# Properties

 Label 2240.2.k.b Level $2240$ Weight $2$ Character orbit 2240.k Analytic conductor $17.886$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1791,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2240.1791");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.k (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: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 140) 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} - 2 \beta_1) q^{7}+O(q^{10})$$ q - b3 * q^3 - b1 * q^5 + (b3 - 2*b1) * q^7 $$q - \beta_{3} q^{3} - \beta_1 q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} + 3 \beta_1) q^{13} + \beta_{2} q^{15} + (2 \beta_{2} + 3 \beta_1) q^{17} + 6 q^{19} + (2 \beta_{2} - 3) q^{21} + ( - 2 \beta_{2} + 2 \beta_1) q^{23} - q^{25} + 3 \beta_{3} q^{27} + ( - 4 \beta_{3} - 1) q^{29} + 6 q^{31} + ( - 2 \beta_{2} + 3 \beta_1) q^{33} + ( - \beta_{2} - 2) q^{35} + (2 \beta_{3} + 6) q^{37} + ( - 3 \beta_{2} + 6 \beta_1) q^{39} + 2 \beta_{2} q^{41} + 2 \beta_1 q^{43} - \beta_{3} q^{47} + ( - 4 \beta_{2} - 1) q^{49} + ( - 3 \beta_{2} - 6 \beta_1) q^{51} - 2 q^{53} + ( - \beta_{3} + 2) q^{55} - 6 \beta_{3} q^{57} - 2 \beta_{3} q^{59} + (2 \beta_{2} + 6 \beta_1) q^{61} + ( - 2 \beta_{3} + 3) q^{65} - 2 \beta_{2} q^{67} + ( - 2 \beta_{2} + 6 \beta_1) q^{69} + ( - 2 \beta_{2} - 4 \beta_1) q^{71} + ( - 4 \beta_{2} - 6 \beta_1) q^{73} + \beta_{3} q^{75} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{77} + ( - 5 \beta_{2} - 6 \beta_1) q^{79} - 9 q^{81} + (2 \beta_{3} + 12) q^{83} + (2 \beta_{3} + 3) q^{85} + (\beta_{3} + 12) q^{87} + (2 \beta_{2} - 6 \beta_1) q^{89} + ( - 4 \beta_{3} + 3 \beta_{2} - 6 \beta_1 + 6) q^{91} - 6 \beta_{3} q^{93} - 6 \beta_1 q^{95} + (6 \beta_{2} + 3 \beta_1) q^{97}+O(q^{100})$$ q - b3 * q^3 - b1 * q^5 + (b3 - 2*b1) * q^7 + (-b2 + 2*b1) * q^11 + (-2*b2 + 3*b1) * q^13 + b2 * q^15 + (2*b2 + 3*b1) * q^17 + 6 * q^19 + (2*b2 - 3) * q^21 + (-2*b2 + 2*b1) * q^23 - q^25 + 3*b3 * q^27 + (-4*b3 - 1) * q^29 + 6 * q^31 + (-2*b2 + 3*b1) * q^33 + (-b2 - 2) * q^35 + (2*b3 + 6) * q^37 + (-3*b2 + 6*b1) * q^39 + 2*b2 * q^41 + 2*b1 * q^43 - b3 * q^47 + (-4*b2 - 1) * q^49 + (-3*b2 - 6*b1) * q^51 - 2 * q^53 + (-b3 + 2) * q^55 - 6*b3 * q^57 - 2*b3 * q^59 + (2*b2 + 6*b1) * q^61 + (-2*b3 + 3) * q^65 - 2*b2 * q^67 + (-2*b2 + 6*b1) * q^69 + (-2*b2 - 4*b1) * q^71 + (-4*b2 - 6*b1) * q^73 + b3 * q^75 + (-2*b3 + 2*b2 - 3*b1 + 4) * q^77 + (-5*b2 - 6*b1) * q^79 - 9 * q^81 + (2*b3 + 12) * q^83 + (2*b3 + 3) * q^85 + (b3 + 12) * q^87 + (2*b2 - 6*b1) * q^89 + (-4*b3 + 3*b2 - 6*b1 + 6) * q^91 - 6*b3 * q^93 - 6*b1 * q^95 + (6*b2 + 3*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 24 q^{19} - 12 q^{21} - 4 q^{25} - 4 q^{29} + 24 q^{31} - 8 q^{35} + 24 q^{37} - 4 q^{49} - 8 q^{53} + 8 q^{55} + 12 q^{65} + 16 q^{77} - 36 q^{81} + 48 q^{83} + 12 q^{85} + 48 q^{87} + 24 q^{91}+O(q^{100})$$ 4 * q + 24 * q^19 - 12 * q^21 - 4 * q^25 - 4 * q^29 + 24 * q^31 - 8 * q^35 + 24 * q^37 - 4 * q^49 - 8 * q^53 + 8 * q^55 + 12 * q^65 + 16 * q^77 - 36 * q^81 + 48 * q^83 + 12 * q^85 + 48 * q^87 + 24 * q^91

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$2\zeta_{12}^{2} - 1$$ 2*v^2 - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 1 ) / 2$$ (b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$$.

 $$n$$ $$897$$ $$1471$$ $$1541$$ $$1921$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1791.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 0 1.00000i 0 1.73205 2.00000i 0 0 0
1791.2 0 −1.73205 0 1.00000i 0 1.73205 + 2.00000i 0 0 0
1791.3 0 1.73205 0 1.00000i 0 −1.73205 2.00000i 0 0 0
1791.4 0 1.73205 0 1.00000i 0 −1.73205 + 2.00000i 0 0 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
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.k.b 4
4.b odd 2 1 2240.2.k.a 4
7.b odd 2 1 2240.2.k.a 4
8.b even 2 1 140.2.g.b yes 4
8.d odd 2 1 140.2.g.a 4
24.f even 2 1 1260.2.c.a 4
24.h odd 2 1 1260.2.c.b 4
28.d even 2 1 inner 2240.2.k.b 4
40.e odd 2 1 700.2.g.f 4
40.f even 2 1 700.2.g.g 4
40.i odd 4 1 700.2.c.c 4
40.i odd 4 1 700.2.c.f 4
40.k even 4 1 700.2.c.b 4
40.k even 4 1 700.2.c.e 4
56.e even 2 1 140.2.g.b yes 4
56.h odd 2 1 140.2.g.a 4
56.j odd 6 1 980.2.o.a 4
56.j odd 6 1 980.2.o.c 4
56.k odd 6 1 980.2.o.a 4
56.k odd 6 1 980.2.o.c 4
56.m even 6 1 980.2.o.b 4
56.m even 6 1 980.2.o.d 4
56.p even 6 1 980.2.o.b 4
56.p even 6 1 980.2.o.d 4
168.e odd 2 1 1260.2.c.b 4
168.i even 2 1 1260.2.c.a 4
280.c odd 2 1 700.2.g.f 4
280.n even 2 1 700.2.g.g 4
280.s even 4 1 700.2.c.b 4
280.s even 4 1 700.2.c.e 4
280.y odd 4 1 700.2.c.c 4
280.y odd 4 1 700.2.c.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 8.d odd 2 1
140.2.g.a 4 56.h odd 2 1
140.2.g.b yes 4 8.b even 2 1
140.2.g.b yes 4 56.e even 2 1
700.2.c.b 4 40.k even 4 1
700.2.c.b 4 280.s even 4 1
700.2.c.c 4 40.i odd 4 1
700.2.c.c 4 280.y odd 4 1
700.2.c.e 4 40.k even 4 1
700.2.c.e 4 280.s even 4 1
700.2.c.f 4 40.i odd 4 1
700.2.c.f 4 280.y odd 4 1
700.2.g.f 4 40.e odd 2 1
700.2.g.f 4 280.c odd 2 1
700.2.g.g 4 40.f even 2 1
700.2.g.g 4 280.n even 2 1
980.2.o.a 4 56.j odd 6 1
980.2.o.a 4 56.k odd 6 1
980.2.o.b 4 56.m even 6 1
980.2.o.b 4 56.p even 6 1
980.2.o.c 4 56.j odd 6 1
980.2.o.c 4 56.k odd 6 1
980.2.o.d 4 56.m even 6 1
980.2.o.d 4 56.p even 6 1
1260.2.c.a 4 24.f even 2 1
1260.2.c.a 4 168.i even 2 1
1260.2.c.b 4 24.h odd 2 1
1260.2.c.b 4 168.e odd 2 1
2240.2.k.a 4 4.b odd 2 1
2240.2.k.a 4 7.b odd 2 1
2240.2.k.b 4 1.a even 1 1 trivial
2240.2.k.b 4 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

 $$T_{3}^{2} - 3$$ T3^2 - 3 $$T_{19} - 6$$ T19 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$(T^{2} + 1)^{2}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} + 14T^{2} + 1$$
$13$ $$T^{4} + 42T^{2} + 9$$
$17$ $$T^{4} + 42T^{2} + 9$$
$19$ $$(T - 6)^{4}$$
$23$ $$T^{4} + 32T^{2} + 64$$
$29$ $$(T^{2} + 2 T - 47)^{2}$$
$31$ $$(T - 6)^{4}$$
$37$ $$(T^{2} - 12 T + 24)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 3)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$T^{4} + 96T^{2} + 576$$
$67$ $$(T^{2} + 12)^{2}$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$T^{4} + 168T^{2} + 144$$
$79$ $$T^{4} + 222T^{2} + 1521$$
$83$ $$(T^{2} - 24 T + 132)^{2}$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$T^{4} + 234T^{2} + 9801$$