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Properties

Label 1425.1.cd.b
Level $1425$
Weight $1$
Character orbit 1425.cd
Analytic conductor $0.711$
Analytic rank $0$
Dimension $24$
Projective image $D_{18}$
CM discriminant -3
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1425,1,Mod(32,1425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1425, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 9, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1425.32");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1425.cd (of order \(36\), degree \(12\), 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: \(0.711167643002\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{36})\)
Coefficient field: \(\Q(\zeta_{72})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{12} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{72}^{31} q^{3} + \zeta_{72}^{2} q^{4} + (\zeta_{72}^{25} + \zeta_{72}^{5}) q^{7} - \zeta_{72}^{26} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{72}^{31} q^{3} + \zeta_{72}^{2} q^{4} + (\zeta_{72}^{25} + \zeta_{72}^{5}) q^{7} - \zeta_{72}^{26} q^{9} + \zeta_{72}^{33} q^{12} + ( - \zeta_{72}^{27} - \zeta_{72}^{19}) q^{13} + \zeta_{72}^{4} q^{16} + \zeta_{72}^{18} q^{19} + ( - \zeta_{72}^{20} - 1) q^{21} + \zeta_{72}^{21} q^{27} + (\zeta_{72}^{27} + \zeta_{72}^{7}) q^{28} + ( - \zeta_{72}^{8} - \zeta_{72}^{4}) q^{31} - \zeta_{72}^{28} q^{36} + (\zeta_{72}^{17} + \zeta_{72}) q^{37} + (\zeta_{72}^{22} + \zeta_{72}^{14}) q^{39} + (\zeta_{72}^{11} - \zeta_{72}^{3}) q^{43} + \zeta_{72}^{35} q^{48} + (\zeta_{72}^{30} + \cdots + \zeta_{72}^{10}) q^{49} + \cdots + \zeta_{72} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 24 q^{21} + 12 q^{61} + 12 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times\).

\(n\) \(476\) \(1027\) \(1351\)
\(\chi(n)\) \(-1\) \(-\zeta_{72}^{18}\) \(\zeta_{72}^{20}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
0.0871557 − 0.996195i
−0.0871557 + 0.996195i
0.573576 − 0.819152i
−0.573576 + 0.819152i
−0.819152 − 0.573576i
0.819152 + 0.573576i
0.0871557 + 0.996195i
−0.0871557 − 0.996195i
−0.819152 + 0.573576i
0.819152 − 0.573576i
0.996195 − 0.0871557i
−0.996195 + 0.0871557i
0.573576 + 0.819152i
−0.573576 − 0.819152i
0.422618 + 0.906308i
−0.422618 − 0.906308i
−0.906308 − 0.422618i
0.906308 + 0.422618i
−0.906308 + 0.422618i
0.906308 − 0.422618i
0 −0.422618 − 0.906308i −0.984808 − 0.173648i 0 0 1.24177 − 0.332731i 0 −0.642788 + 0.766044i 0
32.2 0 0.422618 + 0.906308i −0.984808 − 0.173648i 0 0 −1.24177 + 0.332731i 0 −0.642788 + 0.766044i 0
143.1 0 −0.0871557 + 0.996195i −0.342020 − 0.939693i 0 0 0.509774 + 1.90250i 0 −0.984808 − 0.173648i 0
143.2 0 0.0871557 − 0.996195i −0.342020 − 0.939693i 0 0 −0.509774 − 1.90250i 0 −0.984808 − 0.173648i 0
257.1 0 −0.996195 − 0.0871557i 0.342020 + 0.939693i 0 0 1.90250 − 0.509774i 0 0.984808 + 0.173648i 0
257.2 0 0.996195 + 0.0871557i 0.342020 + 0.939693i 0 0 −1.90250 + 0.509774i 0 0.984808 + 0.173648i 0
668.1 0 −0.422618 + 0.906308i −0.984808 + 0.173648i 0 0 1.24177 + 0.332731i 0 −0.642788 − 0.766044i 0
668.2 0 0.422618 − 0.906308i −0.984808 + 0.173648i 0 0 −1.24177 − 0.332731i 0 −0.642788 − 0.766044i 0
743.1 0 −0.996195 + 0.0871557i 0.342020 − 0.939693i 0 0 1.90250 + 0.509774i 0 0.984808 − 0.173648i 0
743.2 0 0.996195 − 0.0871557i 0.342020 − 0.939693i 0 0 −1.90250 − 0.509774i 0 0.984808 − 0.173648i 0
782.1 0 −0.906308 − 0.422618i 0.984808 − 0.173648i 0 0 0.332731 − 1.24177i 0 0.642788 + 0.766044i 0
782.2 0 0.906308 + 0.422618i 0.984808 − 0.173648i 0 0 −0.332731 + 1.24177i 0 0.642788 + 0.766044i 0
857.1 0 −0.0871557 − 0.996195i −0.342020 + 0.939693i 0 0 0.509774 − 1.90250i 0 −0.984808 + 0.173648i 0
857.2 0 0.0871557 + 0.996195i −0.342020 + 0.939693i 0 0 −0.509774 + 1.90250i 0 −0.984808 + 0.173648i 0
1193.1 0 −0.819152 − 0.573576i −0.642788 + 0.766044i 0 0 −0.177043 − 0.660732i 0 0.342020 + 0.939693i 0
1193.2 0 0.819152 + 0.573576i −0.642788 + 0.766044i 0 0 0.177043 + 0.660732i 0 0.342020 + 0.939693i 0
1268.1 0 −0.573576 − 0.819152i 0.642788 + 0.766044i 0 0 0.660732 + 0.177043i 0 −0.342020 + 0.939693i 0
1268.2 0 0.573576 + 0.819152i 0.642788 + 0.766044i 0 0 −0.660732 − 0.177043i 0 −0.342020 + 0.939693i 0
1307.1 0 −0.573576 + 0.819152i 0.642788 − 0.766044i 0 0 0.660732 − 0.177043i 0 −0.342020 − 0.939693i 0
1307.2 0 0.573576 − 0.819152i 0.642788 − 0.766044i 0 0 −0.660732 + 0.177043i 0 −0.342020 − 0.939693i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner
19.f odd 18 1 inner
57.j even 18 1 inner
95.o odd 18 1 inner
95.r even 36 2 inner
285.bf even 18 1 inner
285.bj odd 36 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1425.1.cd.b ✓ 24
3.b odd 2 1 CM 1425.1.cd.b ✓ 24
5.b even 2 1 inner 1425.1.cd.b ✓ 24
5.c odd 4 2 inner 1425.1.cd.b ✓ 24
15.d odd 2 1 inner 1425.1.cd.b ✓ 24
15.e even 4 2 inner 1425.1.cd.b ✓ 24
19.f odd 18 1 inner 1425.1.cd.b ✓ 24
57.j even 18 1 inner 1425.1.cd.b ✓ 24
95.o odd 18 1 inner 1425.1.cd.b ✓ 24
95.r even 36 2 inner 1425.1.cd.b ✓ 24
285.bf even 18 1 inner 1425.1.cd.b ✓ 24
285.bj odd 36 2 inner 1425.1.cd.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.1.cd.b ✓ 24 1.a even 1 1 trivial
1425.1.cd.b ✓ 24 3.b odd 2 1 CM
1425.1.cd.b ✓ 24 5.b even 2 1 inner
1425.1.cd.b ✓ 24 5.c odd 4 2 inner
1425.1.cd.b ✓ 24 15.d odd 2 1 inner
1425.1.cd.b ✓ 24 15.e even 4 2 inner
1425.1.cd.b ✓ 24 19.f odd 18 1 inner
1425.1.cd.b ✓ 24 57.j even 18 1 inner
1425.1.cd.b ✓ 24 95.o odd 18 1 inner
1425.1.cd.b ✓ 24 95.r even 36 2 inner
1425.1.cd.b ✓ 24 285.bf even 18 1 inner
1425.1.cd.b ✓ 24 285.bj odd 36 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{1}^{\mathrm{new}}(1425, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \) Copy content Toggle raw display
$3$ \( T^{24} - T^{12} + 1 \) Copy content Toggle raw display
$5$ \( T^{24} \) Copy content Toggle raw display
$7$ \( T^{24} - 18 T^{20} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{24} \) Copy content Toggle raw display
$13$ \( T^{24} - 6 T^{20} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{24} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{12} \) Copy content Toggle raw display
$23$ \( T^{24} \) Copy content Toggle raw display
$29$ \( T^{24} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{4} \) Copy content Toggle raw display
$37$ \( (T^{12} + 18 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{24} \) Copy content Toggle raw display
$43$ \( T^{24} - 27 T^{20} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( T^{24} \) Copy content Toggle raw display
$53$ \( T^{24} \) Copy content Toggle raw display
$59$ \( T^{24} \) Copy content Toggle raw display
$61$ \( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{24} + 21 T^{20} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{24} \) Copy content Toggle raw display
$73$ \( T^{24} + 9 T^{20} + \cdots + 81 \) Copy content Toggle raw display
$79$ \( (T^{12} + 6 T^{10} + \cdots + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{24} \) Copy content Toggle raw display
$89$ \( T^{24} \) Copy content Toggle raw display
$97$ \( T^{24} - T^{12} + 1 \) Copy content Toggle raw display
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