# Properties

 Label 2550.2.a.bc Level $2550$ Weight $2$ Character orbit 2550.a Self dual yes Analytic conductor $20.362$ 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] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.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: $$20.3618525154$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} - 2 q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^6 - 2 * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{6} - 2 q^{7} + q^{8} + q^{9} - 4 q^{11} + q^{12} - 4 q^{13} - 2 q^{14} + q^{16} - q^{17} + q^{18} - 4 q^{19} - 2 q^{21} - 4 q^{22} - 8 q^{23} + q^{24} - 4 q^{26} + q^{27} - 2 q^{28} + 2 q^{29} + 4 q^{31} + q^{32} - 4 q^{33} - q^{34} + q^{36} - 6 q^{37} - 4 q^{38} - 4 q^{39} + 8 q^{41} - 2 q^{42} - 6 q^{43} - 4 q^{44} - 8 q^{46} - 8 q^{47} + q^{48} - 3 q^{49} - q^{51} - 4 q^{52} + 2 q^{53} + q^{54} - 2 q^{56} - 4 q^{57} + 2 q^{58} - 6 q^{59} + 14 q^{61} + 4 q^{62} - 2 q^{63} + q^{64} - 4 q^{66} + 2 q^{67} - q^{68} - 8 q^{69} + 2 q^{71} + q^{72} - 4 q^{73} - 6 q^{74} - 4 q^{76} + 8 q^{77} - 4 q^{78} + q^{81} + 8 q^{82} + 16 q^{83} - 2 q^{84} - 6 q^{86} + 2 q^{87} - 4 q^{88} - 2 q^{89} + 8 q^{91} - 8 q^{92} + 4 q^{93} - 8 q^{94} + q^{96} - 3 q^{98} - 4 q^{99}+O(q^{100})$$ q + q^2 + q^3 + q^4 + q^6 - 2 * q^7 + q^8 + q^9 - 4 * q^11 + q^12 - 4 * q^13 - 2 * q^14 + q^16 - q^17 + q^18 - 4 * q^19 - 2 * q^21 - 4 * q^22 - 8 * q^23 + q^24 - 4 * q^26 + q^27 - 2 * q^28 + 2 * q^29 + 4 * q^31 + q^32 - 4 * q^33 - q^34 + q^36 - 6 * q^37 - 4 * q^38 - 4 * q^39 + 8 * q^41 - 2 * q^42 - 6 * q^43 - 4 * q^44 - 8 * q^46 - 8 * q^47 + q^48 - 3 * q^49 - q^51 - 4 * q^52 + 2 * q^53 + q^54 - 2 * q^56 - 4 * q^57 + 2 * q^58 - 6 * q^59 + 14 * q^61 + 4 * q^62 - 2 * q^63 + q^64 - 4 * q^66 + 2 * q^67 - q^68 - 8 * q^69 + 2 * q^71 + q^72 - 4 * q^73 - 6 * q^74 - 4 * q^76 + 8 * q^77 - 4 * q^78 + q^81 + 8 * q^82 + 16 * q^83 - 2 * q^84 - 6 * q^86 + 2 * q^87 - 4 * q^88 - 2 * q^89 + 8 * q^91 - 8 * q^92 + 4 * q^93 - 8 * q^94 + q^96 - 3 * q^98 - 4 * 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 −2.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$17$$ $$1$$

## 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 2550.2.a.bc 1
3.b odd 2 1 7650.2.a.k 1
5.b even 2 1 510.2.a.a 1
5.c odd 4 2 2550.2.d.n 2
15.d odd 2 1 1530.2.a.p 1
20.d odd 2 1 4080.2.a.s 1
85.c even 2 1 8670.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.a 1 5.b even 2 1
1530.2.a.p 1 15.d odd 2 1
2550.2.a.bc 1 1.a even 1 1 trivial
2550.2.d.n 2 5.c odd 4 2
4080.2.a.s 1 20.d odd 2 1
7650.2.a.k 1 3.b odd 2 1
8670.2.a.k 1 85.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2550))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T + 4$$
$17$ $$T + 1$$
$19$ $$T + 4$$
$23$ $$T + 8$$
$29$ $$T - 2$$
$31$ $$T - 4$$
$37$ $$T + 6$$
$41$ $$T - 8$$
$43$ $$T + 6$$
$47$ $$T + 8$$
$53$ $$T - 2$$
$59$ $$T + 6$$
$61$ $$T - 14$$
$67$ $$T - 2$$
$71$ $$T - 2$$
$73$ $$T + 4$$
$79$ $$T$$
$83$ $$T - 16$$
$89$ $$T + 2$$
$97$ $$T$$