# Properties

 Label 380.2.bb.a Level $380$ Weight $2$ Character orbit 380.bb Analytic conductor $3.034$ Analytic rank $0$ Dimension $336$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,2,Mod(59,380)]

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, 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("380.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.bb (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: $$336$$ Relative dimension: $$56$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 18 q^{4} - 12 q^{5} - 18 q^{6} - 24 q^{9}+O(q^{10})$$ 336 * q - 18 * q^4 - 12 * q^5 - 18 * q^6 - 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$336 q - 18 q^{4} - 12 q^{5} - 18 q^{6} - 24 q^{9} - 15 q^{10} + 18 q^{14} - 6 q^{16} - 42 q^{20} + 12 q^{21} + 12 q^{24} - 12 q^{25} + 18 q^{26} - 24 q^{29} - 24 q^{30} + 12 q^{34} - 6 q^{36} - 48 q^{40} - 12 q^{41} - 36 q^{44} - 6 q^{45} - 18 q^{46} - 108 q^{49} - 36 q^{50} + 36 q^{54} - 30 q^{60} - 24 q^{61} + 18 q^{64} - 18 q^{65} - 48 q^{66} - 180 q^{69} - 21 q^{70} - 30 q^{74} - 48 q^{76} + 3 q^{80} - 60 q^{81} + 90 q^{84} - 36 q^{85} + 102 q^{86} - 48 q^{89} - 78 q^{90} + 24 q^{96}+O(q^{100})$$ 336 * q - 18 * q^4 - 12 * q^5 - 18 * q^6 - 24 * q^9 - 15 * q^10 + 18 * q^14 - 6 * q^16 - 42 * q^20 + 12 * q^21 + 12 * q^24 - 12 * q^25 + 18 * q^26 - 24 * q^29 - 24 * q^30 + 12 * q^34 - 6 * q^36 - 48 * q^40 - 12 * q^41 - 36 * q^44 - 6 * q^45 - 18 * q^46 - 108 * q^49 - 36 * q^50 + 36 * q^54 - 30 * q^60 - 24 * q^61 + 18 * q^64 - 18 * q^65 - 48 * q^66 - 180 * q^69 - 21 * q^70 - 30 * q^74 - 48 * q^76 + 3 * q^80 - 60 * q^81 + 90 * q^84 - 36 * q^85 + 102 * q^86 - 48 * q^89 - 78 * q^90 + 24 * q^96

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −1.41344 0.0466259i −1.17302 + 1.39795i 1.99565 + 0.131806i −1.08439 1.95553i 1.72317 1.92123i 1.36812 + 2.36965i −2.81460 0.279350i −0.0573416 0.325200i 1.44155 + 2.81459i
59.2 −1.40916 0.119480i 1.57099 1.87224i 1.97145 + 0.336732i 1.82806 + 1.28771i −2.43747 + 2.45058i −0.614347 1.06408i −2.73785 0.710057i −0.516307 2.92812i −2.42217 2.03300i
59.3 −1.39943 + 0.203928i 0.497064 0.592377i 1.91683 0.570767i −2.20658 + 0.361956i −0.574805 + 0.930358i −0.647033 1.12069i −2.56608 + 1.18965i 0.417106 + 2.36552i 3.01415 0.956516i
59.4 −1.36504 0.369686i 1.57099 1.87224i 1.72666 + 1.00927i 0.572650 2.16150i −2.83661 + 1.97490i −0.614347 1.06408i −1.98385 2.01602i −0.516307 2.92812i −1.58076 + 2.73883i
59.5 −1.34415 0.439613i −1.17302 + 1.39795i 1.61348 + 1.18181i 0.426294 + 2.19506i 2.19126 1.36338i 1.36812 + 2.36965i −1.64922 2.29784i −0.0573416 0.325200i 0.391971 3.13789i
59.6 −1.33052 + 0.479280i −1.75425 + 2.09063i 1.54058 1.27539i −1.37018 + 1.76709i 1.33207 3.62240i −1.59127 2.75617i −1.43851 + 2.43530i −0.772405 4.38052i 0.976131 3.00785i
59.7 −1.31199 + 0.527904i −0.108582 + 0.129403i 1.44263 1.38521i 1.67124 1.48558i 0.0741462 0.227097i 0.176277 + 0.305321i −1.16146 + 2.57895i 0.515989 + 2.92632i −1.40841 + 2.83132i
59.8 −1.31059 + 0.531372i 0.664457 0.791869i 1.43529 1.39282i 0.846472 + 2.06966i −0.450054 + 1.39089i 1.98367 + 3.43582i −1.14097 + 2.58809i 0.335391 + 1.90210i −2.20913 2.26268i
59.9 −1.24529 0.670264i 0.497064 0.592377i 1.10149 + 1.66935i −1.92300 + 1.14109i −1.01604 + 0.404518i −0.647033 1.12069i −0.252775 2.81711i 0.417106 + 2.36552i 3.15952 0.132067i
59.10 −1.08636 0.905441i −1.75425 + 2.09063i 0.360351 + 1.96727i −2.18548 0.472930i 3.79868 0.682806i −1.59127 2.75617i 1.38978 2.46344i −0.772405 4.38052i 1.94601 + 2.49260i
59.11 −1.07107 + 0.923481i 2.07566 2.47367i 0.294366 1.97822i −1.29113 + 1.82565i 0.0612219 + 4.56629i −1.40918 2.44076i 1.51156 + 2.39065i −1.28975 7.31455i −0.303064 3.14772i
59.12 −1.05231 0.944795i −0.108582 + 0.129403i 0.214726 + 1.98844i 2.23516 + 0.0637665i 0.236522 0.0335848i 0.176277 + 0.305321i 1.65271 2.29533i 0.515989 + 2.92632i −2.29184 2.17887i
59.13 −1.04981 0.947574i 0.664457 0.791869i 0.204208 + 1.98955i −0.681915 2.12955i −1.44791 + 0.201691i 1.98367 + 3.43582i 1.67086 2.28215i 0.335391 + 1.90210i −1.30203 + 2.88179i
59.14 −0.997360 + 1.00263i 0.300395 0.357997i −0.0105474 1.99997i −1.26326 1.84504i 0.0593378 + 0.658238i −1.68031 2.91038i 2.01576 + 1.98412i 0.483020 + 2.73934i 3.10982 + 0.573580i
59.15 −0.934972 + 1.06105i −0.912584 + 1.08758i −0.251654 1.98410i 1.86792 + 1.22918i −0.300731 1.98515i −0.869878 1.50667i 2.34052 + 1.58806i 0.170934 + 0.969416i −3.05067 + 0.832706i
59.16 −0.906917 + 1.08513i 1.82738 2.17779i −0.355003 1.96824i 1.74742 1.39518i 0.705894 + 3.95801i 1.71831 + 2.97619i 2.45775 + 1.39981i −0.882491 5.00485i −0.0708181 + 3.16148i
59.17 −0.777543 + 1.18128i −1.85226 + 2.20744i −0.790854 1.83699i −0.543418 2.16903i −1.16739 3.90441i 0.0958109 + 0.165949i 2.78493 + 0.494121i −0.920966 5.22306i 2.98477 + 1.04458i
59.18 −0.690624 1.23411i 2.07566 2.47367i −1.04608 + 1.70462i −2.16257 0.568604i −4.48629 0.853220i −1.40918 2.44076i 2.82614 + 0.113728i −1.28975 7.31455i 0.791797 + 3.06154i
59.19 −0.660906 + 1.25028i 0.742048 0.884338i −1.12641 1.65264i −2.17942 0.500136i 0.615247 + 1.51223i 1.69793 + 2.94091i 2.81071 0.316087i 0.289526 + 1.64198i 2.06570 2.39434i
59.20 −0.594291 1.28328i 0.300395 0.357997i −1.29364 + 1.52529i 0.218254 + 2.22539i −0.637934 0.172738i −1.68031 2.91038i 2.72617 + 0.753640i 0.483020 + 2.73934i 2.72610 1.60261i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.56 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
5.b even 2 1 inner
19.f odd 18 1 inner
20.d odd 2 1 inner
76.k even 18 1 inner
95.o odd 18 1 inner
380.bb 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.bb.a 336
4.b odd 2 1 inner 380.2.bb.a 336
5.b even 2 1 inner 380.2.bb.a 336
19.f odd 18 1 inner 380.2.bb.a 336
20.d odd 2 1 inner 380.2.bb.a 336
76.k even 18 1 inner 380.2.bb.a 336
95.o odd 18 1 inner 380.2.bb.a 336
380.bb even 18 1 inner 380.2.bb.a 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.bb.a 336 1.a even 1 1 trivial
380.2.bb.a 336 4.b odd 2 1 inner
380.2.bb.a 336 5.b even 2 1 inner
380.2.bb.a 336 19.f odd 18 1 inner
380.2.bb.a 336 20.d odd 2 1 inner
380.2.bb.a 336 76.k even 18 1 inner
380.2.bb.a 336 95.o odd 18 1 inner
380.2.bb.a 336 380.bb even 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(380, [\chi])$$.