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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,3,Mod(37,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.55313750685\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} - \beta_1 q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{2} - 3) q^{5} + (\beta_{2} - 2) q^{6} + ( - \beta_{2} + 9) q^{7} + ( - \beta_{3} + 3 \beta_1) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_1) q^{2} - \beta_1 q^{3} + (\beta_{2} - 2) q^{4} + (\beta_{2} - 3) q^{5} + (\beta_{2} - 2) q^{6} + ( - \beta_{2} + 9) q^{7} + ( - \beta_{3} + 3 \beta_1) q^{8} - 3 q^{9} + (4 \beta_{3} + 8 \beta_1) q^{10} + ( - \beta_{2} - 3) q^{11} + (3 \beta_{3} + 3 \beta_1) q^{12} + ( - 2 \beta_{3} - 6 \beta_1) q^{13} + ( - 10 \beta_{3} - 14 \beta_1) q^{14} + (3 \beta_{3} + 4 \beta_1) q^{15} + (\beta_{2} - 6) q^{16} + ( - 5 \beta_{2} + 1) q^{17} + (3 \beta_{3} + 3 \beta_1) q^{18} + (8 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 1) q^{19} + ( - 4 \beta_{2} + 20) q^{20} + ( - 3 \beta_{3} - 10 \beta_1) q^{21} + (2 \beta_{3} - 2 \beta_1) q^{22} + ( - 2 \beta_{2} + 14) q^{23} + (\beta_{2} + 10) q^{24} + ( - 5 \beta_{2} - 2) q^{25} + (6 \beta_{2} - 20) q^{26} + 3 \beta_1 q^{27} + (10 \beta_{2} - 32) q^{28} + (4 \beta_{3} + 12 \beta_1) q^{29} + ( - 4 \beta_{2} + 20) q^{30} + ( - 10 \beta_{3} - 6 \beta_1) q^{31} + (3 \beta_{3} + 23 \beta_1) q^{32} + ( - 3 \beta_{3} + 2 \beta_1) q^{33} + ( - 6 \beta_{3} - 26 \beta_1) q^{34} + (11 \beta_{2} - 41) q^{35} + ( - 3 \beta_{2} + 6) q^{36} + (6 \beta_{3} + 34 \beta_1) q^{37} + ( - 3 \beta_{3} - 4 \beta_{2} - 11 \beta_1 + 40) q^{38} + (2 \beta_{2} - 16) q^{39} + ( - 8 \beta_{3} - 8 \beta_1) q^{40} + (8 \beta_{3} - 4 \beta_1) q^{41} + (10 \beta_{2} - 32) q^{42} + (15 \beta_{2} + 1) q^{43} + ( - 2 \beta_{2} - 8) q^{44} + ( - 3 \beta_{2} + 9) q^{45} + ( - 16 \beta_{3} - 24 \beta_1) q^{46} + (9 \beta_{2} + 17) q^{47} + (3 \beta_{3} + 7 \beta_1) q^{48} + ( - 17 \beta_{2} + 46) q^{49} + ( - 3 \beta_{3} - 23 \beta_1) q^{50} + ( - 15 \beta_{3} - 6 \beta_1) q^{51} + (18 \beta_{3} + 26 \beta_1) q^{52} + (12 \beta_{3} - 24 \beta_1) q^{53} + ( - 3 \beta_{2} + 6) q^{54} + ( - \beta_{2} - 5) q^{55} + (2 \beta_{3} + 26 \beta_1) q^{56} + ( - 6 \beta_{3} - 8 \beta_{2} - 3 \beta_1 + 4) q^{57} + ( - 12 \beta_{2} + 40) q^{58} + (16 \beta_{3} - 12 \beta_1) q^{59} + ( - 12 \beta_{3} - 24 \beta_1) q^{60} + (3 \beta_{2} - 19) q^{61} + (6 \beta_{2} - 52) q^{62} + (3 \beta_{2} - 27) q^{63} + ( - 19 \beta_{2} + 34) q^{64} + (20 \beta_{3} + 32 \beta_1) q^{65} + ( - 2 \beta_{2} - 8) q^{66} + ( - 52 \beta_{3} - 20 \beta_1) q^{67} + (6 \beta_{2} - 72) q^{68} + ( - 6 \beta_{3} - 16 \beta_1) q^{69} + (52 \beta_{3} + 96 \beta_1) q^{70} + (24 \beta_{3} + 44 \beta_1) q^{71} + (3 \beta_{3} - 9 \beta_1) q^{72} + (3 \beta_{2} + 21) q^{73} + ( - 34 \beta_{2} + 92) q^{74} + ( - 15 \beta_{3} - 3 \beta_1) q^{75} + ( - 12 \beta_{3} + 3 \beta_{2} - 44 \beta_1 - 30) q^{76} + ( - 5 \beta_{2} - 13) q^{77} + (18 \beta_{3} + 26 \beta_1) q^{78} + (14 \beta_{3} - 26 \beta_1) q^{79} + ( - 8 \beta_{2} + 32) q^{80} + 9 q^{81} + (4 \beta_{2} + 24) q^{82} + (28 \beta_{2} + 2) q^{83} + (30 \beta_{3} + 42 \beta_1) q^{84} + (11 \beta_{2} - 73) q^{85} + (14 \beta_{3} + 74 \beta_1) q^{86} + ( - 4 \beta_{2} + 32) q^{87} + (14 \beta_{3} - 10 \beta_1) q^{88} + ( - 48 \beta_{3} - 44 \beta_1) q^{89} + ( - 12 \beta_{3} - 24 \beta_1) q^{90} + ( - 32 \beta_{3} - 68 \beta_1) q^{91} + (16 \beta_{2} - 56) q^{92} + (10 \beta_{2} - 8) q^{93} + ( - 8 \beta_{3} + 28 \beta_1) q^{94} + ( - 20 \beta_{3} + 5 \beta_{2} - 48 \beta_1 - 31) q^{95} + ( - 3 \beta_{2} + 66) q^{96} + (16 \beta_{3} - 36 \beta_1) q^{97} + ( - 63 \beta_{3} - 131 \beta_1) q^{98} + (3 \beta_{2} + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 10 q^{5} - 6 q^{6} + 34 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} - 10 q^{5} - 6 q^{6} + 34 q^{7} - 12 q^{9} - 14 q^{11} - 22 q^{16} - 6 q^{17} + 72 q^{20} + 52 q^{23} + 42 q^{24} - 18 q^{25} - 68 q^{26} - 108 q^{28} + 72 q^{30} - 142 q^{35} + 18 q^{36} + 152 q^{38} - 60 q^{39} - 108 q^{42} + 34 q^{43} - 36 q^{44} + 30 q^{45} + 86 q^{47} + 150 q^{49} + 18 q^{54} - 22 q^{55} + 136 q^{58} - 70 q^{61} - 196 q^{62} - 102 q^{63} + 98 q^{64} - 36 q^{66} - 276 q^{68} + 90 q^{73} + 300 q^{74} - 114 q^{76} - 62 q^{77} + 112 q^{80} + 36 q^{81} + 104 q^{82} + 64 q^{83} - 270 q^{85} + 120 q^{87} - 192 q^{92} - 12 q^{93} - 114 q^{95} + 258 q^{96} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{3} - 2\beta _1 + 7 \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \( T_{2}^{4} + 11T_{2}^{2} + 16 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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