# Properties

 Label 5103.2.a.f Level $5103$ Weight $2$ Character orbit 5103.a Self dual yes Analytic conductor $40.748$ Analytic rank $1$ Dimension $27$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5103,2,Mod(1,5103)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5103, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5103.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5103 = 3^{6} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5103.a (trivial)

## 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: yes Analytic conductor: $$40.7476601515$$ Analytic rank: $$1$$ Dimension: $$27$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$27 q - 9 q^{2} + 27 q^{4} - 12 q^{5} + 27 q^{7} - 27 q^{8}+O(q^{10})$$ 27 * q - 9 * q^2 + 27 * q^4 - 12 * q^5 + 27 * q^7 - 27 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$27 q - 9 q^{2} + 27 q^{4} - 12 q^{5} + 27 q^{7} - 27 q^{8} - 24 q^{11} - 9 q^{14} + 27 q^{16} - 30 q^{17} - 30 q^{20} - 39 q^{23} + 27 q^{25} - 9 q^{26} + 27 q^{28} - 39 q^{29} - 63 q^{32} - 12 q^{35} - 9 q^{38} - 42 q^{41} - 42 q^{44} - 27 q^{47} + 27 q^{49} - 36 q^{50} - 66 q^{53} - 27 q^{56} - 18 q^{59} - 36 q^{62} + 27 q^{64} - 69 q^{65} - 21 q^{68} - 72 q^{71} - 54 q^{74} - 24 q^{77} - 21 q^{80} - 39 q^{83} - 27 q^{86} - 42 q^{89} - 75 q^{92} - 78 q^{95} - 9 q^{98}+O(q^{100})$$ 27 * q - 9 * q^2 + 27 * q^4 - 12 * q^5 + 27 * q^7 - 27 * q^8 - 24 * q^11 - 9 * q^14 + 27 * q^16 - 30 * q^17 - 30 * q^20 - 39 * q^23 + 27 * q^25 - 9 * q^26 + 27 * q^28 - 39 * q^29 - 63 * q^32 - 12 * q^35 - 9 * q^38 - 42 * q^41 - 42 * q^44 - 27 * q^47 + 27 * q^49 - 36 * q^50 - 66 * q^53 - 27 * q^56 - 18 * q^59 - 36 * q^62 + 27 * q^64 - 69 * q^65 - 21 * q^68 - 72 * q^71 - 54 * q^74 - 24 * q^77 - 21 * q^80 - 39 * q^83 - 27 * q^86 - 42 * q^89 - 75 * q^92 - 78 * q^95 - 9 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.78307 0 5.74550 2.42866 0 1.00000 −10.4240 0 −6.75913
1.2 −2.68770 0 5.22375 −1.17018 0 1.00000 −8.66449 0 3.14510
1.3 −2.61362 0 4.83099 −3.50881 0 1.00000 −7.39913 0 9.17068
1.4 −2.46956 0 4.09871 2.30541 0 1.00000 −5.18288 0 −5.69333
1.5 −2.31907 0 3.37807 −2.55206 0 1.00000 −3.19584 0 5.91841
1.6 −2.29809 0 3.28121 −3.71728 0 1.00000 −2.94435 0 8.54265
1.7 −1.91771 0 1.67760 2.81660 0 1.00000 0.618268 0 −5.40142
1.8 −1.86596 0 1.48182 −1.49355 0 1.00000 0.966907 0 2.78691
1.9 −1.27285 0 −0.379841 0.744587 0 1.00000 3.02919 0 −0.947751
1.10 −1.26564 0 −0.398155 −0.909945 0 1.00000 3.03520 0 1.15166
1.11 −1.15245 0 −0.671868 0.763383 0 1.00000 3.07918 0 −0.879758
1.12 −1.02676 0 −0.945770 3.79727 0 1.00000 3.02459 0 −3.89888
1.13 −0.954909 0 −1.08815 −4.02967 0 1.00000 2.94890 0 3.84797
1.14 −0.268830 0 −1.92773 −1.98449 0 1.00000 1.05589 0 0.533491
1.15 −0.105569 0 −1.98886 0.0478140 0 1.00000 0.421099 0 −0.00504766
1.16 0.0926744 0 −1.99141 −2.41602 0 1.00000 −0.369901 0 −0.223903
1.17 0.195845 0 −1.96164 2.12393 0 1.00000 −0.775867 0 0.415961
1.18 0.463515 0 −1.78515 0.316473 0 1.00000 −1.75448 0 0.146690
1.19 0.803391 0 −1.35456 −4.10793 0 1.00000 −2.69503 0 −3.30028
1.20 0.961458 0 −1.07560 3.76231 0 1.00000 −2.95706 0 3.61730
See all 27 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.27 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5103.2.a.f 27
3.b odd 2 1 5103.2.a.i 27
27.e even 9 2 567.2.v.b 54
27.f odd 18 2 189.2.v.a 54

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.v.a 54 27.f odd 18 2
567.2.v.b 54 27.e even 9 2
5103.2.a.f 27 1.a even 1 1 trivial
5103.2.a.i 27 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{27} + 9 T_{2}^{26} - 222 T_{2}^{24} - 459 T_{2}^{23} + 2133 T_{2}^{22} + 7362 T_{2}^{21} + \cdots + 27$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5103))$$.