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 \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.455147291521\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

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

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.e.a 2
3.b odd 2 1 171.2.f.a 2
4.b odd 2 1 912.2.q.a 2
12.b even 2 1 2736.2.s.j 2
19.c even 3 1 inner 57.2.e.a 2
19.c even 3 1 1083.2.a.c 1
19.d odd 6 1 1083.2.a.b 1
57.f even 6 1 3249.2.a.f 1
57.h odd 6 1 171.2.f.a 2
57.h odd 6 1 3249.2.a.c 1
76.g odd 6 1 912.2.q.a 2
228.m even 6 1 2736.2.s.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.e.a 2 1.a even 1 1 trivial
57.2.e.a 2 19.c even 3 1 inner
171.2.f.a 2 3.b odd 2 1
171.2.f.a 2 57.h odd 6 1
912.2.q.a 2 4.b odd 2 1
912.2.q.a 2 76.g odd 6 1
1083.2.a.b 1 19.d odd 6 1
1083.2.a.c 1 19.c even 3 1
2736.2.s.j 2 12.b even 2 1
2736.2.s.j 2 228.m even 6 1
3249.2.a.c 1 57.h odd 6 1
3249.2.a.f 1 57.f even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 8T + 19 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( T^{2} - 11T + 121 \) Copy content Toggle raw display
$79$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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