Properties

 Label 2550.2.a.i 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 $$q - q^{2} + q^{3} + q^{4} - q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{12} - 4 q^{13} + 2 q^{14} + q^{16} + q^{17} - q^{18} + 4 q^{19} - 2 q^{21} - 4 q^{23} - q^{24} + 4 q^{26} + q^{27} - 2 q^{28} + 2 q^{29} - q^{32} - q^{34} + q^{36} + 2 q^{37} - 4 q^{38} - 4 q^{39} - 4 q^{41} + 2 q^{42} - 10 q^{43} + 4 q^{46} + 8 q^{47} + q^{48} - 3 q^{49} + q^{51} - 4 q^{52} - 2 q^{53} - q^{54} + 2 q^{56} + 4 q^{57} - 2 q^{58} - 2 q^{59} - 14 q^{61} - 2 q^{63} + q^{64} - 2 q^{67} + q^{68} - 4 q^{69} - 6 q^{71} - q^{72} + 4 q^{73} - 2 q^{74} + 4 q^{76} + 4 q^{78} - 12 q^{79} + q^{81} + 4 q^{82} - 8 q^{83} - 2 q^{84} + 10 q^{86} + 2 q^{87} - 10 q^{89} + 8 q^{91} - 4 q^{92} - 8 q^{94} - q^{96} - 8 q^{97} + 3 q^{98}+O(q^{100})$$ q - q^2 + q^3 + q^4 - q^6 - 2 * q^7 - q^8 + q^9 + q^12 - 4 * q^13 + 2 * q^14 + q^16 + q^17 - q^18 + 4 * q^19 - 2 * q^21 - 4 * q^23 - q^24 + 4 * q^26 + q^27 - 2 * q^28 + 2 * q^29 - q^32 - q^34 + q^36 + 2 * q^37 - 4 * q^38 - 4 * q^39 - 4 * q^41 + 2 * q^42 - 10 * q^43 + 4 * 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 - 2 * q^59 - 14 * q^61 - 2 * q^63 + q^64 - 2 * q^67 + q^68 - 4 * q^69 - 6 * q^71 - q^72 + 4 * q^73 - 2 * q^74 + 4 * q^76 + 4 * q^78 - 12 * q^79 + q^81 + 4 * q^82 - 8 * q^83 - 2 * q^84 + 10 * q^86 + 2 * q^87 - 10 * q^89 + 8 * q^91 - 4 * q^92 - 8 * q^94 - q^96 - 8 * q^97 + 3 * q^98

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.i 1
3.b odd 2 1 7650.2.a.bn 1
5.b even 2 1 510.2.a.d 1
5.c odd 4 2 2550.2.d.f 2
15.d odd 2 1 1530.2.a.h 1
20.d odd 2 1 4080.2.a.r 1
85.c even 2 1 8670.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.d 1 5.b even 2 1
1530.2.a.h 1 15.d odd 2 1
2550.2.a.i 1 1.a even 1 1 trivial
2550.2.d.f 2 5.c odd 4 2
4080.2.a.r 1 20.d odd 2 1
7650.2.a.bn 1 3.b odd 2 1
8670.2.a.y 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 $$T_{11}$$ T11 $$T_{13} + 4$$ T13 + 4

Hecke characteristic polynomials

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