Properties

 Label 378.2.d.b Level $378$ Weight $2$ Character orbit 378.d Analytic conductor $3.018$ Analytic rank $0$ Dimension $4$ Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(377,378)]

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([1, 1]))

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

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

chi := DirichletCharacter("378.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.d (of order $$2$$, degree $$1$$, 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: $$4$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - q^4 - b3 * q^5 + (-b2 + 2) * q^7 + b1 * q^8 $$q - \beta_1 q^{2} - q^{4} - \beta_{3} q^{5} + ( - \beta_{2} + 2) q^{7} + \beta_1 q^{8} + \beta_{2} q^{10} - 3 \beta_1 q^{11} - 2 \beta_{2} q^{13} + ( - \beta_{3} - 2 \beta_1) q^{14} + q^{16} - 4 \beta_{3} q^{17} - \beta_{2} q^{19} + \beta_{3} q^{20} - 3 q^{22} - 3 \beta_1 q^{23} - 2 q^{25} - 2 \beta_{3} q^{26} + (\beta_{2} - 2) q^{28} - 6 \beta_1 q^{29} + 3 \beta_{2} q^{31} - \beta_1 q^{32} + 4 \beta_{2} q^{34} + ( - 2 \beta_{3} + 3 \beta_1) q^{35} + 7 q^{37} - \beta_{3} q^{38} - \beta_{2} q^{40} + 7 \beta_{3} q^{41} - 2 q^{43} + 3 \beta_1 q^{44} - 3 q^{46} + 2 \beta_{3} q^{47} + ( - 4 \beta_{2} + 1) q^{49} + 2 \beta_1 q^{50} + 2 \beta_{2} q^{52} + 12 \beta_1 q^{53} + 3 \beta_{2} q^{55} + (\beta_{3} + 2 \beta_1) q^{56} - 6 q^{58} - 2 \beta_{3} q^{59} - 4 \beta_{2} q^{61} + 3 \beta_{3} q^{62} - q^{64} + 6 \beta_1 q^{65} + 2 q^{67} + 4 \beta_{3} q^{68} + (2 \beta_{2} + 3) q^{70} + 3 \beta_1 q^{71} + 2 \beta_{2} q^{73} - 7 \beta_1 q^{74} + \beta_{2} q^{76} + ( - 3 \beta_{3} - 6 \beta_1) q^{77} - 10 q^{79} - \beta_{3} q^{80} - 7 \beta_{2} q^{82} + 10 \beta_{3} q^{83} + 12 q^{85} + 2 \beta_1 q^{86} + 3 q^{88} + 3 \beta_{3} q^{89} + ( - 4 \beta_{2} - 6) q^{91} + 3 \beta_1 q^{92} - 2 \beta_{2} q^{94} + 3 \beta_1 q^{95} - 8 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - q^4 - b3 * q^5 + (-b2 + 2) * q^7 + b1 * q^8 + b2 * q^10 - 3*b1 * q^11 - 2*b2 * q^13 + (-b3 - 2*b1) * q^14 + q^16 - 4*b3 * q^17 - b2 * q^19 + b3 * q^20 - 3 * q^22 - 3*b1 * q^23 - 2 * q^25 - 2*b3 * q^26 + (b2 - 2) * q^28 - 6*b1 * q^29 + 3*b2 * q^31 - b1 * q^32 + 4*b2 * q^34 + (-2*b3 + 3*b1) * q^35 + 7 * q^37 - b3 * q^38 - b2 * q^40 + 7*b3 * q^41 - 2 * q^43 + 3*b1 * q^44 - 3 * q^46 + 2*b3 * q^47 + (-4*b2 + 1) * q^49 + 2*b1 * q^50 + 2*b2 * q^52 + 12*b1 * q^53 + 3*b2 * q^55 + (b3 + 2*b1) * q^56 - 6 * q^58 - 2*b3 * q^59 - 4*b2 * q^61 + 3*b3 * q^62 - q^64 + 6*b1 * q^65 + 2 * q^67 + 4*b3 * q^68 + (2*b2 + 3) * q^70 + 3*b1 * q^71 + 2*b2 * q^73 - 7*b1 * q^74 + b2 * q^76 + (-3*b3 - 6*b1) * q^77 - 10 * q^79 - b3 * q^80 - 7*b2 * q^82 + 10*b3 * q^83 + 12 * q^85 + 2*b1 * q^86 + 3 * q^88 + 3*b3 * q^89 + (-4*b2 - 6) * q^91 + 3*b1 * q^92 - 2*b2 * q^94 + 3*b1 * q^95 - 8*b2 * q^97 + (-4*b3 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 + 8 * q^7 $$4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} - 8 q^{28} + 28 q^{37} - 8 q^{43} - 12 q^{46} + 4 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} + 12 q^{70} - 40 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100})$$ 4 * q - 4 * q^4 + 8 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 - 8 * q^28 + 28 * q^37 - 8 * q^43 - 12 * q^46 + 4 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 + 12 * q^70 - 40 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

Character values

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

 $$n$$ $$29$$ $$325$$ $$\chi(n)$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
377.1
 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i
1.00000i 0 −1.00000 −1.73205 0 2.00000 1.73205i 1.00000i 0 1.73205i
377.2 1.00000i 0 −1.00000 1.73205 0 2.00000 + 1.73205i 1.00000i 0 1.73205i
377.3 1.00000i 0 −1.00000 −1.73205 0 2.00000 + 1.73205i 1.00000i 0 1.73205i
377.4 1.00000i 0 −1.00000 1.73205 0 2.00000 1.73205i 1.00000i 0 1.73205i
 $$n$$: 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.b odd 2 1 inner
21.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.2.d.b 4
3.b odd 2 1 inner 378.2.d.b 4
4.b odd 2 1 3024.2.k.e 4
7.b odd 2 1 inner 378.2.d.b 4
9.c even 3 1 1134.2.m.b 4
9.c even 3 1 1134.2.m.c 4
9.d odd 6 1 1134.2.m.b 4
9.d odd 6 1 1134.2.m.c 4
12.b even 2 1 3024.2.k.e 4
21.c even 2 1 inner 378.2.d.b 4
28.d even 2 1 3024.2.k.e 4
63.l odd 6 1 1134.2.m.b 4
63.l odd 6 1 1134.2.m.c 4
63.o even 6 1 1134.2.m.b 4
63.o even 6 1 1134.2.m.c 4
84.h odd 2 1 3024.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.2.d.b 4 1.a even 1 1 trivial
378.2.d.b 4 3.b odd 2 1 inner
378.2.d.b 4 7.b odd 2 1 inner
378.2.d.b 4 21.c even 2 1 inner
1134.2.m.b 4 9.c even 3 1
1134.2.m.b 4 9.d odd 6 1
1134.2.m.b 4 63.l odd 6 1
1134.2.m.b 4 63.o even 6 1
1134.2.m.c 4 9.c even 3 1
1134.2.m.c 4 9.d odd 6 1
1134.2.m.c 4 63.l odd 6 1
1134.2.m.c 4 63.o even 6 1
3024.2.k.e 4 4.b odd 2 1
3024.2.k.e 4 12.b even 2 1
3024.2.k.e 4 28.d even 2 1
3024.2.k.e 4 84.h odd 2 1

Hecke kernels

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

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{13}^{2} + 12$$ T13^2 + 12

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$(T^{2} + 9)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$(T^{2} - 48)^{2}$$
$19$ $$(T^{2} + 3)^{2}$$
$23$ $$(T^{2} + 9)^{2}$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$(T^{2} + 27)^{2}$$
$37$ $$(T - 7)^{4}$$
$41$ $$(T^{2} - 147)^{2}$$
$43$ $$(T + 2)^{4}$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 144)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$(T^{2} + 48)^{2}$$
$67$ $$(T - 2)^{4}$$
$71$ $$(T^{2} + 9)^{2}$$
$73$ $$(T^{2} + 12)^{2}$$
$79$ $$(T + 10)^{4}$$
$83$ $$(T^{2} - 300)^{2}$$
$89$ $$(T^{2} - 27)^{2}$$
$97$ $$(T^{2} + 192)^{2}$$