# Properties

 Label 2550.2.a.g 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: 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + 5 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 + 5 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} + 5 q^{7} - q^{8} + q^{9} - q^{11} - q^{12} - 2 q^{13} - 5 q^{14} + q^{16} + q^{17} - q^{18} - 7 q^{19} - 5 q^{21} + q^{22} + 2 q^{23} + q^{24} + 2 q^{26} - q^{27} + 5 q^{28} - 10 q^{29} - 5 q^{31} - q^{32} + q^{33} - q^{34} + q^{36} + 3 q^{37} + 7 q^{38} + 2 q^{39} - 10 q^{41} + 5 q^{42} - 9 q^{43} - q^{44} - 2 q^{46} - 7 q^{47} - q^{48} + 18 q^{49} - q^{51} - 2 q^{52} - 11 q^{53} + q^{54} - 5 q^{56} + 7 q^{57} + 10 q^{58} + 2 q^{61} + 5 q^{62} + 5 q^{63} + q^{64} - q^{66} + 13 q^{67} + q^{68} - 2 q^{69} - 10 q^{71} - q^{72} + 4 q^{73} - 3 q^{74} - 7 q^{76} - 5 q^{77} - 2 q^{78} + 3 q^{79} + q^{81} + 10 q^{82} + 14 q^{83} - 5 q^{84} + 9 q^{86} + 10 q^{87} + q^{88} + 4 q^{89} - 10 q^{91} + 2 q^{92} + 5 q^{93} + 7 q^{94} + q^{96} + 6 q^{97} - 18 q^{98} - q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 + 5 * q^7 - q^8 + q^9 - q^11 - q^12 - 2 * q^13 - 5 * q^14 + q^16 + q^17 - q^18 - 7 * q^19 - 5 * q^21 + q^22 + 2 * q^23 + q^24 + 2 * q^26 - q^27 + 5 * q^28 - 10 * q^29 - 5 * q^31 - q^32 + q^33 - q^34 + q^36 + 3 * q^37 + 7 * q^38 + 2 * q^39 - 10 * q^41 + 5 * q^42 - 9 * q^43 - q^44 - 2 * q^46 - 7 * q^47 - q^48 + 18 * q^49 - q^51 - 2 * q^52 - 11 * q^53 + q^54 - 5 * q^56 + 7 * q^57 + 10 * q^58 + 2 * q^61 + 5 * q^62 + 5 * q^63 + q^64 - q^66 + 13 * q^67 + q^68 - 2 * q^69 - 10 * q^71 - q^72 + 4 * q^73 - 3 * q^74 - 7 * q^76 - 5 * q^77 - 2 * q^78 + 3 * q^79 + q^81 + 10 * q^82 + 14 * q^83 - 5 * q^84 + 9 * q^86 + 10 * q^87 + q^88 + 4 * q^89 - 10 * q^91 + 2 * q^92 + 5 * q^93 + 7 * q^94 + q^96 + 6 * q^97 - 18 * 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 5.00000 −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.g 1
3.b odd 2 1 7650.2.a.cp 1
5.b even 2 1 2550.2.a.ba yes 1
5.c odd 4 2 2550.2.d.p 2
15.d odd 2 1 7650.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.a.g 1 1.a even 1 1 trivial
2550.2.a.ba yes 1 5.b even 2 1
2550.2.d.p 2 5.c odd 4 2
7650.2.a.b 1 15.d odd 2 1
7650.2.a.cp 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{7} - 5$$ T7 - 5 $$T_{11} + 1$$ T11 + 1 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

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