Properties

Label 387.3.r.a
Level $387$
Weight $3$
Character orbit 387.r
Analytic conductor $10.545$
Analytic rank $0$
Dimension $172$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,3,Mod(7,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 387.r (of order \(6\), 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: \(10.5449862307\)
Analytic rank: \(0\)
Dimension: \(172\)
Relative dimension: \(86\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 172 q - 3 q^{3} + 166 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 172 q - 3 q^{3} + 166 q^{4} - 3 q^{5} - 9 q^{6} - 3 q^{7} + 15 q^{9} - 6 q^{10} - 8 q^{11} - 15 q^{12} - 12 q^{13} - 66 q^{14} - 19 q^{15} - 314 q^{16} + 16 q^{17} - 24 q^{18} - 3 q^{19} + q^{21} + 36 q^{22} + q^{23} - 98 q^{24} + 391 q^{25} - 18 q^{26} + 36 q^{27} + 30 q^{28} - 111 q^{29} - 39 q^{30} - 15 q^{31} - 90 q^{32} + 138 q^{33} + 81 q^{34} - 132 q^{35} + 179 q^{36} - 6 q^{37} - 78 q^{38} - 39 q^{39} + 40 q^{41} + 9 q^{42} - 63 q^{43} - 32 q^{44} - 30 q^{45} - 18 q^{46} - 17 q^{47} + 120 q^{48} + 541 q^{49} + 138 q^{51} - 75 q^{52} - 182 q^{53} + 71 q^{54} + 69 q^{55} - 270 q^{56} - 72 q^{57} - 18 q^{58} - 17 q^{59} - 308 q^{60} + 18 q^{61} - 18 q^{62} + 162 q^{63} - 1112 q^{64} + 120 q^{65} - 354 q^{66} - 111 q^{67} + 1210 q^{68} + 225 q^{69} - 297 q^{70} - 150 q^{71} + 84 q^{72} + 48 q^{73} + 36 q^{74} + 117 q^{75} - 15 q^{76} - 2 q^{78} + 18 q^{79} + 489 q^{80} + 515 q^{81} - 122 q^{83} + 131 q^{84} - 75 q^{85} - 177 q^{86} + 339 q^{87} + 180 q^{88} - 402 q^{89} + 835 q^{90} - 429 q^{91} + 574 q^{92} + 588 q^{93} + 252 q^{94} - 390 q^{95} - 680 q^{96} + 107 q^{97} + 333 q^{98} - 122 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −3.33195 1.92370i −2.87889 0.843786i 5.40128 + 9.35529i −5.56892 3.21522i 7.96914 + 8.34960i 11.0230 6.36414i 26.1722i 7.57605 + 4.85834i 12.3703 + 21.4259i
7.2 −3.31385 1.91325i 2.97388 + 0.395039i 5.32107 + 9.21636i −3.68685 2.12860i −9.09917 6.99888i −2.02569 + 1.16953i 25.4162i 8.68789 + 2.34959i 8.14511 + 14.1077i
7.3 −3.26338 1.88411i −0.988151 2.83259i 5.09976 + 8.83304i 3.54051 + 2.04412i −2.11220 + 11.1056i −2.60622 + 1.50470i 23.3612i −7.04711 + 5.59805i −7.70268 13.3414i
7.4 −3.21802 1.85793i −0.00307485 + 3.00000i 4.90377 + 8.49358i 3.25716 + 1.88052i 5.58367 9.64834i −4.57150 + 2.63936i 21.5800i −8.99998 0.0184491i −6.98774 12.1031i
7.5 −3.15952 1.82415i −2.09171 + 2.15053i 4.65504 + 8.06277i −4.21054 2.43095i 10.5317 2.97905i −6.38869 + 3.68851i 19.3728i −0.249528 8.99654i 8.86885 + 15.3613i
7.6 −3.12567 1.80461i 1.46355 2.61878i 4.51320 + 7.81709i −0.264425 0.152666i −9.30044 + 5.54431i 7.65018 4.41683i 18.1413i −4.71604 7.66544i 0.551003 + 0.954365i
7.7 −3.11003 1.79558i −2.85359 + 0.925757i 4.44820 + 7.70451i 6.98116 + 4.03058i 10.5370 + 2.24471i 4.86998 2.81168i 17.5838i 7.28595 5.28346i −14.4744 25.0705i
7.8 −2.88886 1.66788i 0.527498 2.95326i 3.56367 + 6.17245i −7.21156 4.16359i −6.44956 + 7.65174i −8.45660 + 4.88242i 10.4320i −8.44349 3.11568i 13.8888 + 24.0561i
7.9 −2.86728 1.65542i 2.96172 0.477749i 3.48086 + 6.02902i 7.89462 + 4.55796i −9.28294 3.53306i 8.80554 5.08388i 9.80579i 8.54351 2.82991i −15.0907 26.1379i
7.10 −2.82415 1.63052i 2.33397 + 1.88483i 3.31720 + 5.74556i 0.737130 + 0.425582i −3.51821 9.12863i 2.72591 1.57381i 8.59090i 1.89481 + 8.79828i −1.38784 2.40381i
7.11 −2.79671 1.61468i −2.76326 1.16807i 3.21438 + 5.56746i 2.89880 + 1.67363i 5.84196 + 7.72854i −8.94913 + 5.16678i 7.84331i 6.27121 + 6.45538i −5.40474 9.36128i
7.12 −2.79056 1.61113i 0.481819 + 2.96106i 3.19150 + 5.52784i −6.91867 3.99450i 3.42611 9.03929i 5.55234 3.20565i 7.67867i −8.53570 + 2.85339i 12.8713 + 22.2938i
7.13 −2.57467 1.48649i 2.09092 + 2.15129i 2.41930 + 4.19035i 6.34282 + 3.66203i −2.18557 8.64700i −5.59057 + 3.22772i 2.49313i −0.256102 + 8.99636i −10.8871 18.8571i
7.14 −2.55908 1.47749i 2.41765 1.77622i 2.36594 + 4.09793i 2.13001 + 1.22976i −8.81132 + 0.973459i −5.45608 + 3.15007i 2.16269i 2.69006 8.58857i −3.63391 6.29412i
7.15 −2.48598 1.43528i −2.97921 + 0.352563i 2.12006 + 3.67206i −1.19193 0.688159i 7.91229 + 3.39954i −2.78187 + 1.60611i 0.689300i 8.75140 2.10072i 1.97540 + 3.42150i
7.16 −2.39965 1.38544i −1.96256 2.26900i 1.83889 + 3.18505i −5.76682 3.32948i 1.56589 + 8.16382i −0.255369 + 0.147437i 0.892832i −1.29674 + 8.90609i 9.22558 + 15.9792i
7.17 −2.39615 1.38342i −1.28201 + 2.71228i 1.82768 + 3.16563i 1.72160 + 0.993964i 6.82409 4.72547i 8.68573 5.01471i 0.953570i −5.71292 6.95432i −2.75013 4.76336i
7.18 −2.30517 1.33089i −1.40078 2.65289i 1.54253 + 2.67174i 1.93397 + 1.11658i −0.301653 + 7.97963i 5.35565 3.09209i 2.43536i −5.07560 + 7.43224i −2.97209 5.14781i
7.19 −2.22575 1.28503i 2.41565 1.77894i 1.30263 + 2.25622i −5.35135 3.08960i −7.66262 + 0.855267i 6.79901 3.92541i 3.58459i 2.67075 8.59460i 7.94049 + 13.7533i
7.20 −2.02913 1.17152i 2.99843 0.0971885i 0.744905 + 1.29021i −1.17017 0.675599i −6.19804 3.31550i −8.94397 + 5.16380i 5.88146i 8.98111 0.582825i 1.58295 + 2.74175i
See next 80 embeddings (of 172 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.86
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
387.r odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 387.3.r.a yes 172
9.c even 3 1 387.3.n.a 172
43.d odd 6 1 387.3.n.a 172
387.r odd 6 1 inner 387.3.r.a yes 172
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
387.3.n.a 172 9.c even 3 1
387.3.n.a 172 43.d odd 6 1
387.3.r.a yes 172 1.a even 1 1 trivial
387.3.r.a yes 172 387.r odd 6 1 inner

Hecke kernels

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