Properties

Label 475.2.l.c
Level $475$
Weight $2$
Character orbit 475.l
Analytic conductor $3.793$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} - 156 x^{9} + 582 x^{8} - 138 x^{7} + 437 x^{6} - 132 x^{5} + 198 x^{4} - 16 x^{3} + 15 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} - 3 x^{17} + 15 x^{16} - 14 x^{15} + 72 x^{14} - 51 x^{13} + 231 x^{12} - 93 x^{11} + 438 x^{10} - 156 x^{9} + 582 x^{8} - 138 x^{7} + 437 x^{6} - 132 x^{5} + 198 x^{4} - 16 x^{3} + 15 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-2362215094 \nu^{17} - 176902925324 \nu^{16} + 523927730528 \nu^{15} - 2732196133634 \nu^{14} + 2458535955743 \nu^{13} - 12964718137852 \nu^{12} + 9051939676612 \nu^{11} - 41495881520992 \nu^{10} + 16563882463922 \nu^{9} - 77292885590832 \nu^{8} + 27960663596715 \nu^{7} - 101945748763594 \nu^{6} + 24300805314434 \nu^{5} - 73622429378663 \nu^{4} + 22913565282806 \nu^{3} - 34340446173148 \nu^{2} + 1348014480151 \nu - 2601214274036\)\()/ 2269131592089 \)
\(\beta_{3}\)\(=\)\((\)\(-17485732004 \nu^{17} - 88297854546 \nu^{16} + 51093388053 \nu^{15} - 1446829381050 \nu^{14} - 1289873408104 \nu^{13} - 6148599460869 \nu^{12} - 6799863255329 \nu^{11} - 19595360087749 \nu^{10} - 25750669196294 \nu^{9} - 32654560143716 \nu^{8} - 45248146421483 \nu^{7} - 41806244150830 \nu^{6} - 53683022385418 \nu^{5} - 21178028146029 \nu^{4} - 24543770185394 \nu^{3} - 6310322562086 \nu^{2} - 4173984429332 \nu + 1249494973861\)\()/ 6807394776267 \)
\(\beta_{4}\)\(=\)\((\)\(31170269891 \nu^{17} + 67213359096 \nu^{16} + 77458832787 \nu^{15} + 1646328025947 \nu^{14} + 1576807416328 \nu^{13} + 7658100229362 \nu^{12} + 7321752947336 \nu^{11} + 23929992121897 \nu^{10} + 25358952088733 \nu^{9} + 39943209775688 \nu^{8} + 44417859093215 \nu^{7} + 44701878155230 \nu^{6} + 51736757215426 \nu^{5} + 24440209046322 \nu^{4} + 25432415690738 \nu^{3} + 6559174797101 \nu^{2} + 4174852913747 \nu + 2036474776337\)\()/ 6807394776267 \)
\(\beta_{5}\)\(=\)\((\)\(-150661038729 \nu^{17} + 22583410048 \nu^{16} - 1207986331145 \nu^{15} - 3440777225226 \nu^{14} - 9048902407408 \nu^{13} - 16935494273702 \nu^{12} - 33578770190668 \nu^{11} - 61173284253370 \nu^{10} - 91399807598455 \nu^{9} - 107347453764758 \nu^{8} - 139987308514533 \nu^{7} - 140150638887798 \nu^{6} - 150604628667508 \nu^{5} - 71368680699863 \nu^{4} - 65671130584969 \nu^{3} - 18574098462028 \nu^{2} - 25239051620405 \nu + 9924998536452\)\()/ 6807394776267 \)
\(\beta_{6}\)\(=\)\((\)\(271722740134 \nu^{17} - 484705722592 \nu^{16} + 3183507161538 \nu^{15} + 840321900394 \nu^{14} + 16364383205847 \nu^{13} + 8550245559592 \nu^{12} + 52145226330891 \nu^{11} + 44506175702300 \nu^{10} + 108555610980184 \nu^{9} + 86539068699317 \nu^{8} + 143914939792586 \nu^{7} + 117052142258774 \nu^{6} + 128192309299746 \nu^{5} + 63296274710146 \nu^{4} + 52866415784337 \nu^{3} + 15495186910162 \nu^{2} + 23094455322128 \nu - 2451247768033\)\()/ 6807394776267 \)
\(\beta_{7}\)\(=\)\((\)\(14393251623 \nu^{17} - 18779546977 \nu^{16} + 163974948404 \nu^{15} + 84439865154 \nu^{14} + 1054848843451 \nu^{13} + 517821809207 \nu^{12} + 3667695351382 \nu^{11} + 2401081552594 \nu^{10} + 8811394522903 \nu^{9} + 4204775784857 \nu^{8} + 12178339879671 \nu^{7} + 5750184114669 \nu^{6} + 11453985494683 \nu^{5} + 2591457050228 \nu^{4} + 3507803893486 \nu^{3} + 907619400199 \nu^{2} + 1344049384007 \nu - 139873473393\)\()/ 358283935593 \)
\(\beta_{8}\)\(=\)\((\)\(-99295558576 \nu^{17} + 144550753822 \nu^{16} - 988488698886 \nu^{15} - 1021310336031 \nu^{14} - 4366731414611 \nu^{13} - 6588252818956 \nu^{12} - 11452733474463 \nu^{11} - 28551737645641 \nu^{10} - 17195400090500 \nu^{9} - 56855271575000 \nu^{8} - 8530715161662 \nu^{7} - 84690957416629 \nu^{6} + 8235247858388 \nu^{5} - 64199831043919 \nu^{4} + 22920186337367 \nu^{3} - 36148782515349 \nu^{2} + 1289655835611 \nu - 2743148781339\)\()/ 2269131592089 \)
\(\beta_{9}\)\(=\)\((\)\(-330462497810 \nu^{17} + 892333940472 \nu^{16} - 4644440262270 \nu^{15} + 3199654083801 \nu^{14} - 22408105306426 \nu^{13} + 10622726640063 \nu^{12} - 69776390534762 \nu^{11} + 10458949198508 \nu^{10} - 128927816160221 \nu^{9} + 14227694965402 \nu^{8} - 154549880397266 \nu^{7} - 9449471861188 \nu^{6} - 99163676407834 \nu^{5} + 934686762195 \nu^{4} - 19842750752306 \nu^{3} - 19018614220118 \nu^{2} + 2451247768033 \nu + 271722740134\)\()/ 6807394776267 \)
\(\beta_{10}\)\(=\)\((\)\(335323421585 \nu^{17} - 1203461029533 \nu^{16} + 5639724390405 \nu^{15} - 7682766129021 \nu^{14} + 26906316005605 \nu^{13} - 30567017722290 \nu^{12} + 85618508403902 \nu^{11} - 73041253362959 \nu^{10} + 156613392824897 \nu^{9} - 126748060068652 \nu^{8} + 199076210749085 \nu^{7} - 141640168905788 \nu^{6} + 135207369075700 \nu^{5} - 109625290547904 \nu^{4} + 54155545522772 \nu^{3} - 30581574139987 \nu^{2} - 422383778863 \nu - 759862203949\)\()/ 6807394776267 \)
\(\beta_{11}\)\(=\)\((\)\(169369527342 \nu^{17} - 512091105045 \nu^{16} + 2467349014493 \nu^{15} - 2179007028305 \nu^{14} + 10956780516730 \nu^{13} - 7678713505092 \nu^{12} + 32807739705559 \nu^{11} - 11980132322574 \nu^{10} + 53548739509621 \nu^{9} - 19086289356322 \nu^{8} + 58786967689962 \nu^{7} - 10254441293754 \nu^{6} + 22431948273852 \nu^{5} - 10070536517332 \nu^{4} - 3857061703044 \nu^{3} + 8766054588512 \nu^{2} - 15266507142983 \nu + 671726483761\)\()/ 2269131592089 \)
\(\beta_{12}\)\(=\)\((\)\(-183989570606 \nu^{17} + 559360956938 \nu^{16} - 2765267144950 \nu^{15} + 2628615442511 \nu^{14} - 13085191107646 \nu^{13} + 9597611363326 \nu^{12} - 41715567524734 \nu^{11} + 17598532675094 \nu^{10} - 77661391145496 \nu^{9} + 29335472781423 \nu^{8} - 102271734446566 \nu^{7} + 25333093310512 \nu^{6} - 73934241771071 \nu^{5} + 23381283871418 \nu^{4} - 34378241614652 \nu^{3} + 1383447706561 \nu^{2} - 332082681947 \nu + 2362215094\)\()/ 2269131592089 \)
\(\beta_{13}\)\(=\)\((\)\(-759862203949 \nu^{17} + 1944263190262 \nu^{16} - 10194472029702 \nu^{15} + 4998346464881 \nu^{14} - 47027312555307 \nu^{13} + 11846656395794 \nu^{12} - 144961151389929 \nu^{11} - 14951323436645 \nu^{10} - 259778391966703 \nu^{9} - 38074889008853 \nu^{8} - 315491742629666 \nu^{7} - 94215226604123 \nu^{6} - 190419614219925 \nu^{5} - 34905558154432 \nu^{4} - 40827425833998 \nu^{3} - 41997750259588 \nu^{2} + 19183641080752 \nu + 422383778863\)\()/ 6807394776267 \)
\(\beta_{14}\)\(=\)\((\)\(-1201752794172 \nu^{17} + 3351021374386 \nu^{16} - 17453288335442 \nu^{15} + 13589938568817 \nu^{14} - 85919693699728 \nu^{13} + 46214882705029 \nu^{12} - 280006541552455 \nu^{11} + 66417646782434 \nu^{10} - 551278539461887 \nu^{9} + 104668494106942 \nu^{8} - 753304634763705 \nu^{7} + 67175092224633 \nu^{6} - 600411869161108 \nu^{5} + 84122081916376 \nu^{4} - 280065299810173 \nu^{3} - 9094600892191 \nu^{2} - 27211156260656 \nu - 12113076116529\)\()/ 6807394776267 \)
\(\beta_{15}\)\(=\)\((\)\(-1764752036203 \nu^{17} + 5655888876310 \nu^{16} - 27296401124421 \nu^{15} + 29428427601899 \nu^{14} - 128615472664470 \nu^{13} + 113987266569107 \nu^{12} - 410622346773594 \nu^{11} + 241220082848977 \nu^{10} - 759490348933525 \nu^{9} + 429588085896622 \nu^{8} - 1001370170259860 \nu^{7} + 442503520486495 \nu^{6} - 717045289804293 \nu^{5} + 383847702402056 \nu^{4} - 325915380884067 \nu^{3} + 73511199022292 \nu^{2} - 893491525826 \nu + 1723605145714\)\()/ 6807394776267 \)
\(\beta_{16}\)\(=\)\((\)\(-2009357177810 \nu^{17} + 5675262379841 \nu^{16} - 29025194492163 \nu^{15} + 22542369486988 \nu^{14} - 138437780054349 \nu^{13} + 74281026654601 \nu^{12} - 439743089812689 \nu^{11} + 94451845877099 \nu^{10} - 826652550605570 \nu^{9} + 131095657717169 \nu^{8} - 1075352377560484 \nu^{7} + 32966933367212 \nu^{6} - 778255161948012 \nu^{5} + 76344756009802 \nu^{4} - 309405458804505 \nu^{3} - 46549600863206 \nu^{2} - 5869106089249 \nu - 3751600650469\)\()/ 6807394776267 \)
\(\beta_{17}\)\(=\)\((\)\(-2169011193168 \nu^{17} + 7176985755886 \nu^{16} - 34139016228998 \nu^{15} + 39022236800535 \nu^{14} - 158773504414729 \nu^{13} + 150026353889968 \nu^{12} - 501373370047309 \nu^{11} + 321726275203346 \nu^{10} - 904040446392028 \nu^{9} + 552412980832141 \nu^{8} - 1163466071525892 \nu^{7} + 559762487588955 \nu^{6} - 790741872567937 \nu^{5} + 454020967394587 \nu^{4} - 354767905666813 \nu^{3} + 83122854265979 \nu^{2} - 620019744989 \nu + 2187209095662\)\()/ 6807394776267 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{12} - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(\beta_{14} - 5 \beta_{12} - 3 \beta_{10} + 3 \beta_{9} + 3 \beta_{7} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3}\)
\(\nu^{4}\)\(=\)\(-3 \beta_{17} + 3 \beta_{14} - 2 \beta_{11} - 3 \beta_{10} + 9 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 3 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} + 2 \beta_{2} - 11 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-9 \beta_{17} + \beta_{15} + 3 \beta_{14} - \beta_{13} + 39 \beta_{12} - 12 \beta_{11} + 25 \beta_{10} - 13 \beta_{9} - 11 \beta_{8} - 23 \beta_{7} - 8 \beta_{5} + 38 \beta_{4} - 38 \beta_{3} + \beta_{2} - 39 \beta_{1} + 1\)
\(\nu^{6}\)\(=\)\(13 \beta_{17} + 2 \beta_{16} - \beta_{15} - 25 \beta_{14} - 3 \beta_{13} + 114 \beta_{12} - 13 \beta_{11} + 128 \beta_{10} - 128 \beta_{9} - 115 \beta_{7} + 3 \beta_{6} - 64 \beta_{5} + 87 \beta_{4} - 41 \beta_{3} + 7\)
\(\nu^{7}\)\(=\)\(128 \beta_{17} + 13 \beta_{16} - 13 \beta_{15} - 128 \beta_{14} - 3 \beta_{13} + 87 \beta_{11} + 142 \beta_{10} - 269 \beta_{9} + 114 \beta_{8} - 128 \beta_{7} + 10 \beta_{6} - 128 \beta_{5} - 142 \beta_{4} + 269 \beta_{3} - 12 \beta_{2} + 375 \beta_{1}\)
\(\nu^{8}\)\(=\)\(269 \beta_{17} + 14 \beta_{16} - 27 \beta_{15} - 142 \beta_{14} + 27 \beta_{13} - 1181 \beta_{12} + 411 \beta_{11} - 883 \beta_{10} + 455 \beta_{9} + 375 \beta_{8} + 786 \beta_{7} - 14 \beta_{6} + 233 \beta_{5} - 1338 \beta_{4} + 1338 \beta_{3} - 46 \beta_{2} + 1181 \beta_{1} - 46\)
\(\nu^{9}\)\(=\)\(-455 \beta_{17} - 97 \beta_{16} + 44 \beta_{15} + 883 \beta_{14} + 141 \beta_{13} - 3823 \beta_{12} + 455 \beta_{11} - 4309 \beta_{10} + 4309 \beta_{9} + 3857 \beta_{7} - 141 \beta_{6} + 2064 \beta_{5} - 2815 \beta_{4} + 1494 \beta_{3} - 126\)
\(\nu^{10}\)\(=\)\(-4309 \beta_{17} - 452 \beta_{16} + 452 \beta_{15} + 4309 \beta_{14} + 156 \beta_{13} - 2815 \beta_{11} - 4808 \beta_{10} + 9112 \beta_{9} - 3823 \beta_{8} + 4309 \beta_{7} - 296 \beta_{6} + 4309 \beta_{5} + 4808 \beta_{4} - 9112 \beta_{3} + 421 \beta_{2} - 12254 \beta_{1}\)
\(\nu^{11}\)\(=\)\(-9112 \beta_{17} - 499 \beta_{16} + 980 \beta_{15} + 4808 \beta_{14} - 980 \beta_{13} + 39530 \beta_{12} - 13920 \beta_{11} + 29285 \beta_{10} - 15570 \beta_{9} - 12254 \beta_{8} - 26174 \beta_{7} + 499 \beta_{6} - 7446 \beta_{5} + 44855 \beta_{4} - 44855 \beta_{3} + 1307 \beta_{2} - 39530 \beta_{1} + 1307\)
\(\nu^{12}\)\(=\)\(15570 \beta_{17} + 3111 \beta_{16} - 1650 \beta_{15} - 29285 \beta_{14} - 4761 \beta_{13} + 127231 \beta_{12} - 15570 \beta_{11} + 144639 \beta_{10} - 144639 \beta_{9} - 129240 \beta_{7} + 4761 \beta_{6} - 68815 \beta_{5} + 94477 \beta_{4} - 50162 \beta_{3} + 4244\)
\(\nu^{13}\)\(=\)\(144639 \beta_{17} + 15399 \beta_{16} - 15399 \beta_{15} - 144639 \beta_{14} - 5307 \beta_{13} + 94477 \beta_{11} + 161859 \beta_{10} - 304266 \beta_{9} + 127231 \beta_{8} - 144639 \beta_{7} + 10092 \beta_{6} - 144639 \beta_{5} - 161859 \beta_{4} + 304266 \beta_{3} - 13561 \beta_{2} + 410121 \beta_{1}\)
\(\nu^{14}\)\(=\)\(304266 \beta_{17} + 17220 \beta_{16} - 32396 \beta_{15} - 161859 \beta_{14} + 32396 \beta_{13} - 1321304 \beta_{12} + 466125 \beta_{11} - 980815 \beta_{10} + 521594 \beta_{9} + 410121 \beta_{8} + 876246 \beta_{7} - 17220 \beta_{6} + 248262 \beta_{5} - 1502409 \beta_{4} + 1502409 \beta_{3} - 43774 \beta_{2} + 1321304 \beta_{1} - 43774\)
\(\nu^{15}\)\(=\)\(-521594 \beta_{17} - 104569 \beta_{16} + 55469 \beta_{15} + 980815 \beta_{14} + 160038 \beta_{13} - 4258423 \beta_{12} + 521594 \beta_{11} - 4841863 \beta_{10} + 4841863 \beta_{9} + 4326122 \beta_{7} - 160038 \beta_{6} + 2302119 \beta_{5} - 3160375 \beta_{4} + 1681488 \beta_{3} - 140792\)
\(\nu^{16}\)\(=\)\(-4841863 \beta_{17} - 515741 \beta_{16} + 515741 \beta_{15} + 4841863 \beta_{14} + 179079 \beta_{13} - 3160375 \beta_{11} - 5418926 \beta_{10} + 10185670 \beta_{9} - 4258423 \beta_{8} + 4841863 \beta_{7} - 336662 \beta_{6} + 4841863 \beta_{5} + 5418926 \beta_{4} - 10185670 \beta_{3} + 453895 \beta_{2} - 13722757 \beta_{1}\)
\(\nu^{17}\)\(=\)\(-10185670 \beta_{17} - 577063 \beta_{16} + 1085384 \beta_{15} + 5418926 \beta_{14} - 1085384 \beta_{13} + 44225341 \beta_{12} - 15604596 \beta_{11} + 32824390 \beta_{10} - 17465163 \beta_{9} - 13722757 \beta_{8} - 29327353 \beta_{7} + 577063 \beta_{6} - 8303831 \beta_{5} + 50289553 \beta_{4} - 50289553 \beta_{3} + 1462096 \beta_{2} - 44225341 \beta_{1} + 1462096\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(\beta_{13} - \beta_{16}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−0.644984 + 1.11715i
0.154946 0.268374i
0.816390 1.41403i
0.653994 + 1.13275i
−0.128481 0.222535i
−0.791558 1.37102i
−0.566185 0.980662i
0.394508 + 0.683308i
1.61137 + 2.79097i
0.653994 1.13275i
−0.128481 + 0.222535i
−0.791558 + 1.37102i
−0.644984 1.11715i
0.154946 + 0.268374i
0.816390 + 1.41403i
−0.566185 + 0.980662i
0.394508 0.683308i
1.61137 2.79097i
−1.75422 + 1.47196i 3.10785 + 1.13116i 0.563307 3.19467i 0 −7.11687 + 2.59033i −1.46955 + 2.54534i 1.42431 + 2.46697i 6.08105 + 5.10261i 0
101.2 −0.528654 + 0.443593i −0.652945 0.237653i −0.264596 + 1.50060i 0 0.450603 0.164006i −1.16732 + 2.02186i −1.21589 2.10598i −1.92827 1.61801i 0
101.3 0.484738 0.406743i 1.80387 + 0.656554i −0.277766 + 1.57529i 0 1.14145 0.415455i 2.04448 3.54114i 1.13887 + 1.97259i 0.524744 + 0.440313i 0
176.1 −0.289414 + 0.105338i 0.285463 1.61894i −1.45942 + 1.22460i 0 0.0879194 + 0.498616i 0.0445979 + 0.0772459i 0.601369 1.04160i 0.279590 + 0.101762i 0
176.2 1.18116 0.429906i −0.523072 + 2.96649i −0.321776 + 0.270002i 0 0.657481 + 3.72876i 1.86196 + 3.22501i −1.52095 + 2.63437i −5.70738 2.07732i 0
176.3 2.42733 0.883478i 0.0430161 0.243956i 3.57933 3.00342i 0 −0.111115 0.630167i −0.200820 0.347830i 3.45167 5.97847i 2.76141 + 1.00507i 0
226.1 −0.370282 + 2.09998i −1.70859 1.43367i −2.39340 0.871127i 0 3.64334 3.05712i 0.742812 + 1.28659i 0.583208 1.01015i 0.342900 + 1.94468i 0
226.2 −0.0366369 + 0.207778i −0.0612035 0.0513559i 1.83756 + 0.668816i 0 0.0129129 0.0108352i −0.843614 1.46118i −0.417271 + 0.722735i −0.519836 2.94814i 0
226.3 0.385975 2.18897i −0.794389 0.666572i −2.76323 1.00573i 0 −1.76572 + 1.48162i −1.01254 1.75377i −1.04532 + 1.81055i −0.334208 1.89539i 0
251.1 −0.289414 0.105338i 0.285463 + 1.61894i −1.45942 1.22460i 0 0.0879194 0.498616i 0.0445979 0.0772459i 0.601369 + 1.04160i 0.279590 0.101762i 0
251.2 1.18116 + 0.429906i −0.523072 2.96649i −0.321776 0.270002i 0 0.657481 3.72876i 1.86196 3.22501i −1.52095 2.63437i −5.70738 + 2.07732i 0
251.3 2.42733 + 0.883478i 0.0430161 + 0.243956i 3.57933 + 3.00342i 0 −0.111115 + 0.630167i −0.200820 + 0.347830i 3.45167 + 5.97847i 2.76141 1.00507i 0
301.1 −1.75422 1.47196i 3.10785 1.13116i 0.563307 + 3.19467i 0 −7.11687 2.59033i −1.46955 2.54534i 1.42431 2.46697i 6.08105 5.10261i 0
301.2 −0.528654 0.443593i −0.652945 + 0.237653i −0.264596 1.50060i 0 0.450603 + 0.164006i −1.16732 2.02186i −1.21589 + 2.10598i −1.92827 + 1.61801i 0
301.3 0.484738 + 0.406743i 1.80387 0.656554i −0.277766 1.57529i 0 1.14145 + 0.415455i 2.04448 + 3.54114i 1.13887 1.97259i 0.524744 0.440313i 0
351.1 −0.370282 2.09998i −1.70859 + 1.43367i −2.39340 + 0.871127i 0 3.64334 + 3.05712i 0.742812 1.28659i 0.583208 + 1.01015i 0.342900 1.94468i 0
351.2 −0.0366369 0.207778i −0.0612035 + 0.0513559i 1.83756 0.668816i 0 0.0129129 + 0.0108352i −0.843614 + 1.46118i −0.417271 0.722735i −0.519836 + 2.94814i 0
351.3 0.385975 + 2.18897i −0.794389 + 0.666572i −2.76323 + 1.00573i 0 −1.76572 1.48162i −1.01254 + 1.75377i −1.04532 1.81055i −0.334208 + 1.89539i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 351.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.l.c 18
5.b even 2 1 95.2.k.a 18
5.c odd 4 2 475.2.u.b 36
15.d odd 2 1 855.2.bs.c 18
19.e even 9 1 inner 475.2.l.c 18
19.e even 9 1 9025.2.a.cc 9
19.f odd 18 1 9025.2.a.cf 9
95.o odd 18 1 1805.2.a.s 9
95.p even 18 1 95.2.k.a 18
95.p even 18 1 1805.2.a.v 9
95.q odd 36 2 475.2.u.b 36
285.bd odd 18 1 855.2.bs.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.k.a 18 5.b even 2 1
95.2.k.a 18 95.p even 18 1
475.2.l.c 18 1.a even 1 1 trivial
475.2.l.c 18 19.e even 9 1 inner
475.2.u.b 36 5.c odd 4 2
475.2.u.b 36 95.q odd 36 2
855.2.bs.c 18 15.d odd 2 1
855.2.bs.c 18 285.bd odd 18 1
1805.2.a.s 9 95.o odd 18 1
1805.2.a.v 9 95.p even 18 1
9025.2.a.cc 9 19.e even 9 1
9025.2.a.cf 9 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 30 T^{2} + 84 T^{3} - 270 T^{5} + 397 T^{6} + 591 T^{7} + 18 T^{8} - 1130 T^{9} + 804 T^{10} - 354 T^{11} + 268 T^{12} - 66 T^{13} + 33 T^{14} - 19 T^{15} + 6 T^{16} - 3 T^{17} + T^{18} \)
$3$ \( 1 + 21 T + 207 T^{2} + 629 T^{3} + 3396 T^{4} + 8718 T^{5} + 9207 T^{6} + 2655 T^{7} - 1098 T^{8} - 831 T^{9} - 141 T^{10} - 576 T^{11} + 371 T^{12} + 105 T^{13} + 42 T^{14} - 26 T^{15} + 3 T^{16} - 3 T^{17} + T^{18} \)
$5$ \( T^{18} \)
$7$ \( 361 - 2907 T + 35778 T^{2} + 110129 T^{3} + 374922 T^{4} + 289026 T^{5} + 331048 T^{6} + 185751 T^{7} + 168603 T^{8} + 77089 T^{9} + 46251 T^{10} + 12705 T^{11} + 5773 T^{12} + 990 T^{13} + 516 T^{14} + 48 T^{15} + 27 T^{16} + T^{18} \)
$11$ \( 597529 + 5683869 T + 69603909 T^{2} - 160133926 T^{3} + 343751106 T^{4} - 191315736 T^{5} + 113515623 T^{6} - 33042279 T^{7} + 13707387 T^{8} - 2855496 T^{9} + 1117941 T^{10} - 161004 T^{11} + 59822 T^{12} - 5157 T^{13} + 2382 T^{14} - 104 T^{15} + 60 T^{16} + T^{18} \)
$13$ \( 2809 - 6360 T + 55119 T^{2} - 188739 T^{3} + 256356 T^{4} + 666120 T^{5} - 925688 T^{6} - 645645 T^{7} + 1557750 T^{8} - 906001 T^{9} + 293502 T^{10} - 116694 T^{11} + 33331 T^{12} - 1134 T^{13} + 1302 T^{14} - 65 T^{15} + 18 T^{16} - 3 T^{17} + T^{18} \)
$17$ \( 2859481 - 13250676 T + 25551021 T^{2} + 1862136 T^{3} - 18635286 T^{4} + 8598084 T^{5} + 31397977 T^{6} + 31383276 T^{7} + 21413613 T^{8} + 11273852 T^{9} + 4711464 T^{10} + 1599729 T^{11} + 443368 T^{12} + 98823 T^{13} + 17721 T^{14} + 2593 T^{15} + 300 T^{16} + 24 T^{17} + T^{18} \)
$19$ \( 322687697779 + 40224228255 T^{2} + 7480295079 T^{3} + 1626797043 T^{4} + 930100977 T^{5} + 119614101 T^{6} + 47513376 T^{7} + 12845007 T^{8} + 2072495 T^{9} + 676053 T^{10} + 131616 T^{11} + 17439 T^{12} + 7137 T^{13} + 657 T^{14} + 159 T^{15} + 45 T^{16} + T^{18} \)
$23$ \( 2085474889 + 14438261388 T + 33399057753 T^{2} + 24003654614 T^{3} + 10804483290 T^{4} + 3171255645 T^{5} + 1190663276 T^{6} + 198578685 T^{7} + 40845360 T^{8} + 4173576 T^{9} + 1719306 T^{10} - 243417 T^{11} + 67065 T^{12} + 4728 T^{13} - 1284 T^{14} + 229 T^{15} + 6 T^{16} - 9 T^{17} + T^{18} \)
$29$ \( 44269422409 - 32549554503 T + 65664230568 T^{2} - 61752839685 T^{3} + 35472073803 T^{4} - 13426732056 T^{5} + 3330120535 T^{6} - 421997046 T^{7} + 4300119 T^{8} - 5952353 T^{9} + 5999655 T^{10} - 1324812 T^{11} + 54046 T^{12} + 22482 T^{13} - 3630 T^{14} - 16 T^{15} + 87 T^{16} - 15 T^{17} + T^{18} \)
$31$ \( 47085094081 + 165291809295 T + 563671907850 T^{2} + 110627499801 T^{3} + 102358812162 T^{4} + 23603956107 T^{5} + 12923623972 T^{6} + 2658850815 T^{7} + 913462053 T^{8} + 167966162 T^{9} + 44629974 T^{10} + 7079073 T^{11} + 1372033 T^{12} + 174075 T^{13} + 26700 T^{14} + 2746 T^{15} + 303 T^{16} + 18 T^{17} + T^{18} \)
$37$ \( ( 11125 + 88800 T + 171945 T^{2} + 122083 T^{3} + 23907 T^{4} - 5517 T^{5} - 1624 T^{6} - 15 T^{7} + 18 T^{8} + T^{9} )^{2} \)
$41$ \( 130321 + 9547728 T + 200209878 T^{2} + 122680017 T^{3} + 1154741655 T^{4} + 1331837127 T^{5} + 876831130 T^{6} + 439414968 T^{7} + 173250606 T^{8} + 53207192 T^{9} + 13293417 T^{10} + 2579616 T^{11} + 389782 T^{12} + 59034 T^{13} + 13047 T^{14} + 2773 T^{15} + 387 T^{16} + 30 T^{17} + T^{18} \)
$43$ \( 2531341458361 - 9362766561459 T + 12737105286036 T^{2} - 7423393018752 T^{3} + 2495058874806 T^{4} - 496039623858 T^{5} + 80608693804 T^{6} - 4633113510 T^{7} - 1295369637 T^{8} + 327646928 T^{9} - 5700123 T^{10} - 13231119 T^{11} + 4161550 T^{12} - 743499 T^{13} + 93546 T^{14} - 9038 T^{15} + 696 T^{16} - 36 T^{17} + T^{18} \)
$47$ \( 2033313439249 - 2706612353103 T + 2004475286151 T^{2} - 1054599687835 T^{3} + 552226883214 T^{4} - 122874634710 T^{5} - 23719831247 T^{6} + 5354614359 T^{7} + 2735894904 T^{8} + 326085625 T^{9} + 5524881 T^{10} - 6315672 T^{11} - 1006103 T^{12} - 3993 T^{13} + 16812 T^{14} + 2346 T^{15} + 267 T^{16} + 21 T^{17} + T^{18} \)
$53$ \( 44884235881 + 213479509491 T + 401990061072 T^{2} - 544835831678 T^{3} + 451379878755 T^{4} - 261124949298 T^{5} + 72923834963 T^{6} - 5620278834 T^{7} + 1663512543 T^{8} + 544996851 T^{9} + 15812070 T^{10} + 1980957 T^{11} + 343152 T^{12} - 129351 T^{13} + 29175 T^{14} + 404 T^{15} - 105 T^{16} - 12 T^{17} + T^{18} \)
$59$ \( 538419880441 - 2401914251064 T + 4958645853570 T^{2} - 5446228954768 T^{3} + 3567887119455 T^{4} - 1545148164114 T^{5} + 490432688273 T^{6} - 114929216988 T^{7} + 19512992850 T^{8} - 2319706938 T^{9} + 175378443 T^{10} - 3718779 T^{11} - 946605 T^{12} + 121671 T^{13} - 1833 T^{14} - 929 T^{15} + 189 T^{16} - 18 T^{17} + T^{18} \)
$61$ \( 3789756721 + 927847392 T + 1261205235 T^{2} + 3010750713 T^{3} + 6515614254 T^{4} + 5101866828 T^{5} + 8037883063 T^{6} + 6706855317 T^{7} + 2583172947 T^{8} + 509284064 T^{9} + 67079811 T^{10} + 1901295 T^{11} - 733733 T^{12} - 140640 T^{13} - 3765 T^{14} + 1558 T^{15} + 342 T^{16} + 30 T^{17} + T^{18} \)
$67$ \( 42166698100921 + 91012273270368 T + 49225079381382 T^{2} - 3612965661481 T^{3} + 8799382341021 T^{4} - 4024790037285 T^{5} + 1421927012604 T^{6} - 68032964796 T^{7} + 17157655479 T^{8} - 2736653202 T^{9} + 18322515 T^{10} - 16854306 T^{11} + 3467702 T^{12} + 10326 T^{13} + 14172 T^{14} - 2309 T^{15} - 63 T^{16} + T^{18} \)
$71$ \( 2834480882664649 - 1478872028245749 T + 386608415825586 T^{2} + 73550953337053 T^{3} + 13677463355220 T^{4} + 9479792874693 T^{5} + 1723910224454 T^{6} + 60217118727 T^{7} + 8934581688 T^{8} + 5193668055 T^{9} + 999819414 T^{10} + 96691116 T^{11} + 5070534 T^{12} - 160575 T^{13} - 40080 T^{14} - 1558 T^{15} + 69 T^{16} + 12 T^{17} + T^{18} \)
$73$ \( 585504633023209 + 856924997835051 T + 578750680499619 T^{2} + 154609539956732 T^{3} + 17010917228757 T^{4} - 2989089033948 T^{5} - 500929987135 T^{6} - 192930124575 T^{7} + 30879347706 T^{8} + 2251995606 T^{9} + 59476194 T^{10} - 10897053 T^{11} - 1546737 T^{12} - 358785 T^{13} - 846 T^{14} + 3646 T^{15} + 381 T^{16} + 24 T^{17} + T^{18} \)
$79$ \( 14691889 - 168115380 T + 780334149 T^{2} + 1551306657 T^{3} + 2022479112 T^{4} + 1771201848 T^{5} + 853411065 T^{6} + 253204869 T^{7} + 145353153 T^{8} + 100070890 T^{9} + 48173469 T^{10} + 16841202 T^{11} + 5475309 T^{12} + 1272408 T^{13} + 184965 T^{14} + 17445 T^{15} + 1185 T^{16} + 51 T^{17} + T^{18} \)
$83$ \( 36030515128141249 + 11487212177062017 T + 6189090104803176 T^{2} + 705729322317319 T^{3} + 366261422525022 T^{4} + 34913765518053 T^{5} + 14271733039817 T^{6} + 848417232372 T^{7} + 312225018681 T^{8} + 18799307889 T^{9} + 4465349472 T^{10} + 173135181 T^{11} + 33374259 T^{12} + 854691 T^{13} + 178113 T^{14} + 2294 T^{15} + 507 T^{16} + T^{18} \)
$89$ \( 916651141561 - 3243174524466 T + 4995676066575 T^{2} - 5405628999657 T^{3} + 3027366261861 T^{4} + 3787422486660 T^{5} + 1527322760305 T^{6} + 236291037885 T^{7} + 10493386530 T^{8} - 3228152354 T^{9} - 678526578 T^{10} - 31704114 T^{11} + 10139938 T^{12} + 2531178 T^{13} + 315477 T^{14} + 25940 T^{15} + 1473 T^{16} + 54 T^{17} + T^{18} \)
$97$ \( 3665788890625 - 670166615625 T + 67553902498125 T^{2} + 93594162257625 T^{3} + 63954401652225 T^{4} + 27827377963620 T^{5} + 8603893981111 T^{6} + 1985120791083 T^{7} + 347831665503 T^{8} + 45858679496 T^{9} + 4443909177 T^{10} + 305514906 T^{11} + 12687004 T^{12} - 175404 T^{13} - 79563 T^{14} - 4658 T^{15} + 129 T^{16} + 27 T^{17} + T^{18} \)
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