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Properties

Label 207.4.e.a
Level $207$
Weight $4$
Character orbit 207.e
Analytic conductor $12.213$
Analytic rank $0$
Dimension $60$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,4,Mod(70,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.70");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 207.e (of order \(3\), degree \(2\), 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: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 2 q^{2} - q^{3} - 108 q^{4} - 28 q^{6} + 62 q^{7} - 186 q^{8} - 73 q^{9} - 200 q^{10} + 21 q^{11} + 349 q^{12} + 196 q^{13} - 25 q^{14} - 18 q^{15} - 336 q^{16} - 66 q^{17} - 295 q^{18} - 230 q^{19}+ \cdots + 6680 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
70.1 −2.57355 + 4.45752i −3.60487 + 3.74231i −9.24630 − 16.0151i 0.902609 + 1.56336i −7.40412 − 25.6998i −3.57930 + 6.19954i 54.0064 −1.00983 − 26.9811i −9.29163
70.2 −2.55137 + 4.41911i 2.73067 + 4.42080i −9.01901 − 15.6214i 7.85281 + 13.6015i −26.5029 + 0.788011i −0.363909 + 0.630308i 51.2214 −12.0869 + 24.1435i −80.1418
70.3 −2.37008 + 4.10510i −0.535333 − 5.16850i −7.23454 − 12.5306i −5.40004 − 9.35314i 22.4860 + 10.0522i 14.9238 − 25.8488i 30.6645 −26.4268 + 5.53374i 51.1941
70.4 −2.30175 + 3.98674i 5.14035 − 0.759451i −6.59607 − 11.4247i 2.09747 + 3.63292i −8.80405 + 22.2413i 8.90896 − 15.4308i 23.9019 25.8465 − 7.80769i −19.3114
70.5 −2.05243 + 3.55491i −4.17639 − 3.09157i −4.42494 − 7.66422i 1.57198 + 2.72275i 19.5620 − 8.50147i −9.77669 + 16.9337i 3.48861 7.88443 + 25.8232i −12.9055
70.6 −1.68372 + 2.91629i −3.83845 + 3.50233i −1.66982 − 2.89222i −9.76462 − 16.9128i −3.75093 − 17.0910i 9.04552 − 15.6673i −15.6935 2.46739 − 26.8870i 65.7635
70.7 −1.60060 + 2.77231i 3.32200 + 3.99554i −1.12381 − 1.94650i −6.25381 − 10.8319i −16.3941 + 2.81439i −8.10483 + 14.0380i −18.4145 −4.92863 + 26.5463i 40.0393
70.8 −1.51967 + 2.63215i 2.06557 − 4.76796i −0.618808 − 1.07181i 9.87501 + 17.1040i 9.41099 + 12.6826i 2.48108 − 4.29736i −20.5532 −18.4669 − 19.6971i −60.0271
70.9 −1.43497 + 2.48544i −5.11636 + 0.907134i −0.118276 − 0.204859i −0.668007 − 1.15702i 5.08719 − 14.0181i 0.0673957 − 0.116733i −22.2806 25.3542 − 9.28244i 3.83428
70.10 −1.01426 + 1.75675i 0.336238 + 5.18526i 1.94255 + 3.36459i 8.32986 + 14.4277i −9.45025 − 4.66852i −6.32052 + 10.9475i −24.1092 −26.7739 + 3.48697i −33.7946
70.11 −0.811649 + 1.40582i −3.66036 − 3.68806i 2.68245 + 4.64614i 0.826466 + 1.43148i 8.15567 − 2.15239i 13.9680 − 24.1934i −21.6952 −0.203560 + 26.9992i −2.68320
70.12 −0.766015 + 1.32678i 5.19611 − 0.0210829i 2.82644 + 4.89554i −7.31529 − 12.6705i −3.95232 + 6.91022i 0.640043 − 1.10859i −20.9166 26.9991 − 0.219098i 22.4145
70.13 −0.722249 + 1.25097i 0.332165 − 5.18552i 2.95671 + 5.12118i 1.42317 + 2.46500i 6.24704 + 4.16077i −13.5745 + 23.5117i −20.0979 −26.7793 − 3.44490i −4.11153
70.14 −0.422111 + 0.731118i 0.0750198 + 5.19561i 3.64364 + 6.31098i −2.08369 − 3.60905i −3.83027 − 2.13828i 2.15206 − 3.72748i −12.9059 −26.9887 + 0.779548i 3.51819
70.15 −0.00508851 + 0.00881355i 4.46871 − 2.65154i 3.99995 + 6.92811i −2.30210 − 3.98736i 0.000630412 0.0528775i −16.3016 + 28.2353i −0.162831 12.9387 − 23.6979i 0.0468570
70.16 0.0289891 − 0.0502106i 5.05505 − 1.20271i 3.99832 + 6.92529i 8.15005 + 14.1163i 0.0861526 − 0.288682i 4.81238 − 8.33529i 0.927457 24.1070 − 12.1595i 0.945050
70.17 0.160401 − 0.277823i −4.88884 + 1.76047i 3.94854 + 6.83908i 7.85383 + 13.6032i −0.295076 + 1.64061i 16.7204 − 28.9605i 5.09982 20.8015 − 17.2133i 5.03905
70.18 0.570058 − 0.987370i 0.819092 − 5.13119i 3.35007 + 5.80249i −9.96374 − 17.2577i −4.59945 − 3.73382i 2.52132 − 4.36706i 16.7599 −25.6582 − 8.40583i −22.7196
70.19 0.707928 − 1.22617i 4.51263 + 2.57607i 2.99768 + 5.19213i −6.22282 − 10.7782i 6.35331 − 3.70957i 15.4063 − 26.6845i 19.8154 13.7277 + 23.2497i −17.6212
70.20 0.932770 − 1.61560i −3.08980 + 4.17770i 2.25988 + 3.91423i 5.34727 + 9.26174i 3.86743 + 8.88872i −9.50172 + 16.4575i 23.3561 −7.90628 − 25.8165i 19.9511
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 70.30
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.4.e.a ✓ 60
9.c even 3 1 inner 207.4.e.a ✓ 60
9.c even 3 1 1863.4.a.g 30
9.d odd 6 1 1863.4.a.h 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
207.4.e.a ✓ 60 1.a even 1 1 trivial
207.4.e.a ✓ 60 9.c even 3 1 inner
1863.4.a.g 30 9.c even 3 1
1863.4.a.h 30 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 2 T_{2}^{59} + 176 T_{2}^{58} - 266 T_{2}^{57} + 17012 T_{2}^{56} - 19960 T_{2}^{55} + \cdots + 46\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(207, [\chi])\). Copy content Toggle raw display