Properties

Label 605.2.g.e
Level $605$
Weight $2$
Character orbit 605.g
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.g (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( -5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1\)
\(\nu^{6}\)\(=\)\(-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6\)
\(\nu^{7}\)\(=\)\(-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
1.69513 1.23158i
−0.386111 + 0.280526i
0.418926 + 1.28932i
−0.227943 0.701538i
1.69513 + 1.23158i
−0.386111 0.280526i
0.418926 1.28932i
−0.227943 + 0.701538i
−1.69513 1.23158i 0.591149 1.81937i 0.738630 + 2.27327i 0.809017 0.587785i −3.24278 + 2.35601i 0.947813 + 2.91707i 0.252684 0.777682i −0.533593 0.387678i −2.09529
81.2 0.386111 + 0.280526i 0.0998345 0.307259i −0.547647 1.68548i 0.809017 0.587785i 0.124741 0.0906300i −0.829779 2.55380i 0.556333 1.71222i 2.34261 + 1.70201i 0.477260
251.1 −0.418926 + 1.28932i −0.465584 + 0.338266i 0.131180 + 0.0953077i −0.309017 0.951057i −0.241089 0.741996i −2.95244 2.14507i −2.37136 + 1.72290i −0.824707 + 2.53819i 1.35567
251.2 0.227943 0.701538i 2.27460 1.65259i 1.17784 + 0.855749i −0.309017 0.951057i −0.640877 1.97242i 0.834404 + 0.606230i 2.06235 1.49838i 1.51569 4.66481i −0.737640
366.1 −1.69513 + 1.23158i 0.591149 + 1.81937i 0.738630 2.27327i 0.809017 + 0.587785i −3.24278 2.35601i 0.947813 2.91707i 0.252684 + 0.777682i −0.533593 + 0.387678i −2.09529
366.2 0.386111 0.280526i 0.0998345 + 0.307259i −0.547647 + 1.68548i 0.809017 + 0.587785i 0.124741 + 0.0906300i −0.829779 + 2.55380i 0.556333 + 1.71222i 2.34261 1.70201i 0.477260
511.1 −0.418926 1.28932i −0.465584 0.338266i 0.131180 0.0953077i −0.309017 + 0.951057i −0.241089 + 0.741996i −2.95244 + 2.14507i −2.37136 1.72290i −0.824707 2.53819i 1.35567
511.2 0.227943 + 0.701538i 2.27460 + 1.65259i 1.17784 0.855749i −0.309017 + 0.951057i −0.640877 + 1.97242i 0.834404 0.606230i 2.06235 + 1.49838i 1.51569 + 4.66481i −0.737640
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 511.2
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 605.2.g.e 8
11.b odd 2 1 605.2.g.m 8
11.c even 5 1 605.2.a.k 4
11.c even 5 1 inner 605.2.g.e 8
11.c even 5 2 605.2.g.k 8
11.d odd 10 2 55.2.g.b 8
11.d odd 10 1 605.2.a.j 4
11.d odd 10 1 605.2.g.m 8
33.f even 10 2 495.2.n.e 8
33.f even 10 1 5445.2.a.bp 4
33.h odd 10 1 5445.2.a.bi 4
44.g even 10 2 880.2.bo.h 8
44.g even 10 1 9680.2.a.cn 4
44.h odd 10 1 9680.2.a.cm 4
55.h odd 10 2 275.2.h.a 8
55.h odd 10 1 3025.2.a.bd 4
55.j even 10 1 3025.2.a.w 4
55.l even 20 4 275.2.z.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 11.d odd 10 2
275.2.h.a 8 55.h odd 10 2
275.2.z.a 16 55.l even 20 4
495.2.n.e 8 33.f even 10 2
605.2.a.j 4 11.d odd 10 1
605.2.a.k 4 11.c even 5 1
605.2.g.e 8 1.a even 1 1 trivial
605.2.g.e 8 11.c even 5 1 inner
605.2.g.k 8 11.c even 5 2
605.2.g.m 8 11.b odd 2 1
605.2.g.m 8 11.d odd 10 1
880.2.bo.h 8 44.g even 10 2
3025.2.a.w 4 55.j even 10 1
3025.2.a.bd 4 55.h odd 10 1
5445.2.a.bi 4 33.h odd 10 1
5445.2.a.bp 4 33.f even 10 1
9680.2.a.cm 4 44.h odd 10 1
9680.2.a.cn 4 44.g even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\(T_{2}^{8} + \cdots\)
\( T_{3}^{8} - 5 T_{3}^{7} + 13 T_{3}^{6} - 15 T_{3}^{5} + 14 T_{3}^{4} + 15 T_{3}^{3} + 7 T_{3}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T + 5 T^{2} - 3 T^{3} + 4 T^{4} + 3 T^{5} + 5 T^{6} + 3 T^{7} + T^{8} \)
$3$ \( 1 + 7 T^{2} + 15 T^{3} + 14 T^{4} - 15 T^{5} + 13 T^{6} - 5 T^{7} + T^{8} \)
$5$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$7$ \( 961 - 1054 T + 467 T^{2} + 78 T^{3} + 155 T^{4} + 42 T^{5} + 17 T^{6} + 4 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 19321 - 4309 T + 2829 T^{2} - 253 T^{3} + 174 T^{4} - T^{5} + 11 T^{6} + 3 T^{7} + T^{8} \)
$17$ \( 361 + 1539 T + 2829 T^{2} + 1973 T^{3} + 1044 T^{4} + 341 T^{5} + 81 T^{6} + 12 T^{7} + T^{8} \)
$19$ \( 625 + 10625 T + 69000 T^{2} - 125 T^{3} + 3575 T^{4} + 475 T^{5} + 60 T^{6} + 5 T^{7} + T^{8} \)
$23$ \( ( -11 + 10 T + 4 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$29$ \( 203401 + 96063 T + 49536 T^{2} + 17799 T^{3} + 5479 T^{4} + 1287 T^{5} + 214 T^{6} + 21 T^{7} + T^{8} \)
$31$ \( 390625 + 234375 T + 78125 T^{2} + 9375 T^{3} + 2500 T^{4} - 375 T^{5} + 125 T^{6} - 15 T^{7} + T^{8} \)
$37$ \( 1324801 + 210633 T + 325466 T^{2} + 161659 T^{3} + 40449 T^{4} + 5737 T^{5} + 534 T^{6} + 31 T^{7} + T^{8} \)
$41$ \( 101761 - 64757 T + 39208 T^{2} - 10179 T^{3} + 1805 T^{4} - 69 T^{5} - 2 T^{6} + 3 T^{7} + T^{8} \)
$43$ \( ( 211 - 289 T + 121 T^{2} - 19 T^{3} + T^{4} )^{2} \)
$47$ \( 28561 - 10985 T + 3887 T^{2} - 845 T^{3} + 584 T^{4} + 65 T^{5} + 23 T^{6} + 5 T^{7} + T^{8} \)
$53$ \( 885481 - 20702 T + 62845 T^{2} + 14278 T^{3} + 1539 T^{4} - 638 T^{5} + 105 T^{6} + 2 T^{7} + T^{8} \)
$59$ \( 687241 + 59688 T + 49385 T^{2} + 15678 T^{3} + 4819 T^{4} - 1638 T^{5} + 255 T^{6} - 18 T^{7} + T^{8} \)
$61$ \( 28561 + 39546 T + 28730 T^{2} + 12714 T^{3} + 3844 T^{4} + 684 T^{5} + 75 T^{6} + 6 T^{7} + T^{8} \)
$67$ \( ( -4079 - 1014 T + 22 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$71$ \( 17161 + 12445 T + 4272 T^{2} - 5 T^{3} + 2709 T^{4} - 875 T^{5} + 158 T^{6} - 15 T^{7} + T^{8} \)
$73$ \( 121 + 451 T + 859 T^{2} + 997 T^{3} + 774 T^{4} + 359 T^{5} + 81 T^{6} - 2 T^{7} + T^{8} \)
$79$ \( 45954841 - 19198128 T + 3194183 T^{2} + 20079 T^{3} + 19180 T^{4} + 1389 T^{5} + 233 T^{6} - 3 T^{7} + T^{8} \)
$83$ \( 2886601 - 3398 T + 903093 T^{2} - 371546 T^{3} + 75555 T^{4} - 9246 T^{5} + 753 T^{6} - 38 T^{7} + T^{8} \)
$89$ \( ( 1861 - 472 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$97$ \( 9066121 + 746728 T + 1630841 T^{2} + 407104 T^{3} + 126029 T^{4} + 18272 T^{5} + 1409 T^{6} + 56 T^{7} + T^{8} \)
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