# Properties

 Label 390.2.i.b Level $390$ Weight $2$ Character orbit 390.i Analytic conductor $3.114$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(61,390)]

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, 4]))

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

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

chi := DirichletCharacter("390.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.i (of order $$3$$, 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: $$2$$ 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: yes 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.

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

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 1.00000 0.500000 + 0.866025i 1.50000 + 2.59808i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
211.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.500000 0.866025i 1.50000 2.59808i 1.00000 −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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.i.b 2
3.b odd 2 1 1170.2.i.j 2
5.b even 2 1 1950.2.i.o 2
5.c odd 4 2 1950.2.z.i 4
13.c even 3 1 inner 390.2.i.b 2
13.c even 3 1 5070.2.a.q 1
13.e even 6 1 5070.2.a.c 1
13.f odd 12 2 5070.2.b.a 2
39.i odd 6 1 1170.2.i.j 2
65.n even 6 1 1950.2.i.o 2
65.q odd 12 2 1950.2.z.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.b 2 1.a even 1 1 trivial
390.2.i.b 2 13.c even 3 1 inner
1170.2.i.j 2 3.b odd 2 1
1170.2.i.j 2 39.i odd 6 1
1950.2.i.o 2 5.b even 2 1
1950.2.i.o 2 65.n even 6 1
1950.2.z.i 4 5.c odd 4 2
1950.2.z.i 4 65.q odd 12 2
5070.2.a.c 1 13.e even 6 1
5070.2.a.q 1 13.c even 3 1
5070.2.b.a 2 13.f odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} - 3T_{7} + 9$$ T7^2 - 3*T7 + 9 $$T_{11}^{2} + T_{11} + 1$$ T11^2 + T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 3T + 9$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2}$$
$31$ $$(T - 10)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} - 2T + 4$$
$47$ $$(T + 9)^{2}$$
$53$ $$(T + 13)^{2}$$
$59$ $$T^{2} + 4T + 16$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} - 12T + 144$$
$71$ $$T^{2} - 2T + 4$$
$73$ $$(T + 16)^{2}$$
$79$ $$(T + 10)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + T + 1$$
$97$ $$T^{2} + 12T + 144$$