# Properties

 Label 5610.2.a.bo 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[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: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - b * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - \beta q^{7} - q^{8} + q^{9} + q^{10} + q^{11} - q^{12} + ( - \beta + 2) q^{13} + \beta q^{14} + q^{15} + q^{16} + q^{17} - q^{18} + (\beta - 2) q^{19} - q^{20} + \beta q^{21} - q^{22} + (\beta - 6) q^{23} + q^{24} + q^{25} + (\beta - 2) q^{26} - q^{27} - \beta q^{28} + 2 q^{29} - q^{30} + ( - \beta - 2) q^{31} - q^{32} - q^{33} - q^{34} + \beta q^{35} + q^{36} + ( - \beta + 8) q^{37} + ( - \beta + 2) q^{38} + (\beta - 2) q^{39} + q^{40} + (2 \beta - 4) q^{41} - \beta q^{42} + (2 \beta - 2) q^{43} + q^{44} - q^{45} + ( - \beta + 6) q^{46} + (2 \beta - 4) q^{47} - q^{48} + (\beta + 3) q^{49} - q^{50} - q^{51} + ( - \beta + 2) q^{52} + 2 q^{53} + q^{54} - q^{55} + \beta q^{56} + ( - \beta + 2) q^{57} - 2 q^{58} + ( - 2 \beta - 6) q^{59} + q^{60} + (\beta + 4) q^{61} + (\beta + 2) q^{62} - \beta q^{63} + q^{64} + (\beta - 2) q^{65} + q^{66} + (\beta - 12) q^{67} + q^{68} + ( - \beta + 6) q^{69} - \beta q^{70} + (4 \beta - 2) q^{71} - q^{72} - 8 q^{73} + (\beta - 8) q^{74} - q^{75} + (\beta - 2) q^{76} - \beta q^{77} + ( - \beta + 2) q^{78} + 2 \beta q^{79} - q^{80} + q^{81} + ( - 2 \beta + 4) q^{82} + ( - \beta - 6) q^{83} + \beta q^{84} - q^{85} + ( - 2 \beta + 2) q^{86} - 2 q^{87} - q^{88} + 14 q^{89} + q^{90} + ( - \beta + 10) q^{91} + (\beta - 6) q^{92} + (\beta + 2) q^{93} + ( - 2 \beta + 4) q^{94} + ( - \beta + 2) q^{95} + q^{96} + (\beta + 6) q^{97} + ( - \beta - 3) q^{98} + q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 - q^5 + q^6 - b * q^7 - q^8 + q^9 + q^10 + q^11 - q^12 + (-b + 2) * q^13 + b * q^14 + q^15 + q^16 + q^17 - q^18 + (b - 2) * q^19 - q^20 + b * q^21 - q^22 + (b - 6) * q^23 + q^24 + q^25 + (b - 2) * q^26 - q^27 - b * q^28 + 2 * q^29 - q^30 + (-b - 2) * q^31 - q^32 - q^33 - q^34 + b * q^35 + q^36 + (-b + 8) * q^37 + (-b + 2) * q^38 + (b - 2) * q^39 + q^40 + (2*b - 4) * q^41 - b * q^42 + (2*b - 2) * q^43 + q^44 - q^45 + (-b + 6) * q^46 + (2*b - 4) * q^47 - q^48 + (b + 3) * q^49 - q^50 - q^51 + (-b + 2) * q^52 + 2 * q^53 + q^54 - q^55 + b * q^56 + (-b + 2) * q^57 - 2 * q^58 + (-2*b - 6) * q^59 + q^60 + (b + 4) * q^61 + (b + 2) * q^62 - b * q^63 + q^64 + (b - 2) * q^65 + q^66 + (b - 12) * q^67 + q^68 + (-b + 6) * q^69 - b * q^70 + (4*b - 2) * q^71 - q^72 - 8 * q^73 + (b - 8) * q^74 - q^75 + (b - 2) * q^76 - b * q^77 + (-b + 2) * q^78 + 2*b * q^79 - q^80 + q^81 + (-2*b + 4) * q^82 + (-b - 6) * q^83 + b * q^84 - q^85 + (-2*b + 2) * q^86 - 2 * q^87 - q^88 + 14 * q^89 + q^90 + (-b + 10) * q^91 + (b - 6) * q^92 + (b + 2) * q^93 + (-2*b + 4) * q^94 + (-b + 2) * q^95 + q^96 + (b + 6) * q^97 + (-b - 3) * q^98 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} + 3 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} - 2 q^{18} - 3 q^{19} - 2 q^{20} + q^{21} - 2 q^{22} - 11 q^{23} + 2 q^{24} + 2 q^{25} - 3 q^{26} - 2 q^{27} - q^{28} + 4 q^{29} - 2 q^{30} - 5 q^{31} - 2 q^{32} - 2 q^{33} - 2 q^{34} + q^{35} + 2 q^{36} + 15 q^{37} + 3 q^{38} - 3 q^{39} + 2 q^{40} - 6 q^{41} - q^{42} - 2 q^{43} + 2 q^{44} - 2 q^{45} + 11 q^{46} - 6 q^{47} - 2 q^{48} + 7 q^{49} - 2 q^{50} - 2 q^{51} + 3 q^{52} + 4 q^{53} + 2 q^{54} - 2 q^{55} + q^{56} + 3 q^{57} - 4 q^{58} - 14 q^{59} + 2 q^{60} + 9 q^{61} + 5 q^{62} - q^{63} + 2 q^{64} - 3 q^{65} + 2 q^{66} - 23 q^{67} + 2 q^{68} + 11 q^{69} - q^{70} - 2 q^{72} - 16 q^{73} - 15 q^{74} - 2 q^{75} - 3 q^{76} - q^{77} + 3 q^{78} + 2 q^{79} - 2 q^{80} + 2 q^{81} + 6 q^{82} - 13 q^{83} + q^{84} - 2 q^{85} + 2 q^{86} - 4 q^{87} - 2 q^{88} + 28 q^{89} + 2 q^{90} + 19 q^{91} - 11 q^{92} + 5 q^{93} + 6 q^{94} + 3 q^{95} + 2 q^{96} + 13 q^{97} - 7 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 + 3 * q^13 + q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 - 2 * q^18 - 3 * q^19 - 2 * q^20 + q^21 - 2 * q^22 - 11 * q^23 + 2 * q^24 + 2 * q^25 - 3 * q^26 - 2 * q^27 - q^28 + 4 * q^29 - 2 * q^30 - 5 * q^31 - 2 * q^32 - 2 * q^33 - 2 * q^34 + q^35 + 2 * q^36 + 15 * q^37 + 3 * q^38 - 3 * q^39 + 2 * q^40 - 6 * q^41 - q^42 - 2 * q^43 + 2 * q^44 - 2 * q^45 + 11 * q^46 - 6 * q^47 - 2 * q^48 + 7 * q^49 - 2 * q^50 - 2 * q^51 + 3 * q^52 + 4 * q^53 + 2 * q^54 - 2 * q^55 + q^56 + 3 * q^57 - 4 * q^58 - 14 * q^59 + 2 * q^60 + 9 * q^61 + 5 * q^62 - q^63 + 2 * q^64 - 3 * q^65 + 2 * q^66 - 23 * q^67 + 2 * q^68 + 11 * q^69 - q^70 - 2 * q^72 - 16 * q^73 - 15 * q^74 - 2 * q^75 - 3 * q^76 - q^77 + 3 * q^78 + 2 * q^79 - 2 * q^80 + 2 * q^81 + 6 * q^82 - 13 * q^83 + q^84 - 2 * q^85 + 2 * q^86 - 4 * q^87 - 2 * q^88 + 28 * q^89 + 2 * q^90 + 19 * q^91 - 11 * q^92 + 5 * q^93 + 6 * q^94 + 3 * q^95 + 2 * q^96 + 13 * q^97 - 7 * 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
 3.70156 −2.70156
−1.00000 −1.00000 1.00000 −1.00000 1.00000 −3.70156 −1.00000 1.00000 1.00000
1.2 −1.00000 −1.00000 1.00000 −1.00000 1.00000 2.70156 −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.bo 2

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

 $$T_{7}^{2} + T_{7} - 10$$ T7^2 + T7 - 10 $$T_{13}^{2} - 3T_{13} - 8$$ T13^2 - 3*T13 - 8 $$T_{19}^{2} + 3T_{19} - 8$$ T19^2 + 3*T19 - 8 $$T_{23}^{2} + 11T_{23} + 20$$ T23^2 + 11*T23 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + T - 10$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} - 3T - 8$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} + 3T - 8$$
$23$ $$T^{2} + 11T + 20$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 5T - 4$$
$37$ $$T^{2} - 15T + 46$$
$41$ $$T^{2} + 6T - 32$$
$43$ $$T^{2} + 2T - 40$$
$47$ $$T^{2} + 6T - 32$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} + 14T + 8$$
$61$ $$T^{2} - 9T + 10$$
$67$ $$T^{2} + 23T + 122$$
$71$ $$T^{2} - 164$$
$73$ $$(T + 8)^{2}$$
$79$ $$T^{2} - 2T - 40$$
$83$ $$T^{2} + 13T + 32$$
$89$ $$(T - 14)^{2}$$
$97$ $$T^{2} - 13T + 32$$