# Properties

 Label 5550.2.a.bf 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 1110) 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 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 3 q^{7} + q^{8} + q^{9} - 5 q^{11} + q^{12} + 2 q^{13} - 3 q^{14} + q^{16} + 7 q^{17} + q^{18} - 2 q^{19} - 3 q^{21} - 5 q^{22} - 4 q^{23} + q^{24} + 2 q^{26} + q^{27} - 3 q^{28} - 5 q^{29} - 7 q^{31} + q^{32} - 5 q^{33} + 7 q^{34} + q^{36} - q^{37} - 2 q^{38} + 2 q^{39} + q^{41} - 3 q^{42} - 9 q^{43} - 5 q^{44} - 4 q^{46} + q^{48} + 2 q^{49} + 7 q^{51} + 2 q^{52} - 3 q^{53} + q^{54} - 3 q^{56} - 2 q^{57} - 5 q^{58} - 14 q^{59} - 7 q^{61} - 7 q^{62} - 3 q^{63} + q^{64} - 5 q^{66} + 4 q^{67} + 7 q^{68} - 4 q^{69} + 8 q^{71} + q^{72} + 12 q^{73} - q^{74} - 2 q^{76} + 15 q^{77} + 2 q^{78} + 4 q^{79} + q^{81} + q^{82} + 10 q^{83} - 3 q^{84} - 9 q^{86} - 5 q^{87} - 5 q^{88} - 14 q^{89} - 6 q^{91} - 4 q^{92} - 7 q^{93} + q^{96} - q^{97} + 2 q^{98} - 5 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 3 * q^7 + q^8 + q^9 - 5 * q^11 + q^12 + 2 * q^13 - 3 * q^14 + q^16 + 7 * q^17 + q^18 - 2 * q^19 - 3 * q^21 - 5 * q^22 - 4 * q^23 + q^24 + 2 * q^26 + q^27 - 3 * q^28 - 5 * q^29 - 7 * q^31 + q^32 - 5 * q^33 + 7 * q^34 + q^36 - q^37 - 2 * q^38 + 2 * q^39 + q^41 - 3 * q^42 - 9 * q^43 - 5 * q^44 - 4 * q^46 + q^48 + 2 * q^49 + 7 * q^51 + 2 * q^52 - 3 * q^53 + q^54 - 3 * q^56 - 2 * q^57 - 5 * q^58 - 14 * q^59 - 7 * q^61 - 7 * q^62 - 3 * q^63 + q^64 - 5 * q^66 + 4 * q^67 + 7 * q^68 - 4 * q^69 + 8 * q^71 + q^72 + 12 * q^73 - q^74 - 2 * q^76 + 15 * q^77 + 2 * q^78 + 4 * q^79 + q^81 + q^82 + 10 * q^83 - 3 * q^84 - 9 * q^86 - 5 * q^87 - 5 * q^88 - 14 * q^89 - 6 * q^91 - 4 * q^92 - 7 * q^93 + q^96 - q^97 + 2 * q^98 - 5 * 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.bf 1
5.b even 2 1 1110.2.a.d 1
15.d odd 2 1 3330.2.a.q 1
20.d odd 2 1 8880.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.d 1 5.b even 2 1
3330.2.a.q 1 15.d odd 2 1
5550.2.a.bf 1 1.a even 1 1 trivial
8880.2.a.z 1 20.d odd 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 $$T_{11} + 5$$ T11 + 5 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 7$$ T17 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 3$$
$11$ $$T + 5$$
$13$ $$T - 2$$
$17$ $$T - 7$$
$19$ $$T + 2$$
$23$ $$T + 4$$
$29$ $$T + 5$$
$31$ $$T + 7$$
$37$ $$T + 1$$
$41$ $$T - 1$$
$43$ $$T + 9$$
$47$ $$T$$
$53$ $$T + 3$$
$59$ $$T + 14$$
$61$ $$T + 7$$
$67$ $$T - 4$$
$71$ $$T - 8$$
$73$ $$T - 12$$
$79$ $$T - 4$$
$83$ $$T - 10$$
$89$ $$T + 14$$
$97$ $$T + 1$$