↑

Properties

Label 633.2.r.b
Level $633$
Weight $2$
Character orbit 633.r
Analytic conductor $5.055$
Analytic rank $0$
Dimension $216$
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] = [633,2,Mod(34,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 40]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.34");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.r (of order \(21\), degree \(12\), 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: \(5.05453044795\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(18\) over \(\Q(\zeta_{21})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 18 q^{3} - 4 q^{4} - 4 q^{5} - 6 q^{6} - q^{7} - 10 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 18 q^{3} - 4 q^{4} - 4 q^{5} - 6 q^{6} - q^{7} - 10 q^{8} + 18 q^{9} - 18 q^{10} + 4 q^{11} - 41 q^{12} - 6 q^{13} - 22 q^{14} - 12 q^{15} - 24 q^{16} - 19 q^{17} - 2 q^{18} - 8 q^{19} - 5 q^{20} + 6 q^{21} - 20 q^{22} - 28 q^{23} + 5 q^{24} - 8 q^{25} - 19 q^{26} - 36 q^{27} + 90 q^{28} - 63 q^{29} + 66 q^{30} - 2 q^{31} + 54 q^{32} - 16 q^{33} - 36 q^{34} + 46 q^{35} + 10 q^{36} + 14 q^{37} + 35 q^{38} + 17 q^{39} - 18 q^{40} - 8 q^{41} + 30 q^{42} - 24 q^{43} - 65 q^{44} + 2 q^{45} + 157 q^{46} - 59 q^{47} + 53 q^{48} - 43 q^{49} - 56 q^{50} - 19 q^{51} + 11 q^{52} - 16 q^{53} + q^{54} + 88 q^{55} - 172 q^{56} - 36 q^{57} + 50 q^{58} + 39 q^{59} + 10 q^{60} - 45 q^{61} - 35 q^{62} + 2 q^{63} + 64 q^{64} - 78 q^{65} - 6 q^{66} + 51 q^{67} - 109 q^{68} - 11 q^{70} + 34 q^{71} + 40 q^{72} - 26 q^{73} - 50 q^{74} + 18 q^{75} + 106 q^{76} + 44 q^{77} - 12 q^{78} + 36 q^{79} + 105 q^{80} + 18 q^{81} + 77 q^{82} - 27 q^{83} + 4 q^{84} + 105 q^{85} - 22 q^{86} + 29 q^{88} + 50 q^{89} - 27 q^{90} - 141 q^{91} + 31 q^{92} - 2 q^{93} + 210 q^{94} - 72 q^{95} + 158 q^{96} - 36 q^{97} - 37 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
34.1 −2.70711 + 0.408031i −0.988831 + 0.149042i 5.25080 − 1.61966i 0.806004 − 1.01070i 2.61606 − 0.806947i 0.928929 + 2.36687i −8.62049 + 4.15141i 0.955573 − 0.294755i −1.76954 + 3.06494i
34.2 −2.36864 + 0.357015i −0.988831 + 0.149042i 3.57186 − 1.10177i −1.89560 + 2.37701i 2.28898 − 0.706056i 0.918597 + 2.34055i −3.75076 + 1.80627i 0.955573 − 0.294755i 3.64137 − 6.30704i
34.3 −1.89005 + 0.284879i −0.988831 + 0.149042i 1.57997 − 0.487357i 0.149741 − 0.187769i 1.82648 − 0.563393i −0.195990 − 0.499374i 0.596834 − 0.287420i 0.955573 − 0.294755i −0.229526 + 0.397550i
34.4 −1.87792 + 0.283050i −0.988831 + 0.149042i 1.53530 − 0.473579i 0.630079 − 0.790094i 1.81475 − 0.559778i −0.515649 − 1.31385i 0.672983 − 0.324091i 0.955573 − 0.294755i −0.959599 + 1.66207i
34.5 −1.52304 + 0.229562i −0.988831 + 0.149042i 0.355819 − 0.109756i 2.44458 − 3.06540i 1.47182 − 0.453996i 1.56762 + 3.99423i 2.25870 − 1.08773i 0.955573 − 0.294755i −3.01950 + 5.22993i
34.6 −1.11008 + 0.167317i −0.988831 + 0.149042i −0.706873 + 0.218042i −2.00738 + 2.51717i 1.07274 − 0.330896i −1.23919 − 3.15740i 2.77108 − 1.33448i 0.955573 − 0.294755i 1.80717 − 3.13012i
34.7 −0.932508 + 0.140553i −0.988831 + 0.149042i −1.06133 + 0.327377i −1.77298 + 2.22325i 0.901144 − 0.277966i 0.409499 + 1.04339i 2.64299 − 1.27279i 0.955573 − 0.294755i 1.34084 − 2.32240i
34.8 −0.533114 + 0.0803540i −0.988831 + 0.149042i −1.63339 + 0.503835i 2.54759 − 3.19458i 0.515183 − 0.158913i −1.71123 − 4.36013i 1.80179 − 0.867695i 0.955573 − 0.294755i −1.10146 + 1.90778i
34.9 −0.276377 + 0.0416572i −0.988831 + 0.149042i −1.83650 + 0.566484i −0.788144 + 0.988302i 0.267082 − 0.0823838i 1.64456 + 4.19028i 0.987607 − 0.475607i 0.955573 − 0.294755i 0.176655 − 0.305976i
34.10 0.347985 − 0.0524504i −0.988831 + 0.149042i −1.79280 + 0.553006i −1.14762 + 1.43907i −0.336281 + 0.103729i −0.115771 − 0.294980i −1.22899 + 0.591852i 0.955573 − 0.294755i −0.323875 + 0.560968i
34.11 0.686163 − 0.103422i −0.988831 + 0.149042i −1.45102 + 0.447581i 0.903249 − 1.13264i −0.663085 + 0.204535i 0.495508 + 1.26254i −2.19974 + 1.05934i 0.955573 − 0.294755i 0.502636 − 0.870591i
34.12 0.731587 − 0.110269i −0.988831 + 0.149042i −1.38808 + 0.428168i −0.490492 + 0.615057i −0.706981 + 0.218075i −1.85259 − 4.72033i −2.30146 + 1.10832i 0.955573 − 0.294755i −0.291016 + 0.504054i
34.13 1.24613 − 0.187825i −0.988831 + 0.149042i −0.393573 + 0.121401i 1.71016 − 2.14447i −1.20422 + 0.371453i −0.206022 − 0.524936i −2.73846 + 1.31877i 0.955573 − 0.294755i 1.72830 − 2.99351i
34.14 1.68271 − 0.253627i −0.988831 + 0.149042i 0.856030 − 0.264050i −1.65738 + 2.07828i −1.62611 + 0.501589i −0.889180 − 2.26559i −1.69290 + 0.815260i 0.955573 − 0.294755i −2.26177 + 3.91750i
34.15 1.87159 − 0.282097i −0.988831 + 0.149042i 1.51214 − 0.466434i 1.34975 − 1.69253i −1.80865 + 0.557893i 1.89233 + 4.82159i −0.712055 + 0.342908i 0.955573 − 0.294755i 2.04872 − 3.54849i
34.16 2.06390 − 0.311082i −0.988831 + 0.149042i 2.25175 − 0.694574i −1.82424 + 2.28753i −1.99448 + 0.615216i 0.518876 + 1.32208i 0.670299 − 0.322799i 0.955573 − 0.294755i −3.05344 + 5.28872i
34.17 2.35591 − 0.355096i −0.988831 + 0.149042i 3.51306 − 1.08363i 1.89701 − 2.37877i −2.27667 + 0.702259i −0.633739 − 1.61474i 3.59849 − 1.73294i 0.955573 − 0.294755i 3.62448 − 6.27778i
34.18 2.44953 − 0.369208i −0.988831 + 0.149042i 3.95275 − 1.21926i 0.392660 − 0.492380i −2.36715 + 0.730168i −1.38191 − 3.52105i 4.76848 − 2.29638i 0.955573 − 0.294755i 0.780043 − 1.35107i
43.1 −0.957524 − 2.43973i 0.365341 + 0.930874i −3.56933 + 3.31185i 2.05351 + 2.57503i 1.92126 − 1.78267i 1.51699 − 0.228649i 6.77505 + 3.26269i −0.733052 + 0.680173i 4.31608 − 7.47567i
43.2 −0.873769 − 2.22633i 0.365341 + 0.930874i −2.72695 + 2.53024i 0.485669 + 0.609010i 1.75320 − 1.62674i −4.08515 + 0.615738i 3.70626 + 1.78484i −0.733052 + 0.680173i 0.931492 − 1.61339i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.j even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.r.b ✓ 216
211.j even 21 1 inner 633.2.r.b ✓ 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.r.b ✓ 216 1.a even 1 1 trivial
633.2.r.b ✓ 216 211.j even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{216} + 6 T_{2}^{215} + 2 T_{2}^{214} - 54 T_{2}^{213} - 115 T_{2}^{212} - 136 T_{2}^{211} + \cdots + 1842469605376 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display