# Properties

 Label 390.2.i.g Level $390$ Weight $2$ Character orbit 390.i Analytic conductor $3.114$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ x^4 - x^3 + 5*x^2 + 4*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} - 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + q^{5} - \beta_{2} q^{6} + ( - \beta_{2} - \beta_1) q^{7} + q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + (b2 - 1) * q^2 + (b2 - 1) * q^3 - b2 * q^4 + q^5 - b2 * q^6 + (-b2 - b1) * q^7 + q^8 - b2 * q^9 $$q + (\beta_{2} - 1) q^{2} + (\beta_{2} - 1) q^{3} - \beta_{2} q^{4} + q^{5} - \beta_{2} q^{6} + ( - \beta_{2} - \beta_1) q^{7} + q^{8} - \beta_{2} q^{9} + (\beta_{2} - 1) q^{10} + ( - 2 \beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{11} + q^{12} + (\beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} + 1) q^{14} + (\beta_{2} - 1) q^{15} + (\beta_{2} - 1) q^{16} - 2 \beta_1 q^{17} + q^{18} + (\beta_{2} + \beta_1) q^{19} - \beta_{2} q^{20} + ( - \beta_{3} + 1) q^{21} + ( - \beta_{2} + 2 \beta_1) q^{22} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} + (\beta_{2} - 1) q^{24} + q^{25} + ( - 2 \beta_{3} - \beta_1 - 1) q^{26} + q^{27} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{28} + (\beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{29} - \beta_{2} q^{30} + ( - 3 \beta_{3} - 2) q^{31} - \beta_{2} q^{32} + ( - \beta_{2} + 2 \beta_1) q^{33} - 2 \beta_{3} q^{34} + ( - \beta_{2} - \beta_1) q^{35} + (\beta_{2} - 1) q^{36} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{37} + (\beta_{3} - 1) q^{38} + ( - 2 \beta_{3} - \beta_1 - 1) q^{39} + q^{40} + ( - 4 \beta_{3} + 6 \beta_{2} + \cdots - 6) q^{41}+ \cdots + (2 \beta_{3} + 1) q^{99}+O(q^{100})$$ q + (b2 - 1) * q^2 + (b2 - 1) * q^3 - b2 * q^4 + q^5 - b2 * q^6 + (-b2 - b1) * q^7 + q^8 - b2 * q^9 + (b2 - 1) * q^10 + (-2*b3 + b2 - 2*b1 - 1) * q^11 + q^12 + (b3 + b2 - b1) * q^13 + (-b3 + 1) * q^14 + (b2 - 1) * q^15 + (b2 - 1) * q^16 - 2*b1 * q^17 + q^18 + (b2 + b1) * q^19 - b2 * q^20 + (-b3 + 1) * q^21 + (-b2 + 2*b1) * q^22 + (-3*b3 - 3*b1) * q^23 + (b2 - 1) * q^24 + q^25 + (-2*b3 - b1 - 1) * q^26 + q^27 + (b3 + b2 + b1 - 1) * q^28 + (b3 + 4*b2 + b1 - 4) * q^29 - b2 * q^30 + (-3*b3 - 2) * q^31 - b2 * q^32 + (-b2 + 2*b1) * q^33 - 2*b3 * q^34 + (-b2 - b1) * q^35 + (b2 - 1) * q^36 + (2*b3 - b2 + 2*b1 + 1) * q^37 + (b3 - 1) * q^38 + (-2*b3 - b1 - 1) * q^39 + q^40 + (-4*b3 + 6*b2 - 4*b1 - 6) * q^41 + (b3 + b2 + b1 - 1) * q^42 + (-2*b2 - b1) * q^43 + (2*b3 + 1) * q^44 - b2 * q^45 + 3*b1 * q^46 + 7 * q^47 - b2 * q^48 + (3*b3 - 2*b2 + 3*b1 + 2) * q^49 + (b2 - 1) * q^50 - 2*b3 * q^51 + (b3 - b2 + 2*b1 + 1) * q^52 + (b3 + 7) * q^53 + (b2 - 1) * q^54 + (-2*b3 + b2 - 2*b1 - 1) * q^55 + (-b2 - b1) * q^56 + (b3 - 1) * q^57 + (-4*b2 - b1) * q^58 + (-8*b2 - b1) * q^59 + q^60 - 6*b2 * q^61 + (3*b3 - 2*b2 + 3*b1 + 2) * q^62 + (b3 + b2 + b1 - 1) * q^63 + q^64 + (b3 + b2 - b1) * q^65 + (2*b3 + 1) * q^66 + (4*b3 + 4*b2 + 4*b1 - 4) * q^67 + (2*b3 + 2*b1) * q^68 + 3*b1 * q^69 + (-b3 + 1) * q^70 + (10*b2 - 2*b1) * q^71 - b2 * q^72 + 6*b3 * q^73 + (b2 - 2*b1) * q^74 + (b2 - 1) * q^75 + (-b3 - b2 - b1 + 1) * q^76 + (3*b3 - 7) * q^77 + (b3 - b2 + 2*b1 + 1) * q^78 + (b3 + 10) * q^79 + (b2 - 1) * q^80 + (b2 - 1) * q^81 + (-6*b2 + 4*b1) * q^82 + (-2*b3 - 4) * q^83 + (-b2 - b1) * q^84 - 2*b1 * q^85 + (-b3 + 2) * q^86 + (-4*b2 - b1) * q^87 + (-2*b3 + b2 - 2*b1 - 1) * q^88 + (5*b3 - 11*b2 + 5*b1 + 11) * q^89 + q^90 + (b3 + 7*b2 + 3*b1 - 3) * q^91 + 3*b3 * q^92 + (3*b3 - 2*b2 + 3*b1 + 2) * q^93 + (7*b2 - 7) * q^94 + (b2 + b1) * q^95 + q^96 + (4*b2 - 2*b1) * q^97 + (2*b2 - 3*b1) * q^98 + (2*b3 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 3 q^{7} + 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 4 * q^5 - 2 * q^6 - 3 * q^7 + 4 * q^8 - 2 * q^9 $$4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 4 q^{5} - 2 q^{6} - 3 q^{7} + 4 q^{8} - 2 q^{9} - 2 q^{10} + 4 q^{12} - q^{13} + 6 q^{14} - 2 q^{15} - 2 q^{16} - 2 q^{17} + 4 q^{18} + 3 q^{19} - 2 q^{20} + 6 q^{21} + 3 q^{23} - 2 q^{24} + 4 q^{25} - q^{26} + 4 q^{27} - 3 q^{28} - 9 q^{29} - 2 q^{30} - 2 q^{31} - 2 q^{32} + 4 q^{34} - 3 q^{35} - 2 q^{36} - 6 q^{38} - q^{39} + 4 q^{40} - 8 q^{41} - 3 q^{42} - 5 q^{43} - 2 q^{45} + 3 q^{46} + 28 q^{47} - 2 q^{48} + q^{49} - 2 q^{50} + 4 q^{51} + 2 q^{52} + 26 q^{53} - 2 q^{54} - 3 q^{56} - 6 q^{57} - 9 q^{58} - 17 q^{59} + 4 q^{60} - 12 q^{61} + q^{62} - 3 q^{63} + 4 q^{64} - q^{65} - 12 q^{67} - 2 q^{68} + 3 q^{69} + 6 q^{70} + 18 q^{71} - 2 q^{72} - 12 q^{73} - 2 q^{75} + 3 q^{76} - 34 q^{77} + 2 q^{78} + 38 q^{79} - 2 q^{80} - 2 q^{81} - 8 q^{82} - 12 q^{83} - 3 q^{84} - 2 q^{85} + 10 q^{86} - 9 q^{87} + 17 q^{89} + 4 q^{90} + 3 q^{91} - 6 q^{92} + q^{93} - 14 q^{94} + 3 q^{95} + 4 q^{96} + 6 q^{97} + q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 4 * q^5 - 2 * q^6 - 3 * q^7 + 4 * q^8 - 2 * q^9 - 2 * q^10 + 4 * q^12 - q^13 + 6 * q^14 - 2 * q^15 - 2 * q^16 - 2 * q^17 + 4 * q^18 + 3 * q^19 - 2 * q^20 + 6 * q^21 + 3 * q^23 - 2 * q^24 + 4 * q^25 - q^26 + 4 * q^27 - 3 * q^28 - 9 * q^29 - 2 * q^30 - 2 * q^31 - 2 * q^32 + 4 * q^34 - 3 * q^35 - 2 * q^36 - 6 * q^38 - q^39 + 4 * q^40 - 8 * q^41 - 3 * q^42 - 5 * q^43 - 2 * q^45 + 3 * q^46 + 28 * q^47 - 2 * q^48 + q^49 - 2 * q^50 + 4 * q^51 + 2 * q^52 + 26 * q^53 - 2 * q^54 - 3 * q^56 - 6 * q^57 - 9 * q^58 - 17 * q^59 + 4 * q^60 - 12 * q^61 + q^62 - 3 * q^63 + 4 * q^64 - q^65 - 12 * q^67 - 2 * q^68 + 3 * q^69 + 6 * q^70 + 18 * q^71 - 2 * q^72 - 12 * q^73 - 2 * q^75 + 3 * q^76 - 34 * q^77 + 2 * q^78 + 38 * q^79 - 2 * q^80 - 2 * q^81 - 8 * q^82 - 12 * q^83 - 3 * q^84 - 2 * q^85 + 10 * q^86 - 9 * q^87 + 17 * q^89 + 4 * q^90 + 3 * q^91 - 6 * q^92 + q^93 - 14 * q^94 + 3 * q^95 + 4 * q^96 + 6 * q^97 + q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5x^{2} + 4x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20$$ (-v^3 + 5*v^2 - 5*v + 16) / 20 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4 ) / 5$$ (v^3 + 4) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4\beta_{2} + \beta _1 - 4$$ b3 + 4*b2 + b1 - 4 $$\nu^{3}$$ $$=$$ $$5\beta_{3} - 4$$ 5*b3 - 4

## Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i −1.78078 3.08440i 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
61.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.280776 + 0.486319i 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 −1.78078 + 3.08440i 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
211.2 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.280776 0.486319i 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.g 4
3.b odd 2 1 1170.2.i.o 4
5.b even 2 1 1950.2.i.bi 4
5.c odd 4 2 1950.2.z.n 8
13.c even 3 1 inner 390.2.i.g 4
13.c even 3 1 5070.2.a.bi 2
13.e even 6 1 5070.2.a.bb 2
13.f odd 12 2 5070.2.b.r 4
39.i odd 6 1 1170.2.i.o 4
65.n even 6 1 1950.2.i.bi 4
65.q odd 12 2 1950.2.z.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g 4 1.a even 1 1 trivial
390.2.i.g 4 13.c even 3 1 inner
1170.2.i.o 4 3.b odd 2 1
1170.2.i.o 4 39.i odd 6 1
1950.2.i.bi 4 5.b even 2 1
1950.2.i.bi 4 65.n even 6 1
1950.2.z.n 8 5.c odd 4 2
1950.2.z.n 8 65.q odd 12 2
5070.2.a.bb 2 13.e even 6 1
5070.2.a.bi 2 13.c even 3 1
5070.2.b.r 4 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 $$S_{2}^{\mathrm{new}}(390, [\chi])$$:

 $$T_{7}^{4} + 3T_{7}^{3} + 11T_{7}^{2} - 6T_{7} + 4$$ T7^4 + 3*T7^3 + 11*T7^2 - 6*T7 + 4 $$T_{11}^{4} + 17T_{11}^{2} + 289$$ T11^4 + 17*T11^2 + 289

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{2}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 3 T^{3} + \cdots + 4$$
$11$ $$T^{4} + 17T^{2} + 289$$
$13$ $$T^{4} + T^{3} + \cdots + 169$$
$17$ $$T^{4} + 2 T^{3} + \cdots + 256$$
$19$ $$T^{4} - 3 T^{3} + \cdots + 4$$
$23$ $$T^{4} - 3 T^{3} + \cdots + 1296$$
$29$ $$T^{4} + 9 T^{3} + \cdots + 256$$
$31$ $$(T^{2} + T - 38)^{2}$$
$37$ $$T^{4} + 17T^{2} + 289$$
$41$ $$T^{4} + 8 T^{3} + \cdots + 2704$$
$43$ $$T^{4} + 5 T^{3} + \cdots + 4$$
$47$ $$(T - 7)^{4}$$
$53$ $$(T^{2} - 13 T + 38)^{2}$$
$59$ $$T^{4} + 17 T^{3} + \cdots + 4624$$
$61$ $$(T^{2} + 6 T + 36)^{2}$$
$67$ $$T^{4} + 12 T^{3} + \cdots + 1024$$
$71$ $$T^{4} - 18 T^{3} + \cdots + 4096$$
$73$ $$(T^{2} + 6 T - 144)^{2}$$
$79$ $$(T^{2} - 19 T + 86)^{2}$$
$83$ $$(T^{2} + 6 T - 8)^{2}$$
$89$ $$T^{4} - 17 T^{3} + \cdots + 1156$$
$97$ $$T^{4} - 6 T^{3} + \cdots + 64$$