↑

Properties

Label 672.2.bu.a
Level $672$
Weight $2$
Character orbit 672.bu
Analytic conductor $5.366$
Analytic rank $0$
Dimension $256$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,2,Mod(139,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 5, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bu (of order \(8\), degree \(4\), 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: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q + 32 q^{14} - 8 q^{16} + 8 q^{18} - 8 q^{22} + 16 q^{23} + 40 q^{28} - 48 q^{35} + 16 q^{43} - 8 q^{44} - 48 q^{50} - 32 q^{53} + 64 q^{58} + 48 q^{60} - 144 q^{64} + 16 q^{67} + 72 q^{70} - 128 q^{71} + 232 q^{74} - 48 q^{78} + 176 q^{88} + 48 q^{91} - 152 q^{92} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 −1.41190 + 0.0809224i −0.923880 + 0.382683i 1.98690 − 0.228508i 2.29760 + 0.951697i 1.27345 − 0.615072i 2.47546 − 0.933871i −2.78681 + 0.483415i 0.707107 − 0.707107i −3.32098 − 1.15777i
139.2 −1.41190 + 0.0809224i 0.923880 − 0.382683i 1.98690 − 0.228508i −2.29760 − 0.951697i −1.27345 + 0.615072i 0.933871 − 2.47546i −2.78681 + 0.483415i 0.707107 − 0.707107i 3.32098 + 1.15777i
139.3 −1.40494 − 0.161727i −0.923880 + 0.382683i 1.94769 + 0.454432i −2.15921 − 0.894373i 1.35988 − 0.388229i −2.64571 + 0.0150051i −2.66288 − 0.953442i 0.707107 − 0.707107i 2.88890 + 1.60574i
139.4 −1.40494 − 0.161727i 0.923880 − 0.382683i 1.94769 + 0.454432i 2.15921 + 0.894373i −1.35988 + 0.388229i −0.0150051 + 2.64571i −2.66288 − 0.953442i 0.707107 − 0.707107i −2.88890 − 1.60574i
139.5 −1.37134 − 0.345600i −0.923880 + 0.382683i 1.76112 + 0.947866i 0.737744 + 0.305584i 1.39920 − 0.205495i 0.158242 + 2.64101i −2.08751 − 1.90849i 0.707107 − 0.707107i −0.906085 − 0.674022i
139.6 −1.37134 − 0.345600i 0.923880 − 0.382683i 1.76112 + 0.947866i −0.737744 − 0.305584i −1.39920 + 0.205495i −2.64101 − 0.158242i −2.08751 − 1.90849i 0.707107 − 0.707107i 0.906085 + 0.674022i
139.7 −1.32153 + 0.503553i −0.923880 + 0.382683i 1.49287 − 1.33092i −3.01870 − 1.25039i 1.02823 − 0.970949i 2.09399 + 1.61716i −1.30268 + 2.51058i 0.707107 − 0.707107i 4.61894 + 0.132344i
139.8 −1.32153 + 0.503553i 0.923880 − 0.382683i 1.49287 − 1.33092i 3.01870 + 1.25039i −1.02823 + 0.970949i −1.61716 − 2.09399i −1.30268 + 2.51058i 0.707107 − 0.707107i −4.61894 − 0.132344i
139.9 −1.29725 + 0.563147i −0.923880 + 0.382683i 1.36573 − 1.46109i −0.547161 − 0.226641i 0.982999 − 1.01672i −2.48925 + 0.896448i −0.948894 + 2.66451i 0.707107 − 0.707107i 0.837439 − 0.0141204i
139.10 −1.29725 + 0.563147i 0.923880 − 0.382683i 1.36573 − 1.46109i 0.547161 + 0.226641i −0.982999 + 1.01672i −0.896448 + 2.48925i −0.948894 + 2.66451i 0.707107 − 0.707107i −0.837439 + 0.0141204i
139.11 −1.20634 − 0.738068i −0.923880 + 0.382683i 0.910512 + 1.78072i 1.60454 + 0.664622i 1.39696 + 0.220239i −1.04060 − 2.43252i 0.215905 − 2.82017i 0.707107 − 0.707107i −1.44508 − 1.98602i
139.12 −1.20634 − 0.738068i 0.923880 − 0.382683i 0.910512 + 1.78072i −1.60454 − 0.664622i −1.39696 − 0.220239i 2.43252 + 1.04060i 0.215905 − 2.82017i 0.707107 − 0.707107i 1.44508 + 1.98602i
139.13 −1.13995 − 0.836970i −0.923880 + 0.382683i 0.598964 + 1.90820i 0.524702 + 0.217339i 1.37347 + 0.337020i 1.65629 + 2.06318i 0.914321 − 2.67657i 0.707107 − 0.707107i −0.416227 − 0.686915i
139.14 −1.13995 − 0.836970i 0.923880 − 0.382683i 0.598964 + 1.90820i −0.524702 − 0.217339i −1.37347 − 0.337020i −2.06318 − 1.65629i 0.914321 − 2.67657i 0.707107 − 0.707107i 0.416227 + 0.686915i
139.15 −1.01364 − 0.986171i −0.923880 + 0.382683i 0.0549322 + 1.99925i −3.53807 − 1.46552i 1.31387 + 0.523200i 1.98161 − 1.75307i 1.91592 − 2.08069i 0.707107 − 0.707107i 2.14108 + 4.97465i
139.16 −1.01364 − 0.986171i 0.923880 − 0.382683i 0.0549322 + 1.99925i 3.53807 + 1.46552i −1.31387 − 0.523200i 1.75307 − 1.98161i 1.91592 − 2.08069i 0.707107 − 0.707107i −2.14108 − 4.97465i
139.17 −0.963992 + 1.03476i −0.923880 + 0.382683i −0.141437 − 1.99499i −0.663808 − 0.274958i 0.494629 − 1.32489i 1.03097 − 2.43662i 2.20067 + 1.77681i 0.707107 − 0.707107i 0.924420 − 0.421821i
139.18 −0.963992 + 1.03476i 0.923880 − 0.382683i −0.141437 − 1.99499i 0.663808 + 0.274958i −0.494629 + 1.32489i 2.43662 − 1.03097i 2.20067 + 1.77681i 0.707107 − 0.707107i −0.924420 + 0.421821i
139.19 −0.918736 + 1.07514i −0.923880 + 0.382683i −0.311849 − 1.97554i 2.17765 + 0.902011i 0.437363 − 1.34488i −2.57812 + 0.594399i 2.41049 + 1.47972i 0.707107 − 0.707107i −2.97047 + 1.51256i
139.20 −0.918736 + 1.07514i 0.923880 − 0.382683i −0.311849 − 1.97554i −2.17765 − 0.902011i −0.437363 + 1.34488i −0.594399 + 2.57812i 2.41049 + 1.47972i 0.707107 − 0.707107i 2.97047 − 1.51256i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
32.h odd 8 1 inner
224.x even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.2.bu.a ✓ 256
7.b odd 2 1 inner 672.2.bu.a ✓ 256
32.h odd 8 1 inner 672.2.bu.a ✓ 256
224.x even 8 1 inner 672.2.bu.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.bu.a ✓ 256 1.a even 1 1 trivial
672.2.bu.a ✓ 256 7.b odd 2 1 inner
672.2.bu.a ✓ 256 32.h odd 8 1 inner
672.2.bu.a ✓ 256 224.x even 8 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(672, [\chi])\).