Properties

Label 819.2.bt.a
Level $819$
Weight $2$
Character orbit 819.bt
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(311,819)]
 
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([5, 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("819.311");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.bt (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.

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{3} - 102 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 216 q - 6 q^{3} - 102 q^{4} - 6 q^{9} - 12 q^{10} - 6 q^{12} - 3 q^{13} - 6 q^{14} - 90 q^{16} - 12 q^{22} - 188 q^{25} - 6 q^{26} + 18 q^{27} - 12 q^{29} + 14 q^{30} + 60 q^{35} - 16 q^{36} - 72 q^{38} - 24 q^{39} + 6 q^{42} + 4 q^{43} + 54 q^{48} + 40 q^{51} - 24 q^{53} - 48 q^{56} - 48 q^{61} - 12 q^{62} + 120 q^{64} - 54 q^{65} + 6 q^{66} - 12 q^{68} - 24 q^{69} - 6 q^{75} + 66 q^{77} + q^{78} + 16 q^{79} - 30 q^{81} - 24 q^{82} - 24 q^{87} + 12 q^{88} - 48 q^{90} - q^{91} + 78 q^{92} - 6 q^{94} + 54 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
311.1 −1.40095 + 2.42652i −0.702444 1.58322i −2.92535 5.06685i 0.894814i 4.82580 + 0.513515i −2.64130 + 0.153392i 10.7893 −2.01314 + 2.22424i −2.17129 1.25359i
311.2 −1.35333 + 2.34403i −1.40179 + 1.01734i −2.66299 4.61243i 3.44110i −0.487594 4.66264i 1.60270 2.10508i 9.00228 0.930039 2.85220i −8.06605 4.65693i
311.3 −1.32765 + 2.29956i 1.59935 0.664889i −2.52533 4.37399i 0.104647i −0.594429 + 4.56055i 0.0645307 + 2.64496i 8.10041 2.11585 2.12678i 0.240642 + 0.138935i
311.4 −1.30953 + 2.26817i −0.0246295 1.73188i −2.42973 4.20841i 3.59259i 3.96044 + 2.21208i 2.64514 + 0.0566951i 7.48906 −2.99879 + 0.0853104i 8.14860 + 4.70460i
311.5 −1.28708 + 2.22930i 1.35648 + 1.07702i −2.31317 4.00654i 3.96056i −4.14690 + 1.63777i −2.07605 1.64013i 6.76067 0.680051 + 2.92191i −8.82926 5.09757i
311.6 −1.28180 + 2.22015i −0.598913 + 1.62521i −2.28603 3.95952i 1.44047i −2.84051 3.41287i −2.34947 + 1.21655i 6.59374 −2.28261 1.94672i 3.19805 + 1.84640i
311.7 −1.27852 + 2.21445i 1.12633 + 1.31582i −2.26920 3.93038i 0.464206i −4.35385 + 0.811922i 0.656915 + 2.56290i 6.49078 −0.462745 + 2.96410i −1.02796 0.593494i
311.8 −1.27410 + 2.20681i −1.62396 + 0.602296i −2.24667 3.89135i 2.25662i 0.739935 4.35115i 1.17180 + 2.37211i 6.35354 2.27448 1.95621i 4.97992 + 2.87516i
311.9 −1.25337 + 2.17089i 1.59163 0.683159i −2.14186 3.70980i 0.964453i −0.511832 + 4.31151i 2.17734 1.50306i 5.72465 2.06659 2.17468i −2.09373 1.20881i
311.10 −1.25223 + 2.16893i −1.69656 0.348837i −2.13616 3.69995i 1.97217i 2.88109 3.24289i −0.596159 2.57771i 5.69096 2.75662 + 1.18365i 4.27749 + 2.46961i
311.11 −1.14095 + 1.97618i 1.26485 1.18328i −1.60353 2.77740i 3.02648i 0.895260 + 3.84964i −1.99344 1.73960i 2.75440 0.199675 2.99335i 5.98088 + 3.45306i
311.12 −1.13737 + 1.96999i −0.0777012 1.73031i −1.58724 2.74918i 2.13073i 3.49706 + 1.81494i 2.18018 + 1.49894i 2.67163 −2.98793 + 0.268894i −4.19751 2.42343i
311.13 −1.11525 + 1.93167i −0.889365 + 1.48628i −1.48756 2.57654i 0.411732i −1.87914 3.37553i 2.63344 0.254908i 2.17503 −1.41806 2.64369i 0.795331 + 0.459185i
311.14 −1.10144 + 1.90775i 1.58497 + 0.698481i −1.42634 2.47050i 3.02757i −3.07828 + 2.25439i −2.52718 + 0.783189i 1.87836 2.02425 + 2.21414i 5.77586 + 3.33469i
311.15 −1.09691 + 1.89991i 0.772711 1.55013i −1.40643 2.43601i 2.63971i 2.09752 + 3.16844i −2.34387 1.22731i 1.78327 −1.80584 2.39561i −5.01521 2.89553i
311.16 −1.07304 + 1.85856i −1.08685 1.34861i −1.30283 2.25657i 2.74290i 3.67271 0.572861i 1.75443 1.98040i 1.29979 −0.637512 + 2.93148i −5.09784 2.94324i
311.17 −1.02705 + 1.77891i 1.58175 + 0.705730i −1.10968 1.92202i 0.337826i −2.87997 + 2.08897i 0.801180 2.52153i 0.450575 2.00389 + 2.23258i 0.600961 + 0.346965i
311.18 −1.02089 + 1.76823i −1.72010 + 0.203153i −1.08442 1.87827i 3.06373i 1.39680 3.24891i −1.72300 + 2.00780i 0.344720 2.91746 0.698885i −5.41737 3.12772i
311.19 −1.00192 + 1.73537i −1.09620 1.34102i −1.00767 1.74534i 3.71228i 3.42547 0.558718i −0.907306 + 2.48532i 0.0307381 −0.596692 + 2.94006i 6.44217 + 3.71939i
311.20 −0.988721 + 1.71252i 0.00938524 + 1.73203i −0.955139 1.65435i 0.999327i −2.97540 1.69642i −1.75401 1.98077i −0.177420 −2.99982 + 0.0325109i −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 311.108
Significant digits:
Format:

Inner twists

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

Hecke kernels

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