# Properties

 Label 2550.2.a.bg Level $2550$ Weight $2$ Character orbit 2550.a Self dual yes Analytic conductor $20.362$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - \beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ q - q^2 - q^3 + q^4 + q^6 - b * q^7 - q^8 + q^9 $$q - q^{2} - q^{3} + q^{4} + q^{6} - \beta q^{7} - q^{8} + q^{9} + ( - \beta + 2) q^{11} - q^{12} - 2 q^{13} + \beta q^{14} + q^{16} + q^{17} - q^{18} + \beta q^{19} + \beta q^{21} + (\beta - 2) q^{22} + ( - \beta - 1) q^{23} + q^{24} + 2 q^{26} - q^{27} - \beta q^{28} + ( - 2 \beta + 4) q^{29} + ( - 3 \beta + 2) q^{31} - q^{32} + (\beta - 2) q^{33} - q^{34} + q^{36} + q^{37} - \beta q^{38} + 2 q^{39} + (\beta + 1) q^{41} - \beta q^{42} + ( - \beta - 6) q^{43} + ( - \beta + 2) q^{44} + (\beta + 1) q^{46} + (\beta + 4) q^{47} - q^{48} + (\beta + 1) q^{49} - q^{51} - 2 q^{52} + (2 \beta - 1) q^{53} + q^{54} + \beta q^{56} - \beta q^{57} + (2 \beta - 4) q^{58} + ( - 3 \beta + 3) q^{59} + (3 \beta - 1) q^{61} + (3 \beta - 2) q^{62} - \beta q^{63} + q^{64} + ( - \beta + 2) q^{66} + (\beta + 2) q^{67} + q^{68} + (\beta + 1) q^{69} + (\beta + 1) q^{71} - q^{72} - 8 q^{73} - q^{74} + \beta q^{76} + ( - \beta + 8) q^{77} - 2 q^{78} + (3 \beta - 4) q^{79} + q^{81} + ( - \beta - 1) q^{82} + (\beta + 1) q^{83} + \beta q^{84} + (\beta + 6) q^{86} + (2 \beta - 4) q^{87} + (\beta - 2) q^{88} + (2 \beta + 2) q^{89} + 2 \beta q^{91} + ( - \beta - 1) q^{92} + (3 \beta - 2) q^{93} + ( - \beta - 4) q^{94} + q^{96} + ( - 2 \beta + 8) q^{97} + ( - \beta - 1) q^{98} + ( - \beta + 2) q^{99} +O(q^{100})$$ q - q^2 - q^3 + q^4 + q^6 - b * q^7 - q^8 + q^9 + (-b + 2) * q^11 - q^12 - 2 * q^13 + b * q^14 + q^16 + q^17 - q^18 + b * q^19 + b * q^21 + (b - 2) * q^22 + (-b - 1) * q^23 + q^24 + 2 * q^26 - q^27 - b * q^28 + (-2*b + 4) * q^29 + (-3*b + 2) * q^31 - q^32 + (b - 2) * q^33 - q^34 + q^36 + q^37 - b * q^38 + 2 * q^39 + (b + 1) * q^41 - b * q^42 + (-b - 6) * q^43 + (-b + 2) * q^44 + (b + 1) * q^46 + (b + 4) * q^47 - q^48 + (b + 1) * q^49 - q^51 - 2 * q^52 + (2*b - 1) * q^53 + q^54 + b * q^56 - b * q^57 + (2*b - 4) * q^58 + (-3*b + 3) * q^59 + (3*b - 1) * q^61 + (3*b - 2) * q^62 - b * q^63 + q^64 + (-b + 2) * q^66 + (b + 2) * q^67 + q^68 + (b + 1) * q^69 + (b + 1) * q^71 - q^72 - 8 * q^73 - q^74 + b * q^76 + (-b + 8) * q^77 - 2 * q^78 + (3*b - 4) * q^79 + q^81 + (-b - 1) * q^82 + (b + 1) * q^83 + b * q^84 + (b + 6) * q^86 + (2*b - 4) * q^87 + (b - 2) * q^88 + (2*b + 2) * q^89 + 2*b * q^91 + (-b - 1) * q^92 + (3*b - 2) * q^93 + (-b - 4) * q^94 + q^96 + (-2*b + 8) * q^97 + (-b - 1) * q^98 + (-b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{11} - 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} + q^{19} + q^{21} - 3 q^{22} - 3 q^{23} + 2 q^{24} + 4 q^{26} - 2 q^{27} - q^{28} + 6 q^{29} + q^{31} - 2 q^{32} - 3 q^{33} - 2 q^{34} + 2 q^{36} + 2 q^{37} - q^{38} + 4 q^{39} + 3 q^{41} - q^{42} - 13 q^{43} + 3 q^{44} + 3 q^{46} + 9 q^{47} - 2 q^{48} + 3 q^{49} - 2 q^{51} - 4 q^{52} + 2 q^{54} + q^{56} - q^{57} - 6 q^{58} + 3 q^{59} + q^{61} - q^{62} - q^{63} + 2 q^{64} + 3 q^{66} + 5 q^{67} + 2 q^{68} + 3 q^{69} + 3 q^{71} - 2 q^{72} - 16 q^{73} - 2 q^{74} + q^{76} + 15 q^{77} - 4 q^{78} - 5 q^{79} + 2 q^{81} - 3 q^{82} + 3 q^{83} + q^{84} + 13 q^{86} - 6 q^{87} - 3 q^{88} + 6 q^{89} + 2 q^{91} - 3 q^{92} - q^{93} - 9 q^{94} + 2 q^{96} + 14 q^{97} - 3 q^{98} + 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 - q^7 - 2 * q^8 + 2 * q^9 + 3 * q^11 - 2 * q^12 - 4 * q^13 + q^14 + 2 * q^16 + 2 * q^17 - 2 * q^18 + q^19 + q^21 - 3 * q^22 - 3 * q^23 + 2 * q^24 + 4 * q^26 - 2 * q^27 - q^28 + 6 * q^29 + q^31 - 2 * q^32 - 3 * q^33 - 2 * q^34 + 2 * q^36 + 2 * q^37 - q^38 + 4 * q^39 + 3 * q^41 - q^42 - 13 * q^43 + 3 * q^44 + 3 * q^46 + 9 * q^47 - 2 * q^48 + 3 * q^49 - 2 * q^51 - 4 * q^52 + 2 * q^54 + q^56 - q^57 - 6 * q^58 + 3 * q^59 + q^61 - q^62 - q^63 + 2 * q^64 + 3 * q^66 + 5 * q^67 + 2 * q^68 + 3 * q^69 + 3 * q^71 - 2 * q^72 - 16 * q^73 - 2 * q^74 + q^76 + 15 * q^77 - 4 * q^78 - 5 * q^79 + 2 * q^81 - 3 * q^82 + 3 * q^83 + q^84 + 13 * q^86 - 6 * q^87 - 3 * q^88 + 6 * q^89 + 2 * q^91 - 3 * q^92 - q^93 - 9 * q^94 + 2 * q^96 + 14 * q^97 - 3 * q^98 + 3 * 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
 3.37228 −2.37228
−1.00000 −1.00000 1.00000 0 1.00000 −3.37228 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 2.37228 −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.bg 2
3.b odd 2 1 7650.2.a.df 2
5.b even 2 1 2550.2.a.bm yes 2
5.c odd 4 2 2550.2.d.v 4
15.d odd 2 1 7650.2.a.cv 2

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

 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{11}^{2} - 3T_{11} - 6$$ T11^2 - 3*T11 - 6 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 8$$
$11$ $$T^{2} - 3T - 6$$
$13$ $$(T + 2)^{2}$$
$17$ $$(T - 1)^{2}$$
$19$ $$T^{2} - T - 8$$
$23$ $$T^{2} + 3T - 6$$
$29$ $$T^{2} - 6T - 24$$
$31$ $$T^{2} - T - 74$$
$37$ $$(T - 1)^{2}$$
$41$ $$T^{2} - 3T - 6$$
$43$ $$T^{2} + 13T + 34$$
$47$ $$T^{2} - 9T + 12$$
$53$ $$T^{2} - 33$$
$59$ $$T^{2} - 3T - 72$$
$61$ $$T^{2} - T - 74$$
$67$ $$T^{2} - 5T - 2$$
$71$ $$T^{2} - 3T - 6$$
$73$ $$(T + 8)^{2}$$
$79$ $$T^{2} + 5T - 68$$
$83$ $$T^{2} - 3T - 6$$
$89$ $$T^{2} - 6T - 24$$
$97$ $$T^{2} - 14T + 16$$