# Properties

 Label 405.2.a.e Level $405$ Weight $2$ Character orbit 405.a Self dual yes Analytic conductor $3.234$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 405.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: $$3.23394128186$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 45) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{4} - q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10})$$ q + q^2 - q^4 - q^5 - 3 * q^7 - 3 * q^8 $$q + q^{2} - q^{4} - q^{5} - 3 q^{7} - 3 q^{8} - q^{10} - 2 q^{11} - 2 q^{13} - 3 q^{14} - q^{16} + 4 q^{17} - 8 q^{19} + q^{20} - 2 q^{22} + 3 q^{23} + q^{25} - 2 q^{26} + 3 q^{28} - q^{29} + 5 q^{32} + 4 q^{34} + 3 q^{35} - 4 q^{37} - 8 q^{38} + 3 q^{40} + 5 q^{41} - 8 q^{43} + 2 q^{44} + 3 q^{46} + 7 q^{47} + 2 q^{49} + q^{50} + 2 q^{52} - 2 q^{53} + 2 q^{55} + 9 q^{56} - q^{58} - 14 q^{59} + 7 q^{61} + 7 q^{64} + 2 q^{65} - 3 q^{67} - 4 q^{68} + 3 q^{70} + 2 q^{71} + 4 q^{73} - 4 q^{74} + 8 q^{76} + 6 q^{77} - 6 q^{79} + q^{80} + 5 q^{82} + 9 q^{83} - 4 q^{85} - 8 q^{86} + 6 q^{88} - 15 q^{89} + 6 q^{91} - 3 q^{92} + 7 q^{94} + 8 q^{95} + 2 q^{97} + 2 q^{98}+O(q^{100})$$ q + q^2 - q^4 - q^5 - 3 * q^7 - 3 * q^8 - q^10 - 2 * q^11 - 2 * q^13 - 3 * q^14 - q^16 + 4 * q^17 - 8 * q^19 + q^20 - 2 * q^22 + 3 * q^23 + q^25 - 2 * q^26 + 3 * q^28 - q^29 + 5 * q^32 + 4 * q^34 + 3 * q^35 - 4 * q^37 - 8 * q^38 + 3 * q^40 + 5 * q^41 - 8 * q^43 + 2 * q^44 + 3 * q^46 + 7 * q^47 + 2 * q^49 + q^50 + 2 * q^52 - 2 * q^53 + 2 * q^55 + 9 * q^56 - q^58 - 14 * q^59 + 7 * q^61 + 7 * q^64 + 2 * q^65 - 3 * q^67 - 4 * q^68 + 3 * q^70 + 2 * q^71 + 4 * q^73 - 4 * q^74 + 8 * q^76 + 6 * q^77 - 6 * q^79 + q^80 + 5 * q^82 + 9 * q^83 - 4 * q^85 - 8 * q^86 + 6 * q^88 - 15 * q^89 + 6 * q^91 - 3 * q^92 + 7 * q^94 + 8 * q^95 + 2 * q^97 + 2 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
1.00000 0 −1.00000 −1.00000 0 −3.00000 −3.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$5$$ $$+1$$

## 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 405.2.a.e 1
3.b odd 2 1 405.2.a.b 1
4.b odd 2 1 6480.2.a.k 1
5.b even 2 1 2025.2.a.b 1
5.c odd 4 2 2025.2.b.c 2
9.c even 3 2 45.2.e.a 2
9.d odd 6 2 135.2.e.a 2
12.b even 2 1 6480.2.a.x 1
15.d odd 2 1 2025.2.a.e 1
15.e even 4 2 2025.2.b.d 2
36.f odd 6 2 720.2.q.d 2
36.h even 6 2 2160.2.q.a 2
45.h odd 6 2 675.2.e.a 2
45.j even 6 2 225.2.e.a 2
45.k odd 12 4 225.2.k.a 4
45.l even 12 4 675.2.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.e.a 2 9.c even 3 2
135.2.e.a 2 9.d odd 6 2
225.2.e.a 2 45.j even 6 2
225.2.k.a 4 45.k odd 12 4
405.2.a.b 1 3.b odd 2 1
405.2.a.e 1 1.a even 1 1 trivial
675.2.e.a 2 45.h odd 6 2
675.2.k.a 4 45.l even 12 4
720.2.q.d 2 36.f odd 6 2
2025.2.a.b 1 5.b even 2 1
2025.2.a.e 1 15.d odd 2 1
2025.2.b.c 2 5.c odd 4 2
2025.2.b.d 2 15.e even 4 2
2160.2.q.a 2 36.h even 6 2
6480.2.a.k 1 4.b odd 2 1
6480.2.a.x 1 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(405))$$:

 $$T_{2} - 1$$ T2 - 1 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 3$$
$11$ $$T + 2$$
$13$ $$T + 2$$
$17$ $$T - 4$$
$19$ $$T + 8$$
$23$ $$T - 3$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T + 4$$
$41$ $$T - 5$$
$43$ $$T + 8$$
$47$ $$T - 7$$
$53$ $$T + 2$$
$59$ $$T + 14$$
$61$ $$T - 7$$
$67$ $$T + 3$$
$71$ $$T - 2$$
$73$ $$T - 4$$
$79$ $$T + 6$$
$83$ $$T - 9$$
$89$ $$T + 15$$
$97$ $$T - 2$$
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