Properties

Label 57.2.e.a
Level 57
Weight 2
Character orbit 57.e
Analytic conductor 0.455
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 0.500000 0.866025i 1.00000 −3.00000 −0.500000 + 0.866025i 0
49.1 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00000 −3.00000 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
19.c Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2}^{2} + T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).