↑

Properties

Label 819.2.eg.a
Level $819$
Weight $2$
Character orbit 819.eg
Analytic conductor $6.540$
Analytic rank $0$
Dimension $216$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(38,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.eg (of order \(6\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(216\)
Relative dimension: \(108\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{3} + 204 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{3} + 204 q^{4} - 6 q^{9} - 12 q^{10} - 18 q^{12} + 3 q^{13} - 6 q^{14} + 180 q^{16} - 12 q^{22} + 18 q^{23} + 94 q^{25} + 6 q^{26} - 18 q^{27} - 12 q^{29} - 52 q^{30} - 60 q^{35} - 16 q^{36} - 36 q^{38} + 21 q^{39} - 60 q^{40} - 42 q^{42} + 4 q^{43} - 54 q^{48} - 38 q^{51} - 15 q^{52} + 24 q^{53} - 60 q^{56} + 12 q^{62} + 120 q^{64} + 30 q^{66} - 6 q^{68} + 24 q^{69} - 78 q^{74} - 12 q^{75} + 42 q^{77} + q^{78} - 32 q^{79} + 66 q^{81} - 24 q^{82} - 54 q^{87} - 6 q^{88} + 48 q^{90} - q^{91} + 78 q^{92}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −2.80191 −1.72233 + 0.183273i 5.85069 0.774932 + 0.447407i 4.82580 − 0.513515i 1.45349 + 2.21074i −10.7893 2.93282 − 0.631313i −2.17129 − 1.25359i
38.2 −2.70665 0.180146 − 1.72266i 5.32598 2.98008 + 1.72055i −0.487594 + 4.66264i −2.62440 − 0.335434i −9.00228 −2.93509 − 0.620661i −8.06605 − 4.65693i
38.3 −2.65531 0.223865 + 1.71752i 5.05065 −0.0906267 − 0.0523234i −0.594429 − 4.56055i 2.25834 − 1.37837i −8.10041 −2.89977 + 0.768985i 0.240642 + 0.138935i
38.4 −2.61906 −1.51216 + 0.844608i 4.85945 −3.11127 − 1.79629i 3.96044 − 2.21208i −1.27347 − 2.31911i −7.48906 1.57327 − 2.55437i 8.14860 + 4.70460i
38.5 −2.57417 1.61097 + 0.636232i 4.62635 3.42994 + 1.98028i −4.14690 − 1.63777i −0.382367 + 2.61798i −6.76067 2.19042 + 2.04989i −8.82926 − 5.09757i
38.6 −2.56360 1.10802 − 1.33128i 4.57206 −1.24748 − 0.720235i −2.84051 + 3.41287i 2.22830 + 1.42642i −6.59374 −0.544603 − 2.95015i 3.19805 + 1.84640i
38.7 −2.55703 1.70270 + 0.317525i 4.53841 0.402014 + 0.232103i −4.35385 − 0.811922i 1.89108 − 1.85036i −6.49078 2.79836 + 1.08130i −1.02796 − 0.593494i
38.8 −2.54820 −0.290375 − 1.70754i 4.49334 −1.95429 − 1.12831i 0.739935 + 4.35115i 1.46840 − 2.20086i −6.35354 −2.83136 + 0.991652i 4.97992 + 2.87516i
38.9 −2.50673 0.204183 + 1.71997i 4.28371 0.835241 + 0.482226i −0.511832 − 4.31151i −2.39036 − 1.13410i −5.72465 −2.91662 + 0.702379i −2.09373 − 1.20881i
38.10 −2.50446 −1.15038 − 1.29484i 4.27233 −1.70795 − 0.986083i 2.88109 + 3.24289i −1.93428 + 1.80514i −5.69096 −0.353244 + 2.97913i 4.27749 + 2.46961i
38.11 −2.28190 −0.392331 + 1.68703i 3.20706 −2.62101 − 1.51324i 0.895260 − 3.84964i −0.509822 + 2.59617i −2.75440 −2.69215 − 1.32375i 5.98088 + 3.45306i
38.12 −2.27475 −1.53734 + 0.797862i 3.17447 1.84526 + 1.06536i 3.49706 − 1.81494i 0.208028 − 2.63756i −2.67163 1.72683 − 2.45317i −4.19751 − 2.42343i
38.13 −2.23050 0.842474 − 1.51335i 2.97513 −0.356571 − 0.205866i −1.87914 + 3.37553i −1.53748 − 2.15317i −2.17503 −1.58047 − 2.54992i 0.795331 + 0.459185i
38.14 −2.20288 1.39739 + 1.02338i 2.85268 −2.62196 − 1.51379i −3.07828 − 2.25439i 1.94185 + 1.79700i −1.87836 0.905377 + 2.86012i 5.77586 + 3.33469i
38.15 −2.19382 −0.956101 + 1.44425i 2.81286 2.28606 + 1.31986i 2.09752 − 3.16844i 0.109054 + 2.64350i −1.78327 −1.17174 − 2.76171i −5.01521 − 2.89553i
38.16 −2.14608 −1.71136 − 0.266934i 2.60566 2.37542 + 1.37145i 3.67271 + 0.572861i −2.59229 − 0.529185i −1.29979 2.85749 + 0.913638i −5.09784 − 2.94324i
38.17 −2.05411 1.40206 + 1.01697i 2.21935 −0.292566 − 0.168913i −2.87997 − 2.08897i −2.58430 + 0.566923i −0.450575 0.931529 + 2.85171i 0.600961 + 0.346965i
38.18 −2.04177 −0.684112 − 1.59122i 2.16883 2.65327 + 1.53186i 1.39680 + 3.24891i 2.60031 + 0.488264i −0.344720 −2.06398 + 2.17715i −5.41737 − 3.12772i
38.19 −2.00383 −1.70946 − 0.278825i 2.01534 −3.21493 − 1.85614i 3.42547 + 0.558718i 2.60600 − 0.456908i −0.0307381 2.84451 + 0.953280i 6.44217 + 3.71939i
38.20 −1.97744 1.50467 − 0.857885i 1.91028 0.865443 + 0.499664i −2.97540 + 1.69642i −0.838393 + 2.50940i 0.177420 1.52807 − 2.58167i −1.71136 − 0.988056i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
63.i even 6 1 inner
819.eg even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.eg.a yes 216
7.d odd 6 1 819.2.bt.a ✓ 216
9.d odd 6 1 819.2.bt.a ✓ 216
13.b even 2 1 inner 819.2.eg.a yes 216
63.i even 6 1 inner 819.2.eg.a yes 216
91.s odd 6 1 819.2.bt.a ✓ 216
117.n odd 6 1 819.2.bt.a ✓ 216
819.eg even 6 1 inner 819.2.eg.a yes 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bt.a ✓ 216 7.d odd 6 1
819.2.bt.a ✓ 216 9.d odd 6 1
819.2.bt.a ✓ 216 91.s odd 6 1
819.2.bt.a ✓ 216 117.n odd 6 1
819.2.eg.a yes 216 1.a even 1 1 trivial
819.2.eg.a yes 216 13.b even 2 1 inner
819.2.eg.a yes 216 63.i even 6 1 inner
819.2.eg.a yes 216 819.eg even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(819, [\chi])\).