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Properties

Label 731.2.bh.a
Level $731$
Weight $2$
Character orbit 731.bh
Analytic conductor $5.837$
Analytic rank $0$
Dimension $1536$
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(13,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([21, 64]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("731.13");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.bh (of order \(84\), degree \(24\), 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: \(1536\)
Relative dimension: \(64\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1536 q - 22 q^{3} + 208 q^{4} - 30 q^{5} - 12 q^{6} - 14 q^{7} - 30 q^{10} - 32 q^{11} - 64 q^{12} - 76 q^{13} - 50 q^{14} - 304 q^{16} - 26 q^{17} - 72 q^{18} - 70 q^{20} - 40 q^{21} - 66 q^{22} - 36 q^{23} + 58 q^{24} - 16 q^{27} + 42 q^{28} - 30 q^{29} - 204 q^{30} - 4 q^{31} - 216 q^{33} + 40 q^{34} - 96 q^{35} - 4 q^{37} + 48 q^{38} - 64 q^{39} - 18 q^{40} - 36 q^{41} - 88 q^{44} + 172 q^{45} - 142 q^{46} - 80 q^{47} - 136 q^{48} + 20 q^{50} - 50 q^{51} - 60 q^{52} + 60 q^{54} + 80 q^{55} - 88 q^{56} + 16 q^{57} - 160 q^{58} - 6 q^{61} - 52 q^{62} - 100 q^{63} + 144 q^{64} - 20 q^{65} - 100 q^{67} + 30 q^{68} - 96 q^{69} - 2 q^{71} + 580 q^{72} + 62 q^{73} - 118 q^{74} - 124 q^{75} + 388 q^{78} - 36 q^{79} - 44 q^{80} - 188 q^{81} + 12 q^{82} + 216 q^{84} + 112 q^{85} - 332 q^{86} - 42 q^{88} + 76 q^{89} - 140 q^{90} + 92 q^{91} - 140 q^{92} + 44 q^{95} - 292 q^{96} - 84 q^{97} - 20 q^{98} + 320 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −2.74144 − 0.625715i −0.430197 − 0.0160968i 5.32202 + 2.56295i −1.96972 − 2.66887i 1.16929 + 0.313309i −0.660349 − 2.46446i −8.58939 − 6.84981i −2.80680 − 0.210341i 3.72990 + 8.54903i
13.2 −2.66167 − 0.607509i 2.00780 + 0.0751267i 4.91349 + 2.36621i 2.09544 + 2.83922i −5.29847 − 1.41972i 1.25466 + 4.68245i −7.37161 − 5.87867i 1.03402 + 0.0774891i −3.85253 − 8.83008i
13.3 −2.57960 − 0.588776i −3.03566 − 0.113587i 4.50572 + 2.16984i 1.25804 + 1.70458i 7.76391 + 2.08033i 0.317974 + 1.18669i −6.20805 − 4.95075i 6.21075 + 0.465431i −2.24162 − 5.13784i
13.4 −2.49240 − 0.568874i 0.335866 + 0.0125672i 4.08650 + 1.96796i −1.01898 − 1.38067i −0.829963 − 0.222388i 1.02117 + 3.81105i −5.06819 − 4.04174i −2.87896 − 0.215748i 1.75429 + 4.02087i
13.5 −2.40696 − 0.549373i 2.99547 + 0.112083i 3.68971 + 1.77687i 0.0489901 + 0.0663792i −7.14841 − 1.91541i −0.797071 − 2.97471i −4.04437 − 3.22528i 5.96869 + 0.447291i −0.0814503 − 0.186686i
13.6 −2.34214 − 0.534578i 0.295194 + 0.0110454i 3.39791 + 1.63635i 0.0899681 + 0.121902i −0.685480 − 0.183674i −0.259001 − 0.966606i −3.32712 − 2.65329i −2.90459 − 0.217669i −0.145551 − 0.333607i
13.7 −2.27687 − 0.519681i 1.99644 + 0.0747016i 3.11213 + 1.49872i −0.407293 − 0.551863i −4.50682 − 1.20760i −0.238104 − 0.888617i −2.65525 − 2.11749i 0.988594 + 0.0740848i 0.640561 + 1.46818i
13.8 −2.22997 − 0.508977i −2.07186 − 0.0775236i 2.91178 + 1.40224i −1.29034 − 1.74835i 4.58074 + 1.22740i 0.658840 + 2.45882i −2.20288 − 1.75674i 1.29499 + 0.0970462i 1.98755 + 4.55551i
13.9 −2.20558 − 0.503410i −1.08397 − 0.0405595i 2.80924 + 1.35286i 1.31943 + 1.78777i 2.37038 + 0.635141i 0.0632641 + 0.236105i −1.97749 − 1.57700i −1.81826 − 0.136259i −2.01014 − 4.60729i
13.10 −2.14644 − 0.489912i −3.15413 − 0.118019i 2.56527 + 1.23537i −0.955672 − 1.29489i 6.71234 + 1.79857i −1.07677 − 4.01857i −1.45835 − 1.16300i 6.94299 + 0.520305i 1.41691 + 3.24760i
13.11 −1.91310 − 0.436653i 1.16771 + 0.0436927i 1.66735 + 0.802954i −2.60787 − 3.53354i −2.21487 − 0.593473i −0.817724 − 3.05179i 0.229177 + 0.182763i −1.62997 − 0.122149i 3.44619 + 7.89875i
13.12 −1.81482 − 0.414221i −1.61287 − 0.0603495i 1.32006 + 0.635709i 0.0618915 + 0.0838599i 2.90208 + 0.777610i −1.28801 − 4.80691i 0.778392 + 0.620747i −0.393892 − 0.0295181i −0.0775854 − 0.177828i
13.13 −1.80551 − 0.412095i 2.45695 + 0.0919327i 1.28809 + 0.620313i −1.62848 − 2.20652i −4.39816 − 1.17848i 0.829958 + 3.09744i 0.825775 + 0.658534i 3.03656 + 0.227559i 2.03094 + 4.65497i
13.14 −1.64571 − 0.375623i 1.18471 + 0.0443288i 0.765333 + 0.368565i 1.97036 + 2.66974i −1.93304 − 0.517957i 0.416191 + 1.55325i 1.51844 + 1.21092i −1.59004 − 0.119157i −2.23982 − 5.13373i
13.15 −1.64304 − 0.375014i 3.30670 + 0.123728i 0.757013 + 0.364558i 1.28377 + 1.73945i −5.38664 − 1.44335i 0.307074 + 1.14602i 1.52814 + 1.21865i 7.92733 + 0.594071i −1.45697 − 3.33941i
13.16 −1.63874 − 0.374032i 0.664766 + 0.0248738i 0.743635 + 0.358116i 1.32754 + 1.79875i −1.08008 − 0.289406i −0.643546 − 2.40175i 1.54366 + 1.23103i −2.55032 − 0.191120i −1.50270 − 3.44423i
13.17 −1.57309 − 0.359048i 1.88273 + 0.0704466i 0.543769 + 0.261865i −0.939998 − 1.27365i −2.93641 − 0.786808i 0.771538 + 2.87942i 1.76167 + 1.40488i 0.548081 + 0.0410730i 1.02140 + 2.34108i
13.18 −1.51328 − 0.345396i −2.72865 − 0.102099i 0.368770 + 0.177590i 0.902833 + 1.22330i 4.09393 + 1.09697i 1.34368 + 5.01468i 1.93039 + 1.53944i 4.44347 + 0.332992i −0.943715 − 2.16302i
13.19 −1.44415 − 0.329617i −1.80522 − 0.0675468i 0.174972 + 0.0842623i −2.00662 − 2.71888i 2.58474 + 0.692580i 0.117241 + 0.437550i 2.09132 + 1.66777i 0.262663 + 0.0196839i 2.00167 + 4.58787i
13.20 −1.26218 − 0.288084i −2.65893 − 0.0994903i −0.291831 − 0.140538i 2.58345 + 3.50045i 3.32739 + 0.891572i −0.554513 − 2.06947i 2.35224 + 1.87585i 4.06842 + 0.304886i −2.25235 − 5.16244i
See next 80 embeddings (of 1536 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.64
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.c even 4 1 inner
43.g even 21 1 inner
731.bh even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 731.2.bh.a ✓ 1536
17.c even 4 1 inner 731.2.bh.a ✓ 1536
43.g even 21 1 inner 731.2.bh.a ✓ 1536
731.bh even 84 1 inner 731.2.bh.a ✓ 1536
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
731.2.bh.a ✓ 1536 1.a even 1 1 trivial
731.2.bh.a ✓ 1536 17.c even 4 1 inner
731.2.bh.a ✓ 1536 43.g even 21 1 inner
731.2.bh.a ✓ 1536 731.bh even 84 1 inner

Hecke kernels

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