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Properties

Label 63.4.g.a
Level $63$
Weight $4$
Character orbit 63.g
Analytic conductor $3.717$
Analytic rank $0$
Dimension $44$
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] = [63,4,Mod(4,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 63.g (of order \(3\), 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: \(3.71712033036\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + q^{2} - q^{3} - 79 q^{4} + 38 q^{5} - 20 q^{6} - 7 q^{7} - 24 q^{8} - 31 q^{9} - 18 q^{10} - 10 q^{11} - 41 q^{12} - 14 q^{13} - 79 q^{14} + 119 q^{15} - 247 q^{16} - 162 q^{17} + 157 q^{18} + 58 q^{19} - 362 q^{20} + 166 q^{21} - 18 q^{22} + 186 q^{23} + 414 q^{24} + 698 q^{25} - 266 q^{26} + 272 q^{27} - 172 q^{28} + 248 q^{29} + 616 q^{30} + 61 q^{31} - 163 q^{32} + 23 q^{33} + 6 q^{34} + 289 q^{35} - 806 q^{36} - 86 q^{37} + 1522 q^{38} - 565 q^{39} + 36 q^{40} - 692 q^{41} + 395 q^{42} - 86 q^{43} - 443 q^{44} - 1483 q^{45} - 270 q^{46} - 1005 q^{47} - 1013 q^{48} - 277 q^{49} + 239 q^{50} - 1719 q^{51} + 670 q^{52} + 258 q^{53} + 910 q^{54} - 870 q^{55} + 714 q^{56} + 566 q^{57} - 474 q^{58} - 1665 q^{59} + 4 q^{60} + 439 q^{61} + 1812 q^{62} + 493 q^{63} + 872 q^{64} - 613 q^{65} + 3073 q^{66} + 295 q^{67} + 2748 q^{68} + 1389 q^{69} - 1044 q^{70} + 636 q^{71} + 981 q^{72} - 338 q^{73} - 2238 q^{74} - 1064 q^{75} + 1006 q^{76} - 2909 q^{77} + 157 q^{78} + 133 q^{79} - 4817 q^{80} + 1325 q^{81} + 6 q^{82} - 1356 q^{83} - 7081 q^{84} + 483 q^{85} + 6686 q^{86} + 2774 q^{87} - 738 q^{88} - 2200 q^{89} + 2665 q^{90} + 1552 q^{91} - 396 q^{92} + 4365 q^{93} - 1191 q^{94} + 3083 q^{95} - 1468 q^{96} - 266 q^{97} + 3601 q^{98} - 5395 q^{99}+O(q^{100}) \) 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 −2.71525 + 4.70295i 4.53482 − 2.53680i −10.7452 − 18.6112i 13.1170 −0.382714 + 28.2151i 12.4520 − 13.7094i 73.2595 14.1292 − 23.0079i −35.6160 + 61.6887i
4.2 −2.51592 + 4.35771i −1.54886 + 4.95994i −8.65974 − 14.9991i 0.150305 −17.7172 − 19.2283i −18.0980 + 3.93229i 46.8942 −22.2021 − 15.3645i −0.378156 + 0.654986i
4.3 −2.10738 + 3.65009i −5.18812 − 0.288776i −4.88211 − 8.45607i −7.82402 11.9874 − 18.3286i 16.1743 − 9.02169i 7.43579 26.8332 + 2.99641i 16.4882 − 28.5584i
4.4 −1.93332 + 3.34860i 4.87234 + 1.80563i −3.47543 − 6.01962i −13.3562 −15.4661 + 12.8247i 1.72288 + 18.4399i −4.05663 20.4794 + 17.5952i 25.8218 − 44.7246i
4.5 −1.83489 + 3.17813i −2.73769 − 4.41645i −2.73366 − 4.73484i 14.7742 19.0594 − 0.597013i 0.242329 + 18.5187i −9.29438 −12.0101 + 24.1818i −27.1090 + 46.9542i
4.6 −1.67658 + 2.90391i 0.949950 − 5.10858i −1.62181 − 2.80905i −8.70652 13.2422 + 11.3235i −14.4030 − 11.6428i −15.9489 −25.1952 − 9.70580i 14.5971 − 25.2830i
4.7 −1.33146 + 2.30616i 0.537708 + 5.16826i 0.454424 + 0.787086i 19.2381 −12.6348 − 5.64129i 18.0786 + 4.02053i −23.7236 −26.4217 + 5.55802i −25.6147 + 44.3660i
4.8 −0.724044 + 1.25408i −4.59485 + 2.42639i 2.95152 + 5.11218i 4.43275 0.283981 − 7.51913i −18.5126 + 0.531848i −20.1328 15.2253 − 22.2978i −3.20951 + 5.55903i
4.9 −0.667800 + 1.15666i 5.07151 + 1.13126i 3.10809 + 5.38337i 9.00469 −4.69524 + 5.11058i −14.2234 − 11.8615i −18.9871 24.4405 + 11.4744i −6.01333 + 10.4154i
4.10 −0.491847 + 0.851904i −0.512079 + 5.17086i 3.51617 + 6.09019i −18.7142 −4.15321 − 2.97951i 10.0203 − 15.5754i −14.7872 −26.4756 − 5.29577i 9.20454 − 15.9427i
4.11 −0.267129 + 0.462682i 3.78900 − 3.55577i 3.85728 + 6.68101i −1.39324 0.633036 + 2.70295i 16.9882 + 7.37568i −8.39564 1.71302 − 26.9456i 0.372176 − 0.644627i
4.12 0.219258 − 0.379765i −3.77328 − 3.57245i 3.90385 + 6.76167i −16.0932 −2.18401 + 0.649674i −1.97176 + 18.4150i 6.93192 1.47525 + 26.9597i −3.52855 + 6.11163i
4.13 0.295387 − 0.511626i −4.63295 − 2.35283i 3.82549 + 6.62595i 10.9845 −2.57228 + 1.67534i 1.43976 − 18.4642i 9.24621 15.9284 + 21.8011i 3.24467 − 5.61993i
4.14 0.904546 − 1.56672i 1.99500 + 4.79791i 2.36359 + 4.09386i −2.09779 9.32155 + 1.21433i −11.3205 + 14.6576i 23.0246 −19.0400 + 19.1437i −1.89755 + 3.28665i
4.15 1.18941 − 2.06012i 2.07198 − 4.76518i 1.17060 + 2.02754i 18.4675 −7.35242 − 9.93628i −16.1997 + 8.97605i 24.5999 −18.4138 − 19.7467i 21.9654 − 38.0453i
4.16 1.32738 − 2.29909i −4.04989 + 3.25551i 0.476130 + 0.824682i 7.35561 2.10896 + 13.6324i 16.6607 + 8.08840i 23.7661 5.80329 − 26.3690i 9.76368 − 16.9112i
4.17 1.46729 − 2.54141i 4.92390 + 1.65989i −0.305860 − 0.529765i −3.69711 11.4432 − 10.0781i 12.3500 − 13.8013i 21.6814 21.4895 + 16.3463i −5.42471 + 9.39588i
4.18 1.68671 − 2.92146i 0.0520064 − 5.19589i −1.68996 − 2.92710i −9.74532 −15.0919 − 8.91589i 0.158327 − 18.5196i 15.5854 −26.9946 − 0.540440i −16.4375 + 28.4706i
4.19 2.15287 − 3.72888i −4.60682 + 2.40358i −5.26968 − 9.12735i −15.9966 −0.955246 + 22.3529i −16.8821 − 7.61550i −10.9338 15.4456 − 22.1457i −34.4385 + 59.6493i
4.20 2.43564 − 4.21866i 4.95298 − 1.57099i −7.86471 − 13.6221i −6.21080 5.43624 − 24.7213i −4.64741 + 17.9277i −37.6522 22.0640 − 15.5621i −15.1273 + 26.2012i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.4.g.a ✓ 44
3.b odd 2 1 189.4.g.a 44
7.c even 3 1 63.4.h.a yes 44
9.c even 3 1 63.4.h.a yes 44
9.d odd 6 1 189.4.h.a 44
21.h odd 6 1 189.4.h.a 44
63.g even 3 1 inner 63.4.g.a ✓ 44
63.n odd 6 1 189.4.g.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.4.g.a ✓ 44 1.a even 1 1 trivial
63.4.g.a ✓ 44 63.g even 3 1 inner
63.4.h.a yes 44 7.c even 3 1
63.4.h.a yes 44 9.c even 3 1
189.4.g.a 44 3.b odd 2 1
189.4.g.a 44 63.n odd 6 1
189.4.h.a 44 9.d odd 6 1
189.4.h.a 44 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(63, [\chi])\).