# Properties

 Label 5610.2.a.bj Level $5610$ Weight $2$ Character orbit 5610.a Self dual yes Analytic conductor $44.796$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 5 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + 5 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 5 q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 4 q^{13} + 5 q^{14} - q^{15} + q^{16} - q^{17} + q^{18} - 4 q^{19} - q^{20} + 5 q^{21} + q^{22} - 3 q^{23} + q^{24} + q^{25} - 4 q^{26} + q^{27} + 5 q^{28} + 9 q^{29} - q^{30} + 5 q^{31} + q^{32} + q^{33} - q^{34} - 5 q^{35} + q^{36} + 2 q^{37} - 4 q^{38} - 4 q^{39} - q^{40} + 5 q^{42} + 11 q^{43} + q^{44} - q^{45} - 3 q^{46} + q^{48} + 18 q^{49} + q^{50} - q^{51} - 4 q^{52} + 12 q^{53} + q^{54} - q^{55} + 5 q^{56} - 4 q^{57} + 9 q^{58} - 6 q^{59} - q^{60} + 2 q^{61} + 5 q^{62} + 5 q^{63} + q^{64} + 4 q^{65} + q^{66} - 4 q^{67} - q^{68} - 3 q^{69} - 5 q^{70} - 6 q^{71} + q^{72} + 2 q^{73} + 2 q^{74} + q^{75} - 4 q^{76} + 5 q^{77} - 4 q^{78} + 8 q^{79} - q^{80} + q^{81} + 12 q^{83} + 5 q^{84} + q^{85} + 11 q^{86} + 9 q^{87} + q^{88} - q^{90} - 20 q^{91} - 3 q^{92} + 5 q^{93} + 4 q^{95} + q^{96} - 7 q^{97} + 18 q^{98} + q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + 5 * q^7 + q^8 + q^9 - q^10 + q^11 + q^12 - 4 * q^13 + 5 * q^14 - q^15 + q^16 - q^17 + q^18 - 4 * q^19 - q^20 + 5 * q^21 + q^22 - 3 * q^23 + q^24 + q^25 - 4 * q^26 + q^27 + 5 * q^28 + 9 * q^29 - q^30 + 5 * q^31 + q^32 + q^33 - q^34 - 5 * q^35 + q^36 + 2 * q^37 - 4 * q^38 - 4 * q^39 - q^40 + 5 * q^42 + 11 * q^43 + q^44 - q^45 - 3 * q^46 + q^48 + 18 * q^49 + q^50 - q^51 - 4 * q^52 + 12 * q^53 + q^54 - q^55 + 5 * q^56 - 4 * q^57 + 9 * q^58 - 6 * q^59 - q^60 + 2 * q^61 + 5 * q^62 + 5 * q^63 + q^64 + 4 * q^65 + q^66 - 4 * q^67 - q^68 - 3 * q^69 - 5 * q^70 - 6 * q^71 + q^72 + 2 * q^73 + 2 * q^74 + q^75 - 4 * q^76 + 5 * q^77 - 4 * q^78 + 8 * q^79 - q^80 + q^81 + 12 * q^83 + 5 * q^84 + q^85 + 11 * q^86 + 9 * q^87 + q^88 - q^90 - 20 * q^91 - 3 * q^92 + 5 * q^93 + 4 * q^95 + q^96 - 7 * q^97 + 18 * 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 5.00000 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.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5610.2.a.bj 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7} - 5$$ T7 - 5 $$T_{13} + 4$$ T13 + 4 $$T_{19} + 4$$ T19 + 4 $$T_{23} + 3$$ T23 + 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T - 5$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T + 3$$
$29$ $$T - 9$$
$31$ $$T - 5$$
$37$ $$T - 2$$
$41$ $$T$$
$43$ $$T - 11$$
$47$ $$T$$
$53$ $$T - 12$$
$59$ $$T + 6$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T + 6$$
$73$ $$T - 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T$$
$97$ $$T + 7$$