# Properties

 Label 5103.2.a.i Level $5103$ Weight $2$ Character orbit 5103.a Self dual yes Analytic conductor $40.748$ Analytic rank $0$ 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: $$0$$ 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 −2.46283 0 4.06555 1.36934 0 1.00000 −5.08709 0 −3.37246
1.2 −2.30531 0 3.31443 1.59289 0 1.00000 −3.03017 0 −3.67209
1.3 −2.08373 0 2.34193 3.69527 0 1.00000 −0.712484 0 −7.69993
1.4 −1.88628 0 1.55803 −0.142020 0 1.00000 0.833669 0 0.267889
1.5 −1.77354 0 1.14546 −1.97901 0 1.00000 1.51557 0 3.50985
1.6 −1.53990 0 0.371288 2.11956 0 1.00000 2.50805 0 −3.26391
1.7 −1.43332 0 0.0543949 −1.43954 0 1.00000 2.78867 0 2.06331
1.8 −0.961458 0 −1.07560 −3.76231 0 1.00000 2.95706 0 3.61730
1.9 −0.803391 0 −1.35456 4.10793 0 1.00000 2.69503 0 −3.30028
1.10 −0.463515 0 −1.78515 −0.316473 0 1.00000 1.75448 0 0.146690
1.11 −0.195845 0 −1.96164 −2.12393 0 1.00000 0.775867 0 0.415961
1.12 −0.0926744 0 −1.99141 2.41602 0 1.00000 0.369901 0 −0.223903
1.13 0.105569 0 −1.98886 −0.0478140 0 1.00000 −0.421099 0 −0.00504766
1.14 0.268830 0 −1.92773 1.98449 0 1.00000 −1.05589 0 0.533491
1.15 0.954909 0 −1.08815 4.02967 0 1.00000 −2.94890 0 3.84797
1.16 1.02676 0 −0.945770 −3.79727 0 1.00000 −3.02459 0 −3.89888
1.17 1.15245 0 −0.671868 −0.763383 0 1.00000 −3.07918 0 −0.879758
1.18 1.26564 0 −0.398155 0.909945 0 1.00000 −3.03520 0 1.15166
1.19 1.27285 0 −0.379841 −0.744587 0 1.00000 −3.02919 0 −0.947751
1.20 1.86596 0 1.48182 1.49355 0 1.00000 −0.966907 0 2.78691
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.i 27
3.b odd 2 1 5103.2.a.f 27
27.e even 9 2 189.2.v.a 54
27.f odd 18 2 567.2.v.b 54

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

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5103))$$.