# Properties

 Label 378.2.f.a Level $378$ Weight $2$ Character orbit 378.f Analytic conductor $3.018$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(127,378)]

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([4, 0]))

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

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

chi := DirichletCharacter("378.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.f (of order $$3$$, degree $$2$$, not 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 - 3*z * q^5 + (z - 1) * q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} - 3 \zeta_{6} q^{5} + (\zeta_{6} - 1) q^{7} + q^{8} + 3 q^{10} + (6 \zeta_{6} - 6) q^{11} - 2 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 6 q^{17} - 7 q^{19} + (3 \zeta_{6} - 3) q^{20} - 6 \zeta_{6} q^{22} + 3 \zeta_{6} q^{23} + (4 \zeta_{6} - 4) q^{25} + 2 q^{26} + q^{28} + ( - 6 \zeta_{6} + 6) q^{29} - 2 \zeta_{6} q^{31} - \zeta_{6} q^{32} + ( - 6 \zeta_{6} + 6) q^{34} + 3 q^{35} + 2 q^{37} + ( - 7 \zeta_{6} + 7) q^{38} - 3 \zeta_{6} q^{40} + (2 \zeta_{6} - 2) q^{43} + 6 q^{44} - 3 q^{46} - \zeta_{6} q^{49} - 4 \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} - 6 q^{53} + 18 q^{55} + (\zeta_{6} - 1) q^{56} + 6 \zeta_{6} q^{58} + (5 \zeta_{6} - 5) q^{61} + 2 q^{62} + q^{64} + (6 \zeta_{6} - 6) q^{65} - 8 \zeta_{6} q^{67} + 6 \zeta_{6} q^{68} + (3 \zeta_{6} - 3) q^{70} - 3 q^{71} + 2 q^{73} + (2 \zeta_{6} - 2) q^{74} + 7 \zeta_{6} q^{76} - 6 \zeta_{6} q^{77} + (5 \zeta_{6} - 5) q^{79} + 3 q^{80} + ( - 12 \zeta_{6} + 12) q^{83} + 18 \zeta_{6} q^{85} - 2 \zeta_{6} q^{86} + (6 \zeta_{6} - 6) q^{88} + 2 q^{91} + ( - 3 \zeta_{6} + 3) q^{92} + 21 \zeta_{6} q^{95} + (2 \zeta_{6} - 2) q^{97} + q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 - 3*z * q^5 + (z - 1) * q^7 + q^8 + 3 * q^10 + (6*z - 6) * q^11 - 2*z * q^13 - z * q^14 + (z - 1) * q^16 - 6 * q^17 - 7 * q^19 + (3*z - 3) * q^20 - 6*z * q^22 + 3*z * q^23 + (4*z - 4) * q^25 + 2 * q^26 + q^28 + (-6*z + 6) * q^29 - 2*z * q^31 - z * q^32 + (-6*z + 6) * q^34 + 3 * q^35 + 2 * q^37 + (-7*z + 7) * q^38 - 3*z * q^40 + (2*z - 2) * q^43 + 6 * q^44 - 3 * q^46 - z * q^49 - 4*z * q^50 + (2*z - 2) * q^52 - 6 * q^53 + 18 * q^55 + (z - 1) * q^56 + 6*z * q^58 + (5*z - 5) * q^61 + 2 * q^62 + q^64 + (6*z - 6) * q^65 - 8*z * q^67 + 6*z * q^68 + (3*z - 3) * q^70 - 3 * q^71 + 2 * q^73 + (2*z - 2) * q^74 + 7*z * q^76 - 6*z * q^77 + (5*z - 5) * q^79 + 3 * q^80 + (-12*z + 12) * q^83 + 18*z * q^85 - 2*z * q^86 + (6*z - 6) * q^88 + 2 * q^91 + (-3*z + 3) * q^92 + 21*z * q^95 + (2*z - 2) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} - 3 q^{5} - q^{7} + 2 q^{8} + 6 q^{10} - 6 q^{11} - 2 q^{13} - q^{14} - q^{16} - 12 q^{17} - 14 q^{19} - 3 q^{20} - 6 q^{22} + 3 q^{23} - 4 q^{25} + 4 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} - q^{32} + 6 q^{34} + 6 q^{35} + 4 q^{37} + 7 q^{38} - 3 q^{40} - 2 q^{43} + 12 q^{44} - 6 q^{46} - q^{49} - 4 q^{50} - 2 q^{52} - 12 q^{53} + 36 q^{55} - q^{56} + 6 q^{58} - 5 q^{61} + 4 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{67} + 6 q^{68} - 3 q^{70} - 6 q^{71} + 4 q^{73} - 2 q^{74} + 7 q^{76} - 6 q^{77} - 5 q^{79} + 6 q^{80} + 12 q^{83} + 18 q^{85} - 2 q^{86} - 6 q^{88} + 4 q^{91} + 3 q^{92} + 21 q^{95} - 2 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 - 3 * q^5 - q^7 + 2 * q^8 + 6 * q^10 - 6 * q^11 - 2 * q^13 - q^14 - q^16 - 12 * q^17 - 14 * q^19 - 3 * q^20 - 6 * q^22 + 3 * q^23 - 4 * q^25 + 4 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 - q^32 + 6 * q^34 + 6 * q^35 + 4 * q^37 + 7 * q^38 - 3 * q^40 - 2 * q^43 + 12 * q^44 - 6 * q^46 - q^49 - 4 * q^50 - 2 * q^52 - 12 * q^53 + 36 * q^55 - q^56 + 6 * q^58 - 5 * q^61 + 4 * q^62 + 2 * q^64 - 6 * q^65 - 8 * q^67 + 6 * q^68 - 3 * q^70 - 6 * q^71 + 4 * q^73 - 2 * q^74 + 7 * q^76 - 6 * q^77 - 5 * q^79 + 6 * q^80 + 12 * q^83 + 18 * q^85 - 2 * q^86 - 6 * q^88 + 4 * q^91 + 3 * q^92 + 21 * q^95 - 2 * q^97 + 2 * q^98

## Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i −1.50000 2.59808i 0 −0.500000 + 0.866025i 1.00000 0 3.00000
253.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i −1.50000 + 2.59808i 0 −0.500000 0.866025i 1.00000 0 3.00000
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.f.a 2
3.b odd 2 1 126.2.f.a 2
4.b odd 2 1 3024.2.r.a 2
7.b odd 2 1 2646.2.f.c 2
7.c even 3 1 2646.2.e.f 2
7.c even 3 1 2646.2.h.e 2
7.d odd 6 1 2646.2.e.j 2
7.d odd 6 1 2646.2.h.a 2
9.c even 3 1 inner 378.2.f.a 2
9.c even 3 1 1134.2.a.h 1
9.d odd 6 1 126.2.f.a 2
9.d odd 6 1 1134.2.a.a 1
12.b even 2 1 1008.2.r.d 2
21.c even 2 1 882.2.f.h 2
21.g even 6 1 882.2.e.d 2
21.g even 6 1 882.2.h.f 2
21.h odd 6 1 882.2.e.b 2
21.h odd 6 1 882.2.h.j 2
36.f odd 6 1 3024.2.r.a 2
36.f odd 6 1 9072.2.a.w 1
36.h even 6 1 1008.2.r.d 2
36.h even 6 1 9072.2.a.c 1
63.g even 3 1 2646.2.e.f 2
63.h even 3 1 2646.2.h.e 2
63.i even 6 1 882.2.h.f 2
63.j odd 6 1 882.2.h.j 2
63.k odd 6 1 2646.2.e.j 2
63.l odd 6 1 2646.2.f.c 2
63.l odd 6 1 7938.2.a.u 1
63.n odd 6 1 882.2.e.b 2
63.o even 6 1 882.2.f.h 2
63.o even 6 1 7938.2.a.l 1
63.s even 6 1 882.2.e.d 2
63.t odd 6 1 2646.2.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 3.b odd 2 1
126.2.f.a 2 9.d odd 6 1
378.2.f.a 2 1.a even 1 1 trivial
378.2.f.a 2 9.c even 3 1 inner
882.2.e.b 2 21.h odd 6 1
882.2.e.b 2 63.n odd 6 1
882.2.e.d 2 21.g even 6 1
882.2.e.d 2 63.s even 6 1
882.2.f.h 2 21.c even 2 1
882.2.f.h 2 63.o even 6 1
882.2.h.f 2 21.g even 6 1
882.2.h.f 2 63.i even 6 1
882.2.h.j 2 21.h odd 6 1
882.2.h.j 2 63.j odd 6 1
1008.2.r.d 2 12.b even 2 1
1008.2.r.d 2 36.h even 6 1
1134.2.a.a 1 9.d odd 6 1
1134.2.a.h 1 9.c even 3 1
2646.2.e.f 2 7.c even 3 1
2646.2.e.f 2 63.g even 3 1
2646.2.e.j 2 7.d odd 6 1
2646.2.e.j 2 63.k odd 6 1
2646.2.f.c 2 7.b odd 2 1
2646.2.f.c 2 63.l odd 6 1
2646.2.h.a 2 7.d odd 6 1
2646.2.h.a 2 63.t odd 6 1
2646.2.h.e 2 7.c even 3 1
2646.2.h.e 2 63.h even 3 1
3024.2.r.a 2 4.b odd 2 1
3024.2.r.a 2 36.f odd 6 1
7938.2.a.l 1 63.o even 6 1
7938.2.a.u 1 63.l odd 6 1
9072.2.a.c 1 36.h even 6 1
9072.2.a.w 1 36.f odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$T^{2} + 2T + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T + 7)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$T^{2} + 2T + 4$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 5T + 25$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$(T + 3)^{2}$$
$73$ $$(T - 2)^{2}$$
$79$ $$T^{2} + 5T + 25$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 2T + 4$$