Properties

Label 819.1.cj.a
Level $819$
Weight $1$
Character orbit 819.cj
Analytic conductor $0.409$
Analytic rank $0$
Dimension $4$
Projective image $A_{4}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,1,Mod(328,819)]
 
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([2, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.328");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 819.cj (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: \(0.408734245346\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.670761.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{12}^{4} q^{2} - \zeta_{12}^{3} q^{3} - \zeta_{12} q^{5} + \zeta_{12} q^{6} + q^{7} - q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12}^{4} q^{2} - \zeta_{12}^{3} q^{3} - \zeta_{12} q^{5} + \zeta_{12} q^{6} + q^{7} - q^{8} - q^{9} - \zeta_{12}^{5} q^{10} - \zeta_{12}^{4} q^{11} - \zeta_{12}^{3} q^{13} + \zeta_{12}^{4} q^{14} + \zeta_{12}^{4} q^{15} - \zeta_{12}^{4} q^{16} + \zeta_{12} q^{17} - \zeta_{12}^{4} q^{18} - \zeta_{12} q^{19} - \zeta_{12}^{3} q^{21} + \zeta_{12}^{2} q^{22} + \zeta_{12}^{3} q^{24} + \zeta_{12} q^{26} + \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{29} - \zeta_{12}^{2} q^{30} + \zeta_{12} q^{31} - \zeta_{12} q^{33} + \zeta_{12}^{5} q^{34} - \zeta_{12} q^{35} - \zeta_{12}^{2} q^{37} - \zeta_{12}^{5} q^{38} - q^{39} + \zeta_{12} q^{40} + \zeta_{12} q^{42} + \zeta_{12} q^{45} - \zeta_{12}^{5} q^{47} - \zeta_{12} q^{48} + q^{49} - \zeta_{12}^{4} q^{51} - \zeta_{12} q^{54} + \zeta_{12}^{5} q^{55} - q^{56} + \zeta_{12}^{4} q^{57} + \zeta_{12}^{2} q^{58} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{3} q^{61} + \zeta_{12}^{5} q^{62} - q^{63} + q^{64} + \zeta_{12}^{4} q^{65} - \zeta_{12}^{5} q^{66} - q^{67} - \zeta_{12}^{5} q^{70} + \zeta_{12}^{4} q^{71} + q^{72} + q^{74} - \zeta_{12}^{4} q^{77} - \zeta_{12}^{4} q^{78} + \zeta_{12}^{2} q^{79} + \zeta_{12}^{5} q^{80} + q^{81} + \zeta_{12}^{5} q^{83} - \zeta_{12}^{2} q^{85} - \zeta_{12} q^{87} + \zeta_{12}^{4} q^{88} + \zeta_{12}^{5} q^{89} + \zeta_{12}^{5} q^{90} - \zeta_{12}^{3} q^{91} - \zeta_{12}^{4} q^{93} + \zeta_{12}^{3} q^{94} + \zeta_{12}^{2} q^{95} + \zeta_{12}^{4} q^{98} + \zeta_{12}^{4} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{7} - 4 q^{8} - 4 q^{9} + 2 q^{11} - 2 q^{14} - 2 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} + 2 q^{29} - 2 q^{30} - 2 q^{37} - 4 q^{39} + 4 q^{49} + 2 q^{51} - 4 q^{56} - 2 q^{57} + 2 q^{58} - 4 q^{63} + 4 q^{64} - 2 q^{65} - 8 q^{67} - 2 q^{71} + 4 q^{72} + 4 q^{74} + 2 q^{77} + 2 q^{78} + 2 q^{79} + 4 q^{81} - 2 q^{85} - 2 q^{88} + 2 q^{93} + 2 q^{95} - 2 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(\zeta_{12}^{4}\) \(\zeta_{12}^{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
328.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.500000 + 0.866025i 1.00000i 0 −0.866025 0.500000i 0.866025 + 0.500000i 1.00000 −1.00000 −1.00000 0.866025 0.500000i
328.2 −0.500000 + 0.866025i 1.00000i 0 0.866025 + 0.500000i −0.866025 0.500000i 1.00000 −1.00000 −1.00000 −0.866025 + 0.500000i
412.1 −0.500000 0.866025i 1.00000i 0 0.866025 0.500000i −0.866025 + 0.500000i 1.00000 −1.00000 −1.00000 −0.866025 0.500000i
412.2 −0.500000 0.866025i 1.00000i 0 −0.866025 + 0.500000i 0.866025 0.500000i 1.00000 −1.00000 −1.00000 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
117.f even 3 1 inner
819.cj odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.1.cj.a 4
3.b odd 2 1 2457.1.cj.a 4
7.b odd 2 1 inner 819.1.cj.a 4
9.c even 3 1 819.1.ek.a yes 4
9.d odd 6 1 2457.1.ek.a 4
13.c even 3 1 819.1.ek.a yes 4
21.c even 2 1 2457.1.cj.a 4
39.i odd 6 1 2457.1.ek.a 4
63.l odd 6 1 819.1.ek.a yes 4
63.o even 6 1 2457.1.ek.a 4
91.n odd 6 1 819.1.ek.a yes 4
117.f even 3 1 inner 819.1.cj.a 4
117.u odd 6 1 2457.1.cj.a 4
273.bn even 6 1 2457.1.ek.a 4
819.cj odd 6 1 inner 819.1.cj.a 4
819.de even 6 1 2457.1.cj.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.1.cj.a 4 1.a even 1 1 trivial
819.1.cj.a 4 7.b odd 2 1 inner
819.1.cj.a 4 117.f even 3 1 inner
819.1.cj.a 4 819.cj odd 6 1 inner
819.1.ek.a yes 4 9.c even 3 1
819.1.ek.a yes 4 13.c even 3 1
819.1.ek.a yes 4 63.l odd 6 1
819.1.ek.a yes 4 91.n odd 6 1
2457.1.cj.a 4 3.b odd 2 1
2457.1.cj.a 4 21.c even 2 1
2457.1.cj.a 4 117.u odd 6 1
2457.1.cj.a 4 819.de even 6 1
2457.1.ek.a 4 9.d odd 6 1
2457.1.ek.a 4 39.i odd 6 1
2457.1.ek.a 4 63.o even 6 1
2457.1.ek.a 4 273.bn even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(819, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$37$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$89$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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