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Properties

Label 148.4.k.a
Level $148$
Weight $4$
Character orbit 148.k
Analytic conductor $8.732$
Analytic rank $0$
Dimension $54$
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] = [148,4,Mod(9,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 148.k (of order \(9\), degree \(6\), 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: \(8.73228268085\)
Analytic rank: \(0\)
Dimension: \(54\)
Relative dimension: \(9\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{3} - 15 q^{5} + 48 q^{7} + 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 6 q^{3} - 15 q^{5} + 48 q^{7} + 48 q^{9} + 66 q^{11} + 48 q^{13} + 96 q^{15} - 51 q^{17} - 42 q^{19} - 414 q^{21} - 252 q^{23} - 525 q^{25} + 222 q^{27} - 60 q^{29} + 600 q^{31} + 264 q^{33} + 906 q^{35} + 390 q^{37} + 66 q^{39} + 3 q^{41} - 144 q^{43} + 1308 q^{45} - 222 q^{47} - 390 q^{49} - 366 q^{53} - 2070 q^{55} + 1254 q^{57} + 180 q^{59} + 1614 q^{61} - 750 q^{63} - 183 q^{65} - 1350 q^{67} + 264 q^{69} - 2892 q^{71} - 3534 q^{73} - 7812 q^{75} + 1896 q^{77} + 330 q^{79} + 2136 q^{81} - 1080 q^{83} + 2601 q^{85} - 1884 q^{87} + 2292 q^{89} + 4158 q^{91} - 12 q^{93} - 4800 q^{95} - 540 q^{97} + 3570 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −1.43908 + 8.16145i 0 −7.63556 − 6.40700i 0 −3.43161 − 2.87946i 0 −39.1666 − 14.2555i 0
9.2 0 −1.11532 + 6.32532i 0 10.8004 + 9.06259i 0 19.6822 + 16.5153i 0 −13.3940 − 4.87503i 0
9.3 0 −0.689648 + 3.91119i 0 7.10709 + 5.96355i 0 −9.05620 − 7.59906i 0 10.5499 + 3.83986i 0
9.4 0 −0.396322 + 2.24766i 0 −10.6372 − 8.92565i 0 5.00199 + 4.19717i 0 20.4768 + 7.45295i 0
9.5 0 −0.0180780 + 0.102525i 0 4.08553 + 3.42817i 0 −26.0588 − 21.8659i 0 25.3615 + 9.23084i 0
9.6 0 0.784276 − 4.44785i 0 11.7138 + 9.82905i 0 3.63914 + 3.05360i 0 6.20341 + 2.25786i 0
9.7 0 0.789505 − 4.47750i 0 −10.3266 − 8.66506i 0 0.238345 + 0.199996i 0 5.94697 + 2.16452i 0
9.8 0 0.834349 − 4.73183i 0 0.0174217 + 0.0146185i 0 21.8740 + 18.3545i 0 3.67763 + 1.33855i 0
9.9 0 1.78242 − 10.1086i 0 −0.730470 − 0.612937i 0 −12.7343 − 10.6854i 0 −73.6347 − 26.8008i 0
33.1 0 −1.43908 − 8.16145i 0 −7.63556 + 6.40700i 0 −3.43161 + 2.87946i 0 −39.1666 + 14.2555i 0
33.2 0 −1.11532 − 6.32532i 0 10.8004 − 9.06259i 0 19.6822 − 16.5153i 0 −13.3940 + 4.87503i 0
33.3 0 −0.689648 − 3.91119i 0 7.10709 − 5.96355i 0 −9.05620 + 7.59906i 0 10.5499 − 3.83986i 0
33.4 0 −0.396322 − 2.24766i 0 −10.6372 + 8.92565i 0 5.00199 − 4.19717i 0 20.4768 − 7.45295i 0
33.5 0 −0.0180780 − 0.102525i 0 4.08553 − 3.42817i 0 −26.0588 + 21.8659i 0 25.3615 − 9.23084i 0
33.6 0 0.784276 + 4.44785i 0 11.7138 − 9.82905i 0 3.63914 − 3.05360i 0 6.20341 − 2.25786i 0
33.7 0 0.789505 + 4.47750i 0 −10.3266 + 8.66506i 0 0.238345 − 0.199996i 0 5.94697 − 2.16452i 0
33.8 0 0.834349 + 4.73183i 0 0.0174217 − 0.0146185i 0 21.8740 − 18.3545i 0 3.67763 − 1.33855i 0
33.9 0 1.78242 + 10.1086i 0 −0.730470 + 0.612937i 0 −12.7343 + 10.6854i 0 −73.6347 + 26.8008i 0
49.1 0 −6.52294 − 5.47340i 0 −13.2446 + 4.82064i 0 −6.59673 + 2.40102i 0 7.90218 + 44.8155i 0
49.2 0 −5.89284 − 4.94468i 0 6.89698 − 2.51029i 0 26.8553 − 9.77454i 0 5.58721 + 31.6866i 0
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.4.k.a ✓ 54
37.f even 9 1 inner 148.4.k.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.4.k.a ✓ 54 1.a even 1 1 trivial
148.4.k.a ✓ 54 37.f even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(148, [\chi])\).