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Properties

Label 2057.4.a.q
Level $2057$
Weight $4$
Character orbit 2057.a
Self dual yes
Analytic conductor $121.367$
Analytic rank $1$
Dimension $40$
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: \(40\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{2} - 8 q^{3} + 160 q^{4} - 28 q^{5} + 86 q^{6} - 160 q^{7} - 96 q^{8} + 300 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 8 q^{2} - 8 q^{3} + 160 q^{4} - 28 q^{5} + 86 q^{6} - 160 q^{7} - 96 q^{8} + 300 q^{9} - 120 q^{10} - 158 q^{12} - 224 q^{13} + 42 q^{14} - 164 q^{15} + 640 q^{16} + 680 q^{17} - 332 q^{18} - 500 q^{19} - 696 q^{20} - 168 q^{21} - 508 q^{23} + 830 q^{24} + 1208 q^{25} + 370 q^{26} - 752 q^{27} - 1704 q^{28} - 228 q^{29} - 1088 q^{30} - 160 q^{31} - 896 q^{32} - 136 q^{34} + 316 q^{35} + 2202 q^{36} - 24 q^{37} + 918 q^{38} - 2280 q^{39} - 1926 q^{40} - 876 q^{41} - 390 q^{42} - 1624 q^{43} + 476 q^{45} - 86 q^{46} - 1144 q^{47} + 1422 q^{48} + 2656 q^{49} - 3004 q^{50} - 136 q^{51} - 5580 q^{52} + 992 q^{53} + 2106 q^{54} + 298 q^{56} - 92 q^{57} + 198 q^{58} - 1484 q^{59} - 1726 q^{60} - 1800 q^{61} - 1376 q^{62} - 4800 q^{63} + 2800 q^{64} - 160 q^{65} + 508 q^{67} + 2720 q^{68} - 476 q^{69} + 4156 q^{70} - 1756 q^{71} - 6296 q^{72} - 2352 q^{73} - 2164 q^{74} - 1824 q^{75} - 2004 q^{76} - 8132 q^{78} - 6596 q^{79} - 1596 q^{80} - 352 q^{81} - 2510 q^{82} - 2408 q^{83} - 1048 q^{84} - 476 q^{85} - 4202 q^{86} - 7304 q^{87} + 2680 q^{89} - 3200 q^{90} + 1752 q^{91} - 5444 q^{92} + 1632 q^{93} - 4540 q^{94} - 5140 q^{95} + 3976 q^{96} + 464 q^{97} - 4208 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −5.52892 −7.62072 22.5689 −9.38827 42.1343 −16.9089 −80.5503 31.0754 51.9070
1.2 −5.40646 6.51605 21.2298 4.62429 −35.2287 −18.7314 −71.5265 15.4588 −25.0011
1.3 −5.08087 6.43870 17.8152 11.1510 −32.7142 −12.9783 −49.8697 14.4569 −56.6569
1.4 −5.00372 −3.83710 17.0372 12.7381 19.1998 33.2461 −45.2198 −12.2766 −63.7379
1.5 −4.81550 −9.01043 15.1891 20.3997 43.3897 −31.0740 −34.6191 54.1878 −98.2348
1.6 −4.62193 2.36708 13.3622 −16.6687 −10.9405 17.6898 −24.7839 −21.3969 77.0415
1.7 −4.31062 −10.0149 10.5815 −19.6115 43.1704 1.49161 −11.1278 73.2978 84.5379
1.8 −4.03325 −3.27417 8.26712 −2.56411 13.2055 −6.65820 −1.07735 −16.2798 10.3417
1.9 −3.98017 −5.08907 7.84178 −11.1514 20.2554 −1.98007 0.629735 −1.10132 44.3843
1.10 −3.84417 4.28144 6.77767 −22.1116 −16.4586 −31.3691 4.69884 −8.66926 85.0009
1.11 −2.99693 8.83709 0.981575 10.3088 −26.4841 −10.0625 21.0337 51.0942 −30.8947
1.12 −2.98748 −1.95809 0.925017 14.5596 5.84975 −34.5534 21.1363 −23.1659 −43.4965
1.13 −2.68044 4.09990 −0.815218 −7.65180 −10.9896 24.0333 23.6287 −10.1908 20.5102
1.14 −2.20372 1.96488 −3.14361 −3.19958 −4.33005 28.4385 24.5574 −23.1392 7.05099
1.15 −1.87772 −4.42431 −4.47418 10.9725 8.30759 12.7220 23.4230 −7.42551 −20.6032
1.16 −1.53655 −3.76749 −5.63902 −12.6530 5.78892 −31.2884 20.9570 −12.8060 19.4420
1.17 −1.40365 −7.02013 −6.02976 16.2316 9.85382 7.75626 19.6929 22.2822 −22.7836
1.18 −0.666697 1.81210 −7.55552 −7.73107 −1.20812 −1.06303 10.3708 −23.7163 5.15428
1.19 −0.482990 8.47555 −7.76672 14.1895 −4.09361 −6.62010 7.61517 44.8350 −6.85337
1.20 −0.477789 −9.58379 −7.77172 0.919320 4.57903 24.3280 7.53556 64.8490 −0.439241
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
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.q ✓ 40
11.b odd 2 1 2057.4.a.r yes 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2057.4.a.q ✓ 40 1.a even 1 1 trivial
2057.4.a.r yes 40 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{40} + 8 T_{2}^{39} - 208 T_{2}^{38} - 1760 T_{2}^{37} + 19488 T_{2}^{36} + 176768 T_{2}^{35} + \cdots - 763300533878784 \) Copy content Toggle raw display
\( T_{3}^{40} + 8 T_{3}^{39} - 658 T_{3}^{38} - 5040 T_{3}^{37} + 198731 T_{3}^{36} + 1447740 T_{3}^{35} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
\( T_{5}^{40} + 28 T_{5}^{39} - 2712 T_{5}^{38} - 77260 T_{5}^{37} + 3349057 T_{5}^{36} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display