Properties

Label 2057.4.a.b
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2057,4,Mod(1,2057)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2057, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2057.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2057 = 11^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2057.a (trivial)

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: yes
Analytic conductor: \(121.366928882\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} + 4 q^{3} - 7 q^{4} + 17 q^{5} - 4 q^{6} - 9 q^{7} + 15 q^{8} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 4 q^{3} - 7 q^{4} + 17 q^{5} - 4 q^{6} - 9 q^{7} + 15 q^{8} - 11 q^{9} - 17 q^{10} - 28 q^{12} - 8 q^{13} + 9 q^{14} + 68 q^{15} + 41 q^{16} - 17 q^{17} + 11 q^{18} + 5 q^{19} - 119 q^{20} - 36 q^{21} + 64 q^{23} + 60 q^{24} + 164 q^{25} + 8 q^{26} - 152 q^{27} + 63 q^{28} - 90 q^{29} - 68 q^{30} - 79 q^{31} - 161 q^{32} + 17 q^{34} - 153 q^{35} + 77 q^{36} - 9 q^{37} - 5 q^{38} - 32 q^{39} + 255 q^{40} - 124 q^{41} + 36 q^{42} + 83 q^{43} - 187 q^{45} - 64 q^{46} + 614 q^{47} + 164 q^{48} - 262 q^{49} - 164 q^{50} - 68 q^{51} + 56 q^{52} + 186 q^{53} + 152 q^{54} - 135 q^{56} + 20 q^{57} + 90 q^{58} + 355 q^{59} - 476 q^{60} - 713 q^{61} + 79 q^{62} + 99 q^{63} - 167 q^{64} - 136 q^{65} - 596 q^{67} + 119 q^{68} + 256 q^{69} + 153 q^{70} + 312 q^{71} - 165 q^{72} - 496 q^{73} + 9 q^{74} + 656 q^{75} - 35 q^{76} + 32 q^{78} + 411 q^{79} + 697 q^{80} - 311 q^{81} + 124 q^{82} - 555 q^{83} + 252 q^{84} - 289 q^{85} - 83 q^{86} - 360 q^{87} - 1022 q^{89} + 187 q^{90} + 72 q^{91} - 448 q^{92} - 316 q^{93} - 614 q^{94} + 85 q^{95} - 644 q^{96} - 674 q^{97} + 262 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 4.00000 −7.00000 17.0000 −4.00000 −9.00000 15.0000 −11.0000 −17.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(17\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2057.4.a.b 1
11.b odd 2 1 2057.4.a.c yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2057.4.a.b 1 1.a even 1 1 trivial
2057.4.a.c yes 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} - 4 \) Copy content Toggle raw display
\( T_{5} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T - 17 \) Copy content Toggle raw display
$7$ \( T + 9 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 8 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T - 64 \) Copy content Toggle raw display
$29$ \( T + 90 \) Copy content Toggle raw display
$31$ \( T + 79 \) Copy content Toggle raw display
$37$ \( T + 9 \) Copy content Toggle raw display
$41$ \( T + 124 \) Copy content Toggle raw display
$43$ \( T - 83 \) Copy content Toggle raw display
$47$ \( T - 614 \) Copy content Toggle raw display
$53$ \( T - 186 \) Copy content Toggle raw display
$59$ \( T - 355 \) Copy content Toggle raw display
$61$ \( T + 713 \) Copy content Toggle raw display
$67$ \( T + 596 \) Copy content Toggle raw display
$71$ \( T - 312 \) Copy content Toggle raw display
$73$ \( T + 496 \) Copy content Toggle raw display
$79$ \( T - 411 \) Copy content Toggle raw display
$83$ \( T + 555 \) Copy content Toggle raw display
$89$ \( T + 1022 \) Copy content Toggle raw display
$97$ \( T + 674 \) Copy content Toggle raw display
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