# Properties

 Label 435.3.b.b Level $435$ Weight $3$ Character orbit 435.b Self dual yes Analytic conductor $11.853$ Analytic rank $0$ Dimension $1$ CM discriminant -435 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [435,3,Mod(434,435)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(435, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("435.434");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$435 = 3 \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 435.b (of order $$2$$, degree $$1$$, minimal)

## 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: $$11.8528914997$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 4 q^{4} + 5 q^{5} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 4 * q^4 + 5 * q^5 + 9 * q^9 $$q - 3 q^{3} + 4 q^{4} + 5 q^{5} + 9 q^{9} + 7 q^{11} - 12 q^{12} - 15 q^{15} + 16 q^{16} + 20 q^{20} - 41 q^{23} + 25 q^{25} - 27 q^{27} + 29 q^{29} - 21 q^{33} + 36 q^{36} + 71 q^{37} - 53 q^{41} + 59 q^{43} + 28 q^{44} + 45 q^{45} - 48 q^{48} + 49 q^{49} + 19 q^{53} + 35 q^{55} - 60 q^{60} + 64 q^{64} + 123 q^{69} - q^{73} - 75 q^{75} + 80 q^{80} + 81 q^{81} + 79 q^{83} - 87 q^{87} - 62 q^{89} - 164 q^{92} - 49 q^{97} + 63 q^{99}+O(q^{100})$$ q - 3 * q^3 + 4 * q^4 + 5 * q^5 + 9 * q^9 + 7 * q^11 - 12 * q^12 - 15 * q^15 + 16 * q^16 + 20 * q^20 - 41 * q^23 + 25 * q^25 - 27 * q^27 + 29 * q^29 - 21 * q^33 + 36 * q^36 + 71 * q^37 - 53 * q^41 + 59 * q^43 + 28 * q^44 + 45 * q^45 - 48 * q^48 + 49 * q^49 + 19 * q^53 + 35 * q^55 - 60 * q^60 + 64 * q^64 + 123 * q^69 - q^73 - 75 * q^75 + 80 * q^80 + 81 * q^81 + 79 * q^83 - 87 * q^87 - 62 * q^89 - 164 * q^92 - 49 * q^97 + 63 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/435\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$146$$ $$262$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
434.1
 0
0 −3.00000 4.00000 5.00000 0 0 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
435.b odd 2 1 CM by $$\Q(\sqrt{-435})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.3.b.b yes 1
3.b odd 2 1 435.3.b.a 1
5.b even 2 1 435.3.b.c yes 1
15.d odd 2 1 435.3.b.d yes 1
29.b even 2 1 435.3.b.d yes 1
87.d odd 2 1 435.3.b.c yes 1
145.d even 2 1 435.3.b.a 1
435.b odd 2 1 CM 435.3.b.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.3.b.a 1 3.b odd 2 1
435.3.b.a 1 145.d even 2 1
435.3.b.b yes 1 1.a even 1 1 trivial
435.3.b.b yes 1 435.b odd 2 1 CM
435.3.b.c yes 1 5.b even 2 1
435.3.b.c yes 1 87.d odd 2 1
435.3.b.d yes 1 15.d odd 2 1
435.3.b.d yes 1 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 $$S_{3}^{\mathrm{new}}(435, [\chi])$$:

 $$T_{2}$$ T2 $$T_{11} - 7$$ T11 - 7 $$T_{23} + 41$$ T23 + 41

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T$$
$11$ $$T - 7$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 41$$
$29$ $$T - 29$$
$31$ $$T$$
$37$ $$T - 71$$
$41$ $$T + 53$$
$43$ $$T - 59$$
$47$ $$T$$
$53$ $$T - 19$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 1$$
$79$ $$T$$
$83$ $$T - 79$$
$89$ $$T + 62$$
$97$ $$T + 49$$