↑

Properties

Label 672.2.bi.c
Level $672$
Weight $2$
Character orbit 672.bi
Analytic conductor $5.366$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(17,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bi (of order \(6\), 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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{7} - 14 q^{9} - 4 q^{15} - 8 q^{25} - 48 q^{31} - 42 q^{33} + 8 q^{39} - 36 q^{49} + 4 q^{57} + 6 q^{63} - 36 q^{73} + 56 q^{79} + 42 q^{81} + 132 q^{87}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.70874 + 0.283220i 0 −2.24840 + 1.29811i 0 2.53678 + 0.751482i 0 2.83957 − 0.967897i 0
17.2 0 −1.70617 − 0.298296i 0 0.337879 − 0.195075i 0 −1.39526 + 2.24795i 0 2.82204 + 1.01789i 0
17.3 0 −1.69273 + 0.366975i 0 2.66818 − 1.54047i 0 1.46307 − 2.20441i 0 2.73066 − 1.24238i 0
17.4 0 −1.52791 + 0.815778i 0 −0.461663 + 0.266541i 0 −0.489180 − 2.60014i 0 1.66901 − 2.49287i 0
17.5 0 −1.26610 + 1.18195i 0 −1.54900 + 0.894317i 0 −2.63573 + 0.230049i 0 0.206000 − 2.99292i 0
17.6 0 −1.11142 − 1.32844i 0 0.337879 − 0.195075i 0 −1.39526 + 2.24795i 0 −0.529502 + 2.95290i 0
17.7 0 −1.09218 + 1.34430i 0 2.46958 − 1.42581i 0 1.02032 + 2.44110i 0 −0.614290 − 2.93643i 0
17.8 0 −0.618109 + 1.61801i 0 −2.46958 + 1.42581i 0 1.02032 + 2.44110i 0 −2.23588 − 2.00021i 0
17.9 0 −0.609093 − 1.62142i 0 −2.24840 + 1.29811i 0 2.53678 + 0.751482i 0 −2.25801 + 1.97519i 0
17.10 0 −0.528554 − 1.64943i 0 2.66818 − 1.54047i 0 1.46307 − 2.20441i 0 −2.44126 + 1.74363i 0
17.11 0 −0.390548 + 1.68745i 0 1.54900 − 0.894317i 0 −2.63573 + 0.230049i 0 −2.69494 − 1.31806i 0
17.12 0 −0.0574700 − 1.73110i 0 −0.461663 + 0.266541i 0 −0.489180 − 2.60014i 0 −2.99339 + 0.198972i 0
17.13 0 0.0574700 + 1.73110i 0 0.461663 − 0.266541i 0 −0.489180 − 2.60014i 0 −2.99339 + 0.198972i 0
17.14 0 0.390548 − 1.68745i 0 −1.54900 + 0.894317i 0 −2.63573 + 0.230049i 0 −2.69494 − 1.31806i 0
17.15 0 0.528554 + 1.64943i 0 −2.66818 + 1.54047i 0 1.46307 − 2.20441i 0 −2.44126 + 1.74363i 0
17.16 0 0.609093 + 1.62142i 0 2.24840 − 1.29811i 0 2.53678 + 0.751482i 0 −2.25801 + 1.97519i 0
17.17 0 0.618109 − 1.61801i 0 2.46958 − 1.42581i 0 1.02032 + 2.44110i 0 −2.23588 − 2.00021i 0
17.18 0 1.09218 − 1.34430i 0 −2.46958 + 1.42581i 0 1.02032 + 2.44110i 0 −0.614290 − 2.93643i 0
17.19 0 1.11142 + 1.32844i 0 −0.337879 + 0.195075i 0 −1.39526 + 2.24795i 0 −0.529502 + 2.95290i 0
17.20 0 1.26610 − 1.18195i 0 1.54900 − 0.894317i 0 −2.63573 + 0.230049i 0 0.206000 − 2.99292i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bi.c 48
3.b odd 2 1 inner 672.2.bi.c 48
4.b odd 2 1 168.2.ba.c ✓ 48
7.d odd 6 1 inner 672.2.bi.c 48
8.b even 2 1 inner 672.2.bi.c 48
8.d odd 2 1 168.2.ba.c ✓ 48
12.b even 2 1 168.2.ba.c ✓ 48
21.g even 6 1 inner 672.2.bi.c 48
24.f even 2 1 168.2.ba.c ✓ 48
24.h odd 2 1 inner 672.2.bi.c 48
28.f even 6 1 168.2.ba.c ✓ 48
56.j odd 6 1 inner 672.2.bi.c 48
56.m even 6 1 168.2.ba.c ✓ 48
84.j odd 6 1 168.2.ba.c ✓ 48
168.ba even 6 1 inner 672.2.bi.c 48
168.be odd 6 1 168.2.ba.c ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.c ✓ 48 4.b odd 2 1
168.2.ba.c ✓ 48 8.d odd 2 1
168.2.ba.c ✓ 48 12.b even 2 1
168.2.ba.c ✓ 48 24.f even 2 1
168.2.ba.c ✓ 48 28.f even 6 1
168.2.ba.c ✓ 48 56.m even 6 1
168.2.ba.c ✓ 48 84.j odd 6 1
168.2.ba.c ✓ 48 168.be odd 6 1
672.2.bi.c 48 1.a even 1 1 trivial
672.2.bi.c 48 3.b odd 2 1 inner
672.2.bi.c 48 7.d odd 6 1 inner
672.2.bi.c 48 8.b even 2 1 inner
672.2.bi.c 48 21.g even 6 1 inner
672.2.bi.c 48 24.h odd 2 1 inner
672.2.bi.c 48 56.j odd 6 1 inner
672.2.bi.c 48 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 28 T_{5}^{22} + 498 T_{5}^{20} - 5472 T_{5}^{18} + 44115 T_{5}^{16} - 241512 T_{5}^{14} + \cdots + 5184 \) acting on \(S_{2}^{\mathrm{new}}(672, [\chi])\). Copy content Toggle raw display