# Properties

 Label 414.2.a.a Level $414$ Weight $2$ Character orbit 414.a Self dual yes Analytic conductor $3.306$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,2,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, 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("414.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$414 = 2 \cdot 3^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 414.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: $$3.30580664368$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 138) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 2 q^{5} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 2 * q^5 - q^8 $$q - q^{2} + q^{4} - 2 q^{5} - q^{8} + 2 q^{10} - 2 q^{13} + q^{16} - 2 q^{17} - 8 q^{19} - 2 q^{20} + q^{23} - q^{25} + 2 q^{26} + 2 q^{29} - 8 q^{31} - q^{32} + 2 q^{34} + 2 q^{37} + 8 q^{38} + 2 q^{40} - 10 q^{41} + 8 q^{43} - q^{46} - 8 q^{47} - 7 q^{49} + q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{58} + 4 q^{59} + 2 q^{61} + 8 q^{62} + q^{64} + 4 q^{65} + 8 q^{67} - 2 q^{68} - 6 q^{73} - 2 q^{74} - 8 q^{76} + 8 q^{79} - 2 q^{80} + 10 q^{82} + 16 q^{83} + 4 q^{85} - 8 q^{86} - 18 q^{89} + q^{92} + 8 q^{94} + 16 q^{95} + 10 q^{97} + 7 q^{98}+O(q^{100})$$ q - q^2 + q^4 - 2 * q^5 - q^8 + 2 * q^10 - 2 * q^13 + q^16 - 2 * q^17 - 8 * q^19 - 2 * q^20 + q^23 - q^25 + 2 * q^26 + 2 * q^29 - 8 * q^31 - q^32 + 2 * q^34 + 2 * q^37 + 8 * q^38 + 2 * q^40 - 10 * q^41 + 8 * q^43 - q^46 - 8 * q^47 - 7 * q^49 + q^50 - 2 * q^52 - 2 * q^53 - 2 * q^58 + 4 * q^59 + 2 * q^61 + 8 * q^62 + q^64 + 4 * q^65 + 8 * q^67 - 2 * q^68 - 6 * q^73 - 2 * q^74 - 8 * q^76 + 8 * q^79 - 2 * q^80 + 10 * q^82 + 16 * q^83 + 4 * q^85 - 8 * q^86 - 18 * q^89 + q^92 + 8 * q^94 + 16 * q^95 + 10 * q^97 + 7 * 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 −2.00000 0 0 −1.00000 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$-1$$

## 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 414.2.a.a 1
3.b odd 2 1 138.2.a.c 1
4.b odd 2 1 3312.2.a.d 1
12.b even 2 1 1104.2.a.g 1
15.d odd 2 1 3450.2.a.k 1
15.e even 4 2 3450.2.d.f 2
21.c even 2 1 6762.2.a.bg 1
23.b odd 2 1 9522.2.a.d 1
24.f even 2 1 4416.2.a.c 1
24.h odd 2 1 4416.2.a.s 1
69.c even 2 1 3174.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.c 1 3.b odd 2 1
414.2.a.a 1 1.a even 1 1 trivial
1104.2.a.g 1 12.b even 2 1
3174.2.a.e 1 69.c even 2 1
3312.2.a.d 1 4.b odd 2 1
3450.2.a.k 1 15.d odd 2 1
3450.2.d.f 2 15.e even 4 2
4416.2.a.c 1 24.f even 2 1
4416.2.a.s 1 24.h odd 2 1
6762.2.a.bg 1 21.c even 2 1
9522.2.a.d 1 23.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(414))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T + 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T + 8$$
$23$ $$T - 1$$
$29$ $$T - 2$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T + 10$$
$43$ $$T - 8$$
$47$ $$T + 8$$
$53$ $$T + 2$$
$59$ $$T - 4$$
$61$ $$T - 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T - 8$$
$83$ $$T - 16$$
$89$ $$T + 18$$
$97$ $$T - 10$$