# Properties

 Label 5550.2.a.n Level $5550$ Weight $2$ Character orbit 5550.a Self dual yes Analytic conductor $44.317$ 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] = [5550,2,Mod(1,5550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5550.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: $$44.3169731218$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 222) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 + q^4 - q^6 - 3 * q^7 - q^8 + q^9 $$q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} + q^{9} + q^{11} + q^{12} - q^{13} + 3 q^{14} + q^{16} + 3 q^{17} - q^{18} + 3 q^{19} - 3 q^{21} - q^{22} + q^{23} - q^{24} + q^{26} + q^{27} - 3 q^{28} - 4 q^{29} - 6 q^{31} - q^{32} + q^{33} - 3 q^{34} + q^{36} + q^{37} - 3 q^{38} - q^{39} - 10 q^{41} + 3 q^{42} - 12 q^{43} + q^{44} - q^{46} + 6 q^{47} + q^{48} + 2 q^{49} + 3 q^{51} - q^{52} + q^{53} - q^{54} + 3 q^{56} + 3 q^{57} + 4 q^{58} + 2 q^{61} + 6 q^{62} - 3 q^{63} + q^{64} - q^{66} - 2 q^{67} + 3 q^{68} + q^{69} - q^{72} + 3 q^{73} - q^{74} + 3 q^{76} - 3 q^{77} + q^{78} + 14 q^{79} + q^{81} + 10 q^{82} - 9 q^{83} - 3 q^{84} + 12 q^{86} - 4 q^{87} - q^{88} - 3 q^{89} + 3 q^{91} + q^{92} - 6 q^{93} - 6 q^{94} - q^{96} + 10 q^{97} - 2 q^{98} + q^{99}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - 3 * q^7 - q^8 + q^9 + q^11 + q^12 - q^13 + 3 * q^14 + q^16 + 3 * q^17 - q^18 + 3 * q^19 - 3 * q^21 - q^22 + q^23 - q^24 + q^26 + q^27 - 3 * q^28 - 4 * q^29 - 6 * q^31 - q^32 + q^33 - 3 * q^34 + q^36 + q^37 - 3 * q^38 - q^39 - 10 * q^41 + 3 * q^42 - 12 * q^43 + q^44 - q^46 + 6 * q^47 + q^48 + 2 * q^49 + 3 * q^51 - q^52 + q^53 - q^54 + 3 * q^56 + 3 * q^57 + 4 * q^58 + 2 * q^61 + 6 * q^62 - 3 * q^63 + q^64 - q^66 - 2 * q^67 + 3 * q^68 + q^69 - q^72 + 3 * q^73 - q^74 + 3 * q^76 - 3 * q^77 + q^78 + 14 * q^79 + q^81 + 10 * q^82 - 9 * q^83 - 3 * q^84 + 12 * q^86 - 4 * q^87 - q^88 - 3 * q^89 + 3 * q^91 + q^92 - 6 * q^93 - 6 * q^94 - q^96 + 10 * q^97 - 2 * 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 0 −1.00000 −3.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$
$$37$$ $$-1$$

## 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 5550.2.a.n 1
5.b even 2 1 222.2.a.d 1
15.d odd 2 1 666.2.a.b 1
20.d odd 2 1 1776.2.a.h 1
40.e odd 2 1 7104.2.a.f 1
40.f even 2 1 7104.2.a.v 1
60.h even 2 1 5328.2.a.h 1
185.d even 2 1 8214.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
222.2.a.d 1 5.b even 2 1
666.2.a.b 1 15.d odd 2 1
1776.2.a.h 1 20.d odd 2 1
5328.2.a.h 1 60.h even 2 1
5550.2.a.n 1 1.a even 1 1 trivial
7104.2.a.f 1 40.e odd 2 1
7104.2.a.v 1 40.f even 2 1
8214.2.a.b 1 185.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5550))$$:

 $$T_{7} + 3$$ T7 + 3 $$T_{11} - 1$$ T11 - 1 $$T_{13} + 1$$ T13 + 1 $$T_{17} - 3$$ T17 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T - 1$$
$13$ $$T + 1$$
$17$ $$T - 3$$
$19$ $$T - 3$$
$23$ $$T - 1$$
$29$ $$T + 4$$
$31$ $$T + 6$$
$37$ $$T - 1$$
$41$ $$T + 10$$
$43$ $$T + 12$$
$47$ $$T - 6$$
$53$ $$T - 1$$
$59$ $$T$$
$61$ $$T - 2$$
$67$ $$T + 2$$
$71$ $$T$$
$73$ $$T - 3$$
$79$ $$T - 14$$
$83$ $$T + 9$$
$89$ $$T + 3$$
$97$ $$T - 10$$