Properties

Label 400.2.s.d
Level $400$
Weight $2$
Character orbit 400.s
Analytic conductor $3.194$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial: \(x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18} + 2 x^{16} - 4 x^{15} - 5 x^{14} - 14 x^{13} - 10 x^{12} + 6 x^{11} + 37 x^{10} + 70 x^{9} + 74 x^{8} + 24 x^{7} - 80 x^{6} - 224 x^{5} - 160 x^{4} - 256 x^{3} + 256 x^{2} + 512\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 71 \nu^{17} + 98 \nu^{16} + 286 \nu^{15} + 144 \nu^{14} - 123 \nu^{13} - 1148 \nu^{12} - 2354 \nu^{11} - 3026 \nu^{10} - 2001 \nu^{9} + 1732 \nu^{8} + 7730 \nu^{7} + 13124 \nu^{6} + 13472 \nu^{5} + 4592 \nu^{4} - 3584 \nu^{3} - 23488 \nu^{2} - 16768 \nu - 24064 \)\()/1280\)
\(\beta_{2}\)\(=\)\((\)\( -129 \nu^{17} - 124 \nu^{16} - 398 \nu^{15} + 116 \nu^{14} + 797 \nu^{13} + 2778 \nu^{12} + 4526 \nu^{11} + 4402 \nu^{10} + 123 \nu^{9} - 9450 \nu^{8} - 21502 \nu^{7} - 29424 \nu^{6} - 25048 \nu^{5} + 368 \nu^{4} + 22176 \nu^{3} + 64960 \nu^{2} + 44800 \nu + 58624 \)\()/1280\)
\(\beta_{3}\)\(=\)\((\)\( -16 \nu^{17} - 19 \nu^{16} - 53 \nu^{15} + 2 \nu^{14} + 88 \nu^{13} + 331 \nu^{12} + 559 \nu^{11} + 580 \nu^{10} + 98 \nu^{9} - 1033 \nu^{8} - 2471 \nu^{7} - 3408 \nu^{6} - 2972 \nu^{5} - 44 \nu^{4} + 2444 \nu^{3} + 7152 \nu^{2} + 4832 \nu + 6048 \)\()/160\)
\(\beta_{4}\)\(=\)\((\)\( -75 \nu^{17} - 89 \nu^{16} - 248 \nu^{15} - 6 \nu^{14} + 375 \nu^{13} + 1487 \nu^{12} + 2550 \nu^{11} + 2676 \nu^{10} + 583 \nu^{9} - 4379 \nu^{8} - 10894 \nu^{7} - 15406 \nu^{6} - 13500 \nu^{5} - 848 \nu^{4} + 9920 \nu^{3} + 31296 \nu^{2} + 21696 \nu + 28416 \)\()/640\)
\(\beta_{5}\)\(=\)\((\)\(-229 \nu^{17} - 258 \nu^{16} - 706 \nu^{15} + 120 \nu^{14} + 1377 \nu^{13} + 4800 \nu^{12} + 7846 \nu^{11} + 7678 \nu^{10} + 331 \nu^{9} - 15784 \nu^{8} - 35846 \nu^{7} - 48500 \nu^{6} - 40528 \nu^{5} + 1200 \nu^{4} + 37376 \nu^{3} + 102976 \nu^{2} + 71296 \nu + 89600\)\()/1280\)
\(\beta_{6}\)\(=\)\((\)\(-178 \nu^{17} - 217 \nu^{16} - 614 \nu^{15} + 6 \nu^{14} + 954 \nu^{13} + 3793 \nu^{12} + 6492 \nu^{11} + 6810 \nu^{10} + 1284 \nu^{9} - 11749 \nu^{8} - 28628 \nu^{7} - 40294 \nu^{6} - 35116 \nu^{5} - 1872 \nu^{4} + 28832 \nu^{3} + 83776 \nu^{2} + 58816 \nu + 75264\)\()/640\)
\(\beta_{7}\)\(=\)\((\)\(545 \nu^{17} + 684 \nu^{16} + 1918 \nu^{15} + 156 \nu^{14} - 2685 \nu^{13} - 11242 \nu^{12} - 19710 \nu^{11} - 21346 \nu^{10} - 5723 \nu^{9} + 32794 \nu^{8} + 84014 \nu^{7} + 120736 \nu^{6} + 108280 \nu^{5} + 10928 \nu^{4} - 79200 \nu^{3} - 245696 \nu^{2} - 175616 \nu - 222976\)\()/1280\)
\(\beta_{8}\)\(=\)\((\)\(298 \nu^{17} + 337 \nu^{16} + 974 \nu^{15} - 166 \nu^{14} - 1874 \nu^{13} - 6713 \nu^{12} - 11092 \nu^{11} - 10970 \nu^{10} - 604 \nu^{9} + 22749 \nu^{8} + 52268 \nu^{7} + 71254 \nu^{6} + 59756 \nu^{5} - 1008 \nu^{4} - 57152 \nu^{3} - 154176 \nu^{2} - 110016 \nu - 133504\)\()/640\)
\(\beta_{9}\)\(=\)\((\)\(159 \nu^{17} + 235 \nu^{16} + 665 \nu^{15} + 358 \nu^{14} - 257 \nu^{13} - 2621 \nu^{12} - 5511 \nu^{11} - 7306 \nu^{10} - 5065 \nu^{9} + 3611 \nu^{8} + 17383 \nu^{7} + 30068 \nu^{6} + 32638 \nu^{5} + 13104 \nu^{4} - 6096 \nu^{3} - 50784 \nu^{2} - 36384 \nu - 54848\)\()/320\)
\(\beta_{10}\)\(=\)\((\)\(871 \nu^{17} + 1090 \nu^{16} + 3110 \nu^{15} + 352 \nu^{14} - 4043 \nu^{13} - 17564 \nu^{12} - 31194 \nu^{11} - 34194 \nu^{10} - 10465 \nu^{9} + 49524 \nu^{8} + 130042 \nu^{7} + 188932 \nu^{6} + 171792 \nu^{5} + 21456 \nu^{4} - 117504 \nu^{3} - 381376 \nu^{2} - 268416 \nu - 351232\)\()/1280\)
\(\beta_{11}\)\(=\)\((\)\(503 \nu^{17} + 761 \nu^{16} + 2162 \nu^{15} + 1230 \nu^{14} - 679 \nu^{13} - 8175 \nu^{12} - 17512 \nu^{11} - 23616 \nu^{10} - 17047 \nu^{9} + 9903 \nu^{8} + 53232 \nu^{7} + 94050 \nu^{6} + 103336 \nu^{5} + 44320 \nu^{4} - 14672 \nu^{3} - 154432 \nu^{2} - 108032 \nu - 168960\)\()/640\)
\(\beta_{12}\)\(=\)\((\)\(-129 \nu^{17} - 186 \nu^{16} - 524 \nu^{15} - 232 \nu^{14} + 321 \nu^{13} + 2280 \nu^{12} + 4568 \nu^{11} + 5762 \nu^{10} + 3435 \nu^{9} - 4196 \nu^{8} - 15672 \nu^{7} - 25600 \nu^{6} - 26316 \nu^{5} - 8624 \nu^{4} + 8592 \nu^{3} + 45504 \nu^{2} + 32320 \nu + 46208\)\()/128\)
\(\beta_{13}\)\(=\)\((\)\(93 \nu^{17} + 133 \nu^{16} + 374 \nu^{15} + 162 \nu^{14} - 237 \nu^{13} - 1639 \nu^{12} - 3268 \nu^{11} - 4104 \nu^{10} - 2417 \nu^{9} + 3063 \nu^{8} + 11252 \nu^{7} + 18318 \nu^{6} + 18760 \nu^{5} + 6024 \nu^{4} - 6240 \nu^{3} - 32672 \nu^{2} - 23040 \nu - 33024\)\()/64\)
\(\beta_{14}\)\(=\)\((\)\(-93 \nu^{17} - 133 \nu^{16} - 374 \nu^{15} - 162 \nu^{14} + 237 \nu^{13} + 1639 \nu^{12} + 3268 \nu^{11} + 4104 \nu^{10} + 2417 \nu^{9} - 3063 \nu^{8} - 11252 \nu^{7} - 18318 \nu^{6} - 18760 \nu^{5} - 6024 \nu^{4} + 6240 \nu^{3} + 32672 \nu^{2} + 23168 \nu + 33024\)\()/64\)
\(\beta_{15}\)\(=\)\((\)\(-981 \nu^{17} - 1344 \nu^{16} - 3828 \nu^{15} - 1348 \nu^{14} + 3013 \nu^{13} + 17886 \nu^{12} + 34564 \nu^{11} + 41990 \nu^{10} + 21883 \nu^{9} - 38378 \nu^{8} - 125876 \nu^{7} - 198348 \nu^{6} - 196852 \nu^{5} - 53344 \nu^{4} + 82784 \nu^{3} + 368512 \nu^{2} + 259392 \nu + 363008\)\()/640\)
\(\beta_{16}\)\(=\)\((\)\(-1039 \nu^{17} - 1484 \nu^{16} - 4198 \nu^{15} - 1844 \nu^{14} + 2547 \nu^{13} + 18238 \nu^{12} + 36566 \nu^{11} + 46182 \nu^{10} + 27653 \nu^{9} - 33350 \nu^{8} - 125222 \nu^{7} - 205144 \nu^{6} - 211368 \nu^{5} - 69792 \nu^{4} + 67376 \nu^{3} + 364800 \nu^{2} + 258560 \nu + 372864\)\()/640\)
\(\beta_{17}\)\(=\)\((\)\(3253 \nu^{17} + 4754 \nu^{16} + 13458 \nu^{15} + 6632 \nu^{14} - 6769 \nu^{13} - 55664 \nu^{12} - 114262 \nu^{11} - 147678 \nu^{10} - 95003 \nu^{9} + 90616 \nu^{8} + 375990 \nu^{7} + 630772 \nu^{6} + 663696 \nu^{5} + 240976 \nu^{4} - 172352 \nu^{3} - 1092544 \nu^{2} - 772224 \nu - 1132032\)\()/1280\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{14} + \beta_{13}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{17} - \beta_{16} + \beta_{15} - 2 \beta_{13} + \beta_{12} + 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{4} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{17} + \beta_{16} - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{12} - 2 \beta_{10} + \beta_{8} - \beta_{7} + 2 \beta_{5} - 3 \beta_{4} - \beta_{2} - \beta_{1} - 1\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{16} - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 3\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(\beta_{17} + \beta_{16} + \beta_{15} + 2 \beta_{14} - 3 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{4} - 3 \beta_{2} - \beta_{1} + 7\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(\beta_{17} - 2 \beta_{16} + 2 \beta_{15} + \beta_{14} + \beta_{13} + 4 \beta_{12} + \beta_{11} + 2 \beta_{10} - 3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_{1} + 6\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-2 \beta_{16} - 2 \beta_{15} + 5 \beta_{14} - 3 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 6 \beta_{6} - 8 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_{1} - 2\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(\beta_{17} + 3 \beta_{16} - \beta_{15} - 10 \beta_{14} + 15 \beta_{12} + 2 \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - 5 \beta_{7} - 4 \beta_{6} + 8 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} + 5 \beta_{2} - 15 \beta_{1} - 3\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(\beta_{17} + 7 \beta_{16} - 7 \beta_{15} + 15 \beta_{14} + 3 \beta_{13} - 25 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + 2 \beta_{9} - 9 \beta_{8} + 5 \beta_{7} - 10 \beta_{6} - 10 \beta_{5} + 3 \beta_{4} + 12 \beta_{3} - 13 \beta_{2} + 3 \beta_{1} + 31\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(-2 \beta_{17} + 3 \beta_{16} + 7 \beta_{15} - \beta_{14} + 9 \beta_{13} + \beta_{12} - \beta_{11} + 18 \beta_{10} + 7 \beta_{9} + \beta_{8} - 12 \beta_{7} + 5 \beta_{6} + 12 \beta_{5} - 13 \beta_{4} + 7 \beta_{3} + 2 \beta_{2} - 4 \beta_{1} + 23\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(13 \beta_{17} + 17 \beta_{16} + 3 \beta_{15} + 18 \beta_{14} + 15 \beta_{12} + 18 \beta_{11} + 8 \beta_{10} + 14 \beta_{9} - 15 \beta_{8} + 53 \beta_{7} + 2 \beta_{6} - 10 \beta_{5} + 7 \beta_{4} + 16 \beta_{3} - \beta_{2} - 11 \beta_{1} + 27\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-3 \beta_{17} - 42 \beta_{16} - 20 \beta_{15} + 9 \beta_{14} - 21 \beta_{13} + 18 \beta_{12} - 9 \beta_{11} + 18 \beta_{10} - 29 \beta_{9} - 6 \beta_{8} + 3 \beta_{7} + 47 \beta_{6} + 18 \beta_{5} - 24 \beta_{4} - 19 \beta_{3} + 55 \beta_{2} - 29 \beta_{1} - 22\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(-8 \beta_{17} + 2 \beta_{16} - 8 \beta_{15} - 19 \beta_{14} - \beta_{13} + 10 \beta_{12} - 14 \beta_{11} - 12 \beta_{10} + 40 \beta_{9} + 22 \beta_{8} - 72 \beta_{7} - 64 \beta_{6} + 30 \beta_{5} - 62 \beta_{4} + 4 \beta_{3} + 30 \beta_{2} - 24 \beta_{1} + 92\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(37 \beta_{17} + 101 \beta_{16} - 7 \beta_{15} + 62 \beta_{14} + 76 \beta_{13} - 35 \beta_{12} - 42 \beta_{11} - 20 \beta_{10} + 70 \beta_{9} - 13 \beta_{8} + 15 \beta_{7} - 56 \beta_{6} - 18 \beta_{5} - 15 \beta_{4} + 100 \beta_{3} - 61 \beta_{2} - 29 \beta_{1} + 13\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(45 \beta_{17} + 43 \beta_{16} + 13 \beta_{15} + 13 \beta_{14} - 117 \beta_{13} - 63 \beta_{12} + 32 \beta_{11} + 138 \beta_{10} - 4 \beta_{9} - 117 \beta_{8} + 117 \beta_{7} + 24 \beta_{6} - 14 \beta_{5} + 19 \beta_{4} + 56 \beta_{3} - 23 \beta_{2} - 35 \beta_{1} + 233\)\()/2\)
\(\nu^{16}\)\(=\)\((\)\(-164 \beta_{17} - 9 \beta_{16} - 103 \beta_{15} + 7 \beta_{14} + 187 \beta_{13} - 53 \beta_{12} + 59 \beta_{11} - 104 \beta_{10} - 33 \beta_{9} + 51 \beta_{8} + 16 \beta_{7} + 139 \beta_{6} + 130 \beta_{5} - 69 \beta_{4} - 35 \beta_{3} + 86 \beta_{2} - 26 \beta_{1} - 43\)\()/2\)
\(\nu^{17}\)\(=\)\((\)\(163 \beta_{17} - 173 \beta_{16} - 5 \beta_{15} + 158 \beta_{14} - 112 \beta_{13} + 439 \beta_{12} - 54 \beta_{11} + 194 \beta_{10} + 134 \beta_{9} - 37 \beta_{8} + 75 \beta_{7} + 46 \beta_{6} - 180 \beta_{5} - 9 \beta_{4} + 76 \beta_{3} + 419 \beta_{2} - 135 \beta_{1} + 265\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{12}\) \(-\beta_{12}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.08900 0.902261i
0.482716 1.32928i
−0.635486 + 1.26339i
1.41303 0.0578659i
0.235136 + 1.39453i
1.41323 + 0.0526497i
−1.37691 + 0.322680i
0.0376504 1.41371i
−0.480367 1.33013i
−1.08900 + 0.902261i
0.482716 + 1.32928i
−0.635486 1.26339i
1.41303 + 0.0578659i
0.235136 1.39453i
1.41323 0.0526497i
−1.37691 0.322680i
0.0376504 + 1.41371i
−0.480367 + 1.33013i
−1.41267 + 0.0660953i 0.496487 1.99126 0.186742i 0 −0.701372 + 0.0328155i −1.55426 1.55426i −2.80065 + 0.395417i −2.75350 0
107.2 −1.19301 0.759419i 1.39319 0.846564 + 1.81200i 0 −1.66209 1.05801i 2.13436 + 2.13436i 0.366101 2.80463i −1.05903 0
107.3 −0.828280 + 1.14628i −0.692712 −0.627905 1.89888i 0 0.573759 0.794040i 0.343872 + 0.343872i 2.69672 + 0.853049i −2.52015 0
107.4 −0.567819 1.29521i −1.96251 −1.35516 + 1.47090i 0 1.11435 + 2.54187i 1.60205 + 1.60205i 2.67461 + 0.920026i 0.851447 0
107.5 −0.430311 + 1.34716i 2.96561 −1.62967 1.15939i 0 −1.27613 + 3.99515i 0.115101 + 0.115101i 2.26315 1.69652i 5.79486 0
107.6 0.516777 + 1.31641i −1.28110 −1.46588 + 1.36058i 0 −0.662041 1.68645i 1.13975 + 1.13975i −2.54862 1.22659i −1.35879 0
107.7 1.23576 0.687667i −0.614566 1.05423 1.69959i 0 −0.759459 + 0.422617i −2.83610 2.83610i 0.134028 2.82525i −2.62231 0
107.8 1.29924 + 0.558542i 2.55161 1.37606 + 1.45136i 0 3.31516 + 1.42518i −2.40368 2.40368i 0.977191 + 2.65426i 3.51070 0
107.9 1.38031 + 0.307817i −2.85601 1.81050 + 0.849763i 0 −3.94217 0.879127i 0.458895 + 0.458895i 2.23747 + 1.73024i 5.15678 0
243.1 −1.41267 0.0660953i 0.496487 1.99126 + 0.186742i 0 −0.701372 0.0328155i −1.55426 + 1.55426i −2.80065 0.395417i −2.75350 0
243.2 −1.19301 + 0.759419i 1.39319 0.846564 1.81200i 0 −1.66209 + 1.05801i 2.13436 2.13436i 0.366101 + 2.80463i −1.05903 0
243.3 −0.828280 1.14628i −0.692712 −0.627905 + 1.89888i 0 0.573759 + 0.794040i 0.343872 0.343872i 2.69672 0.853049i −2.52015 0
243.4 −0.567819 + 1.29521i −1.96251 −1.35516 1.47090i 0 1.11435 2.54187i 1.60205 1.60205i 2.67461 0.920026i 0.851447 0
243.5 −0.430311 1.34716i 2.96561 −1.62967 + 1.15939i 0 −1.27613 3.99515i 0.115101 0.115101i 2.26315 + 1.69652i 5.79486 0
243.6 0.516777 1.31641i −1.28110 −1.46588 1.36058i 0 −0.662041 + 1.68645i 1.13975 1.13975i −2.54862 + 1.22659i −1.35879 0
243.7 1.23576 + 0.687667i −0.614566 1.05423 + 1.69959i 0 −0.759459 0.422617i −2.83610 + 2.83610i 0.134028 + 2.82525i −2.62231 0
243.8 1.29924 0.558542i 2.55161 1.37606 1.45136i 0 3.31516 1.42518i −2.40368 + 2.40368i 0.977191 2.65426i 3.51070 0
243.9 1.38031 0.307817i −2.85601 1.81050 0.849763i 0 −3.94217 + 0.879127i 0.458895 0.458895i 2.23747 1.73024i 5.15678 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 243.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.s.d 18
4.b odd 2 1 1600.2.s.d 18
5.b even 2 1 80.2.s.b yes 18
5.c odd 4 1 80.2.j.b 18
5.c odd 4 1 400.2.j.d 18
15.d odd 2 1 720.2.z.g 18
15.e even 4 1 720.2.bd.g 18
16.e even 4 1 1600.2.j.d 18
16.f odd 4 1 400.2.j.d 18
20.d odd 2 1 320.2.s.b 18
20.e even 4 1 320.2.j.b 18
20.e even 4 1 1600.2.j.d 18
40.e odd 2 1 640.2.s.c 18
40.f even 2 1 640.2.s.d 18
40.i odd 4 1 640.2.j.d 18
40.k even 4 1 640.2.j.c 18
80.i odd 4 1 640.2.s.c 18
80.i odd 4 1 1600.2.s.d 18
80.j even 4 1 80.2.s.b yes 18
80.k odd 4 1 80.2.j.b 18
80.k odd 4 1 640.2.j.d 18
80.q even 4 1 320.2.j.b 18
80.q even 4 1 640.2.j.c 18
80.s even 4 1 inner 400.2.s.d 18
80.s even 4 1 640.2.s.d 18
80.t odd 4 1 320.2.s.b 18
240.t even 4 1 720.2.bd.g 18
240.bd odd 4 1 720.2.z.g 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.b 18 5.c odd 4 1
80.2.j.b 18 80.k odd 4 1
80.2.s.b yes 18 5.b even 2 1
80.2.s.b yes 18 80.j even 4 1
320.2.j.b 18 20.e even 4 1
320.2.j.b 18 80.q even 4 1
320.2.s.b 18 20.d odd 2 1
320.2.s.b 18 80.t odd 4 1
400.2.j.d 18 5.c odd 4 1
400.2.j.d 18 16.f odd 4 1
400.2.s.d 18 1.a even 1 1 trivial
400.2.s.d 18 80.s even 4 1 inner
640.2.j.c 18 40.k even 4 1
640.2.j.c 18 80.q even 4 1
640.2.j.d 18 40.i odd 4 1
640.2.j.d 18 80.k odd 4 1
640.2.s.c 18 40.e odd 2 1
640.2.s.c 18 80.i odd 4 1
640.2.s.d 18 40.f even 2 1
640.2.s.d 18 80.s even 4 1
720.2.z.g 18 15.d odd 2 1
720.2.z.g 18 240.bd odd 4 1
720.2.bd.g 18 15.e even 4 1
720.2.bd.g 18 240.t even 4 1
1600.2.j.d 18 16.e even 4 1
1600.2.j.d 18 20.e even 4 1
1600.2.s.d 18 4.b odd 2 1
1600.2.s.d 18 80.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{9} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 512 - 256 T^{2} - 256 T^{3} + 64 T^{4} + 192 T^{5} + 32 T^{6} - 32 T^{7} - 24 T^{8} + 8 T^{9} - 12 T^{10} - 8 T^{11} + 4 T^{12} + 12 T^{13} + 2 T^{14} - 4 T^{15} - 2 T^{16} + T^{18} \)
$3$ \( ( 16 + 20 T - 72 T^{2} - 104 T^{3} + 40 T^{4} + 76 T^{5} - 4 T^{6} - 16 T^{7} + T^{9} )^{2} \)
$5$ \( T^{18} \)
$7$ \( 288 - 4128 T + 29584 T^{2} - 108160 T^{3} + 239360 T^{4} - 315488 T^{5} + 244448 T^{6} - 85184 T^{7} + 17328 T^{8} - 11648 T^{9} + 14648 T^{10} - 4160 T^{11} + 352 T^{12} + 120 T^{13} + 200 T^{14} - 32 T^{15} + 2 T^{16} + 2 T^{17} + T^{18} \)
$11$ \( 5431808 - 4060672 T + 1517824 T^{2} - 4217856 T^{3} + 18015744 T^{4} - 19945472 T^{5} + 11514112 T^{6} - 2158336 T^{7} + 289472 T^{8} - 209344 T^{9} + 182240 T^{10} - 27968 T^{11} + 1376 T^{12} - 64 T^{13} + 848 T^{14} - 80 T^{15} + 2 T^{16} + 2 T^{17} + T^{18} \)
$13$ \( 67108864 + 131948800 T^{2} + 105268224 T^{4} + 44025600 T^{6} + 10452224 T^{8} + 1437280 T^{10} + 113344 T^{12} + 4976 T^{14} + 112 T^{16} + T^{18} \)
$17$ \( 512 - 1536 T + 2304 T^{2} + 186368 T^{3} + 1493504 T^{4} + 5352960 T^{5} + 11139328 T^{6} + 12464640 T^{7} + 6499264 T^{8} - 984640 T^{9} + 14816 T^{10} + 36480 T^{11} + 66080 T^{12} - 19232 T^{13} + 2768 T^{14} - 32 T^{15} + 18 T^{16} - 6 T^{17} + T^{18} \)
$19$ \( 4608 - 339456 T + 12503296 T^{2} - 7467008 T^{3} + 2215424 T^{4} + 335360 T^{5} + 13924096 T^{6} - 9073664 T^{7} + 2905024 T^{8} + 1873856 T^{9} + 675296 T^{10} - 48384 T^{11} + 16480 T^{12} + 10080 T^{13} + 2864 T^{14} + 64 T^{15} + 2 T^{16} + 2 T^{17} + T^{18} \)
$23$ \( 17700587552 - 17587696352 T + 8737762576 T^{2} + 758796096 T^{3} + 938524160 T^{4} - 784929440 T^{5} + 332892064 T^{6} + 13833888 T^{7} + 11039216 T^{8} - 7905920 T^{9} + 2752216 T^{10} - 54000 T^{11} + 9120 T^{12} - 7896 T^{13} + 3480 T^{14} - 24 T^{15} + 2 T^{16} - 2 T^{17} + T^{18} \)
$29$ \( 82330112 - 372641280 T + 843321600 T^{2} - 687951872 T^{3} + 281020928 T^{4} - 16436736 T^{5} + 70118656 T^{6} - 45759488 T^{7} + 14859968 T^{8} - 1819968 T^{9} + 693088 T^{10} - 360192 T^{11} + 111008 T^{12} - 15584 T^{13} + 1616 T^{14} - 320 T^{15} + 98 T^{16} - 14 T^{17} + T^{18} \)
$31$ \( 16384 + 1150976 T^{2} + 16687104 T^{4} + 32610304 T^{6} + 17532416 T^{8} + 3648384 T^{10} + 327040 T^{12} + 12832 T^{14} + 196 T^{16} + T^{18} \)
$37$ \( 574297214976 + 457920768256 T^{2} + 131428787200 T^{4} + 18630090496 T^{6} + 1481216256 T^{8} + 69964768 T^{10} + 1988288 T^{12} + 32944 T^{14} + 288 T^{16} + T^{18} \)
$41$ \( 242788765696 + 208906121472 T^{2} + 68836776960 T^{4} + 11517059840 T^{6} + 1077600512 T^{8} + 58367328 T^{10} + 1831744 T^{12} + 32176 T^{14} + 288 T^{16} + T^{18} \)
$43$ \( 337207844416 + 627531510928 T^{2} + 248751738624 T^{4} + 38727202720 T^{6} + 3041915424 T^{8} + 132726008 T^{10} + 3310976 T^{12} + 46520 T^{14} + 340 T^{16} + T^{18} \)
$47$ \( 16870640672 + 178795286432 T + 947437476496 T^{2} + 866389389184 T^{3} + 419056561664 T^{4} + 124183363808 T^{5} + 28973153120 T^{6} + 7620275328 T^{7} + 2411169968 T^{8} + 643585920 T^{9} + 126078328 T^{10} + 18512576 T^{11} + 2503968 T^{12} + 394472 T^{13} + 64680 T^{14} + 8272 T^{15} + 722 T^{16} + 38 T^{17} + T^{18} \)
$53$ \( ( -220832 + 334608 T - 44032 T^{2} - 85984 T^{3} + 10768 T^{4} + 5720 T^{5} - 480 T^{6} - 136 T^{7} + 6 T^{8} + T^{9} )^{2} \)
$59$ \( 144166720393728 - 63266193922560 T + 13881883705600 T^{2} + 2799567403008 T^{3} + 1079531100672 T^{4} - 262683543040 T^{5} + 38509938944 T^{6} + 5387559424 T^{7} + 1811701952 T^{8} - 356104896 T^{9} + 37394528 T^{10} + 1999488 T^{11} + 826528 T^{12} - 162272 T^{13} + 16016 T^{14} + 96 T^{15} + 50 T^{16} - 10 T^{17} + T^{18} \)
$61$ \( 121236758528 - 124325191680 T + 63746150400 T^{2} + 35916963840 T^{3} + 28713865216 T^{4} - 15011409920 T^{5} + 5616384000 T^{6} + 2393885696 T^{7} + 974385920 T^{8} - 231668992 T^{9} + 33596800 T^{10} + 3516032 T^{11} + 740800 T^{12} - 135872 T^{13} + 14496 T^{14} + 720 T^{15} + 98 T^{16} - 14 T^{17} + T^{18} \)
$67$ \( 555525752896 + 1362082550416 T^{2} + 742701558272 T^{4} + 154793099680 T^{6} + 14366974496 T^{8} + 616717432 T^{10} + 12278976 T^{12} + 120376 T^{14} + 564 T^{16} + T^{18} \)
$71$ \( ( -27648 - 72640 T + 110336 T^{2} - 8704 T^{3} - 23136 T^{4} + 3344 T^{5} + 1408 T^{6} - 152 T^{7} - 12 T^{8} + T^{9} )^{2} \)
$73$ \( 35535647232 - 3228962304 T + 146700544 T^{2} - 5453305856 T^{3} + 14687183360 T^{4} - 5125904896 T^{5} + 823567616 T^{6} + 175353344 T^{7} + 212641728 T^{8} - 49164608 T^{9} + 5990112 T^{10} + 758144 T^{11} + 628896 T^{12} - 125856 T^{13} + 13072 T^{14} + 544 T^{15} + 98 T^{16} - 14 T^{17} + T^{18} \)
$79$ \( ( 45002752 + 3950848 T - 7267840 T^{2} - 485376 T^{3} + 296064 T^{4} + 27488 T^{5} - 2976 T^{6} - 320 T^{7} + 8 T^{8} + T^{9} )^{2} \)
$83$ \( ( 8413744 + 2612884 T - 2578680 T^{2} - 329896 T^{3} + 197504 T^{4} + 7292 T^{5} - 3612 T^{6} - 136 T^{7} + 20 T^{8} + T^{9} )^{2} \)
$89$ \( ( -251904 + 5727232 T + 4338688 T^{2} - 1356288 T^{3} - 330368 T^{4} + 55104 T^{5} + 2752 T^{6} - 448 T^{7} - 6 T^{8} + T^{9} )^{2} \)
$97$ \( 380349381734912 - 440719049317888 T + 255335343974656 T^{2} - 73928552845312 T^{3} + 10687493231104 T^{4} - 76758095360 T^{5} + 98986508544 T^{6} - 49913260032 T^{7} + 9402360768 T^{8} + 513287616 T^{9} + 21224416 T^{10} - 9460992 T^{11} + 2501024 T^{12} + 238368 T^{13} + 11856 T^{14} - 512 T^{15} + 162 T^{16} + 18 T^{17} + T^{18} \)
show more
show less