# Properties

 Label 5610.2.a.bf Level $5610$ Weight $2$ Character orbit 5610.a Self dual yes Analytic conductor $44.796$ Analytic rank $1$ 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] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5610.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: $$44.7960755339$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 - 4 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 2 q^{13} - 4 q^{14} - q^{15} + q^{16} + q^{17} + q^{18} - q^{20} - 4 q^{21} + q^{22} + q^{24} + q^{25} - 2 q^{26} + q^{27} - 4 q^{28} + 2 q^{29} - q^{30} + 4 q^{31} + q^{32} + q^{33} + q^{34} + 4 q^{35} + q^{36} - 6 q^{37} - 2 q^{39} - q^{40} - 2 q^{41} - 4 q^{42} + 4 q^{43} + q^{44} - q^{45} - 8 q^{47} + q^{48} + 9 q^{49} + q^{50} + q^{51} - 2 q^{52} - 14 q^{53} + q^{54} - q^{55} - 4 q^{56} + 2 q^{58} + 4 q^{59} - q^{60} + 2 q^{61} + 4 q^{62} - 4 q^{63} + q^{64} + 2 q^{65} + q^{66} - 16 q^{67} + q^{68} + 4 q^{70} - 12 q^{71} + q^{72} - 6 q^{73} - 6 q^{74} + q^{75} - 4 q^{77} - 2 q^{78} - 8 q^{79} - q^{80} + q^{81} - 2 q^{82} - 12 q^{83} - 4 q^{84} - q^{85} + 4 q^{86} + 2 q^{87} + q^{88} - 6 q^{89} - q^{90} + 8 q^{91} + 4 q^{93} - 8 q^{94} + q^{96} + 6 q^{97} + 9 q^{98} + q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 - 4 * q^7 + q^8 + q^9 - q^10 + q^11 + q^12 - 2 * q^13 - 4 * q^14 - q^15 + q^16 + q^17 + q^18 - q^20 - 4 * q^21 + q^22 + q^24 + q^25 - 2 * q^26 + q^27 - 4 * q^28 + 2 * q^29 - q^30 + 4 * q^31 + q^32 + q^33 + q^34 + 4 * q^35 + q^36 - 6 * q^37 - 2 * q^39 - q^40 - 2 * q^41 - 4 * q^42 + 4 * q^43 + q^44 - q^45 - 8 * q^47 + q^48 + 9 * q^49 + q^50 + q^51 - 2 * q^52 - 14 * q^53 + q^54 - q^55 - 4 * q^56 + 2 * q^58 + 4 * q^59 - q^60 + 2 * q^61 + 4 * q^62 - 4 * q^63 + q^64 + 2 * q^65 + q^66 - 16 * q^67 + q^68 + 4 * q^70 - 12 * q^71 + q^72 - 6 * q^73 - 6 * q^74 + q^75 - 4 * q^77 - 2 * q^78 - 8 * q^79 - q^80 + q^81 - 2 * q^82 - 12 * q^83 - 4 * q^84 - q^85 + 4 * q^86 + 2 * q^87 + q^88 - 6 * q^89 - q^90 + 8 * q^91 + 4 * q^93 - 8 * q^94 + q^96 + 6 * q^97 + 9 * q^98 + 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
1.00000 1.00000 1.00000 −1.00000 1.00000 −4.00000 1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$
$$17$$ $$-1$$

## 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 5610.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5610.2.a.bf 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{13} + 2$$ T13 + 2 $$T_{19}$$ T19 $$T_{23}$$ T23

## Hecke characteristic polynomials

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