Properties

Label 392.3.r.a
Level $392$
Weight $3$
Character orbit 392.r
Analytic conductor $10.681$
Analytic rank $0$
Dimension $660$
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] = [392,3,Mod(13,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.13");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.r (of order \(14\), degree \(6\), 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: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(660\)
Relative dimension: \(110\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 660 q - 5 q^{2} - 5 q^{4} - 7 q^{6} - 12 q^{7} - 11 q^{8} - 328 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 660 q - 5 q^{2} - 5 q^{4} - 7 q^{6} - 12 q^{7} - 11 q^{8} - 328 q^{9} - 7 q^{10} + 28 q^{12} - 73 q^{14} + 26 q^{15} - 29 q^{16} - 14 q^{17} + 64 q^{18} - 7 q^{20} - 25 q^{22} - 10 q^{23} + 119 q^{24} - 520 q^{25} - 7 q^{26} + 140 q^{28} + 16 q^{30} - 145 q^{32} - 14 q^{33} - 35 q^{34} + 65 q^{36} - 35 q^{38} + 26 q^{39} + 385 q^{40} - 14 q^{41} - 127 q^{42} + 249 q^{44} - 187 q^{46} - 14 q^{47} - 28 q^{49} + 16 q^{50} - 7 q^{52} + 56 q^{54} - 14 q^{55} - 406 q^{56} - 4 q^{57} - 147 q^{58} - 75 q^{60} + 21 q^{62} - 56 q^{63} - 275 q^{64} + 90 q^{65} + 77 q^{66} + 979 q^{70} + 630 q^{71} + 608 q^{72} - 14 q^{73} + 349 q^{74} + 406 q^{76} + 525 q^{78} - 24 q^{79} - 900 q^{81} + 378 q^{82} + 658 q^{84} - 363 q^{86} - 14 q^{87} + 263 q^{88} - 14 q^{89} - 700 q^{90} - 506 q^{92} - 784 q^{94} - 644 q^{95} - 770 q^{96} + 107 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.99978 0.0298008i −1.84727 2.31641i 3.99822 + 0.119190i 4.36230 + 5.47015i 3.62510 + 4.68735i 6.52431 2.53641i −7.99201 0.357504i 0.0493668 0.216290i −8.56062 11.0691i
13.2 −1.99517 + 0.138981i 1.70976 + 2.14397i 3.96137 0.554578i −2.50322 3.13894i −3.70922 4.03995i −6.98044 + 0.522965i −7.82651 + 1.65703i 0.329359 1.44301i 5.43060 + 5.91481i
13.3 −1.99438 0.149795i 1.68073 + 2.10757i 3.95512 + 0.597496i −1.71847 2.15489i −3.03632 4.45506i 6.99470 0.272369i −7.79853 1.78409i 0.385696 1.68984i 3.10449 + 4.55509i
13.4 −1.99383 + 0.156983i −3.05720 3.83361i 3.95071 0.625994i 0.514788 + 0.645524i 6.69735 + 7.16364i 0.812904 + 6.95264i −7.77878 + 1.86832i −3.34740 + 14.6659i −1.12774 1.20625i
13.5 −1.96609 + 0.366726i 0.511085 + 0.640881i 3.73102 1.44203i 3.48690 + 4.37244i −1.23987 1.07260i −6.98216 0.499436i −6.80670 + 4.20343i 1.85317 8.11926i −8.45905 7.31787i
13.6 −1.96241 + 0.385929i −1.88164 2.35950i 3.70212 1.51470i −0.816179 1.02346i 4.60314 + 3.90413i 1.23528 6.89014i −6.68051 + 4.40122i −0.0239853 + 0.105086i 1.99666 + 1.69345i
13.7 −1.95195 0.435775i 2.96624 + 3.71955i 3.62020 + 1.70122i 4.05361 + 5.08307i −4.16906 8.55298i 0.462548 + 6.98470i −6.32509 4.89829i −3.03377 + 13.2918i −5.69737 11.6884i
13.8 −1.94864 0.450319i −1.53176 1.92077i 3.59443 + 1.75502i −4.77246 5.98448i 2.11990 + 4.43268i −4.17425 + 5.61922i −6.21394 5.03855i 0.659631 2.89003i 6.60491 + 13.8107i
13.9 −1.94810 0.452655i 2.06701 + 2.59195i 3.59021 + 1.76364i 2.47925 + 3.10888i −2.85349 5.98503i −3.31785 6.16375i −6.19577 5.06087i −0.442986 + 1.94085i −3.42258 7.17865i
13.10 −1.93661 0.499555i −2.76411 3.46609i 3.50089 + 1.93488i 5.25857 + 6.59403i 3.62150 + 8.09328i −6.83976 1.48919i −5.81326 5.49600i −2.37076 + 10.3870i −6.88969 15.3970i
13.11 −1.90927 + 0.595568i 0.746108 + 0.935590i 3.29060 2.27419i −5.68661 7.13079i −1.98173 1.34193i 0.995644 6.92883i −4.92819 + 6.30182i 1.68404 7.37825i 15.1041 + 10.2278i
13.12 −1.89610 0.636232i −0.400805 0.502593i 3.19042 + 2.41272i −0.300414 0.376707i 0.440202 + 1.20797i 6.96498 + 0.699351i −4.51431 6.60462i 1.91073 8.37147i 0.329943 + 0.905409i
13.13 −1.88791 + 0.660135i 3.44719 + 4.32263i 3.12844 2.49256i −4.19833 5.26454i −9.36152 5.88516i 0.385894 + 6.98936i −4.26081 + 6.77093i −4.79939 + 21.0275i 11.4014 + 7.16753i
13.14 −1.87964 + 0.683340i −0.0855824 0.107317i 3.06609 2.56887i 2.41719 + 3.03106i 0.234198 + 0.143235i −1.93020 + 6.72862i −4.00774 + 6.92373i 1.99850 8.75598i −6.61468 4.04553i
13.15 −1.84922 0.761826i −3.59610 4.50937i 2.83924 + 2.81757i −4.46752 5.60209i 3.21464 + 11.0784i 5.11740 4.77621i −3.10389 7.37332i −5.39978 + 23.6580i 3.99361 + 13.7630i
13.16 −1.81490 + 0.840317i −1.58246 1.98434i 2.58773 3.05019i −3.75053 4.70302i 4.53948 + 2.27161i 4.49567 + 5.36553i −2.13336 + 7.71030i 0.569259 2.49409i 10.7589 + 5.38387i
13.17 −1.81351 0.843320i 3.58008 + 4.48927i 2.57762 + 3.05874i −3.12084 3.91341i −2.70660 11.1605i 2.29482 6.61315i −2.09504 7.72080i −5.33395 + 23.3696i 2.35941 + 9.72886i
13.18 −1.81340 + 0.843560i 2.52002 + 3.16001i 2.57681 3.05942i 3.38083 + 4.23943i −7.23546 3.60456i 6.07141 3.48397i −2.09198 + 7.72163i −1.63245 + 7.15223i −9.70700 4.83583i
13.19 −1.81153 0.847560i −0.878129 1.10114i 2.56328 + 3.07076i −1.30470 1.63604i 0.657476 + 2.73901i −4.87731 5.02114i −2.04081 7.73531i 1.56129 6.84047i 0.976858 + 4.06954i
13.20 −1.70471 + 1.04593i −3.23276 4.05375i 1.81206 3.56601i −0.985719 1.23605i 9.75086 + 3.52923i −6.43628 2.75214i 0.640762 + 7.97430i −3.97949 + 17.4353i 2.97319 + 1.07612i
See next 80 embeddings (of 660 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.110
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
49.f odd 14 1 inner
392.r odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.r.a 660
8.b even 2 1 inner 392.3.r.a 660
49.f odd 14 1 inner 392.3.r.a 660
392.r odd 14 1 inner 392.3.r.a 660
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.3.r.a 660 1.a even 1 1 trivial
392.3.r.a 660 8.b even 2 1 inner
392.3.r.a 660 49.f odd 14 1 inner
392.3.r.a 660 392.r odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(392, [\chi])\).