Properties

Label 57.4.i.b
Level $57$
Weight $4$
Character orbit 57.i
Analytic conductor $3.363$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,4,Mod(4,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 57.i (of order \(9\), degree \(6\), minimal)

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: no
Analytic conductor: \(3.36310887033\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 9 q^{4} + 12 q^{5} - 9 q^{6} - 48 q^{7} - 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 3 q^{2} + 9 q^{4} + 12 q^{5} - 9 q^{6} - 48 q^{7} - 57 q^{8} - 24 q^{10} - 108 q^{11} + 288 q^{12} - 24 q^{13} - 87 q^{14} - 36 q^{15} + 69 q^{16} - 462 q^{17} - 336 q^{19} + 54 q^{20} - 198 q^{21} + 84 q^{22} + 522 q^{24} + 306 q^{25} + 72 q^{26} + 486 q^{27} + 1938 q^{28} + 342 q^{29} - 180 q^{30} + 1032 q^{31} - 141 q^{32} - 270 q^{33} - 867 q^{34} - 642 q^{35} + 81 q^{36} + 264 q^{37} - 4464 q^{38} + 252 q^{39} - 4209 q^{40} + 558 q^{41} + 261 q^{42} + 1344 q^{43} - 1239 q^{44} - 162 q^{45} + 2229 q^{46} + 2628 q^{47} - 342 q^{48} - 1122 q^{49} + 1503 q^{50} - 720 q^{51} - 2463 q^{52} + 1722 q^{53} - 81 q^{54} + 1860 q^{55} + 2238 q^{56} - 720 q^{57} + 1512 q^{58} - 1986 q^{59} + 2061 q^{60} + 1566 q^{61} + 7287 q^{62} - 216 q^{63} - 2679 q^{64} + 1716 q^{65} + 1260 q^{66} - 1044 q^{67} - 4623 q^{68} + 522 q^{69} + 60 q^{70} - 5874 q^{71} - 1566 q^{72} + 3024 q^{73} - 723 q^{74} - 6408 q^{75} - 6942 q^{76} + 2028 q^{77} - 2835 q^{78} - 3696 q^{79} + 8076 q^{80} + 3597 q^{82} - 4764 q^{83} + 2601 q^{84} + 3300 q^{85} - 627 q^{86} + 504 q^{87} + 3012 q^{88} + 3228 q^{89} + 999 q^{90} - 1272 q^{91} + 18183 q^{92} + 3492 q^{93} - 16410 q^{94} - 3780 q^{95} - 5526 q^{96} - 1230 q^{97} - 19761 q^{98} + 810 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −4.48747 1.63331i −0.520945 2.95442i 11.3414 + 9.51654i 13.4789 11.3101i −2.48775 + 14.1088i 2.15951 3.74039i −16.2488 28.1437i −8.45723 + 3.07818i −78.9591 + 28.7388i
4.2 −3.29747 1.20018i −0.520945 2.95442i 3.30450 + 2.77281i −11.1815 + 9.38240i −1.82804 + 10.3673i 8.00421 13.8637i 6.46774 + 11.2025i −8.45723 + 3.07818i 48.1312 17.5183i
4.3 −0.465112 0.169287i −0.520945 2.95442i −5.94068 4.98483i −5.05805 + 4.24421i −0.257847 + 1.46233i −15.4910 + 26.8311i 3.89906 + 6.75337i −8.45723 + 3.07818i 3.07105 1.11777i
4.4 1.20418 + 0.438285i −0.520945 2.95442i −4.87040 4.08675i 1.07347 0.900745i 0.667569 3.78597i 13.8796 24.0401i −9.19951 15.9340i −8.45723 + 3.07818i 1.68743 0.614173i
4.5 3.52088 + 1.28150i −0.520945 2.95442i 4.62603 + 3.88170i 16.1205 13.5267i 1.95190 11.0698i −13.6365 + 23.6191i −3.67406 6.36365i −8.45723 + 3.07818i 74.0930 26.9677i
4.6 4.96468 + 1.80700i −0.520945 2.95442i 15.2545 + 12.8000i −7.48978 + 6.28467i 2.75231 15.6091i 8.01316 13.8792i 31.4708 + 54.5091i −8.45723 + 3.07818i −48.5407 + 17.6674i
16.1 −3.78921 3.17953i 2.81908 1.02606i 2.85956 + 16.2174i 3.37491 19.1401i −13.9445 5.07537i −4.62456 8.00998i 20.9422 36.2730i 6.89440 5.78509i −73.6446 + 61.7951i
16.2 −3.15086 2.64388i 2.81908 1.02606i 1.54860 + 8.78252i −3.48122 + 19.7430i −11.5953 4.22034i 2.91460 + 5.04824i 1.88795 3.27003i 6.89440 5.78509i 63.1670 53.0034i
16.3 −1.08875 0.913570i 2.81908 1.02606i −1.03842 5.88916i 0.0582200 0.330182i −4.00665 1.45830i −7.44302 12.8917i −9.93464 + 17.2073i 6.89440 5.78509i −0.365031 + 0.306298i
16.4 1.05960 + 0.889114i 2.81908 1.02606i −1.05695 5.99424i 1.90436 10.8002i 3.89939 + 1.41926i 1.68783 + 2.92340i 9.74248 16.8745i 6.89440 5.78509i 11.6205 9.75073i
16.5 2.57889 + 2.16395i 2.81908 1.02606i 0.578834 + 3.28273i −1.62337 + 9.20660i 9.49045 + 3.45424i 10.2044 + 17.6746i 7.85512 13.6055i 6.89440 5.78509i −24.1091 + 20.2299i
16.6 4.12428 + 3.46068i 2.81908 1.02606i 3.64417 + 20.6671i 0.929607 5.27206i 15.1775 + 5.52417i −18.0524 31.2677i −34.9573 + 60.5478i 6.89440 5.78509i 22.0789 18.5264i
25.1 −3.78921 + 3.17953i 2.81908 + 1.02606i 2.85956 16.2174i 3.37491 + 19.1401i −13.9445 + 5.07537i −4.62456 + 8.00998i 20.9422 + 36.2730i 6.89440 + 5.78509i −73.6446 61.7951i
25.2 −3.15086 + 2.64388i 2.81908 + 1.02606i 1.54860 8.78252i −3.48122 19.7430i −11.5953 + 4.22034i 2.91460 5.04824i 1.88795 + 3.27003i 6.89440 + 5.78509i 63.1670 + 53.0034i
25.3 −1.08875 + 0.913570i 2.81908 + 1.02606i −1.03842 + 5.88916i 0.0582200 + 0.330182i −4.00665 + 1.45830i −7.44302 + 12.8917i −9.93464 17.2073i 6.89440 + 5.78509i −0.365031 0.306298i
25.4 1.05960 0.889114i 2.81908 + 1.02606i −1.05695 + 5.99424i 1.90436 + 10.8002i 3.89939 1.41926i 1.68783 2.92340i 9.74248 + 16.8745i 6.89440 + 5.78509i 11.6205 + 9.75073i
25.5 2.57889 2.16395i 2.81908 + 1.02606i 0.578834 3.28273i −1.62337 9.20660i 9.49045 3.45424i 10.2044 17.6746i 7.85512 + 13.6055i 6.89440 + 5.78509i −24.1091 20.2299i
25.6 4.12428 3.46068i 2.81908 + 1.02606i 3.64417 20.6671i 0.929607 + 5.27206i 15.1775 5.52417i −18.0524 + 31.2677i −34.9573 60.5478i 6.89440 + 5.78509i 22.0789 + 18.5264i
28.1 −0.810730 4.59788i −2.29813 + 1.92836i −12.9657 + 4.71912i −4.59346 1.67188i 10.7295 + 9.00316i −13.4140 + 23.2337i 13.5344 + 23.4422i 1.56283 8.86327i −3.96306 + 22.4756i
28.2 −0.443311 2.51414i −2.29813 + 1.92836i 1.39316 0.507068i 12.9233 + 4.70368i 5.86696 + 4.92297i 7.81757 13.5404i −12.1041 20.9650i 1.56283 8.86327i 6.09670 34.5761i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.4.i.b 36
3.b odd 2 1 171.4.u.c 36
19.e even 9 1 inner 57.4.i.b 36
19.e even 9 1 1083.4.a.s 18
19.f odd 18 1 1083.4.a.t 18
57.l odd 18 1 171.4.u.c 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.4.i.b 36 1.a even 1 1 trivial
57.4.i.b 36 19.e even 9 1 inner
171.4.u.c 36 3.b odd 2 1
171.4.u.c 36 57.l odd 18 1
1083.4.a.s 18 19.e even 9 1
1083.4.a.t 18 19.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 3 T_{2}^{35} + 28 T_{2}^{33} + 27 T_{2}^{32} - 1005 T_{2}^{31} + 23712 T_{2}^{30} - 15201 T_{2}^{29} - 216288 T_{2}^{28} - 201152 T_{2}^{27} + 3705891 T_{2}^{26} + 13346883 T_{2}^{25} + \cdots + 24\!\cdots\!16 \) acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display