Properties

Label 819.2.q.b
Level $819$
Weight $2$
Character orbit 819.q
Analytic conductor $6.540$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(79,819)]
 
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([4, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.q (of order \(3\), 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: \(96\)
Relative dimension: \(48\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{2} - 48 q^{4} + 8 q^{5} + 6 q^{6} - 36 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 6 q^{2} - 48 q^{4} + 8 q^{5} + 6 q^{6} - 36 q^{8} + 10 q^{9} - 40 q^{11} + 48 q^{13} - 8 q^{14} - 48 q^{16} - 3 q^{17} - 10 q^{18} - 12 q^{20} + 18 q^{21} - 58 q^{23} - 14 q^{24} + 96 q^{25} - 6 q^{26} + 30 q^{28} + 10 q^{29} - 3 q^{30} + 9 q^{31} + 67 q^{32} - 16 q^{35} - 48 q^{36} - 6 q^{37} + 30 q^{38} - 12 q^{41} - 76 q^{42} + 29 q^{44} + 35 q^{45} - 6 q^{46} - 15 q^{47} - 86 q^{48} - 12 q^{49} + 2 q^{50} - 28 q^{51} - 96 q^{52} + 50 q^{53} - 8 q^{54} - 36 q^{55} + 10 q^{56} + 55 q^{57} - 12 q^{58} - 35 q^{59} + 34 q^{60} - 6 q^{61} - 38 q^{62} + 4 q^{63} + 96 q^{64} + 4 q^{65} + 40 q^{66} - 18 q^{68} + 50 q^{69} - 9 q^{70} - 58 q^{71} + 48 q^{72} - 160 q^{74} + 8 q^{75} + 13 q^{77} + 3 q^{78} + 6 q^{79} - 60 q^{80} - 6 q^{81} - 30 q^{82} - 34 q^{83} + 91 q^{84} + 27 q^{85} - 100 q^{86} - 43 q^{87} - 29 q^{89} - 43 q^{90} + 63 q^{92} + 7 q^{93} - 12 q^{94} + 66 q^{95} + 47 q^{96} - 64 q^{98} - 67 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 −1.30642 2.26278i −1.71591 0.235914i −2.41345 + 4.18023i −0.409745 1.70787 + 4.19093i 2.62346 + 0.342704i 7.38625 2.88869 + 0.809612i 0.535298 + 0.927164i
79.2 −1.28061 2.21808i 1.63464 0.572663i −2.27991 + 3.94892i 3.88979 −3.36355 2.89241i 1.21756 2.34895i 6.55624 2.34411 1.87220i −4.98129 8.62785i
79.3 −1.23220 2.13423i 1.15780 1.28822i −2.03662 + 3.52753i −1.72333 −4.17599 0.883668i −2.48312 0.913305i 5.10930 −0.319007 2.98299i 2.12348 + 3.67797i
79.4 −1.23208 2.13403i −1.12814 1.31427i −2.03606 + 3.52656i 2.20096 −1.41473 + 4.02678i −2.44344 + 1.01469i 5.10606 −0.454599 + 2.96536i −2.71177 4.69692i
79.5 −1.16734 2.02190i −0.0130928 + 1.73200i −1.72537 + 2.98844i −1.75883 3.51721 1.99537i −0.457521 + 2.60589i 3.38704 −2.99966 0.0453535i 2.05316 + 3.55617i
79.6 −1.11130 1.92482i 1.65382 + 0.514665i −1.46996 + 2.54604i −2.34763 −0.847245 3.75525i −0.424108 2.61154i 2.08904 2.47024 + 1.70233i 2.60891 + 4.51876i
79.7 −1.10741 1.91809i −1.45241 + 0.943673i −1.45271 + 2.51616i 1.29021 3.41845 + 1.74081i 0.190765 2.63887i 2.00533 1.21896 2.74119i −1.42879 2.47473i
79.8 −0.965178 1.67174i −0.461029 1.66957i −0.863138 + 1.49500i −3.28745 −2.34610 + 2.38215i 2.50497 + 0.851542i −0.528383 −2.57490 + 1.53944i 3.17297 + 5.49575i
79.9 −0.936148 1.62146i 1.61108 + 0.635949i −0.752744 + 1.30379i 3.05727 −0.477042 3.20763i −0.901056 + 2.48759i −0.925871 2.19114 + 2.04913i −2.86206 4.95723i
79.10 −0.871435 1.50937i −0.149918 + 1.72555i −0.518798 + 0.898585i 3.07299 2.73514 1.27742i −2.63585 0.228635i −1.67734 −2.95505 0.517383i −2.67791 4.63828i
79.11 −0.797019 1.38048i 1.23146 + 1.21799i −0.270480 + 0.468484i −2.27464 0.699905 2.67077i 1.37143 2.26256i −2.32577 0.0330102 + 2.99982i 1.81293 + 3.14009i
79.12 −0.725750 1.25704i −1.70576 + 0.300649i −0.0534274 + 0.0925389i 1.67650 1.61588 + 1.92600i 0.491302 + 2.59974i −2.74790 2.81922 1.02567i −1.21672 2.10743i
79.13 −0.712041 1.23329i 1.33346 1.10539i −0.0140047 + 0.0242568i −0.0771886 −2.31275 0.857454i 2.64248 0.131501i −2.80828 0.556210 2.94799i 0.0549615 + 0.0951961i
79.14 −0.691944 1.19848i −1.54607 0.780820i 0.0424270 0.0734858i −2.75753 0.133993 + 2.39322i −0.989701 + 2.45367i −2.88520 1.78064 + 2.41440i 1.90806 + 3.30485i
79.15 −0.666890 1.15509i −1.17774 + 1.27001i 0.110515 0.191417i −3.70643 2.25240 + 0.513440i −2.40597 1.10059i −2.96237 −0.225845 2.99149i 2.47178 + 4.28126i
79.16 −0.601202 1.04131i 0.371418 + 1.69176i 0.277112 0.479971i 2.56337 1.53835 1.40385i 2.53790 0.747695i −3.07121 −2.72410 + 1.25670i −1.54110 2.66927i
79.17 −0.301506 0.522224i −0.286088 1.70826i 0.818188 1.41714i 3.51406 −0.805836 + 0.664453i 1.42443 + 2.22957i −2.19278 −2.83631 + 0.977427i −1.05951 1.83512i
79.18 −0.276008 0.478060i −0.0310116 1.73177i 0.847639 1.46815i −1.57072 −0.819332 + 0.492809i −2.63406 0.248479i −2.03985 −2.99808 + 0.107410i 0.433531 + 0.750897i
79.19 −0.275079 0.476450i 1.69564 0.353295i 0.848663 1.46993i 1.36525 −0.634761 0.710703i −1.69022 2.03548i −2.03411 2.75037 1.19812i −0.375551 0.650473i
79.20 −0.208396 0.360952i −1.38547 1.03945i 0.913143 1.58161i 3.62518 −0.0864652 + 0.716707i −1.74876 1.98541i −1.59476 0.839080 + 2.88027i −0.755471 1.30851i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.q.b 96
7.c even 3 1 819.2.r.a yes 96
9.c even 3 1 819.2.r.a yes 96
63.g even 3 1 inner 819.2.q.b 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.q.b 96 1.a even 1 1 trivial
819.2.q.b 96 63.g even 3 1 inner
819.2.r.a yes 96 7.c even 3 1
819.2.r.a yes 96 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{96} - 6 T_{2}^{95} + 90 T_{2}^{94} - 440 T_{2}^{93} + 3954 T_{2}^{92} - 16835 T_{2}^{91} + 115310 T_{2}^{90} - 440438 T_{2}^{89} + 2514469 T_{2}^{88} - 8770004 T_{2}^{87} + 43669833 T_{2}^{86} + \cdots + 502681 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display