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Properties

Label 168.4.i.c
Level $168$
Weight $4$
Character orbit 168.i
Analytic conductor $9.912$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [168,4,Mod(125,168)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(168, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("168.125");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.i (of order \(2\), degree \(1\), 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: \(9.91232088096\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 28 q^{4} + 64 q^{7} + 104 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 28 q^{4} + 64 q^{7} + 104 q^{9} - 8 q^{15} - 892 q^{16} + 692 q^{18} + 128 q^{22} - 976 q^{25} + 612 q^{28} - 332 q^{30} + 1544 q^{36} + 568 q^{39} + 780 q^{42} + 208 q^{46} - 4048 q^{49} - 1448 q^{57} - 1760 q^{58} + 4156 q^{60} - 2152 q^{63} + 2764 q^{64} + 1968 q^{70} - 2740 q^{72} - 3620 q^{78} + 4992 q^{79} + 1568 q^{81} + 2484 q^{84} - 2072 q^{88}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1 −2.81762 − 0.246982i −3.70908 + 3.63906i 7.87800 + 1.39180i − 9.02015i 11.3496 − 9.33741i 5.01951 + 17.8271i −21.8535 − 5.86730i 0.514533 − 26.9951i −2.22781 + 25.4154i
125.2 −2.81762 − 0.246982i 3.70908 − 3.63906i 7.87800 + 1.39180i 9.02015i −11.3496 + 9.33741i 5.01951 − 17.8271i −21.8535 − 5.86730i 0.514533 − 26.9951i 2.22781 − 25.4154i
125.3 −2.81762 + 0.246982i −3.70908 − 3.63906i 7.87800 − 1.39180i 9.02015i 11.3496 + 9.33741i 5.01951 − 17.8271i −21.8535 + 5.86730i 0.514533 + 26.9951i −2.22781 − 25.4154i
125.4 −2.81762 + 0.246982i 3.70908 + 3.63906i 7.87800 − 1.39180i − 9.02015i −11.3496 − 9.33741i 5.01951 + 17.8271i −21.8535 + 5.86730i 0.514533 + 26.9951i 2.22781 + 25.4154i
125.5 −2.65881 − 0.964755i −5.10794 − 0.953394i 6.13849 + 5.13019i 17.0023i 12.6612 + 7.46280i −5.04990 + 17.8185i −11.3717 − 19.5623i 25.1821 + 9.73975i 16.4031 − 45.2059i
125.6 −2.65881 − 0.964755i 5.10794 + 0.953394i 6.13849 + 5.13019i − 17.0023i −12.6612 − 7.46280i −5.04990 − 17.8185i −11.3717 − 19.5623i 25.1821 + 9.73975i −16.4031 + 45.2059i
125.7 −2.65881 + 0.964755i −5.10794 + 0.953394i 6.13849 − 5.13019i − 17.0023i 12.6612 − 7.46280i −5.04990 − 17.8185i −11.3717 + 19.5623i 25.1821 − 9.73975i 16.4031 + 45.2059i
125.8 −2.65881 + 0.964755i 5.10794 − 0.953394i 6.13849 − 5.13019i 17.0023i −12.6612 + 7.46280i −5.04990 + 17.8185i −11.3717 + 19.5623i 25.1821 − 9.73975i −16.4031 − 45.2059i
125.9 −2.60932 − 1.09154i −4.05635 − 3.24747i 5.61708 + 5.69635i − 13.2397i 7.03955 + 12.9013i 18.4748 + 1.29729i −8.43895 − 20.9949i 5.90788 + 26.3457i −14.4516 + 34.5465i
125.10 −2.60932 − 1.09154i 4.05635 + 3.24747i 5.61708 + 5.69635i 13.2397i −7.03955 − 12.9013i 18.4748 − 1.29729i −8.43895 − 20.9949i 5.90788 + 26.3457i 14.4516 − 34.5465i
125.11 −2.60932 + 1.09154i −4.05635 + 3.24747i 5.61708 − 5.69635i 13.2397i 7.03955 − 12.9013i 18.4748 − 1.29729i −8.43895 + 20.9949i 5.90788 − 26.3457i −14.4516 − 34.5465i
125.12 −2.60932 + 1.09154i 4.05635 − 3.24747i 5.61708 − 5.69635i − 13.2397i −7.03955 + 12.9013i 18.4748 + 1.29729i −8.43895 + 20.9949i 5.90788 − 26.3457i 14.4516 + 34.5465i
125.13 −2.39518 − 1.50436i −4.72061 + 2.17161i 3.47378 + 7.20645i − 2.27823i 14.5736 + 1.90012i −13.7109 − 12.4503i 2.52080 − 22.4866i 17.5682 − 20.5026i −3.42729 + 5.45677i
125.14 −2.39518 − 1.50436i 4.72061 − 2.17161i 3.47378 + 7.20645i 2.27823i −14.5736 − 1.90012i −13.7109 + 12.4503i 2.52080 − 22.4866i 17.5682 − 20.5026i 3.42729 − 5.45677i
125.15 −2.39518 + 1.50436i −4.72061 − 2.17161i 3.47378 − 7.20645i 2.27823i 14.5736 − 1.90012i −13.7109 + 12.4503i 2.52080 + 22.4866i 17.5682 + 20.5026i −3.42729 − 5.45677i
125.16 −2.39518 + 1.50436i 4.72061 + 2.17161i 3.47378 − 7.20645i − 2.27823i −14.5736 + 1.90012i −13.7109 − 12.4503i 2.52080 + 22.4866i 17.5682 + 20.5026i 3.42729 + 5.45677i
125.17 −2.24057 − 1.72622i −1.04870 + 5.08923i 2.04030 + 7.73545i − 3.71228i 11.1348 − 9.59247i 15.6071 − 9.97081i 8.78168 − 20.8538i −24.8005 − 10.6741i −6.40823 + 8.31763i
125.18 −2.24057 − 1.72622i 1.04870 − 5.08923i 2.04030 + 7.73545i 3.71228i −11.1348 + 9.59247i 15.6071 + 9.97081i 8.78168 − 20.8538i −24.8005 − 10.6741i 6.40823 − 8.31763i
125.19 −2.24057 + 1.72622i −1.04870 − 5.08923i 2.04030 − 7.73545i 3.71228i 11.1348 + 9.59247i 15.6071 + 9.97081i 8.78168 + 20.8538i −24.8005 + 10.6741i −6.40823 − 8.31763i
125.20 −2.24057 + 1.72622i 1.04870 + 5.08923i 2.04030 − 7.73545i − 3.71228i −11.1348 − 9.59247i 15.6071 − 9.97081i 8.78168 + 20.8538i −24.8005 + 10.6741i 6.40823 + 8.31763i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
24.h odd 2 1 inner
56.h odd 2 1 inner
168.i even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.4.i.c ✓ 80
3.b odd 2 1 inner 168.4.i.c ✓ 80
4.b odd 2 1 672.4.i.c 80
7.b odd 2 1 inner 168.4.i.c ✓ 80
8.b even 2 1 inner 168.4.i.c ✓ 80
8.d odd 2 1 672.4.i.c 80
12.b even 2 1 672.4.i.c 80
21.c even 2 1 inner 168.4.i.c ✓ 80
24.f even 2 1 672.4.i.c 80
24.h odd 2 1 inner 168.4.i.c ✓ 80
28.d even 2 1 672.4.i.c 80
56.e even 2 1 672.4.i.c 80
56.h odd 2 1 inner 168.4.i.c ✓ 80
84.h odd 2 1 672.4.i.c 80
168.e odd 2 1 672.4.i.c 80
168.i even 2 1 inner 168.4.i.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.i.c ✓ 80 1.a even 1 1 trivial
168.4.i.c ✓ 80 3.b odd 2 1 inner
168.4.i.c ✓ 80 7.b odd 2 1 inner
168.4.i.c ✓ 80 8.b even 2 1 inner
168.4.i.c ✓ 80 21.c even 2 1 inner
168.4.i.c ✓ 80 24.h odd 2 1 inner
168.4.i.c ✓ 80 56.h odd 2 1 inner
168.4.i.c ✓ 80 168.i even 2 1 inner
672.4.i.c 80 4.b odd 2 1
672.4.i.c 80 8.d odd 2 1
672.4.i.c 80 12.b even 2 1
672.4.i.c 80 24.f even 2 1
672.4.i.c 80 28.d even 2 1
672.4.i.c 80 56.e even 2 1
672.4.i.c 80 84.h odd 2 1
672.4.i.c 80 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 1372 T_{5}^{18} + 782932 T_{5}^{16} + 239880544 T_{5}^{14} + 42479046160 T_{5}^{12} + \cdots + 39\!\cdots\!12 \) acting on \(S_{4}^{\mathrm{new}}(168, [\chi])\). Copy content Toggle raw display