Properties

Label 625.2.d.o
Level $625$
Weight $2$
Character orbit 625.d
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} - \beta_{11} - \beta_1) q^{2} + (\beta_{13} - \beta_{12} + \beta_{11} - \beta_{6}) q^{3} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{4} + (\beta_{8} + 2 \beta_{7} + 2 \beta_{5} + 2) q^{6} + ( - \beta_{13} - \beta_{10}) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12}) q^{8} + ( - \beta_{8} + \beta_{4} - \beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} - \beta_{11} - \beta_1) q^{2} + (\beta_{13} - \beta_{12} + \beta_{11} - \beta_{6}) q^{3} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{4} + (\beta_{8} + 2 \beta_{7} + 2 \beta_{5} + 2) q^{6} + ( - \beta_{13} - \beta_{10}) q^{7} + ( - \beta_{15} + \beta_{13} + \beta_{12}) q^{8} + ( - \beta_{8} + \beta_{4} - \beta_{3} - \beta_{2}) q^{9} + 2 \beta_{5} q^{11} + ( - 2 \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - \beta_1) q^{12} + ( - \beta_{15} - \beta_{12} + \beta_{10} - \beta_{6} - \beta_1) q^{13} + ( - \beta_{8} - 2 \beta_{7} - 2 \beta_{4} - \beta_{2}) q^{14} + (\beta_{8} - \beta_{4} + \beta_{3}) q^{16} + ( - \beta_{15} + \beta_{13} - \beta_{12} + \beta_{6}) q^{17} + (\beta_{14} + 2 \beta_{13}) q^{18} + ( - \beta_{8} - 2 \beta_{5} - 2) q^{19} + (\beta_{9} + \beta_{8}) q^{21} + (2 \beta_{14} - 2 \beta_{12} + 2 \beta_{10} - 2 \beta_{6}) q^{22} + ( - 2 \beta_{15} - \beta_{14} + 2 \beta_{12} + \beta_{6} - \beta_1) q^{23} + ( - \beta_{9} + 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 4) q^{24} + ( - \beta_{9} - 3 \beta_{5} - 3 \beta_{4} - \beta_{2}) q^{26} + (\beta_{14} + 2 \beta_{11} - \beta_{6} + \beta_1) q^{27} + (\beta_{14} - \beta_{12} - 3 \beta_{11} + \beta_{10} + 2 \beta_{6} - 3 \beta_1) q^{28} + (2 \beta_{9} + 2 \beta_{8} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 3) q^{29} + (\beta_{9} - 2 \beta_{8} + 2 \beta_{7} + \beta_{3}) q^{31} + ( - 4 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{10} - 4 \beta_1) q^{32} + (2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{6}) q^{33} + (\beta_{8} + 3 \beta_{7} + \beta_{4} + \beta_{3} + 3) q^{34} + (\beta_{8} + 4 \beta_{7} + \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{36} + ( - 2 \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} + \beta_1) q^{37} + (2 \beta_{15} - \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} - 2 \beta_{10} + 2 \beta_{6} + \beta_1) q^{38} + (\beta_{9} + \beta_{8} + 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{39} + ( - 2 \beta_{8} - \beta_{7} + 4 \beta_{4} - 2 \beta_{3} - \beta_{2} - 1) q^{41} + ( - 2 \beta_{15} + 2 \beta_{13} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{6}) q^{42} + ( - 3 \beta_{15} - \beta_{14} + 2 \beta_{13} - \beta_{10} - 3 \beta_1) q^{43} + (2 \beta_{8} + 2 \beta_{7}) q^{44} + ( - \beta_{9} - \beta_{8} - 4 \beta_{7} - 4 \beta_{5} + \beta_{3} + \beta_{2}) q^{46} + ( - 3 \beta_{14} - \beta_{13} + 4 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + 5 \beta_{6} + \cdots - \beta_1) q^{47}+ \cdots + ( - 2 \beta_{9} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 12 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 12 q^{6} - 2 q^{9} - 8 q^{11} + 14 q^{14} - 4 q^{16} - 20 q^{19} + 2 q^{21} + 40 q^{24} + 12 q^{26} - 30 q^{29} + 2 q^{31} + 24 q^{34} - 34 q^{36} - 24 q^{39} - 18 q^{41} - 16 q^{44} + 32 q^{46} - 28 q^{49} - 8 q^{51} + 20 q^{54} - 30 q^{56} - 30 q^{59} + 12 q^{61} + 52 q^{64} - 36 q^{66} + 26 q^{69} - 58 q^{71} + 24 q^{74} - 40 q^{76} + 20 q^{79} - 24 q^{81} + 54 q^{84} + 32 q^{86} - 80 q^{89} + 2 q^{91} + 14 q^{94} + 22 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 22849 \nu^{15} - 2422021 \nu^{13} + 6573709 \nu^{11} - 1538146 \nu^{9} + 97097069 \nu^{7} - 420964990 \nu^{5} - 210825325 \nu^{3} - 35093875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + 77053830 \nu^{4} + 27131400 \nu^{2} + \cdots + 181387875 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + 54634025 \nu^{4} + 77779875 \nu^{2} + 52412250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + 424939915 \nu^{4} + 771306350 \nu^{2} + \cdots + 563878625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} - 83450271 \nu^{4} - 152170830 \nu^{2} - 167096475 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} - 137248467 \nu^{7} - 474206065 \nu^{5} + 222668250 \nu^{3} + \cdots + 846395125 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} - 135351515 \nu^{4} - 246968035 \nu^{2} + \cdots - 215410900 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} - 716617430 \nu^{4} - 884417625 \nu^{2} + \cdots - 860602375 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} - 210462216 \nu^{4} - 281904075 \nu^{2} + \cdots - 222107450 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 257689 \nu^{15} - 260409 \nu^{13} + 1184711 \nu^{11} + 13501191 \nu^{9} - 4440499 \nu^{7} - 113117275 \nu^{5} - 254856700 \nu^{3} + \cdots - 189940875 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3711283 \nu^{15} + 2442853 \nu^{13} - 18411087 \nu^{11} - 175485672 \nu^{9} + 137417033 \nu^{7} + 1546922530 \nu^{5} + 2678832450 \nu^{3} + \cdots + 1573606875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11296137 \nu^{15} - 2500148 \nu^{13} - 48798533 \nu^{11} - 491452273 \nu^{9} + 752782897 \nu^{7} + 3788808280 \nu^{5} + 5075395150 \nu^{3} + \cdots + 4625990250 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1033909 \nu^{15} - 651386 \nu^{13} - 3517956 \nu^{11} - 44316636 \nu^{9} + 84515054 \nu^{7} + 285294535 \nu^{5} + 432663800 \nu^{3} + \cdots + 416073000 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1468861 \nu^{15} - 201274 \nu^{13} - 5426879 \nu^{11} - 66369199 \nu^{9} + 90901286 \nu^{7} + 466969935 \nu^{5} + 843562400 \nu^{3} + \cdots + 722601875 \nu ) / 78081875 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18299323 \nu^{15} - 8550017 \nu^{13} - 66698807 \nu^{11} - 789938892 \nu^{9} + 1385305413 \nu^{7} + 5454301045 \nu^{5} + 7862490225 \nu^{3} + \cdots + 7417196625 \nu ) / 858900625 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{15} - 5\beta_{14} + 4\beta_{13} + 4\beta_{12} - 5\beta_{10} + 5\beta_{6} + \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{9} + 2\beta_{8} + 3\beta_{7} + 2\beta_{5} + 11\beta_{4} - 4\beta_{3} - \beta_{2} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{15} + 14\beta_{13} - 6\beta_{12} + 5\beta_{11} - 9\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{9} - 13\beta_{8} + 3\beta_{7} + 7\beta_{5} + 6\beta_{4} - 19\beta_{3} - 26\beta_{2} + 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 39\beta_{15} - 20\beta_{14} - 6\beta_{13} - 36\beta_{12} - 30\beta_{10} - 29\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23\beta_{9} - 23\beta_{8} + 63\beta_{7} + 52\beta_{5} + 156\beta_{4} + 11\beta_{3} - 11\beta_{2} + 144 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 26 \beta_{15} - 190 \beta_{14} + 184 \beta_{13} + 49 \beta_{12} + 190 \beta_{11} - 115 \beta_{10} + 115 \beta_{6} + 66 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -57\beta_{9} - 18\beta_{8} + 363\beta_{7} - 208\beta_{5} + 381\beta_{4} - 114\beta_{3} - 96\beta_{2} + 39 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -\beta_{15} + 284\beta_{13} - 286\beta_{12} + 285\beta_{11} + 285\beta_{10} + 190\beta_{6} - 479\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 208\beta_{9} - 688\beta_{8} + 208\beta_{7} - 493\beta_{5} - 149\beta_{4} - 344\beta_{3} - 896\beta_{2} - 606 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2344 \beta_{15} - 290 \beta_{14} - 1651 \beta_{13} - 1846 \beta_{12} + 195 \beta_{11} - 290 \beta_{10} + 95 \beta_{6} - 624 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2208 \beta_{9} - 683 \beta_{8} + 2888 \beta_{7} - 683 \beta_{5} + 3956 \beta_{4} + 2891 \beta_{3} + 1104 \beta_{2} + 1784 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 906 \beta_{15} - 4675 \beta_{14} + 2364 \beta_{13} + 3769 \beta_{12} + 7530 \beta_{11} + 4675 \beta_{6} + 5166 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 393 \beta_{9} + 1012 \beta_{8} + 10778 \beta_{7} - 18118 \beta_{5} - 619 \beta_{4} + 1631 \beta_{3} + 2024 \beta_{2} - 18511 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1639\beta_{15} + 19130\beta_{14} - 12431\beta_{13} - 9336\beta_{12} + 30920\beta_{10} - 13429\beta_1 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 - \beta_{4} - \beta_{5} - \beta_{7}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
−0.917186 + 1.66637i
−1.86824 + 0.357358i
1.86824 0.357358i
0.917186 1.66637i
−0.0566033 + 1.17421i
−0.644389 + 0.983224i
0.644389 0.983224i
0.0566033 1.17421i
−0.0566033 1.17421i
−0.644389 0.983224i
0.644389 + 0.983224i
0.0566033 + 1.17421i
−0.917186 1.66637i
−1.86824 0.357358i
1.86824 + 0.357358i
0.917186 + 1.66637i
−0.713605 2.19625i 0.384204 0.279141i −2.69625 + 1.95894i 0 −0.887234 0.644613i −3.03582 2.48990 + 1.80902i −0.857358 + 2.63868i 0
126.2 −0.350334 1.07822i −2.10569 + 1.52988i 0.578217 0.420099i 0 2.38723 + 1.73443i −0.407162 −2.48990 1.80902i 1.16637 3.58973i 0
126.3 0.350334 + 1.07822i 2.10569 1.52988i 0.578217 0.420099i 0 2.38723 + 1.73443i 0.407162 2.48990 + 1.80902i 1.16637 3.58973i 0
126.4 0.713605 + 2.19625i −0.384204 + 0.279141i −2.69625 + 1.95894i 0 −0.887234 0.644613i 3.03582 −2.48990 1.80902i −0.857358 + 2.63868i 0
251.1 −1.68703 + 1.22570i 0.679371 2.09089i 0.725700 2.23347i 0 1.41668 + 4.36010i −0.992398 0.224514 + 0.690983i −1.48322 1.07763i 0
251.2 −0.148189 + 0.107666i 0.454857 1.39991i −0.607666 + 1.87020i 0 0.0833172 + 0.256424i 3.26086 −0.224514 0.690983i 0.674207 + 0.489840i 0
251.3 0.148189 0.107666i −0.454857 + 1.39991i −0.607666 + 1.87020i 0 0.0833172 + 0.256424i −3.26086 0.224514 + 0.690983i 0.674207 + 0.489840i 0
251.4 1.68703 1.22570i −0.679371 + 2.09089i 0.725700 2.23347i 0 1.41668 + 4.36010i 0.992398 −0.224514 0.690983i −1.48322 1.07763i 0
376.1 −1.68703 1.22570i 0.679371 + 2.09089i 0.725700 + 2.23347i 0 1.41668 4.36010i −0.992398 0.224514 0.690983i −1.48322 + 1.07763i 0
376.2 −0.148189 0.107666i 0.454857 + 1.39991i −0.607666 1.87020i 0 0.0833172 0.256424i 3.26086 −0.224514 + 0.690983i 0.674207 0.489840i 0
376.3 0.148189 + 0.107666i −0.454857 1.39991i −0.607666 1.87020i 0 0.0833172 0.256424i −3.26086 0.224514 0.690983i 0.674207 0.489840i 0
376.4 1.68703 + 1.22570i −0.679371 2.09089i 0.725700 + 2.23347i 0 1.41668 4.36010i 0.992398 −0.224514 + 0.690983i −1.48322 + 1.07763i 0
501.1 −0.713605 + 2.19625i 0.384204 + 0.279141i −2.69625 1.95894i 0 −0.887234 + 0.644613i −3.03582 2.48990 1.80902i −0.857358 2.63868i 0
501.2 −0.350334 + 1.07822i −2.10569 1.52988i 0.578217 + 0.420099i 0 2.38723 1.73443i −0.407162 −2.48990 + 1.80902i 1.16637 + 3.58973i 0
501.3 0.350334 1.07822i 2.10569 + 1.52988i 0.578217 + 0.420099i 0 2.38723 1.73443i 0.407162 2.48990 1.80902i 1.16637 + 3.58973i 0
501.4 0.713605 2.19625i −0.384204 0.279141i −2.69625 1.95894i 0 −0.887234 + 0.644613i 3.03582 −2.48990 + 1.80902i −0.857358 2.63868i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.d.o 16
5.b even 2 1 inner 625.2.d.o 16
5.c odd 4 1 625.2.e.a 8
5.c odd 4 1 625.2.e.i 8
25.d even 5 2 125.2.d.b 16
25.d even 5 1 625.2.a.f 8
25.d even 5 1 inner 625.2.d.o 16
25.e even 10 2 125.2.d.b 16
25.e even 10 1 625.2.a.f 8
25.e even 10 1 inner 625.2.d.o 16
25.f odd 20 2 25.2.e.a 8
25.f odd 20 2 125.2.e.b 8
25.f odd 20 2 625.2.b.c 8
25.f odd 20 1 625.2.e.a 8
25.f odd 20 1 625.2.e.i 8
75.h odd 10 1 5625.2.a.x 8
75.j odd 10 1 5625.2.a.x 8
75.l even 20 2 225.2.m.a 8
100.h odd 10 1 10000.2.a.bj 8
100.j odd 10 1 10000.2.a.bj 8
100.l even 20 2 400.2.y.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.e.a 8 25.f odd 20 2
125.2.d.b 16 25.d even 5 2
125.2.d.b 16 25.e even 10 2
125.2.e.b 8 25.f odd 20 2
225.2.m.a 8 75.l even 20 2
400.2.y.c 8 100.l even 20 2
625.2.a.f 8 25.d even 5 1
625.2.a.f 8 25.e even 10 1
625.2.b.c 8 25.f odd 20 2
625.2.d.o 16 1.a even 1 1 trivial
625.2.d.o 16 5.b even 2 1 inner
625.2.d.o 16 25.d even 5 1 inner
625.2.d.o 16 25.e even 10 1 inner
625.2.e.a 8 5.c odd 4 1
625.2.e.a 8 25.f odd 20 1
625.2.e.i 8 5.c odd 4 1
625.2.e.i 8 25.f odd 20 1
5625.2.a.x 8 75.h odd 10 1
5625.2.a.x 8 75.j odd 10 1
10000.2.a.bj 8 100.h odd 10 1
10000.2.a.bj 8 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{16} + 8T_{2}^{14} + 38T_{2}^{12} + 146T_{2}^{10} + 755T_{2}^{8} + 1246T_{2}^{6} + 863T_{2}^{4} - 17T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{16} + 7T_{3}^{14} + 53T_{3}^{12} + 399T_{3}^{10} + 2105T_{3}^{8} + 4704T_{3}^{6} + 4448T_{3}^{4} - 448T_{3}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 8 T^{14} + 38 T^{12} + 146 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{16} + 7 T^{14} + 53 T^{12} + 399 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 21 T^{6} + 121 T^{4} - 116 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} - 3 T^{14} + 108 T^{12} + 479 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} + 53 T^{14} + 1178 T^{12} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + 50 T^{6} + 135 T^{5} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 7 T^{14} + 333 T^{12} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{8} + 15 T^{7} + 150 T^{6} + \cdots + 483025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - T^{7} - 3 T^{6} + 67 T^{5} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 23 T^{14} + \cdots + 13521270961 \) Copy content Toggle raw display
$41$ \( (T^{8} + 9 T^{7} + 62 T^{6} + 382 T^{5} + \cdots + 13456)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 129 T^{6} + 4421 T^{4} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 173 T^{14} + \cdots + 4294967296 \) Copy content Toggle raw display
$53$ \( T^{16} + 127 T^{14} + \cdots + 76661949773761 \) Copy content Toggle raw display
$59$ \( (T^{8} + 15 T^{7} + 220 T^{6} + \cdots + 4080400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + 252 T^{6} + \cdots + 116281)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 8 T^{14} + \cdots + 60523872256 \) Copy content Toggle raw display
$71$ \( (T^{8} + 29 T^{7} + 462 T^{6} + \cdots + 24245776)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 48 T^{14} + 6078 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + 270 T^{6} + \cdots + 33408400)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 247 T^{14} + \cdots + 9971220736 \) Copy content Toggle raw display
$89$ \( (T^{8} + 40 T^{7} + 740 T^{6} + \cdots + 1392400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 338 T^{14} + \cdots + 90\!\cdots\!61 \) Copy content Toggle raw display
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