Properties

Label 4017.2.a.f
Level $4017$
Weight $2$
Character orbit 4017.a
Self dual yes
Analytic conductor $32.076$
Analytic rank $1$
Dimension $19$
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] = [4017,2,Mod(1,4017)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4017, 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("4017.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4017.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: \(32.0759064919\)
Analytic rank: \(1\)
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: \( x^{19} - 4 x^{18} - 16 x^{17} + 77 x^{16} + 88 x^{15} - 594 x^{14} - 154 x^{13} + 2388 x^{12} - 278 x^{11} - 5460 x^{10} + 1491 x^{9} + 7285 x^{8} - 2223 x^{7} - 5579 x^{6} + 1430 x^{5} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{18}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 4 x^{18} - 16 x^{17} + 77 x^{16} + 88 x^{15} - 594 x^{14} - 154 x^{13} + 2388 x^{12} - 278 x^{11} - 5460 x^{10} + 1491 x^{9} + 7285 x^{8} - 2223 x^{7} - 5579 x^{6} + 1430 x^{5} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 94088 \nu^{18} - 626428 \nu^{17} + 3886444 \nu^{16} + 9779883 \nu^{15} - 46409280 \nu^{14} - 53503587 \nu^{13} + 230717285 \nu^{12} + 115056639 \nu^{11} + \cdots - 30277713 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 193859 \nu^{18} - 2142369 \nu^{17} - 4314 \nu^{16} + 40884782 \nu^{15} - 39656466 \nu^{14} - 310348080 \nu^{13} + 377207518 \nu^{12} + 1213875850 \nu^{11} + \cdots + 8487263 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 301002 \nu^{18} + 2086417 \nu^{17} + 830096 \nu^{16} - 36097151 \nu^{15} + 49187691 \nu^{14} + 235624707 \nu^{13} - 523591869 \nu^{12} + \cdots - 18473720 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 505639 \nu^{18} + 943097 \nu^{17} + 11786836 \nu^{16} - 20247569 \nu^{15} - 112671033 \nu^{14} + 173915817 \nu^{13} + 571733266 \nu^{12} + \cdots - 7102352 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 522452 \nu^{18} - 173763 \nu^{17} - 14520768 \nu^{16} + 6574721 \nu^{15} + 162869580 \nu^{14} - 84744672 \nu^{13} - 963497294 \nu^{12} + 519460555 \nu^{11} + \cdots - 73754683 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 683216 \nu^{18} + 1334850 \nu^{17} - 24023172 \nu^{16} - 16582591 \nu^{15} + 300102768 \nu^{14} + 37272603 \nu^{13} - 1828942592 \nu^{12} + 288782608 \nu^{11} + \cdots + 8797055 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 878636 \nu^{18} - 2477135 \nu^{17} - 17742577 \nu^{16} + 51743466 \nu^{15} + 142470306 \nu^{14} - 438788274 \nu^{13} - 579561356 \nu^{12} + 1949589303 \nu^{11} + \cdots + 9751737 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 898500 \nu^{18} + 445742 \nu^{17} + 25408444 \nu^{16} - 16785145 \nu^{15} - 284915496 \nu^{14} + 214341540 \nu^{13} + 1654523163 \nu^{12} + \cdots - 11793478 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 977686 \nu^{18} - 2109846 \nu^{17} - 19576350 \nu^{16} + 40045288 \nu^{15} + 159886860 \nu^{14} - 304111437 \nu^{13} - 701472610 \nu^{12} + 1199138972 \nu^{11} + \cdots - 3170015 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1006596 \nu^{18} + 4839121 \nu^{17} + 13408784 \nu^{16} - 90912362 \nu^{15} - 38552868 \nu^{14} + 676127358 \nu^{13} - 210763494 \nu^{12} + \cdots - 20840648 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 340719 \nu^{18} - 2041117 \nu^{17} - 2297555 \nu^{16} + 35421487 \nu^{15} - 28351011 \nu^{14} - 233045118 \nu^{13} + 371781427 \nu^{12} + 725834826 \nu^{11} + \cdots - 1711033 ) / 2775667 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1173795 \nu^{18} + 4091147 \nu^{17} + 20237422 \nu^{16} - 78194119 \nu^{15} - 131370000 \nu^{14} + 595242228 \nu^{13} + 398312019 \nu^{12} + \cdots + 33234527 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1656344 \nu^{18} + 2850571 \nu^{17} + 37818140 \nu^{16} - 60186352 \nu^{15} - 356551428 \nu^{14} + 518525517 \nu^{13} + 1803133061 \nu^{12} + \cdots + 21257525 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 2105177 \nu^{18} - 11038000 \nu^{17} - 22037027 \nu^{16} + 196981321 \nu^{15} - 25058403 \nu^{14} - 1357359756 \nu^{13} + 1150153099 \nu^{12} + \cdots + 6865267 ) / 8327001 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 855796 \nu^{18} + 3359640 \nu^{17} + 13123579 \nu^{16} - 62837734 \nu^{15} - 64697412 \nu^{14} + 464512414 \nu^{13} + 54323182 \nu^{12} + \cdots - 10034449 ) / 2775667 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 2617292 \nu^{18} + 11645805 \nu^{17} + 34882692 \nu^{16} - 210313979 \nu^{15} - 106884618 \nu^{14} + 1474350357 \nu^{13} - 434695399 \nu^{12} + \cdots - 25074710 ) / 8327001 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{17} + \beta_{14} + \beta_{12} - \beta_{11} + \beta_{9} - \beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{17} + \beta_{14} - \beta_{13} - \beta_{8} - \beta_{6} - \beta_{4} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{18} - 9 \beta_{17} - \beta_{16} + 8 \beta_{14} - 2 \beta_{13} + 9 \beta_{12} - 7 \beta_{11} + \beta_{10} + 10 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} - 10 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} + 19 \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 14 \beta_{17} - 2 \beta_{16} - \beta_{15} + 12 \beta_{14} - 11 \beta_{13} + 2 \beta_{12} + 3 \beta_{10} + \beta_{9} - 12 \beta_{8} + \beta_{7} - 12 \beta_{6} + \beta_{5} - 11 \beta_{4} + 2 \beta_{3} + 46 \beta_{2} + \beta _1 + 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 12 \beta_{18} - 72 \beta_{17} - 14 \beta_{16} + 58 \beta_{14} - 26 \beta_{13} + 67 \beta_{12} - 44 \beta_{11} + 14 \beta_{10} + 78 \beta_{9} - 25 \beta_{8} - 10 \beta_{7} - 13 \beta_{6} - \beta_{5} - 81 \beta_{4} + 69 \beta_{3} + 93 \beta_{2} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4 \beta_{18} - 138 \beta_{17} - 29 \beta_{16} - 11 \beta_{15} + 109 \beta_{14} - 98 \beta_{13} + 33 \beta_{12} - 2 \beta_{11} + 40 \beta_{10} + 21 \beta_{9} - 108 \beta_{8} + 11 \beta_{7} - 104 \beta_{6} + 11 \beta_{5} - 100 \beta_{4} + \cdots + 470 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 106 \beta_{18} - 557 \beta_{17} - 135 \beta_{16} - 2 \beta_{15} + 424 \beta_{14} - 247 \beta_{13} + 479 \beta_{12} - 280 \beta_{11} + 138 \beta_{10} + 569 \beta_{9} - 233 \beta_{8} - 74 \beta_{7} - 127 \beta_{6} - 13 \beta_{5} + \cdots + 673 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 69 \beta_{18} - 1198 \beta_{17} - 294 \beta_{16} - 90 \beta_{15} + 905 \beta_{14} - 809 \beta_{13} + 372 \beta_{12} - 47 \beta_{11} + 385 \beta_{10} + 275 \beta_{9} - 881 \beta_{8} + 88 \beta_{7} - 811 \beta_{6} + 88 \beta_{5} + \cdots + 3082 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 842 \beta_{18} - 4248 \beta_{17} - 1138 \beta_{16} - 39 \beta_{15} + 3141 \beta_{14} - 2099 \beta_{13} + 3406 \beta_{12} - 1834 \beta_{11} + 1194 \beta_{10} + 4067 \beta_{9} - 1964 \beta_{8} - 495 \beta_{7} + \cdots + 5298 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 795 \beta_{18} - 9808 \beta_{17} - 2598 \beta_{16} - 675 \beta_{15} + 7229 \beta_{14} - 6434 \beta_{13} + 3585 \beta_{12} - 655 \beta_{11} + 3293 \beta_{10} + 2928 \beta_{9} - 6887 \beta_{8} + 618 \beta_{7} + \cdots + 21211 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6394 \beta_{18} - 32191 \beta_{17} - 9047 \beta_{16} - 487 \beta_{15} + 23457 \beta_{14} - 16956 \beta_{13} + 24330 \beta_{12} - 12333 \beta_{11} + 9727 \beta_{10} + 28948 \beta_{9} - 15834 \beta_{8} + \cdots + 41335 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7734 \beta_{18} - 77881 \beta_{17} - 21490 \beta_{16} - 4932 \beta_{15} + 56619 \beta_{14} - 50140 \beta_{13} + 31827 \beta_{12} - 7254 \beta_{11} + 26683 \beta_{10} + 27859 \beta_{9} - 52759 \beta_{8} + \cdots + 150690 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 47601 \beta_{18} - 243283 \beta_{17} - 69979 \beta_{16} - 5003 \beta_{15} + 175946 \beta_{14} - 133442 \beta_{13} + 175055 \beta_{12} - 84696 \beta_{11} + 76823 \beta_{10} + 206548 \beta_{9} + \cdots + 320588 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 68849 \beta_{18} - 607884 \beta_{17} - 171603 \beta_{16} - 35856 \beta_{15} + 438532 \beta_{14} - 386211 \beta_{13} + 269172 \beta_{12} - 70882 \beta_{11} + 210265 \beta_{10} + 247615 \beta_{9} + \cdots + 1092865 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 351573 \beta_{18} - 1836887 \beta_{17} - 534255 \beta_{16} - 46235 \beta_{15} + 1323034 \beta_{14} - 1035008 \beta_{13} + 1268980 \beta_{12} - 591374 \beta_{11} + 596476 \beta_{10} + \cdots + 2475244 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 581913 \beta_{18} - 4696666 \beta_{17} - 1343260 \beta_{16} - 261250 \beta_{15} + 3373548 \beta_{14} - 2954518 \beta_{13} + 2207094 \beta_{12} - 641881 \beta_{11} + 1631484 \beta_{10} + \cdots + 8033073 \) 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
2.75283
2.17652
2.10013
2.02307
1.82121
1.44458
1.21153
0.950625
0.725598
0.116858
−0.0952190
−0.569180
−0.810100
−0.929067
−1.09685
−1.49588
−1.78092
−2.08649
−2.45923
−2.75283 1.00000 5.57808 1.28742 −2.75283 −1.78893 −9.84984 1.00000 −3.54405
1.2 −2.17652 1.00000 2.73724 −0.898923 −2.17652 0.611535 −1.60461 1.00000 1.95652
1.3 −2.10013 1.00000 2.41053 −2.56145 −2.10013 −4.38780 −0.862160 1.00000 5.37937
1.4 −2.02307 1.00000 2.09279 −0.298497 −2.02307 0.750849 −0.187724 1.00000 0.603878
1.5 −1.82121 1.00000 1.31679 3.98129 −1.82121 −0.917766 1.24426 1.00000 −7.25075
1.6 −1.44458 1.00000 0.0867978 −1.93878 −1.44458 1.10404 2.76376 1.00000 2.80071
1.7 −1.21153 1.00000 −0.532186 2.05730 −1.21153 −4.52276 3.06783 1.00000 −2.49249
1.8 −0.950625 1.00000 −1.09631 −0.0164301 −0.950625 3.21492 2.94343 1.00000 0.0156189
1.9 −0.725598 1.00000 −1.47351 −3.68550 −0.725598 −0.281235 2.52037 1.00000 2.67419
1.10 −0.116858 1.00000 −1.98634 −1.39349 −0.116858 −4.81215 0.465837 1.00000 0.162841
1.11 0.0952190 1.00000 −1.99093 2.07279 0.0952190 −1.92020 −0.380013 1.00000 0.197369
1.12 0.569180 1.00000 −1.67603 1.84267 0.569180 1.78912 −2.09233 1.00000 1.04881
1.13 0.810100 1.00000 −1.34374 −3.50747 0.810100 −0.645139 −2.70876 1.00000 −2.84140
1.14 0.929067 1.00000 −1.13683 3.14855 0.929067 −4.69509 −2.91433 1.00000 2.92521
1.15 1.09685 1.00000 −0.796918 0.842996 1.09685 0.628687 −3.06780 1.00000 0.924641
1.16 1.49588 1.00000 0.237660 −0.502630 1.49588 −1.16688 −2.63625 1.00000 −0.751875
1.17 1.78092 1.00000 1.17168 −1.00354 1.78092 −0.242679 −1.47518 1.00000 −1.78722
1.18 2.08649 1.00000 2.35342 0.146235 2.08649 −2.80287 0.737414 1.00000 0.305116
1.19 2.45923 1.00000 4.04782 −2.57255 2.45923 −2.91567 5.03608 1.00000 −6.32649
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)
\(103\) \(1\)

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 4017.2.a.f 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4017.2.a.f 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4017))\):

\( T_{2}^{19} + 4 T_{2}^{18} - 16 T_{2}^{17} - 77 T_{2}^{16} + 88 T_{2}^{15} + 594 T_{2}^{14} - 154 T_{2}^{13} - 2388 T_{2}^{12} - 278 T_{2}^{11} + 5460 T_{2}^{10} + 1491 T_{2}^{9} - 7285 T_{2}^{8} - 2223 T_{2}^{7} + 5579 T_{2}^{6} + 1430 T_{2}^{5} + \cdots - 4 \) Copy content Toggle raw display
\( T_{23}^{19} + 23 T_{23}^{18} + 26 T_{23}^{17} - 2877 T_{23}^{16} - 15622 T_{23}^{15} + 125118 T_{23}^{14} + 1002200 T_{23}^{13} - 2302565 T_{23}^{12} - 27230040 T_{23}^{11} + 18626126 T_{23}^{10} + \cdots - 8242928272 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} + 4 T^{18} - 16 T^{17} - 77 T^{16} + \cdots - 4 \) Copy content Toggle raw display
$3$ \( (T - 1)^{19} \) Copy content Toggle raw display
$5$ \( T^{19} + 3 T^{18} - 39 T^{17} - 120 T^{16} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{19} + 23 T^{18} + 201 T^{17} + \cdots + 1088 \) Copy content Toggle raw display
$11$ \( T^{19} + 15 T^{18} + 7 T^{17} + \cdots - 632432 \) Copy content Toggle raw display
$13$ \( (T - 1)^{19} \) Copy content Toggle raw display
$17$ \( T^{19} - 162 T^{17} + 150 T^{16} + \cdots - 55508 \) Copy content Toggle raw display
$19$ \( T^{19} + 32 T^{18} + 315 T^{17} + \cdots + 28718856 \) Copy content Toggle raw display
$23$ \( T^{19} + 23 T^{18} + \cdots - 8242928272 \) Copy content Toggle raw display
$29$ \( T^{19} - 4 T^{18} + \cdots - 19048836484 \) Copy content Toggle raw display
$31$ \( T^{19} + 50 T^{18} + \cdots - 46165679272 \) Copy content Toggle raw display
$37$ \( T^{19} + 38 T^{18} + \cdots - 110331716848 \) Copy content Toggle raw display
$41$ \( T^{19} + 11 T^{18} + \cdots + 9667900448 \) Copy content Toggle raw display
$43$ \( T^{19} + 17 T^{18} + \cdots - 792081856 \) Copy content Toggle raw display
$47$ \( T^{19} + 38 T^{18} + \cdots - 171387548 \) Copy content Toggle raw display
$53$ \( T^{19} + 12 T^{18} + \cdots - 44112229350416 \) Copy content Toggle raw display
$59$ \( T^{19} + 8 T^{18} + \cdots - 3018303526104 \) Copy content Toggle raw display
$61$ \( T^{19} + \cdots - 108557810930792 \) Copy content Toggle raw display
$67$ \( T^{19} + 48 T^{18} + \cdots - 2099730866432 \) Copy content Toggle raw display
$71$ \( T^{19} + 14 T^{18} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$73$ \( T^{19} + 50 T^{18} + \cdots + 36183209616 \) Copy content Toggle raw display
$79$ \( T^{19} + 21 T^{18} + \cdots - 60268093454944 \) Copy content Toggle raw display
$83$ \( T^{19} + 15 T^{18} + \cdots + 11023063766152 \) Copy content Toggle raw display
$89$ \( T^{19} + \cdots + 291089244833936 \) Copy content Toggle raw display
$97$ \( T^{19} + 42 T^{18} + \cdots + 2033562327768 \) Copy content Toggle raw display
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