# Properties

 Label 2550.2.a.c 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 102) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - 4 q^{11} - q^{12} + 2 q^{13} + q^{16} - q^{17} - q^{18} + 4 q^{19} + 4 q^{22} + q^{24} - 2 q^{26} - q^{27} - 10 q^{29} + 8 q^{31} - q^{32} + 4 q^{33} + q^{34} + q^{36} + 2 q^{37} - 4 q^{38} - 2 q^{39} + 10 q^{41} - 12 q^{43} - 4 q^{44} - q^{48} - 7 q^{49} + q^{51} + 2 q^{52} - 6 q^{53} + q^{54} - 4 q^{57} + 10 q^{58} + 12 q^{59} - 10 q^{61} - 8 q^{62} + q^{64} - 4 q^{66} + 12 q^{67} - q^{68} - q^{72} - 10 q^{73} - 2 q^{74} + 4 q^{76} + 2 q^{78} - 8 q^{79} + q^{81} - 10 q^{82} - 4 q^{83} + 12 q^{86} + 10 q^{87} + 4 q^{88} - 6 q^{89} - 8 q^{93} + q^{96} + 14 q^{97} + 7 q^{98} - 4 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 - q^8 + q^9 - 4 * q^11 - q^12 + 2 * q^13 + q^16 - q^17 - q^18 + 4 * q^19 + 4 * q^22 + q^24 - 2 * q^26 - q^27 - 10 * q^29 + 8 * q^31 - q^32 + 4 * q^33 + q^34 + q^36 + 2 * q^37 - 4 * q^38 - 2 * q^39 + 10 * q^41 - 12 * q^43 - 4 * q^44 - q^48 - 7 * q^49 + q^51 + 2 * q^52 - 6 * q^53 + q^54 - 4 * q^57 + 10 * q^58 + 12 * q^59 - 10 * q^61 - 8 * q^62 + q^64 - 4 * q^66 + 12 * q^67 - q^68 - q^72 - 10 * q^73 - 2 * q^74 + 4 * q^76 + 2 * q^78 - 8 * q^79 + q^81 - 10 * q^82 - 4 * q^83 + 12 * q^86 + 10 * q^87 + 4 * q^88 - 6 * q^89 - 8 * q^93 + q^96 + 14 * q^97 + 7 * 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 0 −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.c 1
3.b odd 2 1 7650.2.a.ca 1
5.b even 2 1 102.2.a.c 1
5.c odd 4 2 2550.2.d.m 2
15.d odd 2 1 306.2.a.b 1
20.d odd 2 1 816.2.a.b 1
35.c odd 2 1 4998.2.a.be 1
40.e odd 2 1 3264.2.a.bc 1
40.f even 2 1 3264.2.a.m 1
60.h even 2 1 2448.2.a.p 1
85.c even 2 1 1734.2.a.j 1
85.j even 4 2 1734.2.b.b 2
85.m even 8 4 1734.2.f.e 4
120.i odd 2 1 9792.2.a.k 1
120.m even 2 1 9792.2.a.l 1
255.h odd 2 1 5202.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 5.b even 2 1
306.2.a.b 1 15.d odd 2 1
816.2.a.b 1 20.d odd 2 1
1734.2.a.j 1 85.c even 2 1
1734.2.b.b 2 85.j even 4 2
1734.2.f.e 4 85.m even 8 4
2448.2.a.p 1 60.h even 2 1
2550.2.a.c 1 1.a even 1 1 trivial
2550.2.d.m 2 5.c odd 4 2
3264.2.a.m 1 40.f even 2 1
3264.2.a.bc 1 40.e odd 2 1
4998.2.a.be 1 35.c odd 2 1
5202.2.a.c 1 255.h odd 2 1
7650.2.a.ca 1 3.b odd 2 1
9792.2.a.k 1 120.i odd 2 1
9792.2.a.l 1 120.m even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T - 2$$
$17$ $$T + 1$$
$19$ $$T - 4$$
$23$ $$T$$
$29$ $$T + 10$$
$31$ $$T - 8$$
$37$ $$T - 2$$
$41$ $$T - 10$$
$43$ $$T + 12$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T + 10$$
$67$ $$T - 12$$
$71$ $$T$$
$73$ $$T + 10$$
$79$ $$T + 8$$
$83$ $$T + 4$$
$89$ $$T + 6$$
$97$ $$T - 14$$