Properties

Label 9405.2.a.bu
Level $9405$
Weight $2$
Character orbit 9405.a
Self dual yes
Analytic conductor $75.099$
Analytic rank $0$
Dimension $19$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 3 x^{18} - 27 x^{17} + 83 x^{16} + 294 x^{15} - 935 x^{14} - 1658 x^{13} + 5548 x^{12} + 5125 x^{11} - 18696 x^{10} - 8186 x^{9} + 35807 x^{8} + 4532 x^{7} - 36401 x^{6} + \cdots - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{8} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - q^{5} + ( - \beta_{8} + 1) q^{7} + ( - \beta_{3} - \beta_1) q^{8} + \beta_1 q^{10} - q^{11} + (\beta_{5} + 1) q^{13} + (\beta_{6} - \beta_1) q^{14} + (\beta_{4} + \beta_{2} + 2) q^{16} + (\beta_{11} - 1) q^{17} + q^{19} + ( - \beta_{2} - 1) q^{20} + \beta_1 q^{22} + ( - \beta_{15} - 1) q^{23} + q^{25} + ( - \beta_{18} + \beta_{17} - \beta_{16} + \beta_{15} + \beta_{14} + 2 \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + \cdots + 1) q^{26}+ \cdots + ( - \beta_{17} - \beta_{16} + \beta_{13} - 2 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{4} - 3 \beta_{3} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 3 q^{2} + 25 q^{4} - 19 q^{5} + 12 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 3 q^{2} + 25 q^{4} - 19 q^{5} + 12 q^{7} - 9 q^{8} + 3 q^{10} - 19 q^{11} + 13 q^{13} + 37 q^{16} - 23 q^{17} + 19 q^{19} - 25 q^{20} + 3 q^{22} - 28 q^{23} + 19 q^{25} + 6 q^{26} + 27 q^{28} - 6 q^{29} + 16 q^{31} - 16 q^{32} + 8 q^{34} - 12 q^{35} + 6 q^{37} - 3 q^{38} + 9 q^{40} - 14 q^{41} + 32 q^{43} - 25 q^{44} + 9 q^{46} - 14 q^{47} + 47 q^{49} - 3 q^{50} + 32 q^{52} - 5 q^{53} + 19 q^{55} + 43 q^{56} - 5 q^{59} + 30 q^{61} + 2 q^{62} + 55 q^{64} - 13 q^{65} + 26 q^{67} - 42 q^{68} + 7 q^{71} + 28 q^{73} + q^{74} + 25 q^{76} - 12 q^{77} + 13 q^{79} - 37 q^{80} + 8 q^{82} - 23 q^{83} + 23 q^{85} + 31 q^{86} + 9 q^{88} + 7 q^{89} + 38 q^{91} - 68 q^{92} + 28 q^{94} - 19 q^{95} + 16 q^{97} + 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 3 x^{18} - 27 x^{17} + 83 x^{16} + 294 x^{15} - 935 x^{14} - 1658 x^{13} + 5548 x^{12} + 5125 x^{11} - 18696 x^{10} - 8186 x^{9} + 35807 x^{8} + 4532 x^{7} - 36401 x^{6} + \cdots - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 7\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 47709 \nu^{18} + 68129 \nu^{17} + 1542306 \nu^{16} - 2216118 \nu^{15} - 20615678 \nu^{14} + 29842543 \nu^{13} + 146926362 \nu^{12} - 214915743 \nu^{11} + \cdots - 8410864 ) / 261194 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37737 \nu^{18} + 7006 \nu^{17} + 1065468 \nu^{16} - 128394 \nu^{15} - 12237643 \nu^{14} + 698457 \nu^{13} + 73815294 \nu^{12} - 81758 \nu^{11} - 251650134 \nu^{10} + \cdots + 287688 ) / 130597 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38200 \nu^{18} - 61221 \nu^{17} - 1208068 \nu^{16} + 1921045 \nu^{15} + 15806054 \nu^{14} - 24981239 \nu^{13} - 110361206 \nu^{12} + 174098185 \nu^{11} + \cdots + 6191487 ) / 130597 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47948 \nu^{18} - 181581 \nu^{17} - 1287590 \nu^{16} + 5045152 \nu^{15} + 13968318 \nu^{14} - 57069023 \nu^{13} - 78799327 \nu^{12} + 339830798 \nu^{11} + \cdots + 5677427 ) / 130597 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 211301 \nu^{18} - 542081 \nu^{17} - 5684608 \nu^{16} + 14777608 \nu^{15} + 61604228 \nu^{14} - 163476031 \nu^{13} - 345641896 \nu^{12} + 948864349 \nu^{11} + \cdots + 11859440 ) / 522388 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 233268 \nu^{18} - 264021 \nu^{17} - 7010593 \nu^{16} + 7842406 \nu^{15} + 86912884 \nu^{14} - 96749604 \nu^{13} - 574504533 \nu^{12} + 643292186 \nu^{11} + \cdots + 14559170 ) / 522388 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 278252 \nu^{18} + 825983 \nu^{17} + 7900915 \nu^{16} - 23469130 \nu^{15} - 91998956 \nu^{14} + 273231804 \nu^{13} + 567722275 \nu^{12} - 1688135938 \nu^{11} + \cdots - 36011222 ) / 522388 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 189239 \nu^{18} + 379429 \nu^{17} + 5484428 \nu^{16} - 10857456 \nu^{15} - 65303122 \nu^{14} + 127800027 \nu^{13} + 412839684 \nu^{12} - 802579203 \nu^{11} + \cdots - 19998376 ) / 261194 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3261 \nu^{18} + 12231 \nu^{17} + 85888 \nu^{16} - 337816 \nu^{15} - 905474 \nu^{14} + 3793347 \nu^{13} + 4890844 \nu^{12} - 22384809 \nu^{11} - 14211235 \nu^{10} + \cdots - 359084 ) / 3578 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 530077 \nu^{18} - 1386063 \nu^{17} - 14671342 \nu^{16} + 38573424 \nu^{15} + 165222968 \nu^{14} - 438037199 \nu^{13} - 976957798 \nu^{12} + 2627904017 \nu^{11} + \cdots + 50323980 ) / 522388 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 531298 \nu^{18} + 1364591 \nu^{17} + 14017573 \nu^{16} - 36729598 \nu^{15} - 147664696 \nu^{14} + 399408630 \nu^{13} + 793944393 \nu^{12} + \cdots - 27329662 ) / 522388 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 774061 \nu^{18} + 2284045 \nu^{17} + 20717236 \nu^{16} - 62668556 \nu^{15} - 223068916 \nu^{14} + 698453963 \nu^{13} + 1240898580 \nu^{12} + \cdots - 62632232 ) / 522388 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 447116 \nu^{18} - 1449593 \nu^{17} - 12047319 \nu^{16} + 40113416 \nu^{15} + 131048934 \nu^{14} - 451753020 \nu^{13} - 740340481 \nu^{12} + 2677739166 \nu^{11} + \cdots + 43990450 ) / 261194 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 250049 \nu^{18} - 591312 \nu^{17} - 6866547 \nu^{16} + 16278421 \nu^{15} + 76490592 \nu^{14} - 182454417 \nu^{13} - 445681099 \nu^{12} + 1077740403 \nu^{11} + \cdots + 17128106 ) / 130597 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 7\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{18} + \beta_{17} + \beta_{11} + \beta_{10} - \beta_{6} + \beta_{4} + 10\beta_{3} + 30\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{16} + \beta_{15} + \beta_{10} + \beta_{9} - 2\beta_{8} - \beta_{7} + \beta_{6} + 11\beta_{4} + 47\beta_{2} + 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 14 \beta_{18} + 15 \beta_{17} + \beta_{15} + \beta_{14} + 2 \beta_{13} + 13 \beta_{11} + 12 \beta_{10} + \beta_{9} - 2 \beta_{7} - 15 \beta_{6} - 2 \beta_{5} + 13 \beta_{4} + 82 \beta_{3} + \beta_{2} + 197 \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{18} + \beta_{17} - 16 \beta_{16} + 18 \beta_{15} + 3 \beta_{14} + \beta_{13} - \beta_{12} + 3 \beta_{11} + 15 \beta_{10} + 17 \beta_{9} - 30 \beta_{8} - 17 \beta_{7} + 14 \beta_{6} + 98 \beta_{4} - \beta_{3} + 320 \beta_{2} - 2 \beta _1 + 673 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 142 \beta_{18} + 160 \beta_{17} + 2 \beta_{16} + 16 \beta_{15} + 17 \beta_{14} + 33 \beta_{13} + \beta_{12} + 125 \beta_{11} + 109 \beta_{10} + 19 \beta_{9} - 3 \beta_{8} - 34 \beta_{7} - 158 \beta_{6} - 35 \beta_{5} + 125 \beta_{4} + 641 \beta_{3} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 20 \beta_{18} + 20 \beta_{17} - 178 \beta_{16} + 217 \beta_{15} + 57 \beta_{14} + 18 \beta_{13} - 18 \beta_{12} + 56 \beta_{11} + 163 \beta_{10} + 197 \beta_{9} - 320 \beta_{8} - 198 \beta_{7} + 139 \beta_{6} + 3 \beta_{5} + 820 \beta_{4} + \cdots + 4721 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1277 \beta_{18} + 1498 \beta_{17} + 40 \beta_{16} + 181 \beta_{15} + 196 \beta_{14} + 376 \beta_{13} + 19 \beta_{12} + 1082 \beta_{11} + 903 \beta_{10} + 240 \beta_{9} - 62 \beta_{8} - 395 \beta_{7} - 1456 \beta_{6} + \cdots - 121 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 257 \beta_{18} + 258 \beta_{17} - 1714 \beta_{16} + 2209 \beta_{15} + 711 \beta_{14} + 223 \beta_{13} - 214 \beta_{12} + 680 \beta_{11} + 1551 \beta_{10} + 1957 \beta_{9} - 2993 \beta_{8} - 1971 \beta_{7} + \cdots + 34014 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 10846 \beta_{18} + 13149 \beta_{17} + 514 \beta_{16} + 1789 \beta_{15} + 1930 \beta_{14} + 3688 \beta_{13} + 238 \beta_{12} + 8948 \beta_{11} + 7207 \beta_{10} + 2544 \beta_{9} - 833 \beta_{8} - 3918 \beta_{7} + \cdots - 3276 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2711 \beta_{18} + 2745 \beta_{17} - 15326 \beta_{16} + 20551 \beta_{15} + 7395 \beta_{14} + 2355 \beta_{13} - 2129 \beta_{12} + 6864 \beta_{11} + 13739 \beta_{10} + 17955 \beta_{9} - 26225 \beta_{8} + \cdots + 249750 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 89329 \beta_{18} + 111259 \beta_{17} + 5447 \beta_{16} + 16491 \beta_{15} + 17529 \beta_{14} + 33467 \beta_{13} + 2499 \beta_{12} + 72367 \beta_{11} + 56600 \beta_{10} + 24498 \beta_{9} - 9240 \beta_{8} + \cdots - 42363 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 25631 \beta_{18} + 26275 \beta_{17} - 131227 \beta_{16} + 181186 \beta_{15} + 69727 \beta_{14} + 22737 \beta_{13} - 19261 \beta_{12} + 62795 \beta_{11} + 116543 \beta_{10} + 157172 \beta_{9} + \cdots + 1859973 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 722622 \beta_{18} + 920088 \beta_{17} + 52039 \beta_{16} + 145657 \beta_{15} + 151916 \beta_{14} + 290082 \beta_{13} + 23891 \beta_{12} + 578267 \beta_{11} + 441378 \beta_{10} + 222323 \beta_{9} + \cdots - 447433 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 226505 \beta_{18} + 235679 \beta_{17} - 1093214 \beta_{16} + 1543313 \beta_{15} + 619779 \beta_{14} + 207270 \beta_{13} - 164800 \beta_{12} + 542142 \beta_{11} + 961451 \beta_{10} + \cdots + 14003623 \) 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
2.79030
2.49488
2.49176
2.43267
1.83662
1.76187
1.34299
0.782000
0.550952
0.0717974
0.0620223
−0.243897
−1.28325
−1.29032
−1.50739
−1.90182
−2.20542
−2.38335
−2.80241
−2.79030 0 5.78575 −1.00000 0 0.0343818 −10.5634 0 2.79030
1.2 −2.49488 0 4.22443 −1.00000 0 −3.95562 −5.54968 0 2.49488
1.3 −2.49176 0 4.20889 −1.00000 0 3.24031 −5.50404 0 2.49176
1.4 −2.43267 0 3.91787 −1.00000 0 3.47772 −4.66554 0 2.43267
1.5 −1.83662 0 1.37316 −1.00000 0 −2.40451 1.15127 0 1.83662
1.6 −1.76187 0 1.10419 −1.00000 0 3.77397 1.57830 0 1.76187
1.7 −1.34299 0 −0.196387 −1.00000 0 −0.573489 2.94972 0 1.34299
1.8 −0.782000 0 −1.38848 −1.00000 0 3.37314 2.64979 0 0.782000
1.9 −0.550952 0 −1.69645 −1.00000 0 −0.686188 2.03657 0 0.550952
1.10 −0.0717974 0 −1.99485 −1.00000 0 4.90896 0.286819 0 0.0717974
1.11 −0.0620223 0 −1.99615 −1.00000 0 −1.86422 0.247850 0 0.0620223
1.12 0.243897 0 −1.94051 −1.00000 0 −2.47000 −0.961081 0 −0.243897
1.13 1.28325 0 −0.353268 −1.00000 0 2.81638 −3.01983 0 −1.28325
1.14 1.29032 0 −0.335086 −1.00000 0 2.78147 −3.01300 0 −1.29032
1.15 1.50739 0 0.272224 −1.00000 0 −2.37262 −2.60443 0 −1.50739
1.16 1.90182 0 1.61692 −1.00000 0 −0.183659 −0.728558 0 −1.90182
1.17 2.20542 0 2.86389 −1.00000 0 −4.95142 1.90525 0 −2.20542
1.18 2.38335 0 3.68035 −1.00000 0 1.92398 4.00486 0 −2.38335
1.19 2.80241 0 5.85351 −1.00000 0 5.13142 10.7991 0 −2.80241
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

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 9405.2.a.bu 19
3.b odd 2 1 9405.2.a.bv yes 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9405.2.a.bu 19 1.a even 1 1 trivial
9405.2.a.bv yes 19 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9405))\):

\( T_{2}^{19} + 3 T_{2}^{18} - 27 T_{2}^{17} - 83 T_{2}^{16} + 294 T_{2}^{15} + 935 T_{2}^{14} - 1658 T_{2}^{13} - 5548 T_{2}^{12} + 5125 T_{2}^{11} + 18696 T_{2}^{10} - 8186 T_{2}^{9} - 35807 T_{2}^{8} + 4532 T_{2}^{7} + 36401 T_{2}^{6} + \cdots + 6 \) Copy content Toggle raw display
\( T_{7}^{19} - 12 T_{7}^{18} - 18 T_{7}^{17} + 712 T_{7}^{16} - 1346 T_{7}^{15} - 15360 T_{7}^{14} + 50804 T_{7}^{13} + 152838 T_{7}^{12} - 707113 T_{7}^{11} - 730492 T_{7}^{10} + 4999113 T_{7}^{9} + 1489164 T_{7}^{8} + \cdots + 69632 \) Copy content Toggle raw display
\( T_{13}^{19} - 13 T_{13}^{18} - 70 T_{13}^{17} + 1514 T_{13}^{16} - 461 T_{13}^{15} - 67431 T_{13}^{14} + 162107 T_{13}^{13} + 1407915 T_{13}^{12} - 5501304 T_{13}^{11} - 12613300 T_{13}^{10} + 79881972 T_{13}^{9} + \cdots - 374969344 \) Copy content Toggle raw display
\( T_{17}^{19} + 23 T_{17}^{18} + 62 T_{17}^{17} - 2036 T_{17}^{16} - 12177 T_{17}^{15} + 71037 T_{17}^{14} + 567803 T_{17}^{13} - 1341729 T_{17}^{12} - 12585112 T_{17}^{11} + 16763516 T_{17}^{10} + 144545392 T_{17}^{9} + \cdots - 3186432 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} + 3 T^{18} - 27 T^{17} - 83 T^{16} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( T^{19} \) Copy content Toggle raw display
$5$ \( (T + 1)^{19} \) Copy content Toggle raw display
$7$ \( T^{19} - 12 T^{18} - 18 T^{17} + \cdots + 69632 \) Copy content Toggle raw display
$11$ \( (T + 1)^{19} \) Copy content Toggle raw display
$13$ \( T^{19} - 13 T^{18} + \cdots - 374969344 \) Copy content Toggle raw display
$17$ \( T^{19} + 23 T^{18} + 62 T^{17} + \cdots - 3186432 \) Copy content Toggle raw display
$19$ \( (T - 1)^{19} \) Copy content Toggle raw display
$23$ \( T^{19} + 28 T^{18} + \cdots - 365282769408 \) Copy content Toggle raw display
$29$ \( T^{19} + 6 T^{18} + \cdots + 958684686336 \) Copy content Toggle raw display
$31$ \( T^{19} - 16 T^{18} + \cdots + 669160050688 \) Copy content Toggle raw display
$37$ \( T^{19} - 6 T^{18} + \cdots + 2372552572928 \) Copy content Toggle raw display
$41$ \( T^{19} + 14 T^{18} + \cdots - 2522320896 \) Copy content Toggle raw display
$43$ \( T^{19} - 32 T^{18} + \cdots + 1529375054848 \) Copy content Toggle raw display
$47$ \( T^{19} + 14 T^{18} + \cdots - 187487698944 \) Copy content Toggle raw display
$53$ \( T^{19} + 5 T^{18} + \cdots + 28730720256 \) Copy content Toggle raw display
$59$ \( T^{19} + 5 T^{18} + \cdots + 702613042752 \) Copy content Toggle raw display
$61$ \( T^{19} - 30 T^{18} + \cdots - 15961563136 \) Copy content Toggle raw display
$67$ \( T^{19} - 26 T^{18} + \cdots - 64136331788288 \) Copy content Toggle raw display
$71$ \( T^{19} - 7 T^{18} + \cdots - 4969451356224 \) Copy content Toggle raw display
$73$ \( T^{19} + \cdots - 528613129060352 \) Copy content Toggle raw display
$79$ \( T^{19} - 13 T^{18} + \cdots - 59\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{19} + 23 T^{18} + \cdots - 8365393993728 \) Copy content Toggle raw display
$89$ \( T^{19} - 7 T^{18} + \cdots + 29289364598016 \) Copy content Toggle raw display
$97$ \( T^{19} - 16 T^{18} + \cdots - 146370583552 \) Copy content Toggle raw display
show more
show less