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Properties

Label 731.2.n.a
Level $731$
Weight $2$
Character orbit 731.n
Analytic conductor $5.837$
Analytic rank $0$
Dimension $256$
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(208,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.208");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.n (of order \(12\), degree \(4\), 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: \(256\)
Relative dimension: \(64\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6} + 2 q^{10} + 4 q^{11} + 8 q^{12} - 8 q^{13} - 6 q^{14} + 248 q^{16} - 2 q^{17} + 16 q^{18} - 14 q^{20} - 16 q^{21} - 4 q^{22} + 8 q^{23} + 12 q^{24} - 12 q^{27} - 14 q^{28} + 2 q^{29} + 8 q^{30} - 24 q^{31} + 20 q^{33} + 16 q^{34} + 40 q^{35} + 18 q^{37} + 8 q^{38} + 36 q^{39} - 10 q^{40} + 8 q^{41} - 80 q^{44} - 4 q^{45} + 2 q^{46} + 24 q^{47} + 24 q^{48} + 92 q^{50} - 20 q^{51} + 4 q^{52} - 88 q^{54} - 80 q^{55} + 60 q^{56} - 44 q^{57} + 34 q^{58} - 8 q^{61} + 24 q^{62} - 26 q^{63} - 200 q^{64} - 8 q^{65} + 44 q^{67} - 58 q^{68} + 40 q^{69} - 26 q^{71} - 48 q^{72} + 36 q^{73} + 90 q^{74} - 156 q^{75} - 24 q^{78} + 22 q^{79} + 30 q^{80} + 132 q^{81} + 156 q^{82} - 160 q^{84} - 28 q^{85} + 52 q^{86} + 28 q^{88} - 20 q^{89} + 28 q^{90} + 34 q^{91} - 70 q^{92} + 40 q^{95} - 16 q^{96} - 92 q^{98} + 30 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
208.1 − 2.76205i 1.78973 + 0.479556i −5.62893 −4.22937 − 1.13326i 1.32456 − 4.94332i −0.560575 + 0.150206i 10.0233i 0.375075 + 0.216550i −3.13011 + 11.6817i
208.2 − 2.67881i −0.393175 − 0.105351i −5.17600 3.35902 + 0.900047i −0.282214 + 1.05324i −1.82716 + 0.489587i 8.50788i −2.45459 − 1.41716i 2.41105 − 8.99816i
208.3 − 2.67476i −2.11059 − 0.565531i −5.15436 2.13934 + 0.573235i −1.51266 + 5.64533i 0.737688 − 0.197663i 8.43716i 1.53669 + 0.887206i 1.53327 − 5.72224i
208.4 − 2.61030i 2.69586 + 0.722353i −4.81365 1.96090 + 0.525421i 1.88556 − 7.03699i −1.40725 + 0.377070i 7.34447i 4.14778 + 2.39472i 1.37151 − 5.11853i
208.5 − 2.53263i −2.42847 − 0.650707i −4.41419 −2.11001 − 0.565376i −1.64800 + 6.15041i −4.08395 + 1.09429i 6.11424i 2.87598 + 1.66045i −1.43189 + 5.34387i
208.6 − 2.49716i 0.186349 + 0.0499322i −4.23579 0.0534817 + 0.0143304i 0.124688 − 0.465344i 4.60333 − 1.23346i 5.58312i −2.56584 − 1.48139i 0.0357852 − 0.133552i
208.7 − 2.28700i 1.18102 + 0.316454i −3.23035 −0.126627 − 0.0339295i 0.723730 − 2.70100i −4.66179 + 1.24912i 2.81381i −1.30340 − 0.752519i −0.0775966 + 0.289595i
208.8 − 2.26939i 1.08136 + 0.289751i −3.15015 1.08055 + 0.289533i 0.657559 − 2.45404i −0.970179 + 0.259959i 2.61014i −1.51268 − 0.873348i 0.657064 − 2.45220i
208.9 − 2.24302i −2.58532 − 0.692735i −3.03116 −2.60177 − 0.697143i −1.55382 + 5.79894i 2.01210 − 0.539141i 2.31291i 3.60594 + 2.08189i −1.56371 + 5.83584i
208.10 − 2.21617i −0.0692057 − 0.0185436i −2.91143 −1.25645 − 0.336665i −0.0410959 + 0.153372i 0.0334492 − 0.00896268i 2.01988i −2.59363 − 1.49743i −0.746107 + 2.78451i
208.11 − 2.16713i −3.18659 − 0.853843i −2.69647 2.45506 + 0.657832i −1.85039 + 6.90576i 2.85934 − 0.766159i 1.50935i 6.82721 + 3.94169i 1.42561 − 5.32045i
208.12 − 2.13900i 3.06744 + 0.821917i −2.57533 −0.243980 − 0.0653743i 1.75808 − 6.56125i 3.12655 − 0.837757i 1.23062i 6.13555 + 3.54236i −0.139836 + 0.521874i
208.13 − 1.83727i 1.97172 + 0.528320i −1.37557 −3.10860 − 0.832946i 0.970668 − 3.62258i −0.484923 + 0.129935i − 1.14725i 1.01047 + 0.583396i −1.53035 + 5.71134i
208.14 − 1.81631i −1.48008 − 0.396586i −1.29900 −3.73299 − 1.00025i −0.720326 + 2.68829i 4.01589 − 1.07605i − 1.27324i −0.564718 − 0.326040i −1.81677 + 6.78028i
208.15 − 1.79720i −1.99454 − 0.534434i −1.22993 0.473839 + 0.126965i −0.960485 + 3.58458i −3.30462 + 0.885471i − 1.38398i 1.09448 + 0.631896i 0.228181 − 0.851582i
208.16 − 1.73334i 2.03902 + 0.546353i −1.00446 3.75820 + 1.00701i 0.947015 − 3.53431i 0.573555 − 0.153684i − 1.72560i 1.26102 + 0.728048i 1.74548 − 6.51423i
208.17 − 1.62839i 0.474953 + 0.127263i −0.651670 2.37962 + 0.637618i 0.207235 − 0.773411i 4.72094 − 1.26497i − 2.19561i −2.38869 − 1.37911i 1.03829 − 3.87497i
208.18 − 1.62403i −0.958554 − 0.256844i −0.637473 3.71240 + 0.994733i −0.417122 + 1.55672i −0.920765 + 0.246718i − 2.21278i −1.74522 − 1.00760i 1.61548 − 6.02904i
208.19 − 1.44677i −2.41714 − 0.647670i −0.0931316 0.138660 + 0.0371537i −0.937027 + 3.49703i 0.626167 − 0.167781i − 2.75879i 2.82501 + 1.63102i 0.0537528 − 0.200608i
208.20 − 1.24128i 2.55087 + 0.683505i 0.459214 1.75642 + 0.470632i 0.848423 − 3.16636i −1.74616 + 0.467882i − 3.05258i 3.44170 + 1.98707i 0.584188 − 2.18022i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 208.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner
43.c even 3 1 inner
731.n even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.n.a ✓ 256
17.c even 4 1 inner 731.2.n.a ✓ 256
43.c even 3 1 inner 731.2.n.a ✓ 256
731.n even 12 1 inner 731.2.n.a ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.n.a ✓ 256 1.a even 1 1 trivial
731.2.n.a ✓ 256 17.c even 4 1 inner
731.2.n.a ✓ 256 43.c even 3 1 inner
731.2.n.a ✓ 256 731.n even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).