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.

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

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 \(T_{2}^{2} \) \(\mathstrut +\mathstrut T_{2} \) \(\mathstrut +\mathstrut 1 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).