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Properties

Label 671.2.bw.a
Level $671$
Weight $2$
Character orbit 671.bw
Analytic conductor $5.358$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [671,2,Mod(98,671)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(671, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("671.98");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 671 = 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 671.bw (of order \(20\), degree \(8\), 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: \(480\)
Relative dimension: \(60\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 20 q^{3} - 20 q^{4} - 20 q^{5} + 100 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 20 q^{3} - 20 q^{4} - 20 q^{5} + 100 q^{9} - 6 q^{11} - 36 q^{12} - 36 q^{15} + 128 q^{16} + 4 q^{20} - 6 q^{22} - 96 q^{23} + 84 q^{25} - 20 q^{26} - 80 q^{27} - 40 q^{31} - 8 q^{33} - 12 q^{34} - 20 q^{36} - 8 q^{37} - 20 q^{38} + 24 q^{42} - 30 q^{44} + 80 q^{45} - 20 q^{49} - 16 q^{53} - 20 q^{55} - 36 q^{56} - 4 q^{58} - 20 q^{59} - 20 q^{64} - 130 q^{66} - 124 q^{67} - 8 q^{69} - 64 q^{70} + 72 q^{71} + 64 q^{77} + 16 q^{78} - 320 q^{80} - 92 q^{81} - 60 q^{82} - 20 q^{86} - 170 q^{88} - 4 q^{89} + 44 q^{91} + 216 q^{92} - 4 q^{93} - 20 q^{97} - 154 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1 −2.48860 − 1.26801i −0.468497 + 0.152224i 3.40974 + 4.69310i 0.186560 + 0.256778i 1.35893 + 0.215233i 0.874200 + 1.71571i −1.66075 − 10.4856i −2.23073 + 1.62072i −0.138678 − 0.875577i
98.2 −2.41178 − 1.22886i 2.07708 − 0.674883i 3.13099 + 4.30944i −1.52572 − 2.09998i −5.83878 − 0.924772i 0.426803 + 0.837648i −1.40867 − 8.89401i 1.43173 − 1.04021i 1.09912 + 6.93957i
98.3 −2.23979 − 1.14123i −1.41006 + 0.458156i 2.53867 + 3.49417i −1.50993 − 2.07824i 3.68109 + 0.583028i −1.41798 − 2.78294i −0.911939 − 5.75776i −0.648687 + 0.471299i 1.01017 + 6.37798i
98.4 −2.15546 − 1.09826i −1.48696 + 0.483143i 2.26425 + 3.11648i 1.43058 + 1.96903i 3.73570 + 0.591677i 0.147708 + 0.289894i −0.700928 − 4.42548i −0.449421 + 0.326523i −0.921056 − 5.81532i
98.5 −2.14272 − 1.09177i 1.90221 − 0.618064i 2.22372 + 3.06069i 0.387835 + 0.533808i −4.75068 − 0.752434i −0.469341 − 0.921134i −0.670845 − 4.23555i 0.809335 − 0.588016i −0.248225 − 1.56723i
98.6 −2.12020 − 1.08029i 1.44369 − 0.469082i 2.15263 + 2.96284i 1.58283 + 2.17858i −3.56765 − 0.565060i 1.57322 + 3.08761i −0.618773 − 3.90678i −0.562857 + 0.408939i −1.00241 − 6.32895i
98.7 −2.05137 − 1.04522i −2.42667 + 0.788474i 1.94005 + 2.67025i −2.15303 − 2.96339i 5.80214 + 0.918968i −0.102778 − 0.201713i −0.468428 − 2.95754i 2.84000 − 2.06338i 1.31925 + 8.32940i
98.8 −2.02197 − 1.03024i −3.18205 + 1.03391i 1.85138 + 2.54821i 1.84897 + 2.54488i 7.49918 + 1.18775i −1.97207 − 3.87041i −0.408164 − 2.57705i 6.62941 − 4.81655i −1.11670 − 7.05056i
98.9 −1.99089 − 1.01441i 1.03710 − 0.336973i 1.75905 + 2.42112i −0.856452 − 1.17881i −2.40658 − 0.381164i −1.85571 − 3.64203i −0.346980 − 2.19075i −1.46503 + 1.06441i 0.509312 + 3.21567i
98.10 −1.84687 − 0.941029i −2.17884 + 0.707948i 1.34984 + 1.85789i −0.0779410 − 0.107277i 4.69024 + 0.742862i 1.68706 + 3.31104i −0.0961350 − 0.606972i 1.81911 − 1.32166i 0.0429968 + 0.271471i
98.11 −1.74735 − 0.890321i 2.46689 − 0.801542i 1.08500 + 1.49337i −2.53296 − 3.48632i −5.02416 − 0.795749i 0.332296 + 0.652168i 0.0472716 + 0.298461i 3.01604 − 2.19128i 1.32203 + 8.34698i
98.12 −1.70563 − 0.869060i 0.00252967 0.000821940i 0.978327 + 1.34655i −0.840861 − 1.15735i −0.00502899 0.000796514i 2.11792 + 4.15666i 0.100489 + 0.634464i −2.42705 + 1.76335i 0.428392 + 2.70476i
98.13 −1.62952 − 0.830281i 1.26524 − 0.411100i 0.790393 + 1.08788i 2.48239 + 3.41671i −2.40305 − 0.380606i 0.388582 + 0.762635i 0.187479 + 1.18370i −0.995232 + 0.723078i −1.20826 − 7.62868i
98.14 −1.43787 − 0.732630i 0.306682 − 0.0996469i 0.355147 + 0.488818i −1.47103 − 2.02469i −0.513972 − 0.0814052i 0.394499 + 0.774247i 0.352363 + 2.22473i −2.34293 + 1.70224i 0.631789 + 3.98896i
98.15 −1.43559 − 0.731472i 1.33143 − 0.432609i 0.350309 + 0.482159i 0.706009 + 0.971738i −2.22784 − 0.352855i −1.64721 − 3.23282i 0.353880 + 2.23431i −0.841484 + 0.611374i −0.302743 − 1.91145i
98.16 −1.36503 − 0.695519i −0.524500 + 0.170420i 0.203996 + 0.280776i 1.20659 + 1.66073i 0.834490 + 0.132170i −0.448015 − 0.879279i 0.396143 + 2.50115i −2.18099 + 1.58458i −0.491967 − 3.10616i
98.17 −1.27921 − 0.651792i −1.55213 + 0.504317i 0.0359835 + 0.0495271i −0.568553 − 0.782546i 2.31421 + 0.366536i −0.936934 − 1.83884i 0.435435 + 2.74923i −0.272281 + 0.197824i 0.217243 + 1.37162i
98.18 −1.22315 − 0.623227i 3.09508 − 1.00565i −0.0678803 − 0.0934292i 0.326859 + 0.449883i −4.41250 − 0.698872i −1.41286 − 2.77289i 0.454299 + 2.86833i 6.14113 − 4.46179i −0.119419 − 0.753984i
98.19 −1.18411 − 0.603334i 2.41859 − 0.785848i −0.137466 − 0.189205i 0.0804878 + 0.110782i −3.33801 − 0.528688i 1.85019 + 3.63121i 0.464410 + 2.93217i 2.80497 − 2.03793i −0.0284679 − 0.179739i
98.20 −1.06364 − 0.541952i −2.47514 + 0.804221i −0.337952 − 0.465151i 2.00512 + 2.75981i 3.06851 + 0.486004i 1.32683 + 2.60405i 0.480858 + 3.03602i 3.05249 − 2.21776i −0.637042 − 4.02213i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 98.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
61.j odd 20 1 inner
671.bw even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 671.2.bw.a ✓ 480
11.b odd 2 1 inner 671.2.bw.a ✓ 480
61.j odd 20 1 inner 671.2.bw.a ✓ 480
671.bw even 20 1 inner 671.2.bw.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
671.2.bw.a ✓ 480 1.a even 1 1 trivial
671.2.bw.a ✓ 480 11.b odd 2 1 inner
671.2.bw.a ✓ 480 61.j odd 20 1 inner
671.2.bw.a ✓ 480 671.bw even 20 1 inner

Hecke kernels

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