# Properties

 Label 5610.2.a.bt Level $5610$ Weight $2$ Character orbit 5610.a Self dual yes Analytic conductor $44.796$ Analytic rank $1$ Dimension $2$ 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 # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$44.7960755339$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{2}$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - 2 q^{13} + q^{15} + q^{16} + q^{17} + q^{18} - q^{20} - q^{22} + \beta q^{23} - q^{24} + q^{25} - 2 q^{26} - q^{27} + ( - \beta - 2) q^{29} + q^{30} - \beta q^{31} + q^{32} + q^{33} + q^{34} + q^{36} + (\beta + 6) q^{37} + 2 q^{39} - q^{40} + ( - \beta + 2) q^{41} - \beta q^{43} - q^{44} - q^{45} + \beta q^{46} - 4 q^{47} - q^{48} - 7 q^{49} + q^{50} - q^{51} - 2 q^{52} - 10 q^{53} - q^{54} + q^{55} + ( - \beta - 2) q^{58} + 2 \beta q^{59} + q^{60} + (\beta + 2) q^{61} - \beta q^{62} + q^{64} + 2 q^{65} + q^{66} - 4 q^{67} + q^{68} - \beta q^{69} + \beta q^{71} + q^{72} + (2 \beta - 2) q^{73} + (\beta + 6) q^{74} - q^{75} + 2 q^{78} - 2 \beta q^{79} - q^{80} + q^{81} + ( - \beta + 2) q^{82} + 4 q^{83} - q^{85} - \beta q^{86} + (\beta + 2) q^{87} - q^{88} - 14 q^{89} - q^{90} + \beta q^{92} + \beta q^{93} - 4 q^{94} - q^{96} + (\beta - 6) q^{97} - 7 q^{98} - q^{99} +O(q^{100})$$ q + q^2 - q^3 + q^4 - q^5 - q^6 + q^8 + q^9 - q^10 - q^11 - q^12 - 2 * q^13 + q^15 + q^16 + q^17 + q^18 - q^20 - q^22 + b * q^23 - q^24 + q^25 - 2 * q^26 - q^27 + (-b - 2) * q^29 + q^30 - b * q^31 + q^32 + q^33 + q^34 + q^36 + (b + 6) * q^37 + 2 * q^39 - q^40 + (-b + 2) * q^41 - b * q^43 - q^44 - q^45 + b * q^46 - 4 * q^47 - q^48 - 7 * q^49 + q^50 - q^51 - 2 * q^52 - 10 * q^53 - q^54 + q^55 + (-b - 2) * q^58 + 2*b * q^59 + q^60 + (b + 2) * q^61 - b * q^62 + q^64 + 2 * q^65 + q^66 - 4 * q^67 + q^68 - b * q^69 + b * q^71 + q^72 + (2*b - 2) * q^73 + (b + 6) * q^74 - q^75 + 2 * q^78 - 2*b * q^79 - q^80 + q^81 + (-b + 2) * q^82 + 4 * q^83 - q^85 - b * q^86 + (b + 2) * q^87 - q^88 - 14 * q^89 - q^90 + b * q^92 + b * q^93 - 4 * q^94 - q^96 + (b - 6) * q^97 - 7 * q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 2 q^{22} - 2 q^{24} + 2 q^{25} - 4 q^{26} - 2 q^{27} - 4 q^{29} + 2 q^{30} + 2 q^{32} + 2 q^{33} + 2 q^{34} + 2 q^{36} + 12 q^{37} + 4 q^{39} - 2 q^{40} + 4 q^{41} - 2 q^{44} - 2 q^{45} - 8 q^{47} - 2 q^{48} - 14 q^{49} + 2 q^{50} - 2 q^{51} - 4 q^{52} - 20 q^{53} - 2 q^{54} + 2 q^{55} - 4 q^{58} + 2 q^{60} + 4 q^{61} + 2 q^{64} + 4 q^{65} + 2 q^{66} - 8 q^{67} + 2 q^{68} + 2 q^{72} - 4 q^{73} + 12 q^{74} - 2 q^{75} + 4 q^{78} - 2 q^{80} + 2 q^{81} + 4 q^{82} + 8 q^{83} - 2 q^{85} + 4 q^{87} - 2 q^{88} - 28 q^{89} - 2 q^{90} - 8 q^{94} - 2 q^{96} - 12 q^{97} - 14 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^20 - 2 * q^22 - 2 * q^24 + 2 * q^25 - 4 * q^26 - 2 * q^27 - 4 * q^29 + 2 * q^30 + 2 * q^32 + 2 * q^33 + 2 * q^34 + 2 * q^36 + 12 * q^37 + 4 * q^39 - 2 * q^40 + 4 * q^41 - 2 * q^44 - 2 * q^45 - 8 * q^47 - 2 * q^48 - 14 * q^49 + 2 * q^50 - 2 * q^51 - 4 * q^52 - 20 * q^53 - 2 * q^54 + 2 * q^55 - 4 * q^58 + 2 * q^60 + 4 * q^61 + 2 * q^64 + 4 * q^65 + 2 * q^66 - 8 * q^67 + 2 * q^68 + 2 * q^72 - 4 * q^73 + 12 * q^74 - 2 * q^75 + 4 * q^78 - 2 * q^80 + 2 * q^81 + 4 * q^82 + 8 * q^83 - 2 * q^85 + 4 * q^87 - 2 * q^88 - 28 * q^89 - 2 * q^90 - 8 * q^94 - 2 * q^96 - 12 * q^97 - 14 * q^98 - 2 * 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
 −1.41421 1.41421
1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 0 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.bt 2

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

 $$T_{7}$$ T7 $$T_{13} + 2$$ T13 + 2 $$T_{19}$$ T19 $$T_{23}^{2} - 32$$ T23^2 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 32$$
$29$ $$T^{2} + 4T - 28$$
$31$ $$T^{2} - 32$$
$37$ $$T^{2} - 12T + 4$$
$41$ $$T^{2} - 4T - 28$$
$43$ $$T^{2} - 32$$
$47$ $$(T + 4)^{2}$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} - 128$$
$61$ $$T^{2} - 4T - 28$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} - 32$$
$73$ $$T^{2} + 4T - 124$$
$79$ $$T^{2} - 128$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 12T + 4$$