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Properties

Label 4334.2.a.d
Level $4334$
Weight $2$
Character orbit 4334.a
Self dual yes
Analytic conductor $34.607$
Analytic rank $1$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} - 6021 x^{9} - 8879 x^{8} + 13530 x^{7} + 11150 x^{6} - 15676 x^{5} - 8037 x^{4} + \cdots - 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 7 x^{16} - 7 x^{15} + 137 x^{14} - 98 x^{13} - 1048 x^{12} + 1313 x^{11} + 4085 x^{10} - 6021 x^{9} - 8879 x^{8} + 13530 x^{7} + 11150 x^{6} - 15676 x^{5} - 8037 x^{4} + \cdots - 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 917115519 \nu^{16} - 5212700619 \nu^{15} - 13267205103 \nu^{14} + 108089258473 \nu^{13} + 52303712884 \nu^{12} - 890592629248 \nu^{11} + \cdots + 275653971112 ) / 100018832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2118122577 \nu^{16} - 11979762949 \nu^{15} - 30898783617 \nu^{14} + 248436835479 \nu^{13} + 126180705388 \nu^{12} - 2046326225264 \nu^{11} + \cdots + 623351680584 ) / 200037664 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1190055619 \nu^{16} - 6765107123 \nu^{15} - 17236648943 \nu^{14} + 140423501145 \nu^{13} + 68116872216 \nu^{12} - 1158620144112 \nu^{11} + \cdots + 363827038920 ) / 100018832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3747780633 \nu^{16} - 21227672221 \nu^{15} - 54617142073 \nu^{14} + 440646194303 \nu^{13} + 221531905052 \nu^{12} - 3634702369104 \nu^{11} + \cdots + 1126215265736 ) / 200037664 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9954510309 \nu^{16} + 56495783281 \nu^{15} + 144505682861 \nu^{14} - 1172270244043 \nu^{13} - 577280311748 \nu^{12} + \cdots - 3005730994568 ) / 400075328 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2707538429 \nu^{16} - 15377659701 \nu^{15} - 39287625049 \nu^{14} + 319260268927 \nu^{13} + 156408278384 \nu^{12} - 2634675505232 \nu^{11} + \cdots + 826000500232 ) / 100018832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2837634319 \nu^{16} - 16108690395 \nu^{15} - 41184831247 \nu^{14} + 334303865353 \nu^{13} + 164304960564 \nu^{12} - 2757232092864 \nu^{11} + \cdots + 858999681160 ) / 100018832 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 728303375 \nu^{16} + 4131017823 \nu^{15} + 10592530399 \nu^{14} - 85773093645 \nu^{13} - 42576437488 \nu^{12} + 707832557708 \nu^{11} + \cdots - 220825196084 ) / 25004708 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11821377509 \nu^{16} + 67109376065 \nu^{15} + 171605413149 \nu^{14} - 1392946997659 \nu^{13} - 684830985492 \nu^{12} + \cdots - 3585384415688 ) / 400075328 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6485269861 \nu^{16} - 36814938225 \nu^{15} - 94066526173 \nu^{14} + 763681201211 \nu^{13} + 374698521364 \nu^{12} - 6294804448000 \nu^{11} + \cdots + 1949477027720 ) / 200037664 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 6607866089 \nu^{16} + 37544828365 \nu^{15} + 95697548569 \nu^{14} - 778806039487 \nu^{13} - 378710389548 \nu^{12} + \cdots - 1994818528520 ) / 200037664 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14135303493 \nu^{16} - 80212263505 \nu^{15} - 205268091629 \nu^{14} + 1664519690699 \nu^{13} + 821012038244 \nu^{12} + \cdots + 4274328485320 ) / 400075328 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 3708772897 \nu^{16} + 21035949917 \nu^{15} + 53921864145 \nu^{14} - 436648931271 \nu^{13} - 216606080428 \nu^{12} + \cdots - 1121904694504 ) / 100018832 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2361132485 \nu^{16} + 13408798839 \nu^{15} + 34223642511 \nu^{14} - 278142921505 \nu^{13} - 135934710226 \nu^{12} + \cdots - 711627008544 ) / 50009416 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 2\beta_{9} - \beta_{4} + 2\beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 2 \beta_{14} - \beta_{13} + \beta_{11} - 2 \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 11 \beta_{2} + 17 \beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} + \beta_{15} + 13 \beta_{14} - 4 \beta_{13} + 12 \beta_{12} + 15 \beta_{11} - 13 \beta_{10} - 29 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + \beta_{5} - 10 \beta_{4} - 5 \beta_{3} + 30 \beta_{2} + 67 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5 \beta_{16} + 15 \beta_{15} + 37 \beta_{14} - 25 \beta_{13} + 11 \beta_{12} + 31 \beta_{11} - 40 \beta_{10} - 89 \beta_{9} + 16 \beta_{8} - 18 \beta_{7} - 16 \beta_{6} + 15 \beta_{5} + 8 \beta_{4} - 29 \beta_{3} + 125 \beta_{2} + 214 \beta _1 + 42 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 35 \beta_{16} + 32 \beta_{15} + 166 \beta_{14} - 94 \beta_{13} + 134 \beta_{12} + 198 \beta_{11} - 172 \beta_{10} - 378 \beta_{9} + 55 \beta_{8} - 56 \beta_{7} - 20 \beta_{6} + 34 \beta_{5} - 76 \beta_{4} - 111 \beta_{3} + 380 \beta_{2} + \cdots - 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 138 \beta_{16} + 211 \beta_{15} + 538 \beta_{14} - 420 \beta_{13} + 241 \beta_{12} + 534 \beta_{11} - 596 \beta_{10} - 1249 \beta_{9} + 211 \beta_{8} - 290 \beta_{7} - 215 \beta_{6} + 213 \beta_{5} + 57 \beta_{4} - 495 \beta_{3} + \cdots - 95 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 672 \beta_{16} + 609 \beta_{15} + 2112 \beta_{14} - 1530 \beta_{13} + 1543 \beta_{12} + 2502 \beta_{11} - 2261 \beta_{10} - 4775 \beta_{9} + 749 \beta_{8} - 1022 \beta_{7} - 470 \beta_{6} + 640 \beta_{5} - 519 \beta_{4} + \cdots - 1857 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2509 \beta_{16} + 2914 \beta_{15} + 7164 \beta_{14} - 6056 \beta_{13} + 3804 \beta_{12} + 7601 \beta_{11} - 8014 \beta_{10} - 16229 \beta_{9} + 2609 \beta_{8} - 4347 \beta_{7} - 2773 \beta_{6} + 2958 \beta_{5} + \cdots - 5269 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10525 \beta_{16} + 9480 \beta_{15} + 26534 \beta_{14} - 21687 \beta_{13} + 18321 \beta_{12} + 30895 \beta_{11} - 29035 \beta_{10} - 59267 \beta_{9} + 9234 \beta_{8} - 15727 \beta_{7} - 7678 \beta_{6} + \cdots - 28609 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 38461 \beta_{16} + 39190 \beta_{15} + 91457 \beta_{14} - 81103 \beta_{13} + 53112 \beta_{12} + 99568 \beta_{11} - 102797 \beta_{10} - 203738 \beta_{9} + 31252 \beta_{8} - 61343 \beta_{7} - 35157 \beta_{6} + \cdots - 93861 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 149613 \beta_{16} + 133692 \beta_{15} + 329175 \beta_{14} - 287489 \beta_{13} + 221332 \beta_{12} + 376455 \beta_{11} - 365136 \beta_{10} - 727165 \beta_{9} + 109437 \beta_{8} - 221752 \beta_{7} + \cdots - 398621 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 539818 \beta_{16} + 513491 \beta_{15} + 1140848 \beta_{14} - 1043078 \beta_{13} + 698651 \beta_{12} + 1251606 \beta_{11} - 1285859 \beta_{10} - 2511541 \beta_{9} + 368192 \beta_{8} + \cdots - 1362389 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2013096 \beta_{16} + 1783860 \beta_{15} + 4043297 \beta_{14} - 3672837 \beta_{13} + 2692678 \beta_{12} + 4551428 \beta_{11} - 4522535 \beta_{10} - 8852491 \beta_{9} + 1277286 \beta_{8} + \cdots - 5256266 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 7196125 \beta_{16} + 6583561 \beta_{15} + 14035487 \beta_{14} - 13096144 \beta_{13} + 8890642 \beta_{12} + 15392025 \beta_{11} - 15848240 \beta_{10} - 30633520 \beta_{9} + \cdots - 18209145 \) 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
3.45762
3.00096
2.69188
2.63673
1.67442
1.32994
1.32368
1.31815
0.577346
−0.216237
−0.673774
−1.02051
−1.23460
−1.29570
−1.87628
−2.06015
−2.63348
1.00000 −3.45762 1.00000 −2.31501 −3.45762 2.74192 1.00000 8.95516 −2.31501
1.2 1.00000 −3.00096 1.00000 4.03995 −3.00096 −2.78062 1.00000 6.00574 4.03995
1.3 1.00000 −2.69188 1.00000 1.31987 −2.69188 2.55905 1.00000 4.24624 1.31987
1.4 1.00000 −2.63673 1.00000 −3.59467 −2.63673 −1.69075 1.00000 3.95235 −3.59467
1.5 1.00000 −1.67442 1.00000 −0.572222 −1.67442 −0.229301 1.00000 −0.196306 −0.572222
1.6 1.00000 −1.32994 1.00000 2.68374 −1.32994 −0.271741 1.00000 −1.23125 2.68374
1.7 1.00000 −1.32368 1.00000 −0.966345 −1.32368 1.91821 1.00000 −1.24787 −0.966345
1.8 1.00000 −1.31815 1.00000 1.46276 −1.31815 −4.53584 1.00000 −1.26249 1.46276
1.9 1.00000 −0.577346 1.00000 0.543375 −0.577346 4.26235 1.00000 −2.66667 0.543375
1.10 1.00000 0.216237 1.00000 0.737633 0.216237 0.342001 1.00000 −2.95324 0.737633
1.11 1.00000 0.673774 1.00000 2.41156 0.673774 −3.55111 1.00000 −2.54603 2.41156
1.12 1.00000 1.02051 1.00000 −4.21122 1.02051 2.47064 1.00000 −1.95855 −4.21122
1.13 1.00000 1.23460 1.00000 −1.34398 1.23460 −1.90837 1.00000 −1.47576 −1.34398
1.14 1.00000 1.29570 1.00000 1.60714 1.29570 −2.75710 1.00000 −1.32117 1.60714
1.15 1.00000 1.87628 1.00000 −2.40821 1.87628 1.70446 1.00000 0.520431 −2.40821
1.16 1.00000 2.06015 1.00000 −2.29448 2.06015 0.0148240 1.00000 1.24422 −2.29448
1.17 1.00000 2.63348 1.00000 −1.09989 2.63348 −3.28860 1.00000 3.93521 −1.09989
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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.d ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4334.2.a.d ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} + 7 T_{3}^{16} - 7 T_{3}^{15} - 137 T_{3}^{14} - 98 T_{3}^{13} + 1048 T_{3}^{12} + 1313 T_{3}^{11} - 4085 T_{3}^{10} - 6021 T_{3}^{9} + 8879 T_{3}^{8} + 13530 T_{3}^{7} - 11150 T_{3}^{6} - 15676 T_{3}^{5} + 8037 T_{3}^{4} + \cdots + 400 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4334))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + 7 T^{16} - 7 T^{15} - 137 T^{14} + \cdots + 400 \) Copy content Toggle raw display
$5$ \( T^{17} + 4 T^{16} - 36 T^{15} + \cdots + 5147 \) Copy content Toggle raw display
$7$ \( T^{17} + 5 T^{16} - 43 T^{15} - 228 T^{14} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( (T + 1)^{17} \) Copy content Toggle raw display
$13$ \( T^{17} + 18 T^{16} + 63 T^{15} + \cdots + 772532 \) Copy content Toggle raw display
$17$ \( T^{17} + 10 T^{16} + \cdots + 140740076 \) Copy content Toggle raw display
$19$ \( T^{17} + 31 T^{16} + 284 T^{15} + \cdots - 89994944 \) Copy content Toggle raw display
$23$ \( T^{17} + 6 T^{16} - 179 T^{15} + \cdots - 726928 \) Copy content Toggle raw display
$29$ \( T^{17} + 16 T^{16} + \cdots - 1614622436 \) Copy content Toggle raw display
$31$ \( T^{17} + 30 T^{16} + 190 T^{15} + \cdots - 22065715 \) Copy content Toggle raw display
$37$ \( T^{17} + 23 T^{16} + \cdots + 1740996688 \) Copy content Toggle raw display
$41$ \( T^{17} + 7 T^{16} + \cdots - 4085969468 \) Copy content Toggle raw display
$43$ \( T^{17} + 23 T^{16} - 65 T^{15} + \cdots - 92908 \) Copy content Toggle raw display
$47$ \( T^{17} + 19 T^{16} - 122 T^{15} + \cdots - 10194992 \) Copy content Toggle raw display
$53$ \( T^{17} + 30 T^{16} + \cdots + 365178260368 \) Copy content Toggle raw display
$59$ \( T^{17} + 28 T^{16} - 14 T^{15} + \cdots - 7289225 \) Copy content Toggle raw display
$61$ \( T^{17} + 19 T^{16} + \cdots - 1699473732784 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 386291825862400 \) Copy content Toggle raw display
$71$ \( T^{17} - T^{16} + \cdots + 51038766188513 \) Copy content Toggle raw display
$73$ \( T^{17} + 10 T^{16} + \cdots - 1774258011932 \) Copy content Toggle raw display
$79$ \( T^{17} + 27 T^{16} + \cdots - 9468083562500 \) Copy content Toggle raw display
$83$ \( T^{17} + 40 T^{16} + \cdots + 25755751965776 \) Copy content Toggle raw display
$89$ \( T^{17} + 17 T^{16} + \cdots + 1574641256000 \) Copy content Toggle raw display
$97$ \( T^{17} + 34 T^{16} + \cdots + 14372578999 \) Copy content Toggle raw display
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