Properties

 Label 1840.2.a.e Level $1840$ Weight $2$ Character orbit 1840.a Self dual yes Analytic conductor $14.692$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) 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 $$q - q^{5} + q^{7} - 3 q^{9} - 6 q^{11} + 6 q^{13} + 7 q^{17} - 2 q^{19} + q^{23} + q^{25} - 5 q^{29} - q^{31} - q^{35} - 5 q^{37} - 7 q^{41} - 8 q^{43} + 3 q^{45} - 8 q^{47} - 6 q^{49} + 3 q^{53} + 6 q^{55} - 13 q^{59} - 8 q^{61} - 3 q^{63} - 6 q^{65} + 9 q^{67} - 7 q^{71} - 2 q^{73} - 6 q^{77} + 12 q^{79} + 9 q^{81} + 5 q^{83} - 7 q^{85} - 12 q^{89} + 6 q^{91} + 2 q^{95} + 2 q^{97} + 18 q^{99}+O(q^{100})$$ q - q^5 + q^7 - 3 * q^9 - 6 * q^11 + 6 * q^13 + 7 * q^17 - 2 * q^19 + q^23 + q^25 - 5 * q^29 - q^31 - q^35 - 5 * q^37 - 7 * q^41 - 8 * q^43 + 3 * q^45 - 8 * q^47 - 6 * q^49 + 3 * q^53 + 6 * q^55 - 13 * q^59 - 8 * q^61 - 3 * q^63 - 6 * q^65 + 9 * q^67 - 7 * q^71 - 2 * q^73 - 6 * q^77 + 12 * q^79 + 9 * q^81 + 5 * q^83 - 7 * q^85 - 12 * q^89 + 6 * q^91 + 2 * q^95 + 2 * q^97 + 18 * 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$23$$ $$-1$$

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 1840.2.a.e 1
4.b odd 2 1 460.2.a.b 1
5.b even 2 1 9200.2.a.q 1
8.b even 2 1 7360.2.a.r 1
8.d odd 2 1 7360.2.a.m 1
12.b even 2 1 4140.2.a.h 1
20.d odd 2 1 2300.2.a.e 1
20.e even 4 2 2300.2.c.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.2.a.b 1 4.b odd 2 1
1840.2.a.e 1 1.a even 1 1 trivial
2300.2.a.e 1 20.d odd 2 1
2300.2.c.g 2 20.e even 4 2
4140.2.a.h 1 12.b even 2 1
7360.2.a.m 1 8.d odd 2 1
7360.2.a.r 1 8.b even 2 1
9200.2.a.q 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}$$ T3 $$T_{7} - 1$$ T7 - 1 $$T_{11} + 6$$ T11 + 6

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 1$$
$7$ $$T - 1$$
$11$ $$T + 6$$
$13$ $$T - 6$$
$17$ $$T - 7$$
$19$ $$T + 2$$
$23$ $$T - 1$$
$29$ $$T + 5$$
$31$ $$T + 1$$
$37$ $$T + 5$$
$41$ $$T + 7$$
$43$ $$T + 8$$
$47$ $$T + 8$$
$53$ $$T - 3$$
$59$ $$T + 13$$
$61$ $$T + 8$$
$67$ $$T - 9$$
$71$ $$T + 7$$
$73$ $$T + 2$$
$79$ $$T - 12$$
$83$ $$T - 5$$
$89$ $$T + 12$$
$97$ $$T - 2$$
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