# Properties

 Label 55.2.e.a Level $55$ Weight $2$ Character orbit 55.e Analytic conductor $0.439$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,2,Mod(32,55)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(55, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([1, 2]))

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

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

chi := DirichletCharacter("55.32");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 55.e (of order $$4$$, degree $$2$$, 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: $$0.439177211117$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{2} - 2) q^{5} + ( - \beta_{3} - \beta_1) q^{6} + \beta_{3} q^{8} - \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - 1) * q^3 + 3*b2 * q^4 + (-b2 - 2) * q^5 + (-b3 - b1) * q^6 + b3 * q^8 - b2 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - 1) q^{3} + 3 \beta_{2} q^{4} + ( - \beta_{2} - 2) q^{5} + ( - \beta_{3} - \beta_1) q^{6} + \beta_{3} q^{8} - \beta_{2} q^{9} + ( - \beta_{3} - 2 \beta_1) q^{10} + (\beta_{3} + \beta_1 + 1) q^{11} + ( - 3 \beta_{2} + 3) q^{12} + 2 \beta_{3} q^{13} + (3 \beta_{2} + 1) q^{15} + q^{16} - 2 \beta_1 q^{17} - \beta_{3} q^{18} + ( - 2 \beta_{3} + 2 \beta_1) q^{19} + ( - 6 \beta_{2} + 3) q^{20} + (5 \beta_{2} + \beta_1 - 5) q^{22} + ( - \beta_{2} - 1) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + (4 \beta_{2} + 3) q^{25} - 10 q^{26} + (4 \beta_{2} - 4) q^{27} + (2 \beta_{3} - 2 \beta_1) q^{29} + (3 \beta_{3} + \beta_1) q^{30} + 2 q^{31} + 3 \beta_1 q^{32} + ( - 2 \beta_{3} - \beta_{2} - 1) q^{33} - 10 \beta_{2} q^{34} + 3 q^{36} + ( - 3 \beta_{2} + 3) q^{37} + (10 \beta_{2} + 10) q^{38} + ( - 2 \beta_{3} + 2 \beta_1) q^{39} + ( - 2 \beta_{3} + \beta_1) q^{40} + ( - 2 \beta_{3} - 2 \beta_1) q^{41} + (3 \beta_{3} + 3 \beta_{2} - 3 \beta_1) q^{44} + (2 \beta_{2} - 1) q^{45} + ( - \beta_{3} - \beta_1) q^{46} + ( - 3 \beta_{2} + 3) q^{47} + ( - \beta_{2} - 1) q^{48} - 7 \beta_{2} q^{49} + (4 \beta_{3} + 3 \beta_1) q^{50} + (2 \beta_{3} + 2 \beta_1) q^{51} - 6 \beta_1 q^{52} + ( - \beta_{2} - 1) q^{53} + (4 \beta_{3} - 4 \beta_1) q^{54} + ( - 3 \beta_{3} - \beta_{2} - \beta_1 - 2) q^{55} - 4 \beta_1 q^{57} + ( - 10 \beta_{2} - 10) q^{58} + 6 \beta_{2} q^{59} + (3 \beta_{2} - 9) q^{60} + (2 \beta_{3} + 2 \beta_1) q^{61} + 2 \beta_1 q^{62} + 13 \beta_{2} q^{64} + ( - 4 \beta_{3} + 2 \beta_1) q^{65} + ( - \beta_{3} - \beta_1 + 10) q^{66} + ( - 3 \beta_{2} + 3) q^{67} - 6 \beta_{3} q^{68} + 2 \beta_{2} q^{69} - 8 q^{71} + \beta_1 q^{72} - 2 \beta_{3} q^{73} + ( - 3 \beta_{3} + 3 \beta_1) q^{74} + ( - 7 \beta_{2} + 1) q^{75} + (6 \beta_{3} + 6 \beta_1) q^{76} + (10 \beta_{2} + 10) q^{78} + (2 \beta_{3} - 2 \beta_1) q^{79} + ( - \beta_{2} - 2) q^{80} + 5 q^{81} + ( - 10 \beta_{2} + 10) q^{82} + 4 \beta_{3} q^{83} + (2 \beta_{3} + 4 \beta_1) q^{85} + 4 \beta_1 q^{87} + (\beta_{3} - 5 \beta_{2} - 5) q^{88} + 6 \beta_{2} q^{89} + (2 \beta_{3} - \beta_1) q^{90} + ( - 3 \beta_{2} + 3) q^{92} + ( - 2 \beta_{2} - 2) q^{93} + ( - 3 \beta_{3} + 3 \beta_1) q^{94} + (2 \beta_{3} - 6 \beta_1) q^{95} + ( - 3 \beta_{3} - 3 \beta_1) q^{96} + (7 \beta_{2} - 7) q^{97} - 7 \beta_{3} q^{98} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - 1) * q^3 + 3*b2 * q^4 + (-b2 - 2) * q^5 + (-b3 - b1) * q^6 + b3 * q^8 - b2 * q^9 + (-b3 - 2*b1) * q^10 + (b3 + b1 + 1) * q^11 + (-3*b2 + 3) * q^12 + 2*b3 * q^13 + (3*b2 + 1) * q^15 + q^16 - 2*b1 * q^17 - b3 * q^18 + (-2*b3 + 2*b1) * q^19 + (-6*b2 + 3) * q^20 + (5*b2 + b1 - 5) * q^22 + (-b2 - 1) * q^23 + (-b3 + b1) * q^24 + (4*b2 + 3) * q^25 - 10 * q^26 + (4*b2 - 4) * q^27 + (2*b3 - 2*b1) * q^29 + (3*b3 + b1) * q^30 + 2 * q^31 + 3*b1 * q^32 + (-2*b3 - b2 - 1) * q^33 - 10*b2 * q^34 + 3 * q^36 + (-3*b2 + 3) * q^37 + (10*b2 + 10) * q^38 + (-2*b3 + 2*b1) * q^39 + (-2*b3 + b1) * q^40 + (-2*b3 - 2*b1) * q^41 + (3*b3 + 3*b2 - 3*b1) * q^44 + (2*b2 - 1) * q^45 + (-b3 - b1) * q^46 + (-3*b2 + 3) * q^47 + (-b2 - 1) * q^48 - 7*b2 * q^49 + (4*b3 + 3*b1) * q^50 + (2*b3 + 2*b1) * q^51 - 6*b1 * q^52 + (-b2 - 1) * q^53 + (4*b3 - 4*b1) * q^54 + (-3*b3 - b2 - b1 - 2) * q^55 - 4*b1 * q^57 + (-10*b2 - 10) * q^58 + 6*b2 * q^59 + (3*b2 - 9) * q^60 + (2*b3 + 2*b1) * q^61 + 2*b1 * q^62 + 13*b2 * q^64 + (-4*b3 + 2*b1) * q^65 + (-b3 - b1 + 10) * q^66 + (-3*b2 + 3) * q^67 - 6*b3 * q^68 + 2*b2 * q^69 - 8 * q^71 + b1 * q^72 - 2*b3 * q^73 + (-3*b3 + 3*b1) * q^74 + (-7*b2 + 1) * q^75 + (6*b3 + 6*b1) * q^76 + (10*b2 + 10) * q^78 + (2*b3 - 2*b1) * q^79 + (-b2 - 2) * q^80 + 5 * q^81 + (-10*b2 + 10) * q^82 + 4*b3 * q^83 + (2*b3 + 4*b1) * q^85 + 4*b1 * q^87 + (b3 - 5*b2 - 5) * q^88 + 6*b2 * q^89 + (2*b3 - b1) * q^90 + (-3*b2 + 3) * q^92 + (-2*b2 - 2) * q^93 + (-3*b3 + 3*b1) * q^94 + (2*b3 - 6*b1) * q^95 + (-3*b3 - 3*b1) * q^96 + (7*b2 - 7) * q^97 - 7*b3 * q^98 + (-b3 - b2 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{3} - 8 q^{5}+O(q^{10})$$ 4 * q - 4 * q^3 - 8 * q^5 $$4 q - 4 q^{3} - 8 q^{5} + 4 q^{11} + 12 q^{12} + 4 q^{15} + 4 q^{16} + 12 q^{20} - 20 q^{22} - 4 q^{23} + 12 q^{25} - 40 q^{26} - 16 q^{27} + 8 q^{31} - 4 q^{33} + 12 q^{36} + 12 q^{37} + 40 q^{38} - 4 q^{45} + 12 q^{47} - 4 q^{48} - 4 q^{53} - 8 q^{55} - 40 q^{58} - 36 q^{60} + 40 q^{66} + 12 q^{67} - 32 q^{71} + 4 q^{75} + 40 q^{78} - 8 q^{80} + 20 q^{81} + 40 q^{82} - 20 q^{88} + 12 q^{92} - 8 q^{93} - 28 q^{97}+O(q^{100})$$ 4 * q - 4 * q^3 - 8 * q^5 + 4 * q^11 + 12 * q^12 + 4 * q^15 + 4 * q^16 + 12 * q^20 - 20 * q^22 - 4 * q^23 + 12 * q^25 - 40 * q^26 - 16 * q^27 + 8 * q^31 - 4 * q^33 + 12 * q^36 + 12 * q^37 + 40 * q^38 - 4 * q^45 + 12 * q^47 - 4 * q^48 - 4 * q^53 - 8 * q^55 - 40 * q^58 - 36 * q^60 + 40 * q^66 + 12 * q^67 - 32 * q^71 + 4 * q^75 + 40 * q^78 - 8 * q^80 + 20 * q^81 + 40 * q^82 - 20 * q^88 + 12 * q^92 - 8 * q^93 - 28 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 5$$ (v^2) / 5 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$5\beta_{2}$$ 5*b2 $$\nu^{3}$$ $$=$$ $$5\beta_{3}$$ 5*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/55\mathbb{Z}\right)^\times$$.

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-\beta_{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −1.58114 + 1.58114i 1.58114 − 1.58114i −1.58114 − 1.58114i 1.58114 + 1.58114i
−1.58114 + 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 1.58114 + 1.58114i 1.00000i 1.58114 4.74342i
32.2 1.58114 1.58114i −1.00000 + 1.00000i 3.00000i −2.00000 + 1.00000i 3.16228i 0 −1.58114 1.58114i 1.00000i −1.58114 + 4.74342i
43.1 −1.58114 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 1.58114 1.58114i 1.00000i 1.58114 + 4.74342i
43.2 1.58114 + 1.58114i −1.00000 1.00000i 3.00000i −2.00000 1.00000i 3.16228i 0 −1.58114 + 1.58114i 1.00000i −1.58114 4.74342i
 $$n$$: 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.c odd 4 1 inner
11.b odd 2 1 inner
55.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.e.a 4
3.b odd 2 1 495.2.k.b 4
4.b odd 2 1 880.2.bd.e 4
5.b even 2 1 275.2.e.b 4
5.c odd 4 1 inner 55.2.e.a 4
5.c odd 4 1 275.2.e.b 4
11.b odd 2 1 inner 55.2.e.a 4
11.c even 5 4 605.2.m.b 16
11.d odd 10 4 605.2.m.b 16
15.e even 4 1 495.2.k.b 4
20.e even 4 1 880.2.bd.e 4
33.d even 2 1 495.2.k.b 4
44.c even 2 1 880.2.bd.e 4
55.d odd 2 1 275.2.e.b 4
55.e even 4 1 inner 55.2.e.a 4
55.e even 4 1 275.2.e.b 4
55.k odd 20 4 605.2.m.b 16
55.l even 20 4 605.2.m.b 16
165.l odd 4 1 495.2.k.b 4
220.i odd 4 1 880.2.bd.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.a 4 1.a even 1 1 trivial
55.2.e.a 4 5.c odd 4 1 inner
55.2.e.a 4 11.b odd 2 1 inner
55.2.e.a 4 55.e even 4 1 inner
275.2.e.b 4 5.b even 2 1
275.2.e.b 4 5.c odd 4 1
275.2.e.b 4 55.d odd 2 1
275.2.e.b 4 55.e even 4 1
495.2.k.b 4 3.b odd 2 1
495.2.k.b 4 15.e even 4 1
495.2.k.b 4 33.d even 2 1
495.2.k.b 4 165.l odd 4 1
605.2.m.b 16 11.c even 5 4
605.2.m.b 16 11.d odd 10 4
605.2.m.b 16 55.k odd 20 4
605.2.m.b 16 55.l even 20 4
880.2.bd.e 4 4.b odd 2 1
880.2.bd.e 4 20.e even 4 1
880.2.bd.e 4 44.c even 2 1
880.2.bd.e 4 220.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 25$$ acting on $$S_{2}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 25$$
$3$ $$(T^{2} + 2 T + 2)^{2}$$
$5$ $$(T^{2} + 4 T + 5)^{2}$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 2 T + 11)^{2}$$
$13$ $$T^{4} + 400$$
$17$ $$T^{4} + 400$$
$19$ $$(T^{2} - 40)^{2}$$
$23$ $$(T^{2} + 2 T + 2)^{2}$$
$29$ $$(T^{2} - 40)^{2}$$
$31$ $$(T - 2)^{4}$$
$37$ $$(T^{2} - 6 T + 18)^{2}$$
$41$ $$(T^{2} + 40)^{2}$$
$43$ $$T^{4}$$
$47$ $$(T^{2} - 6 T + 18)^{2}$$
$53$ $$(T^{2} + 2 T + 2)^{2}$$
$59$ $$(T^{2} + 36)^{2}$$
$61$ $$(T^{2} + 40)^{2}$$
$67$ $$(T^{2} - 6 T + 18)^{2}$$
$71$ $$(T + 8)^{4}$$
$73$ $$T^{4} + 400$$
$79$ $$(T^{2} - 40)^{2}$$
$83$ $$T^{4} + 6400$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T^{2} + 14 T + 98)^{2}$$