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Properties

Label 232.3.w.a
Level $232$
Weight $3$
Character orbit 232.w
Analytic conductor $6.322$
Analytic rank $0$
Dimension $84$
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] = [232,3,Mod(73,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 0, 27]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 232.w (of order \(28\), degree \(12\), 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: \(6.32154213316\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 4 q^{3} - 28 q^{5} + 34 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 4 q^{3} - 28 q^{5} + 34 q^{7} + 56 q^{11} - 64 q^{15} + 34 q^{17} + 6 q^{19} + 28 q^{21} - 58 q^{23} + 102 q^{25} - 256 q^{27} - 6 q^{29} + 38 q^{31} + 224 q^{33} + 46 q^{37} + 128 q^{39} - 18 q^{41} + 104 q^{43} + 286 q^{45} + 150 q^{47} - 290 q^{49} - 126 q^{51} + 46 q^{53} + 2 q^{55} - 228 q^{59} - 358 q^{61} - 28 q^{63} - 36 q^{65} - 70 q^{67} + 684 q^{69} - 224 q^{71} - 882 q^{73} + 504 q^{75} + 188 q^{77} - 222 q^{79} - 330 q^{81} - 164 q^{83} - 416 q^{85} - 612 q^{87} + 838 q^{89} + 742 q^{91} - 1036 q^{93} - 214 q^{95} + 208 q^{97} - 1296 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 0 −3.98422 + 2.50345i 0 −1.23708 − 0.986539i 0 −2.20644 + 9.66706i 0 5.70178 − 11.8399i 0
73.2 0 −3.73362 + 2.34599i 0 2.42997 + 1.93784i 0 2.92299 − 12.8065i 0 4.53129 − 9.40931i 0
73.3 0 −0.949609 + 0.596679i 0 −1.83736 − 1.46524i 0 0.839369 − 3.67751i 0 −3.35922 + 6.97550i 0
73.4 0 −0.00308849 + 0.00194063i 0 −4.99707 − 3.98503i 0 −0.892484 + 3.91023i 0 −3.90495 + 8.10871i 0
73.5 0 0.347708 − 0.218479i 0 5.98221 + 4.77065i 0 −0.984501 + 4.31338i 0 −3.83179 + 7.95679i 0
73.6 0 2.92151 − 1.83571i 0 −4.02866 − 3.21275i 0 1.98195 − 8.68349i 0 1.26045 − 2.61736i 0
73.7 0 3.90812 − 2.45563i 0 2.16263 + 1.72464i 0 0.0396906 − 0.173896i 0 5.33831 − 11.0851i 0
89.1 0 −3.98422 − 2.50345i 0 −1.23708 + 0.986539i 0 −2.20644 − 9.66706i 0 5.70178 + 11.8399i 0
89.2 0 −3.73362 − 2.34599i 0 2.42997 − 1.93784i 0 2.92299 + 12.8065i 0 4.53129 + 9.40931i 0
89.3 0 −0.949609 − 0.596679i 0 −1.83736 + 1.46524i 0 0.839369 + 3.67751i 0 −3.35922 − 6.97550i 0
89.4 0 −0.00308849 − 0.00194063i 0 −4.99707 + 3.98503i 0 −0.892484 − 3.91023i 0 −3.90495 − 8.10871i 0
89.5 0 0.347708 + 0.218479i 0 5.98221 − 4.77065i 0 −0.984501 − 4.31338i 0 −3.83179 − 7.95679i 0
89.6 0 2.92151 + 1.83571i 0 −4.02866 + 3.21275i 0 1.98195 + 8.68349i 0 1.26045 + 2.61736i 0
89.7 0 3.90812 + 2.45563i 0 2.16263 − 1.72464i 0 0.0396906 + 0.173896i 0 5.33831 + 11.0851i 0
97.1 0 −4.81126 − 1.68353i 0 4.64564 + 1.06034i 0 −0.0986052 + 0.0474857i 0 13.2775 + 10.5884i 0
97.2 0 −2.98506 − 1.04452i 0 −4.85817 − 1.10885i 0 5.27316 − 2.53942i 0 0.783062 + 0.624471i 0
97.3 0 −0.497216 − 0.173983i 0 −0.146130 − 0.0333533i 0 1.88398 − 0.907279i 0 −6.81953 − 5.43839i 0
97.4 0 −0.0224870 − 0.00786855i 0 2.35740 + 0.538062i 0 −11.4901 + 5.53332i 0 −7.03604 − 5.61105i 0
97.5 0 1.91568 + 0.670325i 0 8.65906 + 1.97637i 0 7.38603 − 3.55692i 0 −3.81599 − 3.04315i 0
97.6 0 2.34989 + 0.822262i 0 −8.98362 − 2.05045i 0 0.317971 − 0.153127i 0 −2.19062 − 1.74696i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.3.w.a ✓ 84
29.f odd 28 1 inner 232.3.w.a ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.3.w.a ✓ 84 1.a even 1 1 trivial
232.3.w.a ✓ 84 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} + 4 T_{3}^{83} + 8 T_{3}^{82} + 156 T_{3}^{81} - 203 T_{3}^{80} - 4048 T_{3}^{79} + \cdots + 68\!\cdots\!56 \) acting on \(S_{3}^{\mathrm{new}}(232, [\chi])\). Copy content Toggle raw display