# Properties

 Label 55.17.d.b Level $55$ Weight $17$ Character orbit 55.d Self dual yes Analytic conductor $89.278$ Analytic rank $0$ Dimension $1$ CM discriminant -55 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [55,17,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, 17, names="a")

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

chi := DirichletCharacter("55.54");

S:= CuspForms(chi, 17);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$55 = 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$17$$ 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: yes Analytic conductor: $$89.2784991211$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 17 q^{2} - 65247 q^{4} + 390625 q^{5} - 9886078 q^{7} - 2223311 q^{8} + 43046721 q^{9}+O(q^{10})$$ q + 17 * q^2 - 65247 * q^4 + 390625 * q^5 - 9886078 * q^7 - 2223311 * q^8 + 43046721 * q^9 $$q + 17 q^{2} - 65247 q^{4} + 390625 q^{5} - 9886078 q^{7} - 2223311 q^{8} + 43046721 q^{9} + 6640625 q^{10} + 214358881 q^{11} - 678010558 q^{13} - 168063326 q^{14} + 4238231105 q^{16} - 13921943038 q^{17} + 731794257 q^{18} - 25487109375 q^{20} + 3644100977 q^{22} + 152587890625 q^{25} - 11526179486 q^{26} + 645036931266 q^{28} - 1206552215038 q^{31} + 217756838481 q^{32} - 236673031646 q^{34} - 3861749218750 q^{35} - 2808669405087 q^{36} - 868480859375 q^{40} + 7613774646722 q^{43} - 13986273908607 q^{44} + 16815125390625 q^{45} + 64501607652483 q^{49} + 2593994140625 q^{50} + 44238154877826 q^{52} + 83733937890625 q^{55} + 21979825964258 q^{56} - 140214236988478 q^{59} - 20511387655646 q^{62} - 425563241450238 q^{63} - 274054847443103 q^{64} - 264847874218750 q^{65} + 908365017400386 q^{68} - 65649736718750 q^{70} - 77726196639358 q^{71} - 95706248313231 q^{72} + 15\!\cdots\!82 q^{73}+ \cdots + 92\!\cdots\!01 q^{99}+O(q^{100})$$ q + 17 * q^2 - 65247 * q^4 + 390625 * q^5 - 9886078 * q^7 - 2223311 * q^8 + 43046721 * q^9 + 6640625 * q^10 + 214358881 * q^11 - 678010558 * q^13 - 168063326 * q^14 + 4238231105 * q^16 - 13921943038 * q^17 + 731794257 * q^18 - 25487109375 * q^20 + 3644100977 * q^22 + 152587890625 * q^25 - 11526179486 * q^26 + 645036931266 * q^28 - 1206552215038 * q^31 + 217756838481 * q^32 - 236673031646 * q^34 - 3861749218750 * q^35 - 2808669405087 * q^36 - 868480859375 * q^40 + 7613774646722 * q^43 - 13986273908607 * q^44 + 16815125390625 * q^45 + 64501607652483 * q^49 + 2593994140625 * q^50 + 44238154877826 * q^52 + 83733937890625 * q^55 + 21979825964258 * q^56 - 140214236988478 * q^59 - 20511387655646 * q^62 - 425563241450238 * q^63 - 274054847443103 * q^64 - 264847874218750 * q^65 + 908365017400386 * q^68 - 65649736718750 * q^70 - 77726196639358 * q^71 - 95706248313231 * q^72 + 1564720076407682 * q^73 - 2119168617558718 * q^77 + 1655559025390625 * q^80 + 1853020188851841 * q^81 + 3613022253130562 * q^83 - 5438258999218750 * q^85 + 129434168994274 * q^86 - 476586458074991 * q^88 + 7841390882244482 * q^89 + 285857131640625 * q^90 + 6702865261211524 * q^91 + 1096527330092211 * q^98 + 9227446944279201 * q^99

## 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
 0
17.0000 0 −65247.0 390625. 0 −9.88608e6 −2.22331e6 4.30467e7 6.64062e6
 $$n$$: 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
55.d odd 2 1 CM by $$\Q(\sqrt{-55})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.17.d.b yes 1
5.b even 2 1 55.17.d.a 1
11.b odd 2 1 55.17.d.a 1
55.d odd 2 1 CM 55.17.d.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.17.d.a 1 5.b even 2 1
55.17.d.a 1 11.b odd 2 1
55.17.d.b yes 1 1.a even 1 1 trivial
55.17.d.b yes 1 55.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 17$$ acting on $$S_{17}^{\mathrm{new}}(55, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 17$$
$3$ $$T$$
$5$ $$T - 390625$$
$7$ $$T + 9886078$$
$11$ $$T - 214358881$$
$13$ $$T + 678010558$$
$17$ $$T + 13921943038$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$T + 1206552215038$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T - 7613774646722$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T + 140214236988478$$
$61$ $$T$$
$67$ $$T$$
$71$ $$T + 77726196639358$$
$73$ $$T - 1564720076407682$$
$79$ $$T$$
$83$ $$T - 3613022253130562$$
$89$ $$T - 7841390882244482$$
$97$ $$T$$