# 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 $4$

# Related objects

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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 9 q^{3} - 18 q^{4} - 9 q^{6} - 6 q^{7} + 3 q^{9}+O(q^{10})$$ 24 * q - 9 * q^3 - 18 * q^4 - 9 * q^6 - 6 * q^7 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 9 q^{3} - 18 q^{4} - 9 q^{6} - 6 q^{7} + 3 q^{9} - 6 q^{10} + 9 q^{12} + 6 q^{13} - 9 q^{15} + 30 q^{16} - 12 q^{19} - 6 q^{21} + 24 q^{22} - 21 q^{24} + 12 q^{25} - 27 q^{27} - 48 q^{28} + 42 q^{30} - 18 q^{31} + 21 q^{33} - 42 q^{34} + 63 q^{36} + 30 q^{40} + 105 q^{42} + 54 q^{43} + 6 q^{45} - 54 q^{46} - 33 q^{48} + 6 q^{49} + 3 q^{51} - 48 q^{52} - 87 q^{54} - 90 q^{55} - 6 q^{57} + 24 q^{58} - 66 q^{60} - 6 q^{61} - 9 q^{63} + 18 q^{64} - 57 q^{66} - 36 q^{69} + 18 q^{70} + 24 q^{72} + 90 q^{73} + 12 q^{76} + 9 q^{78} + 30 q^{79} + 3 q^{81} + 126 q^{82} + 99 q^{84} - 6 q^{85} + 15 q^{87} + 54 q^{88} + 24 q^{90} + 66 q^{91} + 33 q^{93} - 18 q^{96} - 78 q^{97} + 12 q^{99}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$T_{2}^{24} + 9 T_{2}^{22} + 6 T_{2}^{20} - 14 T_{2}^{18} + 1053 T_{2}^{16} + 2646 T_{2}^{14} + 7629 T_{2}^{12} + 70641 T_{2}^{10} + 100512 T_{2}^{8} - 210569 T_{2}^{6} + 357876 T_{2}^{4} - 269166 T_{2}^{2} + \cdots + 157609$$ acting on $$S_{2}^{\mathrm{new}}(57, [\chi])$$.