# Properties

 Label 342.2.g.c Level $342$ Weight $2$ Character orbit 342.g Analytic conductor $2.731$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,2,Mod(163,342)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("342.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$342 = 2 \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 342.g (of order $$3$$, degree $$2$$, 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: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 114) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + q^{7} + q^{8} +O(q^{10})$$ q + (z - 1) * q^2 - z * q^4 + q^7 + q^8 $$q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + q^{7} + q^{8} + 2 q^{11} + 3 \zeta_{6} q^{13} + (\zeta_{6} - 1) q^{14} + (\zeta_{6} - 1) q^{16} + ( - 4 \zeta_{6} + 4) q^{17} + (2 \zeta_{6} + 3) q^{19} + (2 \zeta_{6} - 2) q^{22} + 4 \zeta_{6} q^{23} + 5 \zeta_{6} q^{25} - 3 q^{26} - \zeta_{6} q^{28} - 3 q^{31} - \zeta_{6} q^{32} + 4 \zeta_{6} q^{34} - 5 q^{37} + (3 \zeta_{6} - 5) q^{38} + ( - 4 \zeta_{6} + 4) q^{41} + ( - 9 \zeta_{6} + 9) q^{43} - 2 \zeta_{6} q^{44} - 4 q^{46} + 10 \zeta_{6} q^{47} - 6 q^{49} - 5 q^{50} + ( - 3 \zeta_{6} + 3) q^{52} - 4 \zeta_{6} q^{53} + q^{56} + (14 \zeta_{6} - 14) q^{59} - 11 \zeta_{6} q^{61} + ( - 3 \zeta_{6} + 3) q^{62} + q^{64} - 3 \zeta_{6} q^{67} - 4 q^{68} + ( - 14 \zeta_{6} + 14) q^{71} + ( - 11 \zeta_{6} + 11) q^{73} + ( - 5 \zeta_{6} + 5) q^{74} + ( - 5 \zeta_{6} + 2) q^{76} + 2 q^{77} + (\zeta_{6} - 1) q^{79} + 4 \zeta_{6} q^{82} - 8 q^{83} + 9 \zeta_{6} q^{86} + 2 q^{88} - 14 \zeta_{6} q^{89} + 3 \zeta_{6} q^{91} + ( - 4 \zeta_{6} + 4) q^{92} - 10 q^{94} + (2 \zeta_{6} - 2) q^{97} + ( - 6 \zeta_{6} + 6) q^{98} +O(q^{100})$$ q + (z - 1) * q^2 - z * q^4 + q^7 + q^8 + 2 * q^11 + 3*z * q^13 + (z - 1) * q^14 + (z - 1) * q^16 + (-4*z + 4) * q^17 + (2*z + 3) * q^19 + (2*z - 2) * q^22 + 4*z * q^23 + 5*z * q^25 - 3 * q^26 - z * q^28 - 3 * q^31 - z * q^32 + 4*z * q^34 - 5 * q^37 + (3*z - 5) * q^38 + (-4*z + 4) * q^41 + (-9*z + 9) * q^43 - 2*z * q^44 - 4 * q^46 + 10*z * q^47 - 6 * q^49 - 5 * q^50 + (-3*z + 3) * q^52 - 4*z * q^53 + q^56 + (14*z - 14) * q^59 - 11*z * q^61 + (-3*z + 3) * q^62 + q^64 - 3*z * q^67 - 4 * q^68 + (-14*z + 14) * q^71 + (-11*z + 11) * q^73 + (-5*z + 5) * q^74 + (-5*z + 2) * q^76 + 2 * q^77 + (z - 1) * q^79 + 4*z * q^82 - 8 * q^83 + 9*z * q^86 + 2 * q^88 - 14*z * q^89 + 3*z * q^91 + (-4*z + 4) * q^92 - 10 * q^94 + (2*z - 2) * q^97 + (-6*z + 6) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8 $$2 q - q^{2} - q^{4} + 2 q^{7} + 2 q^{8} + 4 q^{11} + 3 q^{13} - q^{14} - q^{16} + 4 q^{17} + 8 q^{19} - 2 q^{22} + 4 q^{23} + 5 q^{25} - 6 q^{26} - q^{28} - 6 q^{31} - q^{32} + 4 q^{34} - 10 q^{37} - 7 q^{38} + 4 q^{41} + 9 q^{43} - 2 q^{44} - 8 q^{46} + 10 q^{47} - 12 q^{49} - 10 q^{50} + 3 q^{52} - 4 q^{53} + 2 q^{56} - 14 q^{59} - 11 q^{61} + 3 q^{62} + 2 q^{64} - 3 q^{67} - 8 q^{68} + 14 q^{71} + 11 q^{73} + 5 q^{74} - q^{76} + 4 q^{77} - q^{79} + 4 q^{82} - 16 q^{83} + 9 q^{86} + 4 q^{88} - 14 q^{89} + 3 q^{91} + 4 q^{92} - 20 q^{94} - 2 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - q^2 - q^4 + 2 * q^7 + 2 * q^8 + 4 * q^11 + 3 * q^13 - q^14 - q^16 + 4 * q^17 + 8 * q^19 - 2 * q^22 + 4 * q^23 + 5 * q^25 - 6 * q^26 - q^28 - 6 * q^31 - q^32 + 4 * q^34 - 10 * q^37 - 7 * q^38 + 4 * q^41 + 9 * q^43 - 2 * q^44 - 8 * q^46 + 10 * q^47 - 12 * q^49 - 10 * q^50 + 3 * q^52 - 4 * q^53 + 2 * q^56 - 14 * q^59 - 11 * q^61 + 3 * q^62 + 2 * q^64 - 3 * q^67 - 8 * q^68 + 14 * q^71 + 11 * q^73 + 5 * q^74 - q^76 + 4 * q^77 - q^79 + 4 * q^82 - 16 * q^83 + 9 * q^86 + 4 * q^88 - 14 * q^89 + 3 * q^91 + 4 * q^92 - 20 * q^94 - 2 * q^97 + 6 * q^98

## Character values

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

 $$n$$ $$191$$ $$325$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 1.00000 1.00000 0 0
235.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 1.00000 1.00000 0 0
 $$n$$: 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.g.c 2
3.b odd 2 1 114.2.e.b 2
4.b odd 2 1 2736.2.s.k 2
12.b even 2 1 912.2.q.b 2
19.c even 3 1 inner 342.2.g.c 2
19.c even 3 1 6498.2.a.u 1
19.d odd 6 1 6498.2.a.g 1
57.f even 6 1 2166.2.a.h 1
57.h odd 6 1 114.2.e.b 2
57.h odd 6 1 2166.2.a.b 1
76.g odd 6 1 2736.2.s.k 2
228.m even 6 1 912.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.2.e.b 2 3.b odd 2 1
114.2.e.b 2 57.h odd 6 1
342.2.g.c 2 1.a even 1 1 trivial
342.2.g.c 2 19.c even 3 1 inner
912.2.q.b 2 12.b even 2 1
912.2.q.b 2 228.m even 6 1
2166.2.a.b 1 57.h odd 6 1
2166.2.a.h 1 57.f even 6 1
2736.2.s.k 2 4.b odd 2 1
2736.2.s.k 2 76.g odd 6 1
6498.2.a.g 1 19.d odd 6 1
6498.2.a.u 1 19.c even 3 1

## Hecke kernels

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

 $$T_{5}$$ T5 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T - 1)^{2}$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} - 3T + 9$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} - 8T + 19$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$T^{2}$$
$31$ $$(T + 3)^{2}$$
$37$ $$(T + 5)^{2}$$
$41$ $$T^{2} - 4T + 16$$
$43$ $$T^{2} - 9T + 81$$
$47$ $$T^{2} - 10T + 100$$
$53$ $$T^{2} + 4T + 16$$
$59$ $$T^{2} + 14T + 196$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} + 3T + 9$$
$71$ $$T^{2} - 14T + 196$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T + 8)^{2}$$
$89$ $$T^{2} + 14T + 196$$
$97$ $$T^{2} + 2T + 4$$