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Properties

Label 543.1.y.a
Level $543$
Weight $1$
Character orbit 543.y
Analytic conductor $0.271$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [543,1,Mod(5,543)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(543, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 26]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("543.5");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 543 = 3 \cdot 181 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 543.y (of order \(30\), degree \(8\), 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: \(0.270992301860\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{30}^{11} q^{3} + \zeta_{30}^{2} q^{4} + (\zeta_{30}^{12} - \zeta_{30}^{3}) q^{7} - \zeta_{30}^{7} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{30}^{11} q^{3} + \zeta_{30}^{2} q^{4} + (\zeta_{30}^{12} - \zeta_{30}^{3}) q^{7} - \zeta_{30}^{7} q^{9} - \zeta_{30}^{13} q^{12} + ( - \zeta_{30}^{11} + \zeta_{30}^{2}) q^{13} + \zeta_{30}^{4} q^{16} - q^{19} + (\zeta_{30}^{14} + \zeta_{30}^{8}) q^{21} + \zeta_{30}^{12} q^{25} - \zeta_{30}^{3} q^{27} + (\zeta_{30}^{14} - \zeta_{30}^{5}) q^{28} + (\zeta_{30}^{10} + \zeta_{30}^{8}) q^{31} - \zeta_{30}^{9} q^{36} + \zeta_{30}^{11} q^{37} + ( - \zeta_{30}^{13} - \zeta_{30}^{7}) q^{39} + ( - \zeta_{30}^{9} - \zeta_{30}) q^{43} + q^{48} + ( - \zeta_{30}^{9} + \zeta_{30}^{6} + 1) q^{49} + ( - \zeta_{30}^{13} + \zeta_{30}^{4}) q^{52} + \zeta_{30}^{11} q^{57} + ( - \zeta_{30}^{9} - \zeta_{30}) q^{61} + (\zeta_{30}^{10} + \zeta_{30}^{4}) q^{63} + \zeta_{30}^{6} q^{64} + ( - \zeta_{30}^{5} + \zeta_{30}^{4}) q^{67} + ( - \zeta_{30}^{7} - \zeta_{30}^{3}) q^{73} + \zeta_{30}^{8} q^{75} - \zeta_{30}^{2} q^{76} + (\zeta_{30}^{8} + \zeta_{30}^{6}) q^{79} + \zeta_{30}^{14} q^{81} + (\zeta_{30}^{10} - \zeta_{30}) q^{84} + (2 \zeta_{30}^{14} + \cdots - \zeta_{30}^{5}) q^{91} + \cdots + ( - \zeta_{30}^{11} + 1) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} + q^{4} - 4 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} + q^{4} - 4 q^{7} + q^{9} + q^{12} + 2 q^{13} + q^{16} - 8 q^{19} + 2 q^{21} - 2 q^{25} - 2 q^{27} - 3 q^{28} - 3 q^{31} - 2 q^{36} - q^{37} + 2 q^{39} - q^{43} + 8 q^{48} + 4 q^{49} + 2 q^{52} - q^{57} - q^{61} - 3 q^{63} - 2 q^{64} - 3 q^{67} - q^{73} + q^{75} - q^{76} - q^{79} + q^{81} - 3 q^{84} - q^{91} - q^{93} + 9 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/543\mathbb{Z}\right)^\times\).

\(n\) \(182\) \(364\)
\(\chi(n)\) \(-1\) \(\zeta_{30}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.913545 − 0.406737i
0.669131 + 0.743145i
0.669131 − 0.743145i
−0.104528 − 0.994522i
0.913545 + 0.406737i
−0.978148 + 0.207912i
−0.978148 − 0.207912i
−0.104528 + 0.994522i
0 −0.104528 + 0.994522i 0.669131 − 0.743145i 0 0 0.618034 0 −0.978148 − 0.207912i 0
29.1 0 −0.978148 + 0.207912i −0.104528 + 0.994522i 0 0 −1.61803 0 0.913545 − 0.406737i 0
206.1 0 −0.978148 − 0.207912i −0.104528 − 0.994522i 0 0 −1.61803 0 0.913545 + 0.406737i 0
263.1 0 0.913545 + 0.406737i −0.978148 + 0.207912i 0 0 0.618034 0 0.669131 + 0.743145i 0
326.1 0 −0.104528 − 0.994522i 0.669131 + 0.743145i 0 0 0.618034 0 −0.978148 + 0.207912i 0
389.1 0 0.669131 + 0.743145i 0.913545 − 0.406737i 0 0 −1.61803 0 −0.104528 + 0.994522i 0
476.1 0 0.669131 − 0.743145i 0.913545 + 0.406737i 0 0 −1.61803 0 −0.104528 − 0.994522i 0
479.1 0 0.913545 − 0.406737i −0.978148 − 0.207912i 0 0 0.618034 0 0.669131 − 0.743145i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
181.j even 15 1 inner
543.y odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 543.1.y.a ✓ 8
3.b odd 2 1 CM 543.1.y.a ✓ 8
181.j even 15 1 inner 543.1.y.a ✓ 8
543.y odd 30 1 inner 543.1.y.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
543.1.y.a ✓ 8 1.a even 1 1 trivial
543.1.y.a ✓ 8 3.b odd 2 1 CM
543.1.y.a ✓ 8 181.j even 15 1 inner
543.1.y.a ✓ 8 543.y odd 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(543, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + T - 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T + 1)^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{8} + T^{7} - T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} + 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} + T^{7} + 4 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} - 9 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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