Properties

 Label 2760.2.k.a Level $2760$ Weight $2$ Character orbit 2760.k Analytic conductor $22.039$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2760,2,Mod(2209,2760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2760, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("2760.2209");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2760 = 2^{3} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2760.k (of order $$2$$, degree $$1$$, 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: $$22.0387109579$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + (i - 2) * q^5 + 2*i * q^7 - q^9 $$q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} - 4 q^{11} + (2 i + 1) q^{15} - 2 i q^{17} + 4 q^{19} + 2 q^{21} - i q^{23} + ( - 4 i + 3) q^{25} + i q^{27} - 6 q^{29} - 4 q^{31} + 4 i q^{33} + ( - 4 i - 2) q^{35} + 6 i q^{37} - 2 q^{41} + 2 i q^{43} + ( - i + 2) q^{45} + 3 q^{49} - 2 q^{51} - 6 i q^{53} + ( - 4 i + 8) q^{55} - 4 i q^{57} + 14 q^{59} + 10 q^{61} - 2 i q^{63} - 14 i q^{67} - q^{69} + 10 q^{71} + ( - 3 i - 4) q^{75} - 8 i q^{77} + 16 q^{79} + q^{81} - 16 i q^{83} + (4 i + 2) q^{85} + 6 i q^{87} + 8 q^{89} + 4 i q^{93} + (4 i - 8) q^{95} - 10 i q^{97} + 4 q^{99} +O(q^{100})$$ q - i * q^3 + (i - 2) * q^5 + 2*i * q^7 - q^9 - 4 * q^11 + (2*i + 1) * q^15 - 2*i * q^17 + 4 * q^19 + 2 * q^21 - i * q^23 + (-4*i + 3) * q^25 + i * q^27 - 6 * q^29 - 4 * q^31 + 4*i * q^33 + (-4*i - 2) * q^35 + 6*i * q^37 - 2 * q^41 + 2*i * q^43 + (-i + 2) * q^45 + 3 * q^49 - 2 * q^51 - 6*i * q^53 + (-4*i + 8) * q^55 - 4*i * q^57 + 14 * q^59 + 10 * q^61 - 2*i * q^63 - 14*i * q^67 - q^69 + 10 * q^71 + (-3*i - 4) * q^75 - 8*i * q^77 + 16 * q^79 + q^81 - 16*i * q^83 + (4*i + 2) * q^85 + 6*i * q^87 + 8 * q^89 + 4*i * q^93 + (4*i - 8) * q^95 - 10*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^5 - 2 * q^9 $$2 q - 4 q^{5} - 2 q^{9} - 8 q^{11} + 2 q^{15} + 8 q^{19} + 4 q^{21} + 6 q^{25} - 12 q^{29} - 8 q^{31} - 4 q^{35} - 4 q^{41} + 4 q^{45} + 6 q^{49} - 4 q^{51} + 16 q^{55} + 28 q^{59} + 20 q^{61} - 2 q^{69} + 20 q^{71} - 8 q^{75} + 32 q^{79} + 2 q^{81} + 4 q^{85} + 16 q^{89} - 16 q^{95} + 8 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 - 2 * q^9 - 8 * q^11 + 2 * q^15 + 8 * q^19 + 4 * q^21 + 6 * q^25 - 12 * q^29 - 8 * q^31 - 4 * q^35 - 4 * q^41 + 4 * q^45 + 6 * q^49 - 4 * q^51 + 16 * q^55 + 28 * q^59 + 20 * q^61 - 2 * q^69 + 20 * q^71 - 8 * q^75 + 32 * q^79 + 2 * q^81 + 4 * q^85 + 16 * q^89 - 16 * q^95 + 8 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2760\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$1381$$ $$1657$$ $$1841$$ $$2071$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2209.1
 1.00000i − 1.00000i
0 1.00000i 0 −2.00000 + 1.00000i 0 2.00000i 0 −1.00000 0
2209.2 0 1.00000i 0 −2.00000 1.00000i 0 2.00000i 0 −1.00000 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2760.2.k.a 2
5.b even 2 1 inner 2760.2.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2760.2.k.a 2 1.a even 1 1 trivial
2760.2.k.a 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(2760, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 4$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T + 6)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 36$$
$41$ $$(T + 2)^{2}$$
$43$ $$T^{2} + 4$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 14)^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 16)^{2}$$
$83$ $$T^{2} + 256$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 100$$
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