# Properties

 Label 378.2.a.a 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} - 4 q^{5} - q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 - 4 * q^5 - q^7 - q^8 $$q - q^{2} + q^{4} - 4 q^{5} - q^{7} - q^{8} + 4 q^{10} + 4 q^{11} + 3 q^{13} + q^{14} + q^{16} + 7 q^{17} + 2 q^{19} - 4 q^{20} - 4 q^{22} + q^{23} + 11 q^{25} - 3 q^{26} - q^{28} - q^{29} - 9 q^{31} - q^{32} - 7 q^{34} + 4 q^{35} + 2 q^{37} - 2 q^{38} + 4 q^{40} - 6 q^{41} + 11 q^{43} + 4 q^{44} - q^{46} + 6 q^{47} + q^{49} - 11 q^{50} + 3 q^{52} + 9 q^{53} - 16 q^{55} + q^{56} + q^{58} + 5 q^{59} - 6 q^{61} + 9 q^{62} + q^{64} - 12 q^{65} + 7 q^{67} + 7 q^{68} - 4 q^{70} + 7 q^{71} - 14 q^{73} - 2 q^{74} + 2 q^{76} - 4 q^{77} - 6 q^{79} - 4 q^{80} + 6 q^{82} + 4 q^{83} - 28 q^{85} - 11 q^{86} - 4 q^{88} + 3 q^{89} - 3 q^{91} + q^{92} - 6 q^{94} - 8 q^{95} - 8 q^{97} - q^{98}+O(q^{100})$$ q - q^2 + q^4 - 4 * q^5 - q^7 - q^8 + 4 * q^10 + 4 * q^11 + 3 * q^13 + q^14 + q^16 + 7 * q^17 + 2 * q^19 - 4 * q^20 - 4 * q^22 + q^23 + 11 * q^25 - 3 * q^26 - q^28 - q^29 - 9 * q^31 - q^32 - 7 * q^34 + 4 * q^35 + 2 * q^37 - 2 * q^38 + 4 * q^40 - 6 * q^41 + 11 * q^43 + 4 * q^44 - q^46 + 6 * q^47 + q^49 - 11 * q^50 + 3 * q^52 + 9 * q^53 - 16 * q^55 + q^56 + q^58 + 5 * q^59 - 6 * q^61 + 9 * q^62 + q^64 - 12 * q^65 + 7 * q^67 + 7 * q^68 - 4 * q^70 + 7 * q^71 - 14 * q^73 - 2 * q^74 + 2 * q^76 - 4 * q^77 - 6 * q^79 - 4 * q^80 + 6 * q^82 + 4 * q^83 - 28 * q^85 - 11 * q^86 - 4 * q^88 + 3 * q^89 - 3 * q^91 + q^92 - 6 * q^94 - 8 * q^95 - 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 −4.00000 0 −1.00000 −1.00000 0 4.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$1$$

## 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.a 1
3.b odd 2 1 378.2.a.h yes 1
4.b odd 2 1 3024.2.a.a 1
5.b even 2 1 9450.2.a.dv 1
7.b odd 2 1 2646.2.a.o 1
9.c even 3 2 1134.2.f.p 2
9.d odd 6 2 1134.2.f.a 2
12.b even 2 1 3024.2.a.bd 1
15.d odd 2 1 9450.2.a.bc 1
21.c even 2 1 2646.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.a.a 1 1.a even 1 1 trivial
378.2.a.h yes 1 3.b odd 2 1
1134.2.f.a 2 9.d odd 6 2
1134.2.f.p 2 9.c even 3 2
2646.2.a.o 1 7.b odd 2 1
2646.2.a.p 1 21.c even 2 1
3024.2.a.a 1 4.b odd 2 1
3024.2.a.bd 1 12.b even 2 1
9450.2.a.bc 1 15.d odd 2 1
9450.2.a.dv 1 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(378))$$:

 $$T_{5} + 4$$ T5 + 4 $$T_{17} - 7$$ T17 - 7

## Hecke characteristic polynomials

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