Properties

 Label 3042.2.a.l Level $3042$ Weight $2$ Character orbit 3042.a Self dual yes Analytic conductor $24.290$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8}+O(q^{10})$$ q + q^2 + q^4 - q^5 - q^7 + q^8 $$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - 2 q^{11} - q^{14} + q^{16} + 3 q^{17} - 6 q^{19} - q^{20} - 2 q^{22} + 4 q^{23} - 4 q^{25} - q^{28} - 2 q^{29} - 4 q^{31} + q^{32} + 3 q^{34} + q^{35} - 3 q^{37} - 6 q^{38} - q^{40} - 5 q^{43} - 2 q^{44} + 4 q^{46} + 13 q^{47} - 6 q^{49} - 4 q^{50} - 12 q^{53} + 2 q^{55} - q^{56} - 2 q^{58} - 10 q^{59} - 8 q^{61} - 4 q^{62} + q^{64} + 2 q^{67} + 3 q^{68} + q^{70} - 5 q^{71} + 10 q^{73} - 3 q^{74} - 6 q^{76} + 2 q^{77} - 4 q^{79} - q^{80} - 3 q^{85} - 5 q^{86} - 2 q^{88} + 6 q^{89} + 4 q^{92} + 13 q^{94} + 6 q^{95} - 14 q^{97} - 6 q^{98}+O(q^{100})$$ q + q^2 + q^4 - q^5 - q^7 + q^8 - q^10 - 2 * q^11 - q^14 + q^16 + 3 * q^17 - 6 * q^19 - q^20 - 2 * q^22 + 4 * q^23 - 4 * q^25 - q^28 - 2 * q^29 - 4 * q^31 + q^32 + 3 * q^34 + q^35 - 3 * q^37 - 6 * q^38 - q^40 - 5 * q^43 - 2 * q^44 + 4 * q^46 + 13 * q^47 - 6 * q^49 - 4 * q^50 - 12 * q^53 + 2 * q^55 - q^56 - 2 * q^58 - 10 * q^59 - 8 * q^61 - 4 * q^62 + q^64 + 2 * q^67 + 3 * q^68 + q^70 - 5 * q^71 + 10 * q^73 - 3 * q^74 - 6 * q^76 + 2 * q^77 - 4 * q^79 - q^80 - 3 * q^85 - 5 * q^86 - 2 * q^88 + 6 * q^89 + 4 * q^92 + 13 * q^94 + 6 * q^95 - 14 * q^97 - 6 * 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 −1.00000 0 −1.00000 1.00000 0 −1.00000
 $$n$$: 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.l 1
3.b odd 2 1 338.2.a.a 1
12.b even 2 1 2704.2.a.n 1
13.b even 2 1 234.2.a.b 1
13.d odd 4 2 3042.2.b.f 2
15.d odd 2 1 8450.2.a.y 1
39.d odd 2 1 26.2.a.b 1
39.f even 4 2 338.2.b.a 2
39.h odd 6 2 338.2.c.c 2
39.i odd 6 2 338.2.c.g 2
39.k even 12 4 338.2.e.d 4
52.b odd 2 1 1872.2.a.m 1
65.d even 2 1 5850.2.a.bn 1
65.h odd 4 2 5850.2.e.v 2
104.e even 2 1 7488.2.a.w 1
104.h odd 2 1 7488.2.a.v 1
117.n odd 6 2 2106.2.e.h 2
117.t even 6 2 2106.2.e.t 2
156.h even 2 1 208.2.a.d 1
156.l odd 4 2 2704.2.f.j 2
195.e odd 2 1 650.2.a.g 1
195.s even 4 2 650.2.b.a 2
273.g even 2 1 1274.2.a.o 1
273.w odd 6 2 1274.2.f.l 2
273.ba even 6 2 1274.2.f.a 2
312.b odd 2 1 832.2.a.j 1
312.h even 2 1 832.2.a.a 1
429.e even 2 1 3146.2.a.a 1
624.v even 4 2 3328.2.b.k 2
624.bi odd 4 2 3328.2.b.g 2
663.g odd 2 1 7514.2.a.i 1
741.d even 2 1 9386.2.a.f 1
780.d even 2 1 5200.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.b 1 39.d odd 2 1
208.2.a.d 1 156.h even 2 1
234.2.a.b 1 13.b even 2 1
338.2.a.a 1 3.b odd 2 1
338.2.b.a 2 39.f even 4 2
338.2.c.c 2 39.h odd 6 2
338.2.c.g 2 39.i odd 6 2
338.2.e.d 4 39.k even 12 4
650.2.a.g 1 195.e odd 2 1
650.2.b.a 2 195.s even 4 2
832.2.a.a 1 312.h even 2 1
832.2.a.j 1 312.b odd 2 1
1274.2.a.o 1 273.g even 2 1
1274.2.f.a 2 273.ba even 6 2
1274.2.f.l 2 273.w odd 6 2
1872.2.a.m 1 52.b odd 2 1
2106.2.e.h 2 117.n odd 6 2
2106.2.e.t 2 117.t even 6 2
2704.2.a.n 1 12.b even 2 1
2704.2.f.j 2 156.l odd 4 2
3042.2.a.l 1 1.a even 1 1 trivial
3042.2.b.f 2 13.d odd 4 2
3146.2.a.a 1 429.e even 2 1
3328.2.b.g 2 624.bi odd 4 2
3328.2.b.k 2 624.v even 4 2
5200.2.a.c 1 780.d even 2 1
5850.2.a.bn 1 65.d even 2 1
5850.2.e.v 2 65.h odd 4 2
7488.2.a.v 1 104.h odd 2 1
7488.2.a.w 1 104.e even 2 1
7514.2.a.i 1 663.g odd 2 1
8450.2.a.y 1 15.d odd 2 1
9386.2.a.f 1 741.d even 2 1

Hecke kernels

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

 $$T_{5} + 1$$ T5 + 1 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 2$$ T11 + 2 $$T_{17} - 3$$ T17 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T + 1$$
$11$ $$T + 2$$
$13$ $$T$$
$17$ $$T - 3$$
$19$ $$T + 6$$
$23$ $$T - 4$$
$29$ $$T + 2$$
$31$ $$T + 4$$
$37$ $$T + 3$$
$41$ $$T$$
$43$ $$T + 5$$
$47$ $$T - 13$$
$53$ $$T + 12$$
$59$ $$T + 10$$
$61$ $$T + 8$$
$67$ $$T - 2$$
$71$ $$T + 5$$
$73$ $$T - 10$$
$79$ $$T + 4$$
$83$ $$T$$
$89$ $$T - 6$$
$97$ $$T + 14$$