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Properties

Label 153.4.l.c
Level $153$
Weight $4$
Character orbit 153.l
Analytic conductor $9.027$
Analytic rank $0$
Dimension $40$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [153,4,Mod(19,153)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(153, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("153.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 153 = 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 153.l (of order \(8\), degree \(4\), 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: \(9.02729223088\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 51)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 32 q^{5} + 128 q^{10} - 112 q^{11} - 256 q^{14} - 1024 q^{16} + 112 q^{17} - 32 q^{19} + 640 q^{20} + 728 q^{22} - 208 q^{23} + 296 q^{25} - 1472 q^{26} - 328 q^{28} + 1272 q^{29} - 192 q^{31} + 960 q^{32}+ \cdots + 1008 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −3.75396 − 3.75396i 0 20.1845i 0.615614 + 1.48622i 0 8.32655 − 20.1021i 45.7401 − 45.7401i 0 3.26824 − 7.89022i
19.2 −3.18525 − 3.18525i 0 12.2916i 0.657844 + 1.58818i 0 −2.66469 + 6.43312i 13.6698 − 13.6698i 0 2.96334 − 7.15413i
19.3 −1.98341 − 1.98341i 0 − 0.132152i −6.98831 − 16.8713i 0 −9.61579 + 23.2146i −16.1294 + 16.1294i 0 −19.6020 + 47.3234i
19.4 −0.865877 − 0.865877i 0 − 6.50051i −6.89196 − 16.6387i 0 6.38012 − 15.4030i −12.5557 + 12.5557i 0 −8.43944 + 20.3746i
19.5 −0.511853 − 0.511853i 0 − 7.47601i 6.64023 + 16.0309i 0 −5.63793 + 13.6112i −7.92144 + 7.92144i 0 4.80665 − 11.6043i
19.6 −0.320127 − 0.320127i 0 − 7.79504i 2.81499 + 6.79598i 0 6.62120 − 15.9850i −5.05641 + 5.05641i 0 1.27442 − 3.07673i
19.7 1.89364 + 1.89364i 0 − 0.828264i 1.56537 + 3.77915i 0 −11.5719 + 27.9371i 16.7175 − 16.7175i 0 −4.19208 + 10.1206i
19.8 2.07008 + 2.07008i 0 0.570424i 1.15618 + 2.79128i 0 7.11409 − 17.1749i 15.3798 − 15.3798i 0 −3.38476 + 8.17154i
19.9 2.85730 + 2.85730i 0 8.32833i −7.19045 − 17.3593i 0 10.1833 − 24.5847i −0.938143 + 0.938143i 0 29.0554 − 70.1460i
19.10 3.79946 + 3.79946i 0 20.8719i −4.62215 − 11.1589i 0 −11.9634 + 28.8822i −48.9062 + 48.9062i 0 24.8360 − 59.9594i
100.1 −3.81517 − 3.81517i 0 21.1111i −1.33820 + 0.554300i 0 8.84293 + 3.66286i 50.0210 − 50.0210i 0 7.22021 + 2.99071i
100.2 −3.10148 − 3.10148i 0 11.2384i −14.1902 + 5.87778i 0 −30.3663 − 12.5781i 10.0437 − 10.0437i 0 62.2405 + 25.7809i
100.3 −2.73634 − 2.73634i 0 6.97513i 12.5970 − 5.21785i 0 19.5561 + 8.10040i −2.80440 + 2.80440i 0 −48.7476 − 20.1919i
100.4 −1.41071 − 1.41071i 0 − 4.01979i −9.58757 + 3.97130i 0 5.26522 + 2.18092i −16.9564 + 16.9564i 0 19.1277 + 7.92293i
100.5 −0.378339 − 0.378339i 0 − 7.71372i −7.33680 + 3.03900i 0 24.6775 + 10.2218i −5.94511 + 5.94511i 0 3.92557 + 1.62602i
100.6 0.0147863 + 0.0147863i 0 − 7.99956i 18.9786 − 7.86121i 0 −5.95742 − 2.46764i 0.236574 − 0.236574i 0 0.396861 + 0.164385i
100.7 2.15816 + 2.15816i 0 1.31528i −13.0882 + 5.42131i 0 −26.4168 − 10.9422i 14.4267 − 14.4267i 0 −39.9464 − 16.5464i
100.8 2.21747 + 2.21747i 0 1.83439i 12.9753 − 5.37456i 0 −7.27652 − 3.01403i 13.6721 − 13.6721i 0 40.6904 + 16.8545i
100.9 3.45749 + 3.45749i 0 15.9085i −13.3151 + 5.51528i 0 7.75774 + 3.21336i −27.3435 + 27.3435i 0 −65.1057 − 26.9677i
100.10 3.59414 + 3.59414i 0 17.8356i 10.5477 − 4.36900i 0 6.74598 + 2.79428i −35.3506 + 35.3506i 0 53.6127 + 22.2071i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 153.4.l.c 40
3.b odd 2 1 51.4.h.a ✓ 40
17.d even 8 1 inner 153.4.l.c 40
51.g odd 8 1 51.4.h.a ✓ 40
51.i even 16 1 867.4.a.v 20
51.i even 16 1 867.4.a.w 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.4.h.a ✓ 40 3.b odd 2 1
51.4.h.a ✓ 40 51.g odd 8 1
153.4.l.c 40 1.a even 1 1 trivial
153.4.l.c 40 17.d even 8 1 inner
867.4.a.v 20 51.i even 16 1
867.4.a.w 20 51.i even 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{40} + 2688 T_{2}^{36} - 192 T_{2}^{35} + 5184 T_{2}^{33} + 2794362 T_{2}^{32} + \cdots + 4964982194176 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display