# Properties

 Label 390.2.i.d 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} + 5 \zeta_{6} q^{7} - q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^2 + (z - 1) * q^3 - z * q^4 - q^5 + z * q^6 + 5*z * q^7 - q^8 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{6} + 5 \zeta_{6} q^{7} - q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + ( - 3 \zeta_{6} + 3) q^{11} + q^{12} + (3 \zeta_{6} - 4) q^{13} + 5 q^{14} + ( - \zeta_{6} + 1) q^{15} + (\zeta_{6} - 1) q^{16} + 8 \zeta_{6} q^{17} - q^{18} + 5 \zeta_{6} q^{19} + \zeta_{6} q^{20} - 5 q^{21} - 3 \zeta_{6} q^{22} + ( - 4 \zeta_{6} + 4) q^{23} + ( - \zeta_{6} + 1) q^{24} + q^{25} + (4 \zeta_{6} - 1) q^{26} + q^{27} + ( - 5 \zeta_{6} + 5) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} - \zeta_{6} q^{30} - 2 q^{31} + \zeta_{6} q^{32} + 3 \zeta_{6} q^{33} + 8 q^{34} - 5 \zeta_{6} q^{35} + (\zeta_{6} - 1) q^{36} + ( - 7 \zeta_{6} + 7) q^{37} + 5 q^{38} + ( - 4 \zeta_{6} + 1) q^{39} + q^{40} + (6 \zeta_{6} - 6) q^{41} + (5 \zeta_{6} - 5) q^{42} - 6 \zeta_{6} q^{43} - 3 q^{44} + \zeta_{6} q^{45} - 4 \zeta_{6} q^{46} - 3 q^{47} - \zeta_{6} q^{48} + (18 \zeta_{6} - 18) q^{49} + ( - \zeta_{6} + 1) q^{50} - 8 q^{51} + (\zeta_{6} + 3) q^{52} + q^{53} + ( - \zeta_{6} + 1) q^{54} + (3 \zeta_{6} - 3) q^{55} - 5 \zeta_{6} q^{56} - 5 q^{57} - 4 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} - q^{60} - 2 \zeta_{6} q^{61} + (2 \zeta_{6} - 2) q^{62} + ( - 5 \zeta_{6} + 5) q^{63} + q^{64} + ( - 3 \zeta_{6} + 4) q^{65} + 3 q^{66} + (8 \zeta_{6} - 8) q^{67} + ( - 8 \zeta_{6} + 8) q^{68} + 4 \zeta_{6} q^{69} - 5 q^{70} - 2 \zeta_{6} q^{71} + \zeta_{6} q^{72} - 7 \zeta_{6} q^{74} + (\zeta_{6} - 1) q^{75} + ( - 5 \zeta_{6} + 5) q^{76} + 15 q^{77} + ( - \zeta_{6} - 3) q^{78} - 2 q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 6 \zeta_{6} q^{82} + 8 q^{83} + 5 \zeta_{6} q^{84} - 8 \zeta_{6} q^{85} - 6 q^{86} + 4 \zeta_{6} q^{87} + (3 \zeta_{6} - 3) q^{88} + ( - 11 \zeta_{6} + 11) q^{89} + q^{90} + ( - 5 \zeta_{6} - 15) q^{91} - 4 q^{92} + ( - 2 \zeta_{6} + 2) q^{93} + (3 \zeta_{6} - 3) q^{94} - 5 \zeta_{6} q^{95} - q^{96} + 18 \zeta_{6} q^{98} - 3 q^{99} +O(q^{100})$$ q + (-z + 1) * q^2 + (z - 1) * q^3 - z * q^4 - q^5 + z * q^6 + 5*z * q^7 - q^8 - z * q^9 + (z - 1) * q^10 + (-3*z + 3) * q^11 + q^12 + (3*z - 4) * q^13 + 5 * q^14 + (-z + 1) * q^15 + (z - 1) * q^16 + 8*z * q^17 - q^18 + 5*z * q^19 + z * q^20 - 5 * q^21 - 3*z * q^22 + (-4*z + 4) * q^23 + (-z + 1) * q^24 + q^25 + (4*z - 1) * q^26 + q^27 + (-5*z + 5) * q^28 + (-4*z + 4) * q^29 - z * q^30 - 2 * q^31 + z * q^32 + 3*z * q^33 + 8 * q^34 - 5*z * q^35 + (z - 1) * q^36 + (-7*z + 7) * q^37 + 5 * q^38 + (-4*z + 1) * q^39 + q^40 + (6*z - 6) * q^41 + (5*z - 5) * q^42 - 6*z * q^43 - 3 * q^44 + z * q^45 - 4*z * q^46 - 3 * q^47 - z * q^48 + (18*z - 18) * q^49 + (-z + 1) * q^50 - 8 * q^51 + (z + 3) * q^52 + q^53 + (-z + 1) * q^54 + (3*z - 3) * q^55 - 5*z * q^56 - 5 * q^57 - 4*z * q^58 - 12*z * q^59 - q^60 - 2*z * q^61 + (2*z - 2) * q^62 + (-5*z + 5) * q^63 + q^64 + (-3*z + 4) * q^65 + 3 * q^66 + (8*z - 8) * q^67 + (-8*z + 8) * q^68 + 4*z * q^69 - 5 * q^70 - 2*z * q^71 + z * q^72 - 7*z * q^74 + (z - 1) * q^75 + (-5*z + 5) * q^76 + 15 * q^77 + (-z - 3) * q^78 - 2 * q^79 + (-z + 1) * q^80 + (z - 1) * q^81 + 6*z * q^82 + 8 * q^83 + 5*z * q^84 - 8*z * q^85 - 6 * q^86 + 4*z * q^87 + (3*z - 3) * q^88 + (-11*z + 11) * q^89 + q^90 + (-5*z - 15) * q^91 - 4 * q^92 + (-2*z + 2) * q^93 + (3*z - 3) * q^94 - 5*z * q^95 - q^96 + 18*z * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q + q^2 - q^3 - q^4 - 2 * q^5 + q^6 + 5 * q^7 - 2 * q^8 - q^9 $$2 q + q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + 5 q^{7} - 2 q^{8} - q^{9} - q^{10} + 3 q^{11} + 2 q^{12} - 5 q^{13} + 10 q^{14} + q^{15} - q^{16} + 8 q^{17} - 2 q^{18} + 5 q^{19} + q^{20} - 10 q^{21} - 3 q^{22} + 4 q^{23} + q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 5 q^{28} + 4 q^{29} - q^{30} - 4 q^{31} + q^{32} + 3 q^{33} + 16 q^{34} - 5 q^{35} - q^{36} + 7 q^{37} + 10 q^{38} - 2 q^{39} + 2 q^{40} - 6 q^{41} - 5 q^{42} - 6 q^{43} - 6 q^{44} + q^{45} - 4 q^{46} - 6 q^{47} - q^{48} - 18 q^{49} + q^{50} - 16 q^{51} + 7 q^{52} + 2 q^{53} + q^{54} - 3 q^{55} - 5 q^{56} - 10 q^{57} - 4 q^{58} - 12 q^{59} - 2 q^{60} - 2 q^{61} - 2 q^{62} + 5 q^{63} + 2 q^{64} + 5 q^{65} + 6 q^{66} - 8 q^{67} + 8 q^{68} + 4 q^{69} - 10 q^{70} - 2 q^{71} + q^{72} - 7 q^{74} - q^{75} + 5 q^{76} + 30 q^{77} - 7 q^{78} - 4 q^{79} + q^{80} - q^{81} + 6 q^{82} + 16 q^{83} + 5 q^{84} - 8 q^{85} - 12 q^{86} + 4 q^{87} - 3 q^{88} + 11 q^{89} + 2 q^{90} - 35 q^{91} - 8 q^{92} + 2 q^{93} - 3 q^{94} - 5 q^{95} - 2 q^{96} + 18 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 - q^3 - q^4 - 2 * q^5 + q^6 + 5 * q^7 - 2 * q^8 - q^9 - q^10 + 3 * q^11 + 2 * q^12 - 5 * q^13 + 10 * q^14 + q^15 - q^16 + 8 * q^17 - 2 * q^18 + 5 * q^19 + q^20 - 10 * q^21 - 3 * q^22 + 4 * q^23 + q^24 + 2 * q^25 + 2 * q^26 + 2 * q^27 + 5 * q^28 + 4 * q^29 - q^30 - 4 * q^31 + q^32 + 3 * q^33 + 16 * q^34 - 5 * q^35 - q^36 + 7 * q^37 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 6 * q^41 - 5 * q^42 - 6 * q^43 - 6 * q^44 + q^45 - 4 * q^46 - 6 * q^47 - q^48 - 18 * q^49 + q^50 - 16 * q^51 + 7 * q^52 + 2 * q^53 + q^54 - 3 * q^55 - 5 * q^56 - 10 * q^57 - 4 * q^58 - 12 * q^59 - 2 * q^60 - 2 * q^61 - 2 * q^62 + 5 * q^63 + 2 * q^64 + 5 * q^65 + 6 * q^66 - 8 * q^67 + 8 * q^68 + 4 * q^69 - 10 * q^70 - 2 * q^71 + q^72 - 7 * q^74 - q^75 + 5 * q^76 + 30 * q^77 - 7 * q^78 - 4 * q^79 + q^80 - q^81 + 6 * q^82 + 16 * q^83 + 5 * q^84 - 8 * q^85 - 12 * q^86 + 4 * q^87 - 3 * q^88 + 11 * q^89 + 2 * q^90 - 35 * q^91 - 8 * q^92 + 2 * q^93 - 3 * q^94 - 5 * q^95 - 2 * q^96 + 18 * q^98 - 6 * 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 2.50000 + 4.33013i −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 2.50000 4.33013i −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.d 2
3.b odd 2 1 1170.2.i.g 2
5.b even 2 1 1950.2.i.i 2
5.c odd 4 2 1950.2.z.j 4
13.c even 3 1 inner 390.2.i.d 2
13.c even 3 1 5070.2.a.i 1
13.e even 6 1 5070.2.a.x 1
13.f odd 12 2 5070.2.b.l 2
39.i odd 6 1 1170.2.i.g 2
65.n even 6 1 1950.2.i.i 2
65.q odd 12 2 1950.2.z.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.d 2 1.a even 1 1 trivial
390.2.i.d 2 13.c even 3 1 inner
1170.2.i.g 2 3.b odd 2 1
1170.2.i.g 2 39.i odd 6 1
1950.2.i.i 2 5.b even 2 1
1950.2.i.i 2 65.n even 6 1
1950.2.z.j 4 5.c odd 4 2
1950.2.z.j 4 65.q odd 12 2
5070.2.a.i 1 13.c even 3 1
5070.2.a.x 1 13.e even 6 1
5070.2.b.l 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{2} - 5T_{7} + 25$$ T7^2 - 5*T7 + 25 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - 5T + 25$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} - 8T + 64$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2} - 4T + 16$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} + 6T + 36$$
$47$ $$(T + 3)^{2}$$
$53$ $$(T - 1)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} + 2T + 4$$
$67$ $$T^{2} + 8T + 64$$
$71$ $$T^{2} + 2T + 4$$
$73$ $$T^{2}$$
$79$ $$(T + 2)^{2}$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} - 11T + 121$$
$97$ $$T^{2}$$