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Properties

Label 798.2.bj.a
Level $798$
Weight $2$
Character orbit 798.bj
Analytic conductor $6.372$
Analytic rank $0$
Dimension $24$
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] = [798,2,Mod(559,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.bj (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: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 12 q^{4} + 6 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{3} + 12 q^{4} + 6 q^{7} - 12 q^{9} + 2 q^{10} - 24 q^{12} + 10 q^{13} + 4 q^{14} - 12 q^{16} - 10 q^{19} + 6 q^{21} + 6 q^{22} + 6 q^{25} + 24 q^{27} + 12 q^{28} - 4 q^{30} - 36 q^{31} + 2 q^{34} - 16 q^{35} + 12 q^{36} + 12 q^{38} - 20 q^{39} - 2 q^{40} - 20 q^{41} - 8 q^{42} - 6 q^{43} - 12 q^{48} + 2 q^{49} - 10 q^{52} + 12 q^{53} - 30 q^{55} - 4 q^{56} + 2 q^{57} - 8 q^{58} + 8 q^{59} - 42 q^{61} - 12 q^{62} - 12 q^{63} - 24 q^{64} - 6 q^{66} + 6 q^{67} + 6 q^{70} - 24 q^{71} + 66 q^{73} - 12 q^{75} - 8 q^{76} + 16 q^{77} - 24 q^{78} + 30 q^{79} - 12 q^{81} - 6 q^{82} - 6 q^{84} - 2 q^{85} + 12 q^{86} - 8 q^{89} + 2 q^{90} - 66 q^{91} + 18 q^{93} + 20 q^{95} + 36 q^{97} - 32 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −3.28802 − 1.89834i 0.866025 − 0.500000i 1.34257 − 2.27981i − 1.00000i −0.500000 − 0.866025i 1.89834 + 3.28802i
559.2 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.02194 − 0.590018i 0.866025 − 0.500000i −2.59272 + 0.527083i − 1.00000i −0.500000 − 0.866025i 0.590018 + 1.02194i
559.3 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −0.397289 − 0.229375i 0.866025 − 0.500000i 1.97350 + 1.76218i − 1.00000i −0.500000 − 0.866025i 0.229375 + 0.397289i
559.4 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.343006 + 0.198035i 0.866025 − 0.500000i 2.64565 − 0.0230826i − 1.00000i −0.500000 − 0.866025i −0.198035 − 0.343006i
559.5 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 0.941592 + 0.543629i 0.866025 − 0.500000i −2.32183 − 1.26851i − 1.00000i −0.500000 − 0.866025i −0.543629 − 0.941592i
559.6 −0.866025 − 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.55662 + 1.47607i 0.866025 − 0.500000i −1.27923 − 2.31594i − 1.00000i −0.500000 − 0.866025i −1.47607 − 2.55662i
559.7 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −2.91202 − 1.68126i −0.866025 + 0.500000i 2.58296 − 0.572995i 1.00000i −0.500000 − 0.866025i −1.68126 − 2.91202i
559.8 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.71446 − 0.989846i −0.866025 + 0.500000i −0.233055 + 2.63547i 1.00000i −0.500000 − 0.866025i −0.989846 − 1.71446i
559.9 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i −1.48037 − 0.854693i −0.866025 + 0.500000i −0.893607 − 2.49027i 1.00000i −0.500000 − 0.866025i −0.854693 − 1.48037i
559.10 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.08023 + 1.20102i −0.866025 + 0.500000i 2.22571 − 1.43046i 1.00000i −0.500000 − 0.866025i 1.20102 + 2.08023i
559.11 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.30671 + 1.33178i −0.866025 + 0.500000i −1.54299 − 2.14923i 1.00000i −0.500000 − 0.866025i 1.33178 + 2.30671i
559.12 0.866025 + 0.500000i −0.500000 + 0.866025i 0.500000 + 0.866025i 2.58595 + 1.49300i −0.866025 + 0.500000i 1.09303 + 2.40941i 1.00000i −0.500000 − 0.866025i 1.49300 + 2.58595i
601.1 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i −3.28802 + 1.89834i 0.866025 + 0.500000i 1.34257 + 2.27981i 1.00000i −0.500000 + 0.866025i 1.89834 − 3.28802i
601.2 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i −1.02194 + 0.590018i 0.866025 + 0.500000i −2.59272 − 0.527083i 1.00000i −0.500000 + 0.866025i 0.590018 − 1.02194i
601.3 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i −0.397289 + 0.229375i 0.866025 + 0.500000i 1.97350 − 1.76218i 1.00000i −0.500000 + 0.866025i 0.229375 − 0.397289i
601.4 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.343006 − 0.198035i 0.866025 + 0.500000i 2.64565 + 0.0230826i 1.00000i −0.500000 + 0.866025i −0.198035 + 0.343006i
601.5 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i 0.941592 − 0.543629i 0.866025 + 0.500000i −2.32183 + 1.26851i 1.00000i −0.500000 + 0.866025i −0.543629 + 0.941592i
601.6 −0.866025 + 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i 2.55662 − 1.47607i 0.866025 + 0.500000i −1.27923 + 2.31594i 1.00000i −0.500000 + 0.866025i −1.47607 + 2.55662i
601.7 0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i −2.91202 + 1.68126i −0.866025 − 0.500000i 2.58296 + 0.572995i − 1.00000i −0.500000 + 0.866025i −1.68126 + 2.91202i
601.8 0.866025 − 0.500000i −0.500000 − 0.866025i 0.500000 − 0.866025i −1.71446 + 0.989846i −0.866025 − 0.500000i −0.233055 − 2.63547i − 1.00000i −0.500000 + 0.866025i −0.989846 + 1.71446i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.bj.a ✓ 24
7.b odd 2 1 798.2.bj.b yes 24
19.d odd 6 1 798.2.bj.b yes 24
133.p even 6 1 inner 798.2.bj.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.bj.a ✓ 24 1.a even 1 1 trivial
798.2.bj.a ✓ 24 133.p even 6 1 inner
798.2.bj.b yes 24 7.b odd 2 1
798.2.bj.b yes 24 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 33 T_{5}^{22} + 695 T_{5}^{20} - 204 T_{5}^{19} - 8806 T_{5}^{18} + 4032 T_{5}^{17} + \cdots + 322624 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display