Properties

Label 882.2.t.c
Level $882$
Weight $2$
Character orbit 882.t
Analytic conductor $7.043$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(803,882)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("882.803"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.t (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0,0,24,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} + 16 q^{9} + 48 q^{15} - 24 q^{16} - 32 q^{18} + 48 q^{25} + 32 q^{30} - 16 q^{36} + 32 q^{39} - 48 q^{44} - 48 q^{50} + 48 q^{51} - 96 q^{53} - 80 q^{57} + 48 q^{60} - 48 q^{64} - 16 q^{72}+ \cdots + 104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
803.1 −0.866025 + 0.500000i −1.68502 − 0.400894i 0.500000 − 0.866025i −1.99040 1.65972 − 0.495324i 0 1.00000i 2.67857 + 1.35103i 1.72374 − 0.995200i
803.2 −0.866025 + 0.500000i −1.62990 − 0.586023i 0.500000 − 0.866025i 3.10134 1.70455 − 0.307440i 0 1.00000i 2.31315 + 1.91032i −2.68584 + 1.55067i
803.3 −0.866025 + 0.500000i −1.47241 + 0.912143i 0.500000 − 0.866025i −3.98682 0.819074 − 1.52614i 0 1.00000i 1.33599 − 2.68610i 3.45268 − 1.99341i
803.4 −0.866025 + 0.500000i −1.41984 − 0.991995i 0.500000 − 0.866025i −1.89776 1.72562 + 0.149173i 0 1.00000i 1.03189 + 2.81695i 1.64351 − 0.948881i
803.5 −0.866025 + 0.500000i −1.31069 + 1.13230i 0.500000 − 0.866025i 3.10777 0.568941 − 1.63594i 0 1.00000i 0.435807 − 2.96818i −2.69141 + 1.55389i
803.6 −0.866025 + 0.500000i −0.578227 + 1.63268i 0.500000 − 0.866025i 0.440174 −0.315583 − 1.70306i 0 1.00000i −2.33131 − 1.88812i −0.381202 + 0.220087i
803.7 −0.866025 + 0.500000i 0.578227 − 1.63268i 0.500000 − 0.866025i −0.440174 0.315583 + 1.70306i 0 1.00000i −2.33131 − 1.88812i 0.381202 − 0.220087i
803.8 −0.866025 + 0.500000i 1.31069 − 1.13230i 0.500000 − 0.866025i −3.10777 −0.568941 + 1.63594i 0 1.00000i 0.435807 − 2.96818i 2.69141 − 1.55389i
803.9 −0.866025 + 0.500000i 1.41984 + 0.991995i 0.500000 − 0.866025i 1.89776 −1.72562 − 0.149173i 0 1.00000i 1.03189 + 2.81695i −1.64351 + 0.948881i
803.10 −0.866025 + 0.500000i 1.47241 − 0.912143i 0.500000 − 0.866025i 3.98682 −0.819074 + 1.52614i 0 1.00000i 1.33599 − 2.68610i −3.45268 + 1.99341i
803.11 −0.866025 + 0.500000i 1.62990 + 0.586023i 0.500000 − 0.866025i −3.10134 −1.70455 + 0.307440i 0 1.00000i 2.31315 + 1.91032i 2.68584 − 1.55067i
803.12 −0.866025 + 0.500000i 1.68502 + 0.400894i 0.500000 − 0.866025i 1.99040 −1.65972 + 0.495324i 0 1.00000i 2.67857 + 1.35103i −1.72374 + 0.995200i
803.13 0.866025 − 0.500000i −1.72827 − 0.114384i 0.500000 − 0.866025i −0.949113 −1.55392 + 0.765075i 0 − 1.00000i 2.97383 + 0.395374i −0.821956 + 0.474556i
803.14 0.866025 − 0.500000i −1.35874 + 1.07416i 0.500000 − 0.866025i −2.70053 −0.639623 + 1.60962i 0 − 1.00000i 0.692351 − 2.91902i −2.33872 + 1.35026i
803.15 0.866025 − 0.500000i −1.24917 + 1.19983i 0.500000 − 0.866025i −1.42597 −0.481898 + 1.66366i 0 − 1.00000i 0.120839 − 2.99757i −1.23492 + 0.712984i
803.16 0.866025 − 0.500000i −1.04239 + 1.38327i 0.500000 − 0.866025i 1.16972 −0.211098 + 1.71914i 0 − 1.00000i −0.826865 − 2.88380i 1.01301 − 0.584859i
803.17 0.866025 − 0.500000i −0.883109 − 1.49001i 0.500000 − 0.866025i −3.92134 −1.50980 − 0.848829i 0 − 1.00000i −1.44024 + 2.63168i −3.39598 + 1.96067i
803.18 0.866025 − 0.500000i −0.0893856 − 1.72974i 0.500000 − 0.866025i −1.44900 −0.942282 − 1.45331i 0 − 1.00000i −2.98402 + 0.309228i −1.25487 + 0.724499i
803.19 0.866025 − 0.500000i 0.0893856 + 1.72974i 0.500000 − 0.866025i 1.44900 0.942282 + 1.45331i 0 − 1.00000i −2.98402 + 0.309228i 1.25487 − 0.724499i
803.20 0.866025 − 0.500000i 0.883109 + 1.49001i 0.500000 − 0.866025i 3.92134 1.50980 + 0.848829i 0 − 1.00000i −1.44024 + 2.63168i 3.39598 − 1.96067i
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 803.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.t.c 48
3.b odd 2 1 2646.2.t.c 48
7.b odd 2 1 inner 882.2.t.c 48
7.c even 3 1 882.2.l.c 48
7.c even 3 1 882.2.m.c ✓ 48
7.d odd 6 1 882.2.l.c 48
7.d odd 6 1 882.2.m.c ✓ 48
9.c even 3 1 2646.2.l.c 48
9.d odd 6 1 882.2.l.c 48
21.c even 2 1 2646.2.t.c 48
21.g even 6 1 2646.2.l.c 48
21.g even 6 1 2646.2.m.c 48
21.h odd 6 1 2646.2.l.c 48
21.h odd 6 1 2646.2.m.c 48
63.g even 3 1 2646.2.t.c 48
63.h even 3 1 2646.2.m.c 48
63.i even 6 1 882.2.m.c ✓ 48
63.j odd 6 1 882.2.m.c ✓ 48
63.k odd 6 1 2646.2.t.c 48
63.l odd 6 1 2646.2.l.c 48
63.n odd 6 1 inner 882.2.t.c 48
63.o even 6 1 882.2.l.c 48
63.s even 6 1 inner 882.2.t.c 48
63.t odd 6 1 2646.2.m.c 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.l.c 48 7.c even 3 1
882.2.l.c 48 7.d odd 6 1
882.2.l.c 48 9.d odd 6 1
882.2.l.c 48 63.o even 6 1
882.2.m.c ✓ 48 7.c even 3 1
882.2.m.c ✓ 48 7.d odd 6 1
882.2.m.c ✓ 48 63.i even 6 1
882.2.m.c ✓ 48 63.j odd 6 1
882.2.t.c 48 1.a even 1 1 trivial
882.2.t.c 48 7.b odd 2 1 inner
882.2.t.c 48 63.n odd 6 1 inner
882.2.t.c 48 63.s even 6 1 inner
2646.2.l.c 48 9.c even 3 1
2646.2.l.c 48 21.g even 6 1
2646.2.l.c 48 21.h odd 6 1
2646.2.l.c 48 63.l odd 6 1
2646.2.m.c 48 21.g even 6 1
2646.2.m.c 48 21.h odd 6 1
2646.2.m.c 48 63.h even 3 1
2646.2.m.c 48 63.t odd 6 1
2646.2.t.c 48 3.b odd 2 1
2646.2.t.c 48 21.c even 2 1
2646.2.t.c 48 63.g even 3 1
2646.2.t.c 48 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 2208 T_{5}^{20} - 37880 T_{5}^{18} + 402114 T_{5}^{16} - 2762952 T_{5}^{14} + \cdots + 2408704 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display