# Properties

 Label 39.2.a.b Level $39$ Weight $2$ Character orbit 39.a Self dual yes Analytic conductor $0.311$ 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] = [39,2,Mod(1,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("39.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.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: $$0.311416567883$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
−2.41421 1.00000 3.82843 2.82843 −2.41421 −2.82843 −4.41421 1.00000 −6.82843
1.2 0.414214 1.00000 −1.82843 −2.82843 0.414214 2.82843 −1.58579 1.00000 −1.17157
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 39.2.a.b 2
3.b odd 2 1 117.2.a.c 2
4.b odd 2 1 624.2.a.k 2
5.b even 2 1 975.2.a.l 2
5.c odd 4 2 975.2.c.h 4
7.b odd 2 1 1911.2.a.h 2
8.b even 2 1 2496.2.a.bf 2
8.d odd 2 1 2496.2.a.bi 2
9.c even 3 2 1053.2.e.m 4
9.d odd 6 2 1053.2.e.e 4
11.b odd 2 1 4719.2.a.p 2
12.b even 2 1 1872.2.a.w 2
13.b even 2 1 507.2.a.h 2
13.c even 3 2 507.2.e.h 4
13.d odd 4 2 507.2.b.e 4
13.e even 6 2 507.2.e.d 4
13.f odd 12 4 507.2.j.f 8
15.d odd 2 1 2925.2.a.v 2
15.e even 4 2 2925.2.c.u 4
21.c even 2 1 5733.2.a.u 2
24.f even 2 1 7488.2.a.co 2
24.h odd 2 1 7488.2.a.cl 2
39.d odd 2 1 1521.2.a.f 2
39.f even 4 2 1521.2.b.j 4
52.b odd 2 1 8112.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.b 2 1.a even 1 1 trivial
117.2.a.c 2 3.b odd 2 1
507.2.a.h 2 13.b even 2 1
507.2.b.e 4 13.d odd 4 2
507.2.e.d 4 13.e even 6 2
507.2.e.h 4 13.c even 3 2
507.2.j.f 8 13.f odd 12 4
624.2.a.k 2 4.b odd 2 1
975.2.a.l 2 5.b even 2 1
975.2.c.h 4 5.c odd 4 2
1053.2.e.e 4 9.d odd 6 2
1053.2.e.m 4 9.c even 3 2
1521.2.a.f 2 39.d odd 2 1
1521.2.b.j 4 39.f even 4 2
1872.2.a.w 2 12.b even 2 1
1911.2.a.h 2 7.b odd 2 1
2496.2.a.bf 2 8.b even 2 1
2496.2.a.bi 2 8.d odd 2 1
2925.2.a.v 2 15.d odd 2 1
2925.2.c.u 4 15.e even 4 2
4719.2.a.p 2 11.b odd 2 1
5733.2.a.u 2 21.c even 2 1
7488.2.a.cl 2 24.h odd 2 1
7488.2.a.co 2 24.f even 2 1
8112.2.a.bm 2 52.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(39))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} - 8$$
$7$ $$T^{2} - 8$$
$11$ $$(T + 2)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 4T - 28$$
$19$ $$T^{2} - 8$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 8T + 8$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$T^{2} - 16T + 56$$
$43$ $$T^{2} - 8T - 16$$
$47$ $$T^{2} + 12T + 4$$
$53$ $$(T + 2)^{2}$$
$59$ $$T^{2} - 4T - 28$$
$61$ $$T^{2} - 4T - 124$$
$67$ $$T^{2} - 8T + 8$$
$71$ $$(T - 2)^{2}$$
$73$ $$T^{2} - 12T + 4$$
$79$ $$T^{2} - 128$$
$83$ $$T^{2} + 4T - 28$$
$89$ $$T^{2} - 24T + 136$$
$97$ $$T^{2} + 4T - 28$$