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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 $4$

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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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{9} - 12 q^{11} + 18 q^{14} + 24 q^{16} - 48 q^{19} - 72 q^{21} - 90 q^{24} - 36 q^{26} - 36 q^{29} + 24 q^{31} + 138 q^{34} - 84 q^{36} - 12 q^{39} - 150 q^{41} - 24 q^{44} + 60 q^{46} + 72 q^{49} + 42 q^{51} - 36 q^{54} + 60 q^{56} + 54 q^{59} - 24 q^{61} - 54 q^{64} + 156 q^{66} + 234 q^{69} + 24 q^{71} - 60 q^{76} - 108 q^{79} - 54 q^{81} - 90 q^{84} + 36 q^{86} - 18 q^{89} + 102 q^{91} - 30 q^{94} - 30 q^{96} - 246 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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:

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 \( T_{2}^{60} - 6 T_{2}^{56} - 289 T_{2}^{54} - 117 T_{2}^{52} + 2370 T_{2}^{50} + 69805 T_{2}^{48} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display