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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Newspace parameters

Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

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.

\(\operatorname{Tr}(f)(q) = \) \( 640q - 4q^{2} + 36q^{4} + 24q^{8} - 32q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 640q - 4q^{2} + 36q^{4} + 24q^{8} - 32q^{9} + 4q^{18} - 28q^{23} + 56q^{25} - 84q^{26} - 176q^{28} - 24q^{29} + 12q^{31} + 36q^{32} - 76q^{35} + 28q^{36} + 44q^{37} - 110q^{42} - 88q^{43} + 154q^{44} + 8q^{46} + 12q^{47} - 8q^{49} - 212q^{50} + 44q^{51} + 108q^{52} - 110q^{56} - 88q^{57} + 2q^{58} - 36q^{59} - 168q^{64} - 48q^{70} + 16q^{71} + 12q^{72} - 48q^{73} - 22q^{74} + 48q^{75} + 32q^{78} - 44q^{79} - 594q^{80} + 32q^{81} + 24q^{82} + 352q^{85} - 36q^{87} - 330q^{88} + 244q^{92} - 24q^{93} - 486q^{94} - 154q^{95} - 60q^{96} - 24q^{98} - 44q^{99} + O(q^{100}) \)

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.

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 481.32
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 \(S_{2}^{\mathrm{new}}(483, [\chi])\).