# Properties

 Label 2550.2.d.f Level $2550$ Weight $2$ Character orbit 2550.d Analytic conductor $20.362$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(2449,2550)]

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, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2550.2449");

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.d (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 2 i q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q + i * q^2 + i * q^3 - q^4 - q^6 + 2*i * q^7 - i * q^8 - q^9 $$q + i q^{2} + i q^{3} - q^{4} - q^{6} + 2 i q^{7} - i q^{8} - q^{9} - i q^{12} - 4 i q^{13} - 2 q^{14} + q^{16} - i q^{17} - i q^{18} - 4 q^{19} - 2 q^{21} - 4 i q^{23} + q^{24} + 4 q^{26} - i q^{27} - 2 i q^{28} - 2 q^{29} + i q^{32} + q^{34} + q^{36} - 2 i q^{37} - 4 i q^{38} + 4 q^{39} - 4 q^{41} - 2 i q^{42} - 10 i q^{43} + 4 q^{46} - 8 i q^{47} + i q^{48} + 3 q^{49} + q^{51} + 4 i q^{52} - 2 i q^{53} + q^{54} + 2 q^{56} - 4 i q^{57} - 2 i q^{58} + 2 q^{59} - 14 q^{61} - 2 i q^{63} - q^{64} + 2 i q^{67} + i q^{68} + 4 q^{69} - 6 q^{71} + i q^{72} + 4 i q^{73} + 2 q^{74} + 4 q^{76} + 4 i q^{78} + 12 q^{79} + q^{81} - 4 i q^{82} - 8 i q^{83} + 2 q^{84} + 10 q^{86} - 2 i q^{87} + 10 q^{89} + 8 q^{91} + 4 i q^{92} + 8 q^{94} - q^{96} + 8 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 + i * q^3 - q^4 - q^6 + 2*i * q^7 - i * q^8 - q^9 - i * q^12 - 4*i * q^13 - 2 * q^14 + q^16 - i * q^17 - i * q^18 - 4 * q^19 - 2 * q^21 - 4*i * q^23 + q^24 + 4 * q^26 - i * q^27 - 2*i * q^28 - 2 * q^29 + i * q^32 + q^34 + q^36 - 2*i * q^37 - 4*i * q^38 + 4 * q^39 - 4 * q^41 - 2*i * q^42 - 10*i * q^43 + 4 * q^46 - 8*i * q^47 + i * q^48 + 3 * q^49 + q^51 + 4*i * q^52 - 2*i * q^53 + q^54 + 2 * q^56 - 4*i * q^57 - 2*i * q^58 + 2 * q^59 - 14 * q^61 - 2*i * q^63 - q^64 + 2*i * q^67 + i * q^68 + 4 * q^69 - 6 * q^71 + i * q^72 + 4*i * q^73 + 2 * q^74 + 4 * q^76 + 4*i * q^78 + 12 * q^79 + q^81 - 4*i * q^82 - 8*i * q^83 + 2 * q^84 + 10 * q^86 - 2*i * q^87 + 10 * q^89 + 8 * q^91 + 4*i * q^92 + 8 * q^94 - q^96 + 8*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} - 4 q^{14} + 2 q^{16} - 8 q^{19} - 4 q^{21} + 2 q^{24} + 8 q^{26} - 4 q^{29} + 2 q^{34} + 2 q^{36} + 8 q^{39} - 8 q^{41} + 8 q^{46} + 6 q^{49} + 2 q^{51} + 2 q^{54} + 4 q^{56} + 4 q^{59} - 28 q^{61} - 2 q^{64} + 8 q^{69} - 12 q^{71} + 4 q^{74} + 8 q^{76} + 24 q^{79} + 2 q^{81} + 4 q^{84} + 20 q^{86} + 20 q^{89} + 16 q^{91} + 16 q^{94} - 2 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 - 4 * q^14 + 2 * q^16 - 8 * q^19 - 4 * q^21 + 2 * q^24 + 8 * q^26 - 4 * q^29 + 2 * q^34 + 2 * q^36 + 8 * q^39 - 8 * q^41 + 8 * q^46 + 6 * q^49 + 2 * q^51 + 2 * q^54 + 4 * q^56 + 4 * q^59 - 28 * q^61 - 2 * q^64 + 8 * q^69 - 12 * q^71 + 4 * q^74 + 8 * q^76 + 24 * q^79 + 2 * q^81 + 4 * q^84 + 20 * q^86 + 20 * q^89 + 16 * q^91 + 16 * q^94 - 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$$.

 $$n$$ $$751$$ $$851$$ $$1327$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2449.1
 − 1.00000i 1.00000i
1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i −1.00000 0
2449.2 1.00000i 1.00000i −1.00000 0 −1.00000 2.00000i 1.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2550.2.d.f 2
5.b even 2 1 inner 2550.2.d.f 2
5.c odd 4 1 510.2.a.d 1
5.c odd 4 1 2550.2.a.i 1
15.e even 4 1 1530.2.a.h 1
15.e even 4 1 7650.2.a.bn 1
20.e even 4 1 4080.2.a.r 1
85.g odd 4 1 8670.2.a.y 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.a.d 1 5.c odd 4 1
1530.2.a.h 1 15.e even 4 1
2550.2.a.i 1 5.c odd 4 1
2550.2.d.f 2 1.a even 1 1 trivial
2550.2.d.f 2 5.b even 2 1 inner
4080.2.a.r 1 20.e even 4 1
7650.2.a.bn 1 15.e even 4 1
8670.2.a.y 1 85.g odd 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}$$ T11 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{19} + 4$$ T19 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 1$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T + 2)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T + 4)^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 4$$
$59$ $$(T - 2)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 4$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 12)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$