Properties

Label 2057.4.a.u
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $0$
Dimension $52$
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: \(0\)
Dimension: \(52\)
Twist minimal: no (minimal twist has level 187)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} + 24 q^{6} + 42 q^{7} - 45 q^{8} + 572 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 52 q - q^{2} + 20 q^{3} + 235 q^{4} + 40 q^{5} + 24 q^{6} + 42 q^{7} - 45 q^{8} + 572 q^{9} - 33 q^{10} + 233 q^{12} - 12 q^{13} + 73 q^{14} + 400 q^{15} + 1223 q^{16} + 884 q^{17} - 201 q^{18} + 44 q^{19} + 655 q^{20} - 260 q^{21} + 572 q^{23} + 104 q^{24} + 1858 q^{25} + 465 q^{26} + 1070 q^{27} + 577 q^{28} - 322 q^{29} + 320 q^{30} + 1110 q^{31} - 481 q^{32} - 17 q^{34} - 102 q^{35} + 2507 q^{36} + 1678 q^{37} - 360 q^{38} + 1282 q^{39} - 1791 q^{40} + 826 q^{41} + 2133 q^{42} - 270 q^{43} + 710 q^{45} - 2158 q^{46} + 2464 q^{47} + 2201 q^{48} + 3224 q^{49} - 2379 q^{50} + 340 q^{51} + 3664 q^{52} + 992 q^{53} - 1202 q^{54} + 1731 q^{56} - 1016 q^{57} + 1358 q^{58} + 1442 q^{59} + 1444 q^{60} + 140 q^{61} - 464 q^{62} + 766 q^{63} + 8427 q^{64} + 1268 q^{65} + 5766 q^{67} + 3995 q^{68} + 2460 q^{69} + 2422 q^{70} + 2704 q^{71} - 5455 q^{72} - 4 q^{73} + 4008 q^{74} + 5204 q^{75} + 1935 q^{76} + 4092 q^{78} + 2180 q^{79} + 5040 q^{80} + 7192 q^{81} + 3197 q^{82} - 4200 q^{83} - 7951 q^{84} + 680 q^{85} + 3091 q^{86} + 752 q^{87} - 240 q^{89} + 4495 q^{90} + 5494 q^{91} + 6902 q^{92} + 6266 q^{93} - 5990 q^{94} - 3168 q^{95} + 9467 q^{96} + 5322 q^{97} + 4610 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.58278 1.54543 23.1674 18.5030 −8.62782 4.56305 −84.6764 −24.6116 −103.298
1.2 −5.53450 −6.28501 22.6307 −12.2792 34.7844 31.7914 −80.9739 12.5013 67.9590
1.3 −5.49302 −3.83066 22.1733 17.2968 21.0419 −17.8053 −77.8542 −12.3261 −95.0116
1.4 −5.36660 9.52348 20.8004 −15.8955 −51.1087 −21.2549 −68.6944 63.6966 85.3050
1.5 −4.78655 5.72685 14.9110 0.431714 −27.4118 −26.0630 −33.0800 5.79678 −2.06642
1.6 −4.66855 9.98201 13.7954 20.2462 −46.6016 10.9859 −27.0561 72.6405 −94.5205
1.7 −4.59808 −2.80460 13.1424 1.26573 12.8958 −24.2666 −23.6449 −19.1342 −5.81993
1.8 −4.56499 −9.42960 12.8392 6.53233 43.0460 20.4259 −22.0907 61.9173 −29.8200
1.9 −4.20383 8.49638 9.67219 12.9101 −35.7173 −6.90657 −7.02962 45.1885 −54.2720
1.10 −4.06732 0.0694426 8.54311 −17.2565 −0.282445 17.0158 −2.20900 −26.9952 70.1876
1.11 −3.65613 −8.25912 5.36730 3.13044 30.1964 14.4272 9.62551 41.2131 −11.4453
1.12 −3.43003 −1.96781 3.76513 6.18310 6.74964 26.3490 14.5258 −23.1277 −21.2082
1.13 −3.32824 −1.32929 3.07716 −9.90267 4.42418 −6.00243 16.3844 −25.2330 32.9584
1.14 −3.11011 7.45275 1.67277 −8.38904 −23.1788 34.4760 19.6784 28.5434 26.0908
1.15 −3.04850 −6.77034 1.29337 −1.02951 20.6394 −3.20599 20.4452 18.8374 3.13845
1.16 −2.97728 3.86255 0.864168 −21.2217 −11.4999 4.42148 21.2453 −12.0807 63.1828
1.17 −2.80404 −0.374518 −0.137383 21.1506 1.05016 −7.42020 22.8175 −26.8597 −59.3071
1.18 −2.36540 4.98259 −2.40490 −9.01519 −11.7858 1.33216 24.6117 −2.17379 21.3245
1.19 −2.16938 −6.51073 −3.29381 −10.9160 14.1242 −25.2888 24.5005 15.3896 23.6809
1.20 −1.69820 1.47404 −5.11610 3.94546 −2.50323 −10.0206 22.2738 −24.8272 −6.70020
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.52
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.u 52
11.b odd 2 1 2057.4.a.v 52
11.d odd 10 2 187.4.g.b 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.g.b 104 11.d odd 10 2
2057.4.a.u 52 1.a even 1 1 trivial
2057.4.a.v 52 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{52} + T_{2}^{51} - 325 T_{2}^{50} - 305 T_{2}^{49} + 49453 T_{2}^{48} + 43381 T_{2}^{47} - 4683173 T_{2}^{46} - 3821674 T_{2}^{45} + 309481915 T_{2}^{44} + 233567301 T_{2}^{43} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
\( T_{3}^{52} - 20 T_{3}^{51} - 788 T_{3}^{50} + 17710 T_{3}^{49} + 279137 T_{3}^{48} - 7303284 T_{3}^{47} - 57851124 T_{3}^{46} + 1863625082 T_{3}^{45} + 7516015457 T_{3}^{44} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
\( T_{5}^{52} - 40 T_{5}^{51} - 3379 T_{5}^{50} + 150766 T_{5}^{49} + 5137542 T_{5}^{48} - 263620290 T_{5}^{47} - 4608509171 T_{5}^{46} + 284409116428 T_{5}^{45} + 2664627365542 T_{5}^{44} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display