Properties

Label 2009.2.a.r
Level $2009$
Weight $2$
Character orbit 2009.a
Self dual yes
Analytic conductor $16.042$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2009,2,Mod(1,2009)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2009, 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("2009.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2009 = 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2009.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: \(16.0419457661\)
Analytic rank: \(0\)
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: \( x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + \cdots - 464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 287)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{13} q^{5} + (\beta_{16} - \beta_{15} + \beta_{12} + \beta_{4}) q^{6} + ( - \beta_{15} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{14} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{2} + 1) q^{4} + \beta_{13} q^{5} + (\beta_{16} - \beta_{15} + \beta_{12} + \beta_{4}) q^{6} + ( - \beta_{15} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{14} + 2) q^{9} + (\beta_{16} - \beta_{15} + \beta_{13} + \beta_{6} + \beta_{4} + \beta_1) q^{10} + ( - \beta_{11} + 1) q^{11} + (\beta_{16} + 2 \beta_{15} - 2 \beta_{13} + 2 \beta_{12} + \beta_{11} - \beta_{8} - \beta_{7} - \beta_{4} + \beta_{3} + \cdots + \beta_1) q^{12}+ \cdots + (\beta_{16} - \beta_{15} - 2 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} - \beta_{11} + 3 \beta_{10} - \beta_{9} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} - q^{3} + 25 q^{4} + q^{5} - 2 q^{6} + 9 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} - q^{3} + 25 q^{4} + q^{5} - 2 q^{6} + 9 q^{8} + 26 q^{9} + 2 q^{10} + 15 q^{11} - 4 q^{12} + 5 q^{13} + 24 q^{15} + 33 q^{16} - 4 q^{17} + 10 q^{18} - 5 q^{19} + 26 q^{20} + 16 q^{22} + 12 q^{23} - 16 q^{24} + 24 q^{25} - 31 q^{26} + 11 q^{27} + 14 q^{29} - 33 q^{30} + 3 q^{31} + 16 q^{32} - 4 q^{33} + 24 q^{34} + 57 q^{36} + 24 q^{37} + 45 q^{39} - 36 q^{40} - 17 q^{41} + 14 q^{43} - 9 q^{44} + 21 q^{45} + 44 q^{46} - 19 q^{47} + 60 q^{48} - 4 q^{50} + 2 q^{51} + 25 q^{52} + 4 q^{53} - 68 q^{54} - 9 q^{55} - 12 q^{57} - q^{58} + 27 q^{59} + 66 q^{60} + q^{61} + 23 q^{62} + 75 q^{64} + 22 q^{65} + 16 q^{66} + 49 q^{67} - 45 q^{68} - 12 q^{69} + 40 q^{71} - 23 q^{72} + 14 q^{73} + 33 q^{74} - 27 q^{75} + 9 q^{76} - 12 q^{78} + 61 q^{79} + 82 q^{80} + 53 q^{81} - 3 q^{82} + 18 q^{83} - 13 q^{85} - 4 q^{86} + 17 q^{87} + 74 q^{88} - 18 q^{89} + 20 q^{90} + 28 q^{92} - 36 q^{93} + 5 q^{94} + 20 q^{95} - 148 q^{96} - 26 q^{97} + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 25 x^{15} + 77 x^{14} + 247 x^{13} - 790 x^{12} - 1231 x^{11} + 4173 x^{10} + 3251 x^{9} - 12183 x^{8} - 4259 x^{7} + 19567 x^{6} + 2029 x^{5} - 16136 x^{4} + \cdots - 464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1012 \nu^{16} - 2475 \nu^{15} - 26247 \nu^{14} + 61458 \nu^{13} + 275296 \nu^{12} - 601415 \nu^{11} - 1515086 \nu^{10} + 2962988 \nu^{9} + 4740015 \nu^{8} - 7769389 \nu^{7} + \cdots + 369456 ) / 10256 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1180 \nu^{16} + 2217 \nu^{15} + 31795 \nu^{14} - 55182 \nu^{13} - 346840 \nu^{12} + 539297 \nu^{11} + 1973136 \nu^{10} - 2633450 \nu^{9} - 6263143 \nu^{8} + \cdots - 376144 ) / 10256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1754 \nu^{16} + 3579 \nu^{15} + 46691 \nu^{14} - 88224 \nu^{13} - 504106 \nu^{12} + 853001 \nu^{11} + 2854300 \nu^{10} - 4121206 \nu^{9} - 9121531 \nu^{8} + \cdots - 600696 ) / 10256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1867 \nu^{16} - 3932 \nu^{15} - 49217 \nu^{14} + 95678 \nu^{13} + 526771 \nu^{12} - 909773 \nu^{11} - 2966850 \nu^{10} + 4303909 \nu^{9} + 9491036 \nu^{8} - 10671527 \nu^{7} + \cdots + 660992 ) / 10256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2143 \nu^{16} + 3966 \nu^{15} + 58881 \nu^{14} - 98920 \nu^{13} - 658959 \nu^{12} + 969895 \nu^{11} + 3876772 \nu^{10} - 4764225 \nu^{9} - 12861598 \nu^{8} + \cdots - 888088 ) / 10256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1364 \nu^{16} + 2667 \nu^{15} + 36742 \nu^{14} - 65249 \nu^{13} - 402488 \nu^{12} + 623821 \nu^{11} + 2316727 \nu^{10} - 2964258 \nu^{9} - 7522617 \nu^{8} + \cdots - 441220 ) / 5128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3287 \nu^{16} + 5231 \nu^{15} + 91227 \nu^{14} - 129349 \nu^{13} - 1033873 \nu^{12} + 1253924 \nu^{11} + 6178111 \nu^{10} - 6069903 \nu^{9} - 20897723 \nu^{8} + \cdots - 1559124 ) / 10256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{16} - 6 \nu^{15} - 80 \nu^{14} + 149 \nu^{13} + 865 \nu^{12} - 1455 \nu^{11} - 4903 \nu^{10} + 7123 \nu^{9} + 15686 \nu^{8} - 18420 \nu^{7} - 28247 \nu^{6} + 24079 \nu^{5} + 26846 \nu^{4} + \cdots + 1060 ) / 8 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3916 \nu^{16} + 7264 \nu^{15} + 106581 \nu^{14} - 180711 \nu^{13} - 1181046 \nu^{12} + 1767956 \nu^{11} + 6885565 \nu^{10} - 8673558 \nu^{9} - 22700848 \nu^{8} + \cdots - 1550588 ) / 10256 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4299 \nu^{16} + 7706 \nu^{15} + 117474 \nu^{14} - 190807 \nu^{13} - 1309169 \nu^{12} + 1855339 \nu^{11} + 7693197 \nu^{10} - 9032891 \nu^{9} - 25637738 \nu^{8} + \cdots - 1836276 ) / 10256 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5598 \nu^{16} - 10221 \nu^{15} - 151810 \nu^{14} + 252761 \nu^{13} + 1674636 \nu^{12} - 2452715 \nu^{11} - 9707837 \nu^{10} + 11899406 \nu^{9} + 31773615 \nu^{8} + \cdots + 2094940 ) / 10256 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 6218 \nu^{16} + 11375 \nu^{15} + 167462 \nu^{14} - 280147 \nu^{13} - 1831712 \nu^{12} + 2704861 \nu^{11} + 10510963 \nu^{10} - 13042958 \nu^{9} - 33991461 \nu^{8} + \cdots - 2091540 ) / 10256 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 6461 \nu^{16} - 11391 \nu^{15} - 176082 \nu^{14} + 282012 \nu^{13} + 1955341 \nu^{12} - 2742090 \nu^{11} - 11438576 \nu^{10} + 13351919 \nu^{9} + 37917107 \nu^{8} + \cdots + 2647712 ) / 10256 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 6749 \nu^{16} - 11315 \nu^{15} - 185135 \nu^{14} + 280685 \nu^{13} + 2070479 \nu^{12} - 2735598 \nu^{11} - 12198073 \nu^{10} + 13356571 \nu^{9} + 40687701 \nu^{8} + \cdots + 2865396 ) / 10256 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} - \beta_{11} + \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{9} + \beta_{3} + 8\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 10 \beta_{15} + 10 \beta_{13} - \beta_{12} - 9 \beta_{11} + 9 \beta_{8} + \beta_{7} + 8 \beta_{6} - \beta_{5} + 10 \beta_{4} - 9 \beta_{3} + 8 \beta_{2} + 29 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2 \beta_{16} + 2 \beta_{15} + 9 \beta_{12} - \beta_{11} - \beta_{10} - 10 \beta_{9} + 2 \beta_{7} - \beta_{6} - 2 \beta_{4} + 10 \beta_{3} + 57 \beta_{2} - 2 \beta _1 + 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2 \beta_{16} - 85 \beta_{15} - \beta_{14} + 85 \beta_{13} - 17 \beta_{12} - 70 \beta_{11} + \beta_{10} - \beta_{9} + 71 \beta_{8} + 15 \beta_{7} + 56 \beta_{6} - 10 \beta_{5} + 84 \beta_{4} - 70 \beta_{3} + 56 \beta_{2} + 181 \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30 \beta_{16} + 33 \beta_{15} - 4 \beta_{13} + 63 \beta_{12} - 12 \beta_{11} - 17 \beta_{10} - 79 \beta_{9} + 29 \beta_{7} - 16 \beta_{6} - 5 \beta_{5} - 36 \beta_{4} + 81 \beta_{3} + 397 \beta_{2} - 38 \beta _1 + 590 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 37 \beta_{16} - 687 \beta_{15} - 17 \beta_{14} + 694 \beta_{13} - 199 \beta_{12} - 525 \beta_{11} + 15 \beta_{10} - 14 \beta_{9} + 549 \beta_{8} + 159 \beta_{7} + 386 \beta_{6} - 72 \beta_{5} + 670 \beta_{4} - 529 \beta_{3} + \cdots + 31 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 317 \beta_{16} + 372 \beta_{15} + \beta_{14} - 71 \beta_{13} + 404 \beta_{12} - 107 \beta_{11} - 199 \beta_{10} - 580 \beta_{9} - \beta_{8} + 304 \beta_{7} - 177 \beta_{6} - 89 \beta_{5} - 433 \beta_{4} + 621 \beta_{3} + 2761 \beta_{2} + \cdots + 4053 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 449 \beta_{16} - 5431 \beta_{15} - 189 \beta_{14} + 5576 \beta_{13} - 1984 \beta_{12} - 3893 \beta_{11} + 162 \beta_{10} - 135 \beta_{9} + 4226 \beta_{8} + 1478 \beta_{7} + 2679 \beta_{6} - 447 \beta_{5} + 5232 \beta_{4} + \cdots - 95 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2913 \beta_{16} + 3597 \beta_{15} + 30 \beta_{14} - 844 \beta_{13} + 2485 \beta_{12} - 857 \beta_{11} - 1987 \beta_{10} - 4141 \beta_{9} - 35 \beta_{8} + 2816 \beta_{7} - 1690 \beta_{6} - 1050 \beta_{5} + \cdots + 28587 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4545 \beta_{16} - 42440 \beta_{15} - 1756 \beta_{14} + 44376 \beta_{13} - 18137 \beta_{12} - 28787 \beta_{11} + 1541 \beta_{10} - 1105 \beta_{9} + 32458 \beta_{8} + 12868 \beta_{7} + 18811 \beta_{6} + \cdots - 3431 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 24926 \beta_{16} + 32170 \beta_{15} + 512 \beta_{14} - 8489 \beta_{13} + 14921 \beta_{12} - 6486 \beta_{11} - 18198 \beta_{10} - 29250 \beta_{9} - 641 \beta_{8} + 24485 \beta_{7} - 14964 \beta_{6} + \cdots + 204915 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 41693 \beta_{16} - 329388 \beta_{15} - 14858 \beta_{14} + 350609 \beta_{13} - 157302 \beta_{12} - 212980 \beta_{11} + 13730 \beta_{10} - 8196 \beta_{9} + 248831 \beta_{8} + 107815 \beta_{7} + \cdots - 46076 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 204679 \beta_{16} + 275110 \beta_{15} + 6695 \beta_{14} - 78272 \beta_{13} + 88141 \beta_{12} - 47204 \beta_{11} - 158101 \beta_{10} - 205953 \beta_{9} - 8799 \beta_{8} + 204928 \beta_{7} + \cdots + 1484844 \) 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
−2.76801
−2.48154
−1.98553
−1.77896
−1.66890
−0.972044
−0.902479
−0.290774
0.381923
1.11543
1.17557
1.20954
1.75743
2.36949
2.52950
2.57424
2.73510
−2.76801 2.85049 5.66188 3.02107 −7.89019 0 −10.1361 5.12530 −8.36234
1.2 −2.48154 −1.40932 4.15802 1.01447 3.49728 0 −5.35521 −1.01382 −2.51746
1.3 −1.98553 −2.88783 1.94233 −3.91440 5.73388 0 0.114499 5.33958 7.77216
1.4 −1.77896 −1.20779 1.16468 2.53283 2.14861 0 1.48599 −1.54123 −4.50580
1.5 −1.66890 0.213548 0.785231 −1.00334 −0.356391 0 2.02733 −2.95440 1.67447
1.6 −0.972044 −1.50665 −1.05513 −2.83631 1.46453 0 2.96972 −0.730003 2.75702
1.7 −0.902479 3.32518 −1.18553 0.938362 −3.00091 0 2.87488 8.05685 −0.846852
1.8 −0.290774 1.26632 −1.91545 −3.23790 −0.368212 0 1.13851 −1.39645 0.941496
1.9 0.381923 0.350486 −1.85413 2.79025 0.133859 0 −1.47198 −2.87716 1.06566
1.10 1.11543 2.75409 −0.755825 1.34462 3.07199 0 −3.07392 4.58503 1.49982
1.11 1.17557 −2.72860 −0.618039 2.82853 −3.20766 0 −3.07768 4.44528 3.32513
1.12 1.20954 −0.150404 −0.537006 −3.32743 −0.181921 0 −3.06862 −2.97738 −4.02467
1.13 1.75743 −3.22128 1.08856 −1.92164 −5.66118 0 −1.60179 7.37665 −3.37714
1.14 2.36949 1.64632 3.61450 4.02976 3.90094 0 3.82554 −0.289637 9.54849
1.15 2.52950 2.69617 4.39838 −0.644502 6.81996 0 6.06670 4.26931 −1.63027
1.16 2.57424 −0.468252 4.62673 −2.24224 −1.20539 0 6.76185 −2.78074 −5.77207
1.17 2.73510 −2.52246 5.48079 1.62786 −6.89919 0 9.52032 3.36281 4.45235
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(41\) \(1\)

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 2009.2.a.r 17
7.b odd 2 1 2009.2.a.s 17
7.d odd 6 2 287.2.e.d 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.e.d 34 7.d odd 6 2
2009.2.a.r 17 1.a even 1 1 trivial
2009.2.a.s 17 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2009))\):

\( T_{2}^{17} - 3 T_{2}^{16} - 25 T_{2}^{15} + 77 T_{2}^{14} + 247 T_{2}^{13} - 790 T_{2}^{12} - 1231 T_{2}^{11} + 4173 T_{2}^{10} + 3251 T_{2}^{9} - 12183 T_{2}^{8} - 4259 T_{2}^{7} + 19567 T_{2}^{6} + 2029 T_{2}^{5} - 16136 T_{2}^{4} + \cdots - 464 \) Copy content Toggle raw display
\( T_{3}^{17} + T_{3}^{16} - 38 T_{3}^{15} - 40 T_{3}^{14} + 572 T_{3}^{13} + 631 T_{3}^{12} - 4316 T_{3}^{11} - 4973 T_{3}^{10} + 16946 T_{3}^{9} + 20338 T_{3}^{8} - 32352 T_{3}^{7} - 40386 T_{3}^{6} + 24596 T_{3}^{5} + 32237 T_{3}^{4} + \cdots + 127 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 3 T^{16} - 25 T^{15} + 77 T^{14} + \cdots - 464 \) Copy content Toggle raw display
$3$ \( T^{17} + T^{16} - 38 T^{15} - 40 T^{14} + \cdots + 127 \) Copy content Toggle raw display
$5$ \( T^{17} - T^{16} - 54 T^{15} + \cdots - 169001 \) Copy content Toggle raw display
$7$ \( T^{17} \) Copy content Toggle raw display
$11$ \( T^{17} - 15 T^{16} + 3 T^{15} + \cdots - 62868571 \) Copy content Toggle raw display
$13$ \( T^{17} - 5 T^{16} - 103 T^{15} + \cdots - 1087488 \) Copy content Toggle raw display
$17$ \( T^{17} + 4 T^{16} - 186 T^{15} + \cdots - 153257984 \) Copy content Toggle raw display
$19$ \( T^{17} + 5 T^{16} - 160 T^{15} + \cdots - 105370047 \) Copy content Toggle raw display
$23$ \( T^{17} - 12 T^{16} - 152 T^{15} + \cdots + 7445248 \) Copy content Toggle raw display
$29$ \( T^{17} - 14 T^{16} + \cdots + 313574400 \) Copy content Toggle raw display
$31$ \( T^{17} - 3 T^{16} - 180 T^{15} + \cdots + 430336 \) Copy content Toggle raw display
$37$ \( T^{17} - 24 T^{16} + \cdots + 8594641728 \) Copy content Toggle raw display
$41$ \( (T + 1)^{17} \) Copy content Toggle raw display
$43$ \( T^{17} - 14 T^{16} + \cdots - 9456975872 \) Copy content Toggle raw display
$47$ \( T^{17} + 19 T^{16} + \cdots - 134296840532 \) Copy content Toggle raw display
$53$ \( T^{17} - 4 T^{16} + \cdots - 351924508928 \) Copy content Toggle raw display
$59$ \( T^{17} - 27 T^{16} + \cdots - 12653127175424 \) Copy content Toggle raw display
$61$ \( T^{17} - T^{16} - 477 T^{15} + \cdots + 44785569988 \) Copy content Toggle raw display
$67$ \( T^{17} - 49 T^{16} + \cdots - 40677894128 \) Copy content Toggle raw display
$71$ \( T^{17} - 40 T^{16} + \cdots + 22487137032128 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 144806948824176 \) Copy content Toggle raw display
$79$ \( T^{17} - 61 T^{16} + \cdots - 15425861826749 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 558878703263744 \) Copy content Toggle raw display
$89$ \( T^{17} + 18 T^{16} + \cdots + 770084080128 \) Copy content Toggle raw display
$97$ \( T^{17} + 26 T^{16} + \cdots + 44\!\cdots\!04 \) Copy content Toggle raw display
show more
show less