# Properties

 Label 378.2.a.e Level $378$ Weight $2$ Character orbit 378.a Self dual yes Analytic conductor $3.018$ 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] = [378,2,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.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: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + q^7 + q^8 $$q + q^{2} + q^{4} + q^{7} + q^{8} + 5 q^{13} + q^{14} + q^{16} + 3 q^{17} + 2 q^{19} - 9 q^{23} - 5 q^{25} + 5 q^{26} + q^{28} - 3 q^{29} + 5 q^{31} + q^{32} + 3 q^{34} + 2 q^{37} + 2 q^{38} - 6 q^{41} - q^{43} - 9 q^{46} - 6 q^{47} + q^{49} - 5 q^{50} + 5 q^{52} + 3 q^{53} + q^{56} - 3 q^{58} - 3 q^{59} - 10 q^{61} + 5 q^{62} + q^{64} - 13 q^{67} + 3 q^{68} + 9 q^{71} + 2 q^{73} + 2 q^{74} + 2 q^{76} - 10 q^{79} - 6 q^{82} - 12 q^{83} - q^{86} + 15 q^{89} + 5 q^{91} - 9 q^{92} - 6 q^{94} + 8 q^{97} + q^{98}+O(q^{100})$$ q + q^2 + q^4 + q^7 + q^8 + 5 * q^13 + q^14 + q^16 + 3 * q^17 + 2 * q^19 - 9 * q^23 - 5 * q^25 + 5 * q^26 + q^28 - 3 * q^29 + 5 * q^31 + q^32 + 3 * q^34 + 2 * q^37 + 2 * q^38 - 6 * q^41 - q^43 - 9 * q^46 - 6 * q^47 + q^49 - 5 * q^50 + 5 * q^52 + 3 * q^53 + q^56 - 3 * q^58 - 3 * q^59 - 10 * q^61 + 5 * q^62 + q^64 - 13 * q^67 + 3 * q^68 + 9 * q^71 + 2 * q^73 + 2 * q^74 + 2 * q^76 - 10 * q^79 - 6 * q^82 - 12 * q^83 - q^86 + 15 * q^89 + 5 * q^91 - 9 * q^92 - 6 * q^94 + 8 * q^97 + 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 0 0 1.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 378.2.a.e yes 1
3.b odd 2 1 378.2.a.d 1
4.b odd 2 1 3024.2.a.p 1
5.b even 2 1 9450.2.a.l 1
7.b odd 2 1 2646.2.a.y 1
9.c even 3 2 1134.2.f.e 2
9.d odd 6 2 1134.2.f.k 2
12.b even 2 1 3024.2.a.o 1
15.d odd 2 1 9450.2.a.cl 1
21.c even 2 1 2646.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.d 1 3.b odd 2 1
378.2.a.e yes 1 1.a even 1 1 trivial
1134.2.f.e 2 9.c even 3 2
1134.2.f.k 2 9.d odd 6 2
2646.2.a.f 1 21.c even 2 1
2646.2.a.y 1 7.b odd 2 1
3024.2.a.o 1 12.b even 2 1
3024.2.a.p 1 4.b odd 2 1
9450.2.a.l 1 5.b even 2 1
9450.2.a.cl 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(378))$$:

 $$T_{5}$$ T5 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 5$$
$17$ $$T - 3$$
$19$ $$T - 2$$
$23$ $$T + 9$$
$29$ $$T + 3$$
$31$ $$T - 5$$
$37$ $$T - 2$$
$41$ $$T + 6$$
$43$ $$T + 1$$
$47$ $$T + 6$$
$53$ $$T - 3$$
$59$ $$T + 3$$
$61$ $$T + 10$$
$67$ $$T + 13$$
$71$ $$T - 9$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T + 12$$
$89$ $$T - 15$$
$97$ $$T - 8$$