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Properties

Label 57.4.i.a
Level $57$
Weight $4$
Character orbit 57.i
Analytic conductor $3.363$
Analytic rank $0$
Dimension $24$
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: \(24\)
Relative dimension: \(4\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} + 9 q^{4} - 12 q^{5} - 9 q^{6} + 36 q^{7} + 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 3 q^{2} + 9 q^{4} - 12 q^{5} - 9 q^{6} + 36 q^{7} + 57 q^{8} - 48 q^{10} + 24 q^{11} - 72 q^{12} + 102 q^{13} - 63 q^{14} - 36 q^{15} - 555 q^{16} + 282 q^{17} - 126 q^{19} + 318 q^{20} - 180 q^{21} - 36 q^{22} + 288 q^{23} - 234 q^{24} + 282 q^{25} + 384 q^{26} - 324 q^{27} + 282 q^{28} + 546 q^{29} + 180 q^{30} + 144 q^{31} + 219 q^{32} - 18 q^{33} - 75 q^{34} - 1794 q^{35} + 81 q^{36} - 648 q^{37} - 780 q^{38} - 252 q^{39} - 201 q^{40} + 174 q^{41} - 189 q^{42} - 1854 q^{43} + 2547 q^{44} + 378 q^{45} + 573 q^{46} - 1380 q^{47} + 1278 q^{48} + 936 q^{49} + 639 q^{50} - 396 q^{51} - 111 q^{52} + 978 q^{53} - 81 q^{54} - 1392 q^{55} + 642 q^{56} - 324 q^{57} + 1296 q^{58} + 342 q^{59} + 495 q^{60} + 474 q^{61} - 2739 q^{62} + 756 q^{63} + 3081 q^{64} - 456 q^{65} - 540 q^{66} + 2334 q^{67} + 2301 q^{68} + 342 q^{69} + 288 q^{70} + 2142 q^{71} - 702 q^{72} - 4158 q^{73} - 7077 q^{74} + 108 q^{75} + 174 q^{76} - 5508 q^{77} - 405 q^{78} + 4764 q^{79} - 4002 q^{80} - 927 q^{82} + 840 q^{83} + 315 q^{84} - 1008 q^{85} + 2811 q^{86} + 864 q^{87} - 1020 q^{88} + 4752 q^{89} + 783 q^{90} - 3312 q^{91} - 12423 q^{92} + 1296 q^{93} + 4662 q^{94} + 2460 q^{95} + 54 q^{96} + 3498 q^{97} + 5433 q^{98} - 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −3.31532 − 1.20668i 0.520945 + 2.95442i 3.40693 + 2.85876i −3.62980 + 3.04577i 1.83794 − 10.4235i 14.0936 − 24.4109i 6.26689 + 10.8546i −8.45723 + 3.07818i 15.7092 − 5.71769i
4.2 −2.61616 − 0.952206i 0.520945 + 2.95442i −0.190739 − 0.160049i 4.95536 − 4.15804i 1.45034 − 8.22530i −12.2245 + 21.1735i 11.4829 + 19.8889i −8.45723 + 3.07818i −16.9233 + 6.15959i
4.3 0.446300 + 0.162440i 0.520945 + 2.95442i −5.95556 − 4.99731i −15.7754 + 13.2372i −0.247418 + 1.40318i −2.31623 + 4.01183i −3.74597 − 6.48821i −8.45723 + 3.07818i −9.19081 + 3.34518i
4.4 4.04549 + 1.47244i 0.520945 + 2.95442i 8.06959 + 6.77119i 1.37796 − 1.15625i −2.24273 + 12.7192i 0.604600 − 1.04720i 5.45480 + 9.44800i −8.45723 + 3.07818i 7.27704 − 2.64863i
16.1 −3.72498 − 3.12563i −2.81908 + 1.02606i 2.71673 + 15.4073i −0.0491934 + 0.278990i 13.7081 + 4.98934i 6.77896 + 11.7415i 18.5874 − 32.1943i 6.89440 − 5.78509i 1.05526 − 0.885470i
16.2 −0.244204 − 0.204912i −2.81908 + 1.02606i −1.37154 − 7.77838i −2.93595 + 16.6506i 0.898683 + 0.327094i 15.4635 + 26.7835i −2.53409 + 4.38917i 6.89440 − 5.78509i 4.12887 − 3.46453i
16.3 0.725153 + 0.608476i −2.81908 + 1.02606i −1.23358 − 6.99599i 1.66580 − 9.44724i −2.66860 − 0.971290i −3.76790 − 6.52619i 7.14884 − 12.3821i 6.89440 − 5.78509i 6.95638 − 5.83710i
16.4 3.51007 + 2.94530i −2.81908 + 1.02606i 2.25663 + 12.7980i −1.23235 + 6.98902i −12.9172 − 4.70149i 1.86595 + 3.23192i −11.4447 + 19.8228i 6.89440 − 5.78509i −24.9104 + 20.9023i
25.1 −3.72498 + 3.12563i −2.81908 − 1.02606i 2.71673 − 15.4073i −0.0491934 − 0.278990i 13.7081 − 4.98934i 6.77896 − 11.7415i 18.5874 + 32.1943i 6.89440 + 5.78509i 1.05526 + 0.885470i
25.2 −0.244204 + 0.204912i −2.81908 − 1.02606i −1.37154 + 7.77838i −2.93595 − 16.6506i 0.898683 − 0.327094i 15.4635 − 26.7835i −2.53409 − 4.38917i 6.89440 + 5.78509i 4.12887 + 3.46453i
25.3 0.725153 − 0.608476i −2.81908 − 1.02606i −1.23358 + 6.99599i 1.66580 + 9.44724i −2.66860 + 0.971290i −3.76790 + 6.52619i 7.14884 + 12.3821i 6.89440 + 5.78509i 6.95638 + 5.83710i
25.4 3.51007 − 2.94530i −2.81908 − 1.02606i 2.25663 − 12.7980i −1.23235 − 6.98902i −12.9172 + 4.70149i 1.86595 − 3.23192i −11.4447 − 19.8228i 6.89440 + 5.78509i −24.9104 − 20.9023i
28.1 −0.683528 − 3.87648i 2.29813 − 1.92836i −7.04233 + 2.56320i −11.9714 − 4.35724i −9.04609 − 7.59057i 1.49885 − 2.59608i −0.995306 − 1.72392i 1.56283 − 8.86327i −8.70796 + 49.3853i
28.2 −0.417212 − 2.36613i 2.29813 − 1.92836i 2.09305 − 0.761807i 15.8726 + 5.77717i −5.52156 − 4.63314i −5.33281 + 9.23670i −12.2863 − 21.2805i 1.56283 − 8.86327i 7.04726 − 39.9670i
28.3 0.135759 + 0.769928i 2.29813 − 1.92836i 6.94318 − 2.52711i −4.76970 − 1.73603i 1.79669 + 1.50761i 7.97794 − 13.8182i 6.01552 + 10.4192i 1.56283 − 8.86327i 0.689087 − 3.90801i
28.4 0.638629 + 3.62184i 2.29813 − 1.92836i −5.19236 + 1.88987i 10.4921 + 3.81881i 8.45188 + 7.09197i −6.64192 + 11.5041i 4.55008 + 7.88096i 1.56283 − 8.86327i −7.13058 + 40.4395i
43.1 −3.31532 + 1.20668i 0.520945 − 2.95442i 3.40693 − 2.85876i −3.62980 − 3.04577i 1.83794 + 10.4235i 14.0936 + 24.4109i 6.26689 − 10.8546i −8.45723 − 3.07818i 15.7092 + 5.71769i
43.2 −2.61616 + 0.952206i 0.520945 − 2.95442i −0.190739 + 0.160049i 4.95536 + 4.15804i 1.45034 + 8.22530i −12.2245 − 21.1735i 11.4829 − 19.8889i −8.45723 − 3.07818i −16.9233 − 6.15959i
43.3 0.446300 − 0.162440i 0.520945 − 2.95442i −5.95556 + 4.99731i −15.7754 − 13.2372i −0.247418 − 1.40318i −2.31623 − 4.01183i −3.74597 + 6.48821i −8.45723 − 3.07818i −9.19081 − 3.34518i
43.4 4.04549 − 1.47244i 0.520945 − 2.95442i 8.06959 − 6.77119i 1.37796 + 1.15625i −2.24273 − 12.7192i 0.604600 + 1.04720i 5.45480 − 9.44800i −8.45723 − 3.07818i 7.27704 + 2.64863i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.4
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.a ✓ 24
3.b odd 2 1 171.4.u.a 24
19.e even 9 1 inner 57.4.i.a ✓ 24
19.e even 9 1 1083.4.a.p 12
19.f odd 18 1 1083.4.a.o 12
57.l odd 18 1 171.4.u.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.4.i.a ✓ 24 1.a even 1 1 trivial
57.4.i.a ✓ 24 19.e even 9 1 inner
171.4.u.a 24 3.b odd 2 1
171.4.u.a 24 57.l odd 18 1
1083.4.a.o 12 19.f odd 18 1
1083.4.a.p 12 19.e even 9 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 3 T_{2}^{23} - 28 T_{2}^{21} + 183 T_{2}^{20} + 651 T_{2}^{19} + 6334 T_{2}^{18} + \cdots + 13483584 \) acting on \(S_{4}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display