# Properties

 Label 55.3.d.d Level $55$ Weight $3$ Character orbit 55.d Analytic conductor $1.499$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,3,Mod(54,55)]

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([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 55.d (of order $$2$$, degree $$1$$, 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: $$1.49864145398$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{-21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} + 41x^{2} - 40x + 505$$ x^4 - 2*x^3 + 41*x^2 - 40*x + 505 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} - \beta_1 q^{3} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{3} q^{6} - 5 \beta_{2} q^{7} + 3 \beta_{2} q^{8} - 12 q^{9}+O(q^{10})$$ q - b2 * q^2 - b1 * q^3 + q^4 + (b1 - 2) * q^5 - b3 * q^6 - 5*b2 * q^7 + 3*b2 * q^8 - 12 * q^9 $$q - \beta_{2} q^{2} - \beta_1 q^{3} + q^{4} + (\beta_1 - 2) q^{5} - \beta_{3} q^{6} - 5 \beta_{2} q^{7} + 3 \beta_{2} q^{8} - 12 q^{9} + (\beta_{3} + 2 \beta_{2}) q^{10} + (\beta_{3} - 4) q^{11} - \beta_1 q^{12} - 4 \beta_{2} q^{13} + 25 q^{14} + (2 \beta_1 + 21) q^{15} - 19 q^{16} + 7 \beta_{2} q^{17} + 12 \beta_{2} q^{18} + \beta_{3} q^{19} + (\beta_1 - 2) q^{20} - 5 \beta_{3} q^{21} + (4 \beta_{2} + 5 \beta_1) q^{22} - 6 \beta_1 q^{23} + 3 \beta_{3} q^{24} + ( - 4 \beta_1 - 17) q^{25} + 20 q^{26} + 3 \beta_1 q^{27} - 5 \beta_{2} q^{28} - \beta_{3} q^{29} + (2 \beta_{3} - 21 \beta_{2}) q^{30} - 3 q^{31} + 7 \beta_{2} q^{32} + ( - 21 \beta_{2} + 4 \beta_1) q^{33} - 35 q^{34} + (5 \beta_{3} + 10 \beta_{2}) q^{35} - 12 q^{36} - \beta_1 q^{37} + 5 \beta_1 q^{38} - 4 \beta_{3} q^{39} + ( - 3 \beta_{3} - 6 \beta_{2}) q^{40} - 2 \beta_{3} q^{41} - 25 \beta_1 q^{42} - 10 \beta_{2} q^{43} + (\beta_{3} - 4) q^{44} + ( - 12 \beta_1 + 24) q^{45} - 6 \beta_{3} q^{46} + 14 \beta_1 q^{47} + 19 \beta_1 q^{48} + 76 q^{49} + ( - 4 \beta_{3} + 17 \beta_{2}) q^{50} + 7 \beta_{3} q^{51} - 4 \beta_{2} q^{52} - \beta_1 q^{53} + 3 \beta_{3} q^{54} + ( - 2 \beta_{3} + 21 \beta_{2} + \cdots + 8) q^{55}+ \cdots + ( - 12 \beta_{3} + 48) q^{99}+O(q^{100})$$ q - b2 * q^2 - b1 * q^3 + q^4 + (b1 - 2) * q^5 - b3 * q^6 - 5*b2 * q^7 + 3*b2 * q^8 - 12 * q^9 + (b3 + 2*b2) * q^10 + (b3 - 4) * q^11 - b1 * q^12 - 4*b2 * q^13 + 25 * q^14 + (2*b1 + 21) * q^15 - 19 * q^16 + 7*b2 * q^17 + 12*b2 * q^18 + b3 * q^19 + (b1 - 2) * q^20 - 5*b3 * q^21 + (4*b2 + 5*b1) * q^22 - 6*b1 * q^23 + 3*b3 * q^24 + (-4*b1 - 17) * q^25 + 20 * q^26 + 3*b1 * q^27 - 5*b2 * q^28 - b3 * q^29 + (2*b3 - 21*b2) * q^30 - 3 * q^31 + 7*b2 * q^32 + (-21*b2 + 4*b1) * q^33 - 35 * q^34 + (5*b3 + 10*b2) * q^35 - 12 * q^36 - b1 * q^37 + 5*b1 * q^38 - 4*b3 * q^39 + (-3*b3 - 6*b2) * q^40 - 2*b3 * q^41 - 25*b1 * q^42 - 10*b2 * q^43 + (b3 - 4) * q^44 + (-12*b1 + 24) * q^45 - 6*b3 * q^46 + 14*b1 * q^47 + 19*b1 * q^48 + 76 * q^49 + (-4*b3 + 17*b2) * q^50 + 7*b3 * q^51 - 4*b2 * q^52 - b1 * q^53 + 3*b3 * q^54 + (-2*b3 + 21*b2 - 4*b1 + 8) * q^55 - 75 * q^56 - 21*b2 * q^57 - 5*b1 * q^58 - 18 * q^59 + (2*b1 + 21) * q^60 + 7*b3 * q^61 + 3*b2 * q^62 + 60*b2 * q^63 + 41 * q^64 + (4*b3 + 8*b2) * q^65 + (4*b3 + 105) * q^66 - 6*b1 * q^67 + 7*b2 * q^68 - 126 * q^69 + (25*b1 - 50) * q^70 + 27 * q^71 - 36*b2 * q^72 - 26*b2 * q^73 - b3 * q^74 + (17*b1 - 84) * q^75 + b3 * q^76 + (20*b2 + 25*b1) * q^77 - 20*b1 * q^78 - 6*b3 * q^79 + (-19*b1 + 38) * q^80 - 45 * q^81 - 10*b1 * q^82 + 32*b2 * q^83 - 5*b3 * q^84 + (-7*b3 - 14*b2) * q^85 + 50 * q^86 + 21*b2 * q^87 + (-12*b2 - 15*b1) * q^88 + 37 * q^89 + (-12*b3 - 24*b2) * q^90 + 100 * q^91 - 6*b1 * q^92 + 3*b1 * q^93 + 14*b3 * q^94 + (-2*b3 + 21*b2) * q^95 + 7*b3 * q^96 - 26*b1 * q^97 - 76*b2 * q^98 + (-12*b3 + 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} - 8 q^{5} - 48 q^{9}+O(q^{10})$$ 4 * q + 4 * q^4 - 8 * q^5 - 48 * q^9 $$4 q + 4 q^{4} - 8 q^{5} - 48 q^{9} - 16 q^{11} + 100 q^{14} + 84 q^{15} - 76 q^{16} - 8 q^{20} - 68 q^{25} + 80 q^{26} - 12 q^{31} - 140 q^{34} - 48 q^{36} - 16 q^{44} + 96 q^{45} + 304 q^{49} + 32 q^{55} - 300 q^{56} - 72 q^{59} + 84 q^{60} + 164 q^{64} + 420 q^{66} - 504 q^{69} - 200 q^{70} + 108 q^{71} - 336 q^{75} + 152 q^{80} - 180 q^{81} + 200 q^{86} + 148 q^{89} + 400 q^{91} + 192 q^{99}+O(q^{100})$$ 4 * q + 4 * q^4 - 8 * q^5 - 48 * q^9 - 16 * q^11 + 100 * q^14 + 84 * q^15 - 76 * q^16 - 8 * q^20 - 68 * q^25 + 80 * q^26 - 12 * q^31 - 140 * q^34 - 48 * q^36 - 16 * q^44 + 96 * q^45 + 304 * q^49 + 32 * q^55 - 300 * q^56 - 72 * q^59 + 84 * q^60 + 164 * q^64 + 420 * q^66 - 504 * q^69 - 200 * q^70 + 108 * q^71 - 336 * q^75 + 152 * q^80 - 180 * q^81 + 200 * q^86 + 148 * q^89 + 400 * q^91 + 192 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} + 41x^{2} - 40x + 505$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 3\nu^{2} - 125\nu + 62 ) / 89$$ (-2*v^3 + 3*v^2 - 125*v + 62) / 89 $$\beta_{2}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} - 72\nu + 35 ) / 89$$ (-4*v^3 + 6*v^2 - 72*v + 35) / 89 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu + 20$$ v^2 - v + 20
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta _1 + 1 ) / 2$$ (b2 - 2*b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{3} + \beta_{2} - 2\beta _1 - 39 ) / 2$$ (2*b3 + b2 - 2*b1 - 39) / 2 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} - 61\beta_{2} + 33\beta _1 - 59 ) / 2$$ (3*b3 - 61*b2 + 33*b1 - 59) / 2

## Character values

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

 $$n$$ $$12$$ $$46$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
54.1
 1.61803 − 4.58258i 1.61803 + 4.58258i −0.618034 − 4.58258i −0.618034 + 4.58258i
−2.23607 4.58258i 1.00000 −2.00000 + 4.58258i 10.2470i −11.1803 6.70820 −12.0000 4.47214 10.2470i
54.2 −2.23607 4.58258i 1.00000 −2.00000 4.58258i 10.2470i −11.1803 6.70820 −12.0000 4.47214 + 10.2470i
54.3 2.23607 4.58258i 1.00000 −2.00000 + 4.58258i 10.2470i 11.1803 −6.70820 −12.0000 −4.47214 + 10.2470i
54.4 2.23607 4.58258i 1.00000 −2.00000 4.58258i 10.2470i 11.1803 −6.70820 −12.0000 −4.47214 10.2470i
 $$n$$: 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
11.b odd 2 1 inner
55.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.3.d.d 4
3.b odd 2 1 495.3.h.d 4
4.b odd 2 1 880.3.i.d 4
5.b even 2 1 inner 55.3.d.d 4
5.c odd 4 2 275.3.c.e 4
11.b odd 2 1 inner 55.3.d.d 4
15.d odd 2 1 495.3.h.d 4
20.d odd 2 1 880.3.i.d 4
33.d even 2 1 495.3.h.d 4
44.c even 2 1 880.3.i.d 4
55.d odd 2 1 inner 55.3.d.d 4
55.e even 4 2 275.3.c.e 4
165.d even 2 1 495.3.h.d 4
220.g even 2 1 880.3.i.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.3.d.d 4 1.a even 1 1 trivial
55.3.d.d 4 5.b even 2 1 inner
55.3.d.d 4 11.b odd 2 1 inner
55.3.d.d 4 55.d odd 2 1 inner
275.3.c.e 4 5.c odd 4 2
275.3.c.e 4 55.e even 4 2
495.3.h.d 4 3.b odd 2 1
495.3.h.d 4 15.d odd 2 1
495.3.h.d 4 33.d even 2 1
495.3.h.d 4 165.d even 2 1
880.3.i.d 4 4.b odd 2 1
880.3.i.d 4 20.d odd 2 1
880.3.i.d 4 44.c even 2 1
880.3.i.d 4 220.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 $$S_{3}^{\mathrm{new}}(55, [\chi])$$:

 $$T_{2}^{2} - 5$$ T2^2 - 5 $$T_{3}^{2} + 21$$ T3^2 + 21

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 5)^{2}$$
$3$ $$(T^{2} + 21)^{2}$$
$5$ $$(T^{2} + 4 T + 25)^{2}$$
$7$ $$(T^{2} - 125)^{2}$$
$11$ $$(T^{2} + 8 T + 121)^{2}$$
$13$ $$(T^{2} - 80)^{2}$$
$17$ $$(T^{2} - 245)^{2}$$
$19$ $$(T^{2} + 105)^{2}$$
$23$ $$(T^{2} + 756)^{2}$$
$29$ $$(T^{2} + 105)^{2}$$
$31$ $$(T + 3)^{4}$$
$37$ $$(T^{2} + 21)^{2}$$
$41$ $$(T^{2} + 420)^{2}$$
$43$ $$(T^{2} - 500)^{2}$$
$47$ $$(T^{2} + 4116)^{2}$$
$53$ $$(T^{2} + 21)^{2}$$
$59$ $$(T + 18)^{4}$$
$61$ $$(T^{2} + 5145)^{2}$$
$67$ $$(T^{2} + 756)^{2}$$
$71$ $$(T - 27)^{4}$$
$73$ $$(T^{2} - 3380)^{2}$$
$79$ $$(T^{2} + 3780)^{2}$$
$83$ $$(T^{2} - 5120)^{2}$$
$89$ $$(T - 37)^{4}$$
$97$ $$(T^{2} + 14196)^{2}$$