# Properties

 Label 39.2.a.a Level $39$ Weight $2$ Character orbit 39.a Self dual yes Analytic conductor $0.311$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^7 - 3 * q^8 + q^9 $$q + q^{2} - q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + q^{13} - 4 q^{14} - 2 q^{15} - q^{16} + 2 q^{17} + q^{18} - 2 q^{20} + 4 q^{21} + 4 q^{22} + 3 q^{24} - q^{25} + q^{26} - q^{27} + 4 q^{28} - 10 q^{29} - 2 q^{30} + 4 q^{31} + 5 q^{32} - 4 q^{33} + 2 q^{34} - 8 q^{35} - q^{36} - 2 q^{37} - q^{39} - 6 q^{40} + 6 q^{41} + 4 q^{42} - 12 q^{43} - 4 q^{44} + 2 q^{45} + q^{48} + 9 q^{49} - q^{50} - 2 q^{51} - q^{52} + 6 q^{53} - q^{54} + 8 q^{55} + 12 q^{56} - 10 q^{58} + 12 q^{59} + 2 q^{60} - 2 q^{61} + 4 q^{62} - 4 q^{63} + 7 q^{64} + 2 q^{65} - 4 q^{66} - 8 q^{67} - 2 q^{68} - 8 q^{70} - 3 q^{72} + 2 q^{73} - 2 q^{74} + q^{75} - 16 q^{77} - q^{78} + 8 q^{79} - 2 q^{80} + q^{81} + 6 q^{82} + 4 q^{83} - 4 q^{84} + 4 q^{85} - 12 q^{86} + 10 q^{87} - 12 q^{88} - 2 q^{89} + 2 q^{90} - 4 q^{91} - 4 q^{93} - 5 q^{96} + 10 q^{97} + 9 q^{98} + 4 q^{99}+O(q^{100})$$ q + q^2 - q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^7 - 3 * q^8 + q^9 + 2 * q^10 + 4 * q^11 + q^12 + q^13 - 4 * q^14 - 2 * q^15 - q^16 + 2 * q^17 + q^18 - 2 * q^20 + 4 * q^21 + 4 * q^22 + 3 * q^24 - q^25 + q^26 - q^27 + 4 * q^28 - 10 * q^29 - 2 * q^30 + 4 * q^31 + 5 * q^32 - 4 * q^33 + 2 * q^34 - 8 * q^35 - q^36 - 2 * q^37 - q^39 - 6 * q^40 + 6 * q^41 + 4 * q^42 - 12 * q^43 - 4 * q^44 + 2 * q^45 + q^48 + 9 * q^49 - q^50 - 2 * q^51 - q^52 + 6 * q^53 - q^54 + 8 * q^55 + 12 * q^56 - 10 * q^58 + 12 * q^59 + 2 * q^60 - 2 * q^61 + 4 * q^62 - 4 * q^63 + 7 * q^64 + 2 * q^65 - 4 * q^66 - 8 * q^67 - 2 * q^68 - 8 * q^70 - 3 * q^72 + 2 * q^73 - 2 * q^74 + q^75 - 16 * q^77 - q^78 + 8 * q^79 - 2 * q^80 + q^81 + 6 * q^82 + 4 * q^83 - 4 * q^84 + 4 * q^85 - 12 * q^86 + 10 * q^87 - 12 * q^88 - 2 * q^89 + 2 * q^90 - 4 * q^91 - 4 * q^93 - 5 * q^96 + 10 * q^97 + 9 * 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
 0
1.00000 −1.00000 −1.00000 2.00000 −1.00000 −4.00000 −3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.a 1
3.b odd 2 1 117.2.a.a 1
4.b odd 2 1 624.2.a.i 1
5.b even 2 1 975.2.a.f 1
5.c odd 4 2 975.2.c.f 2
7.b odd 2 1 1911.2.a.f 1
8.b even 2 1 2496.2.a.q 1
8.d odd 2 1 2496.2.a.e 1
9.c even 3 2 1053.2.e.b 2
9.d odd 6 2 1053.2.e.d 2
11.b odd 2 1 4719.2.a.c 1
12.b even 2 1 1872.2.a.h 1
13.b even 2 1 507.2.a.a 1
13.c even 3 2 507.2.e.a 2
13.d odd 4 2 507.2.b.a 2
13.e even 6 2 507.2.e.b 2
13.f odd 12 4 507.2.j.e 4
15.d odd 2 1 2925.2.a.p 1
15.e even 4 2 2925.2.c.e 2
21.c even 2 1 5733.2.a.e 1
24.f even 2 1 7488.2.a.by 1
24.h odd 2 1 7488.2.a.bl 1
39.d odd 2 1 1521.2.a.e 1
39.f even 4 2 1521.2.b.b 2
52.b odd 2 1 8112.2.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 1.a even 1 1 trivial
117.2.a.a 1 3.b odd 2 1
507.2.a.a 1 13.b even 2 1
507.2.b.a 2 13.d odd 4 2
507.2.e.a 2 13.c even 3 2
507.2.e.b 2 13.e even 6 2
507.2.j.e 4 13.f odd 12 4
624.2.a.i 1 4.b odd 2 1
975.2.a.f 1 5.b even 2 1
975.2.c.f 2 5.c odd 4 2
1053.2.e.b 2 9.c even 3 2
1053.2.e.d 2 9.d odd 6 2
1521.2.a.e 1 39.d odd 2 1
1521.2.b.b 2 39.f even 4 2
1872.2.a.h 1 12.b even 2 1
1911.2.a.f 1 7.b odd 2 1
2496.2.a.e 1 8.d odd 2 1
2496.2.a.q 1 8.b even 2 1
2925.2.a.p 1 15.d odd 2 1
2925.2.c.e 2 15.e even 4 2
4719.2.a.c 1 11.b odd 2 1
5733.2.a.e 1 21.c even 2 1
7488.2.a.bl 1 24.h odd 2 1
7488.2.a.by 1 24.f even 2 1
8112.2.a.s 1 52.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T + 4$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T - 4$$
$37$ $$T + 2$$
$41$ $$T - 6$$
$43$ $$T + 12$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T + 8$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 4$$
$89$ $$T + 2$$
$97$ $$T - 10$$