Properties

Label 4010.2.a.k
Level $4010$
Weight $2$
Character orbit 4010.a
Self dual yes
Analytic conductor $32.020$
Analytic rank $1$
Dimension $15$
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] = [4010,2,Mod(1,4010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4010.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: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} - 5452 x^{7} - 4098 x^{6} + 9986 x^{5} + 850 x^{4} - 7216 x^{3} + 1688 x^{2} + \cdots - 512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} - 7 x^{13} + 133 x^{12} - 99 x^{11} - 941 x^{10} + 1290 x^{9} + 3031 x^{8} - 5452 x^{7} - 4098 x^{6} + 9986 x^{5} + 850 x^{4} - 7216 x^{3} + 1688 x^{2} + \cdots - 512 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 58437627 \nu^{14} - 534122576 \nu^{13} + 115180567 \nu^{12} + 10329491164 \nu^{11} - 16005728457 \nu^{10} - 76582202190 \nu^{9} + \cdots + 117611134796 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 20542792 \nu^{14} - 146543981 \nu^{13} - 164096893 \nu^{12} + 2932664074 \nu^{11} - 1662959242 \nu^{10} - 22602084955 \nu^{9} + \cdots + 23702683016 ) / 255927730 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129009841 \nu^{14} - 883037733 \nu^{13} - 1197107414 \nu^{12} + 17651323112 \nu^{11} - 7047514031 \nu^{10} - 135332605845 \nu^{9} + \cdots + 125216059318 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 58921429 \nu^{14} + 416412812 \nu^{13} + 480260011 \nu^{12} - 8324831388 \nu^{11} + 4686009699 \nu^{10} + 63902469990 \nu^{9} + \cdots - 67211679492 ) / 511855460 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 355634499 \nu^{14} - 2493292912 \nu^{13} - 2946774221 \nu^{12} + 49230821268 \nu^{11} - 25215809709 \nu^{10} - 372735449930 \nu^{9} + \cdots + 330809703252 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 357030859 \nu^{14} + 2078381242 \nu^{13} + 4813725111 \nu^{12} - 41262341138 \nu^{11} - 10365332281 \nu^{10} + 312320702630 \nu^{9} + \cdots - 192795588882 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 91199013 \nu^{14} + 582879604 \nu^{13} + 1011640657 \nu^{12} - 11599830506 \nu^{11} + 1677789343 \nu^{10} + 88300145100 \nu^{9} + \cdots - 72900927894 ) / 255927730 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1161424079 \nu^{14} + 7214691402 \nu^{13} + 13595344791 \nu^{12} - 142730708728 \nu^{11} + 6317989589 \nu^{10} + 1078958237680 \nu^{9} + \cdots - 835598044392 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1173296833 \nu^{14} + 7231451704 \nu^{13} + 14132908607 \nu^{12} - 143965640456 \nu^{11} + 153207403 \nu^{10} + 1095593363710 \nu^{9} + \cdots - 845081247084 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 371844021 \nu^{14} + 2450736548 \nu^{13} + 3766297784 \nu^{12} - 48621859147 \nu^{11} + 13776069811 \nu^{10} + 369500503845 \nu^{9} + \cdots - 316360212508 ) / 639819325 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2202353061 \nu^{14} + 13831059568 \nu^{13} + 25351582419 \nu^{12} - 274524650952 \nu^{11} + 20706458751 \nu^{10} + \cdots - 1574486438628 ) / 2559277300 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1275103963 \nu^{14} + 8052863369 \nu^{13} + 14434590052 \nu^{12} - 159745442766 \nu^{11} + 17211325133 \nu^{10} + \cdots - 960771925824 ) / 1279638650 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1426055511 \nu^{14} + 8868477043 \nu^{13} + 16772024394 \nu^{12} - 176015990502 \nu^{11} + 6774287801 \nu^{10} + \cdots - 1018945552978 ) / 1279638650 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{14} + \beta_{12} + 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + 3 \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 8 \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 17 \beta_{14} - 3 \beta_{13} + 5 \beta_{12} + 13 \beta_{11} + 5 \beta_{10} + 15 \beta_{9} + 16 \beta_{8} - 2 \beta_{7} + 12 \beta_{6} - 12 \beta_{5} + 7 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 18 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 64 \beta_{14} - 8 \beta_{13} + 26 \beta_{12} + 38 \beta_{11} + 25 \beta_{10} + 47 \beta_{9} + 62 \beta_{8} - 16 \beta_{7} + 36 \beta_{6} - 36 \beta_{5} + 45 \beta_{4} + 14 \beta_{3} + 34 \beta_{2} + 92 \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 291 \beta_{14} - 63 \beta_{13} + 113 \beta_{12} + 190 \beta_{11} + 118 \beta_{10} + 239 \beta_{9} + 272 \beta_{8} - 46 \beta_{7} + 166 \beta_{6} - 176 \beta_{5} + 175 \beta_{4} + 36 \beta_{3} + 148 \beta_{2} + 305 \beta _1 + 165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1180 \beta_{14} - 218 \beta_{13} + 503 \beta_{12} + 688 \beta_{11} + 503 \beta_{10} + 896 \beta_{9} + 1128 \beta_{8} - 240 \beta_{7} + 614 \beta_{6} - 648 \beta_{5} + 850 \beta_{4} + 180 \beta_{3} + 552 \beta_{2} + \cdots + 468 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 5098 \beta_{14} - 1146 \beta_{13} + 2134 \beta_{12} + 3105 \beta_{11} + 2248 \beta_{10} + 4040 \beta_{9} + 4785 \beta_{8} - 843 \beta_{7} + 2647 \beta_{6} - 2917 \beta_{5} + 3467 \beta_{4} + 569 \beta_{3} + \cdots + 2333 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21215 \beta_{14} - 4459 \beta_{13} + 9167 \beta_{12} + 12348 \beta_{11} + 9418 \beta_{10} + 16284 \beta_{9} + 20152 \beta_{8} - 3831 \beta_{7} + 10663 \beta_{6} - 11692 \beta_{5} + 15430 \beta_{4} + \cdots + 8640 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 90328 \beta_{14} - 20460 \beta_{13} + 38737 \beta_{12} + 53525 \beta_{11} + 40822 \beta_{10} + 70440 \beta_{9} + 85094 \beta_{8} - 14902 \beta_{7} + 45240 \beta_{6} - 50749 \beta_{5} + \cdots + 38650 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 379511 \beta_{14} - 83664 \beta_{13} + 164653 \beta_{12} + 220960 \beta_{11} + 171450 \beta_{10} + 292222 \beta_{9} + 359487 \beta_{8} - 64557 \beta_{7} + 187961 \beta_{6} - 210223 \beta_{5} + \cdots + 156303 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1608230 \beta_{14} - 364938 \beta_{13} + 695590 \beta_{12} + 942964 \beta_{11} + 732462 \beta_{10} + 1245514 \beta_{9} + 1518069 \beta_{8} - 263329 \beta_{7} + 794776 \beta_{6} + \cdots + 674056 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 6781632 \beta_{14} - 1522673 \beta_{13} + 2946121 \beta_{12} + 3949413 \beta_{11} + 3086141 \beta_{10} + 5225929 \beta_{9} + 6416439 \beta_{8} - 1122120 \beta_{7} + \cdots + 2805548 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 28690546 \beta_{14} - 6514636 \beta_{13} + 12448701 \beta_{12} + 16753801 \beta_{11} + 13101570 \beta_{10} + 22155352 \beta_{9} + 27107353 \beta_{8} - 4672966 \beta_{7} + \cdots + 11945030 \) 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
−2.07374
0.679191
2.09866
4.22659
−2.62108
−1.71302
1.95650
1.81559
2.79590
−1.99921
0.698269
−0.560392
−1.05167
2.25248
0.495927
−1.00000 −3.34293 1.00000 −1.00000 3.34293 3.45446 −1.00000 8.17517 1.00000
1.2 −1.00000 −3.00420 1.00000 −1.00000 3.00420 −4.38135 −1.00000 6.02524 1.00000
1.3 −1.00000 −2.86623 1.00000 −1.00000 2.86623 0.981408 −1.00000 5.21525 1.00000
1.4 −1.00000 −2.24378 1.00000 −1.00000 2.24378 −2.96737 −1.00000 2.03457 1.00000
1.5 −1.00000 −1.88769 1.00000 −1.00000 1.88769 1.11871 −1.00000 0.563364 1.00000
1.6 −1.00000 −1.21181 1.00000 −1.00000 1.21181 0.441519 −1.00000 −1.53151 1.00000
1.7 −1.00000 −0.918908 1.00000 −1.00000 0.918908 −3.92602 −1.00000 −2.15561 1.00000
1.8 −1.00000 −0.876107 1.00000 −1.00000 0.876107 4.75824 −1.00000 −2.23244 1.00000
1.9 −1.00000 −0.154802 1.00000 −1.00000 0.154802 −4.41535 −1.00000 −2.97604 1.00000
1.10 −1.00000 0.271531 1.00000 −1.00000 −0.271531 1.75980 −1.00000 −2.92627 1.00000
1.11 −1.00000 1.23027 1.00000 −1.00000 −1.23027 3.50539 −1.00000 −1.48643 1.00000
1.12 −1.00000 1.40727 1.00000 −1.00000 −1.40727 −2.31473 −1.00000 −1.01959 1.00000
1.13 −1.00000 1.72412 1.00000 −1.00000 −1.72412 −1.94532 −1.00000 −0.0274146 1.00000
1.14 −1.00000 2.71977 1.00000 −1.00000 −2.71977 −0.594829 −1.00000 4.39715 1.00000
1.15 −1.00000 3.15350 1.00000 −1.00000 −3.15350 −0.474552 −1.00000 6.94456 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(401\) \(1\)

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 4010.2.a.k 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.k 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\( T_{3}^{15} + 6 T_{3}^{14} - 14 T_{3}^{13} - 140 T_{3}^{12} - 35 T_{3}^{11} + 1126 T_{3}^{10} + 1250 T_{3}^{9} - 3717 T_{3}^{8} - 6016 T_{3}^{7} + 4668 T_{3}^{6} + 11022 T_{3}^{5} - 300 T_{3}^{4} - 7376 T_{3}^{3} - 2304 T_{3}^{2} + \cdots + 128 \) Copy content Toggle raw display
\( T_{7}^{15} + 5 T_{7}^{14} - 50 T_{7}^{13} - 266 T_{7}^{12} + 845 T_{7}^{11} + 5014 T_{7}^{10} - 5611 T_{7}^{9} - 41345 T_{7}^{8} + 11984 T_{7}^{7} + 148096 T_{7}^{6} - 10432 T_{7}^{5} - 223212 T_{7}^{4} + 23616 T_{7}^{3} + \cdots - 14080 \) Copy content Toggle raw display
\( T_{11}^{15} + 2 T_{11}^{14} - 72 T_{11}^{13} - 56 T_{11}^{12} + 1978 T_{11}^{11} - 320 T_{11}^{10} - 24792 T_{11}^{9} + 19212 T_{11}^{8} + 141453 T_{11}^{7} - 166978 T_{11}^{6} - 306328 T_{11}^{5} + 463084 T_{11}^{4} + 114928 T_{11}^{3} + \cdots - 2560 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + 6 T^{14} - 14 T^{13} - 140 T^{12} + \cdots + 128 \) Copy content Toggle raw display
$5$ \( (T + 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 5 T^{14} - 50 T^{13} + \cdots - 14080 \) Copy content Toggle raw display
$11$ \( T^{15} + 2 T^{14} - 72 T^{13} + \cdots - 2560 \) Copy content Toggle raw display
$13$ \( T^{15} + 13 T^{14} - 19 T^{13} + \cdots - 265268 \) Copy content Toggle raw display
$17$ \( T^{15} - 11 T^{14} - 54 T^{13} + \cdots - 50096 \) Copy content Toggle raw display
$19$ \( T^{15} + 15 T^{14} - 26 T^{13} + \cdots + 4323968 \) Copy content Toggle raw display
$23$ \( T^{15} + 3 T^{14} - 221 T^{13} + \cdots + 133743104 \) Copy content Toggle raw display
$29$ \( T^{15} - 28 T^{14} + \cdots + 352010240 \) Copy content Toggle raw display
$31$ \( T^{15} + 12 T^{14} + \cdots - 182380960 \) Copy content Toggle raw display
$37$ \( T^{15} + 23 T^{14} + 76 T^{13} + \cdots + 4181512 \) Copy content Toggle raw display
$41$ \( T^{15} - 24 T^{14} + \cdots + 119996288 \) Copy content Toggle raw display
$43$ \( T^{15} + 24 T^{14} + \cdots + 195278336 \) Copy content Toggle raw display
$47$ \( T^{15} + 3 T^{14} - 304 T^{13} + \cdots + 185155072 \) Copy content Toggle raw display
$53$ \( T^{15} - 10 T^{14} + \cdots + 125904188 \) Copy content Toggle raw display
$59$ \( T^{15} - 2 T^{14} + \cdots + 49410713920 \) Copy content Toggle raw display
$61$ \( T^{15} - 15 T^{14} + \cdots - 6149301808576 \) Copy content Toggle raw display
$67$ \( T^{15} + 48 T^{14} + \cdots - 157282369024 \) Copy content Toggle raw display
$71$ \( T^{15} - 15 T^{14} + \cdots + 31935740224 \) Copy content Toggle raw display
$73$ \( T^{15} + 47 T^{14} + \cdots - 34713267712 \) Copy content Toggle raw display
$79$ \( T^{15} + 34 T^{14} + \cdots - 7250505143072 \) Copy content Toggle raw display
$83$ \( T^{15} + 32 T^{14} + \cdots + 9471377325856 \) Copy content Toggle raw display
$89$ \( T^{15} - 25 T^{14} + \cdots + 30055051552 \) Copy content Toggle raw display
$97$ \( T^{15} + 34 T^{14} + \cdots - 22082901745448 \) Copy content Toggle raw display
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