↑

Properties

Label 153.4.l.b
Level $153$
Weight $4$
Character orbit 153.l
Analytic conductor $9.027$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 80 q^{10} - 128 q^{16} + 128 q^{19} + 280 q^{22} + 16 q^{25} + 184 q^{28} - 192 q^{31} + 24 q^{34} - 416 q^{37} - 488 q^{40} + 672 q^{43} - 384 q^{46} + 944 q^{49} - 4032 q^{52} - 1576 q^{58} + 4816 q^{61} + 2464 q^{67} + 5944 q^{70} + 3024 q^{73} - 3384 q^{76} - 4992 q^{79} - 1176 q^{82} - 2544 q^{85} - 1480 q^{88} - 4016 q^{91} + 4672 q^{94} + 1008 q^{97}+O(q^{100}) \) 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 −3.22560 − 3.22560i 0 12.8090i −2.92988 − 7.07335i 0 −4.41551 + 10.6600i 15.5118 − 15.5118i 0 −13.3652 + 32.2664i
19.2 −2.60260 − 2.60260i 0 5.54705i 7.10732 + 17.1586i 0 0.546150 − 1.31852i −6.38404 + 6.38404i 0 26.1594 − 63.1545i
19.3 −1.56233 − 1.56233i 0 − 3.11822i −3.46961 − 8.37637i 0 11.4295 − 27.5933i −17.3704 + 17.3704i 0 −7.66601 + 18.5074i
19.4 −0.509685 − 0.509685i 0 − 7.48044i 3.53692 + 8.53887i 0 −6.14596 + 14.8377i −7.89014 + 7.89014i 0 2.54942 − 6.15484i
19.5 0.509685 + 0.509685i 0 − 7.48044i −3.53692 − 8.53887i 0 −6.14596 + 14.8377i 7.89014 − 7.89014i 0 2.54942 − 6.15484i
19.6 1.56233 + 1.56233i 0 − 3.11822i 3.46961 + 8.37637i 0 11.4295 − 27.5933i 17.3704 − 17.3704i 0 −7.66601 + 18.5074i
19.7 2.60260 + 2.60260i 0 5.54705i −7.10732 − 17.1586i 0 0.546150 − 1.31852i 6.38404 − 6.38404i 0 26.1594 − 63.1545i
19.8 3.22560 + 3.22560i 0 12.8090i 2.92988 + 7.07335i 0 −4.41551 + 10.6600i −15.5118 + 15.5118i 0 −13.3652 + 32.2664i
100.1 −3.60798 − 3.60798i 0 18.0350i 10.0551 − 4.16494i 0 −24.1664 − 10.0100i 36.2060 − 36.2060i 0 −51.3054 − 21.2514i
100.2 −2.93590 − 2.93590i 0 9.23901i −10.1803 + 4.21680i 0 16.5093 + 6.83839i 3.63760 − 3.63760i 0 42.2683 + 17.5081i
100.3 −1.39664 − 1.39664i 0 − 4.09877i 0.571583 − 0.236758i 0 −12.7805 − 5.29386i −16.8977 + 16.8977i 0 −1.12896 − 0.467632i
100.4 −0.730557 − 0.730557i 0 − 6.93257i 16.9495 − 7.02071i 0 19.0233 + 7.87972i −10.9091 + 10.9091i 0 −17.5116 − 7.25354i
100.5 0.730557 + 0.730557i 0 − 6.93257i −16.9495 + 7.02071i 0 19.0233 + 7.87972i 10.9091 − 10.9091i 0 −17.5116 − 7.25354i
100.6 1.39664 + 1.39664i 0 − 4.09877i −0.571583 + 0.236758i 0 −12.7805 − 5.29386i 16.8977 − 16.8977i 0 −1.12896 − 0.467632i
100.7 2.93590 + 2.93590i 0 9.23901i 10.1803 − 4.21680i 0 16.5093 + 6.83839i −3.63760 + 3.63760i 0 42.2683 + 17.5081i
100.8 3.60798 + 3.60798i 0 18.0350i −10.0551 + 4.16494i 0 −24.1664 − 10.0100i −36.2060 + 36.2060i 0 −51.3054 − 21.2514i
127.1 −3.60798 + 3.60798i 0 − 18.0350i 10.0551 + 4.16494i 0 −24.1664 + 10.0100i 36.2060 + 36.2060i 0 −51.3054 + 21.2514i
127.2 −2.93590 + 2.93590i 0 − 9.23901i −10.1803 − 4.21680i 0 16.5093 − 6.83839i 3.63760 + 3.63760i 0 42.2683 − 17.5081i
127.3 −1.39664 + 1.39664i 0 4.09877i 0.571583 + 0.236758i 0 −12.7805 + 5.29386i −16.8977 − 16.8977i 0 −1.12896 + 0.467632i
127.4 −0.730557 + 0.730557i 0 6.93257i 16.9495 + 7.02071i 0 19.0233 − 7.87972i −10.9091 − 10.9091i 0 −17.5116 + 7.25354i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
17.d even 8 1 inner
51.g odd 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.b ✓ 32
3.b odd 2 1 inner 153.4.l.b ✓ 32
17.d even 8 1 inner 153.4.l.b ✓ 32
51.g odd 8 1 inner 153.4.l.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
153.4.l.b ✓ 32 1.a even 1 1 trivial
153.4.l.b ✓ 32 3.b odd 2 1 inner
153.4.l.b ✓ 32 17.d even 8 1 inner
153.4.l.b ✓ 32 51.g odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} + 1632 T_{2}^{28} + 946842 T_{2}^{24} + 238027740 T_{2}^{20} + 24537184105 T_{2}^{16} + \cdots + 1785793904896 \) acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\). Copy content Toggle raw display