# Properties

 Label 414.2.a.d Level $414$ Weight $2$ Character orbit 414.a Self dual yes Analytic conductor $3.306$ Analytic rank $0$ 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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 $$q + q^{2} + q^{4} + 2 q^{5} - 2 q^{7} + q^{8} + 2 q^{10} + 6 q^{11} - 2 q^{13} - 2 q^{14} + q^{16} + 2 q^{20} + 6 q^{22} + q^{23} - q^{25} - 2 q^{26} - 2 q^{28} - 6 q^{29} + 8 q^{31} + q^{32} - 4 q^{35} + 2 q^{40} - 10 q^{41} - 12 q^{43} + 6 q^{44} + q^{46} + 8 q^{47} - 3 q^{49} - q^{50} - 2 q^{52} - 2 q^{53} + 12 q^{55} - 2 q^{56} - 6 q^{58} + 12 q^{59} + 4 q^{61} + 8 q^{62} + q^{64} - 4 q^{65} - 12 q^{67} - 4 q^{70} - 10 q^{73} - 12 q^{77} - 6 q^{79} + 2 q^{80} - 10 q^{82} - 14 q^{83} - 12 q^{86} + 6 q^{88} + 4 q^{91} + q^{92} + 8 q^{94} - 6 q^{97} - 3 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 2 * q^5 - 2 * q^7 + q^8 + 2 * q^10 + 6 * q^11 - 2 * q^13 - 2 * q^14 + q^16 + 2 * q^20 + 6 * q^22 + q^23 - q^25 - 2 * q^26 - 2 * q^28 - 6 * q^29 + 8 * q^31 + q^32 - 4 * q^35 + 2 * q^40 - 10 * q^41 - 12 * q^43 + 6 * q^44 + q^46 + 8 * q^47 - 3 * q^49 - q^50 - 2 * q^52 - 2 * q^53 + 12 * q^55 - 2 * q^56 - 6 * q^58 + 12 * q^59 + 4 * q^61 + 8 * q^62 + q^64 - 4 * q^65 - 12 * q^67 - 4 * q^70 - 10 * q^73 - 12 * q^77 - 6 * q^79 + 2 * q^80 - 10 * q^82 - 14 * q^83 - 12 * q^86 + 6 * q^88 + 4 * q^91 + q^92 + 8 * q^94 - 6 * q^97 - 3 * 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 −2.00000 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.d 1
3.b odd 2 1 138.2.a.a 1
4.b odd 2 1 3312.2.a.n 1
12.b even 2 1 1104.2.a.e 1
15.d odd 2 1 3450.2.a.y 1
15.e even 4 2 3450.2.d.j 2
21.c even 2 1 6762.2.a.q 1
23.b odd 2 1 9522.2.a.i 1
24.f even 2 1 4416.2.a.m 1
24.h odd 2 1 4416.2.a.z 1
69.c even 2 1 3174.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.a 1 3.b odd 2 1
414.2.a.d 1 1.a even 1 1 trivial
1104.2.a.e 1 12.b even 2 1
3174.2.a.b 1 69.c even 2 1
3312.2.a.n 1 4.b odd 2 1
3450.2.a.y 1 15.d odd 2 1
3450.2.d.j 2 15.e even 4 2
4416.2.a.m 1 24.f even 2 1
4416.2.a.z 1 24.h odd 2 1
6762.2.a.q 1 21.c even 2 1
9522.2.a.i 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

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T + 2$$
$11$ $$T - 6$$
$13$ $$T + 2$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T$$
$41$ $$T + 10$$
$43$ $$T + 12$$
$47$ $$T - 8$$
$53$ $$T + 2$$
$59$ $$T - 12$$
$61$ $$T - 4$$
$67$ $$T + 12$$
$71$ $$T$$
$73$ $$T + 10$$
$79$ $$T + 6$$
$83$ $$T + 14$$
$89$ $$T$$
$97$ $$T + 6$$