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Properties

Label 819.2.y.i
Level $819$
Weight $2$
Character orbit 819.y
Analytic conductor $6.540$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(307,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.y (of order \(4\), 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: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 104 q^{16} - 64 q^{22} + 40 q^{28} - 40 q^{37} - 104 q^{46} - 64 q^{58} - 64 q^{67} - 8 q^{70} + 56 q^{79} - 72 q^{85} - 72 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.88005 − 1.88005i 0 5.06915i −2.87732 + 2.87732i 0 0.825413 − 2.51370i 5.77013 − 5.77013i 0 10.8190
307.2 −1.88005 − 1.88005i 0 5.06915i 2.87732 − 2.87732i 0 2.51370 − 0.825413i 5.77013 − 5.77013i 0 −10.8190
307.3 −1.57361 − 1.57361i 0 2.95247i −1.38061 + 1.38061i 0 0.115976 + 2.64321i 1.49881 − 1.49881i 0 4.34506
307.4 −1.57361 − 1.57361i 0 2.95247i 1.38061 − 1.38061i 0 −2.64321 − 0.115976i 1.49881 − 1.49881i 0 −4.34506
307.5 −0.544573 − 0.544573i 0 − 1.40688i −1.70192 + 1.70192i 0 −1.73650 + 1.99613i −1.85530 + 1.85530i 0 1.85364
307.6 −0.544573 − 0.544573i 0 − 1.40688i 1.70192 − 1.70192i 0 −1.99613 + 1.73650i −1.85530 + 1.85530i 0 −1.85364
307.7 −0.438899 − 0.438899i 0 − 1.61474i −1.84890 + 1.84890i 0 2.62969 − 0.291065i −1.58650 + 1.58650i 0 1.62296
307.8 −0.438899 − 0.438899i 0 − 1.61474i 1.84890 − 1.84890i 0 0.291065 − 2.62969i −1.58650 + 1.58650i 0 −1.62296
307.9 0.438899 + 0.438899i 0 − 1.61474i −1.84890 + 1.84890i 0 0.291065 − 2.62969i 1.58650 − 1.58650i 0 −1.62296
307.10 0.438899 + 0.438899i 0 − 1.61474i 1.84890 − 1.84890i 0 2.62969 − 0.291065i 1.58650 − 1.58650i 0 1.62296
307.11 0.544573 + 0.544573i 0 − 1.40688i −1.70192 + 1.70192i 0 −1.99613 + 1.73650i 1.85530 − 1.85530i 0 −1.85364
307.12 0.544573 + 0.544573i 0 − 1.40688i 1.70192 − 1.70192i 0 −1.73650 + 1.99613i 1.85530 − 1.85530i 0 1.85364
307.13 1.57361 + 1.57361i 0 2.95247i −1.38061 + 1.38061i 0 −2.64321 − 0.115976i −1.49881 + 1.49881i 0 −4.34506
307.14 1.57361 + 1.57361i 0 2.95247i 1.38061 − 1.38061i 0 0.115976 + 2.64321i −1.49881 + 1.49881i 0 4.34506
307.15 1.88005 + 1.88005i 0 5.06915i −2.87732 + 2.87732i 0 2.51370 − 0.825413i −5.77013 + 5.77013i 0 −10.8190
307.16 1.88005 + 1.88005i 0 5.06915i 2.87732 − 2.87732i 0 0.825413 − 2.51370i −5.77013 + 5.77013i 0 10.8190
811.1 −1.88005 + 1.88005i 0 − 5.06915i −2.87732 − 2.87732i 0 0.825413 + 2.51370i 5.77013 + 5.77013i 0 10.8190
811.2 −1.88005 + 1.88005i 0 − 5.06915i 2.87732 + 2.87732i 0 2.51370 + 0.825413i 5.77013 + 5.77013i 0 −10.8190
811.3 −1.57361 + 1.57361i 0 − 2.95247i −1.38061 − 1.38061i 0 0.115976 − 2.64321i 1.49881 + 1.49881i 0 4.34506
811.4 −1.57361 + 1.57361i 0 − 2.95247i 1.38061 + 1.38061i 0 −2.64321 + 0.115976i 1.49881 + 1.49881i 0 −4.34506
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
13.d odd 4 1 inner
21.c even 2 1 inner
39.f even 4 1 inner
91.i even 4 1 inner
273.o odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.y.i ✓ 32
3.b odd 2 1 inner 819.2.y.i ✓ 32
7.b odd 2 1 inner 819.2.y.i ✓ 32
13.d odd 4 1 inner 819.2.y.i ✓ 32
21.c even 2 1 inner 819.2.y.i ✓ 32
39.f even 4 1 inner 819.2.y.i ✓ 32
91.i even 4 1 inner 819.2.y.i ✓ 32
273.o odd 4 1 inner 819.2.y.i ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.y.i ✓ 32 1.a even 1 1 trivial
819.2.y.i ✓ 32 3.b odd 2 1 inner
819.2.y.i ✓ 32 7.b odd 2 1 inner
819.2.y.i ✓ 32 13.d odd 4 1 inner
819.2.y.i ✓ 32 21.c even 2 1 inner
819.2.y.i ✓ 32 39.f even 4 1 inner
819.2.y.i ✓ 32 91.i even 4 1 inner
819.2.y.i ✓ 32 273.o odd 4 1 inner

Hecke kernels

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

\( T_{2}^{16} + 75T_{2}^{12} + 1263T_{2}^{8} + 617T_{2}^{4} + 64 \) Copy content Toggle raw display
\( T_{5}^{16} + 369T_{5}^{12} + 28736T_{5}^{8} + 772816T_{5}^{4} + 6250000 \) Copy content Toggle raw display