Properties

Label 99.2.m.a
Level 99
Weight 2
Character orbit 99.m
Analytic conductor 0.791
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.m (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{15}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1 + \zeta_{15}^{2} - \zeta_{15}^{3} - \zeta_{15}^{6} + \zeta_{15}^{7}\) \(-1 - \zeta_{15}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.669131 + 0.743145i
−0.104528 + 0.994522i
0.669131 0.743145i
−0.104528 0.994522i
0.913545 0.406737i
−0.978148 0.207912i
−0.978148 + 0.207912i
0.913545 + 0.406737i
0.273659 2.60369i −1.28716 + 1.15897i −4.74803 1.00922i −0.338261 3.21834i 2.66535 + 3.66854i −0.669131 + 0.743145i −2.30902 + 7.10642i 0.313585 2.98357i −8.47214
16.1 0.373619 + 0.0794152i 1.72256 + 0.181049i −1.69381 0.754131i 1.20906 0.256993i 0.629204 + 0.204441i 0.104528 + 0.994522i −1.19098 0.865300i 2.93444 + 0.623735i 0.472136
25.1 0.273659 + 2.60369i −1.28716 1.15897i −4.74803 + 1.00922i −0.338261 + 3.21834i 2.66535 3.66854i −0.669131 0.743145i −2.30902 7.10642i 0.313585 + 2.98357i −8.47214
31.1 0.373619 0.0794152i 1.72256 0.181049i −1.69381 + 0.754131i 1.20906 + 0.256993i 0.629204 0.204441i 0.104528 0.994522i −1.19098 + 0.865300i 2.93444 0.623735i 0.472136
49.1 −0.255585 + 0.283856i 0.704489 + 1.58231i 0.193806 + 1.84395i −0.827091 0.918578i −0.629204 0.204441i −0.913545 0.406737i −1.19098 0.865300i −2.00739 + 2.22943i 0.472136
58.1 −2.39169 1.06485i 0.360114 1.69420i 3.24803 + 3.60730i 2.95630 1.31623i −2.66535 + 3.66854i 0.978148 0.207912i −2.30902 7.10642i −2.74064 1.22021i −8.47214
70.1 −2.39169 + 1.06485i 0.360114 + 1.69420i 3.24803 3.60730i 2.95630 + 1.31623i −2.66535 3.66854i 0.978148 + 0.207912i −2.30902 + 7.10642i −2.74064 + 1.22021i −8.47214
97.1 −0.255585 0.283856i 0.704489 1.58231i 0.193806 1.84395i −0.827091 + 0.918578i −0.629204 + 0.204441i −0.913545 + 0.406737i −1.19098 + 0.865300i −2.00739 2.22943i 0.472136
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
11.c even 5 1 inner
99.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.m.a 8
3.b odd 2 1 297.2.n.a 8
9.c even 3 1 inner 99.2.m.a 8
9.c even 3 1 891.2.f.b 4
9.d odd 6 1 297.2.n.a 8
9.d odd 6 1 891.2.f.a 4
11.c even 5 1 inner 99.2.m.a 8
11.c even 5 1 1089.2.e.g 4
11.d odd 10 1 1089.2.e.d 4
33.h odd 10 1 297.2.n.a 8
99.m even 15 1 inner 99.2.m.a 8
99.m even 15 1 891.2.f.b 4
99.m even 15 1 1089.2.e.g 4
99.m even 15 1 9801.2.a.n 2
99.n odd 30 1 297.2.n.a 8
99.n odd 30 1 891.2.f.a 4
99.n odd 30 1 9801.2.a.bb 2
99.o odd 30 1 1089.2.e.d 4
99.o odd 30 1 9801.2.a.bc 2
99.p even 30 1 9801.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.m.a 8 1.a even 1 1 trivial
99.2.m.a 8 9.c even 3 1 inner
99.2.m.a 8 11.c even 5 1 inner
99.2.m.a 8 99.m even 15 1 inner
297.2.n.a 8 3.b odd 2 1
297.2.n.a 8 9.d odd 6 1
297.2.n.a 8 33.h odd 10 1
297.2.n.a 8 99.n odd 30 1
891.2.f.a 4 9.d odd 6 1
891.2.f.a 4 99.n odd 30 1
891.2.f.b 4 9.c even 3 1
891.2.f.b 4 99.m even 15 1
1089.2.e.d 4 11.d odd 10 1
1089.2.e.d 4 99.o odd 30 1
1089.2.e.g 4 11.c even 5 1
1089.2.e.g 4 99.m even 15 1
9801.2.a.m 2 99.p even 30 1
9801.2.a.n 2 99.m even 15 1
9801.2.a.bb 2 99.n odd 30 1
9801.2.a.bc 2 99.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + 4 T^{2} - 2 T^{3} + 9 T^{4} - 4 T^{5} + 16 T^{6} - 8 T^{7} + 16 T^{8} )( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 50 T^{5} + 52 T^{6} + 40 T^{7} + 16 T^{8} ) \)
$3$ \( 1 - 3 T + 6 T^{2} - 9 T^{3} + 9 T^{4} - 27 T^{5} + 54 T^{6} - 81 T^{7} + 81 T^{8} \)
$5$ \( 1 - 6 T + 25 T^{2} - 54 T^{3} + 84 T^{4} - 84 T^{5} + 395 T^{6} - 1746 T^{7} + 5171 T^{8} - 8730 T^{9} + 9875 T^{10} - 10500 T^{11} + 52500 T^{12} - 168750 T^{13} + 390625 T^{14} - 468750 T^{15} + 390625 T^{16} \)
$7$ \( ( 1 - 4 T + 9 T^{2} - 8 T^{3} - 31 T^{4} - 56 T^{5} + 441 T^{6} - 1372 T^{7} + 2401 T^{8} )( 1 + 5 T + 18 T^{2} + 55 T^{3} + 149 T^{4} + 385 T^{5} + 882 T^{6} + 1715 T^{7} + 2401 T^{8} ) \)
$11$ \( 1 - T - 20 T^{2} + T^{3} + 309 T^{4} + 11 T^{5} - 2420 T^{6} - 1331 T^{7} + 14641 T^{8} \)
$13$ \( 1 - 8 T + 13 T^{2} + 80 T^{3} - 400 T^{4} + 1168 T^{5} - 2101 T^{6} - 18940 T^{7} + 139519 T^{8} - 246220 T^{9} - 355069 T^{10} + 2566096 T^{11} - 11424400 T^{12} + 29703440 T^{13} + 62748517 T^{14} - 501988136 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 2244 T^{5} + 5491 T^{6} - 29478 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 703 T^{5} - 6498 T^{6} + 6859 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( ( 1 + 7 T - 8 T^{2} + 77 T^{3} + 1593 T^{4} + 1771 T^{5} - 4232 T^{6} + 85169 T^{7} + 279841 T^{8} )^{2} \)
$29$ \( 1 + 9 T + 74 T^{2} + 129 T^{3} - 54 T^{4} - 8388 T^{5} + 18676 T^{6} + 272682 T^{7} + 3330947 T^{8} + 7907778 T^{9} + 15706516 T^{10} - 204574932 T^{11} - 38193174 T^{12} + 2645938221 T^{13} + 44016925754 T^{14} + 155248886781 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 3 T + 36 T^{2} - 23 T^{3} - 84 T^{4} - 5516 T^{5} - 24056 T^{6} - 63114 T^{7} - 303923 T^{8} - 1956534 T^{9} - 23117816 T^{10} - 164327156 T^{11} - 77575764 T^{12} - 658470473 T^{13} + 31950132516 T^{14} + 82537842333 T^{15} + 852891037441 T^{16} \)
$37$ \( ( 1 - 12 T + 57 T^{2} - 440 T^{3} + 3921 T^{4} - 16280 T^{5} + 78033 T^{6} - 607836 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 3 T - 59 T^{2} - 42 T^{3} + 1326 T^{4} + 9501 T^{5} + 17474 T^{6} - 345486 T^{7} - 294103 T^{8} - 14164926 T^{9} + 29373794 T^{10} + 654818421 T^{11} + 3746959086 T^{12} - 4865960442 T^{13} - 280256150219 T^{14} - 584262821643 T^{15} + 7984925229121 T^{16} \)
$43$ \( ( 1 - 3 T - 68 T^{2} + 27 T^{3} + 3693 T^{4} + 1161 T^{5} - 125732 T^{6} - 238521 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( 1 - 23 T + 327 T^{2} - 2796 T^{3} + 15428 T^{4} - 41679 T^{5} - 80266 T^{6} + 1119532 T^{7} - 8567157 T^{8} + 52618004 T^{9} - 177307594 T^{10} - 4327238817 T^{11} + 75283718468 T^{12} - 641248639572 T^{13} + 3524803412583 T^{14} - 11652331770649 T^{15} + 23811286661761 T^{16} \)
$53$ \( ( 1 - 14 T + 43 T^{2} - 160 T^{3} + 2961 T^{4} - 8480 T^{5} + 120787 T^{6} - 2084278 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 - 3 T + 64 T^{2} + 51 T^{3} - 168 T^{4} + 20412 T^{5} - 184552 T^{6} + 519906 T^{7} - 7825759 T^{8} + 30674454 T^{9} - 642425512 T^{10} + 4192196148 T^{11} - 2035716648 T^{12} + 36461139249 T^{13} + 2699554153024 T^{14} - 7465954454457 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 29 T^{2} - 480 T^{3} - 2160 T^{4} + 21600 T^{5} + 186389 T^{6} - 128880 T^{7} - 11482801 T^{8} - 7861680 T^{9} + 693553469 T^{10} + 4902789600 T^{11} - 29907016560 T^{12} - 405406224480 T^{13} - 1494090856469 T^{14} + 191707312997281 T^{16} \)
$67$ \( ( 1 + 6 T - 31 T^{2} + 402 T^{3} + 4489 T^{4} )^{4} \)
$71$ \( ( 1 - 21 T + 95 T^{2} + 1371 T^{3} - 21536 T^{4} + 97341 T^{5} + 478895 T^{6} - 7516131 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 4 T - 27 T^{2} + 500 T^{3} + 7301 T^{4} + 36500 T^{5} - 143883 T^{6} + 1556068 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( 1 + 22 T + 259 T^{2} + 146 T^{3} - 28378 T^{4} - 420788 T^{5} - 1482397 T^{6} + 19182626 T^{7} + 350855611 T^{8} + 1515427454 T^{9} - 9251639677 T^{10} - 207464894732 T^{11} - 1105325398618 T^{12} + 449250234254 T^{13} + 62959650979939 T^{14} + 422485997695498 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 17 T + 213 T^{2} + 3330 T^{3} + 42290 T^{4} + 490473 T^{5} + 4981814 T^{6} + 51032300 T^{7} + 525354219 T^{8} + 4235680900 T^{9} + 34319716646 T^{10} + 280446085251 T^{11} + 2007012395090 T^{12} + 13117005341190 T^{13} + 69638299527597 T^{14} + 461312866823659 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 18 T + 254 T^{2} + 1602 T^{3} + 7921 T^{4} )^{4} \)
$97$ \( 1 - 6 T + 97 T^{2} - 1530 T^{3} + 9180 T^{4} + 36876 T^{5} + 316511 T^{6} + 9403380 T^{7} - 144949561 T^{8} + 912127860 T^{9} + 2978051999 T^{10} + 33655729548 T^{11} + 812698799580 T^{12} - 13138630593210 T^{13} + 80798284478113 T^{14} - 484789706868678 T^{15} + 7837433594376961 T^{16} \)
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