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Properties

Label 207.4.m.a
Level $207$
Weight $4$
Character orbit 207.m
Analytic conductor $12.213$
Analytic rank $0$
Dimension $1400$
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(4,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([22, 12]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.m (of order \(33\), degree \(20\), 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: \(1400\)
Relative dimension: \(70\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1400 q - 9 q^{2} - 14 q^{3} + 263 q^{4} + 19 q^{5} - 35 q^{6} - 9 q^{7} - 74 q^{8} - 78 q^{9} - 68 q^{10} + 79 q^{11} + 54 q^{12} - 9 q^{13} + 159 q^{14} - 868 q^{15} + 1031 q^{16} - 308 q^{17} - 87 q^{18}+ \cdots + 7484 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −3.26722 + 4.58817i −1.71615 + 4.90457i −7.76003 − 22.4211i −14.9593 + 7.71206i −16.8960 − 23.8983i 12.7056 − 9.99183i 84.9902 + 24.9554i −21.1097 − 16.8339i 13.4911 − 93.8328i
4.2 −3.11000 + 4.36738i 0.691500 − 5.14993i −6.78539 − 19.6051i 6.99819 − 3.60782i 20.3412 + 19.0363i 28.6222 − 22.5088i 65.5705 + 19.2532i −26.0437 − 7.12236i −6.00764 + 41.7841i
4.3 −3.09157 + 4.34150i −4.81737 − 1.94756i −6.67428 − 19.2841i 6.06640 − 3.12745i 23.3485 − 14.8936i −10.1357 + 7.97080i 63.4448 + 18.6291i 19.4141 + 18.7642i −5.17688 + 36.0060i
4.4 −2.95477 + 4.14939i 4.94560 + 1.59406i −5.87025 − 16.9610i 0.0815901 − 0.0420626i −21.2275 + 15.8112i 1.55780 − 1.22507i 48.6222 + 14.2768i 21.9180 + 15.7672i −0.0665456 + 0.462835i
4.5 −2.91301 + 4.09075i 1.41219 + 5.00057i −5.63207 − 16.2728i 10.3413 − 5.33132i −24.5698 − 8.78978i −21.3348 + 16.7779i 44.4262 + 13.0447i −23.0114 + 14.1236i −8.31527 + 57.8340i
4.6 −2.78793 + 3.91510i −2.45036 − 4.58211i −4.93890 − 14.2700i −19.3913 + 9.99691i 24.7708 + 3.18118i −10.7080 + 8.42089i 32.7449 + 9.61476i −14.9914 + 22.4557i 14.9227 − 103.790i
4.7 −2.76588 + 3.88413i 2.37999 − 4.61906i −4.81987 − 13.9261i −2.51163 + 1.29484i 11.3583 + 22.0199i −13.8589 + 10.8987i 30.8209 + 9.04983i −15.6713 − 21.9866i 1.91755 − 13.3369i
4.8 −2.70102 + 3.79305i −3.62300 + 3.72477i −4.47519 − 12.9302i 11.5740 − 5.96683i −4.34245 − 23.8029i 15.3081 − 12.0384i 25.3897 + 7.45510i −0.747791 − 26.9896i −8.62921 + 60.0174i
4.9 −2.69115 + 3.77920i 4.57480 − 2.46398i −4.42348 − 12.7808i −10.4806 + 5.40314i −2.99960 + 23.9200i 4.10886 − 3.23125i 24.5932 + 7.22121i 14.8576 − 22.5445i 7.78546 − 54.1490i
4.10 −2.46695 + 3.46434i −4.27108 + 2.95937i −3.29930 − 9.53272i −2.65977 + 1.37121i 0.284244 − 22.0971i −16.6811 + 13.1182i 8.51848 + 2.50125i 9.48422 − 25.2794i 1.81118 − 12.5970i
4.11 −2.41610 + 3.39293i 4.94150 − 1.60674i −3.05792 − 8.83529i 19.5280 − 10.0674i −6.48755 + 20.6482i −0.748547 + 0.588664i 5.39337 + 1.58363i 21.8367 − 15.8794i −13.0236 + 90.5811i
4.12 −2.33527 + 3.27943i −5.18945 − 0.263780i −2.68463 − 7.75672i −8.48781 + 4.37577i 12.9838 − 16.4025i 13.6708 − 10.7509i 0.804118 + 0.236110i 26.8608 + 2.73775i 5.47131 − 38.0538i
4.13 −2.19970 + 3.08904i −1.31400 − 5.02727i −2.08697 − 6.02992i 7.73363 − 3.98697i 18.4198 + 6.99946i −12.7782 + 10.0489i −5.89140 − 1.72987i −23.5468 + 13.2117i −4.69575 + 32.6597i
4.14 −2.19107 + 3.07693i 0.578436 + 5.16386i −2.05016 − 5.92354i 10.6961 − 5.51421i −17.1562 − 9.53457i 16.0768 − 12.6430i −6.27628 − 1.84288i −26.3308 + 5.97392i −6.46903 + 44.9931i
4.15 −2.18200 + 3.06419i 3.37679 + 3.94934i −2.01161 − 5.81215i −5.70696 + 2.94214i −19.4697 + 1.72968i 19.1429 − 15.0541i −6.67574 − 1.96018i −4.19453 + 26.6722i 3.43731 − 23.9070i
4.16 −2.16862 + 3.04540i 3.13722 + 4.14221i −1.95501 − 5.64863i −16.6844 + 8.60138i −19.4181 + 0.571212i −19.7783 + 15.5539i −7.25554 − 2.13042i −7.31574 + 25.9900i 9.98737 − 69.4637i
4.17 −1.98001 + 2.78054i −4.08591 − 3.21019i −1.19440 − 3.45100i 13.0530 − 6.72929i 17.0162 − 5.00482i 11.9856 − 9.42560i −14.2411 − 4.18156i 6.38935 + 26.2331i −7.13406 + 49.6185i
4.18 −1.82839 + 2.56761i −2.26950 + 4.67433i −0.633090 − 1.82919i −7.14677 + 3.68442i −7.85234 − 14.3737i −9.19892 + 7.23411i −18.3410 − 5.38542i −16.6987 − 21.2168i 3.60692 − 25.0867i
4.19 −1.69742 + 2.38370i 3.62079 − 3.72691i −0.184220 − 0.532269i −15.8263 + 8.15904i 2.73782 + 14.9570i 16.2228 − 12.7578i −20.8807 − 6.13112i −0.779765 − 26.9887i 7.41529 − 51.5745i
4.20 −1.57756 + 2.21538i −1.25640 − 5.04197i 0.197356 + 0.570222i −6.16483 + 3.17819i 13.1519 + 5.17063i 21.8224 − 17.1613i −22.4506 − 6.59209i −23.8429 + 12.6694i 2.68452 − 18.6712i
See next 80 embeddings (of 1400 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.70
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner
23.c even 11 1 inner
207.m even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.m.a ✓ 1400
9.c even 3 1 inner 207.4.m.a ✓ 1400
23.c even 11 1 inner 207.4.m.a ✓ 1400
207.m even 33 1 inner 207.4.m.a ✓ 1400
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.4.m.a ✓ 1400 1.a even 1 1 trivial
207.4.m.a ✓ 1400 9.c even 3 1 inner
207.4.m.a ✓ 1400 23.c even 11 1 inner
207.4.m.a ✓ 1400 207.m even 33 1 inner

Hecke kernels

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