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Properties

Label 731.2.s.b
Level $731$
Weight $2$
Character orbit 731.s
Analytic conductor $5.837$
Analytic rank $0$
Dimension $496$
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] = [731,2,Mod(214,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.s (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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(496\)
Relative dimension: \(62\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9} - 16 q^{10} + 16 q^{13} - 16 q^{14} - 16 q^{15} - 16 q^{17} - 16 q^{21} - 80 q^{23} - 64 q^{24} + 48 q^{31} - 96 q^{35} + 16 q^{36} - 32 q^{38} - 16 q^{40} - 16 q^{41} - 48 q^{43} - 80 q^{44} - 96 q^{47} - 16 q^{49} + 96 q^{52} - 16 q^{53} + 144 q^{54} - 16 q^{56} - 48 q^{57} - 64 q^{58} + 16 q^{59} - 224 q^{60} - 48 q^{64} - 272 q^{66} - 16 q^{68} - 112 q^{74} + 64 q^{78} - 80 q^{79} + 96 q^{81} + 16 q^{83} + 96 q^{86} - 16 q^{87} + 64 q^{90} + 368 q^{92} - 96 q^{95} + 80 q^{96} - 80 q^{97} - 128 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −2.59137 − 1.07338i 1.43157 − 2.14250i 4.14883 + 4.14883i 0.218697 + 1.09946i −6.00945 + 4.01538i 0.509391 − 2.56088i −4.15111 − 10.0217i −1.39286 − 3.36265i 0.613418 − 3.08386i
214.2 −2.49245 − 1.03241i −1.39302 + 2.08480i 3.73222 + 3.73222i 0.435881 + 2.19132i 5.62439 − 3.75810i −0.139368 + 0.700650i −3.38438 − 8.17063i −1.25784 − 3.03670i 1.17592 − 5.91177i
214.3 −2.43313 − 1.00784i −0.227777 + 0.340892i 3.49018 + 3.49018i −0.210988 − 1.06071i 0.897773 − 0.599873i −0.282349 + 1.41946i −2.95886 − 7.14332i 1.08373 + 2.61634i −0.555659 + 2.79349i
214.4 −2.39134 − 0.990524i 1.43792 − 2.15200i 3.32313 + 3.32313i −0.286571 − 1.44069i −5.57015 + 3.72185i −0.790858 + 3.97591i −2.67404 − 6.45570i −1.41543 − 3.41716i −0.741749 + 3.72902i
214.5 −2.24727 − 0.930848i 0.466380 − 0.697987i 2.76951 + 2.76951i 0.751697 + 3.77904i −1.69780 + 1.13443i −0.800996 + 4.02688i −1.78414 − 4.30730i 0.878375 + 2.12058i 1.82845 − 9.19222i
214.6 −2.22769 − 0.922741i −0.269266 + 0.402985i 2.69695 + 2.69695i 0.606709 + 3.05013i 0.971692 − 0.649264i 0.588740 − 2.95980i −1.67391 − 4.04118i 1.05816 + 2.55462i 1.46292 − 7.35460i
214.7 −2.19579 − 0.909528i −1.68196 + 2.51724i 2.58006 + 2.58006i −0.588645 − 2.95932i 5.98274 − 3.99754i 0.916876 − 4.60945i −1.49959 − 3.62032i −2.35943 − 5.69616i −1.39904 + 7.03344i
214.8 −2.07104 − 0.857855i −0.0129146 + 0.0193280i 2.13910 + 2.13910i −0.719326 − 3.61630i 0.0433273 − 0.0289504i −0.318313 + 1.60027i −0.879419 − 2.12311i 1.14784 + 2.77114i −1.61250 + 8.10659i
214.9 −2.04983 − 0.849068i 0.990599 − 1.48254i 2.06668 + 2.06668i 0.292642 + 1.47121i −3.28933 + 2.19786i 0.459436 − 2.30974i −0.783452 − 1.89142i −0.0685766 − 0.165558i 0.649291 − 3.26421i
214.10 −2.03061 − 0.841105i 1.24216 − 1.85903i 2.00170 + 2.00170i −0.565245 − 2.84168i −4.08599 + 2.73017i 0.401952 − 2.02075i −0.698811 − 1.68708i −0.764972 − 1.84681i −1.24236 + 6.24577i
214.11 −1.93285 − 0.800613i 0.211222 − 0.316117i 1.68072 + 1.68072i −0.167026 − 0.839698i −0.661349 + 0.441899i 0.314174 − 1.57946i −0.301744 − 0.728474i 1.09274 + 2.63810i −0.349436 + 1.75673i
214.12 −1.90221 − 0.787922i −1.35593 + 2.02929i 1.58338 + 1.58338i −0.106216 − 0.533986i 4.17819 − 2.79178i −0.224543 + 1.12885i −0.188497 − 0.455072i −1.13143 − 2.73151i −0.218693 + 1.09945i
214.13 −1.67323 − 0.693076i −1.35744 + 2.03155i 0.905142 + 0.905142i 0.494506 + 2.48605i 3.67934 − 2.45845i 0.453283 − 2.27881i 0.498970 + 1.20462i −1.13651 − 2.74379i 0.895597 − 4.50247i
214.14 −1.55591 − 0.644478i −0.529491 + 0.792439i 0.591285 + 0.591285i −0.134545 − 0.676403i 1.33455 − 0.891717i −0.834241 + 4.19401i 0.750042 + 1.81076i 0.800452 + 1.93246i −0.226588 + 1.13913i
214.15 −1.44571 − 0.598831i −0.336754 + 0.503988i 0.317255 + 0.317255i 0.505403 + 2.54083i 0.788652 − 0.526960i 0.214808 − 1.07991i 0.928987 + 2.24277i 1.00745 + 2.43220i 0.790865 − 3.97595i
214.16 −1.35638 − 0.561829i −0.901163 + 1.34869i 0.109890 + 0.109890i −0.614380 − 3.08870i 1.98005 − 1.32302i 0.564221 − 2.83653i 1.03635 + 2.50196i 0.141192 + 0.340868i −0.901990 + 4.53461i
214.17 −1.34652 − 0.557745i 0.877192 − 1.31281i 0.0878103 + 0.0878103i 0.0560935 + 0.282001i −1.91337 + 1.27847i −0.596391 + 2.99826i 1.04623 + 2.52582i 0.194044 + 0.468464i 0.0817538 − 0.411004i
214.18 −1.30999 − 0.542617i 1.88023 − 2.81397i 0.00743503 + 0.00743503i −0.424086 − 2.13202i −3.99000 + 2.66603i 0.264470 − 1.32958i 1.07953 + 2.60621i −3.23509 − 7.81019i −0.601323 + 3.02305i
214.19 −1.25875 − 0.521390i 1.34853 − 2.01822i −0.101617 − 0.101617i 0.174697 + 0.878263i −2.74974 + 1.83732i −0.443684 + 2.23055i 1.11771 + 2.69839i −1.10663 − 2.67164i 0.238018 − 1.19660i
214.20 −1.16026 − 0.480596i −0.484485 + 0.725082i −0.298982 − 0.298982i 0.671308 + 3.37489i 0.910600 − 0.608443i −0.264895 + 1.33172i 1.16440 + 2.81111i 0.857031 + 2.06906i 0.843067 − 4.23838i
See next 80 embeddings (of 496 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.62
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.e odd 16 1 inner
43.b odd 2 1 inner
731.s even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.s.b ✓ 496
17.e odd 16 1 inner 731.2.s.b ✓ 496
43.b odd 2 1 inner 731.2.s.b ✓ 496
731.s even 16 1 inner 731.2.s.b ✓ 496
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.s.b ✓ 496 1.a even 1 1 trivial
731.2.s.b ✓ 496 17.e odd 16 1 inner
731.2.s.b ✓ 496 43.b odd 2 1 inner
731.2.s.b ✓ 496 731.s even 16 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{496} + 8 T_{2}^{494} + 32 T_{2}^{492} - 56 T_{2}^{490} + 32610 T_{2}^{488} + \cdots + 38\!\cdots\!16 \) acting on \(S_{2}^{\mathrm{new}}(731, [\chi])\). Copy content Toggle raw display