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Properties

Label 704.2.bb.a
Level $704$
Weight $2$
Character orbit 704.bb
Analytic conductor $5.621$
Analytic rank $0$
Dimension $752$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,2,Mod(43,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 13, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.bb (of order \(16\), degree \(8\), 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: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(752\)
Relative dimension: \(94\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 752 q - 16 q^{3} - 16 q^{4} - 16 q^{5} - 16 q^{9} - 8 q^{11} - 16 q^{12} - 16 q^{14} - 16 q^{15} - 16 q^{16} - 16 q^{20} - 16 q^{22} - 16 q^{23} - 16 q^{25} - 96 q^{26} - 16 q^{27} - 32 q^{31} + 64 q^{34} - 16 q^{36} - 16 q^{37} - 16 q^{38} - 16 q^{42} - 16 q^{44} - 16 q^{45} - 16 q^{47} - 16 q^{48} - 16 q^{49} - 16 q^{53} + 56 q^{55} + 96 q^{56} - 16 q^{58} + 48 q^{59} - 112 q^{60} + 176 q^{64} - 120 q^{66} - 176 q^{67} - 16 q^{69} - 112 q^{70} + 48 q^{71} - 144 q^{75} - 8 q^{77} + 80 q^{78} - 144 q^{80} - 16 q^{81} - 16 q^{82} - 224 q^{86} + 72 q^{88} - 16 q^{89} - 16 q^{91} + 288 q^{92} - 64 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41421 + 0.00442422i 0.0817065 − 0.0162524i 1.99996 − 0.0125135i −2.92873 + 1.95691i −0.115478 + 0.0233458i −1.30288 − 0.539669i −2.82830 + 0.0265450i −2.76523 + 1.14539i 4.13317 − 2.78044i
43.2 −1.41392 + 0.0289031i −1.82585 + 0.363184i 1.99833 − 0.0817331i 2.94107 − 1.96516i 2.57111 − 0.566285i −2.81046 − 1.16413i −2.82311 + 0.173322i 0.430188 − 0.178190i −4.10164 + 2.86358i
43.3 −1.41355 − 0.0433311i 2.68872 − 0.534819i 1.99624 + 0.122501i −0.572159 + 0.382305i −3.82381 + 0.639488i 2.34658 + 0.971986i −2.81648 − 0.259661i 4.17153 − 1.72790i 0.825341 − 0.515614i
43.4 −1.40845 + 0.127505i −3.24886 + 0.646238i 1.96748 − 0.359171i 0.0197178 − 0.0131750i 4.49347 − 1.32444i 3.48451 + 1.44333i −2.72532 + 0.756740i 7.36583 − 3.05103i −0.0260917 + 0.0210705i
43.5 −1.40529 − 0.158625i 1.54013 − 0.306352i 1.94968 + 0.445829i 3.10682 − 2.07591i −2.21293 + 0.186209i 2.06828 + 0.856709i −2.66914 − 0.935787i −0.493475 + 0.204404i −4.69528 + 2.42444i
43.6 −1.39847 + 0.210411i 1.42138 − 0.282730i 1.91145 − 0.588508i −0.930269 + 0.621586i −1.92827 + 0.694464i −3.03394 − 1.25670i −2.54929 + 1.22520i −0.831254 + 0.344317i 1.17017 − 1.06501i
43.7 −1.38263 − 0.297200i −0.468233 + 0.0931373i 1.82334 + 0.821835i −1.45774 + 0.974032i 0.675074 + 0.0103839i 4.36920 + 1.80978i −2.27677 − 1.67819i −2.56107 + 1.06083i 2.30500 − 0.913488i
43.8 −1.36098 − 0.384345i −2.83862 + 0.564637i 1.70456 + 1.04618i −1.35948 + 0.908374i 4.08033 + 0.322549i −4.52193 − 1.87304i −1.91778 − 2.07897i 4.96731 − 2.05753i 2.19936 − 0.713774i
43.9 −1.35452 − 0.406547i −1.14130 + 0.227020i 1.66944 + 1.10135i 1.75603 − 1.17334i 1.63821 + 0.156491i 1.59784 + 0.661847i −1.81354 − 2.17050i −1.52060 + 0.629854i −2.85559 + 0.875404i
43.10 −1.35328 + 0.410659i 0.534118 − 0.106243i 1.66272 − 1.11147i 1.73015 − 1.15605i −0.679180 + 0.363116i −1.38756 − 0.574745i −1.79368 + 2.18694i −2.49764 + 1.03456i −1.86663 + 2.27496i
43.11 −1.34036 − 0.451053i 0.913978 − 0.181802i 1.59310 + 1.20914i 0.0509126 − 0.0340187i −1.30706 − 0.168574i −2.12931 − 0.881989i −1.58994 − 2.33925i −1.96933 + 0.815725i −0.0835852 + 0.0226329i
43.12 −1.32670 + 0.489775i −1.90389 + 0.378706i 1.52024 − 1.29956i −3.04856 + 2.03698i 2.34039 − 1.43490i −0.0328429 − 0.0136040i −1.38040 + 2.46870i 0.709723 − 0.293977i 3.04685 − 4.19557i
43.13 −1.32045 + 0.506381i 2.75372 − 0.547749i 1.48716 − 1.33730i −0.520544 + 0.347816i −3.35877 + 2.11770i 2.64892 + 1.09722i −1.28653 + 2.51890i 4.51130 − 1.86864i 0.511223 − 0.722867i
43.14 −1.26428 + 0.633722i 0.0763910 − 0.0151951i 1.19679 − 1.60240i 1.16650 − 0.779432i −0.0869500 + 0.0676215i 3.33252 + 1.38037i −0.497603 + 2.78431i −2.76603 + 1.14573i −0.980839 + 1.72466i
43.15 −1.25606 + 0.649856i −1.47360 + 0.293116i 1.15538 − 1.63252i −1.36122 + 0.909540i 1.66044 − 1.32580i 0.265004 + 0.109768i −0.390321 + 2.80137i −0.686073 + 0.284181i 1.11871 − 2.02704i
43.16 −1.21447 − 0.724616i 3.10190 − 0.617007i 0.949863 + 1.76005i −2.95835 + 1.97671i −4.21425 − 1.49835i −2.04164 − 0.845675i 0.121781 − 2.82580i 6.46946 − 2.67974i 5.02517 − 0.256977i
43.17 −1.20138 − 0.746109i −0.989108 + 0.196746i 0.886644 + 1.79272i 0.844270 − 0.564123i 1.33509 + 0.501615i −1.19499 − 0.494980i 0.272368 − 2.81528i −1.83201 + 0.758845i −1.43519 + 0.0478112i
43.18 −1.17347 + 0.789279i −2.31360 + 0.460204i 0.754077 − 1.85239i 1.28547 − 0.858926i 2.35172 − 2.36612i −3.26175 − 1.35106i 0.577167 + 2.76891i 2.36933 − 0.981410i −0.830536 + 2.02252i
43.19 −1.14090 − 0.835669i 2.19475 − 0.436563i 0.603314 + 1.90683i 2.27539 − 1.52037i −2.86881 − 1.33601i 0.620035 + 0.256827i 0.905160 − 2.67968i 1.85469 − 0.768240i −3.86653 − 0.166884i
43.20 −1.11915 + 0.864577i 2.64801 − 0.526722i 0.505012 − 1.93519i −2.50123 + 1.67127i −2.50814 + 2.87889i −3.10969 − 1.28808i 1.10794 + 2.60240i 3.96288 − 1.64148i 1.35432 − 4.03292i
See next 80 embeddings (of 752 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.94
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
64.j odd 16 1 inner
704.bb even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.2.bb.a ✓ 752
11.b odd 2 1 inner 704.2.bb.a ✓ 752
64.j odd 16 1 inner 704.2.bb.a ✓ 752
704.bb even 16 1 inner 704.2.bb.a ✓ 752
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
704.2.bb.a ✓ 752 1.a even 1 1 trivial
704.2.bb.a ✓ 752 11.b odd 2 1 inner
704.2.bb.a ✓ 752 64.j odd 16 1 inner
704.2.bb.a ✓ 752 704.bb even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(704, [\chi])\).