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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(173,819)]
 
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, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.ea (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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 210 q^{4} - 12 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{2} + 210 q^{4} - 12 q^{8} - 6 q^{9} + 3 q^{11} - 6 q^{12} - 3 q^{13} - 6 q^{14} + 3 q^{15} + 198 q^{16} - 24 q^{18} - 12 q^{19} - 6 q^{22} - 9 q^{23} - 24 q^{24} - 188 q^{25} - 24 q^{26} + 18 q^{27} - 6 q^{28} - 6 q^{29} + 29 q^{30} - 12 q^{31} - 30 q^{32} + 6 q^{33} + 6 q^{34} + 33 q^{35} - 40 q^{36} - 9 q^{37} - 72 q^{38} - 3 q^{39} - 15 q^{42} + 4 q^{43} - 18 q^{44} - 3 q^{45} - 6 q^{46} - 54 q^{47} - 30 q^{48} - 6 q^{49} - 6 q^{50} - 2 q^{51} - 36 q^{52} + 12 q^{53} + 9 q^{54} + 15 q^{55} - 12 q^{56} - 36 q^{57} - 3 q^{58} + 15 q^{60} - 24 q^{61} + 24 q^{62} - 15 q^{63} + 156 q^{64} + 12 q^{65} - 12 q^{66} + 24 q^{67} - 3 q^{68} + 15 q^{69} + 21 q^{70} + 48 q^{71} - 60 q^{72} + 3 q^{74} + 3 q^{75} - 24 q^{76} - 42 q^{77} + 28 q^{78} - 5 q^{79} + 30 q^{81} - 69 q^{84} - 3 q^{85} - 156 q^{86} - 27 q^{87} - 21 q^{88} + 45 q^{89} + 96 q^{90} - q^{91} - 12 q^{92} + 9 q^{93} - 3 q^{94} - 48 q^{96} + 6 q^{97} - 3 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
173.1 −2.74949 1.53752 − 0.797525i 5.55972 − 2.89439i −4.22739 + 2.19279i −2.15710 − 1.53196i −9.78743 1.72791 − 2.45241i 7.95812i
173.2 −2.73902 −0.198518 + 1.72064i 5.50221 − 0.638676i 0.543744 − 4.71285i 0.701908 − 2.55095i −9.59260 −2.92118 − 0.683155i 1.74934i
173.3 −2.71372 1.28783 + 1.15823i 5.36427 0.399332i −3.49480 − 3.14312i 1.18926 + 2.36340i −9.12968 0.316992 + 2.98321i − 1.08367i
173.4 −2.71196 −1.18672 − 1.26163i 5.35475 − 2.93327i 3.21833 + 3.42149i −2.31051 + 1.28901i −9.09796 −0.183404 + 2.99439i 7.95492i
173.5 −2.59669 −1.00997 − 1.40711i 4.74280 0.717553i 2.62258 + 3.65383i 2.58940 + 0.543160i −7.12220 −0.959913 + 2.84228i − 1.86326i
173.6 −2.57013 1.64523 − 0.541497i 4.60559 2.79247i −4.22846 + 1.39172i 2.48120 − 0.918514i −6.69671 2.41356 − 1.78177i − 7.17701i
173.7 −2.54451 −1.72287 − 0.178051i 4.47452 3.49774i 4.38387 + 0.453053i −1.76528 − 1.97074i −6.29642 2.93660 + 0.613521i − 8.90001i
173.8 −2.48016 0.892768 − 1.48424i 4.15120 2.25945i −2.21421 + 3.68115i −1.37904 + 2.25793i −5.33532 −1.40593 − 2.65016i − 5.60380i
173.9 −2.47188 −1.16935 + 1.27774i 4.11020 3.30078i 2.89049 − 3.15842i 1.04525 + 2.43053i −5.21616 −0.265244 − 2.98825i − 8.15915i
173.10 −2.46966 −1.72904 + 0.102050i 4.09924 − 0.681593i 4.27015 − 0.252028i 1.87820 − 1.86343i −5.18442 2.97917 − 0.352896i 1.68331i
173.11 −2.44695 −1.21179 + 1.23757i 3.98757 − 4.27896i 2.96518 − 3.02827i 1.88694 + 1.85458i −4.86349 −0.0631517 − 2.99934i 10.4704i
173.12 −2.32769 −0.671655 + 1.59652i 3.41816 1.42049i 1.56341 − 3.71621i −2.02139 + 1.70704i −3.30104 −2.09776 − 2.14462i − 3.30647i
173.13 −2.27860 0.366260 − 1.69288i 3.19204 − 2.27451i −0.834561 + 3.85741i 2.62031 + 0.366026i −2.71618 −2.73171 − 1.24007i 5.18270i
173.14 −2.24398 1.38035 + 1.04624i 3.03546 − 2.09025i −3.09749 − 2.34775i −1.06415 + 2.42231i −2.32356 0.810749 + 2.88837i 4.69048i
173.15 −2.23226 1.73088 + 0.0637037i 2.98299 0.693426i −3.86377 − 0.142203i −2.31475 − 1.28137i −2.19428 2.99188 + 0.220527i − 1.54791i
173.16 −2.21702 0.822677 + 1.52421i 2.91516 4.31146i −1.82389 − 3.37919i −0.432509 − 2.61016i −2.02892 −1.64640 + 2.50786i − 9.55858i
173.17 −2.21662 −0.616362 − 1.61867i 2.91343 1.56070i 1.36624 + 3.58799i −2.55623 + 0.682427i −2.02472 −2.24020 + 1.99538i − 3.45948i
173.18 −2.18061 0.517406 − 1.65296i 2.75504 − 2.35649i −1.12826 + 3.60446i 0.211834 − 2.63726i −1.64644 −2.46458 − 1.71051i 5.13857i
173.19 −2.10435 1.43736 + 0.966432i 2.42830 − 0.735800i −3.02472 − 2.03371i 1.72091 − 2.00959i −0.901291 1.13202 + 2.77823i 1.54838i
173.20 −1.98477 0.397265 + 1.68588i 1.93932 − 3.99970i −0.788481 − 3.34608i −2.59009 − 0.539866i 0.120433 −2.68436 + 1.33948i 7.93850i
See next 80 embeddings (of 216 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 173.108
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
819.ea 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.ea.a yes 216
7.d odd 6 1 819.2.cm.a yes 216
9.d odd 6 1 819.2.cc.a yes 216
13.e even 6 1 819.2.bs.a ✓ 216
63.s even 6 1 819.2.bs.a ✓ 216
91.p odd 6 1 819.2.cc.a yes 216
117.m odd 6 1 819.2.cm.a yes 216
819.ea even 6 1 inner 819.2.ea.a yes 216
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bs.a ✓ 216 13.e even 6 1
819.2.bs.a ✓ 216 63.s even 6 1
819.2.cc.a yes 216 9.d odd 6 1
819.2.cc.a yes 216 91.p odd 6 1
819.2.cm.a yes 216 7.d odd 6 1
819.2.cm.a yes 216 117.m odd 6 1
819.2.ea.a yes 216 1.a even 1 1 trivial
819.2.ea.a yes 216 819.ea even 6 1 inner

Hecke kernels

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