↑

Properties

Label 333.3.r.a
Level $333$
Weight $3$
Character orbit 333.r
Analytic conductor $9.074$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [333,3,Mod(38,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.38");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.r (of order \(6\), 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: \(9.07359280320\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(72\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{3} + 144 q^{4} - 30 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 4 q^{3} + 144 q^{4} - 30 q^{6} + 4 q^{9} - 12 q^{12} + 72 q^{14} - 18 q^{15} - 288 q^{16} - 90 q^{18} - 24 q^{19} + 32 q^{21} - 24 q^{22} + 144 q^{23} - 48 q^{24} + 360 q^{25} - 50 q^{27} - 216 q^{29} + 28 q^{30} + 36 q^{32} - 110 q^{33} + 60 q^{34} - 10 q^{36} + 36 q^{38} + 88 q^{39} - 60 q^{40} + 108 q^{41} + 278 q^{42} - 60 q^{43} + 64 q^{45} - 216 q^{46} + 90 q^{47} - 238 q^{48} - 552 q^{49} - 522 q^{50} + 90 q^{51} - 18 q^{52} + 216 q^{54} + 48 q^{55} + 432 q^{56} - 264 q^{57} + 138 q^{58} - 270 q^{59} - 458 q^{60} + 96 q^{61} + 148 q^{63} - 636 q^{64} - 54 q^{65} - 224 q^{66} + 84 q^{67} - 72 q^{68} + 410 q^{69} - 216 q^{70} - 636 q^{72} - 72 q^{73} + 344 q^{75} + 84 q^{76} + 432 q^{77} - 384 q^{78} + 108 q^{79} + 556 q^{81} - 204 q^{82} - 180 q^{83} - 308 q^{84} + 60 q^{85} + 72 q^{86} + 126 q^{87} + 168 q^{88} - 206 q^{90} + 168 q^{91} - 36 q^{92} + 70 q^{93} - 186 q^{94} - 864 q^{95} + 932 q^{96} - 180 q^{97} + 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −3.39576 − 1.96054i 2.72846 + 1.24720i 5.68747 + 9.85098i −7.85244 + 4.53361i −6.82002 − 9.58445i −4.91855 + 8.51919i − 28.9178i 5.88900 + 6.80586i 35.5534
38.2 −3.33778 − 1.92707i −2.99715 − 0.130794i 5.42719 + 9.40016i −0.798677 + 0.461117i 9.75177 + 6.21227i −0.0140441 + 0.0243250i − 26.4177i 8.96579 + 0.784016i 3.55441
38.3 −3.26513 − 1.88512i 0.779058 − 2.89708i 5.10739 + 8.84625i 3.64869 − 2.10657i −8.00508 + 7.99072i −4.06371 + 7.03855i − 23.4312i −7.78614 − 4.51398i −15.8846
38.4 −3.21682 − 1.85723i 2.98937 − 0.252355i 4.89860 + 8.48463i 4.79142 − 2.76633i −10.0849 − 4.74016i 4.68860 − 8.12090i − 21.5335i 8.87263 − 1.50876i −20.5508
38.5 −3.13647 − 1.81084i 0.123646 + 2.99745i 4.55830 + 7.89521i 7.66187 − 4.42358i 5.04010 − 9.62532i −4.11702 + 7.13088i − 18.5307i −8.96942 + 0.741247i −32.0416
38.6 −3.05082 − 1.76139i 1.42878 − 2.63792i 4.20499 + 7.28326i −5.85593 + 3.38092i −9.00533 + 5.53117i 5.94587 − 10.2986i − 15.5354i −4.91720 − 7.53798i 23.8205
38.7 −2.91757 − 1.68446i −1.41472 − 2.64548i 3.67480 + 6.36493i −0.358825 + 0.207168i −0.328662 + 10.1014i 1.95946 − 3.39388i − 11.2845i −4.99714 + 7.48523i 1.39586
38.8 −2.85115 − 1.64611i −1.28589 + 2.71044i 3.41938 + 5.92254i −5.35542 + 3.09195i 8.12796 − 5.61116i 2.85118 − 4.93840i − 9.34584i −5.69297 − 6.97066i 20.3588
38.9 −2.79455 − 1.61344i −2.00521 + 2.23140i 3.20635 + 5.55356i −0.354063 + 0.204419i 9.20388 − 3.00048i −4.79224 + 8.30040i − 7.78547i −0.958274 − 8.94884i 1.31927
38.10 −2.72612 − 1.57393i −2.63496 + 1.43421i 2.95448 + 5.11732i 7.79669 − 4.50142i 9.44057 + 0.237408i 6.11731 − 10.5955i − 6.00915i 4.88608 − 7.55819i −28.3396
38.11 −2.69621 − 1.55666i 2.54540 + 1.58774i 2.84637 + 4.93006i 1.33123 − 0.768586i −4.39138 − 8.24321i 1.60699 − 2.78338i − 5.27004i 3.95817 + 8.08288i −4.78570
38.12 −2.61598 − 1.51034i 1.43976 + 2.63194i 2.56222 + 4.43790i −0.342383 + 0.197675i 0.208718 − 9.05961i 0.615742 − 1.06650i − 3.39659i −4.85417 + 7.57872i 1.19422
38.13 −2.60837 − 1.50594i −1.83691 − 2.37187i 2.53572 + 4.39199i 4.12121 − 2.37938i 1.21944 + 8.95298i 1.55352 − 2.69077i − 3.22702i −2.25151 + 8.71382i −14.3328
38.14 −2.48524 − 1.43485i 2.59682 − 1.50217i 2.11760 + 3.66779i 0.333280 − 0.192419i −8.60911 + 0.00718270i −4.73790 + 8.20628i − 0.674958i 4.48699 − 7.80173i −1.10437
38.15 −2.45006 − 1.41454i −0.985391 − 2.83355i 2.00185 + 3.46731i −6.74188 + 3.89243i −1.59391 + 8.33624i −4.88663 + 8.46390i − 0.0104923i −7.05801 + 5.58431i 22.0240
38.16 −2.28078 − 1.31681i 2.69783 − 1.31214i 1.46798 + 2.54262i −3.77668 + 2.18047i −7.88101 − 0.559825i 0.802344 − 1.38970i 2.80227i 5.55658 − 7.07986i 11.4851
38.17 −2.10009 − 1.21249i −2.99262 − 0.210365i 0.940265 + 1.62859i −1.80895 + 1.04440i 6.02971 + 4.07030i −5.72627 + 9.91820i 5.13968i 8.91149 + 1.25908i 5.06530
38.18 −1.96391 − 1.13386i −2.98650 − 0.284243i 0.571289 + 0.989501i −6.19349 + 3.57581i 5.54293 + 3.94451i 3.85304 − 6.67366i 6.47985i 8.83841 + 1.69778i 16.2179
38.19 −1.95598 − 1.12928i 1.24781 − 2.72818i 0.550561 + 0.953600i 6.80210 − 3.92719i −5.52158 + 3.92712i 0.996846 − 1.72659i 6.54731i −5.88592 − 6.80852i −17.7396
38.20 −1.80520 − 1.04223i −2.99841 + 0.0977095i 0.172506 + 0.298789i 4.31462 − 2.49105i 5.51457 + 2.94866i −0.990898 + 1.71629i 7.61871i 8.98091 − 0.585946i −10.3850
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.r.a ✓ 144
9.d odd 6 1 inner 333.3.r.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.3.r.a ✓ 144 1.a even 1 1 trivial
333.3.r.a ✓ 144 9.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).