Properties

Label 148.3.b.d
Level $148$
Weight $3$
Character orbit 148.b
Analytic conductor $4.033$
Analytic rank $0$
Dimension $32$
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] = [148,3,Mod(147,148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(148, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("148.147");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 148 = 2^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 148.b (of order \(2\), degree \(1\), 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: \(4.03270791253\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} - 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 4 q^{4} - 136 q^{9} - 26 q^{10} + 26 q^{12} - 76 q^{16} + 80 q^{21} - 56 q^{25} - 42 q^{26} - 32 q^{28} - 96 q^{30} + 32 q^{33} + 140 q^{34} - 218 q^{36} + 80 q^{37} + 324 q^{38} + 182 q^{40} + 280 q^{41} + 66 q^{44} + 270 q^{46} - 66 q^{48} - 464 q^{49} + 272 q^{53} - 286 q^{58} - 130 q^{62} - 260 q^{64} + 208 q^{65} + 444 q^{70} - 136 q^{73} - 8 q^{74} - 32 q^{77} - 238 q^{78} - 112 q^{81} + 700 q^{84} + 48 q^{85} - 392 q^{86} - 172 q^{90}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
147.1 −1.98085 0.276105i 5.42142i 3.84753 + 1.09385i 6.41467i −1.49688 + 10.7390i 11.2188i −7.31936 3.22907i −20.3918 1.77112 12.7065i
147.2 −1.98085 + 0.276105i 5.42142i 3.84753 1.09385i 6.41467i −1.49688 10.7390i 11.2188i −7.31936 + 3.22907i −20.3918 1.77112 + 12.7065i
147.3 −1.80028 0.871193i 4.32601i 2.48205 + 3.13679i 4.81171i 3.76879 7.78804i 7.95088i −1.73564 7.80945i −9.71433 4.19193 8.66244i
147.4 −1.80028 + 0.871193i 4.32601i 2.48205 3.13679i 4.81171i 3.76879 + 7.78804i 7.95088i −1.73564 + 7.80945i −9.71433 4.19193 + 8.66244i
147.5 −1.74750 0.972749i 2.88771i 2.10752 + 3.39976i 8.99975i −2.80901 + 5.04627i 1.82117i −0.375785 7.99117i 0.661152 −8.75450 + 15.7271i
147.6 −1.74750 + 0.972749i 2.88771i 2.10752 3.39976i 8.99975i −2.80901 5.04627i 1.82117i −0.375785 + 7.99117i 0.661152 −8.75450 15.7271i
147.7 −1.60738 1.19010i 2.23022i 1.16732 + 3.82588i 5.95375i −2.65419 + 3.58480i 10.9654i 2.67687 7.53886i 4.02613 7.08557 9.56992i
147.8 −1.60738 + 1.19010i 2.23022i 1.16732 3.82588i 5.95375i −2.65419 3.58480i 10.9654i 2.67687 + 7.53886i 4.02613 7.08557 + 9.56992i
147.9 −1.47472 1.35100i 2.95699i 0.349594 + 3.98469i 1.64530i 3.99490 4.36073i 5.87725i 4.86777 6.34861i 0.256206 −2.22279 + 2.42635i
147.10 −1.47472 + 1.35100i 2.95699i 0.349594 3.98469i 1.64530i 3.99490 + 4.36073i 5.87725i 4.86777 + 6.34861i 0.256206 −2.22279 2.42635i
147.11 −0.933862 1.76859i 1.32495i −2.25580 + 3.30323i 1.38998i −2.34329 + 1.23732i 11.9333i 7.94867 + 0.904826i 7.24450 2.45830 1.29805i
147.12 −0.933862 + 1.76859i 1.32495i −2.25580 3.30323i 1.38998i −2.34329 1.23732i 11.9333i 7.94867 0.904826i 7.24450 2.45830 + 1.29805i
147.13 −0.655926 1.88938i 2.57244i −3.13952 + 2.47859i 5.32643i 4.86032 1.68733i 0.691466i 6.74229 + 4.30599i 2.38255 −10.0637 + 3.49374i
147.14 −0.655926 + 1.88938i 2.57244i −3.13952 2.47859i 5.32643i 4.86032 + 1.68733i 0.691466i 6.74229 4.30599i 2.38255 −10.0637 3.49374i
147.15 −0.469745 1.94405i 5.24065i −3.55868 + 1.82642i 0.496890i −10.1881 + 2.46177i 4.23582i 5.22232 + 6.06031i −18.4644 −0.965980 + 0.233411i
147.16 −0.469745 + 1.94405i 5.24065i −3.55868 1.82642i 0.496890i −10.1881 2.46177i 4.23582i 5.22232 6.06031i −18.4644 −0.965980 0.233411i
147.17 0.469745 1.94405i 5.24065i −3.55868 1.82642i 0.496890i 10.1881 + 2.46177i 4.23582i −5.22232 + 6.06031i −18.4644 −0.965980 0.233411i
147.18 0.469745 + 1.94405i 5.24065i −3.55868 + 1.82642i 0.496890i 10.1881 2.46177i 4.23582i −5.22232 6.06031i −18.4644 −0.965980 + 0.233411i
147.19 0.655926 1.88938i 2.57244i −3.13952 2.47859i 5.32643i −4.86032 1.68733i 0.691466i −6.74229 + 4.30599i 2.38255 −10.0637 3.49374i
147.20 0.655926 + 1.88938i 2.57244i −3.13952 + 2.47859i 5.32643i −4.86032 + 1.68733i 0.691466i −6.74229 4.30599i 2.38255 −10.0637 + 3.49374i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 147.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
37.b even 2 1 inner
148.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 148.3.b.d 32
4.b odd 2 1 inner 148.3.b.d 32
37.b even 2 1 inner 148.3.b.d 32
148.b odd 2 1 inner 148.3.b.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
148.3.b.d 32 1.a even 1 1 trivial
148.3.b.d 32 4.b odd 2 1 inner
148.3.b.d 32 37.b even 2 1 inner
148.3.b.d 32 148.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(148, [\chi])\):

\( T_{3}^{16} + 106 T_{3}^{14} + 4525 T_{3}^{12} + 100754 T_{3}^{10} + 1273980 T_{3}^{8} + 9354466 T_{3}^{6} + 38751853 T_{3}^{4} + 81462714 T_{3}^{2} + 63644625 \) Copy content Toggle raw display
\( T_{5}^{16} + 214 T_{5}^{14} + 17467 T_{5}^{12} + 698102 T_{5}^{10} + 14237025 T_{5}^{8} + 135883168 T_{5}^{6} + 450840064 T_{5}^{4} + 509091840 T_{5}^{2} + 100204544 \) Copy content Toggle raw display
\( T_{19}^{16} - 1912 T_{19}^{14} + 1490544 T_{19}^{12} - 608337792 T_{19}^{10} + 138711081216 T_{19}^{8} - 17413582249984 T_{19}^{6} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display