# Properties

 Label 342.2.b.a Level $342$ Weight $2$ Character orbit 342.b Analytic conductor $2.731$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,2,Mod(341,342)]

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([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("342.341");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.b (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$2.73088374913$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + \beta q^{5} + 2 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + b * q^5 + 2 * q^7 - q^8 $$q - q^{2} + q^{4} + \beta q^{5} + 2 q^{7} - q^{8} - \beta q^{10} + \beta q^{11} - 2 q^{14} + q^{16} + \beta q^{17} + (3 \beta + 1) q^{19} + \beta q^{20} - \beta q^{22} + \beta q^{23} + 3 q^{25} + 2 q^{28} + 6 q^{29} - q^{32} - \beta q^{34} + 2 \beta q^{35} + 6 \beta q^{37} + ( - 3 \beta - 1) q^{38} - \beta q^{40} + 6 q^{41} - 4 q^{43} + \beta q^{44} - \beta q^{46} - 5 \beta q^{47} - 3 q^{49} - 3 q^{50} + 6 q^{53} - 2 q^{55} - 2 q^{56} - 6 q^{58} - 4 q^{61} + q^{64} - 6 \beta q^{67} + \beta q^{68} - 2 \beta q^{70} - 12 q^{71} - 10 q^{73} - 6 \beta q^{74} + (3 \beta + 1) q^{76} + 2 \beta q^{77} - 6 \beta q^{79} + \beta q^{80} - 6 q^{82} - 11 \beta q^{83} - 2 q^{85} + 4 q^{86} - \beta q^{88} + 6 q^{89} + \beta q^{92} + 5 \beta q^{94} + (\beta - 6) q^{95} - 6 \beta q^{97} + 3 q^{98} +O(q^{100})$$ q - q^2 + q^4 + b * q^5 + 2 * q^7 - q^8 - b * q^10 + b * q^11 - 2 * q^14 + q^16 + b * q^17 + (3*b + 1) * q^19 + b * q^20 - b * q^22 + b * q^23 + 3 * q^25 + 2 * q^28 + 6 * q^29 - q^32 - b * q^34 + 2*b * q^35 + 6*b * q^37 + (-3*b - 1) * q^38 - b * q^40 + 6 * q^41 - 4 * q^43 + b * q^44 - b * q^46 - 5*b * q^47 - 3 * q^49 - 3 * q^50 + 6 * q^53 - 2 * q^55 - 2 * q^56 - 6 * q^58 - 4 * q^61 + q^64 - 6*b * q^67 + b * q^68 - 2*b * q^70 - 12 * q^71 - 10 * q^73 - 6*b * q^74 + (3*b + 1) * q^76 + 2*b * q^77 - 6*b * q^79 + b * q^80 - 6 * q^82 - 11*b * q^83 - 2 * q^85 + 4 * q^86 - b * q^88 + 6 * q^89 + b * q^92 + 5*b * q^94 + (b - 6) * q^95 - 6*b * q^97 + 3 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 $$2 q - 2 q^{2} + 2 q^{4} + 4 q^{7} - 2 q^{8} - 4 q^{14} + 2 q^{16} + 2 q^{19} + 6 q^{25} + 4 q^{28} + 12 q^{29} - 2 q^{32} - 2 q^{38} + 12 q^{41} - 8 q^{43} - 6 q^{49} - 6 q^{50} + 12 q^{53} - 4 q^{55} - 4 q^{56} - 12 q^{58} - 8 q^{61} + 2 q^{64} - 24 q^{71} - 20 q^{73} + 2 q^{76} - 12 q^{82} - 4 q^{85} + 8 q^{86} + 12 q^{89} - 12 q^{95} + 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 + 4 * q^7 - 2 * q^8 - 4 * q^14 + 2 * q^16 + 2 * q^19 + 6 * q^25 + 4 * q^28 + 12 * q^29 - 2 * q^32 - 2 * q^38 + 12 * q^41 - 8 * q^43 - 6 * q^49 - 6 * q^50 + 12 * q^53 - 4 * q^55 - 4 * q^56 - 12 * q^58 - 8 * q^61 + 2 * q^64 - 24 * q^71 - 20 * q^73 + 2 * q^76 - 12 * q^82 - 4 * q^85 + 8 * q^86 + 12 * q^89 - 12 * q^95 + 6 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
341.1
 − 1.41421i 1.41421i
−1.00000 0 1.00000 1.41421i 0 2.00000 −1.00000 0 1.41421i
341.2 −1.00000 0 1.00000 1.41421i 0 2.00000 −1.00000 0 1.41421i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.b.a 2
3.b odd 2 1 342.2.b.b yes 2
4.b odd 2 1 2736.2.f.b 2
12.b even 2 1 2736.2.f.a 2
19.b odd 2 1 342.2.b.b yes 2
57.d even 2 1 inner 342.2.b.a 2
76.d even 2 1 2736.2.f.a 2
228.b odd 2 1 2736.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.b.a 2 1.a even 1 1 trivial
342.2.b.a 2 57.d even 2 1 inner
342.2.b.b yes 2 3.b odd 2 1
342.2.b.b yes 2 19.b odd 2 1
2736.2.f.a 2 12.b even 2 1
2736.2.f.a 2 76.d even 2 1
2736.2.f.b 2 4.b odd 2 1
2736.2.f.b 2 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{29} - 6$$ acting on $$S_{2}^{\mathrm{new}}(342, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 2$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 2$$
$19$ $$T^{2} - 2T + 19$$
$23$ $$T^{2} + 2$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 72$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 50$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2}$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 72$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 72$$
$83$ $$T^{2} + 242$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 72$$