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Properties

Label 2112.2.t.a
Level $2112$
Weight $2$
Character orbit 2112.t
Analytic conductor $16.864$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2112,2,Mod(529,2112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2112, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2112.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2112.t (of order \(4\), degree \(2\), 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: \(16.8644049069\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 528)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 24 q^{15} - 8 q^{19} + 56 q^{31} + 40 q^{33} + 24 q^{35} - 64 q^{47} - 40 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{61} + 24 q^{63} + 16 q^{65} - 24 q^{67} - 16 q^{69} - 40 q^{79} - 40 q^{81} + 16 q^{85} - 16 q^{91} - 32 q^{93} + 32 q^{95} + 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1 0 −0.707107 − 0.707107i 0 2.69430 − 2.69430i 0 2.74702i 0 1.00000i 0
529.2 0 −0.707107 − 0.707107i 0 2.31577 − 2.31577i 0 − 4.32386i 0 1.00000i 0
529.3 0 −0.707107 − 0.707107i 0 −1.62061 + 1.62061i 0 − 4.66827i 0 1.00000i 0
529.4 0 −0.707107 − 0.707107i 0 1.09169 − 1.09169i 0 − 2.40742i 0 1.00000i 0
529.5 0 −0.707107 − 0.707107i 0 0.983303 − 0.983303i 0 3.88127i 0 1.00000i 0
529.6 0 −0.707107 − 0.707107i 0 0.232487 − 0.232487i 0 − 0.234470i 0 1.00000i 0
529.7 0 −0.707107 − 0.707107i 0 0.0403720 − 0.0403720i 0 2.61914i 0 1.00000i 0
529.8 0 −0.707107 − 0.707107i 0 −2.10941 + 2.10941i 0 − 2.44976i 0 1.00000i 0
529.9 0 −0.707107 − 0.707107i 0 2.29467 − 2.29467i 0 − 1.84060i 0 1.00000i 0
529.10 0 −0.707107 − 0.707107i 0 −1.67992 + 1.67992i 0 0.676945i 0 1.00000i 0
529.11 0 0.707107 + 0.707107i 0 −2.68689 + 2.68689i 0 3.25700i 0 1.00000i 0
529.12 0 0.707107 + 0.707107i 0 −1.67077 + 1.67077i 0 − 4.47633i 0 1.00000i 0
529.13 0 0.707107 + 0.707107i 0 −0.487980 + 0.487980i 0 0.309539i 0 1.00000i 0
529.14 0 0.707107 + 0.707107i 0 0.288103 − 0.288103i 0 − 4.76851i 0 1.00000i 0
529.15 0 0.707107 + 0.707107i 0 −0.720185 + 0.720185i 0 0.308553i 0 1.00000i 0
529.16 0 0.707107 + 0.707107i 0 −0.814208 + 0.814208i 0 − 0.969522i 0 1.00000i 0
529.17 0 0.707107 + 0.707107i 0 1.12158 − 1.12158i 0 3.79305i 0 1.00000i 0
529.18 0 0.707107 + 0.707107i 0 1.95402 − 1.95402i 0 − 0.896707i 0 1.00000i 0
529.19 0 0.707107 + 0.707107i 0 1.90263 − 1.90263i 0 − 0.451736i 0 1.00000i 0
529.20 0 0.707107 + 0.707107i 0 −3.12894 + 3.12894i 0 − 2.10533i 0 1.00000i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2112.2.t.a 40
4.b odd 2 1 528.2.t.a ✓ 40
16.e even 4 1 inner 2112.2.t.a 40
16.f odd 4 1 528.2.t.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.t.a ✓ 40 4.b odd 2 1
528.2.t.a ✓ 40 16.f odd 4 1
2112.2.t.a 40 1.a even 1 1 trivial
2112.2.t.a 40 16.e even 4 1 inner