Properties

Label 57.3.c.a
Level $57$
Weight $3$
Character orbit 57.c
Analytic conductor $1.553$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.55313750685\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + q^{4} + 4 q^{5} -3 q^{6} -10 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + ( 1 - 2 \zeta_{6} ) q^{3} + q^{4} + 4 q^{5} -3 q^{6} -10 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{10} + 10 q^{11} + ( 1 - 2 \zeta_{6} ) q^{12} + ( -14 + 28 \zeta_{6} ) q^{13} + ( -10 + 20 \zeta_{6} ) q^{14} + ( 4 - 8 \zeta_{6} ) q^{15} -11 q^{16} + 10 q^{17} + ( -3 + 6 \zeta_{6} ) q^{18} + 19 q^{19} + 4 q^{20} + ( -10 + 20 \zeta_{6} ) q^{21} + ( 10 - 20 \zeta_{6} ) q^{22} -20 q^{23} -15 q^{24} -9 q^{25} + 42 q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} -10 q^{28} + ( -20 + 40 \zeta_{6} ) q^{29} -12 q^{30} + ( 10 - 20 \zeta_{6} ) q^{31} + ( 9 - 18 \zeta_{6} ) q^{32} + ( 10 - 20 \zeta_{6} ) q^{33} + ( 10 - 20 \zeta_{6} ) q^{34} -40 q^{35} -3 q^{36} + ( -6 + 12 \zeta_{6} ) q^{37} + ( 19 - 38 \zeta_{6} ) q^{38} + 42 q^{39} + ( 20 - 40 \zeta_{6} ) q^{40} + ( 20 - 40 \zeta_{6} ) q^{41} + 30 q^{42} -10 q^{43} + 10 q^{44} -12 q^{45} + ( -20 + 40 \zeta_{6} ) q^{46} -80 q^{47} + ( -11 + 22 \zeta_{6} ) q^{48} + 51 q^{49} + ( -9 + 18 \zeta_{6} ) q^{50} + ( 10 - 20 \zeta_{6} ) q^{51} + ( -14 + 28 \zeta_{6} ) q^{52} + ( 24 - 48 \zeta_{6} ) q^{53} + 9 q^{54} + 40 q^{55} + ( -50 + 100 \zeta_{6} ) q^{56} + ( 19 - 38 \zeta_{6} ) q^{57} + 60 q^{58} + ( 20 - 40 \zeta_{6} ) q^{59} + ( 4 - 8 \zeta_{6} ) q^{60} -10 q^{61} -30 q^{62} + 30 q^{63} -71 q^{64} + ( -56 + 112 \zeta_{6} ) q^{65} -30 q^{66} + ( -44 + 88 \zeta_{6} ) q^{67} + 10 q^{68} + ( -20 + 40 \zeta_{6} ) q^{69} + ( -40 + 80 \zeta_{6} ) q^{70} + ( 60 - 120 \zeta_{6} ) q^{71} + ( -15 + 30 \zeta_{6} ) q^{72} -10 q^{73} + 18 q^{74} + ( -9 + 18 \zeta_{6} ) q^{75} + 19 q^{76} -100 q^{77} + ( 42 - 84 \zeta_{6} ) q^{78} + ( -10 + 20 \zeta_{6} ) q^{79} -44 q^{80} + 9 q^{81} -60 q^{82} + 70 q^{83} + ( -10 + 20 \zeta_{6} ) q^{84} + 40 q^{85} + ( -10 + 20 \zeta_{6} ) q^{86} + 60 q^{87} + ( 50 - 100 \zeta_{6} ) q^{88} + ( -60 + 120 \zeta_{6} ) q^{89} + ( -12 + 24 \zeta_{6} ) q^{90} + ( 140 - 280 \zeta_{6} ) q^{91} -20 q^{92} -30 q^{93} + ( -80 + 160 \zeta_{6} ) q^{94} + 76 q^{95} -27 q^{96} + ( 44 - 88 \zeta_{6} ) q^{97} + ( 51 - 102 \zeta_{6} ) q^{98} -30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + 8q^{5} - 6q^{6} - 20q^{7} - 6q^{9} + O(q^{10}) \) \( 2q + 2q^{4} + 8q^{5} - 6q^{6} - 20q^{7} - 6q^{9} + 20q^{11} - 22q^{16} + 20q^{17} + 38q^{19} + 8q^{20} - 40q^{23} - 30q^{24} - 18q^{25} + 84q^{26} - 20q^{28} - 24q^{30} - 80q^{35} - 6q^{36} + 84q^{39} + 60q^{42} - 20q^{43} + 20q^{44} - 24q^{45} - 160q^{47} + 102q^{49} + 18q^{54} + 80q^{55} + 120q^{58} - 20q^{61} - 60q^{62} + 60q^{63} - 142q^{64} - 60q^{66} + 20q^{68} - 20q^{73} + 36q^{74} + 38q^{76} - 200q^{77} - 88q^{80} + 18q^{81} - 120q^{82} + 140q^{83} + 80q^{85} + 120q^{87} - 40q^{92} - 60q^{93} + 152q^{95} - 54q^{96} - 60q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(20\) \(40\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 1.73205i 1.00000 4.00000 −3.00000 −10.0000 8.66025i −3.00000 6.92820i
37.2 1.73205i 1.73205i 1.00000 4.00000 −3.00000 −10.0000 8.66025i −3.00000 6.92820i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.c.a 2
3.b odd 2 1 171.3.c.c 2
4.b odd 2 1 912.3.o.a 2
12.b even 2 1 2736.3.o.e 2
19.b odd 2 1 inner 57.3.c.a 2
57.d even 2 1 171.3.c.c 2
76.d even 2 1 912.3.o.a 2
228.b odd 2 1 2736.3.o.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.a 2 1.a even 1 1 trivial
57.3.c.a 2 19.b odd 2 1 inner
171.3.c.c 2 3.b odd 2 1
171.3.c.c 2 57.d even 2 1
912.3.o.a 2 4.b odd 2 1
912.3.o.a 2 76.d even 2 1
2736.3.o.e 2 12.b even 2 1
2736.3.o.e 2 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 3 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T^{2} \)
$3$ \( 3 + T^{2} \)
$5$ \( ( -4 + T )^{2} \)
$7$ \( ( 10 + T )^{2} \)
$11$ \( ( -10 + T )^{2} \)
$13$ \( 588 + T^{2} \)
$17$ \( ( -10 + T )^{2} \)
$19$ \( ( -19 + T )^{2} \)
$23$ \( ( 20 + T )^{2} \)
$29$ \( 1200 + T^{2} \)
$31$ \( 300 + T^{2} \)
$37$ \( 108 + T^{2} \)
$41$ \( 1200 + T^{2} \)
$43$ \( ( 10 + T )^{2} \)
$47$ \( ( 80 + T )^{2} \)
$53$ \( 1728 + T^{2} \)
$59$ \( 1200 + T^{2} \)
$61$ \( ( 10 + T )^{2} \)
$67$ \( 5808 + T^{2} \)
$71$ \( 10800 + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 300 + T^{2} \)
$83$ \( ( -70 + T )^{2} \)
$89$ \( 10800 + T^{2} \)
$97$ \( 5808 + T^{2} \)
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