Properties

Label 2057.4.a.t
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 / 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: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} - 24 q^{6} - 2 q^{7} - 63 q^{8} + 356 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} - 24 q^{6} - 2 q^{7} - 63 q^{8} + 356 q^{9} - 65 q^{10} - 337 q^{12} + 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} + 748 q^{17} - 55 q^{18} - 36 q^{19} - 209 q^{20} - 240 q^{21} - 656 q^{23} - 104 q^{24} + 458 q^{25} - 455 q^{26} - 538 q^{27} + 63 q^{28} - 698 q^{29} + 304 q^{30} - 1418 q^{31} - 83 q^{32} + 17 q^{34} + 102 q^{35} - 85 q^{36} - 1358 q^{37} - 568 q^{38} + 114 q^{39} - 281 q^{40} + 54 q^{41} - 1943 q^{42} + 382 q^{43} - 1226 q^{45} + 232 q^{46} - 544 q^{47} - 2743 q^{48} + 864 q^{49} + 1307 q^{50} - 476 q^{51} + 3112 q^{52} - 2492 q^{53} + 2400 q^{54} + 27 q^{56} + 56 q^{57} - 3746 q^{58} - 662 q^{59} - 1392 q^{60} - 964 q^{61} - 3256 q^{62} + 2322 q^{63} - 3653 q^{64} - 152 q^{65} - 3442 q^{67} + 2363 q^{68} - 1664 q^{69} - 3130 q^{70} - 2936 q^{71} - 1281 q^{72} - 2240 q^{73} + 3494 q^{74} - 1892 q^{75} + 1261 q^{76} + 1120 q^{78} - 1016 q^{79} - 3280 q^{80} + 2140 q^{81} - 3985 q^{82} - 1656 q^{83} - 5401 q^{84} - 408 q^{85} - 1373 q^{86} + 1968 q^{87} - 2992 q^{89} + 1287 q^{90} - 8310 q^{91} - 2442 q^{92} - 4058 q^{93} - 694 q^{94} - 8280 q^{95} + 2203 q^{96} - 4746 q^{97} + 2302 q^{98}+O(q^{100}) \) 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.

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 −5.35848 −7.08785 20.7133 2.40559 37.9801 1.79015 −68.1240 23.2377 −12.8903
1.2 −5.17591 1.82863 18.7901 −2.37867 −9.46484 16.1706 −55.8485 −23.6561 12.3118
1.3 −5.00754 −0.478835 17.0755 −10.2728 2.39778 −33.8426 −45.4457 −26.7707 51.4414
1.4 −4.74167 −8.40276 14.4835 −4.48116 39.8432 6.84168 −30.7424 43.6065 21.2482
1.5 −4.59510 −5.20799 13.1150 19.0975 23.9312 19.5522 −23.5038 0.123151 −87.7548
1.6 −4.48873 7.40887 12.1487 −14.9848 −33.2565 10.8452 −18.6226 27.8914 67.2626
1.7 −4.17775 8.39425 9.45359 −8.30638 −35.0691 10.0314 −6.07275 43.4635 34.7020
1.8 −4.15815 −2.77821 9.29018 17.2462 11.5522 −3.81447 −5.36476 −19.2815 −71.7123
1.9 −3.96050 4.65525 7.68556 8.14307 −18.4371 9.05024 1.24534 −5.32862 −32.2506
1.10 −3.71299 1.08053 5.78632 −1.92298 −4.01199 −27.0782 8.21937 −25.8325 7.14001
1.11 −2.84199 −5.34906 0.0768922 −17.1347 15.2020 7.83596 22.5174 1.61241 48.6966
1.12 −2.61578 −10.1873 −1.15767 −1.97479 26.6478 −29.0117 23.9545 76.7814 5.16562
1.13 −2.47148 4.16424 −1.89179 6.24963 −10.2918 −8.57958 24.4474 −9.65907 −15.4458
1.14 −2.27024 −5.58818 −2.84602 4.32678 12.6865 −4.76061 24.6230 4.22772 −9.82282
1.15 −1.97752 −8.52550 −4.08942 −7.64484 16.8593 34.0014 23.9071 45.6842 15.1178
1.16 −1.92546 7.26305 −4.29259 8.62699 −13.9847 −8.70813 23.6689 25.7519 −16.6110
1.17 −1.86890 3.47806 −4.50721 16.9463 −6.50016 27.1967 23.3747 −14.9031 −31.6709
1.18 −1.28604 5.35520 −6.34610 −17.4555 −6.88701 −12.0664 18.4497 1.67815 22.4485
1.19 −0.556409 0.761089 −7.69041 −17.6266 −0.423477 25.0433 8.73028 −26.4207 9.80761
1.20 −0.515458 −3.56912 −7.73430 2.09077 1.83973 17.6614 8.11038 −14.2614 −1.07771
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
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.t 44
11.b odd 2 1 2057.4.a.s 44
11.c even 5 2 187.4.g.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.g.a 88 11.c even 5 2
2057.4.a.s 44 11.b odd 2 1
2057.4.a.t 44 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2}^{44} - T_{2}^{43} - 245 T_{2}^{42} + 261 T_{2}^{41} + 27677 T_{2}^{40} - 31257 T_{2}^{39} - 1912633 T_{2}^{38} + 2280678 T_{2}^{37} + 90457195 T_{2}^{36} - 113516565 T_{2}^{35} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{44} + 28 T_{3}^{43} - 380 T_{3}^{42} - 17274 T_{3}^{41} + 16330 T_{3}^{40} + 4770576 T_{3}^{39} + 17790024 T_{3}^{38} - 776750576 T_{3}^{37} - 5171824726 T_{3}^{36} + \cdots + 87\!\cdots\!09 \) Copy content Toggle raw display
\( T_{5}^{44} + 24 T_{5}^{43} - 2691 T_{5}^{42} - 68106 T_{5}^{41} + 3224114 T_{5}^{40} + 87105226 T_{5}^{39} - 2265590643 T_{5}^{38} - 66465918588 T_{5}^{37} + 1034498025966 T_{5}^{36} + \cdots - 15\!\cdots\!20 \) Copy content Toggle raw display