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Properties

Label 207.4.k.a
Level $207$
Weight $4$
Character orbit 207.k
Analytic conductor $12.213$
Analytic rank $0$
Dimension $240$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,4,Mod(17,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.k (of order \(22\), degree \(10\), 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: \(12.2133953712\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 120 q^{4} - 96 q^{13} - 960 q^{16} - 648 q^{25} + 96 q^{31} - 3828 q^{34} - 2376 q^{37} + 660 q^{40} + 3432 q^{43} + 6744 q^{46} + 1992 q^{49} + 2940 q^{52} + 1584 q^{55} - 4236 q^{58} - 1056 q^{61}+ \cdots + 12504 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −5.02588 − 2.29524i 0 14.7525 + 17.0253i −10.4096 + 6.68987i 0 −21.0843 − 3.03147i −22.6141 − 77.0164i 0 67.6725 − 9.72984i
17.2 −4.68035 − 2.13744i 0 12.0981 + 13.9619i −11.9438 + 7.67580i 0 15.7604 + 2.26600i −15.1836 − 51.7105i 0 72.3076 − 10.3963i
17.3 −4.46976 − 2.04127i 0 10.5731 + 12.2020i 12.9900 − 8.34816i 0 −15.7159 − 2.25960i −11.2765 − 38.4041i 0 −75.1029 + 10.7982i
17.4 −4.13754 − 1.88955i 0 8.30996 + 9.59021i 5.59605 − 3.59636i 0 31.1534 + 4.47919i −6.00974 − 20.4673i 0 −29.9494 + 4.30607i
17.5 −3.37221 − 1.54004i 0 3.76119 + 4.34065i 12.0953 − 7.77317i 0 −21.2515 − 3.05551i 2.35679 + 8.02650i 0 −52.7588 + 7.58557i
17.6 −3.30139 − 1.50770i 0 3.38716 + 3.90899i −3.53051 + 2.26892i 0 15.8804 + 2.28326i 2.89132 + 9.84694i 0 15.0764 − 2.16766i
17.7 −2.69131 − 1.22908i 0 0.493632 + 0.569681i −11.2770 + 7.24732i 0 3.62983 + 0.521890i 6.04012 + 20.5708i 0 39.2576 − 5.64439i
17.8 −2.68424 − 1.22585i 0 0.463566 + 0.534984i −7.48507 + 4.81036i 0 −33.1778 − 4.77024i 6.06243 + 20.6467i 0 25.9886 − 3.73659i
17.9 −2.29204 − 1.04674i 0 −1.08109 − 1.24764i 13.2608 − 8.52220i 0 17.5008 + 2.51624i 6.85110 + 23.3327i 0 −39.3149 + 5.65263i
17.10 −0.918702 − 0.419557i 0 −4.57090 − 5.27510i −0.902026 + 0.579697i 0 −14.9619 − 2.15120i 4.26242 + 14.5165i 0 1.07191 − 0.154117i
17.11 −0.614837 − 0.280787i 0 −4.93970 − 5.70072i 4.23867 − 2.72403i 0 3.75888 + 0.540445i 2.95985 + 10.0803i 0 −3.37096 + 0.484671i
17.12 −0.212130 − 0.0968765i 0 −5.20327 − 6.00490i 15.1089 − 9.70992i 0 18.5077 + 2.66100i 1.04765 + 3.56796i 0 −4.14572 + 0.596065i
17.13 0.212130 + 0.0968765i 0 −5.20327 − 6.00490i −15.1089 + 9.70992i 0 18.5077 + 2.66100i −1.04765 − 3.56796i 0 −4.14572 + 0.596065i
17.14 0.614837 + 0.280787i 0 −4.93970 − 5.70072i −4.23867 + 2.72403i 0 3.75888 + 0.540445i −2.95985 − 10.0803i 0 −3.37096 + 0.484671i
17.15 0.918702 + 0.419557i 0 −4.57090 − 5.27510i 0.902026 − 0.579697i 0 −14.9619 − 2.15120i −4.26242 − 14.5165i 0 1.07191 − 0.154117i
17.16 2.29204 + 1.04674i 0 −1.08109 − 1.24764i −13.2608 + 8.52220i 0 17.5008 + 2.51624i −6.85110 − 23.3327i 0 −39.3149 + 5.65263i
17.17 2.68424 + 1.22585i 0 0.463566 + 0.534984i 7.48507 − 4.81036i 0 −33.1778 − 4.77024i −6.06243 − 20.6467i 0 25.9886 − 3.73659i
17.18 2.69131 + 1.22908i 0 0.493632 + 0.569681i 11.2770 − 7.24732i 0 3.62983 + 0.521890i −6.04012 − 20.5708i 0 39.2576 − 5.64439i
17.19 3.30139 + 1.50770i 0 3.38716 + 3.90899i 3.53051 − 2.26892i 0 15.8804 + 2.28326i −2.89132 − 9.84694i 0 15.0764 − 2.16766i
17.20 3.37221 + 1.54004i 0 3.76119 + 4.34065i −12.0953 + 7.77317i 0 −21.2515 − 3.05551i −2.35679 − 8.02650i 0 −52.7588 + 7.58557i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.k.a ✓ 240
3.b odd 2 1 inner 207.4.k.a ✓ 240
23.d odd 22 1 inner 207.4.k.a ✓ 240
69.g even 22 1 inner 207.4.k.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.4.k.a ✓ 240 1.a even 1 1 trivial
207.4.k.a ✓ 240 3.b odd 2 1 inner
207.4.k.a ✓ 240 23.d odd 22 1 inner
207.4.k.a ✓ 240 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(207, [\chi])\).