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Properties

Label 48.9.l.a
Level $48$
Weight $9$
Character orbit 48.l
Analytic conductor $19.554$
Analytic rank $0$
Dimension $64$
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] = [48,9,Mod(19,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("48.19");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 48.l (of order \(4\), 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: \(19.5541732829\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(32\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 372 q^{4} + 17460 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 372 q^{4} + 17460 q^{8} - 13784 q^{10} + 39552 q^{11} - 22680 q^{12} + 116772 q^{14} - 161824 q^{16} - 26244 q^{18} - 167552 q^{19} + 690000 q^{20} - 1252768 q^{22} + 1691136 q^{23} - 74844 q^{24} - 1682100 q^{26} + 678600 q^{28} - 2132352 q^{29} - 2054808 q^{30} + 1407840 q^{32} - 3704544 q^{34} - 2415744 q^{35} - 1041012 q^{36} - 4720512 q^{37} + 2413320 q^{38} - 3544848 q^{40} + 9436500 q^{42} - 7244672 q^{43} - 3982056 q^{44} + 11165992 q^{46} - 9467280 q^{48} + 52706752 q^{49} + 17072676 q^{50} + 13862016 q^{51} - 22890864 q^{52} - 5358720 q^{53} - 4960116 q^{54} - 46326784 q^{55} - 9637152 q^{56} + 25732944 q^{58} + 44938752 q^{59} + 40927032 q^{60} + 24476032 q^{61} - 17024868 q^{62} - 116303448 q^{64} + 29941632 q^{65} - 56828952 q^{66} - 44244736 q^{67} - 12614400 q^{68} - 8636544 q^{69} - 60922560 q^{70} + 159664128 q^{71} - 18502020 q^{72} + 140654028 q^{74} + 12918528 q^{75} + 204842760 q^{76} - 94964352 q^{77} + 14001660 q^{78} + 38076648 q^{80} - 306110016 q^{81} - 260917000 q^{82} + 209328000 q^{83} - 127862712 q^{84} + 106960000 q^{85} - 123451824 q^{86} + 104240736 q^{88} + 53520264 q^{90} - 45401472 q^{91} + 351567216 q^{92} + 208488216 q^{94} + 49597920 q^{96} - 681890712 q^{98} + 86500224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −15.9603 − 1.12703i −33.0681 + 33.0681i 253.460 + 35.9753i 156.665 − 156.665i 565.044 − 490.507i −2214.37 −4004.74 − 859.832i − 2187.00i −2676.98 + 2323.85i
19.2 −15.6102 + 3.51009i −33.0681 + 33.0681i 231.359 − 109.587i 801.254 − 801.254i 400.129 − 632.273i 3981.15 −3226.90 + 2522.76i − 2187.00i −9695.29 + 15320.2i
19.3 −15.4815 + 4.04013i 33.0681 − 33.0681i 223.355 − 125.095i 135.282 − 135.282i −378.345 + 645.544i −2730.70 −2952.47 + 2839.04i − 2187.00i −1547.82 + 2640.93i
19.4 −13.4570 + 8.65496i −33.0681 + 33.0681i 106.183 − 232.940i −333.431 + 333.431i 158.796 − 731.202i −1764.79 587.173 + 4053.70i − 2187.00i 1601.16 − 7372.82i
19.5 −13.3280 − 8.85242i 33.0681 − 33.0681i 99.2693 + 235.970i −642.261 + 642.261i −733.463 + 147.998i −3321.99 765.844 − 4023.77i − 2187.00i 14245.6 − 2874.47i
19.6 −13.2763 − 8.92970i 33.0681 − 33.0681i 96.5211 + 237.107i 646.826 − 646.826i −734.311 + 143.734i 1263.53 835.849 − 4009.81i − 2187.00i −14363.4 + 2811.51i
19.7 −12.8440 − 9.54111i −33.0681 + 33.0681i 73.9342 + 245.091i −166.749 + 166.749i 740.232 − 109.219i 1903.82 1388.84 − 3853.36i − 2187.00i 3732.69 − 550.745i
19.8 −11.9758 + 10.6104i 33.0681 − 33.0681i 30.8403 − 254.136i −825.217 + 825.217i −45.1529 + 746.882i 1803.45 2327.13 + 3370.71i − 2187.00i 1126.79 − 18638.5i
19.9 −10.8909 − 11.7213i −33.0681 + 33.0681i −18.7756 + 255.311i 260.136 − 260.136i 747.742 + 27.4575i −368.567 3197.04 − 2560.49i − 2187.00i −5882.23 − 215.999i
19.10 −9.57765 + 12.8167i 33.0681 − 33.0681i −72.5372 − 245.508i 541.894 − 541.894i 107.110 + 740.540i 1146.54 3841.35 + 1421.70i − 2187.00i 1755.24 + 12135.4i
19.11 −7.30885 + 14.2331i −33.0681 + 33.0681i −149.162 − 208.055i −206.104 + 206.104i −228.972 − 712.351i 3383.34 4051.46 − 602.388i − 2187.00i −1427.12 − 4439.88i
19.12 −7.14295 − 14.3171i 33.0681 − 33.0681i −153.957 + 204.532i −24.8301 + 24.8301i −709.642 − 237.235i 2222.06 4028.00 + 743.246i − 2187.00i 532.855 + 178.134i
19.13 −6.68009 + 14.5388i −33.0681 + 33.0681i −166.753 − 194.241i 713.541 − 713.541i −259.872 − 701.668i −3620.00 3937.95 − 1126.84i − 2187.00i 5607.50 + 15140.5i
19.14 −3.18541 − 15.6797i −33.0681 + 33.0681i −235.706 + 99.8926i −349.282 + 349.282i 623.834 + 413.163i −4640.58 2317.11 + 3377.61i − 2187.00i 6589.25 + 4364.04i
19.15 −2.37796 − 15.8223i 33.0681 − 33.0681i −244.691 + 75.2496i 508.668 − 508.668i −601.848 − 444.579i −3253.34 1772.49 + 3692.63i − 2187.00i −9257.88 − 6838.70i
19.16 −1.76142 + 15.9027i 33.0681 − 33.0681i −249.795 − 56.0230i 47.4496 − 47.4496i 467.627 + 584.121i −2246.74 1330.91 − 3873.74i − 2187.00i 670.999 + 838.157i
19.17 1.16854 − 15.9573i −33.0681 + 33.0681i −253.269 − 37.2934i −679.500 + 679.500i 489.036 + 566.318i 4528.78 −891.055 + 3997.90i − 2187.00i 10048.9 + 11637.0i
19.18 1.46245 + 15.9330i 33.0681 − 33.0681i −251.722 + 46.6024i −173.172 + 173.172i 575.235 + 478.515i 3867.21 −1110.65 − 3942.55i − 2187.00i −3012.41 − 2505.90i
19.19 3.42993 − 15.6280i 33.0681 − 33.0681i −232.471 − 107.206i −488.559 + 488.559i −403.369 − 630.211i −90.0632 −2472.78 + 3265.36i − 2187.00i 5959.49 + 9310.93i
19.20 4.16319 + 15.4489i −33.0681 + 33.0681i −221.336 + 128.633i −851.062 + 851.062i −648.534 − 373.196i −2002.62 −2908.70 − 2883.86i − 2187.00i −16691.1 − 9604.82i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 48.9.l.a ✓ 64
4.b odd 2 1 192.9.l.a 64
16.e even 4 1 192.9.l.a 64
16.f odd 4 1 inner 48.9.l.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.l.a ✓ 64 1.a even 1 1 trivial
48.9.l.a ✓ 64 16.f odd 4 1 inner
192.9.l.a 64 4.b odd 2 1
192.9.l.a 64 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(48, [\chi])\).