Properties

Label 57.2.j.a
Level $57$
Weight $2$
Character orbit 57.j
Analytic conductor $0.455$
Analytic rank $0$
Dimension $6$
CM discriminant -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 \(18\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.j.a 6
3.b odd 2 1 CM 57.2.j.a 6
4.b odd 2 1 912.2.cc.a 6
12.b even 2 1 912.2.cc.a 6
19.e even 9 1 1083.2.d.a 6
19.f odd 18 1 inner 57.2.j.a 6
19.f odd 18 1 1083.2.d.a 6
57.j even 18 1 inner 57.2.j.a 6
57.j even 18 1 1083.2.d.a 6
57.l odd 18 1 1083.2.d.a 6
76.k even 18 1 912.2.cc.a 6
228.u odd 18 1 912.2.cc.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.j.a 6 1.a even 1 1 trivial
57.2.j.a 6 3.b odd 2 1 CM
57.2.j.a 6 19.f odd 18 1 inner
57.2.j.a 6 57.j even 18 1 inner
912.2.cc.a 6 4.b odd 2 1
912.2.cc.a 6 12.b even 2 1
912.2.cc.a 6 76.k even 18 1
912.2.cc.a 6 228.u odd 18 1
1083.2.d.a 6 19.e even 9 1
1083.2.d.a 6 19.f odd 18 1
1083.2.d.a 6 57.j even 18 1
1083.2.d.a 6 57.l odd 18 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 9T^{3} + 27 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 21 T^{4} + 34 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 15 T^{5} + 114 T^{4} + \cdots + 8427 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( (T^{2} - T + 19)^{3} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 93 T^{4} + 8649 T^{2} + \cdots + 118803 \) Copy content Toggle raw display
$37$ \( T^{6} + 222 T^{4} + 12321 T^{2} + \cdots + 98283 \) Copy content Toggle raw display
$41$ \( T^{6} \) Copy content Toggle raw display
$43$ \( T^{6} + 39 T^{5} + 636 T^{4} + \cdots + 5041 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} \) Copy content Toggle raw display
$61$ \( T^{6} + 42 T^{5} + 771 T^{4} + \cdots + 516961 \) Copy content Toggle raw display
$67$ \( T^{6} - 33 T^{5} + 564 T^{4} + \cdots + 189003 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 51 T^{5} + 1086 T^{4} + \cdots + 73441 \) Copy content Toggle raw display
$79$ \( T^{6} - 12 T^{5} + 285 T^{4} + \cdots + 48387 \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} \) Copy content Toggle raw display
$97$ \( T^{6} - 243 T^{3} + 19683 \) Copy content Toggle raw display
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