Properties

Label 847.2.f.x
Level $847$
Weight $2$
Character orbit 847.f
Analytic conductor $6.763$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.f (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 77)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 86 x^{12} - 145 x^{11} + 245 x^{10} - 245 x^{9} + 640 x^{8} - 1175 x^{7} + 2135 x^{6} - 2300 x^{5} + 1850 x^{4} - 925 x^{3} + 700 x^{2} + \cdots + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33722118880975 \nu^{15} - 215936836813390 \nu^{14} + 154302737507927 \nu^{13} + \cdots + 88\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 686277383720070 \nu^{15} + \cdots - 15\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 874936745197272 \nu^{15} - 514706296938285 \nu^{14} + \cdots - 92\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!39 \nu^{15} + \cdots + 95\!\cdots\!50 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\!\cdots\!90 \nu^{15} + \cdots - 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16\!\cdots\!90 \nu^{15} + \cdots + 18\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 35\!\cdots\!14 \nu^{15} + \cdots + 82\!\cdots\!05 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\!\cdots\!60 \nu^{15} + \cdots - 29\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 73\!\cdots\!73 \nu^{15} + \cdots - 84\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 54\!\cdots\!19 \nu^{15} + \cdots + 86\!\cdots\!50 ) / 22\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{15} + \cdots + 59\!\cdots\!75 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 16\!\cdots\!01 \nu^{15} + \cdots + 16\!\cdots\!55 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 16\!\cdots\!14 \nu^{15} + \cdots - 10\!\cdots\!35 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 31\!\cdots\!64 \nu^{15} + \cdots + 34\!\cdots\!95 ) / 45\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 2\beta_{10} - 2\beta_{8} + 3\beta_{7} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + \beta_{8} - 5\beta_{6} - \beta_{5} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{15} - \beta_{14} - \beta_{11} - 8 \beta_{10} - 6 \beta_{9} - 15 \beta_{7} + \beta_{5} + \beta_{4} - 6 \beta_{2} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{14} - 7 \beta_{13} - 7 \beta_{12} + 7 \beta_{11} - 8 \beta_{8} + 8 \beta_{7} + 10 \beta_{6} - 10 \beta_{4} - \beta_{3} + 8 \beta_{2} - 18 \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7 \beta_{15} + \beta_{14} - 7 \beta_{13} + 11 \beta_{12} + 54 \beta_{10} + 35 \beta_{9} + 40 \beta_{8} + 47 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 42 \beta_{15} + 54 \beta_{14} + 13 \beta_{12} - 45 \beta_{11} + 12 \beta_{10} - 13 \beta_{9} + 30 \beta_{8} - 86 \beta_{7} + 76 \beta_{5} + 165 \beta_{4} + 42 \beta_{3} - 58 \beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 9 \beta_{15} + 38 \beta_{13} - 210 \beta_{12} + 89 \beta_{11} - 234 \beta_{10} - 89 \beta_{9} - 308 \beta_{8} + 80 \beta_{6} + 28 \beta_{5} + 210 \beta_{2} - 28 \beta _1 - 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 346 \beta_{15} - 346 \beta_{14} + 248 \beta_{13} + 117 \beta_{11} + 38 \beta_{10} + 290 \beta_{9} + 614 \beta_{7} - 524 \beta_{6} - 1000 \beta_{5} - 1000 \beta_{4} - 248 \beta_{3} + 290 \beta_{2} + 476 \beta _1 - 94 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 56 \beta_{14} - 56 \beta_{13} + 1290 \beta_{12} - 1290 \beta_{11} + 1992 \beta_{8} - 1992 \beta_{7} - 272 \beta_{6} + 272 \beta_{4} - 131 \beta_{3} - 1931 \beta_{2} + 328 \beta _1 + 1406 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2174 \beta_{15} + 1477 \beta_{14} - 2174 \beta_{13} - 913 \beta_{12} - 447 \beta_{10} - 1890 \beta_{9} - 1625 \beta_{8} - 2621 \beta_{7} + 6160 \beta_{6} + 6160 \beta_{5} + 3461 \beta_{4} + 697 \beta_{3} + \cdots - 3461 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 284 \beta_{15} + 848 \beta_{14} - 4374 \beta_{12} + 8050 \beta_{11} + 8579 \beta_{10} + 4374 \beta_{9} - 8295 \beta_{8} + 21093 \beta_{7} - 2273 \beta_{5} - 4375 \beta_{4} + 284 \beta_{3} + 12424 \beta_{2} + \cdots - 8669 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 8898 \beta_{15} + 13556 \beta_{13} + 12425 \beta_{12} - 6647 \beta_{11} + 2237 \beta_{10} + 6647 \beta_{9} + 17819 \beta_{8} - 38325 \beta_{6} - 22394 \beta_{5} - 12425 \beta_{2} + 22394 \beta _1 + 5394 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 3382 \beta_{15} - 3382 \beta_{14} + 1131 \beta_{13} - 29041 \beta_{11} - 85364 \beta_{10} - 50750 \beta_{9} - 133987 \beta_{7} + 17557 \beta_{6} + 31350 \beta_{5} + 31350 \beta_{4} - 1131 \beta_{3} + \cdots + 31232 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 54132 \beta_{14} - 54132 \beta_{13} - 82100 \beta_{12} + 82100 \beta_{11} - 120596 \beta_{8} + 120596 \beta_{7} + 143446 \beta_{6} - 143446 \beta_{4} - 30172 \beta_{3} + 128698 \beta_{2} + \cdots - 67213 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(365\)
\(\chi(n)\) \(1\) \(-1 + \beta_{7} - \beta_{8} + \beta_{10}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
148.1
−0.788594 2.42704i
−0.206962 0.636964i
0.435488 + 1.34029i
0.751051 + 2.31150i
−1.38112 1.00344i
0.183009 + 0.132964i
0.901622 + 0.655067i
1.60551 + 1.16647i
−0.788594 + 2.42704i
−0.206962 + 0.636964i
0.435488 1.34029i
0.751051 2.31150i
−1.38112 + 1.00344i
0.183009 0.132964i
0.901622 0.655067i
1.60551 1.16647i
−0.788594 2.42704i 0.332181 + 0.241344i −3.65062 + 2.65233i 1.05961 3.26115i 0.323795 0.996539i 0.809017 0.587785i 5.18703 + 3.76860i −0.874954 2.69283i −8.75055
148.2 −0.206962 0.636964i −2.54013 1.84551i 1.25514 0.911915i 0.662464 2.03885i −0.649815 + 1.99992i 0.809017 0.587785i −1.92429 1.39808i 2.11929 + 6.52251i −1.43578
148.3 0.435488 + 1.34029i 1.75021 + 1.27160i 0.0112975 0.00820814i −0.565930 + 1.74175i −0.942126 + 2.89957i 0.809017 0.587785i 2.29616 + 1.66826i 0.519216 + 1.59798i −2.58091
148.4 0.751051 + 2.31150i −1.16030 0.843005i −3.16091 + 2.29654i 0.388938 1.19703i 1.07716 3.31516i 0.809017 0.587785i −3.74989 2.72445i −0.291419 0.896896i 3.05904
323.1 −1.38112 1.00344i 0.708129 2.17940i 0.282562 + 0.869638i −3.28976 + 2.39015i −3.16491 + 2.29944i −0.309017 0.951057i −0.572703 + 1.76260i −1.82128 1.32323i 6.94194
323.2 0.183009 + 0.132964i −0.0677147 + 0.208405i −0.602221 1.85345i 2.01892 1.46683i −0.0401026 + 0.0291363i −0.309017 0.951057i 0.276036 0.849550i 2.38820 + 1.73513i 0.564516
323.3 0.901622 + 0.655067i −0.883423 + 2.71890i −0.234224 0.720867i −2.79603 + 2.03143i −2.57757 + 1.87272i −0.309017 0.951057i 0.949813 2.92322i −4.18492 3.04052i −3.85168
323.4 1.60551 + 1.16647i 0.861043 2.65002i 0.598967 + 1.84343i 0.0217822 0.0158257i 4.47357 3.25024i −0.309017 0.951057i 0.0378378 0.116453i −3.85415 2.80020i 0.0534317
372.1 −0.788594 + 2.42704i 0.332181 0.241344i −3.65062 2.65233i 1.05961 + 3.26115i 0.323795 + 0.996539i 0.809017 + 0.587785i 5.18703 3.76860i −0.874954 + 2.69283i −8.75055
372.2 −0.206962 + 0.636964i −2.54013 + 1.84551i 1.25514 + 0.911915i 0.662464 + 2.03885i −0.649815 1.99992i 0.809017 + 0.587785i −1.92429 + 1.39808i 2.11929 6.52251i −1.43578
372.3 0.435488 1.34029i 1.75021 1.27160i 0.0112975 + 0.00820814i −0.565930 1.74175i −0.942126 2.89957i 0.809017 + 0.587785i 2.29616 1.66826i 0.519216 1.59798i −2.58091
372.4 0.751051 2.31150i −1.16030 + 0.843005i −3.16091 2.29654i 0.388938 + 1.19703i 1.07716 + 3.31516i 0.809017 + 0.587785i −3.74989 + 2.72445i −0.291419 + 0.896896i 3.05904
729.1 −1.38112 + 1.00344i 0.708129 + 2.17940i 0.282562 0.869638i −3.28976 2.39015i −3.16491 2.29944i −0.309017 + 0.951057i −0.572703 1.76260i −1.82128 + 1.32323i 6.94194
729.2 0.183009 0.132964i −0.0677147 0.208405i −0.602221 + 1.85345i 2.01892 + 1.46683i −0.0401026 0.0291363i −0.309017 + 0.951057i 0.276036 + 0.849550i 2.38820 1.73513i 0.564516
729.3 0.901622 0.655067i −0.883423 2.71890i −0.234224 + 0.720867i −2.79603 2.03143i −2.57757 1.87272i −0.309017 + 0.951057i 0.949813 + 2.92322i −4.18492 + 3.04052i −3.85168
729.4 1.60551 1.16647i 0.861043 + 2.65002i 0.598967 1.84343i 0.0217822 + 0.0158257i 4.47357 + 3.25024i −0.309017 + 0.951057i 0.0378378 + 0.116453i −3.85415 + 2.80020i 0.0534317
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 729.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.f.x 16
11.b odd 2 1 77.2.f.b 16
11.c even 5 1 847.2.a.o 8
11.c even 5 2 847.2.f.v 16
11.c even 5 1 inner 847.2.f.x 16
11.d odd 10 1 77.2.f.b 16
11.d odd 10 1 847.2.a.p 8
11.d odd 10 2 847.2.f.w 16
33.d even 2 1 693.2.m.i 16
33.f even 10 1 693.2.m.i 16
33.f even 10 1 7623.2.a.ct 8
33.h odd 10 1 7623.2.a.cw 8
77.b even 2 1 539.2.f.e 16
77.h odd 6 2 539.2.q.g 32
77.i even 6 2 539.2.q.f 32
77.j odd 10 1 5929.2.a.bs 8
77.l even 10 1 539.2.f.e 16
77.l even 10 1 5929.2.a.bt 8
77.n even 30 2 539.2.q.f 32
77.o odd 30 2 539.2.q.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.f.b 16 11.b odd 2 1
77.2.f.b 16 11.d odd 10 1
539.2.f.e 16 77.b even 2 1
539.2.f.e 16 77.l even 10 1
539.2.q.f 32 77.i even 6 2
539.2.q.f 32 77.n even 30 2
539.2.q.g 32 77.h odd 6 2
539.2.q.g 32 77.o odd 30 2
693.2.m.i 16 33.d even 2 1
693.2.m.i 16 33.f even 10 1
847.2.a.o 8 11.c even 5 1
847.2.a.p 8 11.d odd 10 1
847.2.f.v 16 11.c even 5 2
847.2.f.w 16 11.d odd 10 2
847.2.f.x 16 1.a even 1 1 trivial
847.2.f.x 16 11.c even 5 1 inner
5929.2.a.bs 8 77.j odd 10 1
5929.2.a.bt 8 77.l even 10 1
7623.2.a.ct 8 33.f even 10 1
7623.2.a.cw 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\):

\( T_{2}^{16} - 3 T_{2}^{15} + 14 T_{2}^{14} - 32 T_{2}^{13} + 86 T_{2}^{12} - 145 T_{2}^{11} + 245 T_{2}^{10} - 245 T_{2}^{9} + 640 T_{2}^{8} - 1175 T_{2}^{7} + 2135 T_{2}^{6} - 2300 T_{2}^{5} + 1850 T_{2}^{4} - 925 T_{2}^{3} + 700 T_{2}^{2} + \cdots + 25 \) Copy content Toggle raw display
\( T_{3}^{16} + 2 T_{3}^{15} + 14 T_{3}^{14} + 26 T_{3}^{13} + 124 T_{3}^{12} + 100 T_{3}^{11} + 747 T_{3}^{10} - 178 T_{3}^{9} + 4253 T_{3}^{8} + 872 T_{3}^{7} + 8452 T_{3}^{6} + 14920 T_{3}^{5} + 22464 T_{3}^{4} - 14304 T_{3}^{3} + \cdots + 256 \) Copy content Toggle raw display
\( T_{13}^{16} - 7 T_{13}^{15} + 39 T_{13}^{14} - 81 T_{13}^{13} + 484 T_{13}^{12} - 2465 T_{13}^{11} + 25007 T_{13}^{10} - 87462 T_{13}^{9} + 357453 T_{13}^{8} - 687552 T_{13}^{7} + 2166132 T_{13}^{6} - 2287000 T_{13}^{5} + \cdots + 188897536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 3 T^{15} + 14 T^{14} - 32 T^{13} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + 14 T^{14} + 26 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{16} + 5 T^{15} + 19 T^{14} + 59 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 7 T^{15} + 39 T^{14} + \cdots + 188897536 \) Copy content Toggle raw display
$17$ \( T^{16} - 5 T^{15} + 81 T^{14} + \cdots + 2611456 \) Copy content Toggle raw display
$19$ \( T^{16} + 19 T^{15} + 189 T^{14} + \cdots + 62726400 \) Copy content Toggle raw display
$23$ \( (T^{8} - 16 T^{7} + 40 T^{6} + 342 T^{5} + \cdots + 859)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 3 T^{15} + 116 T^{14} + \cdots + 245025 \) Copy content Toggle raw display
$31$ \( T^{16} + 7 T^{15} + \cdots + 2629638400 \) Copy content Toggle raw display
$37$ \( T^{16} - 4 T^{15} + 84 T^{14} + \cdots + 212521 \) Copy content Toggle raw display
$41$ \( T^{16} - 10 T^{15} + 196 T^{14} + \cdots + 13424896 \) Copy content Toggle raw display
$43$ \( (T^{8} - 4 T^{7} - 105 T^{6} + 268 T^{5} + \cdots - 971)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 23 T^{15} + \cdots + 5345713926400 \) Copy content Toggle raw display
$53$ \( T^{16} - 4 T^{15} + \cdots + 310840815961 \) Copy content Toggle raw display
$59$ \( T^{16} - 17 T^{15} + \cdots + 187142400 \) Copy content Toggle raw display
$61$ \( T^{16} - 7 T^{15} + 224 T^{14} + \cdots + 253446400 \) Copy content Toggle raw display
$67$ \( (T^{8} + 19 T^{7} - 16 T^{6} - 1160 T^{5} + \cdots - 27395)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + 14 T^{15} + \cdots + 379119841 \) Copy content Toggle raw display
$73$ \( T^{16} - 35 T^{15} + \cdots + 105069332736 \) Copy content Toggle raw display
$79$ \( T^{16} + 15 T^{15} + \cdots + 15858514175625 \) Copy content Toggle raw display
$83$ \( T^{16} + 5 T^{15} + 14 T^{14} + \cdots + 756470016 \) Copy content Toggle raw display
$89$ \( (T^{8} - 37 T^{7} + 320 T^{6} + \cdots + 952400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 20 T^{15} + \cdots + 4647025244416 \) Copy content Toggle raw display
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