↑

Properties

Label 819.2.fn.e
Level $819$
Weight $2$
Character orbit 819.fn
Analytic conductor $6.540$
Analytic rank $0$
Dimension $32$
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(73,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.fn (of order \(12\), degree \(4\), 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: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 6 q^{5} - 6 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{2} + 6 q^{5} - 6 q^{7} + 16 q^{8} + 10 q^{11} - 28 q^{14} + 12 q^{16} + 12 q^{19} - 8 q^{22} - 24 q^{26} - 6 q^{28} - 16 q^{29} + 24 q^{31} - 4 q^{32} - 28 q^{35} - 8 q^{37} - 132 q^{40} + 42 q^{44} + 12 q^{46} - 30 q^{47} - 88 q^{50} + 36 q^{52} + 12 q^{53} + 26 q^{58} + 54 q^{59} - 48 q^{61} + 8 q^{65} + 16 q^{67} + 48 q^{68} + 50 q^{70} + 36 q^{71} + 66 q^{73} - 12 q^{74} - 32 q^{79} - 138 q^{80} - 84 q^{85} - 42 q^{86} + 60 q^{89} - 48 q^{92} - 72 q^{94} + 86 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −0.697597 + 2.60347i 0 −4.55935 − 2.63234i −2.44137 − 0.654162i 0 0.722189 + 2.54528i 6.22207 − 6.22207i 0 3.40618 − 5.89968i
73.2 −0.493585 + 1.84208i 0 −1.41759 − 0.818448i 3.30509 + 0.885596i 0 −1.12943 + 2.39257i −0.489646 + 0.489646i 0 −3.26268 + 5.65113i
73.3 −0.423548 + 1.58070i 0 −0.587174 − 0.339005i 2.77274 + 0.742954i 0 1.92960 − 1.81015i −1.52975 + 1.52975i 0 −2.34878 + 4.06820i
73.4 −0.200025 + 0.746505i 0 1.21479 + 0.701360i −1.76272 − 0.472319i 0 −2.63734 + 0.210751i −1.85952 + 1.85952i 0 0.705177 − 1.22140i
73.5 0.186083 − 0.694471i 0 1.28439 + 0.741542i 1.87130 + 0.501414i 0 −0.783278 − 2.52715i 1.77076 − 1.77076i 0 0.696434 − 1.20626i
73.6 0.211401 − 0.788958i 0 1.15429 + 0.666428i −3.03793 − 0.814012i 0 2.28951 − 1.32595i 1.92491 − 1.92491i 0 −1.28444 + 2.22472i
73.7 0.411280 − 1.53492i 0 −0.454770 − 0.262561i 0.0769162 + 0.0206096i 0 1.52171 + 2.16435i 1.65723 − 1.65723i 0 0.0632682 − 0.109584i
73.8 0.639966 − 2.38839i 0 −3.56278 − 2.05697i 1.58199 + 0.423894i 0 −2.54693 + 0.716327i −3.69606 + 3.69606i 0 2.02484 − 3.50713i
460.1 −0.697597 − 2.60347i 0 −4.55935 + 2.63234i −2.44137 + 0.654162i 0 0.722189 − 2.54528i 6.22207 + 6.22207i 0 3.40618 + 5.89968i
460.2 −0.493585 − 1.84208i 0 −1.41759 + 0.818448i 3.30509 − 0.885596i 0 −1.12943 − 2.39257i −0.489646 − 0.489646i 0 −3.26268 − 5.65113i
460.3 −0.423548 − 1.58070i 0 −0.587174 + 0.339005i 2.77274 − 0.742954i 0 1.92960 + 1.81015i −1.52975 − 1.52975i 0 −2.34878 − 4.06820i
460.4 −0.200025 − 0.746505i 0 1.21479 − 0.701360i −1.76272 + 0.472319i 0 −2.63734 − 0.210751i −1.85952 − 1.85952i 0 0.705177 + 1.22140i
460.5 0.186083 + 0.694471i 0 1.28439 − 0.741542i 1.87130 − 0.501414i 0 −0.783278 + 2.52715i 1.77076 + 1.77076i 0 0.696434 + 1.20626i
460.6 0.211401 + 0.788958i 0 1.15429 − 0.666428i −3.03793 + 0.814012i 0 2.28951 + 1.32595i 1.92491 + 1.92491i 0 −1.28444 − 2.22472i
460.7 0.411280 + 1.53492i 0 −0.454770 + 0.262561i 0.0769162 − 0.0206096i 0 1.52171 − 2.16435i 1.65723 + 1.65723i 0 0.0632682 + 0.109584i
460.8 0.639966 + 2.38839i 0 −3.56278 + 2.05697i 1.58199 − 0.423894i 0 −2.54693 − 0.716327i −3.69606 − 3.69606i 0 2.02484 + 3.50713i
577.1 −2.38839 − 0.639966i 0 3.56278 + 2.05697i 0.423894 − 1.58199i 0 −0.716327 − 2.54693i −3.69606 − 3.69606i 0 −2.02484 + 3.50713i
577.2 −1.53492 − 0.411280i 0 0.454770 + 0.262561i 0.0206096 − 0.0769162i 0 −2.16435 + 1.52171i 1.65723 + 1.65723i 0 −0.0632682 + 0.109584i
577.3 −0.788958 − 0.211401i 0 −1.15429 − 0.666428i −0.814012 + 3.03793i 0 1.32595 + 2.28951i 1.92491 + 1.92491i 0 1.28444 − 2.22472i
577.4 −0.694471 − 0.186083i 0 −1.28439 − 0.741542i 0.501414 − 1.87130i 0 2.52715 − 0.783278i 1.77076 + 1.77076i 0 −0.696434 + 1.20626i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
13.d odd 4 1 inner
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.fn.e 32
3.b odd 2 1 91.2.bb.a ✓ 32
7.d odd 6 1 inner 819.2.fn.e 32
13.d odd 4 1 inner 819.2.fn.e 32
21.c even 2 1 637.2.bc.b 32
21.g even 6 1 91.2.bb.a ✓ 32
21.g even 6 1 637.2.i.a 32
21.h odd 6 1 637.2.i.a 32
21.h odd 6 1 637.2.bc.b 32
39.f even 4 1 91.2.bb.a ✓ 32
91.bb even 12 1 inner 819.2.fn.e 32
273.o odd 4 1 637.2.bc.b 32
273.cb odd 12 1 91.2.bb.a ✓ 32
273.cb odd 12 1 637.2.i.a 32
273.cd even 12 1 637.2.i.a 32
273.cd even 12 1 637.2.bc.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.bb.a ✓ 32 3.b odd 2 1
91.2.bb.a ✓ 32 21.g even 6 1
91.2.bb.a ✓ 32 39.f even 4 1
91.2.bb.a ✓ 32 273.cb odd 12 1
637.2.i.a 32 21.g even 6 1
637.2.i.a 32 21.h odd 6 1
637.2.i.a 32 273.cb odd 12 1
637.2.i.a 32 273.cd even 12 1
637.2.bc.b 32 21.c even 2 1
637.2.bc.b 32 21.h odd 6 1
637.2.bc.b 32 273.o odd 4 1
637.2.bc.b 32 273.cd even 12 1
819.2.fn.e 32 1.a even 1 1 trivial
819.2.fn.e 32 7.d odd 6 1 inner
819.2.fn.e 32 13.d odd 4 1 inner
819.2.fn.e 32 91.bb even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\):

\( T_{2}^{32} - 2 T_{2}^{31} + 2 T_{2}^{30} - 12 T_{2}^{29} - 37 T_{2}^{28} + 118 T_{2}^{27} - 90 T_{2}^{26} + \cdots + 50625 \) Copy content Toggle raw display
\( T_{19}^{32} - 12 T_{19}^{31} + 72 T_{19}^{30} - 288 T_{19}^{29} - 364 T_{19}^{28} + \cdots + 338395192861441 \) Copy content Toggle raw display