# Properties

 Label 3042.2.a.h Level $3042$ Weight $2$ Character orbit 3042.a Self dual yes Analytic conductor $24.290$ 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] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.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: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 78) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + 3 q^{5} - 2 q^{7} - q^{8}+O(q^{10})$$ q - q^2 + q^4 + 3 * q^5 - 2 * q^7 - q^8 $$q - q^{2} + q^{4} + 3 q^{5} - 2 q^{7} - q^{8} - 3 q^{10} + 6 q^{11} + 2 q^{14} + q^{16} + 3 q^{17} - 2 q^{19} + 3 q^{20} - 6 q^{22} + 6 q^{23} + 4 q^{25} - 2 q^{28} - 3 q^{29} + 4 q^{31} - q^{32} - 3 q^{34} - 6 q^{35} + 7 q^{37} + 2 q^{38} - 3 q^{40} - 3 q^{41} - 10 q^{43} + 6 q^{44} - 6 q^{46} + 6 q^{47} - 3 q^{49} - 4 q^{50} - 3 q^{53} + 18 q^{55} + 2 q^{56} + 3 q^{58} - 7 q^{61} - 4 q^{62} + q^{64} + 10 q^{67} + 3 q^{68} + 6 q^{70} + 6 q^{71} + 13 q^{73} - 7 q^{74} - 2 q^{76} - 12 q^{77} - 4 q^{79} + 3 q^{80} + 3 q^{82} - 6 q^{83} + 9 q^{85} + 10 q^{86} - 6 q^{88} + 18 q^{89} + 6 q^{92} - 6 q^{94} - 6 q^{95} - 14 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^4 + 3 * q^5 - 2 * q^7 - q^8 - 3 * q^10 + 6 * q^11 + 2 * q^14 + q^16 + 3 * q^17 - 2 * q^19 + 3 * q^20 - 6 * q^22 + 6 * q^23 + 4 * q^25 - 2 * q^28 - 3 * q^29 + 4 * q^31 - q^32 - 3 * q^34 - 6 * q^35 + 7 * q^37 + 2 * q^38 - 3 * q^40 - 3 * q^41 - 10 * q^43 + 6 * q^44 - 6 * q^46 + 6 * q^47 - 3 * q^49 - 4 * q^50 - 3 * q^53 + 18 * q^55 + 2 * q^56 + 3 * q^58 - 7 * q^61 - 4 * q^62 + q^64 + 10 * q^67 + 3 * q^68 + 6 * q^70 + 6 * q^71 + 13 * q^73 - 7 * q^74 - 2 * q^76 - 12 * q^77 - 4 * q^79 + 3 * q^80 + 3 * q^82 - 6 * q^83 + 9 * q^85 + 10 * q^86 - 6 * q^88 + 18 * q^89 + 6 * q^92 - 6 * q^94 - 6 * q^95 - 14 * 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 0 1.00000 3.00000 0 −2.00000 −1.00000 0 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 3042.2.a.h 1
3.b odd 2 1 1014.2.a.f 1
12.b even 2 1 8112.2.a.c 1
13.b even 2 1 3042.2.a.i 1
13.d odd 4 2 3042.2.b.h 2
13.e even 6 2 234.2.h.a 2
39.d odd 2 1 1014.2.a.c 1
39.f even 4 2 1014.2.b.c 2
39.h odd 6 2 78.2.e.a 2
39.i odd 6 2 1014.2.e.a 2
39.k even 12 4 1014.2.i.b 4
52.i odd 6 2 1872.2.t.c 2
156.h even 2 1 8112.2.a.m 1
156.r even 6 2 624.2.q.g 2
195.y odd 6 2 1950.2.i.m 2
195.bf even 12 4 1950.2.z.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 39.h odd 6 2
234.2.h.a 2 13.e even 6 2
624.2.q.g 2 156.r even 6 2
1014.2.a.c 1 39.d odd 2 1
1014.2.a.f 1 3.b odd 2 1
1014.2.b.c 2 39.f even 4 2
1014.2.e.a 2 39.i odd 6 2
1014.2.i.b 4 39.k even 12 4
1872.2.t.c 2 52.i odd 6 2
1950.2.i.m 2 195.y odd 6 2
1950.2.z.g 4 195.bf even 12 4
3042.2.a.h 1 1.a even 1 1 trivial
3042.2.a.i 1 13.b even 2 1
3042.2.b.h 2 13.d odd 4 2
8112.2.a.c 1 12.b even 2 1
8112.2.a.m 1 156.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3042))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 6$$ T11 - 6 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T + 2$$
$11$ $$T - 6$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T + 2$$
$23$ $$T - 6$$
$29$ $$T + 3$$
$31$ $$T - 4$$
$37$ $$T - 7$$
$41$ $$T + 3$$
$43$ $$T + 10$$
$47$ $$T - 6$$
$53$ $$T + 3$$
$59$ $$T$$
$61$ $$T + 7$$
$67$ $$T - 10$$
$71$ $$T - 6$$
$73$ $$T - 13$$
$79$ $$T + 4$$
$83$ $$T + 6$$
$89$ $$T - 18$$
$97$ $$T + 14$$