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Properties

Label 57.2.j.b
Level $57$
Weight $2$
Character orbit 57.j
Analytic conductor $0.455$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [57,2,Mod(2,57)]
 
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([9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("57.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 57 = 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 57.j (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: \(0.455147291521\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{3} - 18 q^{4} - 9 q^{6} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 9 q^{3} - 18 q^{4} - 9 q^{6} - 6 q^{7} + 3 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} - 9 q^{15} + 30 q^{16} - 12 q^{19} - 6 q^{21} + 24 q^{22} - 21 q^{24} + 12 q^{25} - 27 q^{27} - 48 q^{28} + 42 q^{30} - 18 q^{31} + 21 q^{33} - 42 q^{34} + 63 q^{36} + 30 q^{40} + 105 q^{42} + 54 q^{43} + 6 q^{45} - 54 q^{46} - 33 q^{48} + 6 q^{49} + 3 q^{51} - 48 q^{52} - 87 q^{54} - 90 q^{55} - 6 q^{57} + 24 q^{58} - 66 q^{60} - 6 q^{61} - 9 q^{63} + 18 q^{64} - 57 q^{66} - 36 q^{69} + 18 q^{70} + 24 q^{72} + 90 q^{73} + 12 q^{76} + 9 q^{78} + 30 q^{79} + 3 q^{81} + 126 q^{82} + 99 q^{84} - 6 q^{85} + 15 q^{87} + 54 q^{88} + 24 q^{90} + 66 q^{91} + 33 q^{93} - 18 q^{96} - 78 q^{97} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.448036 + 2.54094i −1.71942 − 0.208799i −4.37626 − 1.59283i 0.533860 + 1.46677i 1.30091 − 4.27539i 1.49687 + 2.59266i 3.42787 − 5.93724i 2.91281 + 0.718027i −3.96616 + 0.699341i
2.2 −0.336464 + 1.90818i 1.43743 + 0.966337i −1.64856 − 0.600025i −1.09544 − 3.00968i −2.32759 + 2.41773i −1.23083 − 2.13186i −0.237981 + 0.412196i 1.13238 + 2.77808i 6.11159 − 1.07764i
2.3 0.336464 − 1.90818i −1.20126 − 1.24779i −1.64856 − 0.600025i 1.09544 + 3.00968i −2.78518 + 1.87239i −1.23083 − 2.13186i 0.237981 − 0.412196i −0.113935 + 2.99784i 6.11159 − 1.07764i
2.4 0.448036 − 2.54094i 0.504201 + 1.65704i −4.37626 − 1.59283i −0.533860 − 1.46677i 4.43634 − 0.538732i 1.49687 + 2.59266i −3.42787 + 5.93724i −2.49156 + 1.67096i −3.96616 + 0.699341i
14.1 −2.08555 + 0.759077i −1.47284 − 0.911455i 2.24122 − 1.88060i 1.78018 − 2.12154i 3.76353 + 0.782886i −1.70913 − 2.96030i −1.02725 + 1.77924i 1.33850 + 2.68485i −2.10224 + 5.77587i
14.2 −0.886259 + 0.322572i −0.858393 + 1.50438i −0.850687 + 0.713811i −0.485824 + 0.578982i 0.275488 − 1.61016i 1.38278 + 2.39504i 1.46681 − 2.54059i −1.52632 − 2.58270i 0.243802 − 0.669841i
14.3 0.886259 − 0.322572i −0.292097 − 1.70724i −0.850687 + 0.713811i 0.485824 − 0.578982i −0.809582 − 1.41884i 1.38278 + 2.39504i −1.46681 + 2.54059i −2.82936 + 0.997362i 0.243802 − 0.669841i
14.4 2.08555 − 0.759077i −1.69575 + 0.352748i 2.24122 − 1.88060i −1.78018 + 2.12154i −3.26880 + 2.02288i −1.70913 − 2.96030i 1.02725 − 1.77924i 2.75114 − 1.19634i −2.10224 + 5.77587i
29.1 −0.448036 − 2.54094i −1.71942 + 0.208799i −4.37626 + 1.59283i 0.533860 − 1.46677i 1.30091 + 4.27539i 1.49687 − 2.59266i 3.42787 + 5.93724i 2.91281 − 0.718027i −3.96616 − 0.699341i
29.2 −0.336464 − 1.90818i 1.43743 − 0.966337i −1.64856 + 0.600025i −1.09544 + 3.00968i −2.32759 − 2.41773i −1.23083 + 2.13186i −0.237981 − 0.412196i 1.13238 − 2.77808i 6.11159 + 1.07764i
29.3 0.336464 + 1.90818i −1.20126 + 1.24779i −1.64856 + 0.600025i 1.09544 − 3.00968i −2.78518 − 1.87239i −1.23083 + 2.13186i 0.237981 + 0.412196i −0.113935 − 2.99784i 6.11159 + 1.07764i
29.4 0.448036 + 2.54094i 0.504201 − 1.65704i −4.37626 + 1.59283i −0.533860 + 1.46677i 4.43634 + 0.538732i 1.49687 − 2.59266i −3.42787 − 5.93724i −2.49156 − 1.67096i −3.96616 − 0.699341i
32.1 −1.49833 + 1.25725i 1.40671 + 1.01052i 0.317026 − 1.79794i −0.487091 + 0.0858872i −3.37821 + 0.254491i −0.969730 + 1.67962i −0.170480 − 0.295279i 0.957685 + 2.84303i 0.621842 − 0.741083i
32.2 −0.745719 + 0.625733i 0.0227926 − 1.73190i −0.182741 + 1.03637i 3.79113 − 0.668479i 1.06671 + 1.30577i −0.469963 + 0.814000i −1.48569 − 2.57329i −2.99896 − 0.0789491i −2.40883 + 2.87073i
32.3 0.745719 − 0.625733i 1.09578 − 1.34136i −0.182741 + 1.03637i −3.79113 + 0.668479i −0.0221879 − 1.68595i −0.469963 + 0.814000i 1.48569 + 2.57329i −0.598514 − 2.93969i −2.40883 + 2.87073i
32.4 1.49833 − 1.25725i −1.72716 − 0.130112i 0.317026 − 1.79794i 0.487091 − 0.0858872i −2.75144 + 1.97652i −0.969730 + 1.67962i 0.170480 + 0.295279i 2.96614 + 0.449448i 0.621842 − 0.741083i
41.1 −1.49833 − 1.25725i 1.40671 − 1.01052i 0.317026 + 1.79794i −0.487091 − 0.0858872i −3.37821 − 0.254491i −0.969730 − 1.67962i −0.170480 + 0.295279i 0.957685 − 2.84303i 0.621842 + 0.741083i
41.2 −0.745719 − 0.625733i 0.0227926 + 1.73190i −0.182741 − 1.03637i 3.79113 + 0.668479i 1.06671 − 1.30577i −0.469963 − 0.814000i −1.48569 + 2.57329i −2.99896 + 0.0789491i −2.40883 − 2.87073i
41.3 0.745719 + 0.625733i 1.09578 + 1.34136i −0.182741 − 1.03637i −3.79113 − 0.668479i −0.0221879 + 1.68595i −0.469963 − 0.814000i 1.48569 − 2.57329i −0.598514 + 2.93969i −2.40883 − 2.87073i
41.4 1.49833 + 1.25725i −1.72716 + 0.130112i 0.317026 + 1.79794i 0.487091 + 0.0858872i −2.75144 − 1.97652i −0.969730 − 1.67962i 0.170480 − 0.295279i 2.96614 − 0.449448i 0.621842 + 0.741083i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 57.2.j.b ✓ 24
3.b odd 2 1 inner 57.2.j.b ✓ 24
4.b odd 2 1 912.2.cc.e 24
12.b even 2 1 912.2.cc.e 24
19.e even 9 1 1083.2.d.d 24
19.f odd 18 1 inner 57.2.j.b ✓ 24
19.f odd 18 1 1083.2.d.d 24
57.j even 18 1 inner 57.2.j.b ✓ 24
57.j even 18 1 1083.2.d.d 24
57.l odd 18 1 1083.2.d.d 24
76.k even 18 1 912.2.cc.e 24
228.u odd 18 1 912.2.cc.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.j.b ✓ 24 1.a even 1 1 trivial
57.2.j.b ✓ 24 3.b odd 2 1 inner
57.2.j.b ✓ 24 19.f odd 18 1 inner
57.2.j.b ✓ 24 57.j even 18 1 inner
912.2.cc.e 24 4.b odd 2 1
912.2.cc.e 24 12.b even 2 1
912.2.cc.e 24 76.k even 18 1
912.2.cc.e 24 228.u odd 18 1
1083.2.d.d 24 19.e even 9 1
1083.2.d.d 24 19.f odd 18 1
1083.2.d.d 24 57.j even 18 1
1083.2.d.d 24 57.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 9 T_{2}^{22} + 6 T_{2}^{20} - 14 T_{2}^{18} + 1053 T_{2}^{16} + 2646 T_{2}^{14} + \cdots + 157609 \) acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\). Copy content Toggle raw display