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Properties

Label 2057.4.a.s
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $1$
Dimension $44$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2057,4,Mod(1,2057)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [44,-1,-28,139,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(121.366928882\)
Analytic rank: \(1\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 187)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.14451 −6.70283 18.4660 −17.4930 34.4828 3.29028 −53.8424 17.9279 89.9932
1.2 −5.06085 1.03546 17.6122 6.20042 −5.24030 27.6132 −48.6461 −25.9278 −31.3794
1.3 −4.90010 −9.12402 16.0109 6.33299 44.7086 −22.4833 −39.2544 56.2477 −31.0323
1.4 −4.78098 3.44541 14.8578 0.544918 −16.4724 0.976001 −32.7868 −15.1292 −2.60524
1.5 −4.73079 −1.21482 14.3803 −15.1061 5.74704 −9.37827 −30.1840 −25.5242 71.4635
1.6 −4.35850 5.79852 10.9965 −1.94108 −25.2728 27.7035 −13.0602 6.62282 8.46018
1.7 −4.26818 −5.27578 10.2173 −1.92958 22.5180 −33.3728 −9.46392 0.833892 8.23580
1.8 −3.85832 −2.22238 6.88662 11.4364 8.57465 12.6402 4.29578 −22.0610 −44.1251
1.9 −3.83157 −4.68424 6.68094 19.8105 17.9480 8.51267 5.05408 −5.05794 −75.9053
1.10 −3.38080 5.60751 3.42978 7.08583 −18.9578 −4.98418 15.4510 4.44415 −23.9557
1.11 −3.20225 −7.51392 2.25442 −15.1991 24.0615 8.04966 18.3988 29.4590 48.6713
1.12 −3.19095 9.23633 2.18218 −19.0681 −29.4727 −16.1567 18.5644 58.3098 60.8454
1.13 −1.96488 −0.243424 −4.13924 15.2237 0.478298 −11.1902 23.8522 −26.9407 −29.9127
1.14 −1.94689 10.0150 −4.20960 −3.88091 −19.4981 15.3272 23.7708 73.2992 7.55571
1.15 −1.78136 −3.23043 −4.82675 −6.22684 5.75457 −3.20896 22.8491 −16.5643 11.0923
1.16 −1.63323 −6.45520 −5.33256 3.96216 10.5428 20.2168 21.7751 14.6696 −6.47113
1.17 −1.50580 6.30025 −5.73256 3.99925 −9.48693 11.1510 20.6785 12.6932 −6.02208
1.18 −1.49542 −4.41660 −5.76373 −15.6043 6.60466 31.8747 20.5825 −7.49362 23.3349
1.19 −1.44896 0.883017 −5.90051 −15.0016 −1.27946 −28.3383 20.1413 −26.2203 21.7367
1.20 −1.41506 −9.10302 −5.99760 19.7402 12.8813 −24.5836 19.8075 55.8650 −27.9336
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
Significant digits:
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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.s 44
11.b odd 2 1 2057.4.a.t 44
11.d odd 10 2 187.4.g.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.g.a ✓ 88 11.d odd 10 2
2057.4.a.s 44 1.a even 1 1 trivial
2057.4.a.t 44 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{44} + T_{2}^{43} - 245 T_{2}^{42} - 261 T_{2}^{41} + 27677 T_{2}^{40} + 31257 T_{2}^{39} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{44} + 28 T_{3}^{43} - 380 T_{3}^{42} - 17274 T_{3}^{41} + 16330 T_{3}^{40} + \cdots + 87\!\cdots\!09 \) Copy content Toggle raw display
\( T_{5}^{44} + 24 T_{5}^{43} - 2691 T_{5}^{42} - 68106 T_{5}^{41} + 3224114 T_{5}^{40} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display