Properties

 Label 390.2.f.b Level $390$ Weight $2$ Character orbit 390.f Analytic conductor $3.114$ Analytic rank $0$ Dimension $6$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(259,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("390.259");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$390 = 2 \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 390.f (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: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 4*x^2 - 4*x + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_{4} q^{3} + q^{4} + ( - \beta_{5} - \beta_1) q^{5} + \beta_{4} q^{6} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{7} + q^{8} - q^{9}+O(q^{10})$$ q + q^2 + b4 * q^3 + q^4 + (-b5 - b1) * q^5 + b4 * q^6 + (-b5 - b3 + b2 - 1) * q^7 + q^8 - q^9 $$q + q^{2} + \beta_{4} q^{3} + q^{4} + ( - \beta_{5} - \beta_1) q^{5} + \beta_{4} q^{6} + ( - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{7} + q^{8} - q^{9} + ( - \beta_{5} - \beta_1) q^{10} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{11}+ \cdots + (\beta_{5} - 2 \beta_{4} + \cdots + 2 \beta_1) q^{99}+O(q^{100})$$ q + q^2 + b4 * q^3 + q^4 + (-b5 - b1) * q^5 + b4 * q^6 + (-b5 - b3 + b2 - 1) * q^7 + q^8 - q^9 + (-b5 - b1) * q^10 + (-b5 + 2*b4 + b3 - 2*b1) * q^11 + b4 * q^12 + (b3 + b2 + 2) * q^13 + (-b5 - b3 + b2 - 1) * q^14 + (-b3 + b2) * q^15 + q^16 + (-b4 + b1) * q^17 - q^18 + (3*b4 + b1) * q^19 + (-b5 - b1) * q^20 + (b5 - b4 - b3 + b1) * q^21 + (-b5 + 2*b4 + b3 - 2*b1) * q^22 + (b4 + b1) * q^23 + b4 * q^24 + (b5 - 2*b4 - b3 - 2*b2 - 1) * q^25 + (b3 + b2 + 2) * q^26 - b4 * q^27 + (-b5 - b3 + b2 - 1) * q^28 + (2*b5 + 2*b3 - 3*b2 + 1) * q^29 + (-b3 + b2) * q^30 + (2*b5 - 2*b4 - 2*b3) * q^31 + q^32 + (-b5 - b3 + 2*b2 - 2) * q^33 + (-b4 + b1) * q^34 + (2*b5 - 4*b4 - b2 + 3) * q^35 - q^36 + (-b5 - b3 + 2*b2 - 4) * q^37 + (3*b4 + b1) * q^38 + (-b5 + 2*b4 + b1) * q^39 + (-b5 - b1) * q^40 + (-4*b4 - 2*b1) * q^41 + (b5 - b4 - b3 + b1) * q^42 + (-b5 + 4*b4 + b3 + 2*b1) * q^43 + (-b5 + 2*b4 + b3 - 2*b1) * q^44 + (b5 + b1) * q^45 + (b4 + b1) * q^46 + (2*b5 + 2*b3 - 2*b2 + 6) * q^47 + b4 * q^48 + (3*b5 + 3*b3 - 2*b2 + 1) * q^49 + (b5 - 2*b4 - b3 - 2*b2 - 1) * q^50 + (-b2 + 1) * q^51 + (b3 + b2 + 2) * q^52 + (-2*b5 + 2*b3 + 2*b1) * q^53 - b4 * q^54 + (b5 - 2*b4 - 3*b3 - 6) * q^55 + (-b5 - b3 + b2 - 1) * q^56 + (-b2 - 3) * q^57 + (2*b5 + 2*b3 - 3*b2 + 1) * q^58 + (-3*b5 + 4*b4 + 3*b3) * q^59 + (-b3 + b2) * q^60 + (2*b5 + 2*b3 - 2*b2) * q^61 + (2*b5 - 2*b4 - 2*b3) * q^62 + (b5 + b3 - b2 + 1) * q^63 + q^64 + (-3*b5 - 3*b4 + b3 + 2*b2 - 2*b1 - 4) * q^65 + (-b5 - b3 + 2*b2 - 2) * q^66 + (-b5 - b3 - 2*b2 - 2) * q^67 + (-b4 + b1) * q^68 + (-b2 - 1) * q^69 + (2*b5 - 4*b4 - b2 + 3) * q^70 + (b5 - 6*b4 - b3) * q^71 - q^72 + (3*b5 + 3*b3 - b2 - 5) * q^73 + (-b5 - b3 + 2*b2 - 4) * q^74 + (b5 - b4 + b3 - 2*b1 + 2) * q^75 + (3*b4 + b1) * q^76 + (4*b5 - 10*b4 - 4*b3 + 2*b1) * q^77 + (-b5 + 2*b4 + b1) * q^78 + (-3*b5 - 3*b3 + 8*b2 - 2) * q^79 + (-b5 - b1) * q^80 + q^81 + (-4*b4 - 2*b1) * q^82 + (3*b5 + 3*b3 - 6*b2) * q^83 + (b5 - b4 - b3 + b1) * q^84 + (-b5 - b4 + b3 + b1 + 2) * q^85 + (-b5 + 4*b4 + b3 + 2*b1) * q^86 + (-2*b5 + b4 + 2*b3 - 3*b1) * q^87 + (-b5 + 2*b4 + b3 - 2*b1) * q^88 + (-b5 - 8*b4 + b3) * q^89 + (b5 + b1) * q^90 + (-2*b5 - 3*b4 - 4*b3 + 4*b2 + b1 - 4) * q^91 + (b4 + b1) * q^92 + (2*b5 + 2*b3 + 2) * q^93 + (2*b5 + 2*b3 - 2*b2 + 6) * q^94 + (-b5 - b4 - 3*b3 + 4*b2 + b1 + 2) * q^95 + b4 * q^96 + (b5 + b3 - 5*b2 - 1) * q^97 + (3*b5 + 3*b3 - 2*b2 + 1) * q^98 + (b5 - 2*b4 - b3 + 2*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10})$$ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 - 6 * q^9 $$6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} - 4 q^{7} + 6 q^{8} - 6 q^{9} + 2 q^{10} + 8 q^{13} - 4 q^{14} + 6 q^{16} - 6 q^{18} + 2 q^{20} - 2 q^{25} + 8 q^{26} - 4 q^{28} + 4 q^{29} + 6 q^{32} - 12 q^{33} + 16 q^{35} - 6 q^{36} - 24 q^{37} + 2 q^{39} + 2 q^{40} - 2 q^{45} + 32 q^{47} - 2 q^{49} - 2 q^{50} + 8 q^{51} + 8 q^{52} - 32 q^{55} - 4 q^{56} - 16 q^{57} + 4 q^{58} - 4 q^{61} + 4 q^{63} + 6 q^{64} - 24 q^{65} - 12 q^{66} - 4 q^{67} - 4 q^{69} + 16 q^{70} - 6 q^{72} - 40 q^{73} - 24 q^{74} + 8 q^{75} + 2 q^{78} - 16 q^{79} + 2 q^{80} + 6 q^{81} + 12 q^{85} - 2 q^{90} - 20 q^{91} + 4 q^{93} + 32 q^{94} + 12 q^{95} - 2 q^{98}+O(q^{100})$$ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 - 4 * q^7 + 6 * q^8 - 6 * q^9 + 2 * q^10 + 8 * q^13 - 4 * q^14 + 6 * q^16 - 6 * q^18 + 2 * q^20 - 2 * q^25 + 8 * q^26 - 4 * q^28 + 4 * q^29 + 6 * q^32 - 12 * q^33 + 16 * q^35 - 6 * q^36 - 24 * q^37 + 2 * q^39 + 2 * q^40 - 2 * q^45 + 32 * q^47 - 2 * q^49 - 2 * q^50 + 8 * q^51 + 8 * q^52 - 32 * q^55 - 4 * q^56 - 16 * q^57 + 4 * q^58 - 4 * q^61 + 4 * q^63 + 6 * q^64 - 24 * q^65 - 12 * q^66 - 4 * q^67 - 4 * q^69 + 16 * q^70 - 6 * q^72 - 40 * q^73 - 24 * q^74 + 8 * q^75 + 2 * q^78 - 16 * q^79 + 2 * q^80 + 6 * q^81 + 12 * q^85 - 2 * q^90 - 20 * q^91 + 4 * q^93 + 32 * q^94 + 12 * q^95 - 2 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23$$ (-3*v^5 + v^4 + 11*v^3 - 26*v^2 + 6*v - 1) / 23 $$\beta_{2}$$ $$=$$ $$( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23$$ (-4*v^5 + 9*v^4 - 16*v^3 - 4*v^2 + 8*v - 9) / 23 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23$$ (6*v^5 - 2*v^4 + v^3 + 6*v^2 + 80*v + 2) / 23 $$\beta_{4}$$ $$=$$ $$( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23$$ (7*v^5 - 10*v^4 + 5*v^3 + 30*v^2 + 32*v - 13) / 23 $$\beta_{5}$$ $$=$$ $$( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23$$ (-16*v^5 + 36*v^4 - 41*v^3 - 16*v^2 - 60*v + 56) / 23
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2$$ (b4 + b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2$$ (b5 + 4*b4 - b3 + 2*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2$$ b5 + 2*b4 - 2*b2 + 2*b1 - 2 $$\nu^{4}$$ $$=$$ $$2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7$$ 2*b5 + 2*b3 - 5*b2 - 7 $$\nu^{5}$$ $$=$$ $$-9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9$$ -9*b4 + 5*b3 - 8*b2 - 8*b1 - 9

Character values

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

 $$n$$ $$131$$ $$157$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1
 0.403032 − 0.403032i 1.45161 − 1.45161i −0.854638 + 0.854638i 0.403032 + 0.403032i 1.45161 + 1.45161i −0.854638 − 0.854638i
1.00000 1.00000i 1.00000 −1.48119 1.67513i 1.00000i −4.15633 1.00000 −1.00000 −1.48119 1.67513i
259.2 1.00000 1.00000i 1.00000 0.311108 + 2.21432i 1.00000i 1.52543 1.00000 −1.00000 0.311108 + 2.21432i
259.3 1.00000 1.00000i 1.00000 2.17009 0.539189i 1.00000i 0.630898 1.00000 −1.00000 2.17009 0.539189i
259.4 1.00000 1.00000i 1.00000 −1.48119 + 1.67513i 1.00000i −4.15633 1.00000 −1.00000 −1.48119 + 1.67513i
259.5 1.00000 1.00000i 1.00000 0.311108 2.21432i 1.00000i 1.52543 1.00000 −1.00000 0.311108 2.21432i
259.6 1.00000 1.00000i 1.00000 2.17009 + 0.539189i 1.00000i 0.630898 1.00000 −1.00000 2.17009 + 0.539189i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 259.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.f.b yes 6
3.b odd 2 1 1170.2.f.c 6
4.b odd 2 1 3120.2.r.h 6
5.b even 2 1 390.2.f.a 6
5.c odd 4 1 1950.2.b.l 6
5.c odd 4 1 1950.2.b.m 6
13.b even 2 1 390.2.f.a 6
15.d odd 2 1 1170.2.f.d 6
20.d odd 2 1 3120.2.r.g 6
39.d odd 2 1 1170.2.f.d 6
52.b odd 2 1 3120.2.r.g 6
65.d even 2 1 inner 390.2.f.b yes 6
65.h odd 4 1 1950.2.b.l 6
65.h odd 4 1 1950.2.b.m 6
195.e odd 2 1 1170.2.f.c 6
260.g odd 2 1 3120.2.r.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.f.a 6 5.b even 2 1
390.2.f.a 6 13.b even 2 1
390.2.f.b yes 6 1.a even 1 1 trivial
390.2.f.b yes 6 65.d even 2 1 inner
1170.2.f.c 6 3.b odd 2 1
1170.2.f.c 6 195.e odd 2 1
1170.2.f.d 6 15.d odd 2 1
1170.2.f.d 6 39.d odd 2 1
1950.2.b.l 6 5.c odd 4 1
1950.2.b.l 6 65.h odd 4 1
1950.2.b.m 6 5.c odd 4 1
1950.2.b.m 6 65.h odd 4 1
3120.2.r.g 6 20.d odd 2 1
3120.2.r.g 6 52.b odd 2 1
3120.2.r.h 6 4.b odd 2 1
3120.2.r.h 6 260.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4$$ acting on $$S_{2}^{\mathrm{new}}(390, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{6}$$
$3$ $$(T^{2} + 1)^{3}$$
$5$ $$T^{6} - 2 T^{5} + \cdots + 125$$
$7$ $$(T^{3} + 2 T^{2} - 8 T + 4)^{2}$$
$11$ $$T^{6} + 44 T^{4} + \cdots + 1600$$
$13$ $$T^{6} - 8 T^{5} + \cdots + 2197$$
$17$ $$T^{6} + 16 T^{4} + \cdots + 16$$
$19$ $$T^{6} + 32 T^{4} + \cdots + 16$$
$23$ $$T^{6} + 12 T^{4} + \cdots + 16$$
$29$ $$(T^{3} - 2 T^{2} - 44 T + 20)^{2}$$
$31$ $$T^{6} + 108 T^{4} + \cdots + 1600$$
$37$ $$(T^{3} + 12 T^{2} + \cdots - 16)^{2}$$
$41$ $$T^{6} + 76 T^{4} + \cdots + 1600$$
$43$ $$T^{6} + 128 T^{4} + \cdots + 73984$$
$47$ $$(T^{3} - 16 T^{2} + \cdots - 32)^{2}$$
$53$ $$T^{6} + 236 T^{4} + \cdots + 40000$$
$59$ $$T^{6} + 252 T^{4} + \cdots + 61504$$
$61$ $$(T^{3} + 2 T^{2} + \cdots - 104)^{2}$$
$67$ $$(T^{3} + 2 T^{2} + \cdots - 184)^{2}$$
$71$ $$T^{6} + 112 T^{4} + \cdots + 6400$$
$73$ $$(T^{3} + 20 T^{2} + \cdots - 548)^{2}$$
$79$ $$(T^{3} + 8 T^{2} + \cdots - 2000)^{2}$$
$83$ $$(T^{3} - 144 T + 432)^{2}$$
$89$ $$T^{6} + 252 T^{4} + \cdots + 287296$$
$97$ $$(T^{3} - 100 T + 268)^{2}$$