Properties

Label 77.2.e.a
Level 77
Weight 2
Character orbit 77.e
Analytic conductor 0.615
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.614848095564\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/77\mathbb{Z}\right)^\times\).

\(n\) \(45\) \(57\)
\(\chi(n)\) \(-\zeta_{18}^{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1
−0.766044 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
−0.173648 0.984808i
−0.939693 + 1.62760i −0.326352 0.565258i −0.766044 1.32683i −1.76604 + 3.05888i 1.22668 0.418748 + 2.61240i −0.879385 1.28699 2.22913i −3.31908 5.74881i
23.2 0.173648 0.300767i 0.266044 + 0.460802i 0.939693 + 1.62760i −0.0603074 + 0.104455i 0.184793 −2.47178 0.943555i 1.34730 1.35844 2.35289i 0.0209445 + 0.0362770i
23.3 0.766044 1.32683i −1.43969 2.49362i −0.173648 0.300767i −1.17365 + 2.03282i −4.41147 2.05303 1.66885i 2.53209 −2.64543 + 4.58202i 1.79813 + 3.11446i
67.1 −0.939693 1.62760i −0.326352 + 0.565258i −0.766044 + 1.32683i −1.76604 3.05888i 1.22668 0.418748 2.61240i −0.879385 1.28699 + 2.22913i −3.31908 + 5.74881i
67.2 0.173648 + 0.300767i 0.266044 0.460802i 0.939693 1.62760i −0.0603074 0.104455i 0.184793 −2.47178 + 0.943555i 1.34730 1.35844 + 2.35289i 0.0209445 0.0362770i
67.3 0.766044 + 1.32683i −1.43969 + 2.49362i −0.173648 + 0.300767i −1.17365 2.03282i −4.41147 2.05303 + 1.66885i 2.53209 −2.64543 4.58202i 1.79813 3.11446i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.2.e.a 6
3.b odd 2 1 693.2.i.h 6
4.b odd 2 1 1232.2.q.m 6
7.b odd 2 1 539.2.e.m 6
7.c even 3 1 inner 77.2.e.a 6
7.c even 3 1 539.2.a.j 3
7.d odd 6 1 539.2.a.g 3
7.d odd 6 1 539.2.e.m 6
11.b odd 2 1 847.2.e.c 6
11.c even 5 4 847.2.n.g 24
11.d odd 10 4 847.2.n.f 24
21.g even 6 1 4851.2.a.bk 3
21.h odd 6 1 693.2.i.h 6
21.h odd 6 1 4851.2.a.bj 3
28.f even 6 1 8624.2.a.co 3
28.g odd 6 1 1232.2.q.m 6
28.g odd 6 1 8624.2.a.ch 3
77.h odd 6 1 847.2.e.c 6
77.h odd 6 1 5929.2.a.x 3
77.i even 6 1 5929.2.a.u 3
77.m even 15 4 847.2.n.g 24
77.o odd 30 4 847.2.n.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.e.a 6 1.a even 1 1 trivial
77.2.e.a 6 7.c even 3 1 inner
539.2.a.g 3 7.d odd 6 1
539.2.a.j 3 7.c even 3 1
539.2.e.m 6 7.b odd 2 1
539.2.e.m 6 7.d odd 6 1
693.2.i.h 6 3.b odd 2 1
693.2.i.h 6 21.h odd 6 1
847.2.e.c 6 11.b odd 2 1
847.2.e.c 6 77.h odd 6 1
847.2.n.f 24 11.d odd 10 4
847.2.n.f 24 77.o odd 30 4
847.2.n.g 24 11.c even 5 4
847.2.n.g 24 77.m even 15 4
1232.2.q.m 6 4.b odd 2 1
1232.2.q.m 6 28.g odd 6 1
4851.2.a.bj 3 21.h odd 6 1
4851.2.a.bk 3 21.g even 6 1
5929.2.a.u 3 77.i even 6 1
5929.2.a.x 3 77.h odd 6 1
8624.2.a.ch 3 28.g odd 6 1
8624.2.a.co 3 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3 T_{2}^{4} - 2 T_{2}^{3} + 9 T_{2}^{2} - 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(77, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 3 T^{2} - 2 T^{3} + 3 T^{4} + 3 T^{5} - T^{6} + 6 T^{7} + 12 T^{8} - 16 T^{9} - 48 T^{10} + 64 T^{12} \)
$3$ \( 1 + 3 T - 7 T^{3} + 3 T^{4} + 18 T^{5} + 19 T^{6} + 54 T^{7} + 27 T^{8} - 189 T^{9} + 729 T^{11} + 729 T^{12} \)
$5$ \( 1 + 6 T + 12 T^{2} + 22 T^{3} + 90 T^{4} + 174 T^{5} + 191 T^{6} + 870 T^{7} + 2250 T^{8} + 2750 T^{9} + 7500 T^{10} + 18750 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 17 T^{3} + 343 T^{6} \)
$11$ \( ( 1 + T + T^{2} )^{3} \)
$13$ \( ( 1 - 3 T + 33 T^{2} - 79 T^{3} + 429 T^{4} - 507 T^{5} + 2197 T^{6} )^{2} \)
$17$ \( 1 - 3 T - 6 T^{2} - 95 T^{3} + 45 T^{4} + 1038 T^{5} + 4025 T^{6} + 17646 T^{7} + 13005 T^{8} - 466735 T^{9} - 501126 T^{10} - 4259571 T^{11} + 24137569 T^{12} \)
$19$ \( 1 + 9 T + 6 T^{2} - 27 T^{3} + 1041 T^{4} + 3924 T^{5} + 803 T^{6} + 74556 T^{7} + 375801 T^{8} - 185193 T^{9} + 781926 T^{10} + 22284891 T^{11} + 47045881 T^{12} \)
$23$ \( 1 - 12 T^{2} - 214 T^{3} - 132 T^{4} + 1284 T^{5} + 32471 T^{6} + 29532 T^{7} - 69828 T^{8} - 2603738 T^{9} - 3358092 T^{10} + 148035889 T^{12} \)
$29$ \( ( 1 + 3 T + 51 T^{2} + 225 T^{3} + 1479 T^{4} + 2523 T^{5} + 24389 T^{6} )^{2} \)
$31$ \( 1 + 9 T - 36 T^{2} - 101 T^{3} + 4869 T^{4} + 10872 T^{5} - 109689 T^{6} + 337032 T^{7} + 4679109 T^{8} - 3008891 T^{9} - 33246756 T^{10} + 257662359 T^{11} + 887503681 T^{12} \)
$37$ \( 1 - 75 T^{2} + 144 T^{3} + 2850 T^{4} - 5400 T^{5} - 101635 T^{6} - 199800 T^{7} + 3901650 T^{8} + 7294032 T^{9} - 140562075 T^{10} + 2565726409 T^{12} \)
$41$ \( ( 1 - 9 T + 129 T^{2} - 739 T^{3} + 5289 T^{4} - 15129 T^{5} + 68921 T^{6} )^{2} \)
$43$ \( ( 1 + 120 T^{2} + 9 T^{3} + 5160 T^{4} + 79507 T^{6} )^{2} \)
$47$ \( 1 - 3 T - 54 T^{2} - 271 T^{3} + 1131 T^{4} + 13722 T^{5} + 2903 T^{6} + 644934 T^{7} + 2498379 T^{8} - 28136033 T^{9} - 263502774 T^{10} - 688035021 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 9 T - 24 T^{2} - 45 T^{3} + 1005 T^{4} - 22914 T^{5} - 269075 T^{6} - 1214442 T^{7} + 2823045 T^{8} - 6699465 T^{9} - 189371544 T^{10} + 3763759437 T^{11} + 22164361129 T^{12} \)
$59$ \( 1 - 84 T^{2} + 38 T^{3} + 2100 T^{4} - 1596 T^{5} - 5185 T^{6} - 94164 T^{7} + 7310100 T^{8} + 7804402 T^{9} - 1017858324 T^{10} + 42180533641 T^{12} \)
$61$ \( 1 + 12 T - 3 T^{2} - 1212 T^{3} - 5214 T^{4} + 48180 T^{5} + 814133 T^{6} + 2938980 T^{7} - 19401294 T^{8} - 275100972 T^{9} - 41537523 T^{10} + 10135155612 T^{11} + 51520374361 T^{12} \)
$67$ \( 1 - 189 T^{2} - 16 T^{3} + 23058 T^{4} + 1512 T^{5} - 1791717 T^{6} + 101304 T^{7} + 103507362 T^{8} - 4812208 T^{9} - 3808561869 T^{10} + 90458382169 T^{12} \)
$71$ \( ( 1 + 9 T + 123 T^{2} + 477 T^{3} + 8733 T^{4} + 45369 T^{5} + 357911 T^{6} )^{2} \)
$73$ \( 1 + 6 T - 192 T^{2} - 386 T^{3} + 30078 T^{4} + 32202 T^{5} - 2439513 T^{6} + 2350746 T^{7} + 160285662 T^{8} - 150160562 T^{9} - 5452462272 T^{10} + 12438429558 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 - 3 T - 114 T^{2} + 653 T^{3} + 3879 T^{4} - 23274 T^{5} - 4161 T^{6} - 1838646 T^{7} + 24208839 T^{8} + 321954467 T^{9} - 4440309234 T^{10} - 9231169197 T^{11} + 243087455521 T^{12} \)
$83$ \( ( 1 + 15 T + 267 T^{2} + 2223 T^{3} + 22161 T^{4} + 103335 T^{5} + 571787 T^{6} )^{2} \)
$89$ \( 1 - 15 T - 114 T^{2} + 477 T^{3} + 43035 T^{4} - 156444 T^{5} - 2866295 T^{6} - 13923516 T^{7} + 340880235 T^{8} + 336270213 T^{9} - 7152615474 T^{10} - 83760891735 T^{11} + 496981290961 T^{12} \)
$97$ \( ( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 93411 T^{4} - 423405 T^{5} + 912673 T^{6} )^{2} \)
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