Properties

Label 663.2.z.d
Level $663$
Weight $2$
Character orbit 663.z
Analytic conductor $5.294$
Analytic rank $0$
Dimension $16$
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] = [663,2,Mod(205,663)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(663, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("663.205");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 663 = 3 \cdot 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 663.z (of order \(6\), degree \(2\), 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: \(5.29408165401\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 602x^{10} + 1212x^{8} + 1259x^{6} + 665x^{4} + 168x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{10} - 1) q^{3} + ( - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{6} - \beta_{3}) q^{4} + (\beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{6} + (\beta_{14} + \beta_{11} - \beta_{9} - \beta_{7} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{8} - \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (\beta_{10} - 1) q^{3} + ( - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{6} - \beta_{3}) q^{4} + (\beta_{12} - \beta_{11} + \beta_{9} + \beta_{7} + \beta_{4} + 2 \beta_{3} - \beta_1) q^{5} + \beta_{2} q^{6} + (\beta_{14} + \beta_{11} - \beta_{9} - \beta_{7} - 2 \beta_{3} + \beta_{2} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_1) q^{8} - \beta_{10} q^{9} + (\beta_{5} + \beta_{4} - \beta_1) q^{10} + ( - \beta_{15} + \beta_{14} + 2 \beta_{8} - \beta_{6} - \beta_{5} + \beta_1) q^{11} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{4} + \beta_1 - 1) q^{12} + (\beta_{15} + \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} - \beta_1) q^{13} + ( - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \cdots - 2) q^{14}+ \cdots + ( - \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 4 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 4 q^{4} - 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} - 2 q^{13} - 26 q^{14} - 3 q^{15} - 8 q^{17} + 27 q^{20} + q^{22} - 21 q^{23} + 14 q^{25} + 2 q^{26} + 16 q^{27} - 33 q^{28} + 29 q^{29} + q^{30} - 15 q^{32} - 3 q^{33} + 15 q^{35} + 4 q^{36} - 18 q^{37} + 62 q^{38} + q^{39} + 4 q^{40} + 12 q^{41} + 13 q^{42} - 3 q^{43} + 3 q^{45} - 9 q^{46} + 2 q^{49} - 36 q^{50} + 16 q^{51} - 8 q^{52} - 26 q^{53} + 9 q^{55} - 37 q^{56} + 30 q^{58} - 3 q^{59} + 29 q^{61} - 20 q^{62} + 36 q^{64} - 16 q^{65} - 2 q^{66} + 33 q^{67} + 4 q^{68} - 21 q^{69} + 27 q^{71} + 17 q^{74} - 7 q^{75} - 48 q^{76} - 8 q^{77} - q^{78} - 14 q^{79} - 39 q^{80} - 8 q^{81} - 3 q^{82} + 33 q^{84} + 3 q^{85} + 29 q^{87} - 5 q^{88} - 3 q^{89} - 2 q^{90} - 70 q^{91} - 64 q^{92} - 6 q^{93} - 25 q^{94} - 27 q^{95} + 6 q^{97} + 102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 20x^{14} + 156x^{12} + 602x^{10} + 1212x^{8} + 1259x^{6} + 665x^{4} + 168x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 20\nu^{12} - 156\nu^{10} - 602\nu^{8} - 1208\nu^{6} - 1219\nu^{4} - 553\nu^{2} + 4\nu - 84 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{15} + 2 \nu^{14} + 20 \nu^{13} + 40 \nu^{12} + 156 \nu^{11} + 312 \nu^{10} + 602 \nu^{9} + 1204 \nu^{8} + 1212 \nu^{7} + 2424 \nu^{6} + 1259 \nu^{5} + 2518 \nu^{4} + 665 \nu^{3} + 1314 \nu^{2} + \cdots + 256 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 20 \nu^{13} - 40 \nu^{12} + 156 \nu^{11} - 312 \nu^{10} + 602 \nu^{9} - 1204 \nu^{8} + 1212 \nu^{7} - 2424 \nu^{6} + 1259 \nu^{5} - 2518 \nu^{4} + 665 \nu^{3} - 1314 \nu^{2} + \cdots - 256 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5 \nu^{15} - 22 \nu^{14} - 88 \nu^{13} - 432 \nu^{12} - 556 \nu^{11} - 3272 \nu^{10} - 1426 \nu^{9} - 11996 \nu^{8} - 788 \nu^{7} - 21880 \nu^{6} + 2041 \nu^{5} - 18386 \nu^{4} + 2279 \nu^{3} + \cdots - 552 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} - 11 \nu^{14} - 20 \nu^{13} - 216 \nu^{12} - 156 \nu^{11} - 1636 \nu^{10} - 602 \nu^{9} - 6006 \nu^{8} - 1212 \nu^{7} - 11044 \nu^{6} - 1259 \nu^{5} - 9641 \nu^{4} - 657 \nu^{3} + \cdots - 508 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 11 \nu^{14} + 20 \nu^{13} - 216 \nu^{12} + 156 \nu^{11} - 1636 \nu^{10} + 602 \nu^{9} - 6006 \nu^{8} + 1212 \nu^{7} - 11044 \nu^{6} + 1259 \nu^{5} - 9641 \nu^{4} + 657 \nu^{3} - 3727 \nu^{2} + \cdots - 508 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17 \nu^{14} - 336 \nu^{12} - 2572 \nu^{10} - 9610 \nu^{8} - 18204 \nu^{6} - 16667 \nu^{4} - 6765 \nu^{2} - 8 \nu - 956 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47 \nu^{15} + 90 \nu^{14} + 952 \nu^{13} + 1744 \nu^{12} + 7556 \nu^{11} + 12952 \nu^{10} + 29846 \nu^{9} + 46084 \nu^{8} + 61756 \nu^{7} + 80232 \nu^{6} + 64981 \nu^{5} + 62974 \nu^{4} + \cdots + 2040 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21 \nu^{15} + 416 \nu^{13} + 3196 \nu^{11} + 12018 \nu^{9} + 23044 \nu^{7} + 21607 \nu^{5} + 9089 \nu^{3} + 1316 \nu + 16 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 63 \nu^{15} - 1224 \nu^{13} - 9124 \nu^{11} - 32630 \nu^{9} - 57148 \nu^{7} - 44837 \nu^{5} - 13403 \nu^{3} + 32 \nu^{2} - 932 \nu + 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 189 \nu^{15} - 18 \nu^{14} + 3720 \nu^{13} - 336 \nu^{12} + 28300 \nu^{11} - 2360 \nu^{10} + 104738 \nu^{9} - 7668 \nu^{8} + 195444 \nu^{7} - 11272 \nu^{6} + 174799 \nu^{5} - 5990 \nu^{4} + \cdots + 168 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 195 \nu^{15} - 94 \nu^{14} - 3800 \nu^{13} - 1840 \nu^{12} - 28468 \nu^{11} - 13896 \nu^{10} - 102750 \nu^{9} - 50924 \nu^{8} - 183500 \nu^{7} - 93752 \nu^{6} - 151345 \nu^{5} + \cdots - 4264 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 125 \nu^{15} + 46 \nu^{14} + 2456 \nu^{13} + 912 \nu^{12} + 18636 \nu^{11} + 7016 \nu^{10} + 68706 \nu^{9} + 26444 \nu^{8} + 127476 \nu^{7} + 50936 \nu^{6} + 113199 \nu^{5} + 48250 \nu^{4} + \cdots + 3144 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 125 \nu^{15} - 46 \nu^{14} + 2456 \nu^{13} - 912 \nu^{12} + 18636 \nu^{11} - 7016 \nu^{10} + 68706 \nu^{9} - 26444 \nu^{8} + 127476 \nu^{7} - 50936 \nu^{6} + 113199 \nu^{5} - 48250 \nu^{4} + \cdots - 3144 ) / 64 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{4} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + 5 \beta_{14} + 6 \beta_{13} - 6 \beta_{12} - 6 \beta_{11} - \beta_{8} + 5 \beta_{4} + \beta_{3} - \beta_{2} - 6 \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - 2 \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} + 2 \beta_{10} + 10 \beta_{7} - 10 \beta_{6} - 6 \beta_{4} - 5 \beta_{3} + 25 \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 10 \beta_{15} - 24 \beta_{14} - 34 \beta_{13} + 34 \beta_{12} + 34 \beta_{11} + 10 \beta_{8} - 25 \beta_{4} - 9 \beta_{3} + 12 \beta_{2} + 33 \beta _1 - 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10 \beta_{15} + 21 \beta_{14} + 11 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} - 26 \beta_{10} + 2 \beta_{9} - 72 \beta_{7} + 74 \beta_{6} + 35 \beta_{4} + 26 \beta_{3} - 132 \beta _1 + 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 74 \beta_{15} + 119 \beta_{14} + 193 \beta_{13} - 193 \beta_{12} - 193 \beta_{11} - 72 \beta_{8} - \beta_{7} - \beta_{6} + 130 \beta_{4} + 63 \beta_{3} - 102 \beta_{2} - 178 \beta _1 + 397 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 71 \beta_{15} - 159 \beta_{14} - 88 \beta_{13} + 56 \beta_{12} - 56 \beta_{11} + 232 \beta_{10} - 32 \beta_{9} - \beta_{8} + 461 \beta_{7} - 493 \beta_{6} + 2 \beta_{5} - 208 \beta_{4} - 154 \beta_{3} + \beta_{2} + 727 \beta _1 - 116 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 494 \beta_{15} - 609 \beta_{14} - 1103 \beta_{13} + 1101 \beta_{12} + 1101 \beta_{11} + 2 \beta_{9} + 461 \beta_{8} + 20 \beta_{7} + 18 \beta_{6} - 697 \beta_{4} - 404 \beta_{3} + 757 \beta_{2} + 953 \beta _1 - 2211 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 443 \beta_{15} + 1070 \beta_{14} + 627 \beta_{13} - 295 \beta_{12} + 295 \beta_{11} - 1780 \beta_{10} + 332 \beta_{9} + 20 \beta_{8} - 2800 \beta_{7} + 3132 \beta_{6} - 40 \beta_{5} + 1247 \beta_{4} + 992 \beta_{3} - 20 \beta_{2} + \cdots + 890 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3152 \beta_{15} + 3197 \beta_{14} + 6349 \beta_{13} - 6309 \beta_{12} - 6309 \beta_{11} - 40 \beta_{9} - 2800 \beta_{8} - 244 \beta_{7} - 204 \beta_{6} + 3824 \beta_{4} + 2485 \beta_{3} - 5244 \beta_{2} - 5087 \beta _1 + 12510 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 2596 \beta_{15} - 6822 \beta_{14} - 4226 \beta_{13} + 1374 \beta_{12} - 1374 \beta_{11} + 12636 \beta_{10} - 2852 \beta_{9} - 244 \beta_{8} + 16567 \beta_{7} - 19419 \beta_{6} + 488 \beta_{5} - 7491 \beta_{4} + \cdots - 6318 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 19663 \beta_{15} - 17124 \beta_{14} - 36787 \beta_{13} + 36299 \beta_{12} + 36299 \beta_{11} + 488 \beta_{9} + 16567 \beta_{8} + 2362 \beta_{7} + 1874 \beta_{6} - 21350 \beta_{4} - 14949 \beta_{3} + \cdots - 71585 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 14693 \beta_{15} + 42304 \beta_{14} + 27611 \beta_{13} - 5523 \beta_{12} + 5523 \beta_{11} - 85710 \beta_{10} + 22088 \beta_{9} + 2362 \beta_{8} - 96723 \beta_{7} + 118811 \beta_{6} - 4724 \beta_{5} + \cdots + 42855 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/663\mathbb{Z}\right)^\times\).

\(n\) \(443\) \(547\) \(613\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
205.1
2.29812i
1.96679i
0.729584i
0.598113i
0.513139i
0.978320i
1.64424i
2.45691i
2.29812i
1.96679i
0.729584i
0.598113i
0.513139i
0.978320i
1.64424i
2.45691i
−1.99023 + 1.14906i −0.500000 0.866025i 1.64068 2.84174i 1.20591i 1.99023 + 1.14906i −0.421031 0.243082i 2.94471i −0.500000 + 0.866025i −1.38566 2.40004i
205.2 −1.70329 + 0.983395i −0.500000 0.866025i 0.934132 1.61796i 0.562071i 1.70329 + 0.983395i 2.67180 + 1.54256i 0.259096i −0.500000 + 0.866025i 0.552738 + 0.957371i
205.3 −0.631839 + 0.364792i −0.500000 0.866025i −0.733853 + 1.27107i 0.945208i 0.631839 + 0.364792i −0.308178 0.177927i 2.52998i −0.500000 + 0.866025i −0.344805 0.597219i
205.4 −0.517981 + 0.299056i −0.500000 0.866025i −0.821130 + 1.42224i 1.38228i 0.517981 + 0.299056i −1.15547 0.667109i 2.17848i −0.500000 + 0.866025i 0.413380 + 0.715996i
205.5 0.444391 0.256569i −0.500000 0.866025i −0.868344 + 1.50402i 4.12599i −0.444391 0.256569i 0.528206 + 0.304960i 1.91744i −0.500000 + 0.866025i −1.05860 1.83355i
205.6 0.847250 0.489160i −0.500000 0.866025i −0.521445 + 0.903170i 1.22925i −0.847250 0.489160i 4.03243 + 2.32812i 2.97692i −0.500000 + 0.866025i −0.601301 1.04148i
205.7 1.42396 0.822122i −0.500000 0.866025i 0.351768 0.609281i 3.13210i −1.42396 0.822122i −1.27469 0.735944i 2.13170i −0.500000 + 0.866025i 2.57496 + 4.45997i
205.8 2.12774 1.22845i −0.500000 0.866025i 2.01819 3.49561i 0.284332i −2.12774 1.22845i −4.07306 2.35158i 5.00321i −0.500000 + 0.866025i 0.349288 + 0.604985i
511.1 −1.99023 1.14906i −0.500000 + 0.866025i 1.64068 + 2.84174i 1.20591i 1.99023 1.14906i −0.421031 + 0.243082i 2.94471i −0.500000 0.866025i −1.38566 + 2.40004i
511.2 −1.70329 0.983395i −0.500000 + 0.866025i 0.934132 + 1.61796i 0.562071i 1.70329 0.983395i 2.67180 1.54256i 0.259096i −0.500000 0.866025i 0.552738 0.957371i
511.3 −0.631839 0.364792i −0.500000 + 0.866025i −0.733853 1.27107i 0.945208i 0.631839 0.364792i −0.308178 + 0.177927i 2.52998i −0.500000 0.866025i −0.344805 + 0.597219i
511.4 −0.517981 0.299056i −0.500000 + 0.866025i −0.821130 1.42224i 1.38228i 0.517981 0.299056i −1.15547 + 0.667109i 2.17848i −0.500000 0.866025i 0.413380 0.715996i
511.5 0.444391 + 0.256569i −0.500000 + 0.866025i −0.868344 1.50402i 4.12599i −0.444391 + 0.256569i 0.528206 0.304960i 1.91744i −0.500000 0.866025i −1.05860 + 1.83355i
511.6 0.847250 + 0.489160i −0.500000 + 0.866025i −0.521445 0.903170i 1.22925i −0.847250 + 0.489160i 4.03243 2.32812i 2.97692i −0.500000 0.866025i −0.601301 + 1.04148i
511.7 1.42396 + 0.822122i −0.500000 + 0.866025i 0.351768 + 0.609281i 3.13210i −1.42396 + 0.822122i −1.27469 + 0.735944i 2.13170i −0.500000 0.866025i 2.57496 4.45997i
511.8 2.12774 + 1.22845i −0.500000 + 0.866025i 2.01819 + 3.49561i 0.284332i −2.12774 + 1.22845i −4.07306 + 2.35158i 5.00321i −0.500000 0.866025i 0.349288 0.604985i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 205.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 663.2.z.d 16
13.e even 6 1 inner 663.2.z.d 16
13.f odd 12 2 8619.2.a.bn 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.z.d 16 1.a even 1 1 trivial
663.2.z.d 16 13.e even 6 1 inner
8619.2.a.bn 16 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 10 T_{2}^{14} + 72 T_{2}^{12} + 3 T_{2}^{11} - 238 T_{2}^{10} + 570 T_{2}^{8} - 84 T_{2}^{7} - 505 T_{2}^{6} + 63 T_{2}^{5} + 329 T_{2}^{4} - 84 T_{2}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(663, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 10 T^{14} + 72 T^{12} + 3 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 33 T^{14} + 347 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{16} - 29 T^{14} + 685 T^{12} + \cdots + 196 \) Copy content Toggle raw display
$11$ \( T^{16} - 3 T^{15} - 37 T^{14} + \cdots + 43264 \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{15} + 27 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{16} - 92 T^{14} + 5838 T^{12} + \cdots + 20702500 \) Copy content Toggle raw display
$23$ \( T^{16} + 21 T^{15} + 316 T^{14} + \cdots + 268324 \) Copy content Toggle raw display
$29$ \( T^{16} - 29 T^{15} + \cdots + 60552413476 \) Copy content Toggle raw display
$31$ \( T^{16} + 298 T^{14} + \cdots + 10276687876 \) Copy content Toggle raw display
$37$ \( T^{16} + 18 T^{15} + \cdots + 6305630464 \) Copy content Toggle raw display
$41$ \( T^{16} - 12 T^{15} + \cdots + 420660100 \) Copy content Toggle raw display
$43$ \( T^{16} + 3 T^{15} + \cdots + 3207636496 \) Copy content Toggle raw display
$47$ \( T^{16} + 389 T^{14} + \cdots + 528633693184 \) Copy content Toggle raw display
$53$ \( (T^{8} + 13 T^{7} - 179 T^{6} + \cdots + 36088)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 3 T^{15} + \cdots + 2428161761536 \) Copy content Toggle raw display
$61$ \( T^{16} - 29 T^{15} + \cdots + 776118712576 \) Copy content Toggle raw display
$67$ \( T^{16} - 33 T^{15} + \cdots + 2608383272401 \) Copy content Toggle raw display
$71$ \( T^{16} - 27 T^{15} + \cdots + 23854784466496 \) Copy content Toggle raw display
$73$ \( T^{16} + 365 T^{14} + 41407 T^{12} + \cdots + 6724 \) Copy content Toggle raw display
$79$ \( (T^{8} + 7 T^{7} - 468 T^{6} + \cdots + 17872504)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 315 T^{14} + \cdots + 242861056 \) Copy content Toggle raw display
$89$ \( T^{16} + 3 T^{15} + \cdots + 17701770304 \) Copy content Toggle raw display
$97$ \( T^{16} - 6 T^{15} + \cdots + 16072882810000 \) Copy content Toggle raw display
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