Properties

Label 5915.2.a.bg
Level $5915$
Weight $2$
Character orbit 5915.a
Self dual yes
Analytic conductor $47.232$
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] = [5915,2,Mod(1,5915)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5915, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5915.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5915 = 5 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5915.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: \(47.2315127956\)
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: \( x^{15} - 7 x^{14} + 3 x^{13} + 76 x^{12} - 125 x^{11} - 299 x^{10} + 716 x^{9} + 496 x^{8} - 1774 x^{7} - 244 x^{6} + 2131 x^{5} - 143 x^{4} - 1178 x^{3} + 98 x^{2} + 243 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} + 3 x^{13} + 76 x^{12} - 125 x^{11} - 299 x^{10} + 716 x^{9} + 496 x^{8} - 1774 x^{7} - 244 x^{6} + 2131 x^{5} - 143 x^{4} - 1178 x^{3} + 98 x^{2} + 243 x - 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 127 \nu^{14} - 2610 \nu^{13} + 16010 \nu^{12} + 14805 \nu^{11} - 171874 \nu^{10} + 26473 \nu^{9} + 719619 \nu^{8} - 322414 \nu^{7} - 1400841 \nu^{6} + 730934 \nu^{5} + \cdots - 3742 ) / 7111 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 370 \nu^{14} + 45 \nu^{13} - 13272 \nu^{12} + 16051 \nu^{11} + 125812 \nu^{10} - 198461 \nu^{9} - 516430 \nu^{8} + 913560 \nu^{7} + 1004367 \nu^{6} - 1935760 \nu^{5} + \cdots + 17229 ) / 7111 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 661 \nu^{14} + 5397 \nu^{13} - 4580 \nu^{12} - 58445 \nu^{11} + 109494 \nu^{10} + 234987 \nu^{9} - 559760 \nu^{8} - 433763 \nu^{7} + 1211301 \nu^{6} + 362810 \nu^{5} + \cdots - 27147 ) / 7111 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 770 \nu^{14} + 2597 \nu^{13} + 8209 \nu^{12} - 26869 \nu^{11} - 37348 \nu^{10} + 86484 \nu^{9} + 105907 \nu^{8} - 38687 \nu^{7} - 201526 \nu^{6} - 260997 \nu^{5} + \cdots + 661 ) / 7111 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 788 \nu^{14} + 2787 \nu^{13} + 11430 \nu^{12} - 43640 \nu^{11} - 62380 \nu^{10} + 261460 \nu^{9} + 159859 \nu^{8} - 756177 \nu^{7} - 189540 \nu^{6} + 1093744 \nu^{5} + \cdots - 9556 ) / 7111 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 81 \nu^{14} + 308 \nu^{13} + 1093 \nu^{12} - 4633 \nu^{11} - 5979 \nu^{10} + 28156 \nu^{9} + 16873 \nu^{8} - 87831 \nu^{7} - 24831 \nu^{6} + 146342 \nu^{5} + 15670 \nu^{4} + \cdots + 792 ) / 547 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1431 \nu^{14} - 7994 \nu^{13} - 3629 \nu^{12} + 85314 \nu^{11} - 72146 \nu^{10} - 321471 \nu^{9} + 453853 \nu^{8} + 472450 \nu^{7} - 1009775 \nu^{6} - 101813 \nu^{5} + \cdots + 33597 ) / 7111 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2684 \nu^{14} - 13319 \nu^{13} - 18862 \nu^{12} + 165174 \nu^{11} - 3096 \nu^{10} - 805730 \nu^{9} + 322434 \nu^{8} + 1962582 \nu^{7} - 963950 \nu^{6} - 2480967 \nu^{5} + \cdots + 28578 ) / 7111 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3295 \nu^{14} - 16608 \nu^{13} - 21137 \nu^{12} + 199156 \nu^{11} - 16200 \nu^{10} - 934716 \nu^{9} + 380219 \nu^{8} + 2193709 \nu^{7} - 943358 \nu^{6} - 2717489 \nu^{5} + \cdots - 25593 ) / 7111 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3372 \nu^{14} - 19001 \nu^{13} - 8447 \nu^{12} + 202554 \nu^{11} - 160374 \nu^{10} - 784078 \nu^{9} + 979041 \nu^{8} + 1303725 \nu^{7} - 2100787 \nu^{6} - 755064 \nu^{5} + \cdots - 3615 ) / 7111 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3472 \nu^{14} + 16106 \nu^{13} + 30292 \nu^{12} - 208814 \nu^{11} - 59284 \nu^{10} + 1067190 \nu^{9} - 162575 \nu^{8} - 2718759 \nu^{7} + 774410 \nu^{6} + \cdots - 16801 ) / 7111 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3646 \nu^{14} - 20313 \nu^{13} - 16392 \nu^{12} + 245306 \nu^{11} - 118289 \nu^{10} - 1146798 \nu^{9} + 956574 \nu^{8} + 2616976 \nu^{7} - 2350400 \nu^{6} - 2998458 \nu^{5} + \cdots - 7692 ) / 7111 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3688 \nu^{14} + 18386 \nu^{13} + 26278 \nu^{12} - 232291 \nu^{11} + 10104 \nu^{10} + 1147378 \nu^{9} - 510691 \nu^{8} - 2802550 \nu^{7} + 1522677 \nu^{6} + \cdots + 9926 ) / 7111 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - \beta_{6} + \beta_{5} + 2\beta_{4} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{9} + 2\beta_{8} - 7\beta_{6} + 2\beta_{5} + 8\beta_{4} + 6\beta_{2} + 9\beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 3 \beta_{12} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 9 \beta_{8} - \beta_{7} - 11 \beta_{6} + 10 \beta_{5} + 19 \beta_{4} + 8 \beta_{2} + 31 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{14} + \beta_{13} + 14 \beta_{12} + 3 \beta_{11} - 2 \beta_{10} + 11 \beta_{9} + 21 \beta_{8} - 3 \beta_{7} - 48 \beta_{6} + 25 \beta_{5} + 59 \beta_{4} - \beta_{3} + 33 \beta_{2} + 71 \beta _1 + 69 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{14} + 9 \beta_{13} + 41 \beta_{12} + 16 \beta_{11} - 13 \beta_{10} + 25 \beta_{9} + 70 \beta_{8} - 15 \beta_{7} - 101 \beta_{6} + 88 \beta_{5} + 151 \beta_{4} - 2 \beta_{3} + 55 \beta_{2} + 214 \beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 17 \beta_{14} + 12 \beta_{13} + 142 \beta_{12} + 49 \beta_{11} - 32 \beta_{10} + 94 \beta_{9} + 176 \beta_{8} - 42 \beta_{7} - 353 \beta_{6} + 239 \beta_{5} + 435 \beta_{4} - 12 \beta_{3} + 187 \beta_{2} + 544 \beta _1 + 425 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 54 \beta_{14} + 62 \beta_{13} + 413 \beta_{12} + 183 \beta_{11} - 130 \beta_{10} + 230 \beta_{9} + 529 \beta_{8} - 148 \beta_{7} - 867 \beta_{6} + 746 \beta_{5} + 1155 \beta_{4} - 25 \beta_{3} + 364 \beta_{2} + 1558 \beta _1 + 961 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 212 \beta_{14} + 100 \beta_{13} + 1289 \beta_{12} + 557 \beta_{11} - 351 \beta_{10} + 739 \beta_{9} + 1385 \beta_{8} - 403 \beta_{7} - 2720 \beta_{6} + 2088 \beta_{5} + 3242 \beta_{4} - 94 \beta_{3} + 1105 \beta_{2} + \cdots + 2876 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 675 \beta_{14} + 388 \beta_{13} + 3728 \beta_{12} + 1827 \beta_{11} - 1189 \beta_{10} + 1888 \beta_{9} + 3981 \beta_{8} - 1236 \beta_{7} - 7205 \beta_{6} + 6197 \beta_{5} + 8776 \beta_{4} - 191 \beta_{3} + \cdots + 7177 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2311 \beta_{14} + 699 \beta_{13} + 11114 \beta_{12} + 5492 \beta_{11} - 3340 \beta_{10} + 5605 \beta_{9} + 10664 \beta_{8} - 3315 \beta_{7} - 21450 \beta_{6} + 17518 \beta_{5} + 24413 \beta_{4} + \cdots + 20576 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7242 \beta_{14} + 2265 \beta_{13} + 31972 \beta_{12} + 16979 \beta_{11} - 10415 \beta_{10} + 14678 \beta_{9} + 30018 \beta_{8} - 9481 \beta_{7} - 58874 \beta_{6} + 50781 \beta_{5} + 66802 \beta_{4} + \cdots + 54043 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 23217 \beta_{14} + 4263 \beta_{13} + 93168 \beta_{12} + 50412 \beta_{11} - 29707 \beta_{10} + 41764 \beta_{9} + 81480 \beta_{8} - 25163 \beta_{7} - 170833 \beta_{6} + 143892 \beta_{5} + \cdots + 152129 \) Copy content Toggle raw display

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
2.80810
2.61892
2.22367
2.20957
1.59538
1.40829
1.22968
0.781134
0.00410875
−0.645397
−0.716596
−1.32265
−1.53964
−1.66883
−1.98574
−2.80810 −0.0546445 5.88541 1.00000 0.153447 1.00000 −10.9106 −2.99701 −2.80810
1.2 −2.61892 1.96388 4.85873 1.00000 −5.14324 1.00000 −7.48678 0.856817 −2.61892
1.3 −2.22367 −1.20005 2.94470 1.00000 2.66851 1.00000 −2.10070 −1.55988 −2.22367
1.4 −2.20957 −2.85802 2.88221 1.00000 6.31500 1.00000 −1.94931 5.16827 −2.20957
1.5 −1.59538 0.359098 0.545252 1.00000 −0.572899 1.00000 2.32088 −2.87105 −1.59538
1.6 −1.40829 0.899760 −0.0167169 1.00000 −1.26712 1.00000 2.84012 −2.19043 −1.40829
1.7 −1.22968 1.58287 −0.487878 1.00000 −1.94643 1.00000 3.05930 −0.494522 −1.22968
1.8 −0.781134 −2.25768 −1.38983 1.00000 1.76355 1.00000 2.64791 2.09710 −0.781134
1.9 −0.00410875 −1.70259 −1.99998 1.00000 0.00699554 1.00000 0.0164349 −0.101172 −0.00410875
1.10 0.645397 1.87906 −1.58346 1.00000 1.21274 1.00000 −2.31276 0.530881 0.645397
1.11 0.716596 −1.88253 −1.48649 1.00000 −1.34901 1.00000 −2.49840 0.543904 0.716596
1.12 1.32265 −3.12304 −0.250587 1.00000 −4.13070 1.00000 −2.97675 6.75336 1.32265
1.13 1.53964 0.440775 0.370492 1.00000 0.678635 1.00000 −2.50886 −2.80572 1.53964
1.14 1.66883 0.980122 0.784996 1.00000 1.63566 1.00000 −2.02764 −2.03936 1.66883
1.15 1.98574 −2.02702 1.94316 1.00000 −4.02514 1.00000 −0.112869 1.10882 1.98574
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(-1\)
\(13\) \(1\)

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 5915.2.a.bg 15
13.b even 2 1 5915.2.a.bl yes 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5915.2.a.bg 15 1.a even 1 1 trivial
5915.2.a.bl yes 15 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5915))\):

\( T_{2}^{15} + 7 T_{2}^{14} + 3 T_{2}^{13} - 76 T_{2}^{12} - 125 T_{2}^{11} + 299 T_{2}^{10} + 716 T_{2}^{9} - 496 T_{2}^{8} - 1774 T_{2}^{7} + 244 T_{2}^{6} + 2131 T_{2}^{5} + 143 T_{2}^{4} - 1178 T_{2}^{3} - 98 T_{2}^{2} + 243 T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{15} + 7 T_{3}^{14} + T_{3}^{13} - 85 T_{3}^{12} - 109 T_{3}^{11} + 403 T_{3}^{10} + 670 T_{3}^{9} - 985 T_{3}^{8} - 1678 T_{3}^{7} + 1445 T_{3}^{6} + 1877 T_{3}^{5} - 1374 T_{3}^{4} - 720 T_{3}^{3} + 644 T_{3}^{2} - 91 T_{3} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 7 T^{14} + 3 T^{13} - 76 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{15} + 7 T^{14} + T^{13} - 85 T^{12} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( (T - 1)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + 6 T^{14} - 56 T^{13} - 327 T^{12} + \cdots - 301 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + 6 T^{14} - 90 T^{13} + \cdots - 290471 \) Copy content Toggle raw display
$19$ \( T^{15} + 14 T^{14} - 68 T^{13} + \cdots + 93029 \) Copy content Toggle raw display
$23$ \( T^{15} - 6 T^{14} - 181 T^{13} + \cdots - 29470063 \) Copy content Toggle raw display
$29$ \( T^{15} + 26 T^{14} + \cdots + 858020792 \) Copy content Toggle raw display
$31$ \( T^{15} + 12 T^{14} - 147 T^{13} + \cdots - 89602681 \) Copy content Toggle raw display
$37$ \( T^{15} + 60 T^{14} + \cdots + 18011347423 \) Copy content Toggle raw display
$41$ \( T^{15} + 5 T^{14} - 265 T^{13} + \cdots + 10415231 \) Copy content Toggle raw display
$43$ \( T^{15} - T^{14} - 338 T^{13} + \cdots + 144731819 \) Copy content Toggle raw display
$47$ \( T^{15} + 20 T^{14} + \cdots + 462695285891 \) Copy content Toggle raw display
$53$ \( T^{15} + 4 T^{14} + \cdots + 38517536471 \) Copy content Toggle raw display
$59$ \( T^{15} + 15 T^{14} + \cdots + 240270942379 \) Copy content Toggle raw display
$61$ \( T^{15} + 23 T^{14} + \cdots - 843233089 \) Copy content Toggle raw display
$67$ \( T^{15} + 30 T^{14} + \cdots - 198712892657 \) Copy content Toggle raw display
$71$ \( T^{15} - 5 T^{14} + \cdots + 464571140683 \) Copy content Toggle raw display
$73$ \( T^{15} + 44 T^{14} + \cdots + 2271226384393 \) Copy content Toggle raw display
$79$ \( T^{15} + 6 T^{14} + \cdots + 831428300311 \) Copy content Toggle raw display
$83$ \( T^{15} + 21 T^{14} + \cdots - 27832739776927 \) Copy content Toggle raw display
$89$ \( T^{15} + 18 T^{14} + \cdots - 102553865303 \) Copy content Toggle raw display
$97$ \( T^{15} + 15 T^{14} + \cdots - 2142154453837 \) Copy content Toggle raw display
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