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Properties

Label 633.2.bc.a
Level $633$
Weight $2$
Character orbit 633.bc
Analytic conductor $5.055$
Analytic rank $0$
Dimension $816$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [633,2,Mod(4,633)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(633, base_ring=CyclotomicField(210))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("633.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 633 = 3 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 633.bc (of order \(105\), degree \(48\), 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: \(816\)
Relative dimension: \(17\) over \(\Q(\zeta_{105})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{105}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 816 q + 4 q^{2} + 17 q^{3} - 26 q^{4} - 4 q^{5} + 10 q^{6} + q^{7} - 20 q^{8} - 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 816 q + 4 q^{2} + 17 q^{3} - 26 q^{4} - 4 q^{5} + 10 q^{6} + q^{7} - 20 q^{8} - 17 q^{9} - 17 q^{10} + 8 q^{11} + 115 q^{12} - 12 q^{13} + 69 q^{14} - 48 q^{15} - 18 q^{16} + 91 q^{17} - 9 q^{18} - 44 q^{19} + 11 q^{20} - 24 q^{21} + 22 q^{22} + 12 q^{23} + 5 q^{24} + 8 q^{25} - 43 q^{26} - 34 q^{27} + 133 q^{28} - 65 q^{29} - 54 q^{30} - 49 q^{31} + 91 q^{32} - 10 q^{33} - 66 q^{34} + 62 q^{35} - 12 q^{36} + 30 q^{37} - 85 q^{38} + 22 q^{39} + 34 q^{40} + 24 q^{41} + 10 q^{42} - 75 q^{43} + 49 q^{44} + 2 q^{45} - 49 q^{46} - 100 q^{47} - 22 q^{48} + 158 q^{49} + 62 q^{50} + 4 q^{51} + 46 q^{52} + 73 q^{53} + 3 q^{54} - 32 q^{55} - 122 q^{56} - 39 q^{57} - 12 q^{58} - 301 q^{59} + 12 q^{60} - 44 q^{61} - 71 q^{62} - 12 q^{63} - 52 q^{64} + 22 q^{65} + 15 q^{66} - 184 q^{67} + 26 q^{68} + 22 q^{69} - 21 q^{70} - 8 q^{71} + 16 q^{72} - 35 q^{73} - 23 q^{74} - 27 q^{75} - 163 q^{76} - 216 q^{77} + 13 q^{78} + 108 q^{79} + 30 q^{80} - 17 q^{81} + 15 q^{82} - 20 q^{83} - 12 q^{84} - 97 q^{85} - 162 q^{86} - 2 q^{87} - 367 q^{88} - 16 q^{89} - 19 q^{90} - 133 q^{91} + 100 q^{92} - 11 q^{93} - 172 q^{94} + 98 q^{95} - 182 q^{96} + 117 q^{97} - 232 q^{98} - 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.61959 − 0.0784012i 0.280427 + 0.959875i 4.85966 + 0.291149i −1.51274 − 1.58220i −0.659348 − 2.53646i 0.298293 + 0.482708i −7.48705 − 0.673846i −0.842721 + 0.538351i 3.83871 + 4.26332i
4.2 −2.14251 − 0.0641229i 0.280427 + 0.959875i 2.58982 + 0.155160i 0.613238 + 0.641397i −0.539269 − 2.07453i −1.28582 − 2.08076i −1.26909 − 0.114220i −0.842721 + 0.538351i −1.27274 − 1.41352i
4.3 −1.95299 − 0.0584508i 0.280427 + 0.959875i 1.81434 + 0.108699i 2.28434 + 2.38923i −0.491567 − 1.89102i −0.0867657 − 0.140407i 0.354972 + 0.0319480i −0.842721 + 0.538351i −4.32163 − 4.79966i
4.4 −1.71302 − 0.0512688i 0.280427 + 0.959875i 0.935395 + 0.0560408i −1.07613 − 1.12554i −0.431167 − 1.65866i 1.43801 + 2.32704i 1.81430 + 0.163290i −0.842721 + 0.538351i 1.78573 + 1.98325i
4.5 −1.37553 − 0.0411680i 0.280427 + 0.959875i −0.106037 − 0.00635283i 0.477014 + 0.498918i −0.346220 − 1.33188i −1.34744 − 2.18047i 2.88680 + 0.259817i −0.842721 + 0.538351i −0.635607 − 0.705913i
4.6 −0.924573 − 0.0276714i 0.280427 + 0.959875i −1.14235 − 0.0684398i −1.36991 − 1.43282i −0.232714 − 0.895234i 0.568266 + 0.919588i 2.89682 + 0.260718i −0.842721 + 0.538351i 1.22694 + 1.36265i
4.7 −0.634173 − 0.0189801i 0.280427 + 0.959875i −1.59460 − 0.0955350i 2.19938 + 2.30037i −0.159621 − 0.614050i −1.59993 − 2.58906i 2.27325 + 0.204596i −0.842721 + 0.538351i −1.35113 − 1.50058i
4.8 −0.577692 − 0.0172897i 0.280427 + 0.959875i −1.66299 − 0.0996321i 1.84422 + 1.92891i −0.145405 − 0.559361i 2.70047 + 4.37000i 2.11022 + 0.189924i −0.842721 + 0.538351i −1.03204 − 1.14620i
4.9 −0.405173 − 0.0121264i 0.280427 + 0.959875i −1.83240 − 0.109782i −1.21071 − 1.26631i −0.101982 − 0.392316i −2.65729 − 4.30012i 1.54855 + 0.139372i −0.842721 + 0.538351i 0.475192 + 0.527754i
4.10 0.0776053 + 0.00232264i 0.280427 + 0.959875i −1.99040 − 0.119248i −2.95227 − 3.08783i 0.0195332 + 0.0751427i 1.35814 + 2.19779i −0.308844 − 0.0277964i −0.842721 + 0.538351i −0.221940 − 0.246489i
4.11 0.533788 + 0.0159757i 0.280427 + 0.959875i −1.71175 − 0.102553i 1.73393 + 1.81355i 0.134354 + 0.516850i −0.618994 − 1.00168i −1.97583 − 0.177828i −0.842721 + 0.538351i 0.896581 + 0.995754i
4.12 0.613922 + 0.0183740i 0.280427 + 0.959875i −1.61986 − 0.0970479i −0.666363 − 0.696961i 0.154524 + 0.594441i 0.212715 + 0.344223i −2.21613 − 0.199455i −0.842721 + 0.538351i −0.396289 − 0.440123i
4.13 1.44062 + 0.0431162i 0.280427 + 0.959875i 0.0771109 + 0.00461982i −1.45943 − 1.52645i 0.362604 + 1.39491i −1.79796 − 2.90952i −2.76004 − 0.248408i −0.842721 + 0.538351i −2.03668 − 2.26196i
4.14 1.51452 + 0.0453279i 0.280427 + 0.959875i 0.295305 + 0.0176921i 0.785057 + 0.821105i 0.381205 + 1.46646i 0.912508 + 1.47665i −2.57176 − 0.231462i −0.842721 + 0.538351i 1.15177 + 1.27917i
4.15 2.15549 + 0.0645114i 0.280427 + 0.959875i 2.64556 + 0.158499i 1.85569 + 1.94090i 0.542536 + 2.08709i −0.0958992 − 0.155187i 1.39670 + 0.125705i −0.842721 + 0.538351i 3.87471 + 4.30330i
4.16 2.36591 + 0.0708091i 0.280427 + 0.959875i 3.59612 + 0.215448i 1.45597 + 1.52283i 0.595499 + 2.29084i 0.306812 + 0.496493i 3.77797 + 0.340023i −0.842721 + 0.538351i 3.33687 + 3.70597i
4.17 2.48081 + 0.0742478i 0.280427 + 0.959875i 4.15249 + 0.248781i −1.61916 − 1.69351i 0.624419 + 2.40209i 1.89567 + 3.06764i 5.33921 + 0.480537i −0.842721 + 0.538351i −3.89109 − 4.32149i
16.1 −2.78123 − 0.166627i 0.842721 − 0.538351i 5.72176 + 0.688067i −0.0777252 + 1.73069i −2.43350 + 1.35685i −0.436136 + 0.872030i −10.3160 − 1.87208i 0.420357 − 0.907359i 0.504551 − 4.80048i
16.2 −2.24260 − 0.134358i 0.842721 − 0.538351i 3.02553 + 0.363833i 0.152749 − 3.40123i −1.96222 + 1.09408i −0.583899 + 1.16747i −2.31514 − 0.420135i 0.420357 − 0.907359i −0.799537 + 7.60709i
16.3 −2.07824 − 0.124510i 0.842721 − 0.538351i 2.31788 + 0.278735i −0.0434885 + 0.968347i −1.81841 + 1.01389i −0.223326 + 0.446527i −0.685390 − 0.124380i 0.420357 − 0.907359i 0.210949 − 2.00704i
See next 80 embeddings (of 816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
211.o even 105 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 633.2.bc.a ✓ 816
211.o even 105 1 inner 633.2.bc.a ✓ 816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
633.2.bc.a ✓ 816 1.a even 1 1 trivial
633.2.bc.a ✓ 816 211.o even 105 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{816} - 4 T_{2}^{815} + 38 T_{2}^{814} - 112 T_{2}^{813} + 735 T_{2}^{812} - 1991 T_{2}^{811} + \cdots + 11\!\cdots\!21 \) acting on \(S_{2}^{\mathrm{new}}(633, [\chi])\). Copy content Toggle raw display