# Properties

 Label 2240.2.a.n Level $2240$ Weight $2$ Character orbit 2240.a Self dual yes Analytic conductor $17.886$ Analytic rank $0$ 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] = [2240,2,Mod(1,2240)]

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([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("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.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: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{7} - 3 q^{9}+O(q^{10})$$ q + q^5 - q^7 - 3 * q^9 $$q + q^{5} - q^{7} - 3 q^{9} - 4 q^{11} + 6 q^{13} + 2 q^{17} + q^{25} - 6 q^{29} + 8 q^{31} - q^{35} + 10 q^{37} + 2 q^{41} - 4 q^{43} - 3 q^{45} + 8 q^{47} + q^{49} + 2 q^{53} - 4 q^{55} + 8 q^{59} + 14 q^{61} + 3 q^{63} + 6 q^{65} + 12 q^{67} - 16 q^{71} + 2 q^{73} + 4 q^{77} - 8 q^{79} + 9 q^{81} - 8 q^{83} + 2 q^{85} + 10 q^{89} - 6 q^{91} + 2 q^{97} + 12 q^{99}+O(q^{100})$$ q + q^5 - q^7 - 3 * q^9 - 4 * q^11 + 6 * q^13 + 2 * q^17 + q^25 - 6 * q^29 + 8 * q^31 - q^35 + 10 * q^37 + 2 * q^41 - 4 * q^43 - 3 * q^45 + 8 * q^47 + q^49 + 2 * q^53 - 4 * q^55 + 8 * q^59 + 14 * q^61 + 3 * q^63 + 6 * q^65 + 12 * q^67 - 16 * q^71 + 2 * q^73 + 4 * q^77 - 8 * q^79 + 9 * q^81 - 8 * q^83 + 2 * q^85 + 10 * q^89 - 6 * q^91 + 2 * q^97 + 12 * q^99

## 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
0 0 0 1.00000 0 −1.00000 0 −3.00000 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## 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 2240.2.a.n 1
4.b odd 2 1 2240.2.a.q 1
8.b even 2 1 70.2.a.a 1
8.d odd 2 1 560.2.a.d 1
24.f even 2 1 5040.2.a.bm 1
24.h odd 2 1 630.2.a.d 1
40.e odd 2 1 2800.2.a.m 1
40.f even 2 1 350.2.a.b 1
40.i odd 4 2 350.2.c.b 2
40.k even 4 2 2800.2.g.n 2
56.e even 2 1 3920.2.a.t 1
56.h odd 2 1 490.2.a.h 1
56.j odd 6 2 490.2.e.c 2
56.p even 6 2 490.2.e.d 2
88.b odd 2 1 8470.2.a.j 1
120.i odd 2 1 3150.2.a.bj 1
120.w even 4 2 3150.2.g.c 2
168.i even 2 1 4410.2.a.b 1
280.c odd 2 1 2450.2.a.l 1
280.s even 4 2 2450.2.c.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 8.b even 2 1
350.2.a.b 1 40.f even 2 1
350.2.c.b 2 40.i odd 4 2
490.2.a.h 1 56.h odd 2 1
490.2.e.c 2 56.j odd 6 2
490.2.e.d 2 56.p even 6 2
560.2.a.d 1 8.d odd 2 1
630.2.a.d 1 24.h odd 2 1
2240.2.a.n 1 1.a even 1 1 trivial
2240.2.a.q 1 4.b odd 2 1
2450.2.a.l 1 280.c odd 2 1
2450.2.c.k 2 280.s even 4 2
2800.2.a.m 1 40.e odd 2 1
2800.2.g.n 2 40.k even 4 2
3150.2.a.bj 1 120.i odd 2 1
3150.2.g.c 2 120.w even 4 2
3920.2.a.t 1 56.e even 2 1
4410.2.a.b 1 168.i even 2 1
5040.2.a.bm 1 24.f even 2 1
8470.2.a.j 1 88.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2240))$$:

 $$T_{3}$$ T3 $$T_{11} + 4$$ T11 + 4 $$T_{13} - 6$$ T13 - 6 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T + 1$$
$11$ $$T + 4$$
$13$ $$T - 6$$
$17$ $$T - 2$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 10$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T - 8$$
$53$ $$T - 2$$
$59$ $$T - 8$$
$61$ $$T - 14$$
$67$ $$T - 12$$
$71$ $$T + 16$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T + 8$$
$89$ $$T - 10$$
$97$ $$T - 2$$