# Properties

 Label 4205.2.a.b Level $4205$ Weight $2$ Character orbit 4205.a Self dual yes Analytic conductor $33.577$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - 2 q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} - 2 q^{9}+O(q^{10})$$ q - q^3 - 2 * q^4 - q^5 + (-2*b - 1) * q^7 - 2 * q^9 $$q - q^{3} - 2 q^{4} - q^{5} + ( - 2 \beta - 1) q^{7} - 2 q^{9} + ( - 3 \beta + 3) q^{11} + 2 q^{12} + 2 \beta q^{13} + q^{15} + 4 q^{16} + (3 \beta - 3) q^{17} + 4 \beta q^{19} + 2 q^{20} + (2 \beta + 1) q^{21} + ( - 3 \beta - 3) q^{23} + q^{25} + 5 q^{27} + (4 \beta + 2) q^{28} + ( - 4 \beta + 1) q^{31} + (3 \beta - 3) q^{33} + (2 \beta + 1) q^{35} + 4 q^{36} + ( - 2 \beta + 9) q^{37} - 2 \beta q^{39} + 6 \beta q^{41} + ( - \beta - 2) q^{43} + (6 \beta - 6) q^{44} + 2 q^{45} + 9 q^{47} - 4 q^{48} + (8 \beta - 2) q^{49} + ( - 3 \beta + 3) q^{51} - 4 \beta q^{52} + 3 q^{53} + (3 \beta - 3) q^{55} - 4 \beta q^{57} + ( - 6 \beta + 3) q^{59} - 2 q^{60} + ( - 7 \beta + 1) q^{61} + (4 \beta + 2) q^{63} - 8 q^{64} - 2 \beta q^{65} + (2 \beta + 9) q^{67} + ( - 6 \beta + 6) q^{68} + (3 \beta + 3) q^{69} + (9 \beta - 6) q^{71} + (7 \beta - 9) q^{73} - q^{75} - 8 \beta q^{76} + (3 \beta + 3) q^{77} + ( - 2 \beta - 9) q^{79} - 4 q^{80} + q^{81} + ( - 6 \beta + 12) q^{83} + ( - 4 \beta - 2) q^{84} + ( - 3 \beta + 3) q^{85} + ( - 6 \beta + 3) q^{89} + ( - 6 \beta - 4) q^{91} + (6 \beta + 6) q^{92} + (4 \beta - 1) q^{93} - 4 \beta q^{95} + (8 \beta - 11) q^{97} + (6 \beta - 6) q^{99} +O(q^{100})$$ q - q^3 - 2 * q^4 - q^5 + (-2*b - 1) * q^7 - 2 * q^9 + (-3*b + 3) * q^11 + 2 * q^12 + 2*b * q^13 + q^15 + 4 * q^16 + (3*b - 3) * q^17 + 4*b * q^19 + 2 * q^20 + (2*b + 1) * q^21 + (-3*b - 3) * q^23 + q^25 + 5 * q^27 + (4*b + 2) * q^28 + (-4*b + 1) * q^31 + (3*b - 3) * q^33 + (2*b + 1) * q^35 + 4 * q^36 + (-2*b + 9) * q^37 - 2*b * q^39 + 6*b * q^41 + (-b - 2) * q^43 + (6*b - 6) * q^44 + 2 * q^45 + 9 * q^47 - 4 * q^48 + (8*b - 2) * q^49 + (-3*b + 3) * q^51 - 4*b * q^52 + 3 * q^53 + (3*b - 3) * q^55 - 4*b * q^57 + (-6*b + 3) * q^59 - 2 * q^60 + (-7*b + 1) * q^61 + (4*b + 2) * q^63 - 8 * q^64 - 2*b * q^65 + (2*b + 9) * q^67 + (-6*b + 6) * q^68 + (3*b + 3) * q^69 + (9*b - 6) * q^71 + (7*b - 9) * q^73 - q^75 - 8*b * q^76 + (3*b + 3) * q^77 + (-2*b - 9) * q^79 - 4 * q^80 + q^81 + (-6*b + 12) * q^83 + (-4*b - 2) * q^84 + (-3*b + 3) * q^85 + (-6*b + 3) * q^89 + (-6*b - 4) * q^91 + (6*b + 6) * q^92 + (4*b - 1) * q^93 - 4*b * q^95 + (8*b - 11) * q^97 + (6*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 - 4 * q^7 - 4 * q^9 $$2 q - 2 q^{3} - 4 q^{4} - 2 q^{5} - 4 q^{7} - 4 q^{9} + 3 q^{11} + 4 q^{12} + 2 q^{13} + 2 q^{15} + 8 q^{16} - 3 q^{17} + 4 q^{19} + 4 q^{20} + 4 q^{21} - 9 q^{23} + 2 q^{25} + 10 q^{27} + 8 q^{28} - 2 q^{31} - 3 q^{33} + 4 q^{35} + 8 q^{36} + 16 q^{37} - 2 q^{39} + 6 q^{41} - 5 q^{43} - 6 q^{44} + 4 q^{45} + 18 q^{47} - 8 q^{48} + 4 q^{49} + 3 q^{51} - 4 q^{52} + 6 q^{53} - 3 q^{55} - 4 q^{57} - 4 q^{60} - 5 q^{61} + 8 q^{63} - 16 q^{64} - 2 q^{65} + 20 q^{67} + 6 q^{68} + 9 q^{69} - 3 q^{71} - 11 q^{73} - 2 q^{75} - 8 q^{76} + 9 q^{77} - 20 q^{79} - 8 q^{80} + 2 q^{81} + 18 q^{83} - 8 q^{84} + 3 q^{85} - 14 q^{91} + 18 q^{92} + 2 q^{93} - 4 q^{95} - 14 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^4 - 2 * q^5 - 4 * q^7 - 4 * q^9 + 3 * q^11 + 4 * q^12 + 2 * q^13 + 2 * q^15 + 8 * q^16 - 3 * q^17 + 4 * q^19 + 4 * q^20 + 4 * q^21 - 9 * q^23 + 2 * q^25 + 10 * q^27 + 8 * q^28 - 2 * q^31 - 3 * q^33 + 4 * q^35 + 8 * q^36 + 16 * q^37 - 2 * q^39 + 6 * q^41 - 5 * q^43 - 6 * q^44 + 4 * q^45 + 18 * q^47 - 8 * q^48 + 4 * q^49 + 3 * q^51 - 4 * q^52 + 6 * q^53 - 3 * q^55 - 4 * q^57 - 4 * q^60 - 5 * q^61 + 8 * q^63 - 16 * q^64 - 2 * q^65 + 20 * q^67 + 6 * q^68 + 9 * q^69 - 3 * q^71 - 11 * q^73 - 2 * q^75 - 8 * q^76 + 9 * q^77 - 20 * q^79 - 8 * q^80 + 2 * q^81 + 18 * q^83 - 8 * q^84 + 3 * q^85 - 14 * q^91 + 18 * q^92 + 2 * q^93 - 4 * q^95 - 14 * q^97 - 6 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
0 −1.00000 −2.00000 −1.00000 0 −4.23607 0 −2.00000 0
1.2 0 −1.00000 −2.00000 −1.00000 0 0.236068 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 4205.2.a.b 2
29.b even 2 1 4205.2.a.c yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4205.2.a.b 2 1.a even 1 1 trivial
4205.2.a.c yes 2 29.b even 2 1

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{3} + 1$$ T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 4T - 1$$
$11$ $$T^{2} - 3T - 9$$
$13$ $$T^{2} - 2T - 4$$
$17$ $$T^{2} + 3T - 9$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$T^{2} + 9T + 9$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 2T - 19$$
$37$ $$T^{2} - 16T + 59$$
$41$ $$T^{2} - 6T - 36$$
$43$ $$T^{2} + 5T + 5$$
$47$ $$(T - 9)^{2}$$
$53$ $$(T - 3)^{2}$$
$59$ $$T^{2} - 45$$
$61$ $$T^{2} + 5T - 55$$
$67$ $$T^{2} - 20T + 95$$
$71$ $$T^{2} + 3T - 99$$
$73$ $$T^{2} + 11T - 31$$
$79$ $$T^{2} + 20T + 95$$
$83$ $$T^{2} - 18T + 36$$
$89$ $$T^{2} - 45$$
$97$ $$T^{2} + 14T - 31$$