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Properties

Label 1680.2.cz.e
Level $1680$
Weight $2$
Character orbit 1680.cz
Analytic conductor $13.415$
Analytic rank $0$
Dimension $24$
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] = [1680,2,Mod(97,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.97");
 
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.cz (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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 840)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{11} - 16 q^{13} - 4 q^{15} - 20 q^{17} + 8 q^{19} - 24 q^{23} - 4 q^{25} + 4 q^{37} + 16 q^{43} + 4 q^{45} - 24 q^{47} - 36 q^{49} + 16 q^{53} + 28 q^{55} + 4 q^{57} + 8 q^{59} - 4 q^{63} + 24 q^{65} - 16 q^{67} + 8 q^{69} - 8 q^{71} + 28 q^{73} + 28 q^{77} - 24 q^{81} + 64 q^{83} - 20 q^{85} + 20 q^{87} - 16 q^{89} - 24 q^{91} - 12 q^{93} - 24 q^{95} + 28 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 −0.707107 − 0.707107i 0 1.35806 − 1.77642i 0 −2.59845 + 0.498068i 0 1.00000i 0
97.2 0 −0.707107 − 0.707107i 0 1.98667 + 1.02624i 0 0.630291 + 2.56958i 0 1.00000i 0
97.3 0 −0.707107 − 0.707107i 0 −2.17640 + 0.513127i 0 −0.659108 + 2.56234i 0 1.00000i 0
97.4 0 −0.707107 − 0.707107i 0 −1.75776 − 1.38213i 0 −1.69778 − 2.02918i 0 1.00000i 0
97.5 0 −0.707107 − 0.707107i 0 0.793022 + 2.09072i 0 1.13687 − 2.38904i 0 1.00000i 0
97.6 0 −0.707107 − 0.707107i 0 0.503511 − 2.17864i 0 2.48106 − 0.918874i 0 1.00000i 0
97.7 0 0.707107 + 0.707107i 0 0.0188126 − 2.23599i 0 1.30388 + 2.30215i 0 1.00000i 0
97.8 0 0.707107 + 0.707107i 0 −0.600524 − 2.15392i 0 0.367100 − 2.62016i 0 1.00000i 0
97.9 0 0.707107 + 0.707107i 0 1.43773 + 1.71258i 0 −1.41433 + 2.23600i 0 1.00000i 0
97.10 0 0.707107 + 0.707107i 0 −2.21371 + 0.315413i 0 −2.54973 − 0.706295i 0 1.00000i 0
97.11 0 0.707107 + 0.707107i 0 −1.54016 + 1.62109i 0 1.14733 + 2.38404i 0 1.00000i 0
97.12 0 0.707107 + 0.707107i 0 2.19074 + 0.447937i 0 1.85286 − 1.88862i 0 1.00000i 0
433.1 0 −0.707107 + 0.707107i 0 1.35806 + 1.77642i 0 −2.59845 − 0.498068i 0 − 1.00000i 0
433.2 0 −0.707107 + 0.707107i 0 1.98667 − 1.02624i 0 0.630291 − 2.56958i 0 − 1.00000i 0
433.3 0 −0.707107 + 0.707107i 0 −2.17640 − 0.513127i 0 −0.659108 − 2.56234i 0 − 1.00000i 0
433.4 0 −0.707107 + 0.707107i 0 −1.75776 + 1.38213i 0 −1.69778 + 2.02918i 0 − 1.00000i 0
433.5 0 −0.707107 + 0.707107i 0 0.793022 − 2.09072i 0 1.13687 + 2.38904i 0 − 1.00000i 0
433.6 0 −0.707107 + 0.707107i 0 0.503511 + 2.17864i 0 2.48106 + 0.918874i 0 − 1.00000i 0
433.7 0 0.707107 − 0.707107i 0 0.0188126 + 2.23599i 0 1.30388 − 2.30215i 0 − 1.00000i 0
433.8 0 0.707107 − 0.707107i 0 −0.600524 + 2.15392i 0 0.367100 + 2.62016i 0 − 1.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.cz.e 24
4.b odd 2 1 840.2.bt.b yes 24
5.c odd 4 1 1680.2.cz.f 24
7.b odd 2 1 1680.2.cz.f 24
20.e even 4 1 840.2.bt.a ✓ 24
28.d even 2 1 840.2.bt.a ✓ 24
35.f even 4 1 inner 1680.2.cz.e 24
140.j odd 4 1 840.2.bt.b yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.bt.a ✓ 24 20.e even 4 1
840.2.bt.a ✓ 24 28.d even 2 1
840.2.bt.b yes 24 4.b odd 2 1
840.2.bt.b yes 24 140.j odd 4 1
1680.2.cz.e 24 1.a even 1 1 trivial
1680.2.cz.e 24 35.f even 4 1 inner
1680.2.cz.f 24 5.c odd 4 1
1680.2.cz.f 24 7.b odd 2 1

Hecke kernels

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

\( T_{11}^{12} - 4 T_{11}^{11} - 56 T_{11}^{10} + 184 T_{11}^{9} + 1172 T_{11}^{8} - 2720 T_{11}^{7} + \cdots + 38848 \) Copy content Toggle raw display
\( T_{13}^{24} + 16 T_{13}^{23} + 128 T_{13}^{22} + 544 T_{13}^{21} + 3456 T_{13}^{20} + 34688 T_{13}^{19} + \cdots + 2316304384 \) Copy content Toggle raw display