Properties

Label 48.2.k.a
Level 48
Weight 2
Character orbit 48.k
Analytic conductor 0.383
Analytic rank 0
Dimension 12
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 48.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.383281929702\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: 12.0.163368480538624.2
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{10} - 2 x^{8} + 16 x^{6} - 8 x^{4} - 32 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{8} - 2 \nu^{4} + 4 \nu^{2} + 8 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{9} + 2 \nu^{7} + 8 \nu^{3} \)\()/32\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{11} - 2 \nu^{9} - 2 \nu^{7} + 16 \nu^{5} - 8 \nu^{3} - 32 \nu \)\()/32\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{11} - \nu^{10} - 6 \nu^{7} + 10 \nu^{6} + 4 \nu^{5} - 12 \nu^{4} + 8 \nu^{3} - 32 \nu + 64 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{11} + \nu^{10} + 8 \nu^{7} - 2 \nu^{6} - 16 \nu^{5} - 4 \nu^{4} - 24 \nu^{3} + 16 \nu^{2} + 32 \nu \)\()/32\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{11} + 2 \nu^{9} + 10 \nu^{7} - 16 \nu^{5} - 8 \nu^{3} + 64 \nu \)\()/32\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{11} + \nu^{10} + 6 \nu^{7} - 10 \nu^{6} - 4 \nu^{5} + 12 \nu^{4} - 8 \nu^{3} + 32 \nu^{2} + 32 \nu - 64 \)\()/32\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{9} + 2 \nu^{5} - 4 \nu^{3} - 16 \nu \)\()/8\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{11} - \nu^{10} + 6 \nu^{7} + 10 \nu^{6} - 4 \nu^{5} - 12 \nu^{4} - 8 \nu^{3} - 32 \nu^{2} + 32 \nu + 64 \)\()/32\)
\(\beta_{10}\)\(=\)\((\)\( -2 \nu^{11} - \nu^{10} + 8 \nu^{7} + 2 \nu^{6} - 16 \nu^{5} + 4 \nu^{4} - 24 \nu^{3} - 16 \nu^{2} + 32 \nu \)\()/32\)
\(\beta_{11}\)\(=\)\((\)\( 2 \nu^{11} + 3 \nu^{10} - 8 \nu^{7} - 6 \nu^{6} + 16 \nu^{5} + 20 \nu^{4} + 24 \nu^{3} + 16 \nu^{2} - 32 \nu - 32 \)\()/32\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{4}\)
\(\nu^{3}\)\(=\)\(-\beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - 2 \beta_{3}\)
\(\nu^{4}\)\(=\)\(\beta_{11} + 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} + 1\)
\(\nu^{5}\)\(=\)\(\beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 2 \beta_{3} + \beta_{2}\)
\(\nu^{6}\)\(=\)\(2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 6\)
\(\nu^{7}\)\(=\)\(-2 \beta_{10} + 4 \beta_{8} + 6 \beta_{6} - 2 \beta_{5} + 2 \beta_{2}\)
\(\nu^{8}\)\(=\)\(2 \beta_{11} + 4 \beta_{10} - 2 \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{1} - 6\)
\(\nu^{9}\)\(=\)\(4 \beta_{10} - 10 \beta_{9} - 6 \beta_{8} - 10 \beta_{7} + 6 \beta_{6} + 4 \beta_{5} + 12 \beta_{3} + 10 \beta_{2}\)
\(\nu^{10}\)\(=\)\(8 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} - 8 \beta_{7} + 12 \beta_{5} - 4 \beta_{4} - 8\)
\(\nu^{11}\)\(=\)\(-4 \beta_{10} - 12 \beta_{9} + 4 \beta_{8} - 12 \beta_{7} + 24 \beta_{6} - 4 \beta_{5} + 8 \beta_{3} - 8 \beta_{2}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.204810 1.39930i
1.27715 0.607364i
−1.35164 0.416001i
1.35164 + 0.416001i
−1.27715 + 0.607364i
−0.204810 + 1.39930i
0.204810 + 1.39930i
1.27715 + 0.607364i
−1.35164 + 0.416001i
1.35164 0.416001i
−1.27715 0.607364i
−0.204810 1.39930i
−1.39930 + 0.204810i −0.814141 1.52878i 1.91611 0.573183i 2.08397 2.08397i 1.45234 + 1.97249i −1.14637 −2.56382 + 1.19449i −1.67435 + 2.48929i −2.48929 + 3.34292i
11.2 −0.607364 + 1.27715i 1.73003 + 0.0835731i −1.26222 1.55139i −0.431733 + 0.431733i −1.15749 + 2.15875i −3.10278 2.74798 0.669785i 2.98603 + 0.289169i −0.289169 0.813607i
11.3 −0.416001 1.35164i 0.966579 1.43726i −1.65389 + 1.12457i −1.57184 + 1.57184i −2.34477 0.708570i 2.24914 2.20804 + 1.76765i −1.13145 2.77846i 2.77846 + 1.47068i
11.4 0.416001 + 1.35164i −1.43726 + 0.966579i −1.65389 + 1.12457i 1.57184 1.57184i −1.90437 1.54057i 2.24914 −2.20804 1.76765i 1.13145 2.77846i 2.77846 + 1.47068i
11.5 0.607364 1.27715i 0.0835731 + 1.73003i −1.26222 1.55139i 0.431733 0.431733i 2.26027 + 0.944024i −3.10278 −2.74798 + 0.669785i −2.98603 + 0.289169i −0.289169 0.813607i
11.6 1.39930 0.204810i −1.52878 0.814141i 1.91611 0.573183i −2.08397 + 2.08397i −2.30598 0.826122i −1.14637 2.56382 1.19449i 1.67435 + 2.48929i −2.48929 + 3.34292i
35.1 −1.39930 0.204810i −0.814141 + 1.52878i 1.91611 + 0.573183i 2.08397 + 2.08397i 1.45234 1.97249i −1.14637 −2.56382 1.19449i −1.67435 2.48929i −2.48929 3.34292i
35.2 −0.607364 1.27715i 1.73003 0.0835731i −1.26222 + 1.55139i −0.431733 0.431733i −1.15749 2.15875i −3.10278 2.74798 + 0.669785i 2.98603 0.289169i −0.289169 + 0.813607i
35.3 −0.416001 + 1.35164i 0.966579 + 1.43726i −1.65389 1.12457i −1.57184 1.57184i −2.34477 + 0.708570i 2.24914 2.20804 1.76765i −1.13145 + 2.77846i 2.77846 1.47068i
35.4 0.416001 1.35164i −1.43726 0.966579i −1.65389 1.12457i 1.57184 + 1.57184i −1.90437 + 1.54057i 2.24914 −2.20804 + 1.76765i 1.13145 + 2.77846i 2.77846 1.47068i
35.5 0.607364 + 1.27715i 0.0835731 1.73003i −1.26222 + 1.55139i 0.431733 + 0.431733i 2.26027 0.944024i −3.10278 −2.74798 0.669785i −2.98603 0.289169i −0.289169 + 0.813607i
35.6 1.39930 + 0.204810i −1.52878 + 0.814141i 1.91611 + 0.573183i −2.08397 2.08397i −2.30598 + 0.826122i −1.14637 2.56382 + 1.19449i 1.67435 2.48929i −2.48929 3.34292i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.2.k.a 12
3.b odd 2 1 inner 48.2.k.a 12
4.b odd 2 1 192.2.k.a 12
8.b even 2 1 384.2.k.b 12
8.d odd 2 1 384.2.k.a 12
12.b even 2 1 192.2.k.a 12
16.e even 4 1 192.2.k.a 12
16.e even 4 1 384.2.k.a 12
16.f odd 4 1 inner 48.2.k.a 12
16.f odd 4 1 384.2.k.b 12
24.f even 2 1 384.2.k.a 12
24.h odd 2 1 384.2.k.b 12
48.i odd 4 1 192.2.k.a 12
48.i odd 4 1 384.2.k.a 12
48.k even 4 1 inner 48.2.k.a 12
48.k even 4 1 384.2.k.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.k.a 12 1.a even 1 1 trivial
48.2.k.a 12 3.b odd 2 1 inner
48.2.k.a 12 16.f odd 4 1 inner
48.2.k.a 12 48.k even 4 1 inner
192.2.k.a 12 4.b odd 2 1
192.2.k.a 12 12.b even 2 1
192.2.k.a 12 16.e even 4 1
192.2.k.a 12 48.i odd 4 1
384.2.k.a 12 8.d odd 2 1
384.2.k.a 12 16.e even 4 1
384.2.k.a 12 24.f even 2 1
384.2.k.a 12 48.i odd 4 1
384.2.k.b 12 8.b even 2 1
384.2.k.b 12 16.f odd 4 1
384.2.k.b 12 24.h odd 2 1
384.2.k.b 12 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(48, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - 2 T^{4} - 16 T^{6} - 8 T^{8} + 32 T^{10} + 64 T^{12} \)
$3$ \( 1 + 2 T + 2 T^{2} - 2 T^{3} - 5 T^{4} - 20 T^{5} - 28 T^{6} - 60 T^{7} - 45 T^{8} - 54 T^{9} + 162 T^{10} + 486 T^{11} + 729 T^{12} \)
$5$ \( 1 - 30 T^{4} - 49 T^{8} + 12796 T^{12} - 30625 T^{16} - 11718750 T^{20} + 244140625 T^{24} \)
$7$ \( ( 1 + 2 T + 15 T^{2} + 20 T^{3} + 105 T^{4} + 98 T^{5} + 343 T^{6} )^{4} \)
$11$ \( 1 - 62 T^{4} + 11023 T^{8} - 2631620 T^{12} + 161387743 T^{16} - 13290250622 T^{20} + 3138428376721 T^{24} \)
$13$ \( ( 1 + 2 T + 2 T^{2} - 6 T^{3} - 25 T^{4} + 412 T^{5} + 892 T^{6} + 5356 T^{7} - 4225 T^{8} - 13182 T^{9} + 57122 T^{10} + 742586 T^{11} + 4826809 T^{12} )^{2} \)
$17$ \( ( 1 - 62 T^{2} + 1903 T^{4} - 38180 T^{6} + 549967 T^{8} - 5178302 T^{10} + 24137569 T^{12} )^{2} \)
$19$ \( ( 1 + 6 T + 18 T^{2} + 82 T^{3} + 539 T^{4} + 2636 T^{5} + 9476 T^{6} + 50084 T^{7} + 194579 T^{8} + 562438 T^{9} + 2345778 T^{10} + 14856594 T^{11} + 47045881 T^{12} )^{2} \)
$23$ \( ( 1 - 86 T^{2} + 3791 T^{4} - 105684 T^{6} + 2005439 T^{8} - 24066326 T^{10} + 148035889 T^{12} )^{2} \)
$29$ \( 1 - 830 T^{4} + 2253679 T^{8} - 1165110596 T^{12} + 1593984336799 T^{16} - 415204522757630 T^{20} + 353814783205469041 T^{24} \)
$31$ \( ( 1 - 150 T^{2} + 10019 T^{4} - 392444 T^{6} + 9628259 T^{8} - 138528150 T^{10} + 887503681 T^{12} )^{2} \)
$37$ \( ( 1 + 2 T + 2 T^{2} - 54 T^{3} + 567 T^{4} + 8764 T^{5} + 17852 T^{6} + 324268 T^{7} + 776223 T^{8} - 2735262 T^{9} + 3748322 T^{10} + 138687914 T^{11} + 2565726409 T^{12} )^{2} \)
$41$ \( ( 1 + 138 T^{2} + 7887 T^{4} + 320492 T^{6} + 13258047 T^{8} + 389955018 T^{10} + 4750104241 T^{12} )^{2} \)
$43$ \( ( 1 - 6 T + 18 T^{2} - 226 T^{3} + 4235 T^{4} - 18188 T^{5} + 58436 T^{6} - 782084 T^{7} + 7830515 T^{8} - 17968582 T^{9} + 61538418 T^{10} - 882050658 T^{11} + 6321363049 T^{12} )^{2} \)
$47$ \( ( 1 + 170 T^{2} + 15791 T^{4} + 908172 T^{6} + 34882319 T^{8} + 829545770 T^{10} + 10779215329 T^{12} )^{2} \)
$53$ \( 1 + 7714 T^{4} + 19237903 T^{8} + 30633057916 T^{12} + 151796308101343 T^{16} + 480271251833238754 T^{20} + \)\(49\!\cdots\!41\)\( T^{24} \)
$59$ \( ( 1 - 30 T + 450 T^{2} - 3458 T^{3} + 3915 T^{4} + 231740 T^{5} - 2735068 T^{6} + 13672660 T^{7} + 13628115 T^{8} - 710200582 T^{9} + 5452812450 T^{10} - 21447728970 T^{11} + 42180533641 T^{12} )( 1 + 30 T + 450 T^{2} + 3458 T^{3} + 3915 T^{4} - 231740 T^{5} - 2735068 T^{6} - 13672660 T^{7} + 13628115 T^{8} + 710200582 T^{9} + 5452812450 T^{10} + 21447728970 T^{11} + 42180533641 T^{12} ) \)
$61$ \( ( 1 - 6 T + 18 T^{2} - 430 T^{3} - 121 T^{4} + 30796 T^{5} - 90148 T^{6} + 1878556 T^{7} - 450241 T^{8} - 97601830 T^{9} + 249225138 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} )^{2} \)
$67$ \( ( 1 - 14 T + 98 T^{2} - 706 T^{3} + 9435 T^{4} - 112164 T^{5} + 894884 T^{6} - 7514988 T^{7} + 42353715 T^{8} - 212338678 T^{9} + 1974809858 T^{10} - 18901751498 T^{11} + 90458382169 T^{12} )^{2} \)
$71$ \( ( 1 - 230 T^{2} + 30127 T^{4} - 2527028 T^{6} + 151870207 T^{8} - 5844686630 T^{10} + 128100283921 T^{12} )^{2} \)
$73$ \( ( 1 - 166 T^{2} + 19007 T^{4} - 1414164 T^{6} + 101288303 T^{8} - 4714108006 T^{10} + 151334226289 T^{12} )^{2} \)
$79$ \( ( 1 - 358 T^{2} + 58915 T^{4} - 5817628 T^{6} + 367688515 T^{8} - 13944128998 T^{10} + 243087455521 T^{12} )^{2} \)
$83$ \( 1 - 1374 T^{4} + 18563631 T^{8} - 336062521604 T^{12} + 880998758923551 T^{16} - 3094649526959042334 T^{20} + \)\(10\!\cdots\!61\)\( T^{24} \)
$89$ \( ( 1 + 322 T^{2} + 51919 T^{4} + 5548348 T^{6} + 411250399 T^{8} + 20203001602 T^{10} + 496981290961 T^{12} )^{2} \)
$97$ \( ( 1 + 2 T + 163 T^{2} - 220 T^{3} + 15811 T^{4} + 18818 T^{5} + 912673 T^{6} )^{4} \)
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