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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q + 3 q^{2} - 3 q^{4} + 3 q^{6} - 36 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q + 3 q^{2} - 3 q^{4} + 3 q^{6} - 36 q^{8} + 6 q^{9} + 3 q^{10} - 12 q^{13} - 18 q^{14} + 9 q^{16} + 48 q^{17} - 12 q^{21} - 18 q^{24} - 24 q^{26} + 69 q^{28} - 12 q^{30} - 27 q^{32} + 6 q^{33} - 18 q^{34} - 72 q^{36} - 48 q^{38} - 36 q^{41} - 27 q^{42} - 99 q^{44} + 60 q^{45} + 27 q^{46} - 63 q^{48} + 60 q^{49} + 33 q^{52} - 24 q^{53} + 21 q^{54} - 48 q^{57} + 6 q^{60} - 24 q^{61} + 54 q^{62} + 84 q^{64} - 18 q^{65} - 132 q^{66} + 66 q^{68} - 72 q^{69} + 36 q^{70} - 42 q^{72} + 66 q^{74} + 180 q^{76} + 60 q^{77} + 114 q^{78} + 27 q^{80} - 66 q^{81} + 33 q^{82} - 144 q^{84} - 48 q^{85} + 75 q^{86} + 9 q^{88} - 3 q^{90} - 42 q^{92} - 144 q^{93} + 78 q^{96} + 42 q^{97} - 87 q^{98}+O(q^{100}) \) 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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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:

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 \(S_{2}^{\mathrm{new}}(380, [\chi])\). Copy content Toggle raw display