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Properties

Label 816.2.cr.a
Level $816$
Weight $2$
Character orbit 816.cr
Analytic conductor $6.516$
Analytic rank $0$
Dimension $1120$
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] = [816,2,Mod(5,816)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(816, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([0, 4, 8, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("816.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.cr (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: \(6.51579280494\)
Analytic rank: \(0\)
Dimension: \(1120\)
Relative dimension: \(140\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 8 q^{3} - 16 q^{4} - 8 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1120 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{10} + 16 q^{12} - 32 q^{13} - 16 q^{15} - 16 q^{18} - 48 q^{19} - 16 q^{21} - 16 q^{22} - 8 q^{24} - 8 q^{27} - 64 q^{28} - 8 q^{30} - 32 q^{31} - 16 q^{34} - 8 q^{36} - 16 q^{37} - 48 q^{39} + 48 q^{40} + 8 q^{42} - 16 q^{43} - 8 q^{45} - 16 q^{46} - 8 q^{48} - 32 q^{49} - 88 q^{51} - 32 q^{52} - 8 q^{54} - 80 q^{58} - 8 q^{60} - 80 q^{61} - 16 q^{63} - 16 q^{64} - 24 q^{66} - 16 q^{69} + 48 q^{70} - 112 q^{72} - 8 q^{75} + 48 q^{76} - 56 q^{78} - 32 q^{79} - 16 q^{81} - 112 q^{82} - 64 q^{84} - 16 q^{85} - 16 q^{88} + 24 q^{90} - 128 q^{91} - 8 q^{93} + 64 q^{94} + 96 q^{96} - 32 q^{97} - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41411 − 0.0173937i −0.336444 + 1.69906i 1.99939 + 0.0491931i −0.937320 − 1.40280i 0.505321 − 2.39680i 0.958747 − 0.190707i −2.82650 − 0.104341i −2.77361 − 1.14328i 1.30107 + 2.00001i
5.2 −1.41315 + 0.0549299i −1.18761 + 1.26078i 1.99397 − 0.155248i 1.33017 + 1.99074i 1.60901 − 1.84691i 3.92892 − 0.781510i −2.80924 + 0.328917i −0.179155 − 2.99465i −1.98908 − 2.74014i
5.3 −1.41273 + 0.0646761i 1.33024 + 1.10926i 1.99163 − 0.182740i −2.08474 − 3.12004i −1.95102 − 1.48106i −3.23441 + 0.643364i −2.80183 + 0.386975i 0.539075 + 2.95117i 3.14698 + 4.27295i
5.4 −1.40972 + 0.112593i 1.65146 + 0.522201i 1.97465 − 0.317449i 2.36256 + 3.53582i −2.38689 − 0.550217i −0.276214 + 0.0549424i −2.74796 + 0.669846i 2.45461 + 1.72478i −3.72867 − 4.71853i
5.5 −1.40854 − 0.126537i −0.0384067 − 1.73162i 1.96798 + 0.356466i 1.99938 + 2.99228i −0.165018 + 2.44392i −1.31818 + 0.262202i −2.72687 − 0.751120i −2.99705 + 0.133012i −2.43757 − 4.46775i
5.6 −1.40818 − 0.130455i −1.40859 − 1.00791i 1.96596 + 0.367409i 0.301176 + 0.450742i 1.85206 + 1.60308i −4.45430 + 0.886015i −2.72051 − 0.773849i 0.968236 + 2.83946i −0.365310 − 0.674018i
5.7 −1.40357 − 0.173176i 1.59603 − 0.672819i 1.94002 + 0.486131i 0.0125134 + 0.0187276i −2.35666 + 0.667954i 1.96994 − 0.391845i −2.63877 − 1.01828i 2.09463 − 2.14768i −0.0143202 − 0.0284525i
5.8 −1.40058 + 0.195888i −1.71403 + 0.249173i 1.92326 − 0.548714i −0.927566 − 1.38820i 2.35183 − 0.684745i −1.80889 + 0.359810i −2.58619 + 1.14526i 2.87583 − 0.854180i 1.57106 + 1.76259i
5.9 −1.39172 + 0.251254i 1.56099 − 0.750548i 1.87374 − 0.699348i −0.307954 − 0.460885i −1.98387 + 1.43675i −0.725286 + 0.144268i −2.43200 + 1.44408i 1.87336 − 2.34319i 0.544383 + 0.564046i
5.10 −1.38527 + 0.284647i −1.62347 − 0.603603i 1.83795 − 0.788627i 1.99728 + 2.98914i 2.42076 + 0.374038i 2.50910 − 0.499090i −2.32158 + 1.61563i 2.27133 + 1.95987i −3.61762 − 3.57225i
5.11 −1.36887 − 0.355248i 0.500871 − 1.65805i 1.74760 + 0.972575i −1.53245 − 2.29348i −1.27464 + 2.09172i 9.27252e−6 0 1.84442e-6i −2.04673 − 1.95216i −2.49826 − 1.66094i 1.28297 + 3.68387i
5.12 −1.35703 + 0.398082i −0.945254 − 1.45138i 1.68306 − 1.08042i −1.78082 − 2.66519i 1.86051 + 1.59327i 1.94292 − 0.386472i −1.85387 + 2.13616i −1.21299 + 2.74384i 3.47759 + 2.90783i
5.13 −1.33707 − 0.460698i −1.70444 + 0.308052i 1.57551 + 1.23197i −1.93771 − 2.89999i 2.42087 + 0.373343i 4.81055 − 0.956879i −1.53901 − 2.37307i 2.81021 − 1.05011i 1.25484 + 4.77019i
5.14 −1.33676 − 0.461606i 0.794436 + 1.53911i 1.57384 + 1.23411i 0.755651 + 1.13091i −0.351504 − 2.42414i 2.25574 − 0.448696i −1.53417 − 2.37620i −1.73774 + 2.44545i −0.488087 − 1.86057i
5.15 −1.32045 − 0.506384i −0.305019 + 1.70498i 1.48715 + 1.33730i 1.78999 + 2.67892i 1.26614 − 2.09688i −4.54892 + 0.904836i −1.28651 − 2.51891i −2.81393 − 1.04010i −1.00703 − 4.44379i
5.16 −1.30959 + 0.533841i 0.591364 − 1.62797i 1.43003 − 1.39822i 0.491775 + 0.735993i 0.0946362 + 2.44766i 3.81175 − 0.758205i −1.12632 + 2.59450i −2.30058 − 1.92545i −1.03692 − 0.701316i
5.17 −1.30109 + 0.554231i −1.13732 + 1.30634i 1.38566 − 1.44221i 0.325600 + 0.487296i 0.755734 − 2.32999i −0.970884 + 0.193121i −1.00354 + 2.64441i −0.413027 − 2.97143i −0.693709 − 0.453556i
5.18 −1.29572 + 0.566659i 0.294786 + 1.70678i 1.35779 − 1.46847i 0.891540 + 1.33428i −1.34912 − 2.04447i −3.48130 + 0.692474i −0.927204 + 2.67213i −2.82620 + 1.00627i −1.91127 − 1.22366i
5.19 −1.28514 + 0.590260i 1.48224 + 0.896080i 1.30319 − 1.51714i 0.410721 + 0.614687i −2.43381 − 0.276683i −0.664618 + 0.132201i −0.779277 + 2.71896i 1.39408 + 2.65641i −0.890660 − 0.547529i
5.20 −1.26961 − 0.622973i −1.69403 + 0.360933i 1.22381 + 1.58186i 1.00248 + 1.50032i 2.37560 + 0.597090i −1.37096 + 0.272701i −0.568300 − 2.77075i 2.73945 − 1.22286i −0.338099 − 2.52933i
See next 80 embeddings (of 1120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.140
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
272.bi odd 16 1 inner
816.cr even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 816.2.cr.a yes 1120
3.b odd 2 1 inner 816.2.cr.a yes 1120
16.e even 4 1 816.2.cf.a ✓ 1120
17.e odd 16 1 816.2.cf.a ✓ 1120
48.i odd 4 1 816.2.cf.a ✓ 1120
51.i even 16 1 816.2.cf.a ✓ 1120
272.bi odd 16 1 inner 816.2.cr.a yes 1120
816.cr even 16 1 inner 816.2.cr.a yes 1120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
816.2.cf.a ✓ 1120 16.e even 4 1
816.2.cf.a ✓ 1120 17.e odd 16 1
816.2.cf.a ✓ 1120 48.i odd 4 1
816.2.cf.a ✓ 1120 51.i even 16 1
816.2.cr.a yes 1120 1.a even 1 1 trivial
816.2.cr.a yes 1120 3.b odd 2 1 inner
816.2.cr.a yes 1120 272.bi odd 16 1 inner
816.2.cr.a yes 1120 816.cr even 16 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(816, [\chi])\).