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Properties

Label 819.2.do.h
Level $819$
Weight $2$
Character orbit 819.do
Analytic conductor $6.540$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(361,819)]
 
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([0, 4, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.do (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: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 22 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 22 q^{4} + 4 q^{7} - 16 q^{10} - 26 q^{16} + 2 q^{22} + 22 q^{25} + 14 q^{28} + 18 q^{31} - 6 q^{37} - 34 q^{40} + 4 q^{43} + 72 q^{46} - 70 q^{52} - 22 q^{55} + 108 q^{61} - 100 q^{64} + 72 q^{70} + 30 q^{73} + 42 q^{76} + 40 q^{79} - 36 q^{82} - 24 q^{88} - 30 q^{91} + 112 q^{94} - 36 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1 −2.33665 − 1.34906i 0 2.63995 + 4.57253i −0.933651 + 0.539043i 0 2.27537 + 1.35008i − 8.84961i 0 2.90882
361.2 −2.27495 − 1.31344i 0 2.45026 + 4.24397i 2.99000 − 1.72628i 0 −1.10870 − 2.40225i − 7.61930i 0 −9.06945
361.3 −1.94159 − 1.12098i 0 1.51319 + 2.62092i −0.550805 + 0.318007i 0 −1.65674 + 2.06282i − 2.30110i 0 1.42592
361.4 −1.54694 − 0.893127i 0 0.595353 + 1.03118i 0.972470 − 0.561456i 0 −0.0853404 − 2.64437i 1.44561i 0 −2.00581
361.5 −1.48056 − 0.854802i 0 0.461374 + 0.799123i −1.84265 + 1.06385i 0 2.64237 + 0.133694i 1.84167i 0 3.63754
361.6 −1.26493 − 0.730310i 0 0.0667051 + 0.115537i 2.51646 − 1.45288i 0 −1.58711 + 2.11686i 2.72638i 0 −4.24420
361.7 −1.00552 − 0.580538i 0 −0.325951 − 0.564563i −3.43209 + 1.98152i 0 −2.43773 − 1.02834i 3.07906i 0 4.60139
361.8 −0.356535 − 0.205846i 0 −0.915255 − 1.58527i 1.54520 − 0.892120i 0 0.355695 + 2.62173i 1.57699i 0 −0.734557
361.9 −0.146841 − 0.0847787i 0 −0.985625 − 1.70715i 2.65412 − 1.53236i 0 2.60218 − 0.478169i 0.673355i 0 −0.519645
361.10 0.146841 + 0.0847787i 0 −0.985625 − 1.70715i −2.65412 + 1.53236i 0 2.60218 − 0.478169i − 0.673355i 0 −0.519645
361.11 0.356535 + 0.205846i 0 −0.915255 − 1.58527i −1.54520 + 0.892120i 0 0.355695 + 2.62173i − 1.57699i 0 −0.734557
361.12 1.00552 + 0.580538i 0 −0.325951 − 0.564563i 3.43209 − 1.98152i 0 −2.43773 − 1.02834i − 3.07906i 0 4.60139
361.13 1.26493 + 0.730310i 0 0.0667051 + 0.115537i −2.51646 + 1.45288i 0 −1.58711 + 2.11686i − 2.72638i 0 −4.24420
361.14 1.48056 + 0.854802i 0 0.461374 + 0.799123i 1.84265 − 1.06385i 0 2.64237 + 0.133694i − 1.84167i 0 3.63754
361.15 1.54694 + 0.893127i 0 0.595353 + 1.03118i −0.972470 + 0.561456i 0 −0.0853404 − 2.64437i − 1.44561i 0 −2.00581
361.16 1.94159 + 1.12098i 0 1.51319 + 2.62092i 0.550805 − 0.318007i 0 −1.65674 + 2.06282i 2.30110i 0 1.42592
361.17 2.27495 + 1.31344i 0 2.45026 + 4.24397i −2.99000 + 1.72628i 0 −1.10870 − 2.40225i 7.61930i 0 −9.06945
361.18 2.33665 + 1.34906i 0 2.63995 + 4.57253i 0.933651 − 0.539043i 0 2.27537 + 1.35008i 8.84961i 0 2.90882
667.1 −2.33665 + 1.34906i 0 2.63995 − 4.57253i −0.933651 − 0.539043i 0 2.27537 − 1.35008i 8.84961i 0 2.90882
667.2 −2.27495 + 1.31344i 0 2.45026 − 4.24397i 2.99000 + 1.72628i 0 −1.10870 + 2.40225i 7.61930i 0 −9.06945
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.u even 6 1 inner
273.x odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.do.h yes 36
3.b odd 2 1 inner 819.2.do.h yes 36
7.c even 3 1 819.2.bm.h ✓ 36
13.e even 6 1 819.2.bm.h ✓ 36
21.h odd 6 1 819.2.bm.h ✓ 36
39.h odd 6 1 819.2.bm.h ✓ 36
91.u even 6 1 inner 819.2.do.h yes 36
273.x odd 6 1 inner 819.2.do.h yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.bm.h ✓ 36 7.c even 3 1
819.2.bm.h ✓ 36 13.e even 6 1
819.2.bm.h ✓ 36 21.h odd 6 1
819.2.bm.h ✓ 36 39.h odd 6 1
819.2.do.h yes 36 1.a even 1 1 trivial
819.2.do.h yes 36 3.b odd 2 1 inner
819.2.do.h yes 36 91.u even 6 1 inner
819.2.do.h yes 36 273.x odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 29 T_{2}^{34} + 496 T_{2}^{32} - 5647 T_{2}^{30} + 47934 T_{2}^{28} - 310100 T_{2}^{26} + \cdots + 1089 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display