Properties

Label 57.2.j.a
Level 57
Weight 2
Character orbit 57.j
Analytic conductor 0.455
Analytic rank 0
Dimension 6
CM disc. -3
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 57 = 3 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 57.j (of order \(18\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{3} + 2 \zeta_{18} q^{4} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{7} -3 \zeta_{18}^{4} q^{9} +O(q^{10})\) \( q + ( -\zeta_{18}^{2} + 2 \zeta_{18}^{5} ) q^{3} + 2 \zeta_{18} q^{4} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{7} -3 \zeta_{18}^{4} q^{9} + ( -4 + 2 \zeta_{18}^{3} ) q^{12} + ( -1 - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{13} + 4 \zeta_{18}^{2} q^{16} + ( -2 + 5 \zeta_{18}^{3} ) q^{19} + ( 4 - \zeta_{18} - 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{21} + ( 5 \zeta_{18} - 5 \zeta_{18}^{4} ) q^{25} + ( 3 + 3 \zeta_{18}^{3} ) q^{27} + ( 6 - 6 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{5} ) q^{28} + ( -5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{31} -6 \zeta_{18}^{5} q^{36} + ( 4 \zeta_{18} - 4 \zeta_{18}^{2} + 3 \zeta_{18}^{4} + 7 \zeta_{18}^{5} ) q^{37} + ( 7 \zeta_{18} + 7 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{39} + ( -6 - 6 \zeta_{18} - \zeta_{18}^{3} + 7 \zeta_{18}^{4} ) q^{43} + ( -8 \zeta_{18} + 4 \zeta_{18}^{4} ) q^{48} + ( -7 + 3 \zeta_{18} + 5 \zeta_{18}^{2} + 7 \zeta_{18}^{3} - 8 \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{49} + ( 2 - 2 \zeta_{18} - 8 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{52} + ( -8 \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{57} + ( -9 - 5 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{5} ) q^{61} + ( -3 + 6 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{63} + 8 \zeta_{18}^{3} q^{64} + ( 9 + 9 \zeta_{18}^{2} - 7 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{67} + ( 8 + \zeta_{18} + \zeta_{18}^{3} + 8 \zeta_{18}^{4} ) q^{73} + ( -5 + 10 \zeta_{18}^{3} ) q^{75} + ( -4 \zeta_{18} + 10 \zeta_{18}^{4} ) q^{76} + ( 7 + 10 \zeta_{18} - 10 \zeta_{18}^{3} - 7 \zeta_{18}^{4} ) q^{79} + ( -9 \zeta_{18}^{2} + 9 \zeta_{18}^{5} ) q^{81} + ( 8 \zeta_{18} - 2 \zeta_{18}^{2} - 10 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{84} + ( 5 - \zeta_{18} - 11 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{91} + ( 11 - 7 \zeta_{18} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{93} + ( 6 \zeta_{18}^{2} - 3 \zeta_{18}^{5} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + O(q^{10}) \) \( 6q - 18q^{12} - 15q^{13} + 3q^{19} + 9q^{21} + 27q^{27} + 30q^{28} - 39q^{43} - 21q^{49} - 12q^{52} - 42q^{61} - 36q^{63} + 24q^{64} + 33q^{67} + 51q^{73} + 12q^{79} + 48q^{91} + 54q^{93} + O(q^{100}) \)

Character Values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(-1\) \(\zeta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
0.939693 + 0.342020i
−0.766044 + 0.642788i
0.939693 0.342020i
−0.173648 + 0.984808i
−0.173648 0.984808i
−0.766044 0.642788i
0 −1.11334 + 1.32683i 1.87939 + 0.684040i 0 0 −0.418748 0.725293i 0 −0.520945 2.95442i 0
14.1 0 1.70574 + 0.300767i −1.53209 + 1.28558i 0 0 −2.05303 3.55596i 0 2.81908 + 1.02606i 0
29.1 0 −1.11334 1.32683i 1.87939 0.684040i 0 0 −0.418748 + 0.725293i 0 −0.520945 + 2.95442i 0
32.1 0 −0.592396 + 1.62760i −0.347296 + 1.96962i 0 0 2.47178 4.28125i 0 −2.29813 1.92836i 0
41.1 0 −0.592396 1.62760i −0.347296 1.96962i 0 0 2.47178 + 4.28125i 0 −2.29813 + 1.92836i 0
53.1 0 1.70574 0.300767i −1.53209 1.28558i 0 0 −2.05303 + 3.55596i 0 2.81908 1.02606i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.1
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 CM by \(\Q(\sqrt{-3}) \) yes
19.f Odd 1 yes
57.j Even 1 yes

Hecke kernels

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