Properties

Label 418.2.f.f
Level $418$
Weight $2$
Character orbit 418.f
Analytic conductor $3.338$
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] = [418,2,Mod(115,418)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(418, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("418.115");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 418 = 2 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 418.f (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: \(3.33774680449\)
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: \( x^{16} - 6 x^{15} + 24 x^{14} - 62 x^{13} + 148 x^{12} - 232 x^{11} + 432 x^{10} - 356 x^{9} + 424 x^{8} - 320 x^{7} + 712 x^{6} - 936 x^{5} + 968 x^{4} - 336 x^{3} + 208 x^{2} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{3} + \beta_{9} q^{4} + ( - \beta_{11} - \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{12} + \beta_{8} - \beta_{6} - \beta_{2}) q^{6} + ( - \beta_{15} - \beta_{13} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{13} + \beta_{10} - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{14} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{3} + \beta_{9} q^{4} + ( - \beta_{11} - \beta_{9} - \beta_{4}) q^{5} + ( - \beta_{12} + \beta_{8} - \beta_{6} - \beta_{2}) q^{6} + ( - \beta_{15} - \beta_{13} + \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{9} + 6 \beta_{8} + 4 \beta_{6} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 4 q^{4} + 4 q^{5} + 6 q^{7} - 4 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} - 4 q^{4} + 4 q^{5} + 6 q^{7} - 4 q^{8} - 20 q^{9} + 4 q^{10} + 2 q^{11} - 14 q^{13} + 6 q^{14} - 10 q^{15} - 4 q^{16} + 8 q^{17} + 4 q^{19} + 4 q^{20} + 28 q^{21} + 2 q^{22} + 24 q^{23} - 4 q^{26} - 18 q^{27} - 14 q^{28} - 22 q^{29} - 10 q^{30} - 24 q^{31} + 16 q^{32} + 2 q^{33} + 8 q^{34} - 20 q^{35} + 6 q^{37} + 4 q^{38} + 72 q^{39} - 6 q^{40} - 14 q^{41} - 22 q^{42} + 56 q^{43} + 12 q^{44} - 56 q^{45} + 4 q^{46} - 38 q^{47} - 36 q^{49} + 56 q^{51} - 4 q^{52} + 22 q^{53} - 48 q^{54} - 6 q^{55} + 16 q^{56} - 22 q^{58} - 28 q^{59} + 12 q^{61} + 26 q^{62} - 20 q^{63} - 4 q^{64} - 32 q^{65} + 12 q^{66} - 44 q^{67} + 8 q^{68} + 64 q^{69} - 20 q^{70} + 16 q^{71} - 20 q^{72} + 12 q^{73} + 6 q^{74} + 32 q^{75} - 16 q^{76} - 12 q^{77} - 8 q^{78} + 48 q^{79} - 6 q^{80} - 64 q^{81} + 6 q^{82} + 14 q^{83} + 8 q^{84} + 6 q^{85} - 14 q^{86} + 84 q^{87} - 8 q^{88} - 56 q^{89} + 34 q^{90} + 26 q^{91} - 16 q^{92} - 46 q^{93} + 32 q^{94} + 6 q^{95} - 24 q^{97} + 64 q^{98} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} + 24 x^{14} - 62 x^{13} + 148 x^{12} - 232 x^{11} + 432 x^{10} - 356 x^{9} + 424 x^{8} - 320 x^{7} + 712 x^{6} - 936 x^{5} + 968 x^{4} - 336 x^{3} + 208 x^{2} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2719816147894 \nu^{15} + 17441650885229 \nu^{14} - 72202746279073 \nu^{13} + 198210786464372 \nu^{12} + \cdots - 745208078715200 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10129253398894 \nu^{15} + 64822427251750 \nu^{14} - 270100492494505 \nu^{13} + 735550269713908 \nu^{12} + \cdots + 13\!\cdots\!80 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11724786127521 \nu^{15} + 56209905419431 \nu^{14} - 204651849815486 \nu^{13} + 440517196989001 \nu^{12} + \cdots - 338842957058304 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14176160257280 \nu^{15} + 91820838172799 \nu^{14} - 377637051476985 \nu^{13} + \cdots + 61\!\cdots\!40 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26987880892483 \nu^{15} - 159007095381997 \nu^{14} + 628722781690879 \nu^{13} + \cdots - 796791138145216 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15449512761451 \nu^{15} - 101601301380600 \nu^{14} + 411757188407931 \nu^{13} + \cdots - 35\!\cdots\!64 ) / 12\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 46575504919700 \nu^{15} - 282172845666094 \nu^{14} + \cdots - 40\!\cdots\!12 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49799446134076 \nu^{15} - 271808795911973 \nu^{14} + \cdots - 868706645630000 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55432590829733 \nu^{15} + 282189548836191 \nu^{14} + \cdots - 17\!\cdots\!28 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 114498884611088 \nu^{15} + 686924776100243 \nu^{14} + \cdots + 92\!\cdots\!12 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 12793599648327 \nu^{15} - 76152671395417 \nu^{14} + 303040917435434 \nu^{13} - 777140611293591 \nu^{12} + \cdots - 930541201389856 ) / 226860537499112 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 141938730356411 \nu^{15} + 790744627468916 \nu^{14} + \cdots + 10\!\cdots\!64 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 159073353476567 \nu^{15} - 877113474486394 \nu^{14} + \cdots - 31\!\cdots\!76 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 185478543767947 \nu^{15} + \cdots + 69\!\cdots\!64 ) / 24\!\cdots\!32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{13} - \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{7} + \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} + \beta_{13} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 6 \beta_{6} + \beta_{4} + \beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} + 2 \beta_{12} + 6 \beta_{11} + 2 \beta_{9} + 10 \beta_{8} + 10 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 10 \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 12 \beta_{13} + 8 \beta_{12} + 12 \beta_{11} + 20 \beta_{10} + 12 \beta_{8} - 18 \beta_{7} + 20 \beta_{5} + 12 \beta_{4} - 16 \beta_{3} + 32 \beta_{2} - 16 \beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 28 \beta_{15} - 2 \beta_{14} - 68 \beta_{13} + 28 \beta_{11} + 86 \beta_{10} - 30 \beta_{9} - 66 \beta_{7} - 90 \beta_{6} + 68 \beta_{5} + 46 \beta_{4} - 90 \beta_{3} - 118 \beta _1 - 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 186 \beta_{15} - 30 \beta_{14} - 186 \beta_{13} - 58 \beta_{12} + 68 \beta_{11} + 186 \beta_{10} - 206 \beta_{9} - 122 \beta_{8} - 128 \beta_{7} - 418 \beta_{6} + 118 \beta_{5} - 232 \beta_{3} - 232 \beta_{2} - 418 \beta _1 - 122 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 604 \beta_{15} - 176 \beta_{14} - 304 \beta_{13} - 176 \beta_{12} + 168 \beta_{10} - 660 \beta_{9} - 660 \beta_{8} - 168 \beta_{7} - 1124 \beta_{6} - 328 \beta_{4} - 804 \beta_{2} - 804 \beta _1 - 328 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1108 \beta_{15} - 436 \beta_{14} - 344 \beta_{12} - 620 \beta_{11} - 436 \beta_{10} - 1172 \beta_{9} - 1904 \beta_{8} - 1928 \beta_{6} - 1108 \beta_{5} - 1172 \beta_{4} + 1828 \beta_{3} - 1928 \beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3036 \beta_{13} - 528 \beta_{12} - 3036 \beta_{11} - 3564 \beta_{10} - 3328 \beta_{8} + 2020 \beta_{7} - 5484 \beta_{5} - 3328 \beta_{4} + 7236 \beta_{3} - 3372 \beta_{2} + 7236 \beta _1 + 2640 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10272 \beta_{15} + 3464 \beta_{14} + 16092 \beta_{13} - 10272 \beta_{11} - 16192 \beta_{10} + 11008 \beta_{9} + 12628 \beta_{7} + 18992 \beta_{6} - 16092 \beta_{5} - 6644 \beta_{4} + 18992 \beta_{3} + \cdots + 11008 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 50344 \beta_{15} + 13736 \beta_{14} + 50344 \beta_{13} + 5920 \beta_{12} - 21080 \beta_{11} - 50344 \beta_{10} + 54768 \beta_{9} + 32064 \beta_{8} + 44424 \beta_{7} + 99616 \beta_{6} - 29264 \beta_{5} + \cdots + 32064 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 149960 \beta_{15} + 35184 \beta_{14} + 94968 \beta_{13} + 35184 \beta_{12} - 88680 \beta_{10} + 164048 \beta_{9} + 164048 \beta_{8} + 88680 \beta_{7} + 314856 \beta_{6} + 61536 \beta_{4} + \cdots + 61536 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 277360 \beta_{15} + 61280 \beta_{14} + 123864 \beta_{12} + 187456 \beta_{11} + 61280 \beta_{10} + 303504 \beta_{9} + 505848 \beta_{8} + 601016 \beta_{6} + 277360 \beta_{5} + 303504 \beta_{4} + \cdots + 601016 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 878376 \beta_{13} + 250040 \beta_{12} + 878376 \beta_{11} + 1128416 \beta_{10} + 951992 \beta_{8} - 789776 \beta_{7} + 1397136 \beta_{5} + 951992 \beta_{4} - 1727960 \beta_{3} + \cdots - 574008 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/418\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(343\)
\(\chi(n)\) \(1\) \(\beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
115.1
0.417691 + 1.28552i
−0.628609 1.93466i
−0.151066 0.464934i
0.743950 + 2.28964i
2.46686 + 1.79228i
0.253418 + 0.184119i
0.870631 + 0.632551i
−0.972875 0.706835i
0.417691 1.28552i
−0.628609 + 1.93466i
−0.151066 + 0.464934i
0.743950 2.28964i
2.46686 1.79228i
0.253418 0.184119i
0.870631 0.632551i
−0.972875 + 0.706835i
0.309017 + 0.951057i −1.86150 1.35246i −0.809017 + 0.587785i 0.811598 2.49784i 0.711029 2.18832i −2.21832 + 1.61170i −0.809017 0.587785i 0.708982 + 2.18202i 2.62639
115.2 0.309017 + 0.951057i −0.738622 0.536641i −0.809017 + 0.587785i −0.549198 + 1.69026i 0.282129 0.868303i 4.08354 2.96686i −0.809017 0.587785i −0.669471 2.06042i −1.77724
115.3 0.309017 + 0.951057i −0.128416 0.0932996i −0.809017 + 0.587785i 0.950820 2.92632i 0.0490505 0.150962i −1.09378 + 0.794679i −0.809017 0.587785i −0.919265 2.82921i 3.07692
115.4 0.309017 + 0.951057i 2.72854 + 1.98240i −0.809017 + 0.587785i −0.213220 + 0.656224i −1.04221 + 3.20759i −0.389470 + 0.282966i −0.809017 0.587785i 2.58796 + 7.96492i −0.689994
191.1 −0.809017 0.587785i −0.897497 + 2.76221i 0.309017 + 0.951057i 2.41247 1.75276i 2.34968 1.70714i −0.863737 2.65831i 0.309017 0.951057i −4.39726 3.19480i −2.98197
191.2 −0.809017 0.587785i −0.670482 + 2.06353i 0.309017 + 0.951057i −1.44869 + 1.05254i 1.75534 1.27533i 1.45655 + 4.48281i 0.309017 0.951057i −1.38156 1.00376i 1.79068
191.3 −0.809017 0.587785i 0.636445 1.95878i 0.309017 + 0.951057i −1.58981 + 1.15506i −1.66623 + 1.21059i 1.27470 + 3.92313i 0.309017 0.951057i −1.00469 0.729952i 1.96511
191.4 −0.809017 0.587785i 0.931534 2.86697i 0.309017 + 0.951057i 1.62603 1.18138i −2.43879 + 1.77188i 0.750517 + 2.30985i 0.309017 0.951057i −4.92469 3.57799i −2.00989
229.1 0.309017 0.951057i −1.86150 + 1.35246i −0.809017 0.587785i 0.811598 + 2.49784i 0.711029 + 2.18832i −2.21832 1.61170i −0.809017 + 0.587785i 0.708982 2.18202i 2.62639
229.2 0.309017 0.951057i −0.738622 + 0.536641i −0.809017 0.587785i −0.549198 1.69026i 0.282129 + 0.868303i 4.08354 + 2.96686i −0.809017 + 0.587785i −0.669471 + 2.06042i −1.77724
229.3 0.309017 0.951057i −0.128416 + 0.0932996i −0.809017 0.587785i 0.950820 + 2.92632i 0.0490505 + 0.150962i −1.09378 0.794679i −0.809017 + 0.587785i −0.919265 + 2.82921i 3.07692
229.4 0.309017 0.951057i 2.72854 1.98240i −0.809017 0.587785i −0.213220 0.656224i −1.04221 3.20759i −0.389470 0.282966i −0.809017 + 0.587785i 2.58796 7.96492i −0.689994
267.1 −0.809017 + 0.587785i −0.897497 2.76221i 0.309017 0.951057i 2.41247 + 1.75276i 2.34968 + 1.70714i −0.863737 + 2.65831i 0.309017 + 0.951057i −4.39726 + 3.19480i −2.98197
267.2 −0.809017 + 0.587785i −0.670482 2.06353i 0.309017 0.951057i −1.44869 1.05254i 1.75534 + 1.27533i 1.45655 4.48281i 0.309017 + 0.951057i −1.38156 + 1.00376i 1.79068
267.3 −0.809017 + 0.587785i 0.636445 + 1.95878i 0.309017 0.951057i −1.58981 1.15506i −1.66623 1.21059i 1.27470 3.92313i 0.309017 + 0.951057i −1.00469 + 0.729952i 1.96511
267.4 −0.809017 + 0.587785i 0.931534 + 2.86697i 0.309017 0.951057i 1.62603 + 1.18138i −2.43879 1.77188i 0.750517 2.30985i 0.309017 + 0.951057i −4.92469 + 3.57799i −2.00989
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 115.4
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 418.2.f.f 16
11.c even 5 1 inner 418.2.f.f 16
11.c even 5 1 4598.2.a.ca 8
11.d odd 10 1 4598.2.a.bx 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.f.f 16 1.a even 1 1 trivial
418.2.f.f 16 11.c even 5 1 inner
4598.2.a.bx 8 11.d odd 10 1
4598.2.a.ca 8 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 16 T_{3}^{14} + 6 T_{3}^{13} + 162 T_{3}^{12} + 232 T_{3}^{11} + 1524 T_{3}^{10} + 3660 T_{3}^{9} + 14448 T_{3}^{8} + 29096 T_{3}^{7} + 73576 T_{3}^{6} + 111184 T_{3}^{5} + 184560 T_{3}^{4} + 195936 T_{3}^{3} + \cdots + 1936 \) acting on \(S_{2}^{\mathrm{new}}(418, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 16 T^{14} + 6 T^{13} + \cdots + 1936 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + 18 T^{14} + \cdots + 43681 \) Copy content Toggle raw display
$7$ \( T^{16} - 6 T^{15} + 50 T^{14} + \cdots + 1413721 \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} - 26 T^{14} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} + 14 T^{15} + \cdots + 1111822336 \) Copy content Toggle raw display
$17$ \( T^{16} - 8 T^{15} + 86 T^{14} + \cdots + 18139081 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 12 T^{7} - 34 T^{6} + 740 T^{5} + \cdots - 96331)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 22 T^{15} + 308 T^{14} + \cdots + 3610000 \) Copy content Toggle raw display
$31$ \( T^{16} + 24 T^{15} + 292 T^{14} + \cdots + 25361296 \) Copy content Toggle raw display
$37$ \( T^{16} - 6 T^{15} + \cdots + 657143694736 \) Copy content Toggle raw display
$41$ \( T^{16} + 14 T^{15} + \cdots + 7265159596816 \) Copy content Toggle raw display
$43$ \( (T^{8} - 28 T^{7} + 222 T^{6} + \cdots + 51869)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 38 T^{15} + \cdots + 3590526241 \) Copy content Toggle raw display
$53$ \( T^{16} - 22 T^{15} + \cdots + 665784193936 \) Copy content Toggle raw display
$59$ \( T^{16} + 28 T^{15} + \cdots + 12168376422400 \) Copy content Toggle raw display
$61$ \( T^{16} - 12 T^{15} + \cdots + 22290664247401 \) Copy content Toggle raw display
$67$ \( (T^{8} + 22 T^{7} - 62 T^{6} - 3864 T^{5} + \cdots + 10564)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} - 16 T^{15} + \cdots + 2859602753296 \) Copy content Toggle raw display
$73$ \( T^{16} - 12 T^{15} + \cdots + 1841383864576 \) Copy content Toggle raw display
$79$ \( T^{16} - 48 T^{15} + \cdots + 52063440250000 \) Copy content Toggle raw display
$83$ \( T^{16} - 14 T^{15} + \cdots + 2289526801 \) Copy content Toggle raw display
$89$ \( (T^{8} + 28 T^{7} + 36 T^{6} + \cdots - 31782080)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 385924296241936 \) Copy content Toggle raw display
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