# Properties

 Label 2550.2.d.l 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: yes 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + 3 i q^{7} + i q^{8} - q^{9} +O(q^{10})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + 3*i * q^7 + i * q^8 - q^9 $$q - i q^{2} + i q^{3} - q^{4} + q^{6} + 3 i q^{7} + i q^{8} - q^{9} - 5 q^{11} - i q^{12} - 4 i q^{13} + 3 q^{14} + q^{16} - i q^{17} + i q^{18} + q^{19} - 3 q^{21} + 5 i q^{22} + 4 i q^{23} - q^{24} - 4 q^{26} - i q^{27} - 3 i q^{28} + 4 q^{29} - q^{31} - i q^{32} - 5 i q^{33} - q^{34} + q^{36} - 9 i q^{37} - i q^{38} + 4 q^{39} - 10 q^{41} + 3 i q^{42} - 11 i q^{43} + 5 q^{44} + 4 q^{46} + 9 i q^{47} + i q^{48} - 2 q^{49} + q^{51} + 4 i q^{52} - 3 i q^{53} - q^{54} - 3 q^{56} + i q^{57} - 4 i q^{58} + 8 q^{59} - 14 q^{61} + i q^{62} - 3 i q^{63} - q^{64} - 5 q^{66} - 7 i q^{67} + i q^{68} - 4 q^{69} + 14 q^{71} - i q^{72} - 2 i q^{73} - 9 q^{74} - q^{76} - 15 i q^{77} - 4 i q^{78} + 5 q^{79} + q^{81} + 10 i q^{82} - 8 i q^{83} + 3 q^{84} - 11 q^{86} + 4 i q^{87} - 5 i q^{88} + 2 q^{89} + 12 q^{91} - 4 i q^{92} - i q^{93} + 9 q^{94} + q^{96} - 14 i q^{97} + 2 i q^{98} + 5 q^{99} +O(q^{100})$$ q - i * q^2 + i * q^3 - q^4 + q^6 + 3*i * q^7 + i * q^8 - q^9 - 5 * q^11 - i * q^12 - 4*i * q^13 + 3 * q^14 + q^16 - i * q^17 + i * q^18 + q^19 - 3 * q^21 + 5*i * q^22 + 4*i * q^23 - q^24 - 4 * q^26 - i * q^27 - 3*i * q^28 + 4 * q^29 - q^31 - i * q^32 - 5*i * q^33 - q^34 + q^36 - 9*i * q^37 - i * q^38 + 4 * q^39 - 10 * q^41 + 3*i * q^42 - 11*i * q^43 + 5 * q^44 + 4 * q^46 + 9*i * q^47 + i * q^48 - 2 * q^49 + q^51 + 4*i * q^52 - 3*i * q^53 - q^54 - 3 * q^56 + i * q^57 - 4*i * q^58 + 8 * q^59 - 14 * q^61 + i * q^62 - 3*i * q^63 - q^64 - 5 * q^66 - 7*i * q^67 + i * q^68 - 4 * q^69 + 14 * q^71 - i * q^72 - 2*i * q^73 - 9 * q^74 - q^76 - 15*i * q^77 - 4*i * q^78 + 5 * q^79 + q^81 + 10*i * q^82 - 8*i * q^83 + 3 * q^84 - 11 * q^86 + 4*i * q^87 - 5*i * q^88 + 2 * q^89 + 12 * q^91 - 4*i * q^92 - i * q^93 + 9 * q^94 + q^96 - 14*i * q^97 + 2*i * q^98 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 $$2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 10 q^{11} + 6 q^{14} + 2 q^{16} + 2 q^{19} - 6 q^{21} - 2 q^{24} - 8 q^{26} + 8 q^{29} - 2 q^{31} - 2 q^{34} + 2 q^{36} + 8 q^{39} - 20 q^{41} + 10 q^{44} + 8 q^{46} - 4 q^{49} + 2 q^{51} - 2 q^{54} - 6 q^{56} + 16 q^{59} - 28 q^{61} - 2 q^{64} - 10 q^{66} - 8 q^{69} + 28 q^{71} - 18 q^{74} - 2 q^{76} + 10 q^{79} + 2 q^{81} + 6 q^{84} - 22 q^{86} + 4 q^{89} + 24 q^{91} + 18 q^{94} + 2 q^{96} + 10 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 2 * q^6 - 2 * q^9 - 10 * q^11 + 6 * q^14 + 2 * q^16 + 2 * q^19 - 6 * q^21 - 2 * q^24 - 8 * q^26 + 8 * q^29 - 2 * q^31 - 2 * q^34 + 2 * q^36 + 8 * q^39 - 20 * q^41 + 10 * q^44 + 8 * q^46 - 4 * q^49 + 2 * q^51 - 2 * q^54 - 6 * q^56 + 16 * q^59 - 28 * q^61 - 2 * q^64 - 10 * q^66 - 8 * q^69 + 28 * q^71 - 18 * q^74 - 2 * q^76 + 10 * q^79 + 2 * q^81 + 6 * q^84 - 22 * q^86 + 4 * q^89 + 24 * q^91 + 18 * q^94 + 2 * q^96 + 10 * q^99

## 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 3.00000i 1.00000i −1.00000 0
2449.2 1.00000i 1.00000i −1.00000 0 1.00000 3.00000i 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.l 2
5.b even 2 1 inner 2550.2.d.l 2
5.c odd 4 1 2550.2.a.f 1
5.c odd 4 1 2550.2.a.bb yes 1
15.e even 4 1 7650.2.a.f 1
15.e even 4 1 7650.2.a.ch 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2550.2.a.f 1 5.c odd 4 1
2550.2.a.bb yes 1 5.c odd 4 1
2550.2.d.l 2 1.a even 1 1 trivial
2550.2.d.l 2 5.b even 2 1 inner
7650.2.a.f 1 15.e even 4 1
7650.2.a.ch 1 15.e even 4 1

## Hecke kernels

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

 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} + 5$$ T11 + 5 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{19} - 1$$ T19 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 5)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 1$$
$19$ $$(T - 1)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 4)^{2}$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} + 81$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2} + 121$$
$47$ $$T^{2} + 81$$
$53$ $$T^{2} + 9$$
$59$ $$(T - 8)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 49$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 5)^{2}$$
$83$ $$T^{2} + 64$$
$89$ $$(T - 2)^{2}$$
$97$ $$T^{2} + 196$$