# Properties

 Label 55.6.e.a Level $55$ Weight $6$ Character orbit 55.e Analytic conductor $8.821$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,6,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, 6, names="a")

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

chi := DirichletCharacter("55.32");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$8.82111008971$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 16 \beta_{2} + \beta_1 - 15) q^{3} - 32 \beta_{2} q^{4} + ( - 43 \beta_{2} + 29 \beta_1) q^{5} + ( - 62 \beta_{3} + 243 \beta_{2} - 31) q^{9}+O(q^{10})$$ q + (b3 - 16*b2 + b1 - 15) * q^3 - 32*b2 * q^4 + (-43*b2 + 29*b1) * q^5 + (-62*b3 + 243*b2 - 31) * q^9 $$q + (\beta_{3} - 16 \beta_{2} + \beta_1 - 15) q^{3} - 32 \beta_{2} q^{4} + ( - 43 \beta_{2} + 29 \beta_1) q^{5} + ( - 62 \beta_{3} + 243 \beta_{2} - 31) q^{9} + (121 \beta_{2} - 242 \beta_1) q^{11} + ( - 32 \beta_{3} + 480 \beta_{2} + \cdots - 512) q^{12}+ \cdots + ( - 58806 \beta_{3} + 41261 \beta_{2} - 29403) q^{99}+O(q^{100})$$ q + (b3 - 16*b2 + b1 - 15) * q^3 - 32*b2 * q^4 + (-43*b2 + 29*b1) * q^5 + (-62*b3 + 243*b2 - 31) * q^9 + (121*b2 - 242*b1) * q^11 + (-32*b3 + 480*b2 + 32*b1 - 512) * q^12 + (-478*b3 + 732*b2 - 421*b1 - 601) * q^15 - 1024 * q^16 + (-928*b3 - 1376) * q^20 + (1501*b3 - 1241*b2 + 1501*b1 + 260) * q^23 + (-1653*b3 + 674) * q^25 + (961*b3 + 310*b2 - 961*b1 + 651) * q^27 + (2217*b2 - 4434*b1) * q^31 + (3751*b3 - 2541*b2 + 3751*b1 + 1210) * q^33 + (992*b2 - 1984*b1 + 7776) * q^36 + (-5007*b3 - 1870*b2 + 5007*b1 - 3137) * q^37 + (7744*b3 + 3872) * q^44 + (7047*b3 - 4061*b2 - 1767*b1 + 10449) * q^45 + (-5282*b3 - 14995*b2 + 5282*b1 + 9713) * q^47 + (-1024*b3 + 16384*b2 - 1024*b1 + 15360) * q^48 - 16807*b2 * q^49 + (6476*b3 - 20641*b2 + 6476*b1 - 14165) * q^53 + (6897*b3 - 15851) * q^55 + (-28562*b3 - 14281) * q^59 + (13472*b3 + 19232*b2 - 15296*b1 + 23424) * q^60 + 32768*b2 * q^64 + (-2757*b3 + 35080*b2 + 2757*b1 - 37837) * q^67 + (-47512*b3 + 23461*b2 - 23756) * q^69 - 66273 * q^71 + (27122*b3 - 15743*b2 - 24121*b1 - 5151) * q^75 + (44032*b2 - 29696*b1) * q^80 + (-7533*b2 + 15066*b1 + 48478) * q^81 - 91089*b2 * q^89 + (-48032*b3 - 8320*b2 + 48032*b1 - 39712) * q^92 + (68727*b3 - 46557*b2 + 68727*b1 + 22170) * q^93 + (26493*b3 - 68345*b2 - 26493*b1 + 94838) * q^97 + (-58806*b3 + 41261*b2 - 29403) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 62 q^{3}+O(q^{10})$$ 4 * q - 62 * q^3 $$4 q - 62 q^{3} - 1984 q^{12} - 1448 q^{15} - 4096 q^{16} - 3648 q^{20} - 1962 q^{23} + 6002 q^{25} + 682 q^{27} - 2662 q^{33} + 31104 q^{36} - 2534 q^{37} + 27702 q^{45} + 49416 q^{47} + 63488 q^{48} - 69612 q^{53} - 77198 q^{55} + 66752 q^{60} - 145834 q^{67} - 265092 q^{71} - 74848 q^{75} + 193912 q^{81} - 62784 q^{92} - 48774 q^{93} + 326366 q^{97}+O(q^{100})$$ 4 * q - 62 * q^3 - 1984 * q^12 - 1448 * q^15 - 4096 * q^16 - 3648 * q^20 - 1962 * q^23 + 6002 * q^25 + 682 * q^27 - 2662 * q^33 + 31104 * q^36 - 2534 * q^37 + 27702 * q^45 + 49416 * q^47 + 63488 * q^48 - 69612 * q^53 - 77198 * q^55 + 66752 * q^60 - 145834 * q^67 - 265092 * q^71 - 74848 * q^75 + 193912 * q^81 - 62784 * q^92 - 48774 * q^93 + 326366 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

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

## 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
 −1.65831 − 0.500000i 1.65831 − 0.500000i −1.65831 + 0.500000i 1.65831 + 0.500000i
0 −17.1583 + 17.1583i 32.0000i −48.0911 + 28.5000i 0 0 0 345.815i 0
32.2 0 −13.8417 + 13.8417i 32.0000i 48.0911 + 28.5000i 0 0 0 140.185i 0
43.1 0 −17.1583 17.1583i 32.0000i −48.0911 28.5000i 0 0 0 345.815i 0
43.2 0 −13.8417 13.8417i 32.0000i 48.0911 28.5000i 0 0 0 140.185i 0
 $$n$$: 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
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
5.c odd 4 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 55.6.e.a 4
5.c odd 4 1 inner 55.6.e.a 4
11.b odd 2 1 CM 55.6.e.a 4
55.e even 4 1 inner 55.6.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.6.e.a 4 1.a even 1 1 trivial
55.6.e.a 4 5.c odd 4 1 inner
55.6.e.a 4 11.b odd 2 1 CM
55.6.e.a 4 55.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{6}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 62 T^{3} + \cdots + 225625$$
$5$ $$T^{4} - 3001 T^{2} + 9765625$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 161051)^{2}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + \cdots + 141855841605625$$
$29$ $$T^{4}$$
$31$ $$(T^{2} - 54065979)^{2}$$
$37$ $$T^{4} + \cdots + 18\!\cdots\!25$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + \cdots + 23\!\cdots\!00$$
$53$ $$T^{4} + \cdots + 14\!\cdots\!00$$
$59$ $$(T^{2} + 2243416571)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4} + \cdots + 68\!\cdots\!25$$
$71$ $$(T + 66273)^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$(T^{2} + 8297205921)^{2}$$
$97$ $$T^{4} + \cdots + 89\!\cdots\!25$$