# Properties

 Label 2550.2.a.f 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} + 3 q^{7} - q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 + 3 * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} + 3 q^{7} - q^{8} + q^{9} - 5 q^{11} - q^{12} + 4 q^{13} - 3 q^{14} + q^{16} - q^{17} - q^{18} - q^{19} - 3 q^{21} + 5 q^{22} - 4 q^{23} + q^{24} - 4 q^{26} - q^{27} + 3 q^{28} - 4 q^{29} - q^{31} - q^{32} + 5 q^{33} + q^{34} + q^{36} - 9 q^{37} + q^{38} - 4 q^{39} - 10 q^{41} + 3 q^{42} + 11 q^{43} - 5 q^{44} + 4 q^{46} + 9 q^{47} - q^{48} + 2 q^{49} + q^{51} + 4 q^{52} + 3 q^{53} + q^{54} - 3 q^{56} + q^{57} + 4 q^{58} - 8 q^{59} - 14 q^{61} + q^{62} + 3 q^{63} + q^{64} - 5 q^{66} - 7 q^{67} - q^{68} + 4 q^{69} + 14 q^{71} - q^{72} + 2 q^{73} + 9 q^{74} - q^{76} - 15 q^{77} + 4 q^{78} - 5 q^{79} + q^{81} + 10 q^{82} + 8 q^{83} - 3 q^{84} - 11 q^{86} + 4 q^{87} + 5 q^{88} - 2 q^{89} + 12 q^{91} - 4 q^{92} + q^{93} - 9 q^{94} + q^{96} - 14 q^{97} - 2 q^{98} - 5 q^{99}+O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 + 3 * q^7 - q^8 + q^9 - 5 * q^11 - q^12 + 4 * q^13 - 3 * q^14 + q^16 - q^17 - q^18 - q^19 - 3 * q^21 + 5 * q^22 - 4 * q^23 + q^24 - 4 * q^26 - q^27 + 3 * q^28 - 4 * q^29 - q^31 - q^32 + 5 * q^33 + q^34 + q^36 - 9 * q^37 + q^38 - 4 * q^39 - 10 * q^41 + 3 * q^42 + 11 * q^43 - 5 * q^44 + 4 * q^46 + 9 * q^47 - q^48 + 2 * q^49 + q^51 + 4 * q^52 + 3 * q^53 + q^54 - 3 * q^56 + q^57 + 4 * q^58 - 8 * q^59 - 14 * q^61 + q^62 + 3 * q^63 + q^64 - 5 * q^66 - 7 * q^67 - q^68 + 4 * q^69 + 14 * q^71 - q^72 + 2 * q^73 + 9 * q^74 - q^76 - 15 * q^77 + 4 * q^78 - 5 * q^79 + q^81 + 10 * q^82 + 8 * q^83 - 3 * q^84 - 11 * q^86 + 4 * q^87 + 5 * q^88 - 2 * q^89 + 12 * q^91 - 4 * q^92 + q^93 - 9 * q^94 + q^96 - 14 * 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
$$2$$ $$+1$$
$$3$$ $$+1$$
$$5$$ $$+1$$
$$17$$ $$+1$$

## 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.f 1
3.b odd 2 1 7650.2.a.ch 1
5.b even 2 1 2550.2.a.bb yes 1
5.c odd 4 2 2550.2.d.l 2
15.d odd 2 1 7650.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.a.f 1 1.a even 1 1 trivial
2550.2.a.bb yes 1 5.b even 2 1
2550.2.d.l 2 5.c odd 4 2
7650.2.a.f 1 15.d odd 2 1
7650.2.a.ch 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2550))$$:

 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 5$$ T11 + 5 $$T_{13} - 4$$ T13 - 4

## Hecke characteristic polynomials

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