Properties

 Label 3042.2.b.f Level $3042$ Weight $2$ Character orbit 3042.b Analytic conductor $24.290$ 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] = [3042,2,Mod(1351,3042)]

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, 1]))

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

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

chi := DirichletCharacter("3042.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.b (of order $$2$$, degree $$1$$, not 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: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + i q^{5} - i q^{7} + i q^{8} +O(q^{10})$$ q - i * q^2 - q^4 + i * q^5 - i * q^7 + i * q^8 $$q - i q^{2} - q^{4} + i q^{5} - i q^{7} + i q^{8} + q^{10} - 2 i q^{11} - q^{14} + q^{16} - 3 q^{17} + 6 i q^{19} - i q^{20} - 2 q^{22} - 4 q^{23} + 4 q^{25} + i q^{28} - 2 q^{29} + 4 i q^{31} - i q^{32} + 3 i q^{34} + q^{35} - 3 i q^{37} + 6 q^{38} - q^{40} + 5 q^{43} + 2 i q^{44} + 4 i q^{46} + 13 i q^{47} + 6 q^{49} - 4 i q^{50} - 12 q^{53} + 2 q^{55} + q^{56} + 2 i q^{58} - 10 i q^{59} - 8 q^{61} + 4 q^{62} - q^{64} - 2 i q^{67} + 3 q^{68} - i q^{70} + 5 i q^{71} + 10 i q^{73} - 3 q^{74} - 6 i q^{76} - 2 q^{77} - 4 q^{79} + i q^{80} - 3 i q^{85} - 5 i q^{86} + 2 q^{88} + 6 i q^{89} + 4 q^{92} + 13 q^{94} - 6 q^{95} + 14 i q^{97} - 6 i q^{98} +O(q^{100})$$ q - i * q^2 - q^4 + i * q^5 - i * q^7 + i * q^8 + q^10 - 2*i * q^11 - q^14 + q^16 - 3 * q^17 + 6*i * q^19 - i * q^20 - 2 * q^22 - 4 * q^23 + 4 * q^25 + i * q^28 - 2 * q^29 + 4*i * q^31 - i * q^32 + 3*i * q^34 + q^35 - 3*i * q^37 + 6 * q^38 - q^40 + 5 * q^43 + 2*i * q^44 + 4*i * q^46 + 13*i * q^47 + 6 * q^49 - 4*i * q^50 - 12 * q^53 + 2 * q^55 + q^56 + 2*i * q^58 - 10*i * q^59 - 8 * q^61 + 4 * q^62 - q^64 - 2*i * q^67 + 3 * q^68 - i * q^70 + 5*i * q^71 + 10*i * q^73 - 3 * q^74 - 6*i * q^76 - 2 * q^77 - 4 * q^79 + i * q^80 - 3*i * q^85 - 5*i * q^86 + 2 * q^88 + 6*i * q^89 + 4 * q^92 + 13 * q^94 - 6 * q^95 + 14*i * q^97 - 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 2 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 4 q^{22} - 8 q^{23} + 8 q^{25} - 4 q^{29} + 2 q^{35} + 12 q^{38} - 2 q^{40} + 10 q^{43} + 12 q^{49} - 24 q^{53} + 4 q^{55} + 2 q^{56} - 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 6 q^{74} - 4 q^{77} - 8 q^{79} + 4 q^{88} + 8 q^{92} + 26 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 4 * q^22 - 8 * q^23 + 8 * q^25 - 4 * q^29 + 2 * q^35 + 12 * q^38 - 2 * q^40 + 10 * q^43 + 12 * q^49 - 24 * q^53 + 4 * q^55 + 2 * q^56 - 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 6 * q^74 - 4 * q^77 - 8 * q^79 + 4 * q^88 + 8 * q^92 + 26 * q^94 - 12 * q^95

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
1351.2 1.00000i 0 −1.00000 1.00000i 0 1.00000i 1.00000i 0 1.00000
 $$n$$: 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
13.b even 2 1 inner

Twists

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

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

Hecke kernels

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

 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{17} + 3$$ T17 + 3 $$T_{23} + 4$$ T23 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 4)^{2}$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 9$$
$41$ $$T^{2}$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2} + 169$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 100$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$T^{2} + 25$$
$73$ $$T^{2} + 100$$
$79$ $$(T + 4)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 196$$