# Properties

 Label 138.2.a.c Level $138$ Weight $2$ Character orbit 138.a Self dual yes Analytic conductor $1.102$ 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] = [138,2,Mod(1,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, 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("138.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 138.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: $$1.10193554789$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9} + 2 q^{10} - q^{12} - 2 q^{13} - 2 q^{15} + q^{16} + 2 q^{17} + q^{18} - 8 q^{19} + 2 q^{20} - q^{23} - q^{24} - q^{25} - 2 q^{26} - q^{27} - 2 q^{29} - 2 q^{30} - 8 q^{31} + q^{32} + 2 q^{34} + q^{36} + 2 q^{37} - 8 q^{38} + 2 q^{39} + 2 q^{40} + 10 q^{41} + 8 q^{43} + 2 q^{45} - q^{46} + 8 q^{47} - q^{48} - 7 q^{49} - q^{50} - 2 q^{51} - 2 q^{52} + 2 q^{53} - q^{54} + 8 q^{57} - 2 q^{58} - 4 q^{59} - 2 q^{60} + 2 q^{61} - 8 q^{62} + q^{64} - 4 q^{65} + 8 q^{67} + 2 q^{68} + q^{69} + q^{72} - 6 q^{73} + 2 q^{74} + q^{75} - 8 q^{76} + 2 q^{78} + 8 q^{79} + 2 q^{80} + q^{81} + 10 q^{82} - 16 q^{83} + 4 q^{85} + 8 q^{86} + 2 q^{87} + 18 q^{89} + 2 q^{90} - q^{92} + 8 q^{93} + 8 q^{94} - 16 q^{95} - q^{96} + 10 q^{97} - 7 q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 + 2 * q^5 - q^6 + q^8 + q^9 + 2 * q^10 - q^12 - 2 * q^13 - 2 * q^15 + q^16 + 2 * q^17 + q^18 - 8 * q^19 + 2 * q^20 - q^23 - q^24 - q^25 - 2 * q^26 - q^27 - 2 * q^29 - 2 * q^30 - 8 * q^31 + q^32 + 2 * q^34 + q^36 + 2 * q^37 - 8 * q^38 + 2 * q^39 + 2 * q^40 + 10 * q^41 + 8 * q^43 + 2 * q^45 - q^46 + 8 * q^47 - q^48 - 7 * q^49 - q^50 - 2 * q^51 - 2 * q^52 + 2 * q^53 - q^54 + 8 * q^57 - 2 * q^58 - 4 * q^59 - 2 * q^60 + 2 * q^61 - 8 * q^62 + q^64 - 4 * q^65 + 8 * q^67 + 2 * q^68 + q^69 + q^72 - 6 * q^73 + 2 * q^74 + q^75 - 8 * q^76 + 2 * q^78 + 8 * q^79 + 2 * q^80 + q^81 + 10 * q^82 - 16 * q^83 + 4 * q^85 + 8 * q^86 + 2 * q^87 + 18 * q^89 + 2 * q^90 - q^92 + 8 * q^93 + 8 * q^94 - 16 * q^95 - q^96 + 10 * q^97 - 7 * 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
 0
1.00000 −1.00000 1.00000 2.00000 −1.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$23$$ $$1$$

## 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 138.2.a.c 1
3.b odd 2 1 414.2.a.a 1
4.b odd 2 1 1104.2.a.g 1
5.b even 2 1 3450.2.a.k 1
5.c odd 4 2 3450.2.d.f 2
7.b odd 2 1 6762.2.a.bg 1
8.b even 2 1 4416.2.a.s 1
8.d odd 2 1 4416.2.a.c 1
12.b even 2 1 3312.2.a.d 1
23.b odd 2 1 3174.2.a.e 1
69.c even 2 1 9522.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.c 1 1.a even 1 1 trivial
414.2.a.a 1 3.b odd 2 1
1104.2.a.g 1 4.b odd 2 1
3174.2.a.e 1 23.b odd 2 1
3312.2.a.d 1 12.b even 2 1
3450.2.a.k 1 5.b even 2 1
3450.2.d.f 2 5.c odd 4 2
4416.2.a.c 1 8.d odd 2 1
4416.2.a.s 1 8.b even 2 1
6762.2.a.bg 1 7.b odd 2 1
9522.2.a.d 1 69.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(138))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T - 2$$
$19$ $$T + 8$$
$23$ $$T + 1$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T - 10$$
$43$ $$T - 8$$
$47$ $$T - 8$$
$53$ $$T - 2$$
$59$ $$T + 4$$
$61$ $$T - 2$$
$67$ $$T - 8$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T - 8$$
$83$ $$T + 16$$
$89$ $$T - 18$$
$97$ $$T - 10$$