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Properties

Label 126.4.l.a
Level $126$
Weight $4$
Character orbit 126.l
Analytic conductor $7.434$
Analytic rank $0$
Dimension $48$
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] = [126,4,Mod(5,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.l (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: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} - 12 q^{7} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 192 q^{4} - 12 q^{7} + 60 q^{9} - 24 q^{11} + 72 q^{13} - 132 q^{14} + 132 q^{15} + 768 q^{16} + 144 q^{17} + 120 q^{18} + 168 q^{21} + 408 q^{23} - 600 q^{25} + 120 q^{26} - 450 q^{27} + 48 q^{28} + 42 q^{29} + 360 q^{30} - 780 q^{35} - 240 q^{36} - 168 q^{37} - 138 q^{39} - 618 q^{41} - 480 q^{42} - 42 q^{43} + 96 q^{44} + 366 q^{45} - 252 q^{46} + 396 q^{47} - 42 q^{49} + 1464 q^{50} - 600 q^{51} - 288 q^{52} - 36 q^{53} + 180 q^{54} + 528 q^{56} - 1452 q^{57} - 252 q^{58} - 3000 q^{59} - 528 q^{60} + 2952 q^{62} + 462 q^{63} - 3072 q^{64} - 1200 q^{66} + 1176 q^{67} - 576 q^{68} + 1848 q^{69} - 324 q^{70} - 480 q^{72} + 1260 q^{74} - 534 q^{75} + 6420 q^{77} + 2496 q^{78} - 2460 q^{79} + 1728 q^{81} - 672 q^{84} + 720 q^{85} - 1200 q^{86} - 4344 q^{87} - 4398 q^{89} - 6084 q^{90} - 90 q^{91} - 1632 q^{92} + 840 q^{93} + 1584 q^{97} - 1104 q^{98} - 5076 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 − 2.00000i −5.19138 − 0.222550i −4.00000 −8.49324 − 14.7107i −0.445099 + 10.3828i −0.359342 + 18.5168i 8.00000i 26.9009 + 2.31068i −29.4214 + 16.9865i
5.2 − 2.00000i −4.91310 + 1.69158i −4.00000 1.39685 + 2.41942i 3.38317 + 9.82620i −17.1508 − 6.98919i 8.00000i 21.2771 − 16.6218i 4.83883 − 2.79370i
5.3 − 2.00000i −4.57516 + 2.46331i −4.00000 1.88736 + 3.26901i 4.92662 + 9.15033i 17.9293 − 4.64125i 8.00000i 14.8642 − 22.5401i 6.53802 − 3.77473i
5.4 − 2.00000i −3.17008 − 4.11711i −4.00000 6.43016 + 11.1374i −8.23423 + 6.34015i 11.1140 + 14.8148i 8.00000i −6.90124 + 26.1031i 22.2747 − 12.8603i
5.5 − 2.00000i −2.99672 − 4.24496i −4.00000 −9.84577 − 17.0534i −8.48991 + 5.99344i 4.63716 − 17.9303i 8.00000i −9.03931 + 25.4419i −34.1068 + 19.6915i
5.6 − 2.00000i −0.153472 + 5.19389i −4.00000 9.56569 + 16.5683i 10.3878 + 0.306944i −17.7450 − 5.30225i 8.00000i −26.9529 − 1.59423i 33.1365 − 19.1314i
5.7 − 2.00000i 0.548540 + 5.16712i −4.00000 −1.85594 − 3.21459i 10.3342 − 1.09708i 16.4255 − 8.55580i 8.00000i −26.3982 + 5.66874i −6.42918 + 3.71189i
5.8 − 2.00000i 1.12856 − 5.07211i −4.00000 −0.213260 − 0.369378i −10.1442 − 2.25712i −18.0845 + 3.99406i 8.00000i −24.4527 − 11.4484i −0.738755 + 0.426520i
5.9 − 2.00000i 4.13333 + 3.14890i −4.00000 −7.48096 − 12.9574i 6.29780 − 8.26666i −11.8885 − 14.2008i 8.00000i 7.16886 + 26.0309i −25.9148 + 14.9619i
5.10 − 2.00000i 4.14312 − 3.13601i −4.00000 10.1776 + 17.6281i −6.27202 − 8.28624i 4.40132 − 17.9897i 8.00000i 7.33091 − 25.9857i 35.2562 − 20.3552i
5.11 − 2.00000i 4.44621 − 2.68909i −4.00000 −4.72900 − 8.19087i −5.37818 − 8.89242i 18.4230 − 1.89542i 8.00000i 12.5376 − 23.9125i −16.3817 + 9.45800i
5.12 − 2.00000i 4.86810 + 1.81703i −4.00000 3.16053 + 5.47419i 3.63407 − 9.73620i −1.17581 + 18.4829i 8.00000i 20.3968 + 17.6910i 10.9484 − 6.32106i
5.13 2.00000i −5.18583 − 0.327309i −4.00000 5.48053 + 9.49256i 0.654618 − 10.3717i −1.82034 + 18.4306i − 8.00000i 26.7857 + 3.39474i −18.9851 + 10.9611i
5.14 2.00000i −5.06109 − 1.17703i −4.00000 1.36122 + 2.35771i 2.35405 − 10.1222i 1.29655 − 18.4748i − 8.00000i 24.2292 + 11.9141i −4.71542 + 2.72245i
5.15 2.00000i −4.16916 + 3.10130i −4.00000 −9.06731 − 15.7050i −6.20260 − 8.33833i 15.5669 + 10.0335i − 8.00000i 7.76387 − 25.8597i 31.4101 − 18.1346i
5.16 2.00000i −2.75428 + 4.40613i −4.00000 1.48002 + 2.56347i −8.81226 − 5.50855i −18.4948 − 0.970961i − 8.00000i −11.8279 − 24.2714i −5.12693 + 2.96004i
5.17 2.00000i −2.38024 − 4.61892i −4.00000 −3.58162 − 6.20355i 9.23784 − 4.76049i −7.92464 + 16.7392i − 8.00000i −15.6689 + 21.9883i 12.4071 − 7.16324i
5.18 2.00000i 0.717893 − 5.14632i −4.00000 1.86960 + 3.23823i 10.2926 + 1.43579i −8.88689 − 16.2488i − 8.00000i −25.9693 − 7.38902i −6.47647 + 3.73919i
5.19 2.00000i 1.13140 + 5.07148i −4.00000 −6.21279 − 10.7609i −10.1430 + 2.26280i −2.69645 − 18.3229i − 8.00000i −24.4399 + 11.4758i 21.5217 − 12.4256i
5.20 2.00000i 1.96459 + 4.81044i −4.00000 3.47705 + 6.02243i −9.62089 + 3.92919i 14.1742 + 11.9202i − 8.00000i −19.2807 + 18.9011i −12.0449 + 6.95410i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.i even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.l.a ✓ 48
3.b odd 2 1 378.4.l.a 48
7.d odd 6 1 126.4.t.a yes 48
9.c even 3 1 378.4.t.a 48
9.d odd 6 1 126.4.t.a yes 48
21.g even 6 1 378.4.t.a 48
63.i even 6 1 inner 126.4.l.a ✓ 48
63.t odd 6 1 378.4.l.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.l.a ✓ 48 1.a even 1 1 trivial
126.4.l.a ✓ 48 63.i even 6 1 inner
126.4.t.a yes 48 7.d odd 6 1
126.4.t.a yes 48 9.d odd 6 1
378.4.l.a 48 3.b odd 2 1
378.4.l.a 48 63.t odd 6 1
378.4.t.a 48 9.c even 3 1
378.4.t.a 48 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(126, [\chi])\).