Properties

Label 671.2.f.a
Level $671$
Weight $2$
Character orbit 671.f
Analytic conductor $5.358$
Analytic rank $0$
Dimension $120$
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] = [671,2,Mod(538,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.538");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.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: \(5.35796197563\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 120 q^{9} - 4 q^{11} + 16 q^{12} + 16 q^{15} - 148 q^{16} + 56 q^{20} - 4 q^{22} - 4 q^{23} - 104 q^{25} + 40 q^{26} - 2 q^{33} - 8 q^{34} - 12 q^{37} + 20 q^{38} + 16 q^{42} - 10 q^{44} - 4 q^{53} + 50 q^{55} - 24 q^{56} + 64 q^{58} - 56 q^{67} + 68 q^{69} + 144 q^{70} - 12 q^{71} - 64 q^{77} + 84 q^{78} + 72 q^{81} + 40 q^{82} - 80 q^{86} + 4 q^{89} - 4 q^{91} + 4 q^{92} + 64 q^{93} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
538.1 −1.91401 1.91401i 0.380921i 5.32688i 1.26088i −0.729087 + 0.729087i −2.25350 2.25350i 6.36769 6.36769i 2.85490 2.41334 2.41334i
538.2 −1.91322 1.91322i 0.958783i 5.32085i 2.33392i −1.83437 + 1.83437i −0.592146 0.592146i 6.35353 6.35353i 2.08073 −4.46532 + 4.46532i
538.3 −1.89459 1.89459i 0.936891i 5.17893i 2.40716i 1.77502 1.77502i 3.07176 + 3.07176i 6.02277 6.02277i 2.12224 4.56057 4.56057i
538.4 −1.88881 1.88881i 3.04380i 5.13522i 3.85384i 5.74917 5.74917i −1.76642 1.76642i 5.92184 5.92184i −6.26473 −7.27917 + 7.27917i
538.5 −1.73949 1.73949i 3.20201i 4.05165i 0.618387i −5.56987 + 5.56987i 2.25367 + 2.25367i 3.56882 3.56882i −7.25288 −1.07568 + 1.07568i
538.6 −1.73598 1.73598i 2.48629i 4.02728i 3.17645i −4.31616 + 4.31616i 0.00734923 + 0.00734923i 3.51933 3.51933i −3.18164 5.51426 5.51426i
538.7 −1.70917 1.70917i 2.11174i 3.84252i 0.00911084i 3.60931 3.60931i 1.54773 + 1.54773i 3.14918 3.14918i −1.45943 −0.0155720 + 0.0155720i
538.8 −1.56726 1.56726i 2.67240i 2.91258i 1.99974i 4.18834 4.18834i −1.05898 1.05898i 1.43025 1.43025i −4.14173 3.13411 3.13411i
538.9 −1.52526 1.52526i 2.50389i 2.65282i 3.63890i −3.81907 + 3.81907i −2.43657 2.43657i 0.995712 0.995712i −3.26945 −5.55025 + 5.55025i
538.10 −1.49884 1.49884i 0.414070i 2.49306i 2.38044i 0.620625 0.620625i 0.200846 + 0.200846i 0.739017 0.739017i 2.82855 −3.56790 + 3.56790i
538.11 −1.35252 1.35252i 1.50607i 1.65860i 1.33418i −2.03698 + 2.03698i −0.596912 0.596912i −0.461747 + 0.461747i 0.731760 1.80450 1.80450i
538.12 −1.31077 1.31077i 1.31199i 1.43623i 0.632792i −1.71972 + 1.71972i 2.27656 + 2.27656i −0.738974 + 0.738974i 1.27868 0.829444 0.829444i
538.13 −1.30522 1.30522i 1.39984i 1.40720i 3.41385i 1.82710 1.82710i 1.63770 + 1.63770i −0.773736 + 0.773736i 1.04045 −4.45582 + 4.45582i
538.14 −1.29221 1.29221i 0.775410i 1.33963i 0.709411i 1.00199 1.00199i −2.04981 2.04981i −0.853342 + 0.853342i 2.39874 −0.916710 + 0.916710i
538.15 −1.06517 1.06517i 1.66434i 0.269167i 3.94941i 1.77280 1.77280i 0.741657 + 0.741657i −1.84363 + 1.84363i 0.229976 4.20679 4.20679i
538.16 −1.02458 1.02458i 2.81698i 0.0995254i 1.35679i −2.88621 + 2.88621i −3.70381 3.70381i −1.94719 + 1.94719i −4.93535 1.39014 1.39014i
538.17 −0.903249 0.903249i 2.65283i 0.368284i 3.20713i −2.39617 + 2.39617i 1.19454 + 1.19454i −2.13915 + 2.13915i −4.03751 −2.89684 + 2.89684i
538.18 −0.777546 0.777546i 0.433435i 0.790844i 3.54542i 0.337016 0.337016i −2.52725 2.52725i −2.17001 + 2.17001i 2.81213 2.75673 2.75673i
538.19 −0.774186 0.774186i 1.40798i 0.801272i 0.371549i −1.09004 + 1.09004i 3.30218 + 3.30218i −2.16871 + 2.16871i 1.01760 −0.287648 + 0.287648i
538.20 −0.756136 0.756136i 0.596334i 0.856517i 2.30668i 0.450910 0.450910i 2.13477 + 2.13477i −2.15992 + 2.15992i 2.64439 1.74417 1.74417i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 538.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
61.d odd 4 1 inner
671.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.f.a 120
11.b odd 2 1 inner 671.2.f.a 120
61.d odd 4 1 inner 671.2.f.a 120
671.f even 4 1 inner 671.2.f.a 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.f.a 120 1.a even 1 1 trivial
671.2.f.a 120 11.b odd 2 1 inner
671.2.f.a 120 61.d odd 4 1 inner
671.2.f.a 120 671.f even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).