# Properties

 Label 5610.2.a.bb 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 - 2 * q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + q^{11} - q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + q^{16} + q^{17} + q^{18} - 2 q^{19} + q^{20} + 2 q^{21} + q^{22} + 6 q^{23} - q^{24} + q^{25} - 4 q^{26} - q^{27} - 2 q^{28} - 2 q^{29} - q^{30} - 8 q^{31} + q^{32} - q^{33} + q^{34} - 2 q^{35} + q^{36} - 6 q^{37} - 2 q^{38} + 4 q^{39} + q^{40} + 2 q^{41} + 2 q^{42} - 6 q^{43} + q^{44} + q^{45} + 6 q^{46} + 8 q^{47} - q^{48} - 3 q^{49} + q^{50} - q^{51} - 4 q^{52} - q^{54} + q^{55} - 2 q^{56} + 2 q^{57} - 2 q^{58} + 10 q^{59} - q^{60} - 6 q^{61} - 8 q^{62} - 2 q^{63} + q^{64} - 4 q^{65} - q^{66} - 8 q^{67} + q^{68} - 6 q^{69} - 2 q^{70} - 10 q^{71} + q^{72} + 12 q^{73} - 6 q^{74} - q^{75} - 2 q^{76} - 2 q^{77} + 4 q^{78} - 6 q^{79} + q^{80} + q^{81} + 2 q^{82} + 4 q^{83} + 2 q^{84} + q^{85} - 6 q^{86} + 2 q^{87} + q^{88} - 18 q^{89} + q^{90} + 8 q^{91} + 6 q^{92} + 8 q^{93} + 8 q^{94} - 2 q^{95} - q^{96} - 2 q^{97} - 3 q^{98} + q^{99}+O(q^{100})$$ q + q^2 - q^3 + q^4 + q^5 - q^6 - 2 * q^7 + q^8 + q^9 + q^10 + q^11 - q^12 - 4 * q^13 - 2 * q^14 - q^15 + q^16 + q^17 + q^18 - 2 * q^19 + q^20 + 2 * q^21 + q^22 + 6 * q^23 - q^24 + q^25 - 4 * q^26 - q^27 - 2 * q^28 - 2 * q^29 - q^30 - 8 * q^31 + q^32 - q^33 + q^34 - 2 * q^35 + q^36 - 6 * q^37 - 2 * q^38 + 4 * q^39 + q^40 + 2 * q^41 + 2 * q^42 - 6 * q^43 + q^44 + q^45 + 6 * q^46 + 8 * q^47 - q^48 - 3 * q^49 + q^50 - q^51 - 4 * q^52 - q^54 + q^55 - 2 * q^56 + 2 * q^57 - 2 * q^58 + 10 * q^59 - q^60 - 6 * q^61 - 8 * q^62 - 2 * q^63 + q^64 - 4 * q^65 - q^66 - 8 * q^67 + q^68 - 6 * q^69 - 2 * q^70 - 10 * q^71 + q^72 + 12 * q^73 - 6 * q^74 - q^75 - 2 * q^76 - 2 * q^77 + 4 * q^78 - 6 * q^79 + q^80 + q^81 + 2 * q^82 + 4 * q^83 + 2 * q^84 + q^85 - 6 * q^86 + 2 * q^87 + q^88 - 18 * q^89 + q^90 + 8 * q^91 + 6 * q^92 + 8 * q^93 + 8 * q^94 - 2 * q^95 - q^96 - 2 * q^97 - 3 * 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 −2.00000 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.bb 1

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

 $$T_{7} + 2$$ T7 + 2 $$T_{13} + 4$$ T13 + 4 $$T_{19} + 2$$ T19 + 2 $$T_{23} - 6$$ T23 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T + 2$$
$11$ $$T - 1$$
$13$ $$T + 4$$
$17$ $$T - 1$$
$19$ $$T + 2$$
$23$ $$T - 6$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T - 2$$
$43$ $$T + 6$$
$47$ $$T - 8$$
$53$ $$T$$
$59$ $$T - 10$$
$61$ $$T + 6$$
$67$ $$T + 8$$
$71$ $$T + 10$$
$73$ $$T - 12$$
$79$ $$T + 6$$
$83$ $$T - 4$$
$89$ $$T + 18$$
$97$ $$T + 2$$