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Properties

Label 2592.1.bz.a
Level $2592$
Weight $1$
Character orbit 2592.bz
Analytic conductor $1.294$
Analytic rank $0$
Dimension $18$
Projective image $D_{27}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2592,1,Mod(79,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(54))
 
chi = DirichletCharacter(H, H._module([27, 27, 28]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2592.bz (of order \(54\), degree \(18\), not 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: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\Q(\zeta_{54})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - x^{9} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 648)
Projective image: \(D_{27}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{27} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{54}^{26} q^{3} - \zeta_{54}^{25} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{54}^{26} q^{3} - \zeta_{54}^{25} q^{9} + (\zeta_{54}^{13} + \zeta_{54}^{3}) q^{11} + ( - \zeta_{54}^{19} - \zeta_{54}^{5}) q^{17} + (\zeta_{54}^{23} + \zeta_{54}^{7}) q^{19} + \zeta_{54}^{20} q^{25} - \zeta_{54}^{24} q^{27} + (\zeta_{54}^{12} + \zeta_{54}^{2}) q^{33} + ( - \zeta_{54}^{17} - \zeta_{54}^{15}) q^{41} + (\zeta_{54}^{15} - \zeta_{54}^{10}) q^{43} - \zeta_{54} q^{49} + ( - \zeta_{54}^{18} - \zeta_{54}^{4}) q^{51} + (\zeta_{54}^{22} + \zeta_{54}^{6}) q^{57} + (\zeta_{54}^{9} - \zeta_{54}^{2}) q^{59} + ( - \zeta_{54}^{12} + \zeta_{54}^{11}) q^{67} + (\zeta_{54}^{8} + \zeta_{54}^{4}) q^{73} + \zeta_{54}^{19} q^{75} - \zeta_{54}^{23} q^{81} + ( - \zeta_{54}^{14} - \zeta_{54}^{8}) q^{83} + ( - \zeta_{54}^{21} + \zeta_{54}^{18}) q^{89} + ( - \zeta_{54}^{21} - \zeta_{54}^{13}) q^{97} + (\zeta_{54}^{11} + \zeta_{54}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 9 q^{51} + 9 q^{59} - 9 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(\zeta_{54}^{22}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.286803 − 0.957990i
0.686242 − 0.727374i
0.993238 + 0.116093i
0.0581448 − 0.998308i
0.0581448 + 0.998308i
−0.597159 − 0.802123i
−0.973045 + 0.230616i
−0.973045 − 0.230616i
−0.396080 − 0.918216i
−0.893633 + 0.448799i
0.835488 − 0.549509i
−0.396080 + 0.918216i
0.686242 + 0.727374i
0.286803 + 0.957990i
−0.597159 + 0.802123i
0.835488 + 0.549509i
−0.893633 − 0.448799i
0.993238 − 0.116093i
0 0.286803 + 0.957990i 0 0 0 0 0 −0.835488 + 0.549509i 0
175.1 0 0.686242 + 0.727374i 0 0 0 0 0 −0.0581448 + 0.998308i 0
367.1 0 0.993238 − 0.116093i 0 0 0 0 0 0.973045 − 0.230616i 0
463.1 0 0.0581448 + 0.998308i 0 0 0 0 0 −0.993238 + 0.116093i 0
655.1 0 0.0581448 − 0.998308i 0 0 0 0 0 −0.993238 − 0.116093i 0
751.1 0 −0.597159 + 0.802123i 0 0 0 0 0 −0.286803 − 0.957990i 0
943.1 0 −0.973045 − 0.230616i 0 0 0 0 0 0.893633 + 0.448799i 0
1039.1 0 −0.973045 + 0.230616i 0 0 0 0 0 0.893633 − 0.448799i 0
1231.1 0 −0.396080 + 0.918216i 0 0 0 0 0 −0.686242 − 0.727374i 0
1327.1 0 −0.893633 − 0.448799i 0 0 0 0 0 0.597159 + 0.802123i 0
1519.1 0 0.835488 + 0.549509i 0 0 0 0 0 0.396080 + 0.918216i 0
1615.1 0 −0.396080 − 0.918216i 0 0 0 0 0 −0.686242 + 0.727374i 0
1807.1 0 0.686242 − 0.727374i 0 0 0 0 0 −0.0581448 − 0.998308i 0
1903.1 0 0.286803 − 0.957990i 0 0 0 0 0 −0.835488 − 0.549509i 0
2095.1 0 −0.597159 − 0.802123i 0 0 0 0 0 −0.286803 + 0.957990i 0
2191.1 0 0.835488 − 0.549509i 0 0 0 0 0 0.396080 − 0.918216i 0
2383.1 0 −0.893633 + 0.448799i 0 0 0 0 0 0.597159 − 0.802123i 0
2479.1 0 0.993238 + 0.116093i 0 0 0 0 0 0.973045 + 0.230616i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
81.g even 27 1 inner
648.bf odd 54 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.1.bz.a 18
4.b odd 2 1 648.1.bf.a ✓ 18
8.b even 2 1 648.1.bf.a ✓ 18
8.d odd 2 1 CM 2592.1.bz.a 18
12.b even 2 1 1944.1.bf.a 18
24.h odd 2 1 1944.1.bf.a 18
81.g even 27 1 inner 2592.1.bz.a 18
324.n odd 54 1 648.1.bf.a ✓ 18
324.p even 54 1 1944.1.bf.a 18
648.z odd 54 1 1944.1.bf.a 18
648.bd even 54 1 648.1.bf.a ✓ 18
648.bf odd 54 1 inner 2592.1.bz.a 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.1.bf.a ✓ 18 4.b odd 2 1
648.1.bf.a ✓ 18 8.b even 2 1
648.1.bf.a ✓ 18 324.n odd 54 1
648.1.bf.a ✓ 18 648.bd even 54 1
1944.1.bf.a 18 12.b even 2 1
1944.1.bf.a 18 24.h odd 2 1
1944.1.bf.a 18 324.p even 54 1
1944.1.bf.a 18 648.z odd 54 1
2592.1.bz.a 18 1.a even 1 1 trivial
2592.1.bz.a 18 8.d odd 2 1 CM
2592.1.bz.a 18 81.g even 27 1 inner
2592.1.bz.a 18 648.bf odd 54 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2592, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} - T^{9} + 1 \) Copy content Toggle raw display
$5$ \( T^{18} \) Copy content Toggle raw display
$7$ \( T^{18} \) Copy content Toggle raw display
$11$ \( T^{18} - 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{18} \) Copy content Toggle raw display
$17$ \( T^{18} - 18 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{18} - 9 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{18} \) Copy content Toggle raw display
$29$ \( T^{18} \) Copy content Toggle raw display
$31$ \( T^{18} \) Copy content Toggle raw display
$37$ \( T^{18} \) Copy content Toggle raw display
$41$ \( T^{18} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{18} - 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{18} \) Copy content Toggle raw display
$53$ \( T^{18} \) Copy content Toggle raw display
$59$ \( T^{18} - 9 T^{17} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{18} \) Copy content Toggle raw display
$67$ \( T^{18} - 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{18} \) Copy content Toggle raw display
$73$ \( T^{18} + 9 T^{13} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{18} \) Copy content Toggle raw display
$83$ \( T^{18} - 9 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$97$ \( T^{18} + 3 T^{15} + \cdots + 1 \) Copy content Toggle raw display
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