# Properties

 Label 390.2.bb.a Level $390$ Weight $2$ Character orbit 390.bb Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(121,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("390.121");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.bb (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: $$3.11416567883$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{6} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} +O(q^{10})$$ q + z * q^2 + (-z^2 + 1) * q^3 + z^2 * q^4 + z^3 * q^5 + (-z^3 + z) * q^6 + (-3*z^3 + 3*z) * q^7 + z^3 * q^8 - z^2 * q^9 $$q + \zeta_{12} q^{2} + ( - \zeta_{12}^{2} + 1) q^{3} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{3} q^{5} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{6} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{2} q^{9} + (\zeta_{12}^{2} - 1) q^{10} + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{11} + q^{12} + (3 \zeta_{12}^{3} + \zeta_{12}) q^{13} + 3 q^{14} + \zeta_{12} q^{15} + (\zeta_{12}^{2} - 1) q^{16} - 4 \zeta_{12}^{2} q^{17} - \zeta_{12}^{3} q^{18} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + 4 \zeta_{12} + 2) q^{19} + (\zeta_{12}^{3} - \zeta_{12}) q^{20} - 3 \zeta_{12}^{3} q^{21} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} - \zeta_{12}) q^{22} + (4 \zeta_{12}^{3} - 2 \zeta_{12}) q^{23} + \zeta_{12} q^{24} - q^{25} + (4 \zeta_{12}^{2} - 3) q^{26} - q^{27} + 3 \zeta_{12} q^{28} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{29} + \zeta_{12}^{2} q^{30} + ( - 2 \zeta_{12}^{3} + 8 \zeta_{12}^{2} - 4) q^{31} + (\zeta_{12}^{3} - \zeta_{12}) q^{32} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + 2 \zeta_{12} - 2) q^{33} - 4 \zeta_{12}^{3} q^{34} + 3 \zeta_{12}^{2} q^{35} + ( - \zeta_{12}^{2} + 1) q^{36} + ( - 4 \zeta_{12}^{2} + \zeta_{12} - 4) q^{37} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 4) q^{38} + ( - \zeta_{12}^{3} + 4 \zeta_{12}) q^{39} - q^{40} - 4 \zeta_{12} q^{41} + ( - 3 \zeta_{12}^{2} + 3) q^{42} - 6 \zeta_{12}^{2} q^{43} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{44} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{45} + (2 \zeta_{12}^{2} - 4) q^{46} + ( - 3 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{47} + \zeta_{12}^{2} q^{48} + ( - 2 \zeta_{12}^{2} + 2) q^{49} - \zeta_{12} q^{50} - 4 q^{51} + (4 \zeta_{12}^{3} - 3 \zeta_{12}) q^{52} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 2) q^{53} - \zeta_{12} q^{54} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{55} + 3 \zeta_{12}^{2} q^{56} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{57} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{58} + (8 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 8 \zeta_{12} - 4) q^{59} + \zeta_{12}^{3} q^{60} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12}) q^{61} + (8 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{62} - 3 \zeta_{12} q^{63} - q^{64} + (\zeta_{12}^{2} - 4) q^{65} + (\zeta_{12}^{3} - 2 \zeta_{12} + 2) q^{66} + (2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{67} + ( - 4 \zeta_{12}^{2} + 4) q^{68} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{69} + 3 \zeta_{12}^{3} q^{70} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 6 \zeta_{12} - 8) q^{71} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{72} + ( - 8 \zeta_{12}^{2} + 4) q^{73} + ( - 4 \zeta_{12}^{3} + \zeta_{12}^{2} - 4 \zeta_{12}) q^{74} + (\zeta_{12}^{2} - 1) q^{75} + (\zeta_{12}^{2} + 4 \zeta_{12} + 1) q^{76} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 6) q^{77} + (3 \zeta_{12}^{2} + 1) q^{78} + (4 \zeta_{12}^{3} - 8 \zeta_{12} + 10) q^{79} - \zeta_{12} q^{80} + (\zeta_{12}^{2} - 1) q^{81} - 4 \zeta_{12}^{2} q^{82} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{83} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{84} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{85} - 6 \zeta_{12}^{3} q^{86} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{87} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{88} + ( - 7 \zeta_{12}^{2} - 2 \zeta_{12} - 7) q^{89} + q^{90} + (9 \zeta_{12}^{2} + 3) q^{91} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{92} + (4 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{93} + ( - 4 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12} + 3) q^{94} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + \zeta_{12}) q^{95} + \zeta_{12}^{3} q^{96} + (2 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 12) q^{97} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{98} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{99} +O(q^{100})$$ q + z * q^2 + (-z^2 + 1) * q^3 + z^2 * q^4 + z^3 * q^5 + (-z^3 + z) * q^6 + (-3*z^3 + 3*z) * q^7 + z^3 * q^8 - z^2 * q^9 + (z^2 - 1) * q^10 + (-z^2 + 2*z - 1) * q^11 + q^12 + (3*z^3 + z) * q^13 + 3 * q^14 + z * q^15 + (z^2 - 1) * q^16 - 4*z^2 * q^17 - z^3 * q^18 + (-4*z^3 - z^2 + 4*z + 2) * q^19 + (z^3 - z) * q^20 - 3*z^3 * q^21 + (-z^3 + 2*z^2 - z) * q^22 + (4*z^3 - 2*z) * q^23 + z * q^24 - q^25 + (4*z^2 - 3) * q^26 - q^27 + 3*z * q^28 + (4*z^3 - 2*z^2 - 2*z + 2) * q^29 + z^2 * q^30 + (-2*z^3 + 8*z^2 - 4) * q^31 + (z^3 - z) * q^32 + (-2*z^3 + z^2 + 2*z - 2) * q^33 - 4*z^3 * q^34 + 3*z^2 * q^35 + (-z^2 + 1) * q^36 + (-4*z^2 + z - 4) * q^37 + (-z^3 + 2*z + 4) * q^38 + (-z^3 + 4*z) * q^39 - q^40 - 4*z * q^41 + (-3*z^2 + 3) * q^42 - 6*z^2 * q^43 + (2*z^3 - 2*z^2 + 1) * q^44 + (-z^3 + z) * q^45 + (2*z^2 - 4) * q^46 + (-3*z^3 - 4*z^2 + 2) * q^47 + z^2 * q^48 + (-2*z^2 + 2) * q^49 - z * q^50 - 4 * q^51 + (4*z^3 - 3*z) * q^52 + (-z^3 + 2*z - 2) * q^53 - z * q^54 + (-2*z^3 + 2*z^2 + z - 2) * q^55 + 3*z^2 * q^56 + (-4*z^3 - 2*z^2 + 1) * q^57 + (-2*z^3 + 2*z^2 + 2*z - 4) * q^58 + (8*z^3 + 2*z^2 - 8*z - 4) * q^59 + z^3 * q^60 + (-2*z^3 + 4*z^2 - 2*z) * q^61 + (8*z^3 - 2*z^2 - 4*z + 2) * q^62 - 3*z * q^63 - q^64 + (z^2 - 4) * q^65 + (z^3 - 2*z + 2) * q^66 + (2*z^2 - 2*z + 2) * q^67 + (-4*z^2 + 4) * q^68 + (2*z^3 + 2*z) * q^69 + 3*z^3 * q^70 + (6*z^3 + 4*z^2 - 6*z - 8) * q^71 + (-z^3 + z) * q^72 + (-8*z^2 + 4) * q^73 + (-4*z^3 + z^2 - 4*z) * q^74 + (z^2 - 1) * q^75 + (z^2 + 4*z + 1) * q^76 + (3*z^3 - 6*z + 6) * q^77 + (3*z^2 + 1) * q^78 + (4*z^3 - 8*z + 10) * q^79 - z * q^80 + (z^2 - 1) * q^81 - 4*z^2 * q^82 + (6*z^3 + 4*z^2 - 2) * q^83 + (-3*z^3 + 3*z) * q^84 + (-4*z^3 + 4*z) * q^85 - 6*z^3 * q^86 + (2*z^3 - 2*z^2 + 2*z) * q^87 + (-2*z^3 + 2*z^2 + z - 2) * q^88 + (-7*z^2 - 2*z - 7) * q^89 + q^90 + (9*z^2 + 3) * q^91 + (2*z^3 - 4*z) * q^92 + (4*z^2 - 2*z + 4) * q^93 + (-4*z^3 - 3*z^2 + 2*z + 3) * q^94 + (z^3 + 4*z^2 + z) * q^95 + z^3 * q^96 + (2*z^3 - 6*z^2 - 2*z + 12) * q^97 + (-2*z^3 + 2*z) * q^98 + (-2*z^3 + 2*z^2 - 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 2 q^{4} - 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 2 * q^4 - 2 * q^9 $$4 q + 2 q^{3} + 2 q^{4} - 2 q^{9} - 2 q^{10} - 6 q^{11} + 4 q^{12} + 12 q^{14} - 2 q^{16} - 8 q^{17} + 6 q^{19} + 4 q^{22} - 4 q^{25} - 4 q^{26} - 4 q^{27} + 4 q^{29} + 2 q^{30} - 6 q^{33} + 6 q^{35} + 2 q^{36} - 24 q^{37} + 16 q^{38} - 4 q^{40} + 6 q^{42} - 12 q^{43} - 12 q^{46} + 2 q^{48} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 4 q^{55} + 6 q^{56} - 12 q^{58} - 12 q^{59} + 8 q^{61} + 4 q^{62} - 4 q^{64} - 14 q^{65} + 8 q^{66} + 12 q^{67} + 8 q^{68} - 24 q^{71} + 2 q^{74} - 2 q^{75} + 6 q^{76} + 24 q^{77} + 10 q^{78} + 40 q^{79} - 2 q^{81} - 8 q^{82} - 4 q^{87} - 4 q^{88} - 42 q^{89} + 4 q^{90} + 30 q^{91} + 24 q^{93} + 6 q^{94} + 8 q^{95} + 36 q^{97}+O(q^{100})$$ 4 * q + 2 * q^3 + 2 * q^4 - 2 * q^9 - 2 * q^10 - 6 * q^11 + 4 * q^12 + 12 * q^14 - 2 * q^16 - 8 * q^17 + 6 * q^19 + 4 * q^22 - 4 * q^25 - 4 * q^26 - 4 * q^27 + 4 * q^29 + 2 * q^30 - 6 * q^33 + 6 * q^35 + 2 * q^36 - 24 * q^37 + 16 * q^38 - 4 * q^40 + 6 * q^42 - 12 * q^43 - 12 * q^46 + 2 * q^48 + 4 * q^49 - 16 * q^51 - 8 * q^53 - 4 * q^55 + 6 * q^56 - 12 * q^58 - 12 * q^59 + 8 * q^61 + 4 * q^62 - 4 * q^64 - 14 * q^65 + 8 * q^66 + 12 * q^67 + 8 * q^68 - 24 * q^71 + 2 * q^74 - 2 * q^75 + 6 * q^76 + 24 * q^77 + 10 * q^78 + 40 * q^79 - 2 * q^81 - 8 * q^82 - 4 * q^87 - 4 * q^88 - 42 * q^89 + 4 * q^90 + 30 * q^91 + 24 * q^93 + 6 * q^94 + 8 * q^95 + 36 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/390\mathbb{Z}\right)^\times$$.

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.866025 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i −0.866025 + 0.500000i −2.59808 + 1.50000i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
121.2 0.866025 + 0.500000i 0.500000 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 0.500000i 2.59808 1.50000i 1.00000i −0.500000 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i −0.866025 0.500000i −2.59808 1.50000i 1.00000i −0.500000 + 0.866025i −0.500000 0.866025i
361.2 0.866025 0.500000i 0.500000 + 0.866025i 0.500000 0.866025i 1.00000i 0.866025 + 0.500000i 2.59808 + 1.50000i 1.00000i −0.500000 + 0.866025i −0.500000 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
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bb.a 4
3.b odd 2 1 1170.2.bs.d 4
5.b even 2 1 1950.2.bc.a 4
5.c odd 4 1 1950.2.y.d 4
5.c odd 4 1 1950.2.y.e 4
13.c even 3 1 5070.2.b.p 4
13.e even 6 1 inner 390.2.bb.a 4
13.e even 6 1 5070.2.b.p 4
13.f odd 12 1 5070.2.a.ba 2
13.f odd 12 1 5070.2.a.be 2
39.h odd 6 1 1170.2.bs.d 4
65.l even 6 1 1950.2.bc.a 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a 4 1.a even 1 1 trivial
390.2.bb.a 4 13.e even 6 1 inner
1170.2.bs.d 4 3.b odd 2 1
1170.2.bs.d 4 39.h odd 6 1
1950.2.y.d 4 5.c odd 4 1
1950.2.y.d 4 65.r odd 12 1
1950.2.y.e 4 5.c odd 4 1
1950.2.y.e 4 65.r odd 12 1
1950.2.bc.a 4 5.b even 2 1
1950.2.bc.a 4 65.l even 6 1
5070.2.a.ba 2 13.f odd 12 1
5070.2.a.be 2 13.f odd 12 1
5070.2.b.p 4 13.c even 3 1
5070.2.b.p 4 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 9T_{7}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$(T^{2} - T + 1)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} - 9T^{2} + 81$$
$11$ $$T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1$$
$13$ $$T^{4} + 23T^{2} + 169$$
$17$ $$(T^{2} + 4 T + 16)^{2}$$
$19$ $$T^{4} - 6 T^{3} - T^{2} + 78 T + 169$$
$23$ $$T^{4} + 12T^{2} + 144$$
$29$ $$T^{4} - 4 T^{3} + 24 T^{2} + 32 T + 64$$
$31$ $$T^{4} + 104T^{2} + 1936$$
$37$ $$T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209$$
$41$ $$T^{4} - 16T^{2} + 256$$
$43$ $$(T^{2} + 6 T + 36)^{2}$$
$47$ $$T^{4} + 42T^{2} + 9$$
$53$ $$(T^{2} + 4 T + 1)^{2}$$
$59$ $$T^{4} + 12 T^{3} - 4 T^{2} + \cdots + 2704$$
$61$ $$T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16$$
$67$ $$T^{4} - 12 T^{3} + 56 T^{2} - 96 T + 64$$
$71$ $$T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144$$
$73$ $$(T^{2} + 48)^{2}$$
$79$ $$(T^{2} - 20 T + 52)^{2}$$
$83$ $$T^{4} + 96T^{2} + 576$$
$89$ $$T^{4} + 42 T^{3} + 731 T^{2} + \cdots + 20449$$
$97$ $$T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816$$