Properties

Label 4011.2.a.i
Level $4011$
Weight $2$
Character orbit 4011.a
Self dual yes
Analytic conductor $32.028$
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] = [4011,2,Mod(1,4011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4011, 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("4011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4011 = 3 \cdot 7 \cdot 191 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4011.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: \(32.0279962507\)
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: \( x^{19} - 3 x^{18} - 20 x^{17} + 63 x^{16} + 156 x^{15} - 531 x^{14} - 597 x^{13} + 2313 x^{12} + 1149 x^{11} - 5631 x^{10} - 951 x^{9} + 7819 x^{8} - 87 x^{7} - 6025 x^{6} + 649 x^{5} + \cdots + 18 \) 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - 1) q^{5} - \beta_1 q^{6} + q^{7} + \beta_{3} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{8} - 1) q^{5} - \beta_1 q^{6} + q^{7} + \beta_{3} q^{8} + q^{9} + ( - \beta_{14} - \beta_{13} - \beta_{11} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{10} + (\beta_{15} + \beta_{12}) q^{11} + ( - \beta_{2} - 1) q^{12} + ( - \beta_{17} + \beta_{14} - \beta_{2} - 1) q^{13} + \beta_1 q^{14} + (\beta_{8} + 1) q^{15} + (\beta_{18} - \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + \cdots + 1) q^{16}+ \cdots + (\beta_{15} + \beta_{12}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 3 q^{2} - 19 q^{3} + 11 q^{4} - 12 q^{5} - 3 q^{6} + 19 q^{7} + 6 q^{8} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 3 q^{2} - 19 q^{3} + 11 q^{4} - 12 q^{5} - 3 q^{6} + 19 q^{7} + 6 q^{8} + 19 q^{9} - 12 q^{10} + q^{11} - 11 q^{12} - 25 q^{13} + 3 q^{14} + 12 q^{15} + 3 q^{16} - 9 q^{17} + 3 q^{18} - 29 q^{19} - 14 q^{20} - 19 q^{21} - 5 q^{22} + 18 q^{23} - 6 q^{24} + 3 q^{25} - 15 q^{26} - 19 q^{27} + 11 q^{28} + 2 q^{29} + 12 q^{30} - 24 q^{31} + 15 q^{32} - q^{33} - 16 q^{34} - 12 q^{35} + 11 q^{36} - 24 q^{37} - 26 q^{38} + 25 q^{39} - 44 q^{40} - 14 q^{41} - 3 q^{42} - 17 q^{43} - 6 q^{44} - 12 q^{45} - 16 q^{46} + 7 q^{47} - 3 q^{48} + 19 q^{49} + 7 q^{50} + 9 q^{51} - 64 q^{52} + 4 q^{53} - 3 q^{54} - 15 q^{55} + 6 q^{56} + 29 q^{57} - 15 q^{58} - 23 q^{59} + 14 q^{60} - 38 q^{61} - 4 q^{62} + 19 q^{63} + 33 q^{65} + 5 q^{66} - 20 q^{67} - 27 q^{68} - 18 q^{69} - 12 q^{70} + 14 q^{71} + 6 q^{72} - 19 q^{73} - 11 q^{74} - 3 q^{75} - 33 q^{76} + q^{77} + 15 q^{78} - 16 q^{79} - 10 q^{80} + 19 q^{81} - 25 q^{82} - 11 q^{83} - 11 q^{84} - 5 q^{85} + 5 q^{86} - 2 q^{87} - 25 q^{88} - 19 q^{89} - 12 q^{90} - 25 q^{91} + 22 q^{92} + 24 q^{93} - 35 q^{94} + 26 q^{95} - 15 q^{96} - 57 q^{97} + 3 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{19} - 3 x^{18} - 20 x^{17} + 63 x^{16} + 156 x^{15} - 531 x^{14} - 597 x^{13} + 2313 x^{12} + 1149 x^{11} - 5631 x^{10} - 951 x^{9} + 7819 x^{8} - 87 x^{7} - 6025 x^{6} + 649 x^{5} + \cdots + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 126 \nu^{18} + 629 \nu^{17} + 1563 \nu^{16} - 11974 \nu^{15} - 1212 \nu^{14} + 87336 \nu^{13} - 62019 \nu^{12} - 304737 \nu^{11} + 350865 \nu^{10} + 521643 \nu^{9} - 784776 \nu^{8} + \cdots - 2985 ) / 171 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1269 \nu^{18} - 5351 \nu^{17} - 18840 \nu^{16} + 102814 \nu^{15} + 72252 \nu^{14} - 760680 \nu^{13} + 173127 \nu^{12} + 2716653 \nu^{11} - 1869894 \nu^{10} - 4842423 \nu^{9} + \cdots + 19164 ) / 57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1780 \nu^{18} + 7504 \nu^{17} + 26474 \nu^{16} - 144284 \nu^{15} - 102279 \nu^{14} + 1068687 \nu^{13} - 235665 \nu^{12} - 3823923 \nu^{11} + 2595024 \nu^{10} + 6841065 \nu^{9} + \cdots - 25383 ) / 57 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5521 \nu^{18} - 23299 \nu^{17} - 81950 \nu^{16} + 447707 \nu^{15} + 314193 \nu^{14} - 3313098 \nu^{13} + 752550 \nu^{12} + 11838396 \nu^{11} - 8123088 \nu^{10} - 21132498 \nu^{9} + \cdots + 81363 ) / 171 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 7624 \nu^{18} + 32171 \nu^{17} + 113231 \nu^{16} - 618382 \nu^{15} - 434940 \nu^{14} + 4578093 \nu^{13} - 1033188 \nu^{12} - 16368378 \nu^{11} + 11204607 \nu^{10} + \cdots - 111705 ) / 171 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 7967 \nu^{18} + 33802 \nu^{17} + 117679 \nu^{16} - 649346 \nu^{15} - 441957 \nu^{14} + 4802769 \nu^{13} - 1174287 \nu^{12} - 17143527 \nu^{11} + 12058593 \nu^{10} + \cdots - 120252 ) / 171 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155 \nu^{18} - 654 \nu^{17} - 2302 \nu^{16} + 12570 \nu^{15} + 8844 \nu^{14} - 93051 \nu^{13} + 20970 \nu^{12} + 332658 \nu^{11} - 227541 \nu^{10} - 594279 \nu^{9} + 576699 \nu^{8} + \cdots + 2262 ) / 3 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11264 \nu^{18} - 47452 \nu^{17} - 167536 \nu^{16} + 912113 \nu^{15} + 647616 \nu^{14} - 6753015 \nu^{13} + 1485804 \nu^{12} + 24148659 \nu^{11} - 16389744 \nu^{10} + \cdots + 163713 ) / 171 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11518 \nu^{18} + 48514 \nu^{17} + 171407 \nu^{16} - 932819 \nu^{15} - 663618 \nu^{14} + 6909354 \nu^{13} - 1513104 \nu^{12} - 24723318 \nu^{11} + 16759074 \nu^{10} + \cdots - 168879 ) / 171 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13960 \nu^{18} - 58953 \nu^{17} - 207185 \nu^{16} + 1133190 \nu^{15} + 793296 \nu^{14} - 8389350 \nu^{13} + 1917816 \nu^{12} + 29993256 \nu^{11} - 20626644 \nu^{10} + \cdots + 207495 ) / 171 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 15799 \nu^{18} + 66527 \nu^{17} + 235088 \nu^{16} - 1278925 \nu^{15} - 909951 \nu^{14} + 9470313 \nu^{13} - 2075667 \nu^{12} - 33872595 \nu^{11} + 22977219 \nu^{10} + \cdots - 231162 ) / 171 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 21788 \nu^{18} + 91937 \nu^{17} + 323560 \nu^{16} - 1767049 \nu^{15} - 1242456 \nu^{14} + 13080564 \nu^{13} - 2955387 \nu^{12} - 46759407 \nu^{11} + 32025561 \nu^{10} + \cdots - 320298 ) / 171 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25180 \nu^{18} - 106271 \nu^{17} - 373892 \nu^{16} + 2042584 \nu^{15} + 1435119 \nu^{14} - 15120585 \nu^{13} + 3421032 \nu^{12} + 54054213 \nu^{11} - 37030566 \nu^{10} + \cdots + 371625 ) / 171 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 35718 \nu^{18} + 150752 \nu^{17} + 530343 \nu^{16} - 2897569 \nu^{15} - 2035059 \nu^{14} + 21450105 \nu^{13} - 4859658 \nu^{12} - 76682886 \nu^{11} + 52563780 \nu^{10} + \cdots - 526857 ) / 171 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 19186 \nu^{18} + 80990 \nu^{17} + 284840 \nu^{16} - 1556713 \nu^{15} - 1092381 \nu^{14} + 11524218 \nu^{13} - 2616888 \nu^{12} - 41199006 \nu^{11} + 28263060 \nu^{10} + \cdots - 283416 ) / 57 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} - \beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + \beta_{3} + 5 \beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{18} - 2 \beta_{17} - 2 \beta_{16} + \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{10} - 2 \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 8 \beta_{3} - 3 \beta_{2} + 18 \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9 \beta_{18} - \beta_{17} - 3 \beta_{16} - 9 \beta_{15} + 10 \beta_{14} + 7 \beta_{13} - 21 \beta_{12} + \beta_{11} + 10 \beta_{10} - 11 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} - \beta_{6} + 2 \beta_{5} + 9 \beta_{4} + 11 \beta_{3} + 22 \beta_{2} - 9 \beta _1 + 85 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{18} - 20 \beta_{17} - 23 \beta_{16} - 3 \beta_{15} + 14 \beta_{14} - 11 \beta_{13} - 40 \beta_{12} - \beta_{11} + 16 \beta_{10} - 25 \beta_{9} + 15 \beta_{8} + 14 \beta_{7} - 12 \beta_{6} + 13 \beta_{5} + 13 \beta_{4} + 56 \beta_{3} - 34 \beta_{2} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 66 \beta_{18} - 13 \beta_{17} - 41 \beta_{16} - 65 \beta_{15} + 79 \beta_{14} + 34 \beta_{13} - 178 \beta_{12} + 11 \beta_{11} + 83 \beta_{10} - 96 \beta_{9} + 64 \beta_{8} + 66 \beta_{7} - 13 \beta_{6} + 30 \beta_{5} + 65 \beta_{4} + 93 \beta_{3} + \cdots + 509 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 106 \beta_{18} - 155 \beta_{17} - 203 \beta_{16} - 42 \beta_{15} + 133 \beta_{14} - 102 \beta_{13} - 386 \beta_{12} - 15 \beta_{11} + 168 \beta_{10} - 234 \beta_{9} + 142 \beta_{8} + 131 \beta_{7} - 104 \beta_{6} + 126 \beta_{5} + \cdots + 403 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 461 \beta_{18} - 123 \beta_{17} - 404 \beta_{16} - 446 \beta_{15} + 581 \beta_{14} + 110 \beta_{13} - 1412 \beta_{12} + 84 \beta_{11} + 655 \beta_{10} - 778 \beta_{9} + 433 \beta_{8} + 463 \beta_{7} - 124 \beta_{6} + 311 \beta_{5} + \cdots + 3167 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 843 \beta_{18} - 1111 \beta_{17} - 1641 \beta_{16} - 415 \beta_{15} + 1104 \beta_{14} - 883 \beta_{13} - 3304 \beta_{12} - 148 \beta_{11} + 1503 \beta_{10} - 1974 \beta_{9} + 1143 \beta_{8} + 1065 \beta_{7} - 804 \beta_{6} + \cdots + 3490 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3191 \beta_{18} - 1050 \beta_{17} - 3522 \beta_{16} - 3029 \beta_{15} + 4172 \beta_{14} - 102 \beta_{13} - 10881 \beta_{12} + 553 \beta_{11} + 5056 \beta_{10} - 6096 \beta_{9} + 2934 \beta_{8} + 3236 \beta_{7} + \cdots + 20355 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 6419 \beta_{18} - 7744 \beta_{17} - 12756 \beta_{16} - 3600 \beta_{15} + 8650 \beta_{14} - 7282 \beta_{13} - 26692 \beta_{12} - 1222 \beta_{11} + 12446 \beta_{10} - 15840 \beta_{9} + 8591 \beta_{8} + 8153 \beta_{7} + \cdots + 28398 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 22182 \beta_{18} - 8592 \beta_{17} - 28914 \beta_{16} - 20631 \beta_{15} + 29788 \beta_{14} - 6256 \beta_{13} - 82571 \beta_{12} + 3354 \beta_{11} + 38530 \beta_{10} - 46873 \beta_{9} + 20229 \beta_{8} + \cdots + 134604 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 47953 \beta_{18} - 53542 \beta_{17} - 97153 \beta_{16} - 29348 \beta_{15} + 65942 \beta_{14} - 57967 \beta_{13} - 208851 \beta_{12} - 9183 \beta_{11} + 98794 \beta_{10} - 123587 \beta_{9} + 62650 \beta_{8} + \cdots + 223223 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 155543 \beta_{18} - 68764 \beta_{17} - 229403 \beta_{16} - 141613 \beta_{15} + 213042 \beta_{14} - 78175 \beta_{13} - 620901 \beta_{12} + 19191 \beta_{11} + 291150 \beta_{10} - 356156 \beta_{9} + \cdots + 912120 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 355248 \beta_{18} - 370561 \beta_{17} - 731378 \beta_{16} - 231269 \beta_{15} + 495748 \beta_{14} - 450095 \beta_{13} - 1603067 \beta_{12} - 65350 \beta_{11} + 764846 \beta_{10} - 947772 \beta_{9} + \cdots + 1718868 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 1101124 \beta_{18} - 542225 \beta_{17} - 1781764 \beta_{16} - 981268 \beta_{15} + 1530361 \beta_{14} - 759650 \beta_{13} - 4642095 \beta_{12} + 104044 \beta_{11} + 2187149 \beta_{10} + \cdots + 6307240 \) 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
−2.38901
−2.22223
−2.18973
−1.23863
−1.22088
−1.22041
−0.826538
−0.472873
−0.195659
0.375149
0.570727
0.804207
1.14168
1.26519
1.67892
1.85028
2.26332
2.30694
2.71954
−2.38901 −1.00000 3.70736 2.22462 2.38901 1.00000 −4.07890 1.00000 −5.31464
1.2 −2.22223 −1.00000 2.93833 1.26699 2.22223 1.00000 −2.08518 1.00000 −2.81556
1.3 −2.18973 −1.00000 2.79491 −3.19083 2.18973 1.00000 −1.74065 1.00000 6.98705
1.4 −1.23863 −1.00000 −0.465792 0.329001 1.23863 1.00000 3.05421 1.00000 −0.407511
1.5 −1.22088 −1.00000 −0.509453 −0.195895 1.22088 1.00000 3.06374 1.00000 0.239164
1.6 −1.22041 −1.00000 −0.510601 −2.12455 1.22041 1.00000 3.06396 1.00000 2.59282
1.7 −0.826538 −1.00000 −1.31684 −3.86633 0.826538 1.00000 2.74149 1.00000 3.19567
1.8 −0.472873 −1.00000 −1.77639 1.87834 0.472873 1.00000 1.78575 1.00000 −0.888215
1.9 −0.195659 −1.00000 −1.96172 −1.26181 0.195659 1.00000 0.775146 1.00000 0.246885
1.10 0.375149 −1.00000 −1.85926 −1.67589 −0.375149 1.00000 −1.44780 1.00000 −0.628707
1.11 0.570727 −1.00000 −1.67427 2.41656 −0.570727 1.00000 −2.09700 1.00000 1.37919
1.12 0.804207 −1.00000 −1.35325 2.54735 −0.804207 1.00000 −2.69671 1.00000 2.04859
1.13 1.14168 −1.00000 −0.696573 −2.91945 −1.14168 1.00000 −3.07862 1.00000 −3.33307
1.14 1.26519 −1.00000 −0.399287 −3.83041 −1.26519 1.00000 −3.03556 1.00000 −4.84621
1.15 1.67892 −1.00000 0.818779 1.40365 −1.67892 1.00000 −1.98318 1.00000 2.35662
1.16 1.85028 −1.00000 1.42354 1.08935 −1.85028 1.00000 −1.06661 1.00000 2.01561
1.17 2.26332 −1.00000 3.12263 −3.08794 −2.26332 1.00000 2.54087 1.00000 −6.98900
1.18 2.30694 −1.00000 3.32198 −0.793515 −2.30694 1.00000 3.04973 1.00000 −1.83059
1.19 2.71954 −1.00000 5.39591 −2.20924 −2.71954 1.00000 9.23531 1.00000 −6.00811
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.19
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(7\) \(-1\)
\(191\) \(-1\)

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 4011.2.a.i 19
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4011.2.a.i 19 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{19} - 3 T_{2}^{18} - 20 T_{2}^{17} + 63 T_{2}^{16} + 156 T_{2}^{15} - 531 T_{2}^{14} - 597 T_{2}^{13} + 2313 T_{2}^{12} + 1149 T_{2}^{11} - 5631 T_{2}^{10} - 951 T_{2}^{9} + 7819 T_{2}^{8} - 87 T_{2}^{7} - 6025 T_{2}^{6} + 649 T_{2}^{5} + \cdots + 18 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4011))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{19} - 3 T^{18} - 20 T^{17} + 63 T^{16} + \cdots + 18 \) Copy content Toggle raw display
$3$ \( (T + 1)^{19} \) Copy content Toggle raw display
$5$ \( T^{19} + 12 T^{18} + 23 T^{17} + \cdots + 10776 \) Copy content Toggle raw display
$7$ \( (T - 1)^{19} \) Copy content Toggle raw display
$11$ \( T^{19} - T^{18} - 98 T^{17} + \cdots - 18239400 \) Copy content Toggle raw display
$13$ \( T^{19} + 25 T^{18} + 201 T^{17} + \cdots - 2099374 \) Copy content Toggle raw display
$17$ \( T^{19} + 9 T^{18} - 106 T^{17} + \cdots - 63387976 \) Copy content Toggle raw display
$19$ \( T^{19} + 29 T^{18} + 236 T^{17} + \cdots - 36345900 \) Copy content Toggle raw display
$23$ \( T^{19} - 18 T^{18} + \cdots + 268077800 \) Copy content Toggle raw display
$29$ \( T^{19} - 2 T^{18} + \cdots - 13509026365 \) Copy content Toggle raw display
$31$ \( T^{19} + 24 T^{18} + 44 T^{17} + \cdots + 906276 \) Copy content Toggle raw display
$37$ \( T^{19} + 24 T^{18} + \cdots - 6764609296 \) Copy content Toggle raw display
$41$ \( T^{19} + 14 T^{18} + \cdots - 13316261870592 \) Copy content Toggle raw display
$43$ \( T^{19} + 17 T^{18} + \cdots - 118049304 \) Copy content Toggle raw display
$47$ \( T^{19} + \cdots + 137715328523636 \) Copy content Toggle raw display
$53$ \( T^{19} - 4 T^{18} + \cdots - 5216358741 \) Copy content Toggle raw display
$59$ \( T^{19} + 23 T^{18} + \cdots + 8392841232736 \) Copy content Toggle raw display
$61$ \( T^{19} + 38 T^{18} + \cdots + 15146376289072 \) Copy content Toggle raw display
$67$ \( T^{19} + 20 T^{18} + \cdots + 470556971400 \) Copy content Toggle raw display
$71$ \( T^{19} - 14 T^{18} + \cdots + 28434375813960 \) Copy content Toggle raw display
$73$ \( T^{19} + 19 T^{18} + \cdots + 33316897032988 \) Copy content Toggle raw display
$79$ \( T^{19} + 16 T^{18} + \cdots + 855546240358 \) Copy content Toggle raw display
$83$ \( T^{19} + \cdots + 613790160470608 \) Copy content Toggle raw display
$89$ \( T^{19} + 19 T^{18} + \cdots + 104757107912 \) Copy content Toggle raw display
$97$ \( T^{19} + 57 T^{18} + \cdots - 45\!\cdots\!58 \) Copy content Toggle raw display
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