# Properties

 Label 55.2.a.b Level $55$ Weight $2$ Character orbit 55.a Self dual yes Analytic conductor $0.439$ 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] = [55,2,Mod(1,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, 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("55.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.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: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - 2 \beta q^{3} + (2 \beta + 1) q^{4} - q^{5} + ( - 2 \beta - 4) q^{6} - 2 q^{7} + (\beta + 3) q^{8} + 5 q^{9} +O(q^{10})$$ q + (b + 1) * q^2 - 2*b * q^3 + (2*b + 1) * q^4 - q^5 + (-2*b - 4) * q^6 - 2 * q^7 + (b + 3) * q^8 + 5 * q^9 $$q + (\beta + 1) q^{2} - 2 \beta q^{3} + (2 \beta + 1) q^{4} - q^{5} + ( - 2 \beta - 4) q^{6} - 2 q^{7} + (\beta + 3) q^{8} + 5 q^{9} + ( - \beta - 1) q^{10} + q^{11} + ( - 2 \beta - 8) q^{12} + (2 \beta - 4) q^{13} + ( - 2 \beta - 2) q^{14} + 2 \beta q^{15} + 3 q^{16} + (2 \beta + 4) q^{17} + (5 \beta + 5) q^{18} + ( - 2 \beta - 1) q^{20} + 4 \beta q^{21} + (\beta + 1) q^{22} - 2 \beta q^{23} + ( - 6 \beta - 4) q^{24} + q^{25} - 2 \beta q^{26} - 4 \beta q^{27} + ( - 4 \beta - 2) q^{28} + ( - 4 \beta + 2) q^{29} + (2 \beta + 4) q^{30} + (\beta - 3) q^{32} - 2 \beta q^{33} + (6 \beta + 8) q^{34} + 2 q^{35} + (10 \beta + 5) q^{36} + ( - 4 \beta - 2) q^{37} + (8 \beta - 8) q^{39} + ( - \beta - 3) q^{40} + 6 q^{41} + (4 \beta + 8) q^{42} - 6 q^{43} + (2 \beta + 1) q^{44} - 5 q^{45} + ( - 2 \beta - 4) q^{46} + 2 \beta q^{47} - 6 \beta q^{48} - 3 q^{49} + (\beta + 1) q^{50} + ( - 8 \beta - 8) q^{51} + ( - 6 \beta + 4) q^{52} + (4 \beta + 6) q^{53} + ( - 4 \beta - 8) q^{54} - q^{55} + ( - 2 \beta - 6) q^{56} + ( - 2 \beta - 6) q^{58} + (4 \beta - 4) q^{59} + (2 \beta + 8) q^{60} + ( - 8 \beta + 2) q^{61} - 10 q^{63} + ( - 2 \beta - 7) q^{64} + ( - 2 \beta + 4) q^{65} + ( - 2 \beta - 4) q^{66} + (6 \beta + 4) q^{67} + (10 \beta + 12) q^{68} + 8 q^{69} + (2 \beta + 2) q^{70} + 8 \beta q^{71} + (5 \beta + 15) q^{72} + (2 \beta - 4) q^{73} + ( - 6 \beta - 10) q^{74} - 2 \beta q^{75} - 2 q^{77} + 8 q^{78} + 4 q^{79} - 3 q^{80} + q^{81} + (6 \beta + 6) q^{82} - 6 q^{83} + (4 \beta + 16) q^{84} + ( - 2 \beta - 4) q^{85} + ( - 6 \beta - 6) q^{86} + ( - 4 \beta + 16) q^{87} + (\beta + 3) q^{88} + ( - 8 \beta - 2) q^{89} + ( - 5 \beta - 5) q^{90} + ( - 4 \beta + 8) q^{91} + ( - 2 \beta - 8) q^{92} + (2 \beta + 4) q^{94} + (6 \beta - 4) q^{96} + (4 \beta - 2) q^{97} + ( - 3 \beta - 3) q^{98} + 5 q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - 2*b * q^3 + (2*b + 1) * q^4 - q^5 + (-2*b - 4) * q^6 - 2 * q^7 + (b + 3) * q^8 + 5 * q^9 + (-b - 1) * q^10 + q^11 + (-2*b - 8) * q^12 + (2*b - 4) * q^13 + (-2*b - 2) * q^14 + 2*b * q^15 + 3 * q^16 + (2*b + 4) * q^17 + (5*b + 5) * q^18 + (-2*b - 1) * q^20 + 4*b * q^21 + (b + 1) * q^22 - 2*b * q^23 + (-6*b - 4) * q^24 + q^25 - 2*b * q^26 - 4*b * q^27 + (-4*b - 2) * q^28 + (-4*b + 2) * q^29 + (2*b + 4) * q^30 + (b - 3) * q^32 - 2*b * q^33 + (6*b + 8) * q^34 + 2 * q^35 + (10*b + 5) * q^36 + (-4*b - 2) * q^37 + (8*b - 8) * q^39 + (-b - 3) * q^40 + 6 * q^41 + (4*b + 8) * q^42 - 6 * q^43 + (2*b + 1) * q^44 - 5 * q^45 + (-2*b - 4) * q^46 + 2*b * q^47 - 6*b * q^48 - 3 * q^49 + (b + 1) * q^50 + (-8*b - 8) * q^51 + (-6*b + 4) * q^52 + (4*b + 6) * q^53 + (-4*b - 8) * q^54 - q^55 + (-2*b - 6) * q^56 + (-2*b - 6) * q^58 + (4*b - 4) * q^59 + (2*b + 8) * q^60 + (-8*b + 2) * q^61 - 10 * q^63 + (-2*b - 7) * q^64 + (-2*b + 4) * q^65 + (-2*b - 4) * q^66 + (6*b + 4) * q^67 + (10*b + 12) * q^68 + 8 * q^69 + (2*b + 2) * q^70 + 8*b * q^71 + (5*b + 15) * q^72 + (2*b - 4) * q^73 + (-6*b - 10) * q^74 - 2*b * q^75 - 2 * q^77 + 8 * q^78 + 4 * q^79 - 3 * q^80 + q^81 + (6*b + 6) * q^82 - 6 * q^83 + (4*b + 16) * q^84 + (-2*b - 4) * q^85 + (-6*b - 6) * q^86 + (-4*b + 16) * q^87 + (b + 3) * q^88 + (-8*b - 2) * q^89 + (-5*b - 5) * q^90 + (-4*b + 8) * q^91 + (-2*b - 8) * q^92 + (2*b + 4) * q^94 + (6*b - 4) * q^96 + (4*b - 2) * q^97 + (-3*b - 3) * q^98 + 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 8 * q^6 - 4 * q^7 + 6 * q^8 + 10 * q^9 $$2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 8 q^{6} - 4 q^{7} + 6 q^{8} + 10 q^{9} - 2 q^{10} + 2 q^{11} - 16 q^{12} - 8 q^{13} - 4 q^{14} + 6 q^{16} + 8 q^{17} + 10 q^{18} - 2 q^{20} + 2 q^{22} - 8 q^{24} + 2 q^{25} - 4 q^{28} + 4 q^{29} + 8 q^{30} - 6 q^{32} + 16 q^{34} + 4 q^{35} + 10 q^{36} - 4 q^{37} - 16 q^{39} - 6 q^{40} + 12 q^{41} + 16 q^{42} - 12 q^{43} + 2 q^{44} - 10 q^{45} - 8 q^{46} - 6 q^{49} + 2 q^{50} - 16 q^{51} + 8 q^{52} + 12 q^{53} - 16 q^{54} - 2 q^{55} - 12 q^{56} - 12 q^{58} - 8 q^{59} + 16 q^{60} + 4 q^{61} - 20 q^{63} - 14 q^{64} + 8 q^{65} - 8 q^{66} + 8 q^{67} + 24 q^{68} + 16 q^{69} + 4 q^{70} + 30 q^{72} - 8 q^{73} - 20 q^{74} - 4 q^{77} + 16 q^{78} + 8 q^{79} - 6 q^{80} + 2 q^{81} + 12 q^{82} - 12 q^{83} + 32 q^{84} - 8 q^{85} - 12 q^{86} + 32 q^{87} + 6 q^{88} - 4 q^{89} - 10 q^{90} + 16 q^{91} - 16 q^{92} + 8 q^{94} - 8 q^{96} - 4 q^{97} - 6 q^{98} + 10 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 8 * q^6 - 4 * q^7 + 6 * q^8 + 10 * q^9 - 2 * q^10 + 2 * q^11 - 16 * q^12 - 8 * q^13 - 4 * q^14 + 6 * q^16 + 8 * q^17 + 10 * q^18 - 2 * q^20 + 2 * q^22 - 8 * q^24 + 2 * q^25 - 4 * q^28 + 4 * q^29 + 8 * q^30 - 6 * q^32 + 16 * q^34 + 4 * q^35 + 10 * q^36 - 4 * q^37 - 16 * q^39 - 6 * q^40 + 12 * q^41 + 16 * q^42 - 12 * q^43 + 2 * q^44 - 10 * q^45 - 8 * q^46 - 6 * q^49 + 2 * q^50 - 16 * q^51 + 8 * q^52 + 12 * q^53 - 16 * q^54 - 2 * q^55 - 12 * q^56 - 12 * q^58 - 8 * q^59 + 16 * q^60 + 4 * q^61 - 20 * q^63 - 14 * q^64 + 8 * q^65 - 8 * q^66 + 8 * q^67 + 24 * q^68 + 16 * q^69 + 4 * q^70 + 30 * q^72 - 8 * q^73 - 20 * q^74 - 4 * q^77 + 16 * q^78 + 8 * q^79 - 6 * q^80 + 2 * q^81 + 12 * q^82 - 12 * q^83 + 32 * q^84 - 8 * q^85 - 12 * q^86 + 32 * q^87 + 6 * q^88 - 4 * q^89 - 10 * q^90 + 16 * q^91 - 16 * q^92 + 8 * q^94 - 8 * q^96 - 4 * q^97 - 6 * q^98 + 10 * 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
 −1.41421 1.41421
−0.414214 2.82843 −1.82843 −1.00000 −1.17157 −2.00000 1.58579 5.00000 0.414214
1.2 2.41421 −2.82843 3.82843 −1.00000 −6.82843 −2.00000 4.41421 5.00000 −2.41421
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 55.2.a.b 2
3.b odd 2 1 495.2.a.b 2
4.b odd 2 1 880.2.a.m 2
5.b even 2 1 275.2.a.c 2
5.c odd 4 2 275.2.b.d 4
7.b odd 2 1 2695.2.a.f 2
8.b even 2 1 3520.2.a.bn 2
8.d odd 2 1 3520.2.a.bo 2
11.b odd 2 1 605.2.a.d 2
11.c even 5 4 605.2.g.f 8
11.d odd 10 4 605.2.g.l 8
12.b even 2 1 7920.2.a.ch 2
13.b even 2 1 9295.2.a.g 2
15.d odd 2 1 2475.2.a.x 2
15.e even 4 2 2475.2.c.l 4
20.d odd 2 1 4400.2.a.bn 2
20.e even 4 2 4400.2.b.q 4
33.d even 2 1 5445.2.a.y 2
44.c even 2 1 9680.2.a.bn 2
55.d odd 2 1 3025.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 1.a even 1 1 trivial
275.2.a.c 2 5.b even 2 1
275.2.b.d 4 5.c odd 4 2
495.2.a.b 2 3.b odd 2 1
605.2.a.d 2 11.b odd 2 1
605.2.g.f 8 11.c even 5 4
605.2.g.l 8 11.d odd 10 4
880.2.a.m 2 4.b odd 2 1
2475.2.a.x 2 15.d odd 2 1
2475.2.c.l 4 15.e even 4 2
2695.2.a.f 2 7.b odd 2 1
3025.2.a.o 2 55.d odd 2 1
3520.2.a.bn 2 8.b even 2 1
3520.2.a.bo 2 8.d odd 2 1
4400.2.a.bn 2 20.d odd 2 1
4400.2.b.q 4 20.e even 4 2
5445.2.a.y 2 33.d even 2 1
7920.2.a.ch 2 12.b even 2 1
9295.2.a.g 2 13.b even 2 1
9680.2.a.bn 2 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(55))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 8$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 8T + 8$$
$17$ $$T^{2} - 8T + 8$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} - 4T - 28$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$(T - 6)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 12T + 4$$
$59$ $$T^{2} + 8T - 16$$
$61$ $$T^{2} - 4T - 124$$
$67$ $$T^{2} - 8T - 56$$
$71$ $$T^{2} - 128$$
$73$ $$T^{2} + 8T + 8$$
$79$ $$(T - 4)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 4T - 124$$
$97$ $$T^{2} + 4T - 28$$