Properties

Label 57.3.c.b
Level $57$
Weight $3$
Character orbit 57.c
Analytic conductor $1.553$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Defining polynomial: \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{4} + ( -3 + \beta_{2} ) q^{5} + ( -2 + \beta_{2} ) q^{6} + ( 9 - \beta_{2} ) q^{7} + ( 3 \beta_{1} - \beta_{3} ) q^{8} -3 q^{9} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} -\beta_{1} q^{3} + ( -2 + \beta_{2} ) q^{4} + ( -3 + \beta_{2} ) q^{5} + ( -2 + \beta_{2} ) q^{6} + ( 9 - \beta_{2} ) q^{7} + ( 3 \beta_{1} - \beta_{3} ) q^{8} -3 q^{9} + ( 8 \beta_{1} + 4 \beta_{3} ) q^{10} + ( -3 - \beta_{2} ) q^{11} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{12} + ( -6 \beta_{1} - 2 \beta_{3} ) q^{13} + ( -14 \beta_{1} - 10 \beta_{3} ) q^{14} + ( 4 \beta_{1} + 3 \beta_{3} ) q^{15} + ( -6 + \beta_{2} ) q^{16} + ( 1 - 5 \beta_{2} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{18} + ( 1 + 4 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} ) q^{19} + ( 20 - 4 \beta_{2} ) q^{20} + ( -10 \beta_{1} - 3 \beta_{3} ) q^{21} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{22} + ( 14 - 2 \beta_{2} ) q^{23} + ( 10 + \beta_{2} ) q^{24} + ( -2 - 5 \beta_{2} ) q^{25} + ( -20 + 6 \beta_{2} ) q^{26} + 3 \beta_{1} q^{27} + ( -32 + 10 \beta_{2} ) q^{28} + ( 12 \beta_{1} + 4 \beta_{3} ) q^{29} + ( 20 - 4 \beta_{2} ) q^{30} + ( -6 \beta_{1} - 10 \beta_{3} ) q^{31} + ( 23 \beta_{1} + 3 \beta_{3} ) q^{32} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{33} + ( -26 \beta_{1} - 6 \beta_{3} ) q^{34} + ( -41 + 11 \beta_{2} ) q^{35} + ( 6 - 3 \beta_{2} ) q^{36} + ( 34 \beta_{1} + 6 \beta_{3} ) q^{37} + ( 40 - 11 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{38} + ( -16 + 2 \beta_{2} ) q^{39} + ( -8 \beta_{1} - 8 \beta_{3} ) q^{40} + ( -4 \beta_{1} + 8 \beta_{3} ) q^{41} + ( -32 + 10 \beta_{2} ) q^{42} + ( 1 + 15 \beta_{2} ) q^{43} + ( -8 - 2 \beta_{2} ) q^{44} + ( 9 - 3 \beta_{2} ) q^{45} + ( -24 \beta_{1} - 16 \beta_{3} ) q^{46} + ( 17 + 9 \beta_{2} ) q^{47} + ( 7 \beta_{1} + 3 \beta_{3} ) q^{48} + ( 46 - 17 \beta_{2} ) q^{49} + ( -23 \beta_{1} - 3 \beta_{3} ) q^{50} + ( -6 \beta_{1} - 15 \beta_{3} ) q^{51} + ( 26 \beta_{1} + 18 \beta_{3} ) q^{52} + ( -24 \beta_{1} + 12 \beta_{3} ) q^{53} + ( 6 - 3 \beta_{2} ) q^{54} + ( -5 - \beta_{2} ) q^{55} + ( 26 \beta_{1} + 2 \beta_{3} ) q^{56} + ( 4 - 3 \beta_{1} - 8 \beta_{2} - 6 \beta_{3} ) q^{57} + ( 40 - 12 \beta_{2} ) q^{58} + ( -12 \beta_{1} + 16 \beta_{3} ) q^{59} + ( -24 \beta_{1} - 12 \beta_{3} ) q^{60} + ( -19 + 3 \beta_{2} ) q^{61} + ( -52 + 6 \beta_{2} ) q^{62} + ( -27 + 3 \beta_{2} ) q^{63} + ( 34 - 19 \beta_{2} ) q^{64} + ( 32 \beta_{1} + 20 \beta_{3} ) q^{65} + ( -8 - 2 \beta_{2} ) q^{66} + ( -20 \beta_{1} - 52 \beta_{3} ) q^{67} + ( -72 + 6 \beta_{2} ) q^{68} + ( -16 \beta_{1} - 6 \beta_{3} ) q^{69} + ( 96 \beta_{1} + 52 \beta_{3} ) q^{70} + ( 44 \beta_{1} + 24 \beta_{3} ) q^{71} + ( -9 \beta_{1} + 3 \beta_{3} ) q^{72} + ( 21 + 3 \beta_{2} ) q^{73} + ( 92 - 34 \beta_{2} ) q^{74} + ( -3 \beta_{1} - 15 \beta_{3} ) q^{75} + ( -30 - 44 \beta_{1} + 3 \beta_{2} - 12 \beta_{3} ) q^{76} + ( -13 - 5 \beta_{2} ) q^{77} + ( 26 \beta_{1} + 18 \beta_{3} ) q^{78} + ( -26 \beta_{1} + 14 \beta_{3} ) q^{79} + ( 32 - 8 \beta_{2} ) q^{80} + 9 q^{81} + ( 24 + 4 \beta_{2} ) q^{82} + ( 2 + 28 \beta_{2} ) q^{83} + ( 42 \beta_{1} + 30 \beta_{3} ) q^{84} + ( -73 + 11 \beta_{2} ) q^{85} + ( 74 \beta_{1} + 14 \beta_{3} ) q^{86} + ( 32 - 4 \beta_{2} ) q^{87} + ( -10 \beta_{1} + 14 \beta_{3} ) q^{88} + ( -44 \beta_{1} - 48 \beta_{3} ) q^{89} + ( -24 \beta_{1} - 12 \beta_{3} ) q^{90} + ( -68 \beta_{1} - 32 \beta_{3} ) q^{91} + ( -56 + 16 \beta_{2} ) q^{92} + ( -8 + 10 \beta_{2} ) q^{93} + ( 28 \beta_{1} - 8 \beta_{3} ) q^{94} + ( -31 - 48 \beta_{1} + 5 \beta_{2} - 20 \beta_{3} ) q^{95} + ( 66 - 3 \beta_{2} ) q^{96} + ( -36 \beta_{1} + 16 \beta_{3} ) q^{97} + ( -131 \beta_{1} - 63 \beta_{3} ) q^{98} + ( 9 + 3 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{4} - 10q^{5} - 6q^{6} + 34q^{7} - 12q^{9} + O(q^{10}) \) \( 4q - 6q^{4} - 10q^{5} - 6q^{6} + 34q^{7} - 12q^{9} - 14q^{11} - 22q^{16} - 6q^{17} + 72q^{20} + 52q^{23} + 42q^{24} - 18q^{25} - 68q^{26} - 108q^{28} + 72q^{30} - 142q^{35} + 18q^{36} + 152q^{38} - 60q^{39} - 108q^{42} + 34q^{43} - 36q^{44} + 30q^{45} + 86q^{47} + 150q^{49} + 18q^{54} - 22q^{55} + 136q^{58} - 70q^{61} - 196q^{62} - 102q^{63} + 98q^{64} - 36q^{66} - 276q^{68} + 90q^{73} + 300q^{74} - 114q^{76} - 62q^{77} + 112q^{80} + 36q^{81} + 104q^{82} + 64q^{83} - 270q^{85} + 120q^{87} - 192q^{92} - 12q^{93} - 114q^{95} + 258q^{96} + 42q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 4 x^{2} - 5 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 4 \nu^{2} - 4 \nu - 15 \)\()/10\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 9 \nu + 5 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{3} - 2 \nu^{2} + 2 \nu + 25 \)\()/10\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + \beta_{2} + 5 \beta_{1} + 4\)\()/2\)
\(\nu^{3}\)\(=\)\(-4 \beta_{3} - 2 \beta_{1} + 7\)

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
−1.63746 1.52274i
2.13746 0.656712i
2.13746 + 0.656712i
−1.63746 + 1.52274i
3.04547i 1.73205i −5.27492 −6.27492 −5.27492 12.2749 3.88273i −3.00000 19.1101i
37.2 1.31342i 1.73205i 2.27492 1.27492 2.27492 4.72508 8.24163i −3.00000 1.67451i
37.3 1.31342i 1.73205i 2.27492 1.27492 2.27492 4.72508 8.24163i −3.00000 1.67451i
37.4 3.04547i 1.73205i −5.27492 −6.27492 −5.27492 12.2749 3.88273i −3.00000 19.1101i
\(n\): 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.b 4
3.b odd 2 1 171.3.c.f 4
4.b odd 2 1 912.3.o.b 4
12.b even 2 1 2736.3.o.l 4
19.b odd 2 1 inner 57.3.c.b 4
57.d even 2 1 171.3.c.f 4
76.d even 2 1 912.3.o.b 4
228.b odd 2 1 2736.3.o.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.c.b 4 1.a even 1 1 trivial
57.3.c.b 4 19.b odd 2 1 inner
171.3.c.f 4 3.b odd 2 1
171.3.c.f 4 57.d even 2 1
912.3.o.b 4 4.b odd 2 1
912.3.o.b 4 76.d even 2 1
2736.3.o.l 4 12.b even 2 1
2736.3.o.l 4 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 11 T_{2}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 11 T^{2} + T^{4} \)
$3$ \( ( 3 + T^{2} )^{2} \)
$5$ \( ( -8 + 5 T + T^{2} )^{2} \)
$7$ \( ( 58 - 17 T + T^{2} )^{2} \)
$11$ \( ( -2 + 7 T + T^{2} )^{2} \)
$13$ \( 3136 + 188 T^{2} + T^{4} \)
$17$ \( ( -354 + 3 T + T^{2} )^{2} \)
$19$ \( 130321 + 494 T^{2} + T^{4} \)
$23$ \( ( 112 - 26 T + T^{2} )^{2} \)
$29$ \( 50176 + 752 T^{2} + T^{4} \)
$31$ \( 222784 + 956 T^{2} + T^{4} \)
$37$ \( 7354944 + 6108 T^{2} + T^{4} \)
$41$ \( 12544 + 992 T^{2} + T^{4} \)
$43$ \( ( -3134 - 17 T + T^{2} )^{2} \)
$47$ \( ( -692 - 43 T + T^{2} )^{2} \)
$53$ \( 4064256 + 6768 T^{2} + T^{4} \)
$59$ \( 256 + 4832 T^{2} + T^{4} \)
$61$ \( ( 178 + 35 T + T^{2} )^{2} \)
$67$ \( 162205696 + 25904 T^{2} + T^{4} \)
$71$ \( 112896 + 11616 T^{2} + T^{4} \)
$73$ \( ( 378 - 45 T + T^{2} )^{2} \)
$79$ \( 5456896 + 8396 T^{2} + T^{4} \)
$83$ \( ( -10916 - 32 T + T^{2} )^{2} \)
$89$ \( 94945536 + 24288 T^{2} + T^{4} \)
$97$ \( 21086464 + 14048 T^{2} + T^{4} \)
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