Properties

Label 2057.4.a.d
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: no (minimal twist has level 17)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{2} - 8 q^{3} + q^{4} + 6 q^{5} - 24 q^{6} + 28 q^{7} - 21 q^{8} + 37 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} - 8 q^{3} + q^{4} + 6 q^{5} - 24 q^{6} + 28 q^{7} - 21 q^{8} + 37 q^{9} + 18 q^{10} - 8 q^{12} + 58 q^{13} + 84 q^{14} - 48 q^{15} - 71 q^{16} - 17 q^{17} + 111 q^{18} - 116 q^{19} + 6 q^{20} - 224 q^{21} - 60 q^{23} + 168 q^{24} - 89 q^{25} + 174 q^{26} - 80 q^{27} + 28 q^{28} - 30 q^{29} - 144 q^{30} - 172 q^{31} - 45 q^{32} - 51 q^{34} + 168 q^{35} + 37 q^{36} - 58 q^{37} - 348 q^{38} - 464 q^{39} - 126 q^{40} + 342 q^{41} - 672 q^{42} + 148 q^{43} + 222 q^{45} - 180 q^{46} + 288 q^{47} + 568 q^{48} + 441 q^{49} - 267 q^{50} + 136 q^{51} + 58 q^{52} + 318 q^{53} - 240 q^{54} - 588 q^{56} + 928 q^{57} - 90 q^{58} + 252 q^{59} - 48 q^{60} - 110 q^{61} - 516 q^{62} + 1036 q^{63} + 433 q^{64} + 348 q^{65} - 484 q^{67} - 17 q^{68} + 480 q^{69} + 504 q^{70} - 708 q^{71} - 777 q^{72} - 362 q^{73} - 174 q^{74} + 712 q^{75} - 116 q^{76} - 1392 q^{78} + 484 q^{79} - 426 q^{80} - 359 q^{81} + 1026 q^{82} - 756 q^{83} - 224 q^{84} - 102 q^{85} + 444 q^{86} + 240 q^{87} - 774 q^{89} + 666 q^{90} + 1624 q^{91} - 60 q^{92} + 1376 q^{93} + 864 q^{94} - 696 q^{95} + 360 q^{96} - 382 q^{97} + 1323 q^{98}+O(q^{100}) \) 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
3.00000 −8.00000 1.00000 6.00000 −24.0000 28.0000 −21.0000 37.0000 18.0000
\(n\): 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.d 1
11.b odd 2 1 17.4.a.a 1
33.d even 2 1 153.4.a.d 1
44.c even 2 1 272.4.a.d 1
55.d odd 2 1 425.4.a.d 1
55.e even 4 2 425.4.b.c 2
77.b even 2 1 833.4.a.a 1
88.b odd 2 1 1088.4.a.l 1
88.g even 2 1 1088.4.a.a 1
132.d odd 2 1 2448.4.a.f 1
187.b odd 2 1 289.4.a.a 1
187.f odd 4 2 289.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 11.b odd 2 1
153.4.a.d 1 33.d even 2 1
272.4.a.d 1 44.c even 2 1
289.4.a.a 1 187.b odd 2 1
289.4.b.a 2 187.f odd 4 2
425.4.a.d 1 55.d odd 2 1
425.4.b.c 2 55.e even 4 2
833.4.a.a 1 77.b even 2 1
1088.4.a.a 1 88.g even 2 1
1088.4.a.l 1 88.b odd 2 1
2057.4.a.d 1 1.a even 1 1 trivial
2448.4.a.f 1 132.d odd 2 1

Hecke kernels

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

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{3} + 8 \) Copy content Toggle raw display
\( T_{5} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T + 8 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T - 28 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 58 \) Copy content Toggle raw display
$17$ \( T + 17 \) Copy content Toggle raw display
$19$ \( T + 116 \) Copy content Toggle raw display
$23$ \( T + 60 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T + 172 \) Copy content Toggle raw display
$37$ \( T + 58 \) Copy content Toggle raw display
$41$ \( T - 342 \) Copy content Toggle raw display
$43$ \( T - 148 \) Copy content Toggle raw display
$47$ \( T - 288 \) Copy content Toggle raw display
$53$ \( T - 318 \) Copy content Toggle raw display
$59$ \( T - 252 \) Copy content Toggle raw display
$61$ \( T + 110 \) Copy content Toggle raw display
$67$ \( T + 484 \) Copy content Toggle raw display
$71$ \( T + 708 \) Copy content Toggle raw display
$73$ \( T + 362 \) Copy content Toggle raw display
$79$ \( T - 484 \) Copy content Toggle raw display
$83$ \( T + 756 \) Copy content Toggle raw display
$89$ \( T + 774 \) Copy content Toggle raw display
$97$ \( T + 382 \) Copy content Toggle raw display
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