# Properties

 Label 675.2.u.c Level $675$ Weight $2$ Character orbit 675.u Analytic conductor $5.390$ Analytic rank $0$ Dimension $60$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(49,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([14, 9]))

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

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

chi := DirichletCharacter("675.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.u (of order $$18$$, degree $$6$$, not 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: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$60$$ Relative dimension: $$10$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 6 q^{9}+O(q^{10})$$ 60 * q + 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$60 q + 6 q^{9} - 12 q^{11} + 18 q^{14} + 24 q^{16} - 48 q^{19} - 72 q^{21} - 90 q^{24} - 36 q^{26} - 36 q^{29} + 24 q^{31} + 138 q^{34} - 84 q^{36} - 12 q^{39} - 150 q^{41} - 24 q^{44} + 60 q^{46} + 72 q^{49} + 42 q^{51} - 36 q^{54} + 60 q^{56} + 54 q^{59} - 24 q^{61} - 54 q^{64} + 156 q^{66} + 234 q^{69} + 24 q^{71} - 60 q^{76} - 108 q^{79} - 54 q^{81} - 90 q^{84} + 36 q^{86} - 18 q^{89} + 102 q^{91} - 30 q^{94} - 30 q^{96} - 246 q^{99}+O(q^{100})$$ 60 * q + 6 * q^9 - 12 * q^11 + 18 * q^14 + 24 * q^16 - 48 * q^19 - 72 * q^21 - 90 * q^24 - 36 * q^26 - 36 * q^29 + 24 * q^31 + 138 * q^34 - 84 * q^36 - 12 * q^39 - 150 * q^41 - 24 * q^44 + 60 * q^46 + 72 * q^49 + 42 * q^51 - 36 * q^54 + 60 * q^56 + 54 * q^59 - 24 * q^61 - 54 * q^64 + 156 * q^66 + 234 * q^69 + 24 * q^71 - 60 * q^76 - 108 * q^79 - 54 * q^81 - 90 * q^84 + 36 * q^86 - 18 * q^89 + 102 * q^91 - 30 * q^94 - 30 * q^96 - 246 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1 −1.47962 1.76334i −0.912694 1.47207i −0.572799 + 3.24850i 0 −1.24532 + 3.78748i 4.22769 0.745456i 2.58877 1.49463i −1.33398 + 2.68710i 0
49.2 −1.36249 1.62376i −1.71592 0.235856i −0.432902 + 2.45511i 0 1.95495 + 3.10759i −0.0622407 + 0.0109747i 0.904959 0.522478i 2.88874 + 0.809420i 0
49.3 −1.04972 1.25101i 1.47630 0.905836i −0.115814 + 0.656812i 0 −2.68292 0.895989i −3.26909 + 0.576430i −1.88532 + 1.08849i 1.35892 2.67457i 0
49.4 −0.446338 0.531925i 1.45285 0.942993i 0.263570 1.49478i 0 −1.15006 0.351912i 0.287167 0.0506352i −2.11545 + 1.22136i 1.22153 2.74005i 0
49.5 −0.156523 0.186537i 0.255766 1.71306i 0.337000 1.91122i 0 −0.359583 + 0.220424i 3.90696 0.688903i −0.831029 + 0.479795i −2.86917 0.876286i 0
49.6 0.156523 + 0.186537i −0.255766 + 1.71306i 0.337000 1.91122i 0 −0.359583 + 0.220424i −3.90696 + 0.688903i 0.831029 0.479795i −2.86917 0.876286i 0
49.7 0.446338 + 0.531925i −1.45285 + 0.942993i 0.263570 1.49478i 0 −1.15006 0.351912i −0.287167 + 0.0506352i 2.11545 1.22136i 1.22153 2.74005i 0
49.8 1.04972 + 1.25101i −1.47630 + 0.905836i −0.115814 + 0.656812i 0 −2.68292 0.895989i 3.26909 0.576430i 1.88532 1.08849i 1.35892 2.67457i 0
49.9 1.36249 + 1.62376i 1.71592 + 0.235856i −0.432902 + 2.45511i 0 1.95495 + 3.10759i 0.0622407 0.0109747i −0.904959 + 0.522478i 2.88874 + 0.809420i 0
49.10 1.47962 + 1.76334i 0.912694 + 1.47207i −0.572799 + 3.24850i 0 −1.24532 + 3.78748i −4.22769 + 0.745456i −2.58877 + 1.49463i −1.33398 + 2.68710i 0
124.1 −1.47962 + 1.76334i −0.912694 + 1.47207i −0.572799 3.24850i 0 −1.24532 3.78748i 4.22769 + 0.745456i 2.58877 + 1.49463i −1.33398 2.68710i 0
124.2 −1.36249 + 1.62376i −1.71592 + 0.235856i −0.432902 2.45511i 0 1.95495 3.10759i −0.0622407 0.0109747i 0.904959 + 0.522478i 2.88874 0.809420i 0
124.3 −1.04972 + 1.25101i 1.47630 + 0.905836i −0.115814 0.656812i 0 −2.68292 + 0.895989i −3.26909 0.576430i −1.88532 1.08849i 1.35892 + 2.67457i 0
124.4 −0.446338 + 0.531925i 1.45285 + 0.942993i 0.263570 + 1.49478i 0 −1.15006 + 0.351912i 0.287167 + 0.0506352i −2.11545 1.22136i 1.22153 + 2.74005i 0
124.5 −0.156523 + 0.186537i 0.255766 + 1.71306i 0.337000 + 1.91122i 0 −0.359583 0.220424i 3.90696 + 0.688903i −0.831029 0.479795i −2.86917 + 0.876286i 0
124.6 0.156523 0.186537i −0.255766 1.71306i 0.337000 + 1.91122i 0 −0.359583 0.220424i −3.90696 0.688903i 0.831029 + 0.479795i −2.86917 + 0.876286i 0
124.7 0.446338 0.531925i −1.45285 0.942993i 0.263570 + 1.49478i 0 −1.15006 + 0.351912i −0.287167 0.0506352i 2.11545 + 1.22136i 1.22153 + 2.74005i 0
124.8 1.04972 1.25101i −1.47630 0.905836i −0.115814 0.656812i 0 −2.68292 + 0.895989i 3.26909 + 0.576430i 1.88532 + 1.08849i 1.35892 + 2.67457i 0
124.9 1.36249 1.62376i 1.71592 0.235856i −0.432902 2.45511i 0 1.95495 3.10759i 0.0622407 + 0.0109747i −0.904959 0.522478i 2.88874 0.809420i 0
124.10 1.47962 1.76334i 0.912694 1.47207i −0.572799 3.24850i 0 −1.24532 3.78748i −4.22769 0.745456i −2.58877 1.49463i −1.33398 2.68710i 0
See all 60 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.10 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
27.e even 9 1 inner
135.p even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.u.c 60
5.b even 2 1 inner 675.2.u.c 60
5.c odd 4 1 135.2.k.a 30
5.c odd 4 1 675.2.l.d 30
15.e even 4 1 405.2.k.a 30
27.e even 9 1 inner 675.2.u.c 60
135.p even 18 1 inner 675.2.u.c 60
135.q even 36 1 405.2.k.a 30
135.q even 36 1 3645.2.a.g 15
135.r odd 36 1 135.2.k.a 30
135.r odd 36 1 675.2.l.d 30
135.r odd 36 1 3645.2.a.h 15

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.k.a 30 5.c odd 4 1
135.2.k.a 30 135.r odd 36 1
405.2.k.a 30 15.e even 4 1
405.2.k.a 30 135.q even 36 1
675.2.l.d 30 5.c odd 4 1
675.2.l.d 30 135.r odd 36 1
675.2.u.c 60 1.a even 1 1 trivial
675.2.u.c 60 5.b even 2 1 inner
675.2.u.c 60 27.e even 9 1 inner
675.2.u.c 60 135.p even 18 1 inner
3645.2.a.g 15 135.q even 36 1
3645.2.a.h 15 135.r odd 36 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{60} - 6 T_{2}^{56} - 289 T_{2}^{54} - 117 T_{2}^{52} + 2370 T_{2}^{50} + 69805 T_{2}^{48} + 142875 T_{2}^{46} + 228681 T_{2}^{44} - 4064560 T_{2}^{42} - 11852649 T_{2}^{40} - 11557182 T_{2}^{38} + 202716961 T_{2}^{36} + \cdots + 81$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.