Properties

 Label 390.2.a.h Level $390$ Weight $2$ Character orbit 390.a Self dual yes Analytic conductor $3.114$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(1,390)]

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, 0, 0]))

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

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

chi := DirichletCharacter("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

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: yes Analytic conductor: $$3.11416567883$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
1.00000 1.00000 1.00000 1.00000 1.00000 −2.82843 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 2.82843 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$+1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.a.h 2
3.b odd 2 1 1170.2.a.o 2
4.b odd 2 1 3120.2.a.bc 2
5.b even 2 1 1950.2.a.bd 2
5.c odd 4 2 1950.2.e.o 4
12.b even 2 1 9360.2.a.ch 2
13.b even 2 1 5070.2.a.bc 2
13.d odd 4 2 5070.2.b.q 4
15.d odd 2 1 5850.2.a.cl 2
15.e even 4 2 5850.2.e.bk 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.h 2 1.a even 1 1 trivial
1170.2.a.o 2 3.b odd 2 1
1950.2.a.bd 2 5.b even 2 1
1950.2.e.o 4 5.c odd 4 2
3120.2.a.bc 2 4.b odd 2 1
5070.2.a.bc 2 13.b even 2 1
5070.2.b.q 4 13.d odd 4 2
5850.2.a.cl 2 15.d odd 2 1
5850.2.e.bk 4 15.e even 4 2
9360.2.a.ch 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(390))$$:

 $$T_{7}^{2} - 8$$ T7^2 - 8 $$T_{11}^{2} - 32$$ T11^2 - 32 $$T_{31} - 4$$ T31 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 8$$
$11$ $$T^{2} - 32$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 4T - 4$$
$19$ $$T^{2} - 8$$
$23$ $$T^{2} - 72$$
$29$ $$T^{2} + 12T + 28$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + 12T + 4$$
$41$ $$T^{2} + 4T - 28$$
$43$ $$T^{2} - 8T - 16$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 4T - 124$$
$59$ $$T^{2} + 16T + 32$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} - 32$$
$71$ $$T^{2} - 32$$
$73$ $$T^{2} + 12T - 36$$
$79$ $$T^{2} - 16T + 32$$
$83$ $$T^{2} - 24T + 112$$
$89$ $$T^{2} + 20T + 68$$
$97$ $$T^{2} + 12T + 28$$