# Properties

 Label 550.2.a.n Level $550$ Weight $2$ Character orbit 550.a Self dual yes Analytic conductor $4.392$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [550,2,Mod(1,550)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$550 = 2 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 550.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: $$4.39177211117$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 110) 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + (\beta + 5) q^{9} +O(q^{10})$$ q + q^2 + b * q^3 + q^4 + b * q^6 - b * q^7 + q^8 + (b + 5) * q^9 $$q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} - \beta q^{7} + q^{8} + (\beta + 5) q^{9} - q^{11} + \beta q^{12} - 2 q^{13} - \beta q^{14} + q^{16} + ( - \beta + 2) q^{17} + (\beta + 5) q^{18} + ( - \beta + 4) q^{19} + ( - \beta - 8) q^{21} - q^{22} + ( - 2 \beta + 4) q^{23} + \beta q^{24} - 2 q^{26} + (3 \beta + 8) q^{27} - \beta q^{28} + (\beta - 2) q^{29} + \beta q^{31} + q^{32} - \beta q^{33} + ( - \beta + 2) q^{34} + (\beta + 5) q^{36} + ( - \beta - 6) q^{37} + ( - \beta + 4) q^{38} - 2 \beta q^{39} + ( - 4 \beta + 2) q^{41} + ( - \beta - 8) q^{42} + 4 q^{43} - q^{44} + ( - 2 \beta + 4) q^{46} + ( - 2 \beta + 4) q^{47} + \beta q^{48} + (\beta + 1) q^{49} + (\beta - 8) q^{51} - 2 q^{52} + (3 \beta - 6) q^{53} + (3 \beta + 8) q^{54} - \beta q^{56} + (3 \beta - 8) q^{57} + (\beta - 2) q^{58} + ( - 2 \beta + 4) q^{59} + ( - \beta - 2) q^{61} + \beta q^{62} + ( - 6 \beta - 8) q^{63} + q^{64} - \beta q^{66} - 8 q^{67} + ( - \beta + 2) q^{68} + (2 \beta - 16) q^{69} + 3 \beta q^{71} + (\beta + 5) q^{72} + (4 \beta + 2) q^{73} + ( - \beta - 6) q^{74} + ( - \beta + 4) q^{76} + \beta q^{77} - 2 \beta q^{78} + (2 \beta - 8) q^{79} + (8 \beta + 9) q^{81} + ( - 4 \beta + 2) q^{82} + (2 \beta - 4) q^{83} + ( - \beta - 8) q^{84} + 4 q^{86} + ( - \beta + 8) q^{87} - q^{88} + ( - \beta + 2) q^{89} + 2 \beta q^{91} + ( - 2 \beta + 4) q^{92} + (\beta + 8) q^{93} + ( - 2 \beta + 4) q^{94} + \beta q^{96} + (2 \beta + 6) q^{97} + (\beta + 1) q^{98} + ( - \beta - 5) q^{99} +O(q^{100})$$ q + q^2 + b * q^3 + q^4 + b * q^6 - b * q^7 + q^8 + (b + 5) * q^9 - q^11 + b * q^12 - 2 * q^13 - b * q^14 + q^16 + (-b + 2) * q^17 + (b + 5) * q^18 + (-b + 4) * q^19 + (-b - 8) * q^21 - q^22 + (-2*b + 4) * q^23 + b * q^24 - 2 * q^26 + (3*b + 8) * q^27 - b * q^28 + (b - 2) * q^29 + b * q^31 + q^32 - b * q^33 + (-b + 2) * q^34 + (b + 5) * q^36 + (-b - 6) * q^37 + (-b + 4) * q^38 - 2*b * q^39 + (-4*b + 2) * q^41 + (-b - 8) * q^42 + 4 * q^43 - q^44 + (-2*b + 4) * q^46 + (-2*b + 4) * q^47 + b * q^48 + (b + 1) * q^49 + (b - 8) * q^51 - 2 * q^52 + (3*b - 6) * q^53 + (3*b + 8) * q^54 - b * q^56 + (3*b - 8) * q^57 + (b - 2) * q^58 + (-2*b + 4) * q^59 + (-b - 2) * q^61 + b * q^62 + (-6*b - 8) * q^63 + q^64 - b * q^66 - 8 * q^67 + (-b + 2) * q^68 + (2*b - 16) * q^69 + 3*b * q^71 + (b + 5) * q^72 + (4*b + 2) * q^73 + (-b - 6) * q^74 + (-b + 4) * q^76 + b * q^77 - 2*b * q^78 + (2*b - 8) * q^79 + (8*b + 9) * q^81 + (-4*b + 2) * q^82 + (2*b - 4) * q^83 + (-b - 8) * q^84 + 4 * q^86 + (-b + 8) * q^87 - q^88 + (-b + 2) * q^89 + 2*b * q^91 + (-2*b + 4) * q^92 + (b + 8) * q^93 + (-2*b + 4) * q^94 + b * q^96 + (2*b + 6) * q^97 + (b + 1) * q^98 + (-b - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 11 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 - q^7 + 2 * q^8 + 11 * q^9 $$2 q + 2 q^{2} + q^{3} + 2 q^{4} + q^{6} - q^{7} + 2 q^{8} + 11 q^{9} - 2 q^{11} + q^{12} - 4 q^{13} - q^{14} + 2 q^{16} + 3 q^{17} + 11 q^{18} + 7 q^{19} - 17 q^{21} - 2 q^{22} + 6 q^{23} + q^{24} - 4 q^{26} + 19 q^{27} - q^{28} - 3 q^{29} + q^{31} + 2 q^{32} - q^{33} + 3 q^{34} + 11 q^{36} - 13 q^{37} + 7 q^{38} - 2 q^{39} - 17 q^{42} + 8 q^{43} - 2 q^{44} + 6 q^{46} + 6 q^{47} + q^{48} + 3 q^{49} - 15 q^{51} - 4 q^{52} - 9 q^{53} + 19 q^{54} - q^{56} - 13 q^{57} - 3 q^{58} + 6 q^{59} - 5 q^{61} + q^{62} - 22 q^{63} + 2 q^{64} - q^{66} - 16 q^{67} + 3 q^{68} - 30 q^{69} + 3 q^{71} + 11 q^{72} + 8 q^{73} - 13 q^{74} + 7 q^{76} + q^{77} - 2 q^{78} - 14 q^{79} + 26 q^{81} - 6 q^{83} - 17 q^{84} + 8 q^{86} + 15 q^{87} - 2 q^{88} + 3 q^{89} + 2 q^{91} + 6 q^{92} + 17 q^{93} + 6 q^{94} + q^{96} + 14 q^{97} + 3 q^{98} - 11 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + q^3 + 2 * q^4 + q^6 - q^7 + 2 * q^8 + 11 * q^9 - 2 * q^11 + q^12 - 4 * q^13 - q^14 + 2 * q^16 + 3 * q^17 + 11 * q^18 + 7 * q^19 - 17 * q^21 - 2 * q^22 + 6 * q^23 + q^24 - 4 * q^26 + 19 * q^27 - q^28 - 3 * q^29 + q^31 + 2 * q^32 - q^33 + 3 * q^34 + 11 * q^36 - 13 * q^37 + 7 * q^38 - 2 * q^39 - 17 * q^42 + 8 * q^43 - 2 * q^44 + 6 * q^46 + 6 * q^47 + q^48 + 3 * q^49 - 15 * q^51 - 4 * q^52 - 9 * q^53 + 19 * q^54 - q^56 - 13 * q^57 - 3 * q^58 + 6 * q^59 - 5 * q^61 + q^62 - 22 * q^63 + 2 * q^64 - q^66 - 16 * q^67 + 3 * q^68 - 30 * q^69 + 3 * q^71 + 11 * q^72 + 8 * q^73 - 13 * q^74 + 7 * q^76 + q^77 - 2 * q^78 - 14 * q^79 + 26 * q^81 - 6 * q^83 - 17 * q^84 + 8 * q^86 + 15 * q^87 - 2 * q^88 + 3 * q^89 + 2 * q^91 + 6 * q^92 + 17 * q^93 + 6 * q^94 + q^96 + 14 * q^97 + 3 * q^98 - 11 * 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
 −2.37228 3.37228
1.00000 −2.37228 1.00000 0 −2.37228 2.37228 1.00000 2.62772 0
1.2 1.00000 3.37228 1.00000 0 3.37228 −3.37228 1.00000 8.37228 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$11$$ $$+1$$

## 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 550.2.a.n 2
3.b odd 2 1 4950.2.a.bw 2
4.b odd 2 1 4400.2.a.bl 2
5.b even 2 1 110.2.a.d 2
5.c odd 4 2 550.2.b.f 4
11.b odd 2 1 6050.2.a.cb 2
15.d odd 2 1 990.2.a.m 2
15.e even 4 2 4950.2.c.bc 4
20.d odd 2 1 880.2.a.n 2
20.e even 4 2 4400.2.b.p 4
35.c odd 2 1 5390.2.a.bp 2
40.e odd 2 1 3520.2.a.bj 2
40.f even 2 1 3520.2.a.bq 2
55.d odd 2 1 1210.2.a.r 2
60.h even 2 1 7920.2.a.bq 2
220.g even 2 1 9680.2.a.bt 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.d 2 5.b even 2 1
550.2.a.n 2 1.a even 1 1 trivial
550.2.b.f 4 5.c odd 4 2
880.2.a.n 2 20.d odd 2 1
990.2.a.m 2 15.d odd 2 1
1210.2.a.r 2 55.d odd 2 1
3520.2.a.bj 2 40.e odd 2 1
3520.2.a.bq 2 40.f even 2 1
4400.2.a.bl 2 4.b odd 2 1
4400.2.b.p 4 20.e even 4 2
4950.2.a.bw 2 3.b odd 2 1
4950.2.c.bc 4 15.e even 4 2
5390.2.a.bp 2 35.c odd 2 1
6050.2.a.cb 2 11.b odd 2 1
7920.2.a.bq 2 60.h even 2 1
9680.2.a.bt 2 220.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(550))$$:

 $$T_{3}^{2} - T_{3} - 8$$ T3^2 - T3 - 8 $$T_{7}^{2} + T_{7} - 8$$ T7^2 + T7 - 8 $$T_{13} + 2$$ T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} - T - 8$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 8$$
$11$ $$(T + 1)^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 3T - 6$$
$19$ $$T^{2} - 7T + 4$$
$23$ $$T^{2} - 6T - 24$$
$29$ $$T^{2} + 3T - 6$$
$31$ $$T^{2} - T - 8$$
$37$ $$T^{2} + 13T + 34$$
$41$ $$T^{2} - 132$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} - 6T - 24$$
$53$ $$T^{2} + 9T - 54$$
$59$ $$T^{2} - 6T - 24$$
$61$ $$T^{2} + 5T - 2$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} - 3T - 72$$
$73$ $$T^{2} - 8T - 116$$
$79$ $$T^{2} + 14T + 16$$
$83$ $$T^{2} + 6T - 24$$
$89$ $$T^{2} - 3T - 6$$
$97$ $$T^{2} - 14T + 16$$
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