↑

Properties

Label 400.2.s.d
Level $400$
Weight $2$
Character orbit 400.s
Analytic conductor $3.194$
Analytic rank $0$
Dimension $18$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(107,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.s (of order \(4\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + \cdots + 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 71 \nu^{17} + 98 \nu^{16} + 286 \nu^{15} + 144 \nu^{14} - 123 \nu^{13} - 1148 \nu^{12} - 2354 \nu^{11} + \cdots - 24064 ) / 1280 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 129 \nu^{17} - 124 \nu^{16} - 398 \nu^{15} + 116 \nu^{14} + 797 \nu^{13} + 2778 \nu^{12} + \cdots + 58624 ) / 1280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + \cdots + 6048 ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + \cdots + 28416 ) / 640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + \cdots + 89600 ) / 1280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + \cdots + 75264 ) / 640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} + \cdots - 222976 ) / 1280 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 298 \nu^{17} + 337 \nu^{16} + 974 \nu^{15} - 166 \nu^{14} - 1874 \nu^{13} - 6713 \nu^{12} + \cdots - 133504 ) / 640 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 159 \nu^{17} + 235 \nu^{16} + 665 \nu^{15} + 358 \nu^{14} - 257 \nu^{13} - 2621 \nu^{12} + \cdots - 54848 ) / 320 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} + \cdots - 351232 ) / 1280 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 503 \nu^{17} + 761 \nu^{16} + 2162 \nu^{15} + 1230 \nu^{14} - 679 \nu^{13} - 8175 \nu^{12} + \cdots - 168960 ) / 640 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 129 \nu^{17} - 186 \nu^{16} - 524 \nu^{15} - 232 \nu^{14} + 321 \nu^{13} + 2280 \nu^{12} + \cdots + 46208 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} + \cdots - 33024 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 93 \nu^{17} - 133 \nu^{16} - 374 \nu^{15} - 162 \nu^{14} + 237 \nu^{13} + 1639 \nu^{12} + \cdots + 33024 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + \cdots + 363008 ) / 640 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + \cdots + 372864 ) / 640 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 3253 \nu^{17} + 4754 \nu^{16} + 13458 \nu^{15} + 6632 \nu^{14} - 6769 \nu^{13} - 55664 \nu^{12} + \cdots - 1132032 ) / 1280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{8} + \beta_{7} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - 2 \beta_{10} + \beta_{8} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \cdots + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + \cdots + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{17} - 2 \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} + \cdots + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2 \beta_{16} - 2 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + \cdots - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( \beta_{17} + 3 \beta_{16} - \beta_{15} - 10 \beta_{14} + 15 \beta_{12} + 2 \beta_{10} + 4 \beta_{9} + \cdots - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( \beta_{17} + 7 \beta_{16} - 7 \beta_{15} + 15 \beta_{14} + 3 \beta_{13} - 25 \beta_{12} - 8 \beta_{11} + \cdots + 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2 \beta_{17} + 3 \beta_{16} + 7 \beta_{15} - \beta_{14} + 9 \beta_{13} + \beta_{12} - \beta_{11} + \cdots + 23 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 13 \beta_{17} + 17 \beta_{16} + 3 \beta_{15} + 18 \beta_{14} + 15 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} + \cdots + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3 \beta_{17} - 42 \beta_{16} - 20 \beta_{15} + 9 \beta_{14} - 21 \beta_{13} + 18 \beta_{12} + \cdots - 22 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8 \beta_{17} + 2 \beta_{16} - 8 \beta_{15} - 19 \beta_{14} - \beta_{13} + 10 \beta_{12} - 14 \beta_{11} + \cdots + 92 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 37 \beta_{17} + 101 \beta_{16} - 7 \beta_{15} + 62 \beta_{14} + 76 \beta_{13} - 35 \beta_{12} + \cdots + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 45 \beta_{17} + 43 \beta_{16} + 13 \beta_{15} + 13 \beta_{14} - 117 \beta_{13} - 63 \beta_{12} + \cdots + 233 ) / 2 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 164 \beta_{17} - 9 \beta_{16} - 103 \beta_{15} + 7 \beta_{14} + 187 \beta_{13} - 53 \beta_{12} + \cdots - 43 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 163 \beta_{17} - 173 \beta_{16} - 5 \beta_{15} + 158 \beta_{14} - 112 \beta_{13} + 439 \beta_{12} + \cdots + 265 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{12}\) \(-\beta_{12}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.08900 − 0.902261i
0.482716 − 1.32928i
−0.635486 + 1.26339i
1.41303 − 0.0578659i
0.235136 + 1.39453i
1.41323 + 0.0526497i
−1.37691 + 0.322680i
0.0376504 − 1.41371i
−0.480367 − 1.33013i
−1.08900 + 0.902261i
0.482716 + 1.32928i
−0.635486 − 1.26339i
1.41303 + 0.0578659i
0.235136 − 1.39453i
1.41323 − 0.0526497i
−1.37691 − 0.322680i
0.0376504 + 1.41371i
−0.480367 + 1.33013i
−1.41267 + 0.0660953i 0.496487 1.99126 − 0.186742i 0 −0.701372 + 0.0328155i −1.55426 − 1.55426i −2.80065 + 0.395417i −2.75350 0
107.2 −1.19301 − 0.759419i 1.39319 0.846564 + 1.81200i 0 −1.66209 − 1.05801i 2.13436 + 2.13436i 0.366101 − 2.80463i −1.05903 0
107.3 −0.828280 + 1.14628i −0.692712 −0.627905 − 1.89888i 0 0.573759 − 0.794040i 0.343872 + 0.343872i 2.69672 + 0.853049i −2.52015 0
107.4 −0.567819 − 1.29521i −1.96251 −1.35516 + 1.47090i 0 1.11435 + 2.54187i 1.60205 + 1.60205i 2.67461 + 0.920026i 0.851447 0
107.5 −0.430311 + 1.34716i 2.96561 −1.62967 − 1.15939i 0 −1.27613 + 3.99515i 0.115101 + 0.115101i 2.26315 − 1.69652i 5.79486 0
107.6 0.516777 + 1.31641i −1.28110 −1.46588 + 1.36058i 0 −0.662041 − 1.68645i 1.13975 + 1.13975i −2.54862 − 1.22659i −1.35879 0
107.7 1.23576 − 0.687667i −0.614566 1.05423 − 1.69959i 0 −0.759459 + 0.422617i −2.83610 − 2.83610i 0.134028 − 2.82525i −2.62231 0
107.8 1.29924 + 0.558542i 2.55161 1.37606 + 1.45136i 0 3.31516 + 1.42518i −2.40368 − 2.40368i 0.977191 + 2.65426i 3.51070 0
107.9 1.38031 + 0.307817i −2.85601 1.81050 + 0.849763i 0 −3.94217 − 0.879127i 0.458895 + 0.458895i 2.23747 + 1.73024i 5.15678 0
243.1 −1.41267 − 0.0660953i 0.496487 1.99126 + 0.186742i 0 −0.701372 − 0.0328155i −1.55426 + 1.55426i −2.80065 − 0.395417i −2.75350 0
243.2 −1.19301 + 0.759419i 1.39319 0.846564 − 1.81200i 0 −1.66209 + 1.05801i 2.13436 − 2.13436i 0.366101 + 2.80463i −1.05903 0
243.3 −0.828280 − 1.14628i −0.692712 −0.627905 + 1.89888i 0 0.573759 + 0.794040i 0.343872 − 0.343872i 2.69672 − 0.853049i −2.52015 0
243.4 −0.567819 + 1.29521i −1.96251 −1.35516 − 1.47090i 0 1.11435 − 2.54187i 1.60205 − 1.60205i 2.67461 − 0.920026i 0.851447 0
243.5 −0.430311 − 1.34716i 2.96561 −1.62967 + 1.15939i 0 −1.27613 − 3.99515i 0.115101 − 0.115101i 2.26315 + 1.69652i 5.79486 0
243.6 0.516777 − 1.31641i −1.28110 −1.46588 − 1.36058i 0 −0.662041 + 1.68645i 1.13975 − 1.13975i −2.54862 + 1.22659i −1.35879 0
243.7 1.23576 + 0.687667i −0.614566 1.05423 + 1.69959i 0 −0.759459 − 0.422617i −2.83610 + 2.83610i 0.134028 + 2.82525i −2.62231 0
243.8 1.29924 − 0.558542i 2.55161 1.37606 − 1.45136i 0 3.31516 − 1.42518i −2.40368 + 2.40368i 0.977191 − 2.65426i 3.51070 0
243.9 1.38031 − 0.307817i −2.85601 1.81050 − 0.849763i 0 −3.94217 + 0.879127i 0.458895 − 0.458895i 2.23747 − 1.73024i 5.15678 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.s.d 18
4.b odd 2 1 1600.2.s.d 18
5.b even 2 1 80.2.s.b yes 18
5.c odd 4 1 80.2.j.b ✓ 18
5.c odd 4 1 400.2.j.d 18
15.d odd 2 1 720.2.z.g 18
15.e even 4 1 720.2.bd.g 18
16.e even 4 1 1600.2.j.d 18
16.f odd 4 1 400.2.j.d 18
20.d odd 2 1 320.2.s.b 18
20.e even 4 1 320.2.j.b 18
20.e even 4 1 1600.2.j.d 18
40.e odd 2 1 640.2.s.c 18
40.f even 2 1 640.2.s.d 18
40.i odd 4 1 640.2.j.d 18
40.k even 4 1 640.2.j.c 18
80.i odd 4 1 640.2.s.c 18
80.i odd 4 1 1600.2.s.d 18
80.j even 4 1 80.2.s.b yes 18
80.k odd 4 1 80.2.j.b ✓ 18
80.k odd 4 1 640.2.j.d 18
80.q even 4 1 320.2.j.b 18
80.q even 4 1 640.2.j.c 18
80.s even 4 1 inner 400.2.s.d 18
80.s even 4 1 640.2.s.d 18
80.t odd 4 1 320.2.s.b 18
240.t even 4 1 720.2.bd.g 18
240.bd odd 4 1 720.2.z.g 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b ✓ 18 5.c odd 4 1
80.2.j.b ✓ 18 80.k odd 4 1
80.2.s.b yes 18 5.b even 2 1
80.2.s.b yes 18 80.j even 4 1
320.2.j.b 18 20.e even 4 1
320.2.j.b 18 80.q even 4 1
320.2.s.b 18 20.d odd 2 1
320.2.s.b 18 80.t odd 4 1
400.2.j.d 18 5.c odd 4 1
400.2.j.d 18 16.f odd 4 1
400.2.s.d 18 1.a even 1 1 trivial
400.2.s.d 18 80.s even 4 1 inner
640.2.j.c 18 40.k even 4 1
640.2.j.c 18 80.q even 4 1
640.2.j.d 18 40.i odd 4 1
640.2.j.d 18 80.k odd 4 1
640.2.s.c 18 40.e odd 2 1
640.2.s.c 18 80.i odd 4 1
640.2.s.d 18 40.f even 2 1
640.2.s.d 18 80.s even 4 1
720.2.z.g 18 15.d odd 2 1
720.2.z.g 18 240.bd odd 4 1
720.2.bd.g 18 15.e even 4 1
720.2.bd.g 18 240.t even 4 1
1600.2.j.d 18 16.e even 4 1
1600.2.j.d 18 20.e even 4 1
1600.2.s.d 18 4.b odd 2 1
1600.2.s.d 18 80.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{9} - 16T_{3}^{7} - 4T_{3}^{6} + 76T_{3}^{5} + 40T_{3}^{4} - 104T_{3}^{3} - 72T_{3}^{2} + 20T_{3} + 16 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} - 2 T^{16} + \cdots + 512 \) Copy content Toggle raw display
$3$ \( (T^{9} - 16 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} + 2 T^{17} + \cdots + 288 \) Copy content Toggle raw display
$11$ \( T^{18} + 2 T^{17} + \cdots + 5431808 \) Copy content Toggle raw display
$13$ \( T^{18} + 112 T^{16} + \cdots + 67108864 \) Copy content Toggle raw display
$17$ \( T^{18} - 6 T^{17} + \cdots + 512 \) Copy content Toggle raw display
$19$ \( T^{18} + 2 T^{17} + \cdots + 4608 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 17700587552 \) Copy content Toggle raw display
$29$ \( T^{18} - 14 T^{17} + \cdots + 82330112 \) Copy content Toggle raw display
$31$ \( T^{18} + 196 T^{16} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 574297214976 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 242788765696 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 337207844416 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16870640672 \) Copy content Toggle raw display
$53$ \( (T^{9} + 6 T^{8} + \cdots - 220832)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 144166720393728 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 121236758528 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 555525752896 \) Copy content Toggle raw display
$71$ \( (T^{9} - 12 T^{8} + \cdots - 27648)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 35535647232 \) Copy content Toggle raw display
$79$ \( (T^{9} + 8 T^{8} + \cdots + 45002752)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + 20 T^{8} + \cdots + 8413744)^{2} \) Copy content Toggle raw display
$89$ \( (T^{9} - 6 T^{8} + \cdots - 251904)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 380349381734912 \) Copy content Toggle raw display
show more
show less