601 \) Copy content Toggle raw display
$13$ \( T^{20} - 10 T^{19} + \cdots + 3150625 \) Copy content Toggle raw display
$17$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{5} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 7545354496 \) Copy content Toggle raw display
$23$ \( (T^{10} + 6 T^{9} + \cdots + 137380)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 240100727193856 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 36741222400 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 861184000000 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{10} + 16 T^{9} + \cdots - 704)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 269653182054400 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1016525716441 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 2834239590400 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + 13 T^{9} + \cdots - 68864)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 50\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 1140818176 \) Copy content Toggle raw display
$89$ \( (T^{10} + 8 T^{9} + \cdots - 1287864976)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
show more
show less