Properties

Label 208.4.p.a
Level $208$
Weight $4$
Character orbit 208.p
Analytic conductor $12.272$
Analytic rank $0$
Dimension $164$
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] = [208,4,Mod(77,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.77");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.p (of order \(4\), 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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 4 q^{3} - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 164 q - 4 q^{3} - 4 q^{4} - 4 q^{10} - 184 q^{12} - 2 q^{13} + 172 q^{14} - 4 q^{16} - 8 q^{17} + 284 q^{22} - 284 q^{26} - 160 q^{27} - 4 q^{29} - 380 q^{30} - 504 q^{35} - 768 q^{36} - 224 q^{38} - 744 q^{40} + 792 q^{42} + 428 q^{43} - 2428 q^{48} + 6852 q^{49} - 112 q^{51} - 380 q^{52} - 4 q^{53} - 876 q^{56} - 4 q^{61} - 2192 q^{62} - 544 q^{64} + 484 q^{65} + 4424 q^{66} + 1304 q^{68} + 104 q^{69} - 5544 q^{74} + 1708 q^{75} - 1376 q^{77} + 4232 q^{78} + 3152 q^{79} - 10700 q^{81} - 7200 q^{82} + 3960 q^{88} + 5796 q^{90} + 1960 q^{91} + 3372 q^{92} - 5068 q^{94} + 6072 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −2.82494 + 0.140455i 6.40502 + 6.40502i 7.96054 0.793553i 2.11354 + 2.11354i −18.9934 17.1942i 17.9223 −22.3766 + 3.35983i 55.0486i −6.26748 5.67377i
77.2 −2.82451 + 0.148855i −2.12524 2.12524i 7.95568 0.840882i −12.7715 12.7715i 6.31910 + 5.68639i 12.9494 −22.3457 + 3.55932i 17.9667i 37.9744 + 34.1722i
77.3 −2.81839 0.238103i −3.35598 3.35598i 7.88661 + 1.34213i 4.60987 + 4.60987i 8.65939 + 10.2575i −11.3328 −21.9080 5.66047i 4.47474i −11.8948 14.0900i
77.4 −2.81645 0.260067i 1.73668 + 1.73668i 7.86473 + 1.46493i −5.75097 5.75097i −4.43961 5.34292i −21.8456 −21.7696 6.17126i 20.9679i 14.7016 + 17.6929i
77.5 −2.78961 + 0.466994i −5.87109 5.87109i 7.56383 2.60546i 14.3059 + 14.3059i 19.1198 + 13.6363i 13.5187 −19.8834 + 10.8005i 41.9395i −46.5885 33.2270i
77.6 −2.77687 + 0.537582i −0.574375 0.574375i 7.42201 2.98559i 1.31602 + 1.31602i 1.90374 + 1.28619i 32.3407 −19.0050 + 12.2805i 26.3402i −4.36189 2.94695i
77.7 −2.77335 + 0.555438i 2.56937 + 2.56937i 7.38298 3.08085i 7.06578 + 7.06578i −8.55291 5.69865i −19.8768 −18.7644 + 12.6451i 13.7966i −23.5205 15.6713i
77.8 −2.62224 1.06013i 3.73069 + 3.73069i 5.75225 + 5.55982i −8.80493 8.80493i −5.82774 13.7378i 9.35030 −9.18965 20.6773i 0.836117i 13.7542 + 32.4230i
77.9 −2.62201 1.06070i 0.373419 + 0.373419i 5.74983 + 5.56233i 7.38008 + 7.38008i −0.583021 1.37519i −12.9160 −9.17611 20.6833i 26.7211i −11.5226 27.1787i
77.10 −2.60879 1.09281i −6.01467 6.01467i 5.61154 + 5.70181i −4.12290 4.12290i 9.11812 + 22.2639i −6.88029 −8.40833 21.0071i 45.3526i 6.25022 + 15.2613i
77.11 −2.60768 + 1.09546i −4.82487 4.82487i 5.59994 5.71320i −5.24233 5.24233i 17.8671 + 7.29625i −32.3272 −8.34425 + 21.0327i 19.5588i 19.4130 + 7.92753i
77.12 −2.53500 + 1.25450i 5.09937 + 5.09937i 4.85245 6.36033i −10.2683 10.2683i −19.3241 6.52973i −7.54188 −4.32190 + 22.2108i 25.0072i 38.9116 + 13.1485i
77.13 −2.47917 1.36152i −4.76379 4.76379i 4.29254 + 6.75086i −0.130556 0.130556i 5.32425 + 18.2962i 15.3712 −1.45052 22.5809i 18.3873i 0.145916 + 0.501425i
77.14 −2.47587 1.36750i 1.48929 + 1.48929i 4.25989 + 6.77151i 14.2987 + 14.2987i −1.65068 5.72388i 19.3394 −1.28690 22.5908i 22.5641i −15.8483 54.9554i
77.15 −2.41290 1.47578i 6.93624 + 6.93624i 3.64414 + 7.12182i 7.79313 + 7.79313i −6.50005 26.9728i −35.4139 1.71731 22.5622i 69.2228i −7.30306 30.3050i
77.16 −2.41193 + 1.47737i 3.87772 + 3.87772i 3.63478 7.12659i 12.4454 + 12.4454i −15.0816 3.62397i −4.13238 1.76175 + 22.5587i 3.07347i −48.4036 11.6309i
77.17 −2.37565 + 1.53502i −7.05438 7.05438i 3.28740 7.29335i −7.17035 7.17035i 27.5874 + 5.93007i 18.7503 3.38577 + 22.3727i 72.5286i 28.0409 + 6.02756i
77.18 −2.22790 + 1.74254i −2.07519 2.07519i 1.92708 7.76443i 1.67350 + 1.67350i 8.23941 + 1.00720i −4.91715 9.23652 + 20.6564i 18.3872i −6.64455 0.812245i
77.19 −2.18649 + 1.79423i 0.279568 + 0.279568i 1.56151 7.84613i −9.66462 9.66462i −1.11288 0.109666i 2.26833 10.6635 + 19.9572i 26.8437i 38.4721 + 3.79113i
77.20 −2.05920 1.93899i 4.45439 + 4.45439i 0.480628 + 7.98555i −2.81162 2.81162i −0.535468 17.8095i 15.8553 14.4942 17.3758i 12.6832i 0.337989 + 11.2414i
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
16.e even 4 1 inner
208.p even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.p.a 164
13.b even 2 1 inner 208.4.p.a 164
16.e even 4 1 inner 208.4.p.a 164
208.p even 4 1 inner 208.4.p.a 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
208.4.p.a 164 1.a even 1 1 trivial
208.4.p.a 164 13.b even 2 1 inner
208.4.p.a 164 16.e even 4 1 inner
208.4.p.a 164 208.p even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(208, [\chi])\).