↑

Properties

Label 670.2.f.a
Level $670$
Weight $2$
Character orbit 670.f
Analytic conductor $5.350$
Analytic rank $0$
Dimension $68$
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] = [670,2,Mod(133,670)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(670, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("670.133");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 670 = 2 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 670.f (of order \(4\), degree \(2\), 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: \(5.34997693543\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q + 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q + 8 q^{6} + 8 q^{15} - 68 q^{16} - 16 q^{21} + 8 q^{23} - 24 q^{25} + 24 q^{33} - 24 q^{35} + 76 q^{36} - 48 q^{37} + 72 q^{47} - 8 q^{56} - 24 q^{60} + 40 q^{62} - 72 q^{65} - 4 q^{67} + 64 q^{71} + 32 q^{73} - 32 q^{76} - 64 q^{77} - 20 q^{81} - 48 q^{82} + 8 q^{83} + 40 q^{86} - 32 q^{90} + 64 q^{91} - 8 q^{92} - 72 q^{93} - 8 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
133.1 −0.707107 − 0.707107i −2.40287 + 2.40287i 1.00000i −2.05487 + 0.881768i 3.39817 1.04520 + 1.04520i 0.707107 − 0.707107i − 8.54758i 2.07652 + 0.829508i
133.2 −0.707107 − 0.707107i −1.97542 + 1.97542i 1.00000i 1.76545 + 1.37229i 2.79367 −2.25070 − 2.25070i 0.707107 − 0.707107i − 4.80460i −0.278005 − 2.21872i
133.3 −0.707107 − 0.707107i −1.94425 + 1.94425i 1.00000i −0.148407 − 2.23114i 2.74959 −3.24195 − 3.24195i 0.707107 − 0.707107i − 4.56023i −1.47271 + 1.68259i
133.4 −0.707107 − 0.707107i −1.67029 + 1.67029i 1.00000i 2.18051 − 0.495376i 2.36215 3.73163 + 3.73163i 0.707107 − 0.707107i − 2.57975i −1.89213 − 1.19157i
133.5 −0.707107 − 0.707107i −1.09147 + 1.09147i 1.00000i 1.74910 − 1.39307i 1.54358 −0.382963 − 0.382963i 0.707107 − 0.707107i 0.617367i −2.22185 − 0.251755i
133.6 −0.707107 − 0.707107i −1.07854 + 1.07854i 1.00000i −0.999328 − 2.00034i 1.52529 0.471925 + 0.471925i 0.707107 − 0.707107i 0.673505i −0.707819 + 2.12108i
133.7 −0.707107 − 0.707107i −0.942775 + 0.942775i 1.00000i −0.386645 + 2.20239i 1.33329 1.72769 + 1.72769i 0.707107 − 0.707107i 1.22235i 1.83072 − 1.28392i
133.8 −0.707107 − 0.707107i −0.499940 + 0.499940i 1.00000i −2.17343 + 0.525549i 0.707023 −1.40942 − 1.40942i 0.707107 − 0.707107i 2.50012i 1.90847 + 1.16523i
133.9 −0.707107 − 0.707107i 0.123434 − 0.123434i 1.00000i 1.33097 + 1.79681i −0.174562 1.79292 + 1.79292i 0.707107 − 0.707107i 2.96953i 0.329398 − 2.21167i
133.10 −0.707107 − 0.707107i 0.550030 − 0.550030i 1.00000i 0.390907 + 2.20163i −0.777860 −3.28527 − 3.28527i 0.707107 − 0.707107i 2.39493i 1.28038 − 1.83320i
133.11 −0.707107 − 0.707107i 0.696578 − 0.696578i 1.00000i −1.22597 − 1.87002i −0.985110 3.04344 + 3.04344i 0.707107 − 0.707107i 2.02956i −0.455414 + 2.18920i
133.12 −0.707107 − 0.707107i 0.756584 − 0.756584i 1.00000i −1.81580 − 1.30494i −1.06997 −1.21737 − 1.21737i 0.707107 − 0.707107i 1.85516i 0.361228 + 2.20670i
133.13 −0.707107 − 0.707107i 0.828562 − 0.828562i 1.00000i 0.752785 − 2.10554i −1.17176 −0.156315 − 0.156315i 0.707107 − 0.707107i 1.62697i −2.02114 + 0.956545i
133.14 −0.707107 − 0.707107i 1.68277 − 1.68277i 1.00000i 1.50584 − 1.65301i −2.37980 −3.16109 − 3.16109i 0.707107 − 0.707107i − 2.66344i −2.23365 + 0.104060i
133.15 −0.707107 − 0.707107i 1.71991 − 1.71991i 1.00000i 2.10215 + 0.762204i −2.43232 1.07341 + 1.07341i 0.707107 − 0.707107i − 2.91619i −0.947486 − 2.02541i
133.16 −0.707107 − 0.707107i 1.74202 − 1.74202i 1.00000i −1.65230 + 1.50663i −2.46359 1.95490 + 1.95490i 0.707107 − 0.707107i − 3.06928i 2.23369 + 0.103004i
133.17 −0.707107 − 0.707107i 2.09146 − 2.09146i 1.00000i −1.32097 + 1.80417i −2.95777 −1.15023 − 1.15023i 0.707107 − 0.707107i − 5.74842i 2.20981 − 0.341673i
133.18 0.707107 + 0.707107i −2.09146 + 2.09146i 1.00000i 1.32097 − 1.80417i −2.95777 1.15023 + 1.15023i −0.707107 + 0.707107i − 5.74842i 2.20981 − 0.341673i
133.19 0.707107 + 0.707107i −1.74202 + 1.74202i 1.00000i 1.65230 − 1.50663i −2.46359 −1.95490 − 1.95490i −0.707107 + 0.707107i − 3.06928i 2.23369 + 0.103004i
133.20 0.707107 + 0.707107i −1.71991 + 1.71991i 1.00000i −2.10215 − 0.762204i −2.43232 −1.07341 − 1.07341i −0.707107 + 0.707107i − 2.91619i −0.947486 − 2.02541i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 133.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
67.b odd 2 1 inner
335.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 670.2.f.a ✓ 68
5.c odd 4 1 inner 670.2.f.a ✓ 68
67.b odd 2 1 inner 670.2.f.a ✓ 68
335.f even 4 1 inner 670.2.f.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
670.2.f.a ✓ 68 1.a even 1 1 trivial
670.2.f.a ✓ 68 5.c odd 4 1 inner
670.2.f.a ✓ 68 67.b odd 2 1 inner
670.2.f.a ✓ 68 335.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(670, [\chi])\).