Properties

Label 201.2.e.b
Level $201$
Weight $2$
Character orbit 201.e
Analytic conductor $1.605$
Analytic rank $0$
Dimension $10$
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] = [201,2,Mod(37,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.e (of order \(3\), 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: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 49x^{6} - 39x^{5} + 128x^{4} - 14x^{3} + 119x^{2} - 49x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 49x^{6} - 39x^{5} + 128x^{4} - 14x^{3} + 119x^{2} - 49x + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24307 \nu^{9} + 181995 \nu^{8} - 604465 \nu^{7} + 1626651 \nu^{6} - 3301924 \nu^{5} + 7319233 \nu^{4} - 12172941 \nu^{3} + 14158031 \nu^{2} + \cdots + 6437963 ) / 4414074 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32653 \nu^{9} - 100089 \nu^{8} + 396649 \nu^{7} - 539103 \nu^{6} + 1764532 \nu^{5} - 2509639 \nu^{4} + 5937753 \nu^{3} - 2828441 \nu^{2} + 1453144 \nu - 2618021 ) / 4414074 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53429 \nu^{9} + 74205 \nu^{8} - 434201 \nu^{7} + 30783 \nu^{6} - 2078918 \nu^{5} + 319199 \nu^{4} - 4329273 \nu^{3} - 5189747 \nu^{2} - 3529610 \nu + 1164877 ) / 4414074 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33718 \nu^{9} - 85104 \nu^{8} + 337264 \nu^{7} - 351819 \nu^{6} + 1500352 \nu^{5} - 2133904 \nu^{4} + 4154526 \nu^{3} - 2404976 \nu^{2} + 1235584 \nu - 7130123 ) / 2207037 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31943 \nu^{9} + 110079 \nu^{8} - 436239 \nu^{7} + 663959 \nu^{6} - 1940652 \nu^{5} + 2760129 \nu^{4} - 5655213 \nu^{3} + 3110751 \nu^{2} + \cdots + 1081311 ) / 1471358 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 112429 \nu^{9} - 45435 \nu^{8} - 824203 \nu^{7} - 1270707 \nu^{6} - 5458108 \nu^{5} - 6300671 \nu^{4} - 13242393 \nu^{3} - 19664257 \nu^{2} + \cdots - 11599567 ) / 4414074 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 62752 \nu^{9} + 76248 \nu^{8} - 512362 \nu^{7} - 36093 \nu^{6} - 2500291 \nu^{5} - 400286 \nu^{4} - 5308260 \nu^{3} - 6568339 \nu^{2} - 6992440 \nu - 4073797 ) / 2207037 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 117800 \nu^{9} + 300861 \nu^{8} - 1192301 \nu^{7} + 1489527 \nu^{6} - 5304068 \nu^{5} + 7543811 \nu^{4} - 12249042 \nu^{3} + 8502109 \nu^{2} + \cdots + 12355714 ) / 2207037 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 5\beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{8} + \beta_{7} + \beta_{6} + 13\beta_{4} - \beta_{2} - 8\beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 10\beta_{8} + 2\beta_{7} + 10\beta_{5} + 9\beta_{4} - 29\beta_{3} - 8\beta_{2} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} - 12\beta_{6} + 45\beta_{5} - 56\beta_{3} + 65 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 80\beta_{8} - 22\beta_{7} - 55\beta_{6} - 68\beta_{4} + 55\beta_{2} + 178\beta _1 + 68 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -77\beta_{9} + 288\beta_{8} - 77\beta_{7} - 288\beta_{5} - 352\beta_{4} + 381\beta_{3} + 102\beta_{2} + 381\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -179\beta_{9} + 365\beta_{6} - 585\beta_{5} + 1123\beta_{3} - 490 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/201\mathbb{Z}\right)^\times\).

\(n\) \(68\) \(136\)
\(\chi(n)\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
1.28367 + 2.22339i
0.947050 + 1.64034i
0.330147 + 0.571831i
−0.527336 0.913372i
−1.03354 1.79014i
1.28367 2.22339i
0.947050 1.64034i
0.330147 0.571831i
−0.527336 + 0.913372i
−1.03354 + 1.79014i
−1.28367 2.22339i −1.00000 −2.29564 + 3.97616i −1.02393 1.28367 + 2.22339i −0.850574 + 1.47324i 6.65272 1.00000 1.31439 + 2.27659i
37.2 −0.947050 1.64034i −1.00000 −0.793808 + 1.37492i 1.30648 0.947050 + 1.64034i 2.53710 4.39438i −0.781096 1.00000 −1.23731 2.14308i
37.3 −0.330147 0.571831i −1.00000 0.782006 1.35447i 3.22431 0.330147 + 0.571831i −1.02143 + 1.76916i −2.35330 1.00000 −1.06450 1.84376i
37.4 0.527336 + 0.913372i −1.00000 0.443834 0.768743i 0.832996 −0.527336 0.913372i 1.13029 1.95772i 3.04554 1.00000 0.439269 + 0.760836i
37.5 1.03354 + 1.79014i −1.00000 −1.13639 + 1.96829i −3.33986 −1.03354 1.79014i −2.29539 + 3.97573i −0.563865 1.00000 −3.45186 5.97880i
163.1 −1.28367 + 2.22339i −1.00000 −2.29564 3.97616i −1.02393 1.28367 2.22339i −0.850574 1.47324i 6.65272 1.00000 1.31439 2.27659i
163.2 −0.947050 + 1.64034i −1.00000 −0.793808 1.37492i 1.30648 0.947050 1.64034i 2.53710 + 4.39438i −0.781096 1.00000 −1.23731 + 2.14308i
163.3 −0.330147 + 0.571831i −1.00000 0.782006 + 1.35447i 3.22431 0.330147 0.571831i −1.02143 1.76916i −2.35330 1.00000 −1.06450 + 1.84376i
163.4 0.527336 0.913372i −1.00000 0.443834 + 0.768743i 0.832996 −0.527336 + 0.913372i 1.13029 + 1.95772i 3.04554 1.00000 0.439269 0.760836i
163.5 1.03354 1.79014i −1.00000 −1.13639 1.96829i −3.33986 −1.03354 + 1.79014i −2.29539 3.97573i −0.563865 1.00000 −3.45186 + 5.97880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.e.b 10
3.b odd 2 1 603.2.g.g 10
67.c even 3 1 inner 201.2.e.b 10
201.g odd 6 1 603.2.g.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.e.b 10 1.a even 1 1 trivial
201.2.e.b 10 67.c even 3 1 inner
603.2.g.g 10 3.b odd 2 1
603.2.g.g 10 201.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 2T_{2}^{9} + 10T_{2}^{8} + 8T_{2}^{7} + 49T_{2}^{6} + 39T_{2}^{5} + 128T_{2}^{4} + 14T_{2}^{3} + 119T_{2}^{2} + 49T_{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + 10 T^{8} + 8 T^{7} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} - T^{4} - 12 T^{3} + 13 T^{2} + 12 T - 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + 30 T^{8} + 51 T^{7} + \cdots + 33489 \) Copy content Toggle raw display
$11$ \( T^{10} - 2 T^{9} + 25 T^{8} + 14 T^{7} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} + 42 T^{8} - 57 T^{7} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{10} - 2 T^{9} + 40 T^{8} + \cdots + 219961 \) Copy content Toggle raw display
$19$ \( T^{10} - 3 T^{9} + 32 T^{8} + \cdots + 4489 \) Copy content Toggle raw display
$23$ \( T^{10} + T^{9} + 31 T^{8} - 18 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$29$ \( T^{10} - 12 T^{9} + 180 T^{8} + \cdots + 20421361 \) Copy content Toggle raw display
$31$ \( T^{10} + 12 T^{9} + 109 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( T^{10} - 27 T^{9} + \cdots + 381303729 \) Copy content Toggle raw display
$41$ \( T^{10} + 7 T^{9} + 140 T^{8} + \cdots + 53275401 \) Copy content Toggle raw display
$43$ \( (T^{5} - 6 T^{4} - 171 T^{3} + 1117 T^{2} + \cdots - 41232)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 33 T^{9} + 702 T^{8} + \cdots + 32867289 \) Copy content Toggle raw display
$53$ \( (T^{5} + 12 T^{4} - 89 T^{3} - 1745 T^{2} + \cdots - 9476)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} - 12 T^{4} - 67 T^{3} + 1039 T^{2} + \cdots - 3792)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} - T^{9} + 150 T^{8} + \cdots + 166590649 \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + \cdots + 1350125107 \) Copy content Toggle raw display
$71$ \( T^{10} + T^{9} + 199 T^{8} + \cdots + 379041961 \) Copy content Toggle raw display
$73$ \( T^{10} - 12 T^{9} + \cdots + 3692871361 \) Copy content Toggle raw display
$79$ \( T^{10} - 7 T^{9} + 108 T^{8} + \cdots + 187489 \) Copy content Toggle raw display
$83$ \( T^{10} - 12 T^{9} + 138 T^{8} - 126 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{5} - 6 T^{4} - 49 T^{3} + 227 T^{2} + \cdots + 156)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 9 T^{9} + 192 T^{8} + \cdots + 23902321 \) Copy content Toggle raw display
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