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Properties

Label 961.2.g.t
Level $961$
Weight $2$
Character orbit 961.g
Analytic conductor $7.674$
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] = [961,2,Mod(235,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.235");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.g (of order \(15\), degree \(8\), not 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: \(7.67362363425\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{15})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 19x^{14} + 140x^{12} + 511x^{10} + 979x^{8} + 956x^{6} + 410x^{4} + 44x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} - 4 \nu^{14} + 48 \nu^{13} - 65 \nu^{12} + 458 \nu^{11} - 358 \nu^{10} + 2196 \nu^{9} + \cdots + 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2 \nu^{15} + 4 \nu^{14} + 48 \nu^{13} + 65 \nu^{12} + 458 \nu^{11} + 358 \nu^{10} + 2196 \nu^{9} + \cdots - 255 ) / 186 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6 \nu^{15} - 10 \nu^{14} + 144 \nu^{13} - 209 \nu^{12} + 1374 \nu^{11} - 1732 \nu^{10} + 6619 \nu^{9} + \cdots - 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{15} + 315 \nu^{13} + 2250 \nu^{11} + 7940 \nu^{9} + 14865 \nu^{7} + 14844 \nu^{5} + 7255 \nu^{3} + \cdots + 93 ) / 186 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6 \nu^{15} + 10 \nu^{14} + 144 \nu^{13} + 209 \nu^{12} + 1374 \nu^{11} + 1732 \nu^{10} + 6619 \nu^{9} + \cdots + 463 ) / 186 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{15} + 28 \nu^{14} - 144 \nu^{13} + 517 \nu^{12} - 1374 \nu^{11} + 3653 \nu^{10} - 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8 \nu^{15} + 33 \nu^{14} + 130 \nu^{13} + 575 \nu^{12} + 716 \nu^{11} + 3713 \nu^{10} + 1282 \nu^{9} + \cdots - 143 ) / 186 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6 \nu^{15} + 28 \nu^{14} + 144 \nu^{13} + 517 \nu^{12} + 1374 \nu^{11} + 3653 \nu^{10} + 6619 \nu^{9} + \cdots + 354 ) / 186 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36 \nu^{15} + 38 \nu^{14} - 709 \nu^{13} + 695 \nu^{12} - 5485 \nu^{11} + 4827 \nu^{10} + \cdots + 538 ) / 186 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53 \nu^{15} + 5 \nu^{14} - 962 \nu^{13} + 89 \nu^{12} - 6619 \nu^{11} + 587 \nu^{10} - 21738 \nu^{9} + \cdots + 61 ) / 186 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 50 \nu^{15} - 25 \nu^{14} + 983 \nu^{13} - 445 \nu^{12} + 7575 \nu^{11} - 2966 \nu^{10} + 29263 \nu^{9} + \cdots + 36 ) / 186 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 57 \nu^{15} + 21 \nu^{14} + 1058 \nu^{13} + 380 \nu^{12} + 7535 \nu^{11} + 2608 \nu^{10} + 26161 \nu^{9} + \cdots + 126 ) / 186 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 27 \nu^{15} + 53 \nu^{14} + 524 \nu^{13} + 962 \nu^{12} + 3982 \nu^{11} + 6619 \nu^{10} + 15200 \nu^{9} + \cdots + 597 ) / 186 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 36 \nu^{15} + 68 \nu^{14} + 709 \nu^{13} + 1229 \nu^{12} + 5485 \nu^{11} + 8411 \nu^{10} + 21362 \nu^{9} + \cdots + 470 ) / 186 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 135 \nu^{15} + 34 \nu^{14} + 2558 \nu^{13} + 599 \nu^{12} + 18763 \nu^{11} + 3942 \nu^{10} + \cdots + 359 ) / 186 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{4} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} - 6 \beta_{8} + 6 \beta_{6} + \cdots + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 8 \beta_{14} + 3 \beta_{12} - 6 \beta_{11} + 2 \beta_{10} - 7 \beta_{9} + 12 \beta_{8} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7 \beta_{15} - 6 \beta_{14} - 4 \beta_{13} - 6 \beta_{12} - 14 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 19 \beta_{15} + 55 \beta_{14} - 8 \beta_{12} + 38 \beta_{11} - 19 \beta_{10} + 47 \beta_{9} + \cdots - 77 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 42 \beta_{15} + 35 \beta_{14} + 44 \beta_{13} + 29 \beta_{12} + 86 \beta_{11} + 42 \beta_{10} + \cdots - 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 145 \beta_{15} - 365 \beta_{14} + 14 \beta_{12} - 248 \beta_{11} + 145 \beta_{10} - 309 \beta_{9} + \cdots + 455 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 245 \beta_{15} - 212 \beta_{14} - 356 \beta_{13} - 128 \beta_{12} - 518 \beta_{11} - 245 \beta_{10} + \cdots + 234 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1022 \beta_{15} + 2385 \beta_{14} + 34 \beta_{12} + 1625 \beta_{11} - 1022 \beta_{10} + 2000 \beta_{9} + \cdots - 2780 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1432 \beta_{15} + 1322 \beta_{14} + 2578 \beta_{13} + 518 \beta_{12} + 3129 \beta_{11} + 1432 \beta_{10} + \cdots - 1831 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6922 \beta_{15} - 15445 \beta_{14} - 619 \beta_{12} - 10601 \beta_{11} + 6922 \beta_{10} - 12821 \beta_{9} + \cdots + 17280 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 8460 \beta_{15} - 8372 \beta_{14} - 17726 \beta_{13} - 1817 \beta_{12} - 19052 \beta_{11} - 8460 \beta_{10} + \cdots + 13340 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 45841 \beta_{15} + 99462 \beta_{14} + 5217 \beta_{12} + 68772 \beta_{11} - 45841 \beta_{10} + \cdots - 108434 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 50619 \beta_{15} + 53382 \beta_{14} + 118538 \beta_{13} + 4347 \beta_{12} + 116998 \beta_{11} + \cdots - 93153 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{8} - \beta_{11}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
0.176392i
− 2.16544i
− 0.333129i
1.14660i
1.83925i
− 1.42343i
− 1.83925i
1.42343i
− 0.176392i
2.16544i
0.333129i
− 1.14660i
− 2.52368i
1.03739i
2.52368i
− 1.03739i
−0.996848 + 0.724253i 1.89558 − 0.843968i −0.148869 + 0.458173i 0.772811 + 1.33855i −1.27836 + 2.21419i −2.54521 − 2.82674i −0.944957 − 2.90828i 0.873564 − 0.970191i −1.73982 − 0.774619i
235.2 1.49685 − 1.08752i 0.456183 − 0.203106i 0.439812 − 1.35360i −0.603681 − 1.04561i 0.461954 − 0.800128i 2.49789 + 2.77419i 0.329747 + 1.01486i −1.84054 + 2.04413i −2.04074 − 0.908596i
338.1 −1.67638 + 1.21796i 0.223627 − 2.12767i 0.708788 − 2.18143i −1.17396 + 2.03335i 2.21654 + 3.83916i 3.59424 − 0.763980i 0.188054 + 0.578772i −1.54253 − 0.327876i −0.508547 − 4.83851i
338.2 2.17638 − 1.58123i −0.148343 + 1.41139i 1.61830 − 4.98062i −0.304192 + 0.526876i 1.90889 + 3.30629i 1.68914 − 0.359038i −2.69088 − 8.28167i 0.964427 + 0.204995i 0.171076 + 1.62768i
448.1 −0.213065 + 0.655747i −0.603824 + 0.670615i 1.23343 + 0.896137i −1.85376 + 3.21080i −0.311099 − 0.538840i −0.0797964 + 0.759212i −1.96606 + 1.42843i 0.228465 + 2.17370i −1.71050 − 1.89971i
448.2 0.713065 − 2.19459i −1.70141 + 1.88961i −2.68972 − 1.95420i 1.24923 − 2.16373i 2.93370 + 5.08132i −0.167442 + 1.59310i −2.47295 + 1.79670i −0.362238 − 3.44646i −3.85771 − 4.28442i
547.1 −0.213065 − 0.655747i −0.603824 − 0.670615i 1.23343 − 0.896137i −1.85376 − 3.21080i −0.311099 + 0.538840i −0.0797964 − 0.759212i −1.96606 − 1.42843i 0.228465 − 2.17370i −1.71050 + 1.89971i
547.2 0.713065 + 2.19459i −1.70141 − 1.88961i −2.68972 + 1.95420i 1.24923 + 2.16373i 2.93370 − 5.08132i −0.167442 − 1.59310i −2.47295 − 1.79670i −0.362238 + 3.44646i −3.85771 + 4.28442i
732.1 −0.996848 − 0.724253i 1.89558 + 0.843968i −0.148869 − 0.458173i 0.772811 − 1.33855i −1.27836 − 2.21419i −2.54521 + 2.82674i −0.944957 + 2.90828i 0.873564 + 0.970191i −1.73982 + 0.774619i
732.2 1.49685 + 1.08752i 0.456183 + 0.203106i 0.439812 + 1.35360i −0.603681 + 1.04561i 0.461954 + 0.800128i 2.49789 − 2.77419i 0.329747 − 1.01486i −1.84054 − 2.04413i −2.04074 + 0.908596i
816.1 −1.67638 − 1.21796i 0.223627 + 2.12767i 0.708788 + 2.18143i −1.17396 − 2.03335i 2.21654 − 3.83916i 3.59424 + 0.763980i 0.188054 − 0.578772i −1.54253 + 0.327876i −0.508547 + 4.83851i
816.2 2.17638 + 1.58123i −0.148343 − 1.41139i 1.61830 + 4.98062i −0.304192 − 0.526876i 1.90889 − 3.30629i 1.68914 + 0.359038i −2.69088 + 8.28167i 0.964427 − 0.204995i 0.171076 − 1.62768i
844.1 0.108599 + 0.334232i 2.83007 − 0.601549i 1.51812 − 1.10298i −1.48661 + 2.57489i 0.508398 + 0.880572i −0.988571 − 0.440140i 1.10215 + 0.800755i 4.90678 − 2.18464i −1.02206 − 0.217245i
844.2 0.391401 + 1.20461i −1.45188 + 0.308606i 0.320145 − 0.232599i 1.90016 − 3.29117i −0.940018 − 1.62816i 1.99974 + 0.890342i 2.45490 + 1.78359i −0.727921 + 0.324092i 4.70831 + 1.00078i
846.1 0.108599 − 0.334232i 2.83007 + 0.601549i 1.51812 + 1.10298i −1.48661 − 2.57489i 0.508398 − 0.880572i −0.988571 + 0.440140i 1.10215 − 0.800755i 4.90678 + 2.18464i −1.02206 + 0.217245i
846.2 0.391401 − 1.20461i −1.45188 − 0.308606i 0.320145 + 0.232599i 1.90016 + 3.29117i −0.940018 + 1.62816i 1.99974 − 0.890342i 2.45490 − 1.78359i −0.727921 − 0.324092i 4.70831 − 1.00078i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 235.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.g even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 961.2.g.t 16
31.b odd 2 1 961.2.g.n 16
31.c even 3 1 961.2.d.p 16
31.c even 3 1 961.2.g.s 16
31.d even 5 1 31.2.g.a ✓ 16
31.d even 5 1 961.2.c.j 16
31.d even 5 1 961.2.g.k 16
31.d even 5 1 961.2.g.s 16
31.e odd 6 1 961.2.d.q 16
31.e odd 6 1 961.2.g.m 16
31.f odd 10 1 961.2.c.i 16
31.f odd 10 1 961.2.g.j 16
31.f odd 10 1 961.2.g.l 16
31.f odd 10 1 961.2.g.m 16
31.g even 15 1 31.2.g.a ✓ 16
31.g even 15 1 961.2.a.i 8
31.g even 15 1 961.2.c.j 16
31.g even 15 2 961.2.d.o 16
31.g even 15 1 961.2.d.p 16
31.g even 15 1 961.2.g.k 16
31.g even 15 1 inner 961.2.g.t 16
31.h odd 30 1 961.2.a.j 8
31.h odd 30 1 961.2.c.i 16
31.h odd 30 2 961.2.d.n 16
31.h odd 30 1 961.2.d.q 16
31.h odd 30 1 961.2.g.j 16
31.h odd 30 1 961.2.g.l 16
31.h odd 30 1 961.2.g.n 16
93.l odd 10 1 279.2.y.c 16
93.o odd 30 1 279.2.y.c 16
93.o odd 30 1 8649.2.a.bf 8
93.p even 30 1 8649.2.a.be 8
124.l odd 10 1 496.2.bg.c 16
124.n odd 30 1 496.2.bg.c 16
155.n even 10 1 775.2.bl.a 16
155.s odd 20 2 775.2.ck.a 32
155.u even 30 1 775.2.bl.a 16
155.w odd 60 2 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 31.d even 5 1
31.2.g.a ✓ 16 31.g even 15 1
279.2.y.c 16 93.l odd 10 1
279.2.y.c 16 93.o odd 30 1
496.2.bg.c 16 124.l odd 10 1
496.2.bg.c 16 124.n odd 30 1
775.2.bl.a 16 155.n even 10 1
775.2.bl.a 16 155.u even 30 1
775.2.ck.a 32 155.s odd 20 2
775.2.ck.a 32 155.w odd 60 2
961.2.a.i 8 31.g even 15 1
961.2.a.j 8 31.h odd 30 1
961.2.c.i 16 31.f odd 10 1
961.2.c.i 16 31.h odd 30 1
961.2.c.j 16 31.d even 5 1
961.2.c.j 16 31.g even 15 1
961.2.d.n 16 31.h odd 30 2
961.2.d.o 16 31.g even 15 2
961.2.d.p 16 31.c even 3 1
961.2.d.p 16 31.g even 15 1
961.2.d.q 16 31.e odd 6 1
961.2.d.q 16 31.h odd 30 1
961.2.g.j 16 31.f odd 10 1
961.2.g.j 16 31.h odd 30 1
961.2.g.k 16 31.d even 5 1
961.2.g.k 16 31.g even 15 1
961.2.g.l 16 31.f odd 10 1
961.2.g.l 16 31.h odd 30 1
961.2.g.m 16 31.e odd 6 1
961.2.g.m 16 31.f odd 10 1
961.2.g.n 16 31.b odd 2 1
961.2.g.n 16 31.h odd 30 1
961.2.g.s 16 31.c even 3 1
961.2.g.s 16 31.d even 5 1
961.2.g.t 16 1.a even 1 1 trivial
961.2.g.t 16 31.g even 15 1 inner
8649.2.a.be 8 93.p even 30 1
8649.2.a.bf 8 93.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(961, [\chi])\):

\( T_{2}^{16} - 4 T_{2}^{15} + 9 T_{2}^{14} - 4 T_{2}^{13} + 6 T_{2}^{12} - 22 T_{2}^{11} + 221 T_{2}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{16} - 3 T_{3}^{15} - T_{3}^{14} - 9 T_{3}^{13} + 42 T_{3}^{12} - 18 T_{3}^{11} + 148 T_{3}^{10} + \cdots + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} - 3 T^{15} + \cdots + 961 \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{15} + \cdots + 68121 \) Copy content Toggle raw display
$11$ \( T^{16} + 2 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$13$ \( T^{16} - 18 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$17$ \( T^{16} + T^{15} + \cdots + 74805201 \) Copy content Toggle raw display
$19$ \( T^{16} - 11 T^{15} + \cdots + 361201 \) Copy content Toggle raw display
$23$ \( T^{16} - 21 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$29$ \( T^{16} - 26 T^{15} + \cdots + 77841 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 344807761 \) Copy content Toggle raw display
$41$ \( T^{16} + 3 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{16} - 8 T^{15} + \cdots + 7612081 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3306365001 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 366207732801 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 167728401 \) Copy content Toggle raw display
$61$ \( (T^{8} + 30 T^{7} + \cdots + 38161)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 7485883441 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 214944921 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 17441907675201 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 84609661119201 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1446653267361 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 117957215601 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 7131992195241 \) Copy content Toggle raw display
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