# Properties

 Label 378.2.k.c Level $378$ Weight $2$ Character orbit 378.k Analytic conductor $3.018$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(215,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.k (of order $$6$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$3.01834519640$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8}+O(q^{10})$$ q + z * q^2 + z^2 * q^4 + (-2*z^3 + z) * q^5 + (-2*z^2 + 3) * q^7 + z^3 * q^8 $$q + \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{5} + ( - 2 \zeta_{12}^{2} + 3) q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 2) q^{10} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + (4 \zeta_{12}^{2} + 4) q^{19} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{20} - 6 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{25} + (\zeta_{12}^{2} + 2) q^{28} + 9 \zeta_{12}^{3} q^{29} + ( - 2 \zeta_{12}^{2} + 4) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - 4 \zeta_{12}^{2} + 2) q^{34} + ( - 4 \zeta_{12}^{3} - \zeta_{12}) q^{35} + (4 \zeta_{12}^{2} - 4) q^{37} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{38} + (\zeta_{12}^{2} + 1) q^{40} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{41} - 8 q^{43} - 6 \zeta_{12}^{2} q^{46} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 5) q^{49} + 2 \zeta_{12}^{3} q^{50} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{53} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{56} + (9 \zeta_{12}^{2} - 9) q^{58} + ( - 7 \zeta_{12}^{3} - 7 \zeta_{12}) q^{59} + (2 \zeta_{12}^{2} + 2) q^{61} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{62} - q^{64} - 14 \zeta_{12}^{2} q^{67} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + ( - 5 \zeta_{12}^{2} + 4) q^{70} - 6 \zeta_{12}^{3} q^{71} + (7 \zeta_{12}^{2} - 14) q^{73} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{74} + (8 \zeta_{12}^{2} - 4) q^{76} + (11 \zeta_{12}^{2} - 11) q^{79} + (\zeta_{12}^{3} + \zeta_{12}) q^{80} + ( - 2 \zeta_{12}^{2} - 2) q^{82} + ( - 10 \zeta_{12}^{3} + 20 \zeta_{12}) q^{83} - 6 q^{85} - 8 \zeta_{12} q^{86} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{89} - 6 \zeta_{12}^{3} q^{92} + (2 \zeta_{12}^{2} - 4) q^{94} + ( - 12 \zeta_{12}^{3} + 12 \zeta_{12}) q^{95} + ( - 8 \zeta_{12}^{2} + 4) q^{97} + ( - 8 \zeta_{12}^{3} + 5 \zeta_{12}) q^{98} +O(q^{100})$$ q + z * q^2 + z^2 * q^4 + (-2*z^3 + z) * q^5 + (-2*z^2 + 3) * q^7 + z^3 * q^8 + (-z^2 + 2) * q^10 + (-2*z^3 + 3*z) * q^14 + (z^2 - 1) * q^16 + (-2*z^3 - 2*z) * q^17 + (4*z^2 + 4) * q^19 + (-z^3 + 2*z) * q^20 - 6*z * q^23 + 2*z^2 * q^25 + (z^2 + 2) * q^28 + 9*z^3 * q^29 + (-2*z^2 + 4) * q^31 + (z^3 - z) * q^32 + (-4*z^2 + 2) * q^34 + (-4*z^3 - z) * q^35 + (4*z^2 - 4) * q^37 + (4*z^3 + 4*z) * q^38 + (z^2 + 1) * q^40 + (2*z^3 - 4*z) * q^41 - 8 * q^43 - 6*z^2 * q^46 + (4*z^3 - 2*z) * q^47 + (-8*z^2 + 5) * q^49 + 2*z^3 * q^50 + (3*z^3 - 3*z) * q^53 + (z^3 + 2*z) * q^56 + (9*z^2 - 9) * q^58 + (-7*z^3 - 7*z) * q^59 + (2*z^2 + 2) * q^61 + (-2*z^3 + 4*z) * q^62 - q^64 - 14*z^2 * q^67 + (-4*z^3 + 2*z) * q^68 + (-5*z^2 + 4) * q^70 - 6*z^3 * q^71 + (7*z^2 - 14) * q^73 + (4*z^3 - 4*z) * q^74 + (8*z^2 - 4) * q^76 + (11*z^2 - 11) * q^79 + (z^3 + z) * q^80 + (-2*z^2 - 2) * q^82 + (-10*z^3 + 20*z) * q^83 - 6 * q^85 - 8*z * q^86 + (12*z^3 - 6*z) * q^89 - 6*z^3 * q^92 + (2*z^2 - 4) * q^94 + (-12*z^3 + 12*z) * q^95 + (-8*z^2 + 4) * q^97 + (-8*z^3 + 5*z) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q + 2 * q^4 + 8 * q^7 $$4 q + 2 q^{4} + 8 q^{7} + 6 q^{10} - 2 q^{16} + 24 q^{19} + 4 q^{25} + 10 q^{28} + 12 q^{31} - 8 q^{37} + 6 q^{40} - 32 q^{43} - 12 q^{46} + 4 q^{49} - 18 q^{58} + 12 q^{61} - 4 q^{64} - 28 q^{67} + 6 q^{70} - 42 q^{73} - 22 q^{79} - 12 q^{82} - 24 q^{85} - 12 q^{94}+O(q^{100})$$ 4 * q + 2 * q^4 + 8 * q^7 + 6 * q^10 - 2 * q^16 + 24 * q^19 + 4 * q^25 + 10 * q^28 + 12 * q^31 - 8 * q^37 + 6 * q^40 - 32 * q^43 - 12 * q^46 + 4 * q^49 - 18 * q^58 + 12 * q^61 - 4 * q^64 - 28 * q^67 + 6 * q^70 - 42 * q^73 - 22 * q^79 - 12 * q^82 - 24 * q^85 - 12 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.866025 + 1.50000i 0 2.00000 1.73205i 1.00000i 0 1.50000 0.866025i
215.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 1.50000i 0 2.00000 1.73205i 1.00000i 0 1.50000 0.866025i
269.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.866025 1.50000i 0 2.00000 + 1.73205i 1.00000i 0 1.50000 + 0.866025i
269.2 0.866025 0.500000i 0 0.500000 0.866025i 0.866025 + 1.50000i 0 2.00000 + 1.73205i 1.00000i 0 1.50000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.k.c 4
3.b odd 2 1 inner 378.2.k.c 4
7.c even 3 1 2646.2.d.a 4
7.d odd 6 1 inner 378.2.k.c 4
7.d odd 6 1 2646.2.d.a 4
9.c even 3 1 1134.2.l.b 4
9.c even 3 1 1134.2.t.c 4
9.d odd 6 1 1134.2.l.b 4
9.d odd 6 1 1134.2.t.c 4
21.g even 6 1 inner 378.2.k.c 4
21.g even 6 1 2646.2.d.a 4
21.h odd 6 1 2646.2.d.a 4
63.i even 6 1 1134.2.t.c 4
63.k odd 6 1 1134.2.l.b 4
63.s even 6 1 1134.2.l.b 4
63.t odd 6 1 1134.2.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.k.c 4 1.a even 1 1 trivial
378.2.k.c 4 3.b odd 2 1 inner
378.2.k.c 4 7.d odd 6 1 inner
378.2.k.c 4 21.g even 6 1 inner
1134.2.l.b 4 9.c even 3 1
1134.2.l.b 4 9.d odd 6 1
1134.2.l.b 4 63.k odd 6 1
1134.2.l.b 4 63.s even 6 1
1134.2.t.c 4 9.c even 3 1
1134.2.t.c 4 9.d odd 6 1
1134.2.t.c 4 63.i even 6 1
1134.2.t.c 4 63.t odd 6 1
2646.2.d.a 4 7.c even 3 1
2646.2.d.a 4 7.d odd 6 1
2646.2.d.a 4 21.g even 6 1
2646.2.d.a 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 3T_{5}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(378, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 3T^{2} + 9$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} + 12T^{2} + 144$$
$19$ $$(T^{2} - 12 T + 48)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} + 81)^{2}$$
$31$ $$(T^{2} - 6 T + 12)^{2}$$
$37$ $$(T^{2} + 4 T + 16)^{2}$$
$41$ $$(T^{2} - 12)^{2}$$
$43$ $$(T + 8)^{4}$$
$47$ $$T^{4} + 12T^{2} + 144$$
$53$ $$T^{4} - 9T^{2} + 81$$
$59$ $$T^{4} + 147 T^{2} + 21609$$
$61$ $$(T^{2} - 6 T + 12)^{2}$$
$67$ $$(T^{2} + 14 T + 196)^{2}$$
$71$ $$(T^{2} + 36)^{2}$$
$73$ $$(T^{2} + 21 T + 147)^{2}$$
$79$ $$(T^{2} + 11 T + 121)^{2}$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$T^{4} + 108 T^{2} + 11664$$
$97$ $$(T^{2} + 48)^{2}$$
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