# Properties

 Label 4205.2.a.y Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $24$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4205,2,Mod(1,4205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4205, 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("4205.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4205 = 5 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4205.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: $$33.5770940499$$ Analytic rank: $$1$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 145) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 14 q^{4} + 24 q^{5} - 18 q^{6} - 38 q^{7} + 18 q^{9}+O(q^{10})$$ 24 * q + 14 * q^4 + 24 * q^5 - 18 * q^6 - 38 * q^7 + 18 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 14 q^{4} + 24 q^{5} - 18 q^{6} - 38 q^{7} + 18 q^{9} - 34 q^{13} - 10 q^{16} + 14 q^{20} - 10 q^{22} - 44 q^{23} - 4 q^{24} + 24 q^{25} - 44 q^{28} - 18 q^{30} - 58 q^{33} + 8 q^{34} - 38 q^{35} - 28 q^{36} - 44 q^{38} + 2 q^{42} + 18 q^{45} + 26 q^{49} - 48 q^{51} - 4 q^{52} - 28 q^{53} - 40 q^{54} - 32 q^{57} - 66 q^{59} - 2 q^{62} - 60 q^{63} - 12 q^{64} - 34 q^{65} - 144 q^{67} - 4 q^{71} + 12 q^{74} + 60 q^{78} - 10 q^{80} + 76 q^{81} + 4 q^{82} - 114 q^{83} - 30 q^{86} - 104 q^{88} - 32 q^{91} - 134 q^{92} - 32 q^{93} - 66 q^{94} + 22 q^{96}+O(q^{100})$$ 24 * q + 14 * q^4 + 24 * q^5 - 18 * q^6 - 38 * q^7 + 18 * q^9 - 34 * q^13 - 10 * q^16 + 14 * q^20 - 10 * q^22 - 44 * q^23 - 4 * q^24 + 24 * q^25 - 44 * q^28 - 18 * q^30 - 58 * q^33 + 8 * q^34 - 38 * q^35 - 28 * q^36 - 44 * q^38 + 2 * q^42 + 18 * q^45 + 26 * q^49 - 48 * q^51 - 4 * q^52 - 28 * q^53 - 40 * q^54 - 32 * q^57 - 66 * q^59 - 2 * q^62 - 60 * q^63 - 12 * q^64 - 34 * q^65 - 144 * q^67 - 4 * q^71 + 12 * q^74 + 60 * q^78 - 10 * q^80 + 76 * q^81 + 4 * q^82 - 114 * q^83 - 30 * q^86 - 104 * q^88 - 32 * q^91 - 134 * q^92 - 32 * q^93 - 66 * q^94 + 22 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.52311 −0.367113 4.36608 1.00000 0.926266 −5.12713 −5.96987 −2.86523 −2.52311
1.2 −2.46493 1.17334 4.07587 1.00000 −2.89220 −0.528087 −5.11687 −1.62327 −2.46493
1.3 −2.07250 3.28868 2.29525 1.00000 −6.81579 −4.26791 −0.611896 7.81544 −2.07250
1.4 −1.95283 0.620899 1.81353 1.00000 −1.21251 0.126430 0.364137 −2.61448 −1.95283
1.5 −1.86976 −0.995019 1.49598 1.00000 1.86044 0.907041 0.942385 −2.00994 −1.86976
1.6 −1.34022 −1.23909 −0.203815 1.00000 1.66065 −3.86391 2.95359 −1.46465 −1.34022
1.7 −1.29519 −3.05287 −0.322483 1.00000 3.95404 −3.30555 3.00806 6.31999 −1.29519
1.8 −1.26889 2.62094 −0.389926 1.00000 −3.32568 −0.00237662 3.03255 3.86933 −1.26889
1.9 −0.976160 2.48787 −1.04711 1.00000 −2.42856 2.05638 2.97447 3.18951 −0.976160
1.10 −0.946293 1.03726 −1.10453 1.00000 −0.981552 1.25250 2.93779 −1.92409 −0.946293
1.11 −0.110687 −2.50070 −1.98775 1.00000 0.276795 −2.79886 0.441392 3.25348 −0.110687
1.12 −0.0943500 0.232166 −1.99110 1.00000 −0.0219049 −3.44853 0.376560 −2.94610 −0.0943500
1.13 0.0943500 −0.232166 −1.99110 1.00000 −0.0219049 −3.44853 −0.376560 −2.94610 0.0943500
1.14 0.110687 2.50070 −1.98775 1.00000 0.276795 −2.79886 −0.441392 3.25348 0.110687
1.15 0.946293 −1.03726 −1.10453 1.00000 −0.981552 1.25250 −2.93779 −1.92409 0.946293
1.16 0.976160 −2.48787 −1.04711 1.00000 −2.42856 2.05638 −2.97447 3.18951 0.976160
1.17 1.26889 −2.62094 −0.389926 1.00000 −3.32568 −0.00237662 −3.03255 3.86933 1.26889
1.18 1.29519 3.05287 −0.322483 1.00000 3.95404 −3.30555 −3.00806 6.31999 1.29519
1.19 1.34022 1.23909 −0.203815 1.00000 1.66065 −3.86391 −2.95359 −1.46465 1.34022
1.20 1.86976 0.995019 1.49598 1.00000 1.86044 0.907041 −0.942385 −2.00994 1.86976
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$29$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4205.2.a.y 24
29.b even 2 1 inner 4205.2.a.y 24
29.f odd 28 2 145.2.m.a 24
145.o even 28 2 725.2.p.b 48
145.s odd 28 2 725.2.q.b 24
145.t even 28 2 725.2.p.b 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.m.a 24 29.f odd 28 2
725.2.p.b 48 145.o even 28 2
725.2.p.b 48 145.t even 28 2
725.2.q.b 24 145.s odd 28 2
4205.2.a.y 24 1.a even 1 1 trivial
4205.2.a.y 24 29.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{24} - 31 T_{2}^{22} + 414 T_{2}^{20} - 3129 T_{2}^{18} + 14790 T_{2}^{16} - 45609 T_{2}^{14} + \cdots + 1$$ T2^24 - 31*T2^22 + 414*T2^20 - 3129*T2^18 + 14790*T2^16 - 45609*T2^14 + 92854*T2^12 - 123288*T2^10 + 102402*T2^8 - 48363*T2^6 + 10158*T2^4 - 199*T2^2 + 1 $$T_{3}^{24} - 45 T_{3}^{22} + 845 T_{3}^{20} - 8613 T_{3}^{18} + 52013 T_{3}^{16} - 191573 T_{3}^{14} + \cdots + 169$$ T3^24 - 45*T3^22 + 845*T3^20 - 8613*T3^18 + 52013*T3^16 - 191573*T3^14 + 432173*T3^12 - 595737*T3^10 + 491861*T3^8 - 230569*T3^6 + 54977*T3^4 - 5501*T3^2 + 169