Properties

Label 44.2.g.a
Level $44$
Weight $2$
Character orbit 44.g
Analytic conductor $0.351$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 44 = 2^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 44.g (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.351341768894\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + 120 x^{7} - 8 x^{6} - 240 x^{5} + 560 x^{4} - 800 x^{3} + 832 x^{2} - 640 x + 256\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 5 x^{15} + 13 x^{14} - 25 x^{13} + 35 x^{12} - 30 x^{11} - 2 x^{10} + 60 x^{9} - 116 x^{8} + 120 x^{7} - 8 x^{6} - 240 x^{5} + 560 x^{4} - 800 x^{3} + 832 x^{2} - 640 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{15} - 5 \nu^{14} + 13 \nu^{13} - 25 \nu^{12} + 35 \nu^{11} - 30 \nu^{10} - 2 \nu^{9} + 60 \nu^{8} - 116 \nu^{7} + 120 \nu^{6} - 8 \nu^{5} - 240 \nu^{4} + 560 \nu^{3} - 800 \nu^{2} + 832 \nu - 640 \)\()/128\)
\(\beta_{3}\)\(=\)\((\)\( -35 \nu^{15} + 94 \nu^{14} - 210 \nu^{13} + 334 \nu^{12} - 328 \nu^{11} + 111 \nu^{10} + 476 \nu^{9} - 1018 \nu^{8} + 1368 \nu^{7} - 500 \nu^{6} - 1600 \nu^{5} + 4744 \nu^{4} - 7520 \nu^{3} + 7760 \nu^{2} - 7072 \nu + 2304 \)\()/64\)
\(\beta_{4}\)\(=\)\((\)\( -30 \nu^{15} + 141 \nu^{14} - 309 \nu^{13} + 573 \nu^{12} - 693 \nu^{11} + 429 \nu^{10} + 398 \nu^{9} - 1670 \nu^{8} + 2452 \nu^{7} - 2036 \nu^{6} - 1384 \nu^{5} + 6728 \nu^{4} - 13008 \nu^{3} + 15856 \nu^{2} - 14368 \nu + 9664 \)\()/64\)
\(\beta_{5}\)\(=\)\((\)\( 33 \nu^{15} - 147 \nu^{14} + 323 \nu^{13} - 591 \nu^{12} + 705 \nu^{11} - 424 \nu^{10} - 446 \nu^{9} + 1736 \nu^{8} - 2524 \nu^{7} + 2000 \nu^{6} + 1544 \nu^{5} - 7072 \nu^{4} + 13424 \nu^{3} - 16128 \nu^{2} + 14592 \nu - 9472 \)\()/64\)
\(\beta_{6}\)\(=\)\((\)\( -37 \nu^{15} + 152 \nu^{14} - 334 \nu^{13} + 602 \nu^{12} - 704 \nu^{11} + 405 \nu^{10} + 498 \nu^{9} - 1774 \nu^{8} + 2556 \nu^{7} - 1916 \nu^{6} - 1704 \nu^{5} + 7336 \nu^{4} - 13648 \nu^{3} + 16176 \nu^{2} - 14656 \nu + 9088 \)\()/64\)
\(\beta_{7}\)\(=\)\((\)\( 35 \nu^{15} - 120 \nu^{14} + 265 \nu^{13} - 455 \nu^{12} + 501 \nu^{11} - 248 \nu^{10} - 475 \nu^{9} + 1358 \nu^{8} - 1902 \nu^{7} + 1168 \nu^{6} + 1612 \nu^{5} - 5880 \nu^{4} + 10280 \nu^{3} - 11632 \nu^{2} + 10544 \nu - 5536 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -47 \nu^{15} + 153 \nu^{14} - 338 \nu^{13} + 572 \nu^{12} - 618 \nu^{11} + 291 \nu^{10} + 633 \nu^{9} - 1712 \nu^{8} + 2382 \nu^{7} - 1356 \nu^{6} - 2148 \nu^{5} + 7520 \nu^{4} - 12920 \nu^{3} + 14400 \nu^{2} - 13072 \nu + 6464 \)\()/32\)
\(\beta_{9}\)\(=\)\((\)\( 71 \nu^{15} - 281 \nu^{14} + 619 \nu^{13} - 1107 \nu^{12} + 1281 \nu^{11} - 722 \nu^{10} - 952 \nu^{9} + 3264 \nu^{8} - 4688 \nu^{7} + 3408 \nu^{6} + 3264 \nu^{5} - 13632 \nu^{4} + 25088 \nu^{3} - 29504 \nu^{2} + 26720 \nu - 16128 \)\()/64\)
\(\beta_{10}\)\(=\)\((\)\( -279 \nu^{15} + 987 \nu^{14} - 2179 \nu^{13} + 3775 \nu^{12} - 4213 \nu^{11} + 2162 \nu^{10} + 3766 \nu^{9} - 11244 \nu^{8} + 15852 \nu^{7} - 10168 \nu^{6} - 12808 \nu^{5} + 48240 \nu^{4} - 85456 \nu^{3} + 97632 \nu^{2} - 88384 \nu + 48384 \)\()/128\)
\(\beta_{11}\)\(=\)\((\)\(307 \nu^{15} - 1097 \nu^{14} + 2421 \nu^{13} - 4209 \nu^{12} + 4711 \nu^{11} - 2444 \nu^{10} - 4142 \nu^{9} + 12520 \nu^{8} - 17676 \nu^{7} + 11488 \nu^{6} + 14088 \nu^{5} - 53600 \nu^{4} + 95248 \nu^{3} - 109056 \nu^{2} + 98880 \nu - 54528\)\()/128\)
\(\beta_{12}\)\(=\)\((\)\(-293 \nu^{15} + 1079 \nu^{14} - 2383 \nu^{13} + 4179 \nu^{12} - 4733 \nu^{11} + 2528 \nu^{10} + 3942 \nu^{9} - 12400 \nu^{8} + 17612 \nu^{7} - 11888 \nu^{6} - 13448 \nu^{5} + 52672 \nu^{4} - 94608 \nu^{3} + 109376 \nu^{2} - 99072 \nu + 56320\)\()/128\)
\(\beta_{13}\)\(=\)\((\)\( 162 \nu^{15} - 591 \nu^{14} + 1304 \nu^{13} - 2280 \nu^{12} + 2572 \nu^{11} - 1360 \nu^{10} - 2183 \nu^{9} + 6772 \nu^{8} - 9594 \nu^{7} + 6392 \nu^{6} + 7444 \nu^{5} - 28824 \nu^{4} + 51608 \nu^{3} - 59456 \nu^{2} + 53872 \nu - 30400 \)\()/32\)
\(\beta_{14}\)\(=\)\((\)\(-361 \nu^{15} + 1262 \nu^{14} - 2786 \nu^{13} + 4814 \nu^{12} - 5348 \nu^{11} + 2721 \nu^{10} + 4868 \nu^{9} - 14342 \nu^{8} + 20176 \nu^{7} - 12748 \nu^{6} - 16560 \nu^{5} + 61736 \nu^{4} - 108864 \nu^{3} + 123984 \nu^{2} - 112352 \nu + 60672\)\()/64\)
\(\beta_{15}\)\(=\)\((\)\(-785 \nu^{15} + 2815 \nu^{14} - 6215 \nu^{13} + 10819 \nu^{12} - 12133 \nu^{11} + 6324 \nu^{10} + 10574 \nu^{9} - 32160 \nu^{8} + 45468 \nu^{7} - 29696 \nu^{6} - 36008 \nu^{5} + 137504 \nu^{4} - 244752 \nu^{3} + 280832 \nu^{2} - 254464 \nu + 141056\)\()/128\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(-\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{15} + \beta_{13} + \beta_{12} - \beta_{10} + 2 \beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(-\beta_{15} + \beta_{13} + 3 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{1} + 4\)
\(\nu^{5}\)\(=\)\(\beta_{13} - 2 \beta_{11} + 3 \beta_{6} - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{14} - 2 \beta_{12} + 4 \beta_{11} + \beta_{10} - 5 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 2 \beta_{1}\)
\(\nu^{7}\)\(=\)\(6 \beta_{15} - 3 \beta_{14} + 5 \beta_{13} - \beta_{11} + \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} + 10\)
\(\nu^{8}\)\(=\)\(-5 \beta_{15} - 2 \beta_{14} - \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} - 8 \beta_{9} + 10 \beta_{8} - 12 \beta_{7} + \beta_{6} - 6 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 9 \beta_{2} - 10 \beta_{1} + 6\)
\(\nu^{9}\)\(=\)\(-7 \beta_{15} + 6 \beta_{14} + 3 \beta_{13} + \beta_{12} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} - 4 \beta_{8} - 11 \beta_{7} + 17 \beta_{6} - 3 \beta_{5} - 7 \beta_{4} + 3 \beta_{3} - 12 \beta_{2} - 5 \beta_{1} + 4\)
\(\nu^{10}\)\(=\)\(-2 \beta_{15} + 14 \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{9} - 14 \beta_{8} + 10 \beta_{7} + \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 2 \beta_{1}\)
\(\nu^{11}\)\(=\)\(26 \beta_{15} + 21 \beta_{14} + 40 \beta_{13} - 24 \beta_{12} - 20 \beta_{11} + 19 \beta_{10} + 3 \beta_{9} - 51 \beta_{8} + 39 \beta_{7} - 31 \beta_{6} + \beta_{5} + 40 \beta_{4} + 20 \beta_{3} + 27 \beta_{2} + 42 \beta_{1} + 28\)
\(\nu^{12}\)\(=\)\(20 \beta_{15} - \beta_{14} - \beta_{13} - 30 \beta_{12} + \beta_{11} - 23 \beta_{10} - 21 \beta_{9} - 15 \beta_{8} - 12 \beta_{7} - 16 \beta_{6} + 8 \beta_{5} - 4 \beta_{4} + 2 \beta_{2} - 9 \beta_{1} + 10\)
\(\nu^{13}\)\(=\)\(-21 \beta_{15} + 8 \beta_{14} + 11 \beta_{13} + 31 \beta_{12} + 24 \beta_{11} - 11 \beta_{10} - 38 \beta_{9} + 48 \beta_{8} - 8 \beta_{7} + 95 \beta_{6} - 2 \beta_{5} - 49 \beta_{4} - 8 \beta_{3} + 11 \beta_{2} + 50\)
\(\nu^{14}\)\(=\)\(7 \beta_{15} + 45 \beta_{13} + 19 \beta_{12} - 33 \beta_{11} - 9 \beta_{10} + 52 \beta_{9} + 62 \beta_{8} + 23 \beta_{7} + 85 \beta_{6} - 33 \beta_{5} + 27 \beta_{4} - 9 \beta_{3} + 32 \beta_{2} - 29 \beta_{1} + 32\)
\(\nu^{15}\)\(=\)\(20 \beta_{15} + 20 \beta_{14} + 61 \beta_{13} - 56 \beta_{12} + 22 \beta_{11} + 40 \beta_{10} - 88 \beta_{9} - 24 \beta_{8} - 104 \beta_{7} - 73 \beta_{6} - 144 \beta_{5} - 66 \beta_{4} - 21 \beta_{3} - 46 \beta_{2} + 62 \beta_{1} + 68\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/44\mathbb{Z}\right)^\times\).

\(n\) \(13\) \(23\)
\(\chi(n)\) \(-\beta_{12}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.40958 0.114404i
0.0737040 1.41229i
−0.544389 + 1.30524i
−1.36594 0.366325i
1.40958 + 0.114404i
0.0737040 + 1.41229i
−0.544389 1.30524i
−1.36594 + 0.366325i
1.40874 + 0.124276i
1.06665 0.928579i
0.656642 + 1.25253i
−0.204982 1.39928i
1.40874 0.124276i
1.06665 + 0.928579i
0.656642 1.25253i
−0.204982 + 1.39928i
−1.40958 0.114404i 0.704424 + 0.228881i 1.97382 + 0.322523i 1.09089 0.792578i −0.966756 0.403215i 0.503194 + 1.54867i −2.74536 0.680436i −1.98322 1.44090i −1.62837 + 0.992398i
7.2 −0.0737040 1.41229i 1.70537 + 0.554109i −1.98914 + 0.208183i −2.39991 + 1.74363i 0.656871 2.44932i −0.815620 2.51022i 0.440622 + 2.79390i 0.174207 + 0.126569i 2.63940 + 3.26086i
7.3 0.544389 + 1.30524i −0.704424 0.228881i −1.40728 + 1.42111i 1.09089 0.792578i −0.0847364 1.04404i −0.503194 1.54867i −2.62099 1.06320i −1.98322 1.44090i 1.62837 + 0.992398i
7.4 1.36594 0.366325i −1.70537 0.554109i 1.73161 1.00076i −2.39991 + 1.74363i −2.53243 0.132161i 0.815620 + 2.51022i 1.99868 2.00132i 0.174207 + 0.126569i −2.63940 + 3.26086i
19.1 −1.40958 + 0.114404i 0.704424 0.228881i 1.97382 0.322523i 1.09089 + 0.792578i −0.966756 + 0.403215i 0.503194 1.54867i −2.74536 + 0.680436i −1.98322 + 1.44090i −1.62837 0.992398i
19.2 −0.0737040 + 1.41229i 1.70537 0.554109i −1.98914 0.208183i −2.39991 1.74363i 0.656871 + 2.44932i −0.815620 + 2.51022i 0.440622 2.79390i 0.174207 0.126569i 2.63940 3.26086i
19.3 0.544389 1.30524i −0.704424 + 0.228881i −1.40728 1.42111i 1.09089 + 0.792578i −0.0847364 + 1.04404i −0.503194 + 1.54867i −2.62099 + 1.06320i −1.98322 + 1.44090i 1.62837 0.992398i
19.4 1.36594 + 0.366325i −1.70537 + 0.554109i 1.73161 + 1.00076i −2.39991 1.74363i −2.53243 + 0.132161i 0.815620 2.51022i 1.99868 + 2.00132i 0.174207 0.126569i −2.63940 3.26086i
35.1 −1.40874 + 0.124276i −1.59814 2.19965i 1.96911 0.350146i −0.720859 2.21858i 2.52473 + 2.90013i 1.04462 + 0.758960i −2.73046 + 0.737979i −1.35736 + 4.17752i 1.29122 + 3.03582i
35.2 −1.06665 0.928579i 1.59814 + 2.19965i 0.275480 + 1.98094i −0.720859 2.21858i 0.337896 3.83025i −1.04462 0.758960i 1.54562 2.36877i −1.35736 + 4.17752i −1.29122 + 3.03582i
35.3 −0.656642 + 1.25253i 0.539857 + 0.743049i −1.13764 1.64492i 0.529876 + 1.63079i −1.28518 + 0.188268i −1.93399 1.40513i 2.80733 0.344804i 0.666375 2.05089i −2.39055 0.407162i
35.4 0.204982 1.39928i −0.539857 0.743049i −1.91596 0.573655i 0.529876 + 1.63079i −1.15039 + 0.603098i 1.93399 + 1.40513i −1.19544 + 2.56338i 0.666375 2.05089i 2.39055 0.407162i
39.1 −1.40874 0.124276i −1.59814 + 2.19965i 1.96911 + 0.350146i −0.720859 + 2.21858i 2.52473 2.90013i 1.04462 0.758960i −2.73046 0.737979i −1.35736 4.17752i 1.29122 3.03582i
39.2 −1.06665 + 0.928579i 1.59814 2.19965i 0.275480 1.98094i −0.720859 + 2.21858i 0.337896 + 3.83025i −1.04462 + 0.758960i 1.54562 + 2.36877i −1.35736 4.17752i −1.29122 3.03582i
39.3 −0.656642 1.25253i 0.539857 0.743049i −1.13764 + 1.64492i 0.529876 1.63079i −1.28518 0.188268i −1.93399 + 1.40513i 2.80733 + 0.344804i 0.666375 + 2.05089i −2.39055 + 0.407162i
39.4 0.204982 + 1.39928i −0.539857 + 0.743049i −1.91596 + 0.573655i 0.529876 1.63079i −1.15039 0.603098i 1.93399 1.40513i −1.19544 2.56338i 0.666375 + 2.05089i 2.39055 + 0.407162i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 44.2.g.a 16
3.b odd 2 1 396.2.r.a 16
4.b odd 2 1 inner 44.2.g.a 16
8.b even 2 1 704.2.u.c 16
8.d odd 2 1 704.2.u.c 16
11.b odd 2 1 484.2.g.i 16
11.c even 5 1 484.2.c.d 16
11.c even 5 1 484.2.g.f 16
11.c even 5 1 484.2.g.i 16
11.c even 5 1 484.2.g.j 16
11.d odd 10 1 inner 44.2.g.a 16
11.d odd 10 1 484.2.c.d 16
11.d odd 10 1 484.2.g.f 16
11.d odd 10 1 484.2.g.j 16
12.b even 2 1 396.2.r.a 16
33.f even 10 1 396.2.r.a 16
44.c even 2 1 484.2.g.i 16
44.g even 10 1 inner 44.2.g.a 16
44.g even 10 1 484.2.c.d 16
44.g even 10 1 484.2.g.f 16
44.g even 10 1 484.2.g.j 16
44.h odd 10 1 484.2.c.d 16
44.h odd 10 1 484.2.g.f 16
44.h odd 10 1 484.2.g.i 16
44.h odd 10 1 484.2.g.j 16
88.k even 10 1 704.2.u.c 16
88.p odd 10 1 704.2.u.c 16
132.n odd 10 1 396.2.r.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.g.a 16 1.a even 1 1 trivial
44.2.g.a 16 4.b odd 2 1 inner
44.2.g.a 16 11.d odd 10 1 inner
44.2.g.a 16 44.g even 10 1 inner
396.2.r.a 16 3.b odd 2 1
396.2.r.a 16 12.b even 2 1
396.2.r.a 16 33.f even 10 1
396.2.r.a 16 132.n odd 10 1
484.2.c.d 16 11.c even 5 1
484.2.c.d 16 11.d odd 10 1
484.2.c.d 16 44.g even 10 1
484.2.c.d 16 44.h odd 10 1
484.2.g.f 16 11.c even 5 1
484.2.g.f 16 11.d odd 10 1
484.2.g.f 16 44.g even 10 1
484.2.g.f 16 44.h odd 10 1
484.2.g.i 16 11.b odd 2 1
484.2.g.i 16 11.c even 5 1
484.2.g.i 16 44.c even 2 1
484.2.g.i 16 44.h odd 10 1
484.2.g.j 16 11.c even 5 1
484.2.g.j 16 11.d odd 10 1
484.2.g.j 16 44.g even 10 1
484.2.g.j 16 44.h odd 10 1
704.2.u.c 16 8.b even 2 1
704.2.u.c 16 8.d odd 2 1
704.2.u.c 16 88.k even 10 1
704.2.u.c 16 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(44, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 256 + 640 T + 832 T^{2} + 800 T^{3} + 560 T^{4} + 240 T^{5} - 8 T^{6} - 120 T^{7} - 116 T^{8} - 60 T^{9} - 2 T^{10} + 30 T^{11} + 35 T^{12} + 25 T^{13} + 13 T^{14} + 5 T^{15} + T^{16} \)
$3$ \( 121 - 319 T^{2} + 432 T^{4} - 357 T^{6} + 675 T^{8} - 253 T^{10} + 42 T^{12} - T^{14} + T^{16} \)
$5$ \( ( 256 - 192 T + 128 T^{2} - 24 T^{3} + 25 T^{4} + 6 T^{5} + 8 T^{6} + 3 T^{7} + T^{8} )^{2} \)
$7$ \( 30976 + 11264 T^{2} + 10272 T^{4} + 6572 T^{6} + 2505 T^{8} + 378 T^{10} + 72 T^{12} + 11 T^{14} + T^{16} \)
$11$ \( 214358881 - 62004635 T^{2} + 7598679 T^{4} - 649165 T^{6} + 56496 T^{8} - 5365 T^{10} + 519 T^{12} - 35 T^{14} + T^{16} \)
$13$ \( ( 16 + 120 T + 296 T^{2} + 220 T^{3} + 61 T^{4} + 20 T^{5} + 16 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$17$ \( ( 1 + 15 T + 176 T^{2} - 335 T^{3} + 401 T^{4} - 85 T^{5} - 24 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$19$ \( 1771561 - 247566 T^{2} + 614542 T^{4} + 124182 T^{6} + 47600 T^{8} + 15948 T^{10} + 4387 T^{12} + 106 T^{14} + T^{16} \)
$23$ \( ( 2816 + 10752 T^{2} + 2064 T^{4} + 88 T^{6} + T^{8} )^{2} \)
$29$ \( ( 400 + 1400 T + 2600 T^{2} + 1900 T^{3} + 85 T^{4} - 320 T^{5} + 20 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$31$ \( 428888770816 - 18536176384 T^{2} + 2166038752 T^{4} - 166420932 T^{6} + 8183825 T^{8} - 149838 T^{10} + 5812 T^{12} - 111 T^{14} + T^{16} \)
$37$ \( ( 13456 + 54984 T + 82132 T^{2} - 25278 T^{3} + 5305 T^{4} - 732 T^{5} + 132 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$41$ \( ( 2588881 - 1488325 T + 399696 T^{2} - 48155 T^{3} - 639 T^{4} + 895 T^{5} - 74 T^{6} - 5 T^{7} + T^{8} )^{2} \)
$43$ \( ( 148016 - 58912 T^{2} + 5429 T^{4} - 143 T^{6} + T^{8} )^{2} \)
$47$ \( 1206517709056 - 122161433856 T^{2} + 7676727392 T^{4} - 380460228 T^{6} + 35709505 T^{8} - 297282 T^{10} + 8412 T^{12} - 129 T^{14} + T^{16} \)
$53$ \( ( 15376 - 11656 T + 11112 T^{2} - 4528 T^{3} + 1905 T^{4} - 672 T^{5} + 162 T^{6} - 19 T^{7} + T^{8} )^{2} \)
$59$ \( 17080137481 - 50132283454 T^{2} + 56821159822 T^{4} - 2926349122 T^{6} + 64872880 T^{8} - 181268 T^{10} + 40067 T^{12} + 114 T^{14} + T^{16} \)
$61$ \( ( 633616 - 429840 T + 247784 T^{2} + 61030 T^{3} + 4781 T^{4} - 880 T^{5} - 126 T^{6} + 5 T^{7} + T^{8} )^{2} \)
$67$ \( ( 1760000 + 226000 T^{2} + 10225 T^{4} + 185 T^{6} + T^{8} )^{2} \)
$71$ \( 21908736256 - 4629348416 T^{2} + 460206432 T^{4} - 25650768 T^{6} + 1299225 T^{8} - 62942 T^{10} + 2112 T^{12} + T^{14} + T^{16} \)
$73$ \( ( 6561 - 10935 T + 2916 T^{2} + 1215 T^{3} + 1701 T^{4} - 135 T^{5} + 36 T^{6} + 15 T^{7} + T^{8} )^{2} \)
$79$ \( 28606986496 + 4499694144 T^{2} + 610810832 T^{4} + 79698372 T^{6} + 12605305 T^{8} + 46638 T^{10} + 8112 T^{12} + 141 T^{14} + T^{16} \)
$83$ \( 435963075625 + 72795318750 T^{2} + 6940690750 T^{4} + 533610000 T^{6} + 40502900 T^{8} + 1975500 T^{10} + 46705 T^{12} - 100 T^{14} + T^{16} \)
$89$ \( ( 116 - 186 T - 19 T^{2} + 9 T^{3} + T^{4} )^{4} \)
$97$ \( ( 3721 + 1098 T + 184 T^{2} + 17 T^{3} + T^{4} )^{4} \)
show more
show less