# Properties

 Label 380.2.n.a Level $380$ Weight $2$ Character orbit 380.n Analytic conductor $3.034$ Analytic rank $0$ Dimension $40$ 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(31,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("380.31");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.n (of order $$6$$, degree $$2$$, 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: $$40$$ Relative dimension: $$20$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 3 q^{2} + q^{4} + 20 q^{5} + 3 q^{6} - 22 q^{9}+O(q^{10})$$ 40 * q - 3 * q^2 + q^4 + 20 * q^5 + 3 * q^6 - 22 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$40 q - 3 q^{2} + q^{4} + 20 q^{5} + 3 q^{6} - 22 q^{9} - 3 q^{10} + 12 q^{13} + 18 q^{14} - 7 q^{16} + 4 q^{17} + 2 q^{20} + 12 q^{21} - 8 q^{24} - 20 q^{25} + 2 q^{26} + 8 q^{28} + 6 q^{30} - 18 q^{32} - 6 q^{33} - 27 q^{34} - 14 q^{36} + 38 q^{38} + 36 q^{41} - 21 q^{42} - 8 q^{44} - 44 q^{45} - 18 q^{48} - 60 q^{49} - 33 q^{52} + 42 q^{53} + 9 q^{54} + 12 q^{57} - 62 q^{58} + 3 q^{60} + 12 q^{61} - 23 q^{62} + 64 q^{64} + 2 q^{66} + 72 q^{68} + 18 q^{70} + 42 q^{72} - 18 q^{73} + 6 q^{74} - 62 q^{76} - 28 q^{77} - 24 q^{78} + 7 q^{80} - 48 q^{81} - q^{82} - 4 q^{85} + 78 q^{86} - 18 q^{89} + 39 q^{90} + 16 q^{92} + 8 q^{96} + 30 q^{97} - 12 q^{98}+O(q^{100})$$ 40 * q - 3 * q^2 + q^4 + 20 * q^5 + 3 * q^6 - 22 * q^9 - 3 * q^10 + 12 * q^13 + 18 * q^14 - 7 * q^16 + 4 * q^17 + 2 * q^20 + 12 * q^21 - 8 * q^24 - 20 * q^25 + 2 * q^26 + 8 * q^28 + 6 * q^30 - 18 * q^32 - 6 * q^33 - 27 * q^34 - 14 * q^36 + 38 * q^38 + 36 * q^41 - 21 * q^42 - 8 * q^44 - 44 * q^45 - 18 * q^48 - 60 * q^49 - 33 * q^52 + 42 * q^53 + 9 * q^54 + 12 * q^57 - 62 * q^58 + 3 * q^60 + 12 * q^61 - 23 * q^62 + 64 * q^64 + 2 * q^66 + 72 * q^68 + 18 * q^70 + 42 * q^72 - 18 * q^73 + 6 * q^74 - 62 * q^76 - 28 * q^77 - 24 * q^78 + 7 * q^80 - 48 * q^81 - q^82 - 4 * q^85 + 78 * q^86 - 18 * q^89 + 39 * q^90 + 16 * q^92 + 8 * q^96 + 30 * q^97 - 12 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −1.41224 0.0746014i −1.09575 1.89790i 1.98887 + 0.210711i 0.500000 + 0.866025i 1.40588 + 2.76204i 2.65350i −2.79305 0.445948i −0.901338 + 1.56116i −0.641516 1.26034i
31.2 −1.39096 + 0.255414i 1.57351 + 2.72539i 1.86953 0.710539i 0.500000 + 0.866025i −2.88478 3.38901i 3.91622i −2.41895 + 1.46583i −3.45184 + 5.97877i −0.916674 1.07690i
31.3 −1.38644 + 0.278901i −0.418744 0.725286i 1.84443 0.773359i 0.500000 + 0.866025i 0.782847 + 0.888777i 3.40884i −2.34150 + 1.58663i 1.14931 1.99066i −0.934755 1.06124i
31.4 −1.27347 0.615033i 0.481867 + 0.834618i 1.24347 + 1.56646i 0.500000 + 0.866025i −0.100327 1.35923i 1.00869i −0.620104 2.75961i 1.03561 1.79373i −0.104103 1.41038i
31.5 −0.934755 + 1.06124i 0.418744 + 0.725286i −0.252465 1.98400i 0.500000 + 0.866025i −1.16113 0.233577i 3.40884i 2.34150 + 1.58663i 1.14931 1.99066i −1.38644 0.278901i
31.6 −0.916674 + 1.07690i −1.57351 2.72539i −0.319419 1.97433i 0.500000 + 0.866025i 4.37736 + 0.803789i 3.91622i 2.41895 + 1.46583i −3.45184 + 5.97877i −1.39096 0.255414i
31.7 −0.840559 1.13730i 1.52787 + 2.64635i −0.586922 + 1.91194i 0.500000 + 0.866025i 1.72544 3.96206i 2.58057i 2.66780 0.939591i −3.16877 + 5.48848i 0.564655 1.29660i
31.8 −0.776263 1.18212i −1.45074 2.51275i −0.794830 + 1.83528i 0.500000 + 0.866025i −1.84423 + 3.66550i 1.19935i 2.78652 0.485072i −2.70927 + 4.69260i 0.635617 1.26333i
31.9 −0.641516 + 1.26034i 1.09575 + 1.89790i −1.17692 1.61706i 0.500000 + 0.866025i −3.09494 + 0.163489i 2.65350i 2.79305 0.445948i −0.901338 + 1.56116i −1.41224 + 0.0746014i
31.10 −0.293307 1.38346i −0.256977 0.445098i −1.82794 + 0.811560i 0.500000 + 0.866025i −0.540403 + 0.486069i 1.09058i 1.65891 + 2.29085i 1.36793 2.36932i 1.05146 0.945743i
31.11 −0.104103 + 1.41038i −0.481867 0.834618i −1.97833 0.293648i 0.500000 + 0.866025i 1.22729 0.592728i 1.00869i 0.620104 2.75961i 1.03561 1.79373i −1.27347 + 0.615033i
31.12 0.270280 1.38815i 0.435984 + 0.755146i −1.85390 0.750377i 0.500000 + 0.866025i 1.16609 0.401108i 4.89965i −1.54270 + 2.37067i 1.11984 1.93961i 1.33731 0.460003i
31.13 0.564655 + 1.29660i −1.52787 2.64635i −1.36233 + 1.46426i 0.500000 + 0.866025i 2.56853 3.47531i 2.58057i −2.66780 0.939591i −3.16877 + 5.48848i −0.840559 + 1.13730i
31.14 0.635617 + 1.26333i 1.45074 + 2.51275i −1.19198 + 1.60598i 0.500000 + 0.866025i −2.25231 + 3.42990i 1.19935i −2.78652 0.485072i −2.70927 + 4.69260i −0.776263 + 1.18212i
31.15 0.759006 1.19328i 0.600896 + 1.04078i −0.847818 1.81141i 0.500000 + 0.866025i 1.69802 + 0.0729253i 4.07590i −2.80501 0.363190i 0.777849 1.34727i 1.41291 + 0.0606805i
31.16 1.05146 + 0.945743i 0.256977 + 0.445098i 0.211139 + 1.98882i 0.500000 + 0.866025i −0.150747 + 0.711038i 1.09058i −1.65891 + 2.29085i 1.36793 2.36932i −0.293307 + 1.38346i
31.17 1.15432 0.817038i −1.05340 1.82454i 0.664897 1.88624i 0.500000 + 0.866025i −2.70668 1.24543i 0.279006i −0.773631 2.72057i −0.719298 + 1.24586i 1.28473 + 0.591149i
31.18 1.28473 0.591149i 1.05340 + 1.82454i 1.30109 1.51894i 0.500000 + 0.866025i 2.43191 + 1.72133i 0.279006i 0.773631 2.72057i −0.719298 + 1.24586i 1.15432 + 0.817038i
31.19 1.33731 + 0.460003i −0.435984 0.755146i 1.57679 + 1.23033i 0.500000 + 0.866025i −0.235676 1.21042i 4.89965i 1.54270 + 2.37067i 1.11984 1.93961i 0.270280 + 1.38815i
31.20 1.41291 0.0606805i −0.600896 1.04078i 1.99264 0.171472i 0.500000 + 0.866025i −0.912167 1.43407i 4.07590i 2.80501 0.363190i 0.777849 1.34727i 0.759006 + 1.19328i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 31.20 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.d odd 6 1 inner
76.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.n.a 40
4.b odd 2 1 inner 380.2.n.a 40
19.d odd 6 1 inner 380.2.n.a 40
76.f even 6 1 inner 380.2.n.a 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.n.a 40 1.a even 1 1 trivial
380.2.n.a 40 4.b odd 2 1 inner
380.2.n.a 40 19.d odd 6 1 inner
380.2.n.a 40 76.f even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + 41 T_{3}^{38} + 992 T_{3}^{36} + 15975 T_{3}^{34} + 191688 T_{3}^{32} + 1746767 T_{3}^{30} + 12475873 T_{3}^{28} + 69587636 T_{3}^{26} + 306343209 T_{3}^{24} + 1047905415 T_{3}^{22} + \cdots + 9834496$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.