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Properties

Label 387.3.x.a
Level $387$
Weight $3$
Character orbit 387.x
Analytic conductor $10.545$
Analytic rank $0$
Dimension $168$
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] = [387,3,Mod(35,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.35");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.x (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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 48 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q + 48 q^{4} + 16 q^{7} + 16 q^{10} - 44 q^{13} - 136 q^{16} - 64 q^{19} + 72 q^{22} + 156 q^{25} - 8 q^{28} - 204 q^{31} + 32 q^{34} + 432 q^{37} + 24 q^{40} - 224 q^{43} - 268 q^{46} + 528 q^{49} + 184 q^{52} + 724 q^{55} - 228 q^{58} - 152 q^{61} - 832 q^{64} + 280 q^{67} + 1036 q^{70} + 692 q^{73} - 696 q^{76} - 184 q^{79} - 128 q^{82} - 904 q^{85} + 472 q^{88} + 216 q^{91} - 264 q^{94} - 1044 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −3.70954 − 0.846677i 0 9.43991 + 4.54602i −0.613113 − 0.488941i 0 0.765348 −19.2694 − 15.3668i 0 1.86039 + 2.33285i
35.2 −3.39581 − 0.775072i 0 7.32694 + 3.52847i −7.13113 − 5.68688i 0 −0.414590 −11.2532 − 8.97411i 0 19.8082 + 24.8387i
35.3 −3.35799 − 0.766438i 0 7.08477 + 3.41184i 1.46215 + 1.16603i 0 5.45947 −10.4040 − 8.29691i 0 −4.01619 − 5.03615i
35.4 −2.88548 − 0.658593i 0 4.28840 + 2.06518i 5.60174 + 4.46723i 0 −6.37370 −1.75807 − 1.40202i 0 −13.2216 − 16.5794i
35.5 −2.81230 − 0.641890i 0 3.89314 + 1.87484i 4.79314 + 3.82240i 0 −13.2047 −0.724091 − 0.577443i 0 −11.0262 − 13.8264i
35.6 −2.66349 − 0.607925i 0 3.12075 + 1.50288i −5.22800 − 4.16919i 0 3.50368 1.14536 + 0.913390i 0 11.3902 + 14.2828i
35.7 −2.37238 − 0.541481i 0 1.73113 + 0.833666i 3.95044 + 3.15037i 0 8.46052 3.95454 + 3.15364i 0 −7.66609 − 9.61297i
35.8 −1.81454 − 0.414158i 0 −0.482836 − 0.232522i −3.81085 − 3.03905i 0 −11.1643 6.60043 + 5.26367i 0 5.65630 + 7.09277i
35.9 −1.80080 − 0.411021i 0 −0.529936 − 0.255204i 0.314903 + 0.251127i 0 −3.13917 6.62593 + 5.28400i 0 −0.463858 − 0.581660i
35.10 −1.13869 − 0.259899i 0 −2.37480 − 1.14364i −1.36734 − 1.09042i 0 −0.592310 6.05958 + 4.83236i 0 1.27358 + 1.59702i
35.11 −1.13292 − 0.258581i 0 −2.38724 − 1.14963i −3.68993 − 2.94262i 0 9.50268 6.04139 + 4.81785i 0 3.41948 + 4.28789i
35.12 −0.938476 − 0.214201i 0 −2.76902 − 1.33349i 6.65817 + 5.30971i 0 11.4683 5.32342 + 4.24529i 0 −5.11119 − 6.40923i
35.13 −0.681400 − 0.155525i 0 −3.16376 − 1.52359i 2.97161 + 2.36978i 0 −0.572469 4.10459 + 3.27330i 0 −1.65630 − 2.07693i
35.14 −0.228931 − 0.0522519i 0 −3.55420 − 1.71161i −4.60961 − 3.67604i 0 −8.68666 1.45858 + 1.16318i 0 0.863200 + 1.08242i
35.15 0.228931 + 0.0522519i 0 −3.55420 − 1.71161i 4.60961 + 3.67604i 0 −8.68666 −1.45858 − 1.16318i 0 0.863200 + 1.08242i
35.16 0.681400 + 0.155525i 0 −3.16376 − 1.52359i −2.97161 − 2.36978i 0 −0.572469 −4.10459 − 3.27330i 0 −1.65630 − 2.07693i
35.17 0.938476 + 0.214201i 0 −2.76902 − 1.33349i −6.65817 − 5.30971i 0 11.4683 −5.32342 − 4.24529i 0 −5.11119 − 6.40923i
35.18 1.13292 + 0.258581i 0 −2.38724 − 1.14963i 3.68993 + 2.94262i 0 9.50268 −6.04139 − 4.81785i 0 3.41948 + 4.28789i
35.19 1.13869 + 0.259899i 0 −2.37480 − 1.14364i 1.36734 + 1.09042i 0 −0.592310 −6.05958 − 4.83236i 0 1.27358 + 1.59702i
35.20 1.80080 + 0.411021i 0 −0.529936 − 0.255204i −0.314903 − 0.251127i 0 −3.13917 −6.62593 − 5.28400i 0 −0.463858 − 0.581660i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
43.e even 7 1 inner
129.l odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.x.a ✓ 168
3.b odd 2 1 inner 387.3.x.a ✓ 168
43.e even 7 1 inner 387.3.x.a ✓ 168
129.l odd 14 1 inner 387.3.x.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.3.x.a ✓ 168 1.a even 1 1 trivial
387.3.x.a ✓ 168 3.b odd 2 1 inner
387.3.x.a ✓ 168 43.e even 7 1 inner
387.3.x.a ✓ 168 129.l odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(387, [\chi])\).