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Properties

Label 483.2.bf.a
Level $483$
Weight $2$
Character orbit 483.bf
Analytic conductor $3.857$
Analytic rank $0$
Dimension $640$
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] = [483,2,Mod(10,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 11, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.10");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bf (of order \(66\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9} + 4 q^{18} - 28 q^{23} + 56 q^{25} - 84 q^{26} - 176 q^{28} - 24 q^{29} + 12 q^{31} + 36 q^{32} - 76 q^{35} + 28 q^{36} + 44 q^{37} - 110 q^{42} - 88 q^{43} + 154 q^{44} + 8 q^{46} + 12 q^{47} - 8 q^{49} - 212 q^{50} + 44 q^{51} + 108 q^{52} - 110 q^{56} - 88 q^{57} + 2 q^{58} - 36 q^{59} - 168 q^{64} - 48 q^{70} + 16 q^{71} + 12 q^{72} - 48 q^{73} - 22 q^{74} + 48 q^{75} + 32 q^{78} - 44 q^{79} - 594 q^{80} + 32 q^{81} + 24 q^{82} + 352 q^{85} - 36 q^{87} - 330 q^{88} + 244 q^{92} - 24 q^{93} - 486 q^{94} - 154 q^{95} - 60 q^{96} - 24 q^{98} - 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −0.853118 + 2.46492i 0.458227 + 0.888835i −3.77593 − 2.96942i −2.73649 + 0.261303i −2.58183 + 0.371212i 0.617353 + 2.57272i 6.15210 − 3.95371i −0.580057 + 0.814576i 1.69045 − 6.96815i
10.2 −0.850716 + 2.45798i −0.458227 − 0.888835i −3.74585 − 2.94577i 1.41466 − 0.135083i 2.57456 − 0.370166i −1.58058 + 2.12174i 6.05103 − 3.88876i −0.580057 + 0.814576i −0.871437 + 3.59211i
10.3 −0.838936 + 2.42395i −0.458227 − 0.888835i −3.59960 − 2.83076i −2.60640 + 0.248881i 2.53891 − 0.365041i 2.49779 − 0.872377i 5.56577 − 3.57691i −0.580057 + 0.814576i 1.58333 − 6.52656i
10.4 −0.797371 + 2.30385i 0.458227 + 0.888835i −3.09983 − 2.43773i 1.48893 − 0.142176i −2.41312 + 0.346955i 2.64327 + 0.114618i 3.98605 − 2.56168i −0.580057 + 0.814576i −0.859680 + 3.54365i
10.5 −0.647644 + 1.87125i 0.458227 + 0.888835i −1.51001 − 1.18748i −3.91898 + 0.374217i −1.96000 + 0.281805i −1.01941 − 2.44148i −0.131596 + 0.0845715i −0.580057 + 0.814576i 1.83785 − 7.57573i
10.6 −0.643746 + 1.85998i −0.458227 − 0.888835i −1.47302 − 1.15839i 2.63722 − 0.251824i 1.94820 − 0.280109i −0.893381 − 2.49036i −0.208726 + 0.134140i −0.580057 + 0.814576i −1.22931 + 5.06730i
10.7 −0.632527 + 1.82757i 0.458227 + 0.888835i −1.36781 − 1.07566i 0.843558 − 0.0805501i −1.91425 + 0.275227i −2.38264 + 1.15022i −0.422845 + 0.271746i −0.580057 + 0.814576i −0.386363 + 1.59261i
10.8 −0.544288 + 1.57262i −0.458227 − 0.888835i −0.604770 − 0.475596i 1.92390 − 0.183711i 1.64721 − 0.236832i 0.763953 + 2.53306i −1.72283 + 1.10720i −0.580057 + 0.814576i −0.758252 + 3.12556i
10.9 −0.473440 + 1.36792i −0.458227 − 0.888835i −0.0749403 − 0.0589337i −1.20704 + 0.115258i 1.43279 − 0.206005i 0.607831 − 2.57498i −2.31938 + 1.49058i −0.580057 + 0.814576i 0.413798 − 1.70570i
10.10 −0.340926 + 0.985040i 0.458227 + 0.888835i 0.718033 + 0.564667i 3.21073 − 0.306587i −1.03176 + 0.148345i −1.39165 − 2.25018i −2.55481 + 1.64188i −0.580057 + 0.814576i −0.792618 + 3.26722i
10.11 −0.319328 + 0.922636i 0.458227 + 0.888835i 0.822819 + 0.647072i 2.77405 − 0.264890i −0.966396 + 0.138947i 1.16682 + 2.37456i −2.50245 + 1.60823i −0.580057 + 0.814576i −0.641435 + 2.64403i
10.12 −0.311023 + 0.898642i −0.458227 − 0.888835i 0.861285 + 0.677322i −1.14736 + 0.109560i 0.941264 − 0.135333i −2.64155 − 0.149090i −2.47652 + 1.59156i −0.580057 + 0.814576i 0.258401 − 1.06514i
10.13 −0.184479 + 0.533017i 0.458227 + 0.888835i 1.32203 + 1.03966i −3.44259 + 0.328727i −0.558297 + 0.0802710i 0.528452 + 2.59244i −1.74704 + 1.12275i −0.580057 + 0.814576i 0.459867 − 1.89560i
10.14 −0.175281 + 0.506440i 0.458227 + 0.888835i 1.34635 + 1.05878i −1.28396 + 0.122604i −0.530461 + 0.0762687i 2.45236 − 0.992951i −1.67388 + 1.07574i −0.580057 + 0.814576i 0.162963 − 0.671742i
10.15 −0.131204 + 0.379090i −0.458227 − 0.888835i 1.44561 + 1.13684i 3.76955 − 0.359949i 0.397070 − 0.0570900i 2.39754 − 1.11885i −1.29558 + 0.832618i −0.580057 + 0.814576i −0.358129 + 1.47623i
10.16 −0.0815859 + 0.235727i −0.458227 − 0.888835i 1.52320 + 1.19785i −1.53664 + 0.146731i 0.246907 − 0.0354999i −2.59838 − 0.498437i −0.826333 + 0.531052i −0.580057 + 0.814576i 0.0907797 − 0.374199i
10.17 0.0478877 − 0.138362i 0.458227 + 0.888835i 1.55526 + 1.22307i −2.41931 + 0.231016i 0.144925 − 0.0208370i −2.60435 + 0.466234i 0.490049 − 0.314935i −0.580057 + 0.814576i −0.0838914 + 0.345805i
10.18 0.0503654 − 0.145521i −0.458227 − 0.888835i 1.55347 + 1.22166i −3.87542 + 0.370058i −0.152423 + 0.0219152i 2.22141 − 1.43713i 0.515109 − 0.331040i −0.580057 + 0.814576i −0.141336 + 0.582594i
10.19 0.0597980 − 0.172775i −0.458227 − 0.888835i 1.54583 + 1.21565i 0.371368 − 0.0354614i −0.180970 + 0.0260195i 1.77822 + 1.95906i 0.610086 − 0.392079i −0.580057 + 0.814576i 0.0160803 − 0.0662837i
10.20 0.114790 − 0.331663i 0.458227 + 0.888835i 1.47528 + 1.16017i 0.530378 − 0.0506450i 0.347394 − 0.0499477i 0.390363 − 2.61680i 1.14464 − 0.735614i −0.580057 + 0.814576i 0.0440849 − 0.181721i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
23.d odd 22 1 inner
161.o even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.bf.a ✓ 640
7.d odd 6 1 inner 483.2.bf.a ✓ 640
23.d odd 22 1 inner 483.2.bf.a ✓ 640
161.o even 66 1 inner 483.2.bf.a ✓ 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.bf.a ✓ 640 1.a even 1 1 trivial
483.2.bf.a ✓ 640 7.d odd 6 1 inner
483.2.bf.a ✓ 640 23.d odd 22 1 inner
483.2.bf.a ✓ 640 161.o even 66 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).