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Properties

Label 4334.2.a.b
Level $4334$
Weight $2$
Character orbit 4334.a
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4334,2,Mod(1,4334)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4334, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4334.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4334 = 2 \cdot 11 \cdot 197 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4334.a (trivial)

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: yes
Analytic conductor: \(34.6071642360\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} - 2966 x^{6} + 1621 x^{5} + 2328 x^{4} - 1160 x^{3} - 589 x^{2} + 364 x - 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{13} - 1) q^{7} + q^{8} + (\beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + ( - \beta_{2} - 1) q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{13} - 1) q^{7} + q^{8} + (\beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{9} + ( - \beta_{2} - 1) q^{10} + q^{11} + (\beta_1 - 1) q^{12} + (\beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{2} - 1) q^{13} + ( - \beta_{13} - 1) q^{14} + ( - \beta_{13} - \beta_{10} - \beta_{8} - \beta_{5} - \beta_{4} - \beta_1) q^{15} + q^{16} + ( - \beta_{14} + \beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{17} + (\beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{18} + (\beta_{14} + \beta_{12} + \beta_{10} + \beta_{7} + \beta_1 - 1) q^{19} + ( - \beta_{2} - 1) q^{20} + ( - \beta_{14} + \beta_{13} - \beta_{11} - \beta_{9} + \beta_{6} - \beta_{4} - \beta_1 - 1) q^{21} + q^{22} + (\beta_{14} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{4} - 2 \beta_{3} - 2) q^{23} + (\beta_1 - 1) q^{24} + (\beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{6} + \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} + \beta_{5} + \beta_{2} - 1) q^{26} + (\beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + 2 \beta_1 - 3) q^{27} + ( - \beta_{13} - 1) q^{28} + (\beta_{12} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{13} - \beta_{10} - \beta_{8} - \beta_{5} - \beta_{4} - \beta_1) q^{30} + ( - \beta_{12} - 2 \beta_{11} - 2 \beta_{9} - \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} - 3) q^{31} + q^{32} + (\beta_1 - 1) q^{33} + ( - \beta_{14} + \beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{34} + (\beta_{14} + 2 \beta_{13} - 2 \beta_{11} + \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{35}+ \cdots + (\beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 15 q^{2} - 9 q^{3} + 15 q^{4} - 11 q^{5} - 9 q^{6} - 11 q^{7} + 15 q^{8} + 10 q^{9} - 11 q^{10} + 15 q^{11} - 9 q^{12} - 21 q^{13} - 11 q^{14} - 2 q^{15} + 15 q^{16} - 4 q^{17} + 10 q^{18} - 22 q^{19} - 11 q^{20} - 13 q^{21} + 15 q^{22} - 16 q^{23} - 9 q^{24} + 6 q^{25} - 21 q^{26} - 21 q^{27} - 11 q^{28} - 8 q^{29} - 2 q^{30} - 33 q^{31} + 15 q^{32} - 9 q^{33} - 4 q^{34} - 2 q^{35} + 10 q^{36} - q^{37} - 22 q^{38} + q^{39} - 11 q^{40} - 10 q^{41} - 13 q^{42} - 8 q^{43} + 15 q^{44} - 10 q^{45} - 16 q^{46} - 31 q^{47} - 9 q^{48} + 2 q^{49} + 6 q^{50} + 2 q^{51} - 21 q^{52} - 18 q^{53} - 21 q^{54} - 11 q^{55} - 11 q^{56} + 16 q^{57} - 8 q^{58} - 37 q^{59} - 2 q^{60} - 31 q^{61} - 33 q^{62} - 20 q^{63} + 15 q^{64} - 13 q^{65} - 9 q^{66} + q^{67} - 4 q^{68} - 25 q^{69} - 2 q^{70} - 28 q^{71} + 10 q^{72} - 20 q^{73} - q^{74} - 9 q^{75} - 22 q^{76} - 11 q^{77} + q^{78} - 6 q^{79} - 11 q^{80} + 3 q^{81} - 10 q^{82} - 15 q^{83} - 13 q^{84} - 31 q^{85} - 8 q^{86} - 16 q^{87} + 15 q^{88} - 17 q^{89} - 10 q^{90} - 21 q^{91} - 16 q^{92} + 10 q^{93} - 31 q^{94} - 3 q^{95} - 9 q^{96} - 9 q^{97} + 2 q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 6 x^{14} - 8 x^{13} + 94 x^{12} - 13 x^{11} - 582 x^{10} + 295 x^{9} + 1814 x^{8} - 1056 x^{7} - 2966 x^{6} + 1621 x^{5} + 2328 x^{4} - 1160 x^{3} - 589 x^{2} + 364 x - 45 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 38967 \nu^{14} - 368011 \nu^{13} + 425109 \nu^{12} + 4789161 \nu^{11} - 10763764 \nu^{10} - 23566082 \nu^{9} + 64364521 \nu^{8} + 57985681 \nu^{7} + \cdots + 1206969 ) / 634962 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54577 \nu^{14} + 385559 \nu^{13} + 83031 \nu^{12} - 5421409 \nu^{11} + 5331816 \nu^{10} + 29719730 \nu^{9} - 37764223 \nu^{8} - 83109287 \nu^{7} + \cdots - 6667041 ) / 634962 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 98545 \nu^{14} - 474903 \nu^{13} - 1355241 \nu^{12} + 7723627 \nu^{11} + 7561874 \nu^{10} - 48396178 \nu^{9} - 23976157 \nu^{8} + 144520227 \nu^{7} + \cdots + 9614679 ) / 634962 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80572 \nu^{14} + 418510 \nu^{13} + 889566 \nu^{12} - 6314419 \nu^{11} - 3409969 \nu^{10} + 36652260 \nu^{9} + 6862302 \nu^{8} - 101964871 \nu^{7} + \cdots - 4212711 ) / 317481 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 86087 \nu^{14} + 387355 \nu^{13} + 1322100 \nu^{12} - 6517250 \nu^{11} - 8512356 \nu^{10} + 42707227 \nu^{9} + 30673450 \nu^{8} - 136201942 \nu^{7} + \cdots - 8052534 ) / 317481 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95161 \nu^{14} - 359889 \nu^{13} - 1717950 \nu^{12} + 5892832 \nu^{11} + 13864988 \nu^{10} - 36106162 \nu^{9} - 63023512 \nu^{8} + 98685486 \nu^{7} + \cdots - 2783562 ) / 317481 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 122660 \nu^{14} - 479233 \nu^{13} - 2151627 \nu^{12} + 7881734 \nu^{11} + 16736406 \nu^{10} - 48853678 \nu^{9} - 73469164 \nu^{8} + 137700412 \nu^{7} + \cdots - 327249 ) / 317481 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 315587 \nu^{14} - 1461863 \nu^{13} - 4408503 \nu^{12} + 22996733 \nu^{11} + 26403326 \nu^{10} - 138607224 \nu^{9} - 96384309 \nu^{8} + 394052573 \nu^{7} + \cdots + 6171147 ) / 634962 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 171261 \nu^{14} - 781019 \nu^{13} - 2530686 \nu^{12} + 12684189 \nu^{11} + 16167988 \nu^{10} - 79521244 \nu^{9} - 61404685 \nu^{8} + 238297865 \nu^{7} + \cdots + 9127485 ) / 317481 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 229297 \nu^{14} + 991991 \nu^{13} + 3660657 \nu^{12} - 16438801 \nu^{11} - 25498530 \nu^{10} + 104958869 \nu^{9} + 102423653 \nu^{8} - 318963461 \nu^{7} + \cdots - 12825138 ) / 317481 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 556079 \nu^{14} - 2558707 \nu^{13} - 7971699 \nu^{12} + 40845473 \nu^{11} + 49263864 \nu^{10} - 250923130 \nu^{9} - 183244921 \nu^{8} + 732657025 \nu^{7} + \cdots + 23916789 ) / 634962 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 698419 \nu^{14} + 3263243 \nu^{13} + 9641889 \nu^{12} - 51272989 \nu^{11} - 56674248 \nu^{10} + 308276840 \nu^{9} + 202934273 \nu^{8} + \cdots - 18878145 ) / 634962 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1054983 \nu^{14} + 4782031 \nu^{13} + 15465027 \nu^{12} - 76630953 \nu^{11} - 98195870 \nu^{10} + 471467996 \nu^{9} + 371832821 \nu^{8} + \cdots - 38939397 ) / 634962 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{9} + \beta_{8} + \beta_{4} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + 2 \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + 3 \beta_{4} + 3 \beta_{2} + 5 \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{14} + 12 \beta_{13} + 5 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 12 \beta_{9} + 15 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + 15 \beta_{4} - 2 \beta_{3} + 13 \beta_{2} + 6 \beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{14} + 36 \beta_{13} + 22 \beta_{12} - 16 \beta_{11} - 17 \beta_{10} + 36 \beta_{9} + 56 \beta_{8} - 6 \beta_{7} + 5 \beta_{6} + 7 \beta_{5} + 52 \beta_{4} - 6 \beta_{3} + 49 \beta_{2} + 42 \beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60 \beta_{14} + 155 \beta_{13} + 91 \beta_{12} - 52 \beta_{11} - 61 \beta_{10} + 154 \beta_{9} + 228 \beta_{8} - 29 \beta_{7} + 27 \beta_{6} + 42 \beta_{5} + 212 \beta_{4} - 40 \beta_{3} + 188 \beta_{2} + 109 \beta _1 + 252 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 258 \beta_{14} + 538 \beta_{13} + 355 \beta_{12} - 251 \beta_{11} - 261 \beta_{10} + 538 \beta_{9} + 871 \beta_{8} - 134 \beta_{7} + 117 \beta_{6} + 156 \beta_{5} + 776 \beta_{4} - 141 \beta_{3} + 718 \beta_{2} + 489 \beta _1 + 746 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 955 \beta_{14} + 2110 \beta_{13} + 1385 \beta_{12} - 922 \beta_{11} - 981 \beta_{10} + 2102 \beta_{9} + 3368 \beta_{8} - 548 \beta_{7} + 501 \beta_{6} + 689 \beta_{5} + 2996 \beta_{4} - 646 \beta_{3} + 2712 \beta_{2} + \cdots + 2872 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3780 \beta_{14} + 7710 \beta_{13} + 5276 \beta_{12} - 3810 \beta_{11} - 3868 \beta_{10} + 7724 \beta_{9} + 12810 \beta_{8} - 2257 \beta_{7} + 2017 \beta_{6} + 2604 \beta_{5} + 11172 \beta_{4} - 2398 \beta_{3} + \cdots + 9758 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14182 \beta_{14} + 29379 \beta_{13} + 20143 \beta_{12} - 14367 \beta_{11} - 14643 \beta_{10} + 29391 \beta_{9} + 48717 \beta_{8} - 8812 \beta_{7} + 8025 \beta_{6} + 10429 \beta_{5} + 42395 \beta_{4} + \cdots + 36509 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 54451 \beta_{14} + 109320 \beta_{13} + 76175 \beta_{12} - 56135 \beta_{11} - 56123 \beta_{10} + 109726 \beta_{9} + 184344 \beta_{8} - 34528 \beta_{7} + 31194 \beta_{6} + 39498 \beta_{5} + \cdots + 131448 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 204941 \beta_{14} + 412755 \beta_{13} + 288315 \beta_{12} - 212317 \beta_{11} - 212177 \beta_{10} + 414160 \beta_{9} + 696636 \beta_{8} - 132264 \beta_{7} + 120411 \beta_{6} + 152459 \beta_{5} + \cdots + 490297 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 777391 \beta_{14} + 1545912 \beta_{13} + 1087084 \beta_{12} - 811500 \beta_{11} - 804392 \beta_{10} + 1553778 \beta_{9} + 2627849 \beta_{8} - 506868 \beta_{7} + 460189 \beta_{6} + \cdots + 1808835 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2926659 \beta_{14} + 5819694 \beta_{13} + 4099451 \beta_{12} - 3064525 \beta_{11} - 3034548 \beta_{10} + 5849580 \beta_{9} + 9903740 \beta_{8} - 1923895 \beta_{7} + 1752098 \beta_{6} + \cdots + 6761899 \) Copy content Toggle raw display

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.22455
−1.92605
−1.55193
−1.31874
−1.15333
−0.956062
0.216655
0.311474
0.594746
1.49640
1.50020
1.97073
2.63464
2.64609
3.75972
1.00000 −3.22455 1.00000 1.46351 −3.22455 −3.27885 1.00000 7.39772 1.46351
1.2 1.00000 −2.92605 1.00000 1.09129 −2.92605 3.40685 1.00000 5.56174 1.09129
1.3 1.00000 −2.55193 1.00000 −3.59659 −2.55193 −3.65108 1.00000 3.51234 −3.59659
1.4 1.00000 −2.31874 1.00000 0.745559 −2.31874 −1.91118 1.00000 2.37657 0.745559
1.5 1.00000 −2.15333 1.00000 −3.80220 −2.15333 3.88018 1.00000 1.63685 −3.80220
1.6 1.00000 −1.95606 1.00000 −1.56251 −1.95606 1.47305 1.00000 0.826177 −1.56251
1.7 1.00000 −0.783345 1.00000 1.44668 −0.783345 −0.609332 1.00000 −2.38637 1.44668
1.8 1.00000 −0.688526 1.00000 3.20899 −0.688526 −0.886807 1.00000 −2.52593 3.20899
1.9 1.00000 −0.405254 1.00000 −3.75919 −0.405254 −0.916969 1.00000 −2.83577 −3.75919
1.10 1.00000 0.496403 1.00000 −1.51850 0.496403 −2.22955 1.00000 −2.75358 −1.51850
1.11 1.00000 0.500205 1.00000 −1.03580 0.500205 3.13873 1.00000 −2.74980 −1.03580
1.12 1.00000 0.970726 1.00000 0.208503 0.970726 −0.143082 1.00000 −2.05769 0.208503
1.13 1.00000 1.63464 1.00000 1.52654 1.63464 −4.88584 1.00000 −0.327947 1.52654
1.14 1.00000 1.64609 1.00000 −2.29877 1.64609 −1.96736 1.00000 −0.290376 −2.29877
1.15 1.00000 2.75972 1.00000 −3.11751 2.75972 −2.41878 1.00000 4.61606 −3.11751
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(197\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4334.2.a.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.b ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} + 9 T_{3}^{14} + 13 T_{3}^{13} - 101 T_{3}^{12} - 328 T_{3}^{11} + 188 T_{3}^{10} + 1713 T_{3}^{9} + 785 T_{3}^{8} - 3491 T_{3}^{7} - 2897 T_{3}^{6} + 2876 T_{3}^{5} + 2792 T_{3}^{4} - 876 T_{3}^{3} - 911 T_{3}^{2} + \cdots + 92 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 9 T^{14} + 13 T^{13} - 101 T^{12} + \cdots + 92 \) Copy content Toggle raw display
$5$ \( T^{15} + 11 T^{14} + 20 T^{13} + \cdots - 1593 \) Copy content Toggle raw display
$7$ \( T^{15} + 11 T^{14} + 7 T^{13} + \cdots + 5139 \) Copy content Toggle raw display
$11$ \( (T - 1)^{15} \) Copy content Toggle raw display
$13$ \( T^{15} + 21 T^{14} + 123 T^{13} + \cdots - 5407092 \) Copy content Toggle raw display
$17$ \( T^{15} + 4 T^{14} - 118 T^{13} + \cdots + 6801437 \) Copy content Toggle raw display
$19$ \( T^{15} + 22 T^{14} + 95 T^{13} + \cdots + 29155276 \) Copy content Toggle raw display
$23$ \( T^{15} + 16 T^{14} + \cdots - 1535732732 \) Copy content Toggle raw display
$29$ \( T^{15} + 8 T^{14} + \cdots + 1332415089 \) Copy content Toggle raw display
$31$ \( T^{15} + 33 T^{14} + \cdots + 4002626421 \) Copy content Toggle raw display
$37$ \( T^{15} + T^{14} - 246 T^{13} + \cdots - 123232320 \) Copy content Toggle raw display
$41$ \( T^{15} + 10 T^{14} + \cdots + 916477780 \) Copy content Toggle raw display
$43$ \( T^{15} + 8 T^{14} - 248 T^{13} + \cdots + 124197232 \) Copy content Toggle raw display
$47$ \( T^{15} + 31 T^{14} + \cdots + 196235160860 \) Copy content Toggle raw display
$53$ \( T^{15} + 18 T^{14} + \cdots - 215937896796 \) Copy content Toggle raw display
$59$ \( T^{15} + 37 T^{14} + \cdots - 4362371913 \) Copy content Toggle raw display
$61$ \( T^{15} + 31 T^{14} + \cdots + 1433948561 \) Copy content Toggle raw display
$67$ \( T^{15} - T^{14} - 308 T^{13} + \cdots + 6717348 \) Copy content Toggle raw display
$71$ \( T^{15} + 28 T^{14} + \cdots + 142096755597 \) Copy content Toggle raw display
$73$ \( T^{15} + 20 T^{14} + \cdots + 1326652570743 \) Copy content Toggle raw display
$79$ \( T^{15} + 6 T^{14} + \cdots + 15958972420 \) Copy content Toggle raw display
$83$ \( T^{15} + 15 T^{14} + \cdots + 11887227124 \) Copy content Toggle raw display
$89$ \( T^{15} + 17 T^{14} + \cdots + 17608005935984 \) Copy content Toggle raw display
$97$ \( T^{15} + 9 T^{14} + \cdots - 2778253788055 \) Copy content Toggle raw display
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