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Properties

Label 57.3.k.b
Level $57$
Weight $3$
Character orbit 57.k
Analytic conductor $1.553$
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] = [57,3,Mod(10,57)]
 
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, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 57.k (of order \(18\), 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: \(1.55313750685\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{2} - 3 q^{4} + 9 q^{6} - 9 q^{7} + 27 q^{8} - 6 q^{10} + 15 q^{11} - 108 q^{12} - 33 q^{13} + 33 q^{14} - 18 q^{15} - 3 q^{16} - 30 q^{17} - 15 q^{19} + 186 q^{20} + 18 q^{21} - 84 q^{22} - 21 q^{23} + 72 q^{24} + 30 q^{25} + 48 q^{26} + 108 q^{27} + 90 q^{28} - 90 q^{29} - 288 q^{31} - 417 q^{32} + 9 q^{33} + 75 q^{34} + 54 q^{35} + 9 q^{36} - 24 q^{38} + 18 q^{39} + 237 q^{40} - 6 q^{41} - 99 q^{42} - 141 q^{43} + 93 q^{44} - 9 q^{45} + 549 q^{46} + 615 q^{47} - 81 q^{49} + 135 q^{50} - 9 q^{51} - 339 q^{52} - 54 q^{53} - 27 q^{54} - 51 q^{55} + 99 q^{57} + 168 q^{58} + 18 q^{59} + 171 q^{60} - 129 q^{61} - 873 q^{62} - 99 q^{63} + 345 q^{64} - 189 q^{65} - 108 q^{66} + 111 q^{67} - 603 q^{68} - 396 q^{69} - 312 q^{70} - 144 q^{71} - 54 q^{72} + 408 q^{73} + 105 q^{74} + 318 q^{76} + 108 q^{77} + 207 q^{78} + 6 q^{79} - 1278 q^{80} - 795 q^{82} + 477 q^{83} + 837 q^{84} + 651 q^{85} + 633 q^{86} + 81 q^{87} - 504 q^{88} - 123 q^{89} - 99 q^{90} - 132 q^{91} + 1203 q^{92} + 198 q^{93} - 72 q^{95} - 126 q^{96} + 309 q^{97} + 339 q^{98} - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.75842 + 0.662712i −1.11334 − 1.32683i 9.92780 − 3.61342i −0.758325 − 0.276008i 5.06371 + 4.24896i −5.10471 + 8.84162i −21.6978 + 12.5272i −0.520945 + 2.95442i 3.03302 + 0.534803i
10.2 −1.68828 + 0.297689i −1.11334 − 1.32683i −0.997107 + 0.362917i 8.34658 + 3.03791i 2.27461 + 1.90863i 5.55292 − 9.61794i 7.51394 − 4.33818i −0.520945 + 2.95442i −14.9957 − 2.64415i
10.3 −0.637756 + 0.112454i −1.11334 − 1.32683i −3.36468 + 1.22464i −6.57652 − 2.39366i 0.859247 + 0.720994i −1.12773 + 1.95329i 4.25147 − 2.45459i −0.520945 + 2.95442i 4.46340 + 0.787017i
10.4 2.87872 − 0.507596i −1.11334 − 1.32683i 4.27061 − 1.55438i 1.04130 + 0.379004i −3.87849 − 3.25444i −1.13703 + 1.96939i 1.37889 − 0.796102i −0.520945 + 2.95442i 3.19000 + 0.562484i
13.1 −2.42904 − 2.89482i −0.592396 + 1.62760i −1.78514 + 10.1240i 0.487525 + 2.76489i 6.15055 − 2.23862i −3.07210 + 5.32104i 20.5529 − 11.8662i −2.29813 − 1.92836i 6.81964 − 8.12733i
13.2 −1.01678 − 1.21175i −0.592396 + 1.62760i 0.260091 − 1.47505i −0.649199 − 3.68179i 2.57458 − 0.937072i 6.08998 − 10.5482i −7.53148 + 4.34830i −2.29813 − 1.92836i −3.80133 + 4.53025i
13.3 0.924561 + 1.10185i −0.592396 + 1.62760i 0.335335 − 1.90178i 1.45131 + 8.23078i −2.34107 + 0.852080i 0.258464 − 0.447673i 7.38814 − 4.26554i −2.29813 − 1.92836i −7.72725 + 9.20898i
13.4 2.13460 + 2.54392i −0.592396 + 1.62760i −1.22041 + 6.92131i −0.870886 − 4.93904i −5.40501 + 1.96726i 2.46536 − 4.27012i −8.70859 + 5.02791i −2.29813 − 1.92836i 10.7055 − 12.7584i
22.1 −2.42904 + 2.89482i −0.592396 − 1.62760i −1.78514 − 10.1240i 0.487525 − 2.76489i 6.15055 + 2.23862i −3.07210 − 5.32104i 20.5529 + 11.8662i −2.29813 + 1.92836i 6.81964 + 8.12733i
22.2 −1.01678 + 1.21175i −0.592396 − 1.62760i 0.260091 + 1.47505i −0.649199 + 3.68179i 2.57458 + 0.937072i 6.08998 + 10.5482i −7.53148 − 4.34830i −2.29813 + 1.92836i −3.80133 − 4.53025i
22.3 0.924561 − 1.10185i −0.592396 − 1.62760i 0.335335 + 1.90178i 1.45131 − 8.23078i −2.34107 − 0.852080i 0.258464 + 0.447673i 7.38814 + 4.26554i −2.29813 + 1.92836i −7.72725 − 9.20898i
22.4 2.13460 − 2.54392i −0.592396 − 1.62760i −1.22041 − 6.92131i −0.870886 + 4.93904i −5.40501 − 1.96726i 2.46536 + 4.27012i −8.70859 − 5.02791i −2.29813 + 1.92836i 10.7055 + 12.7584i
34.1 −1.28897 + 3.54141i 1.70574 − 0.300767i −7.81596 − 6.55837i −6.09501 + 5.11432i −1.13350 + 6.42839i −2.17630 + 3.76946i 20.2453 − 11.6886i 2.81908 − 1.02606i −10.2556 − 28.1771i
34.2 −0.691603 + 1.90016i 1.70574 − 0.300767i −0.0681276 − 0.0571659i 6.34346 − 5.32279i −0.608185 + 3.44919i −5.23790 + 9.07231i −6.84906 + 3.95431i 2.81908 − 1.02606i 5.72702 + 15.7349i
34.3 0.0268890 − 0.0738770i 1.70574 − 0.300767i 3.05944 + 2.56718i −2.91128 + 2.44286i 0.0236458 − 0.134102i 2.84408 − 4.92609i 0.544262 − 0.314230i 2.81908 − 1.02606i 0.102189 + 0.280763i
34.4 1.04608 − 2.87407i 1.70574 − 0.300767i −4.10184 − 3.44186i 0.191054 − 0.160314i 0.919905 − 5.21704i −3.85502 + 6.67710i −3.58795 + 2.07151i 2.81908 − 1.02606i −0.260896 − 0.716805i
40.1 −3.75842 − 0.662712i −1.11334 + 1.32683i 9.92780 + 3.61342i −0.758325 + 0.276008i 5.06371 − 4.24896i −5.10471 − 8.84162i −21.6978 − 12.5272i −0.520945 − 2.95442i 3.03302 − 0.534803i
40.2 −1.68828 − 0.297689i −1.11334 + 1.32683i −0.997107 − 0.362917i 8.34658 − 3.03791i 2.27461 − 1.90863i 5.55292 + 9.61794i 7.51394 + 4.33818i −0.520945 − 2.95442i −14.9957 + 2.64415i
40.3 −0.637756 − 0.112454i −1.11334 + 1.32683i −3.36468 − 1.22464i −6.57652 + 2.39366i 0.859247 − 0.720994i −1.12773 − 1.95329i 4.25147 + 2.45459i −0.520945 − 2.95442i 4.46340 − 0.787017i
40.4 2.87872 + 0.507596i −1.11334 + 1.32683i 4.27061 + 1.55438i 1.04130 − 0.379004i −3.87849 + 3.25444i −1.13703 − 1.96939i 1.37889 + 0.796102i −0.520945 − 2.95442i 3.19000 − 0.562484i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.3.k.b ✓ 24
3.b odd 2 1 171.3.ba.d 24
19.f odd 18 1 inner 57.3.k.b ✓ 24
57.j even 18 1 171.3.ba.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.3.k.b ✓ 24 1.a even 1 1 trivial
57.3.k.b ✓ 24 19.f odd 18 1 inner
171.3.ba.d 24 3.b odd 2 1
171.3.ba.d 24 57.j even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 9 T_{2}^{23} + 42 T_{2}^{22} + 126 T_{2}^{21} + 273 T_{2}^{20} + 789 T_{2}^{19} + \cdots + 419904 \) acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display