Properties

Label 93.4.g.a
Level $93$
Weight $4$
Character orbit 93.g
Analytic conductor $5.487$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [93,4,Mod(26,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.26");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 93.g (of order \(6\), 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: \(5.48717763053\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 220 q^{4} + 66 q^{6} + 24 q^{7} + 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 3 q^{3} - 220 q^{4} + 66 q^{6} + 24 q^{7} + 31 q^{9} - 40 q^{10} - 60 q^{12} - 132 q^{13} + 860 q^{16} + 212 q^{18} - 112 q^{19} - 243 q^{21} + 78 q^{22} - 450 q^{24} + 908 q^{25} - 410 q^{28} - 698 q^{31} - 354 q^{33} + 348 q^{34} - 714 q^{36} - 600 q^{37} - 54 q^{39} + 1084 q^{40} - 1242 q^{42} - 432 q^{43} - 379 q^{45} + 2244 q^{48} - 2298 q^{49} - 351 q^{51} + 3780 q^{52} + 642 q^{55} - 357 q^{57} + 1766 q^{63} - 140 q^{64} - 5928 q^{66} + 146 q^{67} + 1440 q^{69} - 2816 q^{70} + 580 q^{72} - 444 q^{73} + 5604 q^{75} - 172 q^{76} + 1776 q^{78} + 906 q^{79} - 2297 q^{81} + 3512 q^{82} + 624 q^{84} + 510 q^{87} + 30 q^{88} + 376 q^{90} - 1299 q^{93} - 9208 q^{94} + 318 q^{96} + 6264 q^{97} + 1353 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1 5.32278i 4.01232 + 3.30171i −20.3320 16.2145 9.36145i 17.5743 21.3567i 10.0069 17.3325i 65.6406i 5.19736 + 26.4950i −49.8290 86.3063i
26.2 5.02024i −4.97128 1.51209i −17.2028 −9.62201 + 5.55527i −7.59107 + 24.9570i 13.9460 24.1553i 46.2002i 22.4272 + 15.0341i 27.8888 + 48.3048i
26.3 4.92894i 1.70332 4.90904i −16.2944 −0.772143 + 0.445797i −24.1964 8.39553i −4.86632 + 8.42871i 40.8827i −21.1974 16.7233i 2.19730 + 3.80584i
26.4 4.88422i −3.69925 + 3.64905i −15.8556 4.82657 2.78662i 17.8227 + 18.0679i −14.4254 + 24.9855i 38.3682i 0.368936 26.9975i −13.6104 23.5740i
26.5 4.38211i 3.16472 + 4.12123i −11.2029 −15.9571 + 9.21282i 18.0597 13.8682i −8.93192 + 15.4705i 14.0356i −6.96906 + 26.0851i 40.3716 + 69.9257i
26.6 3.53784i −1.72942 + 4.89991i −4.51631 −3.07857 + 1.77741i 17.3351 + 6.11840i 11.5958 20.0844i 12.3247i −21.0182 16.9480i 6.28821 + 10.8915i
26.7 3.26617i −1.81950 4.86718i −2.66788 15.0188 8.67113i −15.8970 + 5.94280i 7.66329 13.2732i 17.4156i −20.3788 + 17.7117i −28.3214 49.0541i
26.8 3.17111i 5.09574 1.01660i −2.05595 2.46691 1.42427i −3.22377 16.1591i −1.63919 + 2.83916i 18.8492i 24.9330 10.3607i −4.51653 7.82286i
26.9 2.69884i −3.51955 3.82267i 0.716275 −10.6925 + 6.17331i −10.3168 + 9.49869i −12.8440 + 22.2465i 23.5238i −2.22558 + 26.9081i 16.6608 + 28.8573i
26.10 2.66165i −5.12832 + 0.836861i 0.915642 10.2342 5.90875i 2.22743 + 13.6498i −0.623888 + 1.08061i 23.7303i 25.5993 8.58339i −15.7270 27.2399i
26.11 1.71661i 2.40525 4.60595i 5.05324 −16.7823 + 9.68924i −7.90663 4.12889i 15.8688 27.4856i 22.4074i −15.4295 22.1569i 16.6327 + 28.8086i
26.12 1.30390i 3.55936 + 3.78563i 6.29985 −1.87197 + 1.08078i 4.93607 4.64103i 8.53372 14.7808i 18.6455i −1.66195 + 26.9488i 1.40923 + 2.44085i
26.13 1.27199i 1.79919 + 4.87472i 6.38204 15.8415 9.14608i 6.20060 2.28855i −15.2681 + 26.4450i 18.2938i −20.5258 + 17.5411i −11.6337 20.1502i
26.14 0.445163i −5.18371 + 0.359415i 7.80183 −3.94983 + 2.28043i 0.159999 + 2.30760i 3.17077 5.49193i 7.03440i 26.7416 3.72621i 1.01517 + 1.75832i
26.15 0.202553i −2.31599 + 4.65147i 7.95897 −10.0418 + 5.79761i 0.942168 + 0.469111i −6.18654 + 10.7154i 3.23253i −16.2724 21.5455i 1.17432 + 2.03399i
26.16 0.202553i 2.87029 4.33144i 7.95897 10.0418 5.79761i 0.877346 + 0.581386i −6.18654 + 10.7154i 3.23253i −10.5228 24.8650i 1.17432 + 2.03399i
26.17 0.445163i −2.28059 4.66893i 7.80183 3.94983 2.28043i 2.07844 1.01524i 3.17077 5.49193i 7.03440i −16.5978 + 21.2958i 1.01517 + 1.75832i
26.18 1.27199i 5.12123 0.879218i 6.38204 −15.8415 + 9.14608i 1.11836 + 6.51415i −15.2681 + 26.4450i 18.2938i 25.4540 9.00535i −11.6337 20.1502i
26.19 1.30390i 5.05813 + 1.18968i 6.29985 1.87197 1.08078i −1.55122 + 6.59528i 8.53372 14.7808i 18.6455i 24.1693 + 12.0351i 1.40923 + 2.44085i
26.20 1.71661i −2.78624 + 4.38598i 5.05324 16.7823 9.68924i −7.52904 4.78290i 15.8688 27.4856i 22.4074i −11.4737 24.4408i 16.6327 + 28.8086i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.e odd 6 1 inner
93.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.4.g.a 60
3.b odd 2 1 inner 93.4.g.a 60
31.e odd 6 1 inner 93.4.g.a 60
93.g even 6 1 inner 93.4.g.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.4.g.a 60 1.a even 1 1 trivial
93.4.g.a 60 3.b odd 2 1 inner
93.4.g.a 60 31.e odd 6 1 inner
93.4.g.a 60 93.g even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(93, [\chi])\).