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Properties

Label 2057.4.a.v
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

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 −5.59840 1.97577 23.3420 −6.71579 −11.0611 17.5990 −85.8908 −23.0963 37.5977
1.2 −5.30504 −5.67293 20.1434 9.03130 30.0951 −21.4564 −64.4212 5.18212 −47.9113
1.3 −5.25702 6.71119 19.6363 18.9900 −35.2809 10.5430 −61.1724 18.0401 −99.8311
1.4 −5.04342 −3.05493 17.4361 −21.2606 15.4073 −1.75197 −47.5901 −17.6674 107.226
1.5 −4.98791 7.45663 16.8792 −11.1529 −37.1930 −25.1067 −44.2887 28.6014 55.6297
1.6 −4.89586 9.65367 15.9694 6.25774 −47.2630 −15.8334 −39.0173 66.1934 −30.6370
1.7 −4.81699 1.93840 15.2034 15.7457 −9.33727 −16.8367 −34.6986 −23.2426 −75.8471
1.8 −4.28562 −5.69718 10.3665 −9.89047 24.4160 8.28929 −10.1420 5.45790 42.3868
1.9 −4.16110 −9.53915 9.31472 −1.68857 39.6933 −16.7115 −5.47068 63.9954 7.02630
1.10 −3.88649 0.0923746 7.10482 −3.54222 −0.359013 27.1367 3.47912 −26.9915 13.7668
1.11 −3.81604 −7.73467 6.56212 7.63552 29.5158 36.0008 5.48698 32.8251 −29.1374
1.12 −3.58990 0.612326 4.88735 1.26349 −2.19819 14.6037 11.1741 −26.6251 −4.53581
1.13 −3.47916 6.91663 4.10458 13.6530 −24.0641 −17.5663 13.5528 20.8398 −47.5009
1.14 −2.89521 −2.34508 0.382253 −9.69474 6.78951 −24.6549 22.0550 −21.5006 28.0683
1.15 −2.79893 −4.86297 −0.165993 17.2339 13.6111 −32.4229 22.8560 −3.35150 −48.2364
1.16 −2.58640 5.23582 −1.31053 −13.1678 −13.5419 30.6365 24.0808 0.413836 34.0571
1.17 −2.39242 6.08346 −2.27632 −5.38121 −14.5542 −28.7430 24.5853 10.0085 12.8741
1.18 −2.31426 9.47954 −2.64421 16.2575 −21.9381 32.2052 24.6334 62.8617 −37.6239
1.19 −1.83521 8.06986 −4.63199 −16.4652 −14.8099 −7.61543 23.1824 38.1227 30.2171
1.20 −1.80553 −7.06906 −4.74008 14.7486 12.7634 −11.2757 23.0025 22.9716 −26.6290
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.v 52
11.b odd 2 1 2057.4.a.u 52
11.c even 5 2 187.4.g.b ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
187.4.g.b ✓ 104 11.c even 5 2
2057.4.a.u 52 11.b odd 2 1
2057.4.a.v 52 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2057))\):

\( T_{2}^{52} - T_{2}^{51} - 325 T_{2}^{50} + 305 T_{2}^{49} + 49453 T_{2}^{48} - 43381 T_{2}^{47} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
\( T_{3}^{52} - 20 T_{3}^{51} - 788 T_{3}^{50} + 17710 T_{3}^{49} + 279137 T_{3}^{48} + \cdots + 20\!\cdots\!09 \) Copy content Toggle raw display
\( T_{5}^{52} - 40 T_{5}^{51} - 3379 T_{5}^{50} + 150766 T_{5}^{49} + 5137542 T_{5}^{48} + \cdots + 23\!\cdots\!80 \) Copy content Toggle raw display