# Properties

 Label 342.2.a.f Level $342$ Weight $2$ Character orbit 342.a Self dual yes Analytic conductor $2.731$ 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] = [342,2,Mod(1,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, 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("342.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.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: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 4 q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 + 4 * q^7 + q^8 $$q + q^{2} + q^{4} + 4 q^{7} + q^{8} - 4 q^{11} + 4 q^{14} + q^{16} + 2 q^{17} + q^{19} - 4 q^{22} + 2 q^{23} - 5 q^{25} + 4 q^{28} + 6 q^{29} + 6 q^{31} + q^{32} + 2 q^{34} - 8 q^{37} + q^{38} - 10 q^{41} - 12 q^{43} - 4 q^{44} + 2 q^{46} - 10 q^{47} + 9 q^{49} - 5 q^{50} - 2 q^{53} + 4 q^{56} + 6 q^{58} - 4 q^{59} - 10 q^{61} + 6 q^{62} + q^{64} + 2 q^{68} + 16 q^{71} - 2 q^{73} - 8 q^{74} + q^{76} - 16 q^{77} + 10 q^{79} - 10 q^{82} + 16 q^{83} - 12 q^{86} - 4 q^{88} + 2 q^{89} + 2 q^{92} - 10 q^{94} - 10 q^{97} + 9 q^{98}+O(q^{100})$$ q + q^2 + q^4 + 4 * q^7 + q^8 - 4 * q^11 + 4 * q^14 + q^16 + 2 * q^17 + q^19 - 4 * q^22 + 2 * q^23 - 5 * q^25 + 4 * q^28 + 6 * q^29 + 6 * q^31 + q^32 + 2 * q^34 - 8 * q^37 + q^38 - 10 * q^41 - 12 * q^43 - 4 * q^44 + 2 * q^46 - 10 * q^47 + 9 * q^49 - 5 * q^50 - 2 * q^53 + 4 * q^56 + 6 * q^58 - 4 * q^59 - 10 * q^61 + 6 * q^62 + q^64 + 2 * q^68 + 16 * q^71 - 2 * q^73 - 8 * q^74 + q^76 - 16 * q^77 + 10 * q^79 - 10 * q^82 + 16 * q^83 - 12 * q^86 - 4 * q^88 + 2 * q^89 + 2 * q^92 - 10 * q^94 - 10 * q^97 + 9 * 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 0 1.00000 0 0 4.00000 1.00000 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$19$$ $$-1$$

## 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 342.2.a.f 1
3.b odd 2 1 114.2.a.a 1
4.b odd 2 1 2736.2.a.j 1
5.b even 2 1 8550.2.a.a 1
12.b even 2 1 912.2.a.h 1
15.d odd 2 1 2850.2.a.x 1
15.e even 4 2 2850.2.d.s 2
19.b odd 2 1 6498.2.a.h 1
21.c even 2 1 5586.2.a.p 1
24.f even 2 1 3648.2.a.j 1
24.h odd 2 1 3648.2.a.bb 1
57.d even 2 1 2166.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.a.a 1 3.b odd 2 1
342.2.a.f 1 1.a even 1 1 trivial
912.2.a.h 1 12.b even 2 1
2166.2.a.i 1 57.d even 2 1
2736.2.a.j 1 4.b odd 2 1
2850.2.a.x 1 15.d odd 2 1
2850.2.d.s 2 15.e even 4 2
3648.2.a.j 1 24.f even 2 1
3648.2.a.bb 1 24.h odd 2 1
5586.2.a.p 1 21.c even 2 1
6498.2.a.h 1 19.b odd 2 1
8550.2.a.a 1 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(342))$$:

 $$T_{5}$$ T5 $$T_{7} - 4$$ T7 - 4 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 4$$
$11$ $$T + 4$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T - 1$$
$23$ $$T - 2$$
$29$ $$T - 6$$
$31$ $$T - 6$$
$37$ $$T + 8$$
$41$ $$T + 10$$
$43$ $$T + 12$$
$47$ $$T + 10$$
$53$ $$T + 2$$
$59$ $$T + 4$$
$61$ $$T + 10$$
$67$ $$T$$
$71$ $$T - 16$$
$73$ $$T + 2$$
$79$ $$T - 10$$
$83$ $$T - 16$$
$89$ $$T - 2$$
$97$ $$T + 10$$