Properties

 Label 6080.2.a.ch Level $6080$ Weight $2$ Character orbit 6080.a Self dual yes Analytic conductor $48.549$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.11344.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ x^4 - 2*x^3 - 4*x^2 + 4*x + 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 95) 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9}+O(q^{10})$$ q - b2 * q^3 + q^5 + (b1 - 1) * q^7 + (-b3 + 2) * q^9 $$q - \beta_{2} q^{3} + q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + 2) q^{9} + (\beta_1 + 1) q^{11} + ( - \beta_{2} - \beta_1 - 1) q^{13} - \beta_{2} q^{15} + (2 \beta_{2} + 2) q^{17} + q^{19} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{21} + (2 \beta_{2} + \beta_1 + 3) q^{23} + q^{25} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{27} + ( - 2 \beta_{2} - 2) q^{29} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{31} + ( - \beta_{3} + \beta_1 + 2) q^{33} + (\beta_1 - 1) q^{35} + ( - \beta_{3} - \beta_{2} + 1) q^{37} + ( - \beta_1 + 3) q^{39} + (\beta_{3} - \beta_1 + 4) q^{41} + ( - \beta_1 + 1) q^{43} + ( - \beta_{3} + 2) q^{45} + (2 \beta_{3} - \beta_1 + 3) q^{47} + (2 \beta_{3} - 2 \beta_1 + 5) q^{49} + (2 \beta_{3} - 2 \beta_{2} - 10) q^{51} + (\beta_{3} + \beta_{2} + 3) q^{53} + (\beta_1 + 1) q^{55} - \beta_{2} q^{57} + (\beta_{3} + \beta_1) q^{59} + (\beta_{3} - 2 \beta_1 - 5) q^{61} + ( - 4 \beta_{2} - \beta_1 - 7) q^{63} + ( - \beta_{2} - \beta_1 - 1) q^{65} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{67} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 8) q^{69} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 4) q^{71} + ( - 2 \beta_{2} + 6) q^{73} - \beta_{2} q^{75} + (2 \beta_{3} + 10) q^{77} + ( - \beta_{3} - \beta_1 + 4) q^{79} + ( - \beta_{3} + 2 \beta_1 + 4) q^{81} + (2 \beta_{2} - \beta_1 + 1) q^{83} + (2 \beta_{2} + 2) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2} + 10) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{89} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 8) q^{91} + ( - 4 \beta_{3} + 2 \beta_1 + 10) q^{93} + q^{95} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{97} + ( - 2 \beta_{3} - 4 \beta_{2} - \beta_1 - 3) q^{99}+O(q^{100})$$ q - b2 * q^3 + q^5 + (b1 - 1) * q^7 + (-b3 + 2) * q^9 + (b1 + 1) * q^11 + (-b2 - b1 - 1) * q^13 - b2 * q^15 + (2*b2 + 2) * q^17 + q^19 + (-b3 + 2*b2 + b1 + 2) * q^21 + (2*b2 + b1 + 3) * q^23 + q^25 + (-b3 - 2*b2 + b1 - 2) * q^27 + (-2*b2 - 2) * q^29 + (-b3 - 2*b2 + b1 - 2) * q^31 + (-b3 + b1 + 2) * q^33 + (b1 - 1) * q^35 + (-b3 - b2 + 1) * q^37 + (-b1 + 3) * q^39 + (b3 - b1 + 4) * q^41 + (-b1 + 1) * q^43 + (-b3 + 2) * q^45 + (2*b3 - b1 + 3) * q^47 + (2*b3 - 2*b1 + 5) * q^49 + (2*b3 - 2*b2 - 10) * q^51 + (b3 + b2 + 3) * q^53 + (b1 + 1) * q^55 - b2 * q^57 + (b3 + b1) * q^59 + (b3 - 2*b1 - 5) * q^61 + (-4*b2 - b1 - 7) * q^63 + (-b2 - b1 - 1) * q^65 + (b3 - 3*b2 - b1 - 6) * q^67 + (b3 - 2*b2 + b1 - 8) * q^69 + (b3 - 2*b2 + b1 + 4) * q^71 + (-2*b2 + 6) * q^73 - b2 * q^75 + (2*b3 + 10) * q^77 + (-b3 - b1 + 4) * q^79 + (-b3 + 2*b1 + 4) * q^81 + (2*b2 - b1 + 1) * q^83 + (2*b2 + 2) * q^85 + (-2*b3 + 2*b2 + 10) * q^87 + (-2*b3 - 2*b2) * q^89 + (-3*b3 + 2*b2 + b1 - 8) * q^91 + (-4*b3 + 2*b1 + 10) * q^93 + q^95 + (-b3 - b2 + 2*b1 + 7) * q^97 + (-2*b3 - 4*b2 - b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9 $$4 q + 2 q^{3} + 4 q^{5} - 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 8 q^{23} + 4 q^{25} - 4 q^{27} - 4 q^{29} - 4 q^{31} + 8 q^{33} - 4 q^{35} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} + 8 q^{45} + 12 q^{47} + 20 q^{49} - 36 q^{51} + 10 q^{53} + 4 q^{55} + 2 q^{57} - 20 q^{61} - 20 q^{63} - 2 q^{65} - 18 q^{67} - 28 q^{69} + 20 q^{71} + 28 q^{73} + 2 q^{75} + 40 q^{77} + 16 q^{79} + 16 q^{81} + 4 q^{85} + 36 q^{87} + 4 q^{89} - 36 q^{91} + 40 q^{93} + 4 q^{95} + 30 q^{97} - 4 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 + 4 * q^5 - 4 * q^7 + 8 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 8 * q^23 + 4 * q^25 - 4 * q^27 - 4 * q^29 - 4 * q^31 + 8 * q^33 - 4 * q^35 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 + 8 * q^45 + 12 * q^47 + 20 * q^49 - 36 * q^51 + 10 * q^53 + 4 * q^55 + 2 * q^57 - 20 * q^61 - 20 * q^63 - 2 * q^65 - 18 * q^67 - 28 * q^69 + 20 * q^71 + 28 * q^73 + 2 * q^75 + 40 * q^77 + 16 * q^79 + 16 * q^81 + 4 * q^85 + 36 * q^87 + 4 * q^89 - 36 * q^91 + 40 * q^93 + 4 * q^95 + 30 * q^97 - 4 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 4x^{2} + 4x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2\nu^{2} - 3\nu + 2$$ v^3 - 2*v^2 - 3*v + 2 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 2\nu - 5$$ 2*v^2 - 2*v - 5
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta _1 + 6 ) / 2$$ (b3 + b1 + 6) / 2 $$\nu^{3}$$ $$=$$ $$( 2\beta_{3} + 2\beta_{2} + 5\beta _1 + 11 ) / 2$$ (2*b3 + 2*b2 + 5*b1 + 11) / 2

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
 −0.552409 2.78165 −1.51658 1.28734
0 −2.87834 0 1.00000 0 −3.10482 0 5.28487 0
1.2 0 0.296842 0 1.00000 0 3.56331 0 −2.91188 0
1.3 0 1.53844 0 1.00000 0 −5.03316 0 −0.633188 0
1.4 0 3.04306 0 1.00000 0 0.574672 0 6.26020 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

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 6080.2.a.ch 4
4.b odd 2 1 6080.2.a.cc 4
8.b even 2 1 1520.2.a.t 4
8.d odd 2 1 95.2.a.b 4
24.f even 2 1 855.2.a.m 4
40.e odd 2 1 475.2.a.i 4
40.f even 2 1 7600.2.a.cf 4
40.k even 4 2 475.2.b.e 8
56.e even 2 1 4655.2.a.y 4
120.m even 2 1 4275.2.a.bo 4
152.b even 2 1 1805.2.a.p 4
760.p even 2 1 9025.2.a.bf 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.b 4 8.d odd 2 1
475.2.a.i 4 40.e odd 2 1
475.2.b.e 8 40.k even 4 2
855.2.a.m 4 24.f even 2 1
1520.2.a.t 4 8.b even 2 1
1805.2.a.p 4 152.b even 2 1
4275.2.a.bo 4 120.m even 2 1
4655.2.a.y 4 56.e even 2 1
6080.2.a.cc 4 4.b odd 2 1
6080.2.a.ch 4 1.a even 1 1 trivial
7600.2.a.cf 4 40.f even 2 1
9025.2.a.bf 4 760.p even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6080))$$:

 $$T_{3}^{4} - 2T_{3}^{3} - 8T_{3}^{2} + 16T_{3} - 4$$ T3^4 - 2*T3^3 - 8*T3^2 + 16*T3 - 4 $$T_{7}^{4} + 4T_{7}^{3} - 16T_{7}^{2} - 48T_{7} + 32$$ T7^4 + 4*T7^3 - 16*T7^2 - 48*T7 + 32 $$T_{11}^{4} - 4T_{11}^{3} - 16T_{11}^{2} + 32T_{11} + 48$$ T11^4 - 4*T11^3 - 16*T11^2 + 32*T11 + 48

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2 T^{3} - 8 T^{2} + 16 T - 4$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 4 T^{3} - 16 T^{2} - 48 T + 32$$
$11$ $$T^{4} - 4 T^{3} - 16 T^{2} + 32 T + 48$$
$13$ $$T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 20$$
$17$ $$T^{4} - 4 T^{3} - 32 T^{2} + 16 T + 48$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} - 8 T^{3} - 24 T^{2} + 176 T + 288$$
$29$ $$T^{4} + 4 T^{3} - 32 T^{2} - 16 T + 48$$
$31$ $$T^{4} + 4 T^{3} - 80 T^{2} - 512 T - 640$$
$37$ $$T^{4} - 6 T^{3} - 24 T^{2} + 40 T + 4$$
$41$ $$T^{4} - 16 T^{3} + 56 T^{2} + \cdots - 240$$
$43$ $$T^{4} - 4 T^{3} - 16 T^{2} + 48 T + 32$$
$47$ $$T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 1056$$
$53$ $$T^{4} - 10 T^{3} + 184 T - 348$$
$59$ $$T^{4} - 64 T^{2} - 224 T - 192$$
$61$ $$T^{4} + 20 T^{3} + 56 T^{2} + \cdots - 2656$$
$67$ $$T^{4} + 18 T^{3} + 8 T^{2} + \cdots - 1076$$
$71$ $$T^{4} - 20 T^{3} + 32 T^{2} + \cdots - 4224$$
$73$ $$T^{4} - 28 T^{3} + 256 T^{2} + \cdots + 176$$
$79$ $$T^{4} - 16 T^{3} + 32 T^{2} + \cdots - 1856$$
$83$ $$T^{4} - 72 T^{2} - 112 T + 480$$
$89$ $$T^{4} - 4 T^{3} - 144 T^{2} + \cdots + 240$$
$97$ $$T^{4} - 30 T^{3} + 224 T^{2} + \cdots - 1388$$