# Properties

 Label 378.2.e.a Level $378$ Weight $2$ Character orbit 378.e 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(37,378)]

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([2, 2]))

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

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

chi := DirichletCharacter("378.37");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.e (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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + (3 \zeta_{6} - 3) q^{5} + (2 \zeta_{6} - 3) q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + (3*z - 3) * q^5 + (2*z - 3) * q^7 - q^8 $$q - q^{2} + q^{4} + (3 \zeta_{6} - 3) q^{5} + (2 \zeta_{6} - 3) q^{7} - q^{8} + ( - 3 \zeta_{6} + 3) q^{10} - 3 \zeta_{6} q^{11} - 5 \zeta_{6} q^{13} + ( - 2 \zeta_{6} + 3) q^{14} + q^{16} + ( - 3 \zeta_{6} + 3) q^{17} - 5 \zeta_{6} q^{19} + (3 \zeta_{6} - 3) q^{20} + 3 \zeta_{6} q^{22} + (3 \zeta_{6} - 3) q^{23} - 4 \zeta_{6} q^{25} + 5 \zeta_{6} q^{26} + (2 \zeta_{6} - 3) q^{28} + (3 \zeta_{6} - 3) q^{29} - 4 q^{31} - q^{32} + (3 \zeta_{6} - 3) q^{34} + ( - 9 \zeta_{6} + 3) q^{35} + 7 \zeta_{6} q^{37} + 5 \zeta_{6} q^{38} + ( - 3 \zeta_{6} + 3) q^{40} - 9 \zeta_{6} q^{41} + (11 \zeta_{6} - 11) q^{43} - 3 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{46} + ( - 8 \zeta_{6} + 5) q^{49} + 4 \zeta_{6} q^{50} - 5 \zeta_{6} q^{52} + (3 \zeta_{6} - 3) q^{53} + 9 q^{55} + ( - 2 \zeta_{6} + 3) q^{56} + ( - 3 \zeta_{6} + 3) q^{58} - 12 q^{59} + 2 q^{61} + 4 q^{62} + q^{64} + 15 q^{65} - 4 q^{67} + ( - 3 \zeta_{6} + 3) q^{68} + (9 \zeta_{6} - 3) q^{70} + (11 \zeta_{6} - 11) q^{73} - 7 \zeta_{6} q^{74} - 5 \zeta_{6} q^{76} + (3 \zeta_{6} + 6) q^{77} + 8 q^{79} + (3 \zeta_{6} - 3) q^{80} + 9 \zeta_{6} q^{82} + ( - 3 \zeta_{6} + 3) q^{83} + 9 \zeta_{6} q^{85} + ( - 11 \zeta_{6} + 11) q^{86} + 3 \zeta_{6} q^{88} + 15 \zeta_{6} q^{89} + (5 \zeta_{6} + 10) q^{91} + (3 \zeta_{6} - 3) q^{92} + 15 q^{95} + ( - \zeta_{6} + 1) q^{97} + (8 \zeta_{6} - 5) q^{98} +O(q^{100})$$ q - q^2 + q^4 + (3*z - 3) * q^5 + (2*z - 3) * q^7 - q^8 + (-3*z + 3) * q^10 - 3*z * q^11 - 5*z * q^13 + (-2*z + 3) * q^14 + q^16 + (-3*z + 3) * q^17 - 5*z * q^19 + (3*z - 3) * q^20 + 3*z * q^22 + (3*z - 3) * q^23 - 4*z * q^25 + 5*z * q^26 + (2*z - 3) * q^28 + (3*z - 3) * q^29 - 4 * q^31 - q^32 + (3*z - 3) * q^34 + (-9*z + 3) * q^35 + 7*z * q^37 + 5*z * q^38 + (-3*z + 3) * q^40 - 9*z * q^41 + (11*z - 11) * q^43 - 3*z * q^44 + (-3*z + 3) * q^46 + (-8*z + 5) * q^49 + 4*z * q^50 - 5*z * q^52 + (3*z - 3) * q^53 + 9 * q^55 + (-2*z + 3) * q^56 + (-3*z + 3) * q^58 - 12 * q^59 + 2 * q^61 + 4 * q^62 + q^64 + 15 * q^65 - 4 * q^67 + (-3*z + 3) * q^68 + (9*z - 3) * q^70 + (11*z - 11) * q^73 - 7*z * q^74 - 5*z * q^76 + (3*z + 6) * q^77 + 8 * q^79 + (3*z - 3) * q^80 + 9*z * q^82 + (-3*z + 3) * q^83 + 9*z * q^85 + (-11*z + 11) * q^86 + 3*z * q^88 + 15*z * q^89 + (5*z + 10) * q^91 + (3*z - 3) * q^92 + 15 * q^95 + (-z + 1) * q^97 + (8*z - 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 4 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 4 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} - 3 q^{5} - 4 q^{7} - 2 q^{8} + 3 q^{10} - 3 q^{11} - 5 q^{13} + 4 q^{14} + 2 q^{16} + 3 q^{17} - 5 q^{19} - 3 q^{20} + 3 q^{22} - 3 q^{23} - 4 q^{25} + 5 q^{26} - 4 q^{28} - 3 q^{29} - 8 q^{31} - 2 q^{32} - 3 q^{34} - 3 q^{35} + 7 q^{37} + 5 q^{38} + 3 q^{40} - 9 q^{41} - 11 q^{43} - 3 q^{44} + 3 q^{46} + 2 q^{49} + 4 q^{50} - 5 q^{52} - 3 q^{53} + 18 q^{55} + 4 q^{56} + 3 q^{58} - 24 q^{59} + 4 q^{61} + 8 q^{62} + 2 q^{64} + 30 q^{65} - 8 q^{67} + 3 q^{68} + 3 q^{70} - 11 q^{73} - 7 q^{74} - 5 q^{76} + 15 q^{77} + 16 q^{79} - 3 q^{80} + 9 q^{82} + 3 q^{83} + 9 q^{85} + 11 q^{86} + 3 q^{88} + 15 q^{89} + 25 q^{91} - 3 q^{92} + 30 q^{95} + q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 3 * q^5 - 4 * q^7 - 2 * q^8 + 3 * q^10 - 3 * q^11 - 5 * q^13 + 4 * q^14 + 2 * q^16 + 3 * q^17 - 5 * q^19 - 3 * q^20 + 3 * q^22 - 3 * q^23 - 4 * q^25 + 5 * q^26 - 4 * q^28 - 3 * q^29 - 8 * q^31 - 2 * q^32 - 3 * q^34 - 3 * q^35 + 7 * q^37 + 5 * q^38 + 3 * q^40 - 9 * q^41 - 11 * q^43 - 3 * q^44 + 3 * q^46 + 2 * q^49 + 4 * q^50 - 5 * q^52 - 3 * q^53 + 18 * q^55 + 4 * q^56 + 3 * q^58 - 24 * q^59 + 4 * q^61 + 8 * q^62 + 2 * q^64 + 30 * q^65 - 8 * q^67 + 3 * q^68 + 3 * q^70 - 11 * q^73 - 7 * q^74 - 5 * q^76 + 15 * q^77 + 16 * q^79 - 3 * q^80 + 9 * q^82 + 3 * q^83 + 9 * q^85 + 11 * q^86 + 3 * q^88 + 15 * q^89 + 25 * q^91 - 3 * q^92 + 30 * q^95 + q^97 - 2 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0 1.00000 −1.50000 2.59808i 0 −2.00000 1.73205i −1.00000 0 1.50000 + 2.59808i
235.1 −1.00000 0 1.00000 −1.50000 + 2.59808i 0 −2.00000 + 1.73205i −1.00000 0 1.50000 2.59808i
 $$n$$: 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
63.h 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.e.a 2
3.b odd 2 1 126.2.e.b 2
4.b odd 2 1 3024.2.q.a 2
7.b odd 2 1 2646.2.e.e 2
7.c even 3 1 378.2.h.b 2
7.c even 3 1 2646.2.f.e 2
7.d odd 6 1 2646.2.f.i 2
7.d odd 6 1 2646.2.h.f 2
9.c even 3 1 378.2.h.b 2
9.c even 3 1 1134.2.g.f 2
9.d odd 6 1 126.2.h.a yes 2
9.d odd 6 1 1134.2.g.d 2
12.b even 2 1 1008.2.q.e 2
21.c even 2 1 882.2.e.h 2
21.g even 6 1 882.2.f.a 2
21.g even 6 1 882.2.h.e 2
21.h odd 6 1 126.2.h.a yes 2
21.h odd 6 1 882.2.f.e 2
28.g odd 6 1 3024.2.t.f 2
36.f odd 6 1 3024.2.t.f 2
36.h even 6 1 1008.2.t.c 2
63.g even 3 1 1134.2.g.f 2
63.g even 3 1 2646.2.f.e 2
63.h even 3 1 inner 378.2.e.a 2
63.h even 3 1 7938.2.a.o 1
63.i even 6 1 882.2.e.h 2
63.i even 6 1 7938.2.a.bd 1
63.j odd 6 1 126.2.e.b 2
63.j odd 6 1 7938.2.a.r 1
63.k odd 6 1 2646.2.f.i 2
63.l odd 6 1 2646.2.h.f 2
63.n odd 6 1 882.2.f.e 2
63.n odd 6 1 1134.2.g.d 2
63.o even 6 1 882.2.h.e 2
63.s even 6 1 882.2.f.a 2
63.t odd 6 1 2646.2.e.e 2
63.t odd 6 1 7938.2.a.c 1
84.n even 6 1 1008.2.t.c 2
252.u odd 6 1 3024.2.q.a 2
252.bb even 6 1 1008.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.b 2 3.b odd 2 1
126.2.e.b 2 63.j odd 6 1
126.2.h.a yes 2 9.d odd 6 1
126.2.h.a yes 2 21.h odd 6 1
378.2.e.a 2 1.a even 1 1 trivial
378.2.e.a 2 63.h even 3 1 inner
378.2.h.b 2 7.c even 3 1
378.2.h.b 2 9.c even 3 1
882.2.e.h 2 21.c even 2 1
882.2.e.h 2 63.i even 6 1
882.2.f.a 2 21.g even 6 1
882.2.f.a 2 63.s even 6 1
882.2.f.e 2 21.h odd 6 1
882.2.f.e 2 63.n odd 6 1
882.2.h.e 2 21.g even 6 1
882.2.h.e 2 63.o even 6 1
1008.2.q.e 2 12.b even 2 1
1008.2.q.e 2 252.bb even 6 1
1008.2.t.c 2 36.h even 6 1
1008.2.t.c 2 84.n even 6 1
1134.2.g.d 2 9.d odd 6 1
1134.2.g.d 2 63.n odd 6 1
1134.2.g.f 2 9.c even 3 1
1134.2.g.f 2 63.g even 3 1
2646.2.e.e 2 7.b odd 2 1
2646.2.e.e 2 63.t odd 6 1
2646.2.f.e 2 7.c even 3 1
2646.2.f.e 2 63.g even 3 1
2646.2.f.i 2 7.d odd 6 1
2646.2.f.i 2 63.k odd 6 1
2646.2.h.f 2 7.d odd 6 1
2646.2.h.f 2 63.l odd 6 1
3024.2.q.a 2 4.b odd 2 1
3024.2.q.a 2 252.u odd 6 1
3024.2.t.f 2 28.g odd 6 1
3024.2.t.f 2 36.f odd 6 1
7938.2.a.c 1 63.t odd 6 1
7938.2.a.o 1 63.h even 3 1
7938.2.a.r 1 63.j odd 6 1
7938.2.a.bd 1 63.i even 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

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3T + 9$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 11T + 121$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 3T + 9$$
$89$ $$T^{2} - 15T + 225$$
$97$ $$T^{2} - T + 1$$