Properties

Label 52.2.l.b
Level $52$
Weight $2$
Character orbit 52.l
Analytic conductor $0.415$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 52.l (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.415222090511\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.102930383934669717504.1
Defining polynomial: \(x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{15} + 5 x^{14} - 2 x^{13} + 5 x^{12} - 8 x^{11} - 12 x^{10} + 32 x^{9} - 36 x^{8} + 64 x^{7} - 48 x^{6} - 64 x^{5} + 80 x^{4} - 64 x^{3} + 320 x^{2} - 512 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -19 \nu^{15} + 26 \nu^{14} + 49 \nu^{13} + 12 \nu^{12} - 163 \nu^{11} - 146 \nu^{10} + 172 \nu^{9} + 184 \nu^{8} + 412 \nu^{7} - 408 \nu^{6} - 1072 \nu^{5} - 544 \nu^{4} + 1360 \nu^{3} + 2592 \nu^{2} - 2112 \nu - 1152 \)\()/1408\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{15} - 4 \nu^{14} + 5 \nu^{13} - 2 \nu^{12} + 5 \nu^{11} - 8 \nu^{10} - 12 \nu^{9} + 32 \nu^{8} - 36 \nu^{7} + 64 \nu^{6} - 48 \nu^{5} - 64 \nu^{4} + 80 \nu^{3} - 64 \nu^{2} + 320 \nu - 512 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{15} + 6 \nu^{14} + 31 \nu^{13} + 76 \nu^{12} - 61 \nu^{11} - 182 \nu^{10} + 40 \nu^{8} + 580 \nu^{7} + 152 \nu^{6} - 704 \nu^{5} - 672 \nu^{4} - 16 \nu^{3} + 2336 \nu^{2} - 512 \nu - 384 \)\()/1408\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{15} - 62 \nu^{14} + 27 \nu^{13} + 56 \nu^{12} + 167 \nu^{11} - 146 \nu^{10} - 400 \nu^{9} + 96 \nu^{8} - 28 \nu^{7} + 1352 \nu^{6} + 160 \nu^{5} - 1600 \nu^{4} - 1104 \nu^{3} - 224 \nu^{2} + 5632 \nu - 2560 \)\()/1408\)
\(\beta_{6}\)\(=\)\((\)\( 10 \nu^{15} - 37 \nu^{14} - 12 \nu^{13} + 7 \nu^{12} + 106 \nu^{11} + 87 \nu^{10} - 266 \nu^{9} - 80 \nu^{8} - 264 \nu^{7} + 612 \nu^{6} + 872 \nu^{5} - 480 \nu^{4} - 800 \nu^{3} - 1744 \nu^{2} + 2976 \nu + 1216 \)\()/704\)
\(\beta_{7}\)\(=\)\((\)\( -25 \nu^{15} + 72 \nu^{14} - 13 \nu^{13} - 34 \nu^{12} - 149 \nu^{11} - 28 \nu^{10} + 396 \nu^{9} - 136 \nu^{8} + 404 \nu^{7} - 992 \nu^{6} - 880 \nu^{5} + 1440 \nu^{4} + 688 \nu^{3} + 1984 \nu^{2} - 5440 \nu + 2432 \)\()/1408\)
\(\beta_{8}\)\(=\)\((\)\( -7 \nu^{15} + 17 \nu^{14} + \nu^{13} + 5 \nu^{12} - 57 \nu^{11} - 9 \nu^{10} + 100 \nu^{9} - 14 \nu^{8} + 152 \nu^{7} - 300 \nu^{6} - 216 \nu^{5} + 232 \nu^{4} + 272 \nu^{3} + 992 \nu^{2} - 1888 \nu + 544 \)\()/352\)
\(\beta_{9}\)\(=\)\((\)\( -7 \nu^{15} + 28 \nu^{14} - 21 \nu^{13} - 6 \nu^{12} - 57 \nu^{11} + 24 \nu^{10} + 166 \nu^{9} - 124 \nu^{8} + 152 \nu^{7} - 520 \nu^{6} - 40 \nu^{5} + 672 \nu^{4} + 96 \nu^{3} + 640 \nu^{2} - 2592 \nu + 1600 \)\()/352\)
\(\beta_{10}\)\(=\)\((\)\( -47 \nu^{15} + 78 \nu^{14} + 13 \nu^{13} + 32 \nu^{12} - 191 \nu^{11} - 150 \nu^{10} + 420 \nu^{9} - 160 \nu^{8} + 876 \nu^{7} - 952 \nu^{6} - 1136 \nu^{5} + 992 \nu^{4} + 368 \nu^{3} + 3744 \nu^{2} - 6592 \nu + 3200 \)\()/704\)
\(\beta_{11}\)\(=\)\((\)\( 53 \nu^{15} - 139 \nu^{14} + 61 \nu^{13} + 3 \nu^{12} + 271 \nu^{11} - 59 \nu^{10} - 748 \nu^{9} + 628 \nu^{8} - 948 \nu^{7} + 2140 \nu^{6} + 528 \nu^{5} - 2912 \nu^{4} - 304 \nu^{3} - 3312 \nu^{2} + 12800 \nu - 8704 \)\()/704\)
\(\beta_{12}\)\(=\)\((\)\( 34 \nu^{15} - 83 \nu^{14} + 31 \nu^{13} - 7 \nu^{12} + 173 \nu^{11} - \nu^{10} - 467 \nu^{9} + 340 \nu^{8} - 596 \nu^{7} + 1228 \nu^{6} + 508 \nu^{5} - 1648 \nu^{4} - 192 \nu^{3} - 2480 \nu^{2} + 7568 \nu - 4608 \)\()/352\)
\(\beta_{13}\)\(=\)\((\)\( 73 \nu^{15} - 204 \nu^{14} + 109 \nu^{13} + 6 \nu^{12} + 365 \nu^{11} - 112 \nu^{10} - 1068 \nu^{9} + 960 \nu^{8} - 1252 \nu^{7} + 3072 \nu^{6} + 480 \nu^{5} - 4544 \nu^{4} + 80 \nu^{3} - 4160 \nu^{2} + 17728 \nu - 13568 \)\()/704\)
\(\beta_{14}\)\(=\)\((\)\( -82 \nu^{15} + 201 \nu^{14} - 52 \nu^{13} + 13 \nu^{12} - 434 \nu^{11} - 51 \nu^{10} + 1138 \nu^{9} - 624 \nu^{8} + 1560 \nu^{7} - 2932 \nu^{6} - 1608 \nu^{5} + 3568 \nu^{4} + 992 \nu^{3} + 6416 \nu^{2} - 17696 \nu + 9728 \)\()/704\)
\(\beta_{15}\)\(=\)\((\)\( -46 \nu^{15} + 129 \nu^{14} - 72 \nu^{13} + 3 \nu^{12} - 230 \nu^{11} + 87 \nu^{10} + 654 \nu^{9} - 686 \nu^{8} + 848 \nu^{7} - 1896 \nu^{6} - 112 \nu^{5} + 2744 \nu^{4} - 400 \nu^{3} + 2496 \nu^{2} - 11552 \nu + 9760 \)\()/352\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{5} - \beta_{4} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{14} - \beta_{13} + 2 \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(\beta_{15} + 2 \beta_{12} + \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{5}\)\(=\)\(\beta_{15} + 2 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} + 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} + \beta_{1} - 3\)
\(\nu^{6}\)\(=\)\(3 \beta_{15} + 4 \beta_{14} + 4 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + \beta_{9} + 3 \beta_{8} + 3 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 7 \beta_{3} + \beta_{1} - 2\)
\(\nu^{7}\)\(=\)\(3 \beta_{15} + 3 \beta_{14} + 5 \beta_{13} + 6 \beta_{12} - 3 \beta_{11} + 4 \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} - \beta_{5} + 5 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 4 \beta_{1} - 2\)
\(\nu^{8}\)\(=\)\(-2 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} + 2 \beta_{12} + 8 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - \beta_{8} - 2 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 9 \beta_{1}\)
\(\nu^{9}\)\(=\)\(2 \beta_{15} + 7 \beta_{14} + 5 \beta_{13} - 2 \beta_{12} + 5 \beta_{11} + 4 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 2 \beta_{6} - 5 \beta_{4} + 6 \beta_{3} - \beta_{2} + 5 \beta_{1} + 5\)
\(\nu^{10}\)\(=\)\(-3 \beta_{15} + 2 \beta_{14} + 10 \beta_{13} - 10 \beta_{12} - 6 \beta_{11} - 11 \beta_{9} + 5 \beta_{7} + 4 \beta_{6} - 12 \beta_{5} - 13 \beta_{3} - 12 \beta_{2} + 8 \beta_{1} - 6\)
\(\nu^{11}\)\(=\)\(-13 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} - 22 \beta_{11} - 16 \beta_{9} - 2 \beta_{8} - 17 \beta_{7} - 11 \beta_{6} + 15 \beta_{5} - 22 \beta_{4} - 18 \beta_{3} - 2 \beta_{2} + \beta_{1} + 17\)
\(\nu^{12}\)\(=\)\(-7 \beta_{15} - 20 \beta_{13} - 4 \beta_{12} - 20 \beta_{10} - 27 \beta_{9} + 7 \beta_{8} - 23 \beta_{7} - 16 \beta_{6} - 3 \beta_{5} + 23 \beta_{4} - \beta_{3} - 4 \beta_{2} + 7 \beta_{1} + 20\)
\(\nu^{13}\)\(=\)\(13 \beta_{15} - 13 \beta_{14} + 13 \beta_{13} + 14 \beta_{12} - 35 \beta_{11} + 16 \beta_{10} - 30 \beta_{9} + 27 \beta_{8} - 35 \beta_{7} - 27 \beta_{6} + 9 \beta_{5} - 33 \beta_{4} + 14 \beta_{3} - 13 \beta_{2} + 58 \beta_{1} + 2\)
\(\nu^{14}\)\(=\)\(30 \beta_{14} - 78 \beta_{13} + 54 \beta_{12} + 30 \beta_{11} - 60 \beta_{10} - 78 \beta_{9} + 45 \beta_{8} - 30 \beta_{6} + 77 \beta_{5} - 27 \beta_{4} - 2 \beta_{3} + 56 \beta_{2} + 15 \beta_{1} - 60\)
\(\nu^{15}\)\(=\)\(78 \beta_{15} + 15 \beta_{14} + 45 \beta_{13} + 90 \beta_{12} - 27 \beta_{11} - 72 \beta_{10} + 93 \beta_{8} + 34 \beta_{7} - 78 \beta_{6} + 66 \beta_{5} + 61 \beta_{4} + 90 \beta_{3} - 45 \beta_{2} + 111 \beta_{1} + 61\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/52\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.31256 + 0.526485i
−0.873468 + 1.11223i
1.08916 0.902074i
−1.39427 0.236640i
−0.00757716 1.41419i
1.41121 0.0921725i
1.17605 + 0.785427i
−0.713659 + 1.22094i
1.31256 0.526485i
−0.873468 1.11223i
1.08916 + 0.902074i
−1.39427 + 0.236640i
−0.00757716 + 1.41419i
1.41121 + 0.0921725i
1.17605 0.785427i
−0.713659 1.22094i
−1.11223 0.873468i −2.16981 1.25274i 0.474107 + 1.94299i −2.19962 2.19962i 1.31910 + 3.28859i 0.152604 0.569525i 1.16983 2.57517i 1.63871 + 2.83834i 0.525184 + 4.36778i
7.2 −0.526485 + 1.31256i 2.16981 + 1.25274i −1.44563 1.38209i −2.19962 2.19962i −2.78667 + 2.18846i −0.152604 + 0.569525i 2.57517 1.16983i 1.63871 + 2.83834i 4.04520 1.72907i
7.3 0.236640 1.39427i 0.736159 + 0.425021i −1.88800 0.659882i −0.166404 0.166404i 0.766801 0.925830i −0.684384 + 2.55416i −1.36683 + 2.47624i −1.13871 1.97231i −0.271390 + 0.192635i
7.4 0.902074 + 1.08916i −0.736159 0.425021i −0.372527 + 1.96500i −0.166404 0.166404i −0.201154 1.18519i 0.684384 2.55416i −2.47624 + 1.36683i −1.13871 1.97231i 0.0311314 0.331348i
11.1 −1.22094 0.713659i 1.40004 0.808315i 0.981383 + 1.74267i −1.52798 1.52798i −2.28623 0.0122495i 1.97429 0.529008i 0.0454612 2.82806i −0.193255 + 0.334727i 0.775114 + 2.95603i
11.2 −0.785427 + 1.17605i 1.81380 1.04720i −0.766209 1.84741i 0.894007 + 0.894007i −0.193046 + 2.95563i −4.37156 + 1.17136i 2.77446 + 0.549903i 0.693255 1.20075i −1.75358 + 0.349224i
11.3 0.0921725 + 1.41121i −1.81380 + 1.04720i −1.98301 + 0.260149i 0.894007 + 0.894007i −1.64500 2.46313i 4.37156 1.17136i −0.549903 2.77446i 0.693255 1.20075i −1.17923 + 1.34403i
11.4 1.41419 0.00757716i −1.40004 + 0.808315i 1.99989 0.0214311i −1.52798 1.52798i −1.97381 + 1.15372i −1.97429 + 0.529008i 2.82806 0.0454612i −0.193255 + 0.334727i −2.17244 2.14928i
15.1 −1.11223 + 0.873468i −2.16981 + 1.25274i 0.474107 1.94299i −2.19962 + 2.19962i 1.31910 3.28859i 0.152604 + 0.569525i 1.16983 + 2.57517i 1.63871 2.83834i 0.525184 4.36778i
15.2 −0.526485 1.31256i 2.16981 1.25274i −1.44563 + 1.38209i −2.19962 + 2.19962i −2.78667 2.18846i −0.152604 0.569525i 2.57517 + 1.16983i 1.63871 2.83834i 4.04520 + 1.72907i
15.3 0.236640 + 1.39427i 0.736159 0.425021i −1.88800 + 0.659882i −0.166404 + 0.166404i 0.766801 + 0.925830i −0.684384 2.55416i −1.36683 2.47624i −1.13871 + 1.97231i −0.271390 0.192635i
15.4 0.902074 1.08916i −0.736159 + 0.425021i −0.372527 1.96500i −0.166404 + 0.166404i −0.201154 + 1.18519i 0.684384 + 2.55416i −2.47624 1.36683i −1.13871 + 1.97231i 0.0311314 + 0.331348i
19.1 −1.22094 + 0.713659i 1.40004 + 0.808315i 0.981383 1.74267i −1.52798 + 1.52798i −2.28623 + 0.0122495i 1.97429 + 0.529008i 0.0454612 + 2.82806i −0.193255 0.334727i 0.775114 2.95603i
19.2 −0.785427 1.17605i 1.81380 + 1.04720i −0.766209 + 1.84741i 0.894007 0.894007i −0.193046 2.95563i −4.37156 1.17136i 2.77446 0.549903i 0.693255 + 1.20075i −1.75358 0.349224i
19.3 0.0921725 1.41121i −1.81380 1.04720i −1.98301 0.260149i 0.894007 0.894007i −1.64500 + 2.46313i 4.37156 + 1.17136i −0.549903 + 2.77446i 0.693255 + 1.20075i −1.17923 1.34403i
19.4 1.41419 + 0.00757716i −1.40004 0.808315i 1.99989 + 0.0214311i −1.52798 + 1.52798i −1.97381 1.15372i −1.97429 0.529008i 2.82806 + 0.0454612i −0.193255 0.334727i −2.17244 + 2.14928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
13.f odd 12 1 inner
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 52.2.l.b 16
3.b odd 2 1 468.2.cb.f 16
4.b odd 2 1 inner 52.2.l.b 16
8.b even 2 1 832.2.bu.n 16
8.d odd 2 1 832.2.bu.n 16
12.b even 2 1 468.2.cb.f 16
13.b even 2 1 676.2.l.k 16
13.c even 3 1 676.2.f.h 16
13.c even 3 1 676.2.l.m 16
13.d odd 4 1 676.2.l.i 16
13.d odd 4 1 676.2.l.m 16
13.e even 6 1 676.2.f.i 16
13.e even 6 1 676.2.l.i 16
13.f odd 12 1 inner 52.2.l.b 16
13.f odd 12 1 676.2.f.h 16
13.f odd 12 1 676.2.f.i 16
13.f odd 12 1 676.2.l.k 16
39.k even 12 1 468.2.cb.f 16
52.b odd 2 1 676.2.l.k 16
52.f even 4 1 676.2.l.i 16
52.f even 4 1 676.2.l.m 16
52.i odd 6 1 676.2.f.i 16
52.i odd 6 1 676.2.l.i 16
52.j odd 6 1 676.2.f.h 16
52.j odd 6 1 676.2.l.m 16
52.l even 12 1 inner 52.2.l.b 16
52.l even 12 1 676.2.f.h 16
52.l even 12 1 676.2.f.i 16
52.l even 12 1 676.2.l.k 16
104.u even 12 1 832.2.bu.n 16
104.x odd 12 1 832.2.bu.n 16
156.v odd 12 1 468.2.cb.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.2.l.b 16 1.a even 1 1 trivial
52.2.l.b 16 4.b odd 2 1 inner
52.2.l.b 16 13.f odd 12 1 inner
52.2.l.b 16 52.l even 12 1 inner
468.2.cb.f 16 3.b odd 2 1
468.2.cb.f 16 12.b even 2 1
468.2.cb.f 16 39.k even 12 1
468.2.cb.f 16 156.v odd 12 1
676.2.f.h 16 13.c even 3 1
676.2.f.h 16 13.f odd 12 1
676.2.f.h 16 52.j odd 6 1
676.2.f.h 16 52.l even 12 1
676.2.f.i 16 13.e even 6 1
676.2.f.i 16 13.f odd 12 1
676.2.f.i 16 52.i odd 6 1
676.2.f.i 16 52.l even 12 1
676.2.l.i 16 13.d odd 4 1
676.2.l.i 16 13.e even 6 1
676.2.l.i 16 52.f even 4 1
676.2.l.i 16 52.i odd 6 1
676.2.l.k 16 13.b even 2 1
676.2.l.k 16 13.f odd 12 1
676.2.l.k 16 52.b odd 2 1
676.2.l.k 16 52.l even 12 1
676.2.l.m 16 13.c even 3 1
676.2.l.m 16 13.d odd 4 1
676.2.l.m 16 52.f even 4 1
676.2.l.m 16 52.j odd 6 1
832.2.bu.n 16 8.b even 2 1
832.2.bu.n 16 8.d odd 2 1
832.2.bu.n 16 104.u even 12 1
832.2.bu.n 16 104.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(52, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 256 T + 320 T^{2} + 128 T^{3} + 80 T^{4} - 64 T^{5} - 48 T^{6} - 80 T^{7} - 36 T^{8} - 40 T^{9} - 12 T^{10} - 8 T^{11} + 5 T^{12} + 4 T^{13} + 5 T^{14} + 2 T^{15} + T^{16} \)
$3$ \( 2704 - 5824 T^{2} + 9164 T^{4} - 5824 T^{6} + 2605 T^{8} - 686 T^{10} + 131 T^{12} - 14 T^{14} + T^{16} \)
$5$ \( ( 4 + 24 T + 72 T^{2} + 5 T^{4} + 18 T^{5} + 18 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$7$ \( 43264 + 204672 T^{2} + 303408 T^{4} - 91512 T^{6} - 1399 T^{8} + 2790 T^{10} + 207 T^{12} - 30 T^{14} + T^{16} \)
$11$ \( 692224 - 159744 T^{2} - 124992 T^{4} + 31680 T^{6} + 25241 T^{8} - 2970 T^{10} - 57 T^{12} + 18 T^{14} + T^{16} \)
$13$ \( ( 28561 + 13182 T + 5577 T^{2} + 1326 T^{3} + 428 T^{4} + 102 T^{5} + 33 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$17$ \( ( 1 + 30 T + 290 T^{2} - 300 T^{3} + 39 T^{4} + 60 T^{5} + 2 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$19$ \( 77228944 + 34378656 T^{2} + 4389420 T^{4} - 316872 T^{6} - 72643 T^{8} + 4374 T^{10} + 891 T^{12} - 54 T^{14} + T^{16} \)
$23$ \( 77228944 + 91395200 T^{2} + 87130316 T^{4} + 23024144 T^{6} + 4615261 T^{8} + 232858 T^{10} + 8843 T^{12} + 106 T^{14} + T^{16} \)
$29$ \( ( 51529 + 55388 T + 49094 T^{2} + 13040 T^{3} + 3319 T^{4} + 304 T^{5} + 62 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$31$ \( 1235663104 + 9894433536 T^{4} + 18310880 T^{8} + 9072 T^{12} + T^{16} \)
$37$ \( ( 1 + 2 T + 5 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$41$ \( ( 81 - 54 T + 45 T^{2} - 12 T^{3} + T^{4} )^{4} \)
$43$ \( 5671027857664 + 1531577976448 T^{2} + 359783787440 T^{4} + 13276514728 T^{6} + 337890073 T^{8} + 4728770 T^{10} + 48143 T^{12} + 266 T^{14} + T^{16} \)
$47$ \( 5671027857664 + 39057328896 T^{4} + 70290656 T^{8} + 20976 T^{12} + T^{16} \)
$53$ \( ( -128 - 112 T - 3 T^{2} + 8 T^{3} + T^{4} )^{4} \)
$59$ \( 171720267307264 - 206627151744 T^{2} - 106493647056 T^{4} + 128241144 T^{6} + 53073017 T^{8} + 48798 T^{10} - 8121 T^{12} - 6 T^{14} + T^{16} \)
$61$ \( ( 6889 - 13114 T + 20150 T^{2} - 9496 T^{3} + 3763 T^{4} - 200 T^{5} + 62 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$67$ \( 2205735869584 + 548171044512 T^{2} + 39908056812 T^{4} - 1367500680 T^{6} - 3260179 T^{8} + 466830 T^{10} + 1587 T^{12} - 126 T^{14} + T^{16} \)
$71$ \( 5067731344 + 876466656 T^{2} - 336235956 T^{4} - 66891096 T^{6} + 28929197 T^{8} + 684558 T^{10} - 141 T^{12} - 126 T^{14} + T^{16} \)
$73$ \( ( 676 - 15496 T + 177608 T^{2} - 35424 T^{3} + 3533 T^{4} - 6 T^{5} + 50 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$79$ \( ( 53248 + 51200 T^{2} + 7040 T^{4} + 160 T^{6} + T^{8} )^{2} \)
$83$ \( 538409280507904 + 3819952817664 T^{4} + 891937424 T^{8} + 61704 T^{12} + T^{16} \)
$89$ \( ( 2896804 - 2474708 T + 479246 T^{2} + 5950 T^{3} + 19699 T^{4} + 1736 T^{5} + 215 T^{6} + 26 T^{7} + T^{8} )^{2} \)
$97$ \( ( 2116 + 12236 T + 25694 T^{2} + 23070 T^{3} + 9371 T^{4} + 1812 T^{5} + 323 T^{6} + 14 T^{7} + T^{8} )^{2} \)
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