# Properties

 Label 39.4.a.a Level $39$ Weight $4$ Character orbit 39.a Self dual yes Analytic conductor $2.301$ Analytic rank $1$ 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] = [39,4,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, 4, names="a")

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

chi := DirichletCharacter("39.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$2.30107449022$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 8 q^{4} - 12 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 8 * q^4 - 12 * q^5 + 2 * q^7 + 9 * q^9 $$q - 3 q^{3} - 8 q^{4} - 12 q^{5} + 2 q^{7} + 9 q^{9} - 36 q^{11} + 24 q^{12} + 13 q^{13} + 36 q^{15} + 64 q^{16} - 78 q^{17} + 74 q^{19} + 96 q^{20} - 6 q^{21} - 96 q^{23} + 19 q^{25} - 27 q^{27} - 16 q^{28} + 18 q^{29} - 214 q^{31} + 108 q^{33} - 24 q^{35} - 72 q^{36} - 286 q^{37} - 39 q^{39} - 384 q^{41} + 524 q^{43} + 288 q^{44} - 108 q^{45} + 300 q^{47} - 192 q^{48} - 339 q^{49} + 234 q^{51} - 104 q^{52} + 558 q^{53} + 432 q^{55} - 222 q^{57} + 576 q^{59} - 288 q^{60} + 74 q^{61} + 18 q^{63} - 512 q^{64} - 156 q^{65} + 38 q^{67} + 624 q^{68} + 288 q^{69} - 456 q^{71} - 682 q^{73} - 57 q^{75} - 592 q^{76} - 72 q^{77} + 704 q^{79} - 768 q^{80} + 81 q^{81} - 888 q^{83} + 48 q^{84} + 936 q^{85} - 54 q^{87} - 1020 q^{89} + 26 q^{91} + 768 q^{92} + 642 q^{93} - 888 q^{95} + 110 q^{97} - 324 q^{99}+O(q^{100})$$ q - 3 * q^3 - 8 * q^4 - 12 * q^5 + 2 * q^7 + 9 * q^9 - 36 * q^11 + 24 * q^12 + 13 * q^13 + 36 * q^15 + 64 * q^16 - 78 * q^17 + 74 * q^19 + 96 * q^20 - 6 * q^21 - 96 * q^23 + 19 * q^25 - 27 * q^27 - 16 * q^28 + 18 * q^29 - 214 * q^31 + 108 * q^33 - 24 * q^35 - 72 * q^36 - 286 * q^37 - 39 * q^39 - 384 * q^41 + 524 * q^43 + 288 * q^44 - 108 * q^45 + 300 * q^47 - 192 * q^48 - 339 * q^49 + 234 * q^51 - 104 * q^52 + 558 * q^53 + 432 * q^55 - 222 * q^57 + 576 * q^59 - 288 * q^60 + 74 * q^61 + 18 * q^63 - 512 * q^64 - 156 * q^65 + 38 * q^67 + 624 * q^68 + 288 * q^69 - 456 * q^71 - 682 * q^73 - 57 * q^75 - 592 * q^76 - 72 * q^77 + 704 * q^79 - 768 * q^80 + 81 * q^81 - 888 * q^83 + 48 * q^84 + 936 * q^85 - 54 * q^87 - 1020 * q^89 + 26 * q^91 + 768 * q^92 + 642 * q^93 - 888 * q^95 + 110 * q^97 - 324 * 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
0 −3.00000 −8.00000 −12.0000 0 2.00000 0 9.00000 0
 $$n$$: 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 39.4.a.a 1
3.b odd 2 1 117.4.a.a 1
4.b odd 2 1 624.4.a.g 1
5.b even 2 1 975.4.a.e 1
7.b odd 2 1 1911.4.a.f 1
8.b even 2 1 2496.4.a.o 1
8.d odd 2 1 2496.4.a.f 1
12.b even 2 1 1872.4.a.m 1
13.b even 2 1 507.4.a.c 1
13.d odd 4 2 507.4.b.b 2
39.d odd 2 1 1521.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.a 1 1.a even 1 1 trivial
117.4.a.a 1 3.b odd 2 1
507.4.a.c 1 13.b even 2 1
507.4.b.b 2 13.d odd 4 2
624.4.a.g 1 4.b odd 2 1
975.4.a.e 1 5.b even 2 1
1521.4.a.f 1 39.d odd 2 1
1872.4.a.m 1 12.b even 2 1
1911.4.a.f 1 7.b odd 2 1
2496.4.a.f 1 8.d odd 2 1
2496.4.a.o 1 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(39))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 12$$
$7$ $$T - 2$$
$11$ $$T + 36$$
$13$ $$T - 13$$
$17$ $$T + 78$$
$19$ $$T - 74$$
$23$ $$T + 96$$
$29$ $$T - 18$$
$31$ $$T + 214$$
$37$ $$T + 286$$
$41$ $$T + 384$$
$43$ $$T - 524$$
$47$ $$T - 300$$
$53$ $$T - 558$$
$59$ $$T - 576$$
$61$ $$T - 74$$
$67$ $$T - 38$$
$71$ $$T + 456$$
$73$ $$T + 682$$
$79$ $$T - 704$$
$83$ $$T + 888$$
$89$ $$T + 1020$$
$97$ $$T - 110$$