↑

Properties

Label 633.2.j.b
Level $633$
Weight $2$
Character orbit 633.j
Analytic conductor $5.055$
Analytic rank $0$
Dimension $114$
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(58,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.58");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.j (of order \(7\), degree \(6\), 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: \(114\)
Relative dimension: \(19\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 114 q + 3 q^{2} - 19 q^{3} - 9 q^{4} - 2 q^{5} + 3 q^{6} - 10 q^{7} - 11 q^{8} - 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 114 q + 3 q^{2} - 19 q^{3} - 9 q^{4} - 2 q^{5} + 3 q^{6} - 10 q^{7} - 11 q^{8} - 19 q^{9} - 12 q^{10} - 10 q^{11} - 23 q^{12} - 14 q^{13} - 14 q^{14} + 12 q^{15} - 25 q^{16} + 7 q^{17} - 4 q^{18} + 12 q^{19} - 22 q^{20} + 60 q^{21} - 4 q^{22} - 2 q^{23} - 11 q^{24} - 29 q^{25} + 13 q^{26} - 19 q^{27} + 34 q^{28} + 39 q^{29} + 9 q^{30} + 6 q^{31} - 45 q^{32} + 4 q^{33} + 9 q^{34} - 19 q^{35} - 23 q^{36} - 43 q^{37} - 17 q^{38} + 21 q^{39} + 30 q^{40} - 16 q^{41} - 21 q^{42} + 47 q^{43} - 31 q^{44} - 2 q^{45} + 23 q^{46} - 4 q^{47} - 18 q^{48} - 23 q^{49} + 2 q^{50} + 7 q^{51} - 64 q^{52} + 19 q^{53} - 4 q^{54} - 22 q^{55} + 67 q^{56} - 2 q^{57} - 29 q^{58} + 33 q^{59} - 22 q^{60} + 16 q^{61} + 35 q^{62} - 10 q^{63} - 65 q^{64} - 39 q^{65} + 3 q^{66} + 53 q^{67} - 5 q^{68} - 30 q^{69} - 52 q^{70} + 8 q^{71} - 4 q^{72} + 49 q^{73} - 28 q^{74} - 43 q^{75} - 96 q^{76} + 16 q^{77} + 6 q^{78} - 14 q^{79} + 144 q^{80} - 19 q^{81} - 65 q^{82} - 6 q^{83} - 57 q^{84} - 9 q^{85} + 85 q^{86} - 24 q^{87} + 25 q^{88} - 56 q^{89} + 9 q^{90} + 2 q^{91} + 107 q^{92} + 6 q^{93} + 9 q^{94} - 9 q^{95} + 4 q^{96} - 62 q^{97} + 67 q^{98} + 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
58.1 −2.36289 + 1.13791i −0.900969 + 0.433884i 3.04143 − 3.81383i −3.61863 − 1.74264i 1.63517 − 2.05044i −1.35311 − 0.651625i −1.67960 + 7.35879i 0.623490 − 0.781831i 10.5334
58.2 −2.33717 + 1.12552i −0.900969 + 0.433884i 2.94857 − 3.69739i 1.87837 + 0.904573i 1.61737 − 2.02812i −3.12627 − 1.50553i −1.57535 + 6.90204i 0.623490 − 0.781831i −5.40817
58.3 −2.13940 + 1.03028i −0.900969 + 0.433884i 2.26858 − 2.84471i 2.30099 + 1.10810i 1.48051 − 1.85650i 2.87508 + 1.38456i −0.865776 + 3.79321i 0.623490 − 0.781831i −6.06439
58.4 −1.62239 + 0.781300i −0.900969 + 0.433884i 0.774729 − 0.971480i −1.03795 − 0.499850i 1.12273 − 1.40785i −3.09241 − 1.48923i 0.303500 − 1.32972i 0.623490 − 0.781831i 2.07449
58.5 −1.58319 + 0.762425i −0.900969 + 0.433884i 0.678223 − 0.850465i −2.48416 − 1.19631i 1.09560 − 1.37384i 2.84916 + 1.37209i 0.356690 − 1.56276i 0.623490 − 0.781831i 4.84500
58.6 −1.53363 + 0.738557i −0.900969 + 0.433884i 0.559572 − 0.701682i 1.35119 + 0.650697i 1.06130 − 1.33083i −0.828745 − 0.399102i 0.417605 − 1.82965i 0.623490 − 0.781831i −2.55279
58.7 −0.930504 + 0.448107i −0.900969 + 0.433884i −0.581942 + 0.729732i 1.14733 + 0.552527i 0.643929 − 0.807461i 2.46745 + 1.18826i 0.674132 − 2.95357i 0.623490 − 0.781831i −1.31519
58.8 −0.486617 + 0.234343i −0.900969 + 0.433884i −1.06510 + 1.33559i −3.07499 − 1.48083i 0.336750 − 0.422271i −3.87409 − 1.86566i 0.445679 − 1.95265i 0.623490 − 0.781831i 1.84336
58.9 −0.343608 + 0.165473i −0.900969 + 0.433884i −1.15629 + 1.44995i 3.31754 + 1.59764i 0.237784 − 0.298172i −4.58440 − 2.20773i 0.327113 − 1.43318i 0.623490 − 0.781831i −1.40430
58.10 −0.290285 + 0.139794i −0.900969 + 0.433884i −1.18226 + 1.48250i 1.82875 + 0.880681i 0.200884 − 0.251900i −0.133120 − 0.0641070i 0.279336 − 1.22385i 0.623490 − 0.781831i −0.653974
58.11 0.598358 − 0.288154i −0.900969 + 0.433884i −0.971980 + 1.21882i 0.140546 + 0.0676833i −0.414077 + 0.519236i 2.60605 + 1.25501i −0.525947 + 2.30433i 0.623490 − 0.781831i 0.103600
58.12 0.646807 − 0.311486i −0.900969 + 0.433884i −0.925643 + 1.16072i −3.36537 − 1.62068i −0.447605 + 0.561278i 0.482307 + 0.232267i −0.556661 + 2.43889i 0.623490 − 0.781831i −2.68157
58.13 0.657171 − 0.316477i −0.900969 + 0.433884i −0.915263 + 1.14770i −0.509267 − 0.245250i −0.454776 + 0.570272i −1.39914 − 0.673792i −0.562878 + 2.46613i 0.623490 − 0.781831i −0.412291
58.14 1.46499 − 0.705500i −0.900969 + 0.433884i 0.401475 − 0.503434i 2.86516 + 1.37979i −1.01380 + 1.27127i −0.786327 − 0.378675i −0.490661 + 2.14972i 0.623490 − 0.781831i 5.17085
58.15 1.77070 − 0.852725i −0.900969 + 0.433884i 1.16126 − 1.45618i −0.492664 − 0.237254i −1.22536 + 1.53656i 2.92805 + 1.41008i −0.0601219 + 0.263411i 0.623490 − 0.781831i −1.07467
58.16 1.85753 − 0.894541i −0.900969 + 0.433884i 1.40325 − 1.75961i −1.86627 − 0.898748i −1.28545 + 1.61191i −3.19145 − 1.53692i 0.114983 − 0.503772i 0.623490 − 0.781831i −4.27062
58.17 2.18467 − 1.05208i −0.900969 + 0.433884i 2.41892 − 3.03323i −0.150644 − 0.0725461i −1.51184 + 1.89578i −4.13277 − 1.99024i 1.01420 − 4.44351i 0.623490 − 0.781831i −0.405431
58.18 2.46221 − 1.18574i −0.900969 + 0.433884i 3.40953 − 4.27542i −3.42054 − 1.64724i −1.70390 + 2.13663i 3.55776 + 1.71333i 2.10923 − 9.24114i 0.623490 − 0.781831i −10.3753
58.19 2.48724 − 1.19779i −0.900969 + 0.433884i 3.50467 − 4.39472i 3.38868 + 1.63190i −1.72122 + 2.15834i −0.273724 − 0.131818i 2.22440 − 9.74575i 0.623490 − 0.781831i 10.3831
148.1 −0.620868 + 2.72020i −0.222521 + 0.974928i −5.21207 − 2.51000i −0.553752 − 2.42615i −2.51384 − 1.21060i −0.0131070 − 0.0574257i 6.58444 − 8.25663i −0.900969 − 0.433884i 6.94340
See next 80 embeddings (of 114 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 58.19
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.f even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.j.b ✓ 114
211.f even 7 1 inner 633.2.j.b ✓ 114
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.j.b ✓ 114 1.a even 1 1 trivial
633.2.j.b ✓ 114 211.f even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{114} - 3 T_{2}^{113} + 28 T_{2}^{112} - 66 T_{2}^{111} + 437 T_{2}^{110} - 917 T_{2}^{109} + \cdots + 11758041 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display