# Properties

 Label 138.2.a.b 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 + 2 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{12} + 2 q^{13} - 2 q^{14} + q^{16} - q^{18} + 2 q^{19} + 2 q^{21} - q^{23} - q^{24} - 5 q^{25} - 2 q^{26} + q^{27} + 2 q^{28} - 6 q^{29} - 4 q^{31} - q^{32} + q^{36} - 10 q^{37} - 2 q^{38} + 2 q^{39} - 6 q^{41} - 2 q^{42} + 2 q^{43} + q^{46} + q^{48} - 3 q^{49} + 5 q^{50} + 2 q^{52} + 12 q^{53} - q^{54} - 2 q^{56} + 2 q^{57} + 6 q^{58} + 12 q^{59} - 10 q^{61} + 4 q^{62} + 2 q^{63} + q^{64} + 14 q^{67} - q^{69} - q^{72} + 2 q^{73} + 10 q^{74} - 5 q^{75} + 2 q^{76} - 2 q^{78} - 10 q^{79} + q^{81} + 6 q^{82} + 2 q^{84} - 2 q^{86} - 6 q^{87} + 12 q^{89} + 4 q^{91} - q^{92} - 4 q^{93} - q^{96} - 10 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 + 2 * q^7 - q^8 + q^9 + q^12 + 2 * q^13 - 2 * q^14 + q^16 - q^18 + 2 * q^19 + 2 * q^21 - q^23 - q^24 - 5 * q^25 - 2 * q^26 + q^27 + 2 * q^28 - 6 * q^29 - 4 * q^31 - q^32 + q^36 - 10 * q^37 - 2 * q^38 + 2 * q^39 - 6 * q^41 - 2 * q^42 + 2 * q^43 + q^46 + q^48 - 3 * q^49 + 5 * q^50 + 2 * q^52 + 12 * q^53 - q^54 - 2 * q^56 + 2 * q^57 + 6 * q^58 + 12 * q^59 - 10 * q^61 + 4 * q^62 + 2 * q^63 + q^64 + 14 * q^67 - q^69 - q^72 + 2 * q^73 + 10 * q^74 - 5 * q^75 + 2 * q^76 - 2 * q^78 - 10 * q^79 + q^81 + 6 * q^82 + 2 * q^84 - 2 * q^86 - 6 * q^87 + 12 * q^89 + 4 * q^91 - q^92 - 4 * q^93 - q^96 - 10 * q^97 + 3 * 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 0 −1.00000 2.00000 −1.00000 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$23$$ $$1$$

## 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.b 1
3.b odd 2 1 414.2.a.c 1
4.b odd 2 1 1104.2.a.b 1
5.b even 2 1 3450.2.a.o 1
5.c odd 4 2 3450.2.d.g 2
7.b odd 2 1 6762.2.a.g 1
8.b even 2 1 4416.2.a.i 1
8.d odd 2 1 4416.2.a.t 1
12.b even 2 1 3312.2.a.h 1
23.b odd 2 1 3174.2.a.d 1
69.c even 2 1 9522.2.a.k 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T - 2$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T$$
$19$ $$T - 2$$
$23$ $$T + 1$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T + 10$$
$41$ $$T + 6$$
$43$ $$T - 2$$
$47$ $$T$$
$53$ $$T - 12$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T - 14$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 10$$
$83$ $$T$$
$89$ $$T - 12$$
$97$ $$T + 10$$