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Properties

Label 1680.2.k.i
Level $1680$
Weight $2$
Character orbit 1680.k
Analytic conductor $13.415$
Analytic rank $0$
Dimension $24$
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] = [1680,2,Mod(209,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.k (of order \(2\), degree \(1\), 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{9} + 6 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} + 2 q^{39} - 6 q^{51} - 24 q^{53} - 8 q^{57} + 16 q^{63} - 16 q^{65} - 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} - 32 q^{93} - 24 q^{95} + 6 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1 0 −1.70556 − 0.301745i 0 −2.04579 + 0.902636i 0 1.30232 + 2.30304i 0 2.81790 + 1.02929i 0
209.2 0 −1.70556 + 0.301745i 0 −2.04579 − 0.902636i 0 1.30232 − 2.30304i 0 2.81790 − 1.02929i 0
209.3 0 −1.64602 − 0.539084i 0 1.30213 − 1.81782i 0 −2.19974 − 1.47008i 0 2.41878 + 1.77469i 0
209.4 0 −1.64602 + 0.539084i 0 1.30213 + 1.81782i 0 −2.19974 + 1.47008i 0 2.41878 − 1.77469i 0
209.5 0 −1.37873 − 1.04838i 0 1.84099 + 1.26915i 0 2.63201 + 0.269297i 0 0.801780 + 2.89087i 0
209.6 0 −1.37873 + 1.04838i 0 1.84099 − 1.26915i 0 2.63201 − 0.269297i 0 0.801780 − 2.89087i 0
209.7 0 −0.773053 − 1.54996i 0 −0.194052 + 2.22763i 0 −0.942148 − 2.47232i 0 −1.80478 + 2.39641i 0
209.8 0 −0.773053 + 1.54996i 0 −0.194052 − 2.22763i 0 −0.942148 + 2.47232i 0 −1.80478 − 2.39641i 0
209.9 0 −0.726113 − 1.57250i 0 −2.20987 − 0.341309i 0 −2.06855 + 1.64958i 0 −1.94552 + 2.28363i 0
209.10 0 −0.726113 + 1.57250i 0 −2.20987 + 0.341309i 0 −2.06855 − 1.64958i 0 −1.94552 − 2.28363i 0
209.11 0 −0.325454 − 1.70120i 0 −1.34492 − 1.78639i 0 1.53984 − 2.15149i 0 −2.78816 + 1.10732i 0
209.12 0 −0.325454 + 1.70120i 0 −1.34492 + 1.78639i 0 1.53984 + 2.15149i 0 −2.78816 − 1.10732i 0
209.13 0 0.325454 − 1.70120i 0 1.34492 − 1.78639i 0 −1.53984 + 2.15149i 0 −2.78816 − 1.10732i 0
209.14 0 0.325454 + 1.70120i 0 1.34492 + 1.78639i 0 −1.53984 − 2.15149i 0 −2.78816 + 1.10732i 0
209.15 0 0.726113 − 1.57250i 0 2.20987 − 0.341309i 0 2.06855 − 1.64958i 0 −1.94552 − 2.28363i 0
209.16 0 0.726113 + 1.57250i 0 2.20987 + 0.341309i 0 2.06855 + 1.64958i 0 −1.94552 + 2.28363i 0
209.17 0 0.773053 − 1.54996i 0 0.194052 + 2.22763i 0 0.942148 + 2.47232i 0 −1.80478 − 2.39641i 0
209.18 0 0.773053 + 1.54996i 0 0.194052 − 2.22763i 0 0.942148 − 2.47232i 0 −1.80478 + 2.39641i 0
209.19 0 1.37873 − 1.04838i 0 −1.84099 + 1.26915i 0 −2.63201 − 0.269297i 0 0.801780 − 2.89087i 0
209.20 0 1.37873 + 1.04838i 0 −1.84099 − 1.26915i 0 −2.63201 + 0.269297i 0 0.801780 + 2.89087i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
15.d odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.k.i 24
3.b odd 2 1 1680.2.k.h 24
4.b odd 2 1 840.2.k.a ✓ 24
5.b even 2 1 1680.2.k.h 24
7.b odd 2 1 inner 1680.2.k.i 24
12.b even 2 1 840.2.k.b yes 24
15.d odd 2 1 inner 1680.2.k.i 24
20.d odd 2 1 840.2.k.b yes 24
21.c even 2 1 1680.2.k.h 24
28.d even 2 1 840.2.k.a ✓ 24
35.c odd 2 1 1680.2.k.h 24
60.h even 2 1 840.2.k.a ✓ 24
84.h odd 2 1 840.2.k.b yes 24
105.g even 2 1 inner 1680.2.k.i 24
140.c even 2 1 840.2.k.b yes 24
420.o odd 2 1 840.2.k.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.k.a ✓ 24 4.b odd 2 1
840.2.k.a ✓ 24 28.d even 2 1
840.2.k.a ✓ 24 60.h even 2 1
840.2.k.a ✓ 24 420.o odd 2 1
840.2.k.b yes 24 12.b even 2 1
840.2.k.b yes 24 20.d odd 2 1
840.2.k.b yes 24 84.h odd 2 1
840.2.k.b yes 24 140.c even 2 1
1680.2.k.h 24 3.b odd 2 1
1680.2.k.h 24 5.b even 2 1
1680.2.k.h 24 21.c even 2 1
1680.2.k.h 24 35.c odd 2 1
1680.2.k.i 24 1.a even 1 1 trivial
1680.2.k.i 24 7.b odd 2 1 inner
1680.2.k.i 24 15.d odd 2 1 inner
1680.2.k.i 24 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{12} + 71T_{11}^{10} + 1628T_{11}^{8} + 13408T_{11}^{6} + 38400T_{11}^{4} + 43008T_{11}^{2} + 16384 \) Copy content Toggle raw display
\( T_{13}^{12} - 79T_{13}^{10} + 2368T_{13}^{8} - 32776T_{13}^{6} + 197088T_{13}^{4} - 335872T_{13}^{2} + 8192 \) Copy content Toggle raw display
\( T_{23}^{6} + 4T_{23}^{5} - 68T_{23}^{4} - 64T_{23}^{3} + 864T_{23}^{2} - 128T_{23} - 2048 \) Copy content Toggle raw display