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Properties

Label 232.4.q.a
Level $232$
Weight $4$
Character orbit 232.q
Analytic conductor $13.688$
Analytic rank $0$
Dimension $132$
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] = [232,4,Mod(9,232)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(232, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("232.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 232 = 2^{3} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 232.q (of order \(14\), 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: \(13.6884431213\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 18 q^{5} + 12 q^{7} + 152 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 18 q^{5} + 12 q^{7} + 152 q^{9} + 58 q^{13} + 294 q^{15} + 602 q^{21} - 76 q^{23} - 98 q^{25} - 38 q^{29} + 6 q^{33} - 200 q^{35} + 144 q^{45} - 742 q^{47} - 3164 q^{49} + 600 q^{51} + 666 q^{53} + 4116 q^{55} + 876 q^{57} + 6704 q^{59} + 1260 q^{61} - 446 q^{63} + 42 q^{65} - 268 q^{67} - 5068 q^{69} + 1722 q^{71} - 2814 q^{73} + 2506 q^{77} + 818 q^{81} - 3304 q^{83} - 3432 q^{87} + 2310 q^{89} + 692 q^{91} - 942 q^{93} + 420 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −7.99883 + 6.37885i 0 −1.60826 + 0.774496i 0 −3.96220 − 4.96844i 0 17.2834 − 75.7236i 0
9.2 0 −6.56462 + 5.23511i 0 3.39354 − 1.63424i 0 17.1701 + 21.5306i 0 9.67980 − 42.4100i 0
9.3 0 −5.45667 + 4.35155i 0 −8.29921 + 3.99669i 0 −19.8307 − 24.8669i 0 4.83119 − 21.1668i 0
9.4 0 −4.79317 + 3.82242i 0 −19.4124 + 9.34854i 0 10.0791 + 12.6388i 0 2.35547 − 10.3200i 0
9.5 0 −4.79041 + 3.82022i 0 12.3263 − 5.93601i 0 −6.21401 − 7.79212i 0 2.34584 − 10.2778i 0
9.6 0 −4.37353 + 3.48777i 0 6.37020 − 3.06772i 0 −9.34933 − 11.7237i 0 0.955122 − 4.18466i 0
9.7 0 −3.45920 + 2.75862i 0 −7.10505 + 3.42161i 0 13.5144 + 16.9465i 0 −1.65198 + 7.23778i 0
9.8 0 −3.40585 + 2.71608i 0 17.2545 − 8.30931i 0 6.16684 + 7.73297i 0 −1.78530 + 7.82192i 0
9.9 0 −2.66640 + 2.12638i 0 −16.2461 + 7.82369i 0 −14.7116 − 18.4478i 0 −3.41989 + 14.9835i 0
9.10 0 −0.402449 + 0.320942i 0 2.42065 − 1.16572i 0 −5.90107 − 7.39970i 0 −5.94910 + 26.0647i 0
9.11 0 −0.0600110 + 0.0478572i 0 −9.16686 + 4.41453i 0 −0.530985 − 0.665834i 0 −6.00675 + 26.3173i 0
9.12 0 0.880641 − 0.702288i 0 6.95493 − 3.34932i 0 −18.0786 − 22.6699i 0 −5.72574 + 25.0861i 0
9.13 0 1.05582 − 0.841989i 0 −0.101731 + 0.0489910i 0 13.7177 + 17.2015i 0 −5.60225 + 24.5451i 0
9.14 0 1.98123 − 1.57998i 0 7.24477 − 3.48890i 0 17.7844 + 22.3009i 0 −4.57912 + 20.0625i 0
9.15 0 2.60418 − 2.07676i 0 17.7851 − 8.56485i 0 11.4646 + 14.3761i 0 −3.53926 + 15.5065i 0
9.16 0 3.16921 − 2.52736i 0 −9.00426 + 4.33622i 0 2.79456 + 3.50427i 0 −2.35172 + 10.3036i 0
9.17 0 3.33338 − 2.65828i 0 9.41156 − 4.53237i 0 −11.4843 − 14.4008i 0 −1.96310 + 8.60088i 0
9.18 0 4.53231 − 3.61439i 0 −11.7574 + 5.66209i 0 −13.5822 − 17.0316i 0 1.46990 − 6.44004i 0
9.19 0 6.11357 − 4.87541i 0 −13.4031 + 6.45458i 0 −2.29136 − 2.87328i 0 7.59806 − 33.2893i 0
9.20 0 6.21801 − 4.95870i 0 7.28351 − 3.50755i 0 −9.02046 − 11.3113i 0 8.06692 − 35.3435i 0
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 232.4.q.a ✓ 132
29.e even 14 1 inner 232.4.q.a ✓ 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.q.a ✓ 132 1.a even 1 1 trivial
232.4.q.a ✓ 132 29.e even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(232, [\chi])\).