Properties

 Label 380.2.be.b Level $380$ Weight $2$ Character orbit 380.be Analytic conductor $3.034$ Analytic rank $0$ Dimension $120$ CM no Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(51,380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 0, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("380.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.be (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: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 3 q^{2} - 3 q^{4} + 3 q^{6} - 36 q^{8} + 6 q^{9}+O(q^{10})$$ 120 * q + 3 * q^2 - 3 * q^4 + 3 * q^6 - 36 * q^8 + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$120 q + 3 q^{2} - 3 q^{4} + 3 q^{6} - 36 q^{8} + 6 q^{9} + 3 q^{10} - 12 q^{13} - 18 q^{14} + 9 q^{16} + 48 q^{17} - 12 q^{21} - 18 q^{24} - 24 q^{26} + 69 q^{28} - 12 q^{30} - 27 q^{32} + 6 q^{33} - 18 q^{34} - 72 q^{36} - 48 q^{38} - 36 q^{41} - 27 q^{42} - 99 q^{44} + 60 q^{45} + 27 q^{46} - 63 q^{48} + 60 q^{49} + 33 q^{52} - 24 q^{53} + 21 q^{54} - 48 q^{57} + 6 q^{60} - 24 q^{61} + 54 q^{62} + 84 q^{64} - 18 q^{65} - 132 q^{66} + 66 q^{68} - 72 q^{69} + 36 q^{70} - 42 q^{72} + 66 q^{74} + 180 q^{76} + 60 q^{77} + 114 q^{78} + 27 q^{80} - 66 q^{81} + 33 q^{82} - 144 q^{84} - 48 q^{85} + 75 q^{86} + 9 q^{88} - 3 q^{90} - 42 q^{92} - 144 q^{93} + 78 q^{96} + 42 q^{97} - 87 q^{98}+O(q^{100})$$ 120 * q + 3 * q^2 - 3 * q^4 + 3 * q^6 - 36 * q^8 + 6 * q^9 + 3 * q^10 - 12 * q^13 - 18 * q^14 + 9 * q^16 + 48 * q^17 - 12 * q^21 - 18 * q^24 - 24 * q^26 + 69 * q^28 - 12 * q^30 - 27 * q^32 + 6 * q^33 - 18 * q^34 - 72 * q^36 - 48 * q^38 - 36 * q^41 - 27 * q^42 - 99 * q^44 + 60 * q^45 + 27 * q^46 - 63 * q^48 + 60 * q^49 + 33 * q^52 - 24 * q^53 + 21 * q^54 - 48 * q^57 + 6 * q^60 - 24 * q^61 + 54 * q^62 + 84 * q^64 - 18 * q^65 - 132 * q^66 + 66 * q^68 - 72 * q^69 + 36 * q^70 - 42 * q^72 + 66 * q^74 + 180 * q^76 + 60 * q^77 + 114 * q^78 + 27 * q^80 - 66 * q^81 + 33 * q^82 - 144 * q^84 - 48 * q^85 + 75 * q^86 + 9 * q^88 - 3 * q^90 - 42 * q^92 - 144 * q^93 + 78 * q^96 + 42 * q^97 - 87 * q^98

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1 −1.41410 + 0.0175744i −3.08618 1.12328i 1.99938 0.0497042i −0.173648 0.984808i 4.38393 + 1.53420i 1.26066 + 0.727844i −2.82646 + 0.105425i 5.96464 + 5.00493i 0.262864 + 1.38957i
51.2 −1.38360 + 0.292675i 0.385053 + 0.140148i 1.82868 0.809888i −0.173648 0.984808i −0.573776 0.0812128i −0.799236 0.461439i −2.29313 + 1.65577i −2.16951 1.82043i 0.528488 + 1.31175i
51.3 −1.38069 0.306113i 2.40378 + 0.874903i 1.81259 + 0.845293i −0.173648 0.984808i −3.05104 1.94379i 3.65740 + 2.11160i −2.24386 1.72194i 2.71455 + 2.27778i −0.0617092 + 1.41287i
51.4 −1.06258 + 0.933229i 1.09042 + 0.396882i 0.258166 1.98327i −0.173648 0.984808i −1.52905 + 0.595896i 1.28762 + 0.743405i 1.57652 + 2.34831i −1.26662 1.06282i 1.10357 + 0.884387i
51.5 −1.00655 0.993404i −1.11886 0.407232i 0.0262989 + 1.99983i −0.173648 0.984808i 0.721647 + 1.52138i 0.611057 + 0.352794i 1.96016 2.03906i −1.21212 1.01709i −0.803525 + 1.16376i
51.6 −0.888043 + 1.10063i −2.01002 0.731586i −0.422760 1.95481i −0.173648 0.984808i 2.59019 1.56260i 0.167166 + 0.0965133i 2.52694 + 1.27065i 1.20682 + 1.01264i 1.23811 + 0.683429i
51.7 −0.803525 1.16376i 1.11886 + 0.407232i −0.708694 + 1.87023i −0.173648 0.984808i −0.425111 1.62931i −0.611057 0.352794i 2.74596 0.678023i −1.21212 1.01709i −1.00655 + 0.993404i
51.8 −0.339536 + 1.37285i 1.08575 + 0.395182i −1.76943 0.932264i −0.173648 0.984808i −0.911179 + 1.35640i −4.46754 2.57933i 1.88064 2.11262i −1.27544 1.07022i 1.41095 + 0.0959850i
51.9 −0.0617092 1.41287i −2.40378 0.874903i −1.99238 + 0.174374i −0.173648 0.984808i −1.08779 + 3.45020i −3.65740 2.11160i 0.369315 + 2.80421i 2.71455 + 2.27778i −1.38069 + 0.306113i
51.10 0.0889099 + 1.41142i −1.86185 0.677657i −1.98419 + 0.250978i −0.173648 0.984808i 0.790919 2.68809i 1.89621 + 1.09478i −0.530648 2.77820i 0.709122 + 0.595024i 1.37453 0.332649i
51.11 0.262864 1.38957i 3.08618 + 1.12328i −1.86180 0.730536i −0.173648 0.984808i 2.37212 3.99320i −1.26066 0.727844i −1.50453 + 2.39508i 5.96464 + 5.00493i −1.41410 0.0175744i
51.12 0.528488 1.31175i −0.385053 0.140148i −1.44140 1.38649i −0.173648 0.984808i −0.387335 + 0.431029i 0.799236 + 0.461439i −2.58050 + 1.15802i −2.16951 1.82043i −1.38360 0.292675i
51.13 0.727006 + 1.21304i 2.81895 + 1.02602i −0.942924 + 1.76377i −0.173648 0.984808i 0.804801 + 4.16542i 0.824757 + 0.476173i −2.82504 + 0.138470i 4.59567 + 3.85622i 1.06837 0.926603i
51.14 0.765254 + 1.18928i −0.700287 0.254884i −0.828773 + 1.82020i −0.173648 0.984808i −0.232769 1.02789i −3.13008 1.80715i −2.79895 + 0.407274i −1.87270 1.57138i 1.03833 0.960144i
51.15 1.03833 + 0.960144i 0.700287 + 0.254884i 0.156246 + 1.99389i −0.173648 0.984808i 0.482402 + 0.937029i 3.13008 + 1.80715i −1.75219 + 2.22033i −1.87270 1.57138i 0.765254 1.18928i
51.16 1.06837 + 0.926603i −2.81895 1.02602i 0.282813 + 1.97990i −0.173648 0.984808i −2.06097 3.70821i −0.824757 0.476173i −1.53244 + 2.37732i 4.59567 + 3.85622i 0.727006 1.21304i
51.17 1.10357 0.884387i −1.09042 0.396882i 0.435721 1.95196i −0.173648 0.984808i −1.55435 + 0.526371i −1.28762 0.743405i −1.24544 2.53946i −1.26662 1.06282i −1.06258 0.933229i
51.18 1.23811 0.683429i 2.01002 + 0.731586i 1.06585 1.69233i −0.173648 0.984808i 2.98862 0.467918i −0.167166 0.0965133i 0.163056 2.82372i 1.20682 + 1.01264i −0.888043 1.10063i
51.19 1.37453 + 0.332649i 1.86185 + 0.677657i 1.77869 + 0.914475i −0.173648 0.984808i 2.33375 + 1.55080i −1.89621 1.09478i 2.14067 + 1.84866i 0.709122 + 0.595024i 0.0889099 1.41142i
51.20 1.41095 0.0959850i −1.08575 0.395182i 1.98157 0.270861i −0.173648 0.984808i −1.56988 0.453367i 4.46754 + 2.57933i 2.76991 0.572373i −1.27544 1.07022i −0.339536 1.37285i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 51.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
19.f odd 18 1 inner
76.k even 18 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.be.b 120
4.b odd 2 1 inner 380.2.be.b 120
19.f odd 18 1 inner 380.2.be.b 120
76.k even 18 1 inner 380.2.be.b 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.be.b 120 1.a even 1 1 trivial
380.2.be.b 120 4.b odd 2 1 inner
380.2.be.b 120 19.f odd 18 1 inner
380.2.be.b 120 76.k even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{120} - 3 T_{3}^{118} + 48 T_{3}^{116} + 2456 T_{3}^{114} - 9435 T_{3}^{112} + 100191 T_{3}^{110} + 4248290 T_{3}^{108} - 13998669 T_{3}^{106} + 130308615 T_{3}^{104} + 3762907413 T_{3}^{102} + \cdots + 21447124258816$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.