# Properties

 Label 4205.2.a.z Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $0$ Dimension $36$ 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: $$0$$ Dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 50 q^{4} - 36 q^{5} + 2 q^{6} + 14 q^{7} + 42 q^{9}+O(q^{10})$$ 36 * q + 50 * q^4 - 36 * q^5 + 2 * q^6 + 14 * q^7 + 42 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 50 q^{4} - 36 q^{5} + 2 q^{6} + 14 q^{7} + 42 q^{9} + 10 q^{13} + 58 q^{16} - 50 q^{20} + 82 q^{22} - 20 q^{23} + 40 q^{24} + 36 q^{25} + 28 q^{28} - 2 q^{30} + 18 q^{33} + 72 q^{34} - 14 q^{35} + 100 q^{36} - 8 q^{38} - 14 q^{42} - 42 q^{45} + 42 q^{49} + 104 q^{51} + 8 q^{52} - 28 q^{53} + 80 q^{54} + 88 q^{57} + 86 q^{59} - 42 q^{62} - 12 q^{63} + 136 q^{64} - 10 q^{65} + 96 q^{67} + 28 q^{71} - 12 q^{74} - 24 q^{78} - 58 q^{80} + 32 q^{81} + 28 q^{82} + 10 q^{83} + 46 q^{86} + 212 q^{88} + 152 q^{91} - 22 q^{92} + 8 q^{93} + 114 q^{94} + 122 q^{96}+O(q^{100})$$ 36 * q + 50 * q^4 - 36 * q^5 + 2 * q^6 + 14 * q^7 + 42 * q^9 + 10 * q^13 + 58 * q^16 - 50 * q^20 + 82 * q^22 - 20 * q^23 + 40 * q^24 + 36 * q^25 + 28 * q^28 - 2 * q^30 + 18 * q^33 + 72 * q^34 - 14 * q^35 + 100 * q^36 - 8 * q^38 - 14 * q^42 - 42 * q^45 + 42 * q^49 + 104 * q^51 + 8 * q^52 - 28 * q^53 + 80 * q^54 + 88 * q^57 + 86 * q^59 - 42 * q^62 - 12 * q^63 + 136 * q^64 - 10 * q^65 + 96 * q^67 + 28 * q^71 - 12 * q^74 - 24 * q^78 - 58 * q^80 + 32 * q^81 + 28 * q^82 + 10 * q^83 + 46 * q^86 + 212 * q^88 + 152 * q^91 - 22 * q^92 + 8 * q^93 + 114 * q^94 + 122 * 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 −2.73278 −1.21520 5.46806 −1.00000 3.32088 −1.57483 −9.47743 −1.52328 2.73278
1.2 −2.69299 −3.06406 5.25219 −1.00000 8.25149 −2.38206 −8.75812 6.38847 2.69299
1.3 −2.67123 0.713943 5.13549 −1.00000 −1.90711 2.77457 −8.37562 −2.49029 2.67123
1.4 −2.65705 2.29699 5.05993 −1.00000 −6.10322 4.10952 −8.13038 2.27615 2.65705
1.5 −2.25867 −3.23671 3.10160 −1.00000 7.31067 1.13372 −2.48814 7.47631 2.25867
1.6 −2.19416 0.113339 2.81436 −1.00000 −0.248683 3.93157 −1.78684 −2.98715 2.19416
1.7 −2.04322 1.70781 2.17475 −1.00000 −3.48944 −3.88337 −0.357058 −0.0833797 2.04322
1.8 −2.01118 2.62919 2.04484 −1.00000 −5.28777 −3.87224 −0.0901779 3.91265 2.01118
1.9 −1.77096 −2.16305 1.13629 −1.00000 3.83068 4.83294 1.52959 1.67880 1.77096
1.10 −1.75945 3.29205 1.09567 −1.00000 −5.79221 0.522998 1.59112 7.83761 1.75945
1.11 −1.54851 −0.450009 0.397871 −1.00000 0.696842 1.74706 2.48091 −2.79749 1.54851
1.12 −1.26166 0.253469 −0.408220 −1.00000 −0.319791 −2.75032 3.03835 −2.93575 1.26166
1.13 −1.12063 1.00389 −0.744181 −1.00000 −1.12499 −0.443804 3.07522 −1.99221 1.12063
1.14 −1.00099 −2.52600 −0.998018 −1.00000 2.52851 −1.84414 3.00099 3.38070 1.00099
1.15 −0.688039 −1.31919 −1.52660 −1.00000 0.907653 −2.91509 2.42644 −1.25974 0.688039
1.16 −0.665686 1.73079 −1.55686 −1.00000 −1.15216 3.49242 2.36775 −0.00438097 0.665686
1.17 −0.623107 2.18308 −1.61174 −1.00000 −1.36029 1.60747 2.25050 1.76584 0.623107
1.18 −0.405675 −2.31455 −1.83543 −1.00000 0.938954 2.51358 1.55594 2.35713 0.405675
1.19 0.405675 2.31455 −1.83543 −1.00000 0.938954 2.51358 −1.55594 2.35713 −0.405675
1.20 0.623107 −2.18308 −1.61174 −1.00000 −1.36029 1.60747 −2.25050 1.76584 −0.623107
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.36 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.z 36
29.b even 2 1 inner 4205.2.a.z 36
29.f odd 28 2 145.2.m.b 36
145.o even 28 2 725.2.p.c 72
145.s odd 28 2 725.2.q.c 36
145.t even 28 2 725.2.p.c 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.m.b 36 29.f odd 28 2
725.2.p.c 72 145.o even 28 2
725.2.p.c 72 145.t even 28 2
725.2.q.c 36 145.s odd 28 2
4205.2.a.z 36 1.a even 1 1 trivial
4205.2.a.z 36 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4205))$$:

 $$T_{2}^{36} - 61 T_{2}^{34} + 1699 T_{2}^{32} - 28638 T_{2}^{30} + 326457 T_{2}^{28} - 2664319 T_{2}^{26} + \cdots + 707281$$ T2^36 - 61*T2^34 + 1699*T2^32 - 28638*T2^30 + 326457*T2^28 - 2664319*T2^26 + 16076995*T2^24 - 73047845*T2^22 + 252203978*T2^20 - 663182943*T2^18 + 1323570636*T2^16 - 1987128653*T2^14 + 2211650401*T2^12 - 1786309608*T2^10 + 1015935505*T2^8 - 389586704*T2^6 + 94169095*T2^4 - 12693213*T2^2 + 707281 $$T_{3}^{36} - 75 T_{3}^{34} + 2548 T_{3}^{32} - 51946 T_{3}^{30} + 709680 T_{3}^{28} - 6871220 T_{3}^{26} + \cdots + 57121$$ T3^36 - 75*T3^34 + 2548*T3^32 - 51946*T3^30 + 709680*T3^28 - 6871220*T3^26 + 48619957*T3^24 - 255519977*T3^22 + 1003529653*T3^20 - 2939479405*T3^18 + 6359805205*T3^16 - 9981102085*T3^14 + 11040825297*T3^12 - 8240537532*T3^10 + 3881006374*T3^8 - 1036775977*T3^6 + 131783263*T3^4 - 5976493*T3^2 + 57121