Properties

Label 1155.2.c.f
Level $1155$
Weight $2$
Character orbit 1155.c
Analytic conductor $9.223$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1155,2,Mod(694,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1155.694");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.c (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 33 x^{18} + 456 x^{16} + 3426 x^{14} + 15210 x^{12} + 40640 x^{10} + 63865 x^{8} + 55281 x^{6} + 22984 x^{4} + 3428 x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} + \beta_{7} q^{5} - \beta_{3} q^{6} - \beta_{11} q^{7} + (\beta_{18} + \beta_{17} - \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{11} q^{3} + (\beta_{2} - 1) q^{4} + \beta_{7} q^{5} - \beta_{3} q^{6} - \beta_{11} q^{7} + (\beta_{18} + \beta_{17} - \beta_1) q^{8} - q^{9} + ( - \beta_{13} + \beta_{9}) q^{10} + q^{11} + ( - \beta_{17} + \beta_{11}) q^{12} - \beta_{16} q^{13} - \beta_{3} q^{14} - \beta_{8} q^{15} + (\beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} + 1) q^{16} + ( - \beta_{19} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_1) q^{17} - \beta_1 q^{18} + (\beta_{19} - \beta_{18} + \beta_{16} - \beta_{15} + \beta_{14} + \beta_{9} - 2 \beta_{8} - \beta_{5} - \beta_1 - 2) q^{19} + ( - \beta_{19} + \beta_{18} - \beta_{16} + \beta_{13} - 2 \beta_{11} - \beta_{9} + 2 \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{20}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 26 q^{4} - 2 q^{5} + 6 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 26 q^{4} - 2 q^{5} + 6 q^{6} - 20 q^{9} + 2 q^{10} + 20 q^{11} + 6 q^{14} - 2 q^{15} + 38 q^{16} - 34 q^{19} + 4 q^{20} - 20 q^{21} - 18 q^{24} - 4 q^{25} + 28 q^{26} - 10 q^{29} - 6 q^{30} + 12 q^{31} - 32 q^{34} - 2 q^{35} + 26 q^{36} - 2 q^{40} + 52 q^{41} - 26 q^{44} + 2 q^{45} + 40 q^{46} - 20 q^{49} + 6 q^{50} + 6 q^{51} - 6 q^{54} - 2 q^{55} - 18 q^{56} - 14 q^{59} - 28 q^{60} + 78 q^{61} - 26 q^{64} - 4 q^{65} + 6 q^{66} - 18 q^{69} - 6 q^{70} - 8 q^{74} - 8 q^{75} + 84 q^{76} - 52 q^{79} - 40 q^{80} + 20 q^{81} + 26 q^{84} - 24 q^{85} + 4 q^{86} - 10 q^{89} - 2 q^{90} - 96 q^{94} - 30 q^{95} + 62 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 33 x^{18} + 456 x^{16} + 3426 x^{14} + 15210 x^{12} + 40640 x^{10} + 63865 x^{8} + 55281 x^{6} + 22984 x^{4} + 3428 x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 932 \nu^{18} - 29599 \nu^{16} - 388661 \nu^{14} - 2720792 \nu^{12} - 10903803 \nu^{10} - 24948019 \nu^{8} - 30691092 \nu^{6} - 17834990 \nu^{4} + \cdots - 10112 ) / 171356 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21445 \nu^{19} - 701609 \nu^{17} - 9437765 \nu^{15} - 67184034 \nu^{13} - 271650511 \nu^{11} - 625930321 \nu^{9} - 801263181 \nu^{7} + \cdots - 52934572 \nu ) / 3255764 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17223 \nu^{19} - 39658 \nu^{18} - 539302 \nu^{17} - 1304803 \nu^{16} - 7239623 \nu^{15} - 17627746 \nu^{14} - 55067041 \nu^{13} - 125789839 \nu^{12} + \cdots - 12649508 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 28357 \nu^{18} + 807316 \nu^{16} + 9309882 \nu^{14} + 55840854 \nu^{12} + 186103148 \nu^{10} + 341926714 \nu^{8} + 328984885 \nu^{6} + 163946632 \nu^{4} + \cdots + 1859432 ) / 3255764 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11377 \nu^{19} - 77206 \nu^{18} + 402088 \nu^{17} - 2366551 \nu^{16} + 6106853 \nu^{15} - 29875558 \nu^{14} + 52047565 \nu^{13} - 200454623 \nu^{12} + \cdots - 12324404 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 40638 \nu^{19} + 2923 \nu^{18} - 1359828 \nu^{17} + 68607 \nu^{16} - 19061802 \nu^{15} + 566385 \nu^{14} - 145349694 \nu^{13} + 1509738 \nu^{12} + \cdots - 4197764 ) / 3255764 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 40638 \nu^{19} + 2923 \nu^{18} + 1359828 \nu^{17} + 68607 \nu^{16} + 19061802 \nu^{15} + 566385 \nu^{14} + 145349694 \nu^{13} + 1509738 \nu^{12} + 655436372 \nu^{11} + \cdots - 4197764 ) / 3255764 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11377 \nu^{19} - 77206 \nu^{18} - 402088 \nu^{17} - 2366551 \nu^{16} - 6106853 \nu^{15} - 29875558 \nu^{14} - 52047565 \nu^{13} - 200454623 \nu^{12} + \cdots - 12324404 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2528 \nu^{19} + 82492 \nu^{17} + 1123169 \nu^{15} + 8272267 \nu^{13} + 35730088 \nu^{11} + 91834117 \nu^{9} + 136502701 \nu^{7} + 109059276 \nu^{5} + \cdots + 5043820 \nu ) / 171356 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3503 \nu^{18} - 111664 \nu^{16} - 1471065 \nu^{14} - 10332043 \nu^{12} - 41590554 \nu^{10} - 95909007 \nu^{8} - 119768558 \nu^{6} - 71376026 \nu^{4} + \cdots + 467236 ) / 171356 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 158482 \nu^{19} - 20801 \nu^{18} + 5086207 \nu^{17} - 781727 \nu^{16} + 67999162 \nu^{15} - 11937193 \nu^{14} + 491154011 \nu^{13} - 95841406 \nu^{12} + \cdots - 8350020 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 158482 \nu^{19} - 20801 \nu^{18} - 5086207 \nu^{17} - 781727 \nu^{16} - 67999162 \nu^{15} - 11937193 \nu^{14} - 491154011 \nu^{13} - 95841406 \nu^{12} + \cdots - 8350020 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 192793 \nu^{19} + 21163 \nu^{18} - 6452077 \nu^{17} + 777136 \nu^{16} - 90636213 \nu^{15} + 12008451 \nu^{14} - 694260760 \nu^{13} + 101264279 \nu^{12} + \cdots + 12865832 ) / 6511528 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 110783 \nu^{19} - 3732736 \nu^{17} - 52853791 \nu^{15} - 408736717 \nu^{13} - 1878395562 \nu^{11} - 5231616809 \nu^{9} - 8634683824 \nu^{7} + \cdots - 500823852 \nu ) / 3255764 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 6652 \nu^{19} + 217877 \nu^{17} + 2980846 \nu^{15} + 22096009 \nu^{13} + 96286461 \nu^{11} + 250554332 \nu^{9} + 378817011 \nu^{7} + 309342838 \nu^{5} + \cdots + 15121348 \nu ) / 171356 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 6652 \nu^{19} - 217877 \nu^{17} - 2980846 \nu^{15} - 22096009 \nu^{13} - 96286461 \nu^{11} - 250554332 \nu^{9} - 378817011 \nu^{7} - 309342838 \nu^{5} + \cdots - 14264568 \nu ) / 171356 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 8594 \nu^{19} - 286263 \nu^{17} - 3995054 \nu^{15} - 30327549 \nu^{13} - 136061779 \nu^{11} - 367228218 \nu^{9} - 581983849 \nu^{7} - 506157206 \nu^{5} + \cdots - 32364064 \nu ) / 171356 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{18} + \beta_{17} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{3} - 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{18} - 10\beta_{17} + \beta_{16} + 2\beta_{11} + 2\beta_{9} - 2\beta_{8} + \beta_{4} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{19} + 3 \beta_{18} - \beta_{16} - \beta_{15} - 3 \beta_{14} - 10 \beta_{12} + 9 \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + 9 \beta_{7} + 10 \beta_{6} + 3 \beta_{5} + 12 \beta_{3} + 48 \beta_{2} - \beta _1 - 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3 \beta_{19} + 71 \beta_{18} + 83 \beta_{17} - 13 \beta_{16} - \beta_{14} + \beta_{13} - 32 \beta_{11} + \beta_{10} - 29 \beta_{9} + 29 \beta_{8} - \beta_{7} - 14 \beta_{4} - 165 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14 \beta_{19} - 42 \beta_{18} + 14 \beta_{16} + 14 \beta_{15} + 44 \beta_{14} + 2 \beta_{13} + 81 \beta_{12} - 67 \beta_{10} + 37 \beta_{9} - 33 \beta_{8} - 67 \beta_{7} - 81 \beta_{6} - 42 \beta_{5} - 113 \beta_{3} - 331 \beta_{2} + 14 \beta _1 + 513 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 49 \beta_{19} - 545 \beta_{18} - 648 \beta_{17} + 130 \beta_{16} + 19 \beta_{14} - 19 \beta_{13} + 350 \beta_{11} - 23 \beta_{10} + 300 \beta_{9} - 300 \beta_{8} + 23 \beta_{7} + 141 \beta_{4} + 1006 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 140 \beta_{19} + 421 \beta_{18} - 140 \beta_{16} - 141 \beta_{15} - 464 \beta_{14} - 43 \beta_{13} - 610 \beta_{12} + 478 \beta_{10} - 330 \beta_{9} + 371 \beta_{8} + 478 \beta_{7} + 626 \beta_{6} + 421 \beta_{5} + 960 \beta_{3} + \cdots - 3235 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 555 \beta_{19} + 4157 \beta_{18} + 4919 \beta_{17} - 1181 \beta_{16} - 239 \beta_{14} + 239 \beta_{13} - 3268 \beta_{11} + 317 \beta_{10} - 2732 \beta_{9} + 2732 \beta_{8} - 317 \beta_{7} - 1258 \beta_{4} + \cdots - 6301 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1235 \beta_{19} - 3728 \beta_{18} + 1235 \beta_{16} + 1258 \beta_{15} + 4307 \beta_{14} + 579 \beta_{13} + 4441 \beta_{12} - 3386 \beta_{10} + 2617 \beta_{9} - 3581 \beta_{8} - 3386 \beta_{7} - 4783 \beta_{6} + \cdots + 21117 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5418 \beta_{19} - 31673 \beta_{18} - 36800 \beta_{17} + 10201 \beta_{16} + 2508 \beta_{14} - 2508 \beta_{13} + 28062 \beta_{11} - 3504 \beta_{10} + 23371 \beta_{9} - 23371 \beta_{8} + 3504 \beta_{7} + \cdots + 40376 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 10278 \beta_{19} + 31141 \beta_{18} - 10278 \beta_{16} - 10585 \beta_{15} - 37478 \beta_{14} - 6337 \beta_{13} - 31766 \beta_{12} + 24072 \beta_{10} - 19711 \beta_{9} + 31986 \beta_{8} + \cdots - 141781 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 48908 \beta_{19} + 241331 \beta_{18} + 273167 \beta_{17} - 85364 \beta_{16} - 23814 \beta_{14} + 23814 \beta_{13} - 229332 \beta_{11} + 34382 \beta_{10} - 192948 \beta_{9} + 192948 \beta_{8} + \cdots - 263895 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 82968 \beta_{19} - 252102 \beta_{18} + 82968 \beta_{16} + 86166 \beta_{15} + 313974 \beta_{14} + 61872 \beta_{13} + 225061 \beta_{12} - 172285 \beta_{10} + 144998 \beta_{9} - 273040 \beta_{8} + \cdots + 974063 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 421078 \beta_{19} - 1838815 \beta_{18} - 2019572 \beta_{17} + 698865 \beta_{16} + 212606 \beta_{14} - 212606 \beta_{13} + 1816978 \beta_{11} - 313816 \beta_{10} + 1557768 \beta_{9} + \cdots + 1754916 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 658089 \beta_{19} + 2003251 \beta_{18} - 658089 \beta_{16} - 687073 \beta_{15} - 2566315 \beta_{14} - 563064 \beta_{13} - 1586702 \beta_{12} + 1241773 \beta_{10} - 1055751 \beta_{9} + \cdots - 6817561 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 3513815 \beta_{19} + 14006619 \beta_{18} + 14904295 \beta_{17} - 5630417 \beta_{16} - 1822329 \beta_{14} + 1822329 \beta_{13} - 14109884 \beta_{11} + 2731717 \beta_{10} + \cdots - 11848557 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
2.74150i
2.55795i
2.32980i
2.28323i
1.93467i
1.40171i
1.28094i
0.792050i
0.568210i
0.0342944i
0.0342944i
0.568210i
0.792050i
1.28094i
1.40171i
1.93467i
2.28323i
2.32980i
2.55795i
2.74150i
2.74150i 1.00000i −5.51584 −0.739422 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 + 2.02713i
694.2 2.55795i 1.00000i −4.54312 0.0738813 + 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 0.188985i
694.3 2.32980i 1.00000i −3.42795 1.29334 + 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 3.01322i
694.4 2.28323i 1.00000i −3.21315 −2.23363 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 + 5.09990i
694.5 1.93467i 1.00000i −1.74296 1.72871 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 3.34450i
694.6 1.40171i 1.00000i 0.0352144 −2.22487 + 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 + 3.11862i
694.7 1.28094i 1.00000i 0.359182 2.05729 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 2.63527i
694.8 0.792050i 1.00000i 1.37266 −0.199975 + 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 + 0.158390i
694.9 0.568210i 1.00000i 1.67714 1.08688 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 0.617578i
694.10 0.0342944i 1.00000i 1.99882 −1.84220 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 + 0.0631773i
694.11 0.0342944i 1.00000i 1.99882 −1.84220 + 1.26739i 0.0342944 1.00000i 0.137137i −1.00000 −0.0434646 0.0631773i
694.12 0.568210i 1.00000i 1.67714 1.08688 + 1.95415i −0.568210 1.00000i 2.08939i −1.00000 −1.11037 + 0.617578i
694.13 0.792050i 1.00000i 1.37266 −0.199975 2.22711i 0.792050 1.00000i 2.67131i −1.00000 1.76398 0.158390i
694.14 1.28094i 1.00000i 0.359182 2.05729 + 0.876110i −1.28094 1.00000i 3.02198i −1.00000 −1.12225 + 2.63527i
694.15 1.40171i 1.00000i 0.0352144 −2.22487 0.223473i 1.40171 1.00000i 2.85278i −1.00000 0.313244 3.11862i
694.16 1.93467i 1.00000i −1.74296 1.72871 + 1.41829i 1.93467 1.00000i 0.497289i −1.00000 −2.74393 + 3.34450i
694.17 2.28323i 1.00000i −3.21315 −2.23363 + 0.104355i −2.28323 1.00000i 2.76990i −1.00000 −0.238267 5.09990i
694.18 2.32980i 1.00000i −3.42795 1.29334 1.82408i −2.32980 1.00000i 3.32682i −1.00000 4.24974 + 3.01322i
694.19 2.55795i 1.00000i −4.54312 0.0738813 2.23485i 2.55795 1.00000i 6.50518i −1.00000 5.71663 + 0.188985i
694.20 2.74150i 1.00000i −5.51584 −0.739422 + 2.11027i 2.74150 1.00000i 9.63869i −1.00000 −5.78532 2.02713i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.c.f 20
5.b even 2 1 inner 1155.2.c.f 20
5.c odd 4 1 5775.2.a.cn 10
5.c odd 4 1 5775.2.a.co 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.f 20 1.a even 1 1 trivial
1155.2.c.f 20 5.b even 2 1 inner
5775.2.a.cn 10 5.c odd 4 1
5775.2.a.co 10 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{20} + 33 T_{2}^{18} + 456 T_{2}^{16} + 3426 T_{2}^{14} + 15210 T_{2}^{12} + 40640 T_{2}^{10} + 63865 T_{2}^{8} + 55281 T_{2}^{6} + 22984 T_{2}^{4} + 3428 T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{20} + 152 T_{13}^{18} + 9402 T_{13}^{16} + 310352 T_{13}^{14} + 5968441 T_{13}^{12} + 67891612 T_{13}^{10} + 437647316 T_{13}^{8} + 1403976868 T_{13}^{6} + 1590982304 T_{13}^{4} + \cdots + 58491904 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 33 T^{18} + 456 T^{16} + 3426 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{20} + 2 T^{19} + 4 T^{18} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{10} \) Copy content Toggle raw display
$11$ \( (T - 1)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 152 T^{18} + \cdots + 58491904 \) Copy content Toggle raw display
$17$ \( T^{20} + 191 T^{18} + \cdots + 173817856 \) Copy content Toggle raw display
$19$ \( (T^{10} + 17 T^{9} + 29 T^{8} + \cdots - 1905888)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + 311 T^{18} + \cdots + 3560678424576 \) Copy content Toggle raw display
$29$ \( (T^{10} + 5 T^{9} - 173 T^{8} + \cdots - 2190256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} - 6 T^{9} - 236 T^{8} + \cdots - 117243392)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 464 T^{18} + \cdots + 53257768075264 \) Copy content Toggle raw display
$41$ \( (T^{10} - 26 T^{9} + 134 T^{8} + \cdots - 6464512)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 102729414344704 \) Copy content Toggle raw display
$47$ \( T^{20} + 496 T^{18} + \cdots + 23123460096 \) Copy content Toggle raw display
$53$ \( T^{20} + 671 T^{18} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{10} + 7 T^{9} - 299 T^{8} + \cdots + 104226256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} - 39 T^{9} + 341 T^{8} + \cdots - 5992448)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 480 T^{18} + \cdots + 3899771846656 \) Copy content Toggle raw display
$71$ \( (T^{10} - 434 T^{8} + 778 T^{7} + \cdots - 851968)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + 620 T^{18} + \cdots + 3406823424 \) Copy content Toggle raw display
$79$ \( (T^{10} + 26 T^{9} - 68 T^{8} + \cdots + 40370176)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + 1095 T^{18} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{10} + 5 T^{9} - 467 T^{8} + \cdots - 928874496)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 1195 T^{18} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
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