Properties

Label 2385.1.q.b
Level $2385$
Weight $1$
Character orbit 2385.q
Analytic conductor $1.190$
Analytic rank $0$
Dimension $16$
Projective image $D_{20}$
CM discriminant -159
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2385,1,Mod(847,2385)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2385, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2385.847");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2385 = 3^{2} \cdot 5 \cdot 53 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2385.q (of order \(4\), 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: \(1.19027005513\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{40}^{7} - \zeta_{40}^{3}) q^{2} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{6}) q^{4}+ \cdots + ( - \zeta_{40}^{17} + \cdots - \zeta_{40}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{40}^{7} - \zeta_{40}^{3}) q^{2} + (\zeta_{40}^{14} + \cdots + \zeta_{40}^{6}) q^{4}+ \cdots + ( - \zeta_{40}^{19} + \cdots - \zeta_{40}^{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} + 4 q^{13} - 12 q^{28} - 4 q^{37} - 4 q^{40} + 4 q^{43} + 8 q^{52} + 12 q^{70} - 12 q^{82} + 8 q^{91} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2385\mathbb{Z}\right)^\times\).

\(n\) \(1432\) \(1486\) \(1856\)
\(\chi(n)\) \(\zeta_{40}^{10}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
847.1
0.891007 0.453990i
−0.453990 + 0.891007i
0.987688 + 0.156434i
0.156434 + 0.987688i
−0.156434 0.987688i
−0.987688 0.156434i
0.453990 0.891007i
−0.891007 + 0.453990i
0.891007 + 0.453990i
−0.453990 0.891007i
0.987688 0.156434i
0.156434 0.987688i
−0.156434 + 0.987688i
−0.987688 + 0.156434i
0.453990 + 0.891007i
−0.891007 0.453990i
−1.14412 + 1.14412i 0 1.61803i −0.453990 0.891007i 0 −0.221232 0.221232i 0.707107 + 0.707107i 0 1.53884 + 0.500000i
847.2 −1.14412 + 1.14412i 0 1.61803i 0.891007 + 0.453990i 0 −1.39680 1.39680i 0.707107 + 0.707107i 0 −1.53884 + 0.500000i
847.3 −0.437016 + 0.437016i 0 0.618034i 0.156434 0.987688i 0 1.26007 + 1.26007i −0.707107 0.707107i 0 0.363271 + 0.500000i
847.4 −0.437016 + 0.437016i 0 0.618034i 0.987688 0.156434i 0 −0.642040 0.642040i −0.707107 0.707107i 0 −0.363271 + 0.500000i
847.5 0.437016 0.437016i 0 0.618034i −0.987688 + 0.156434i 0 −0.642040 0.642040i 0.707107 + 0.707107i 0 −0.363271 + 0.500000i
847.6 0.437016 0.437016i 0 0.618034i −0.156434 + 0.987688i 0 1.26007 + 1.26007i 0.707107 + 0.707107i 0 0.363271 + 0.500000i
847.7 1.14412 1.14412i 0 1.61803i −0.891007 0.453990i 0 −1.39680 1.39680i −0.707107 0.707107i 0 −1.53884 + 0.500000i
847.8 1.14412 1.14412i 0 1.61803i 0.453990 + 0.891007i 0 −0.221232 0.221232i −0.707107 0.707107i 0 1.53884 + 0.500000i
2278.1 −1.14412 1.14412i 0 1.61803i −0.453990 + 0.891007i 0 −0.221232 + 0.221232i 0.707107 0.707107i 0 1.53884 0.500000i
2278.2 −1.14412 1.14412i 0 1.61803i 0.891007 0.453990i 0 −1.39680 + 1.39680i 0.707107 0.707107i 0 −1.53884 0.500000i
2278.3 −0.437016 0.437016i 0 0.618034i 0.156434 + 0.987688i 0 1.26007 1.26007i −0.707107 + 0.707107i 0 0.363271 0.500000i
2278.4 −0.437016 0.437016i 0 0.618034i 0.987688 + 0.156434i 0 −0.642040 + 0.642040i −0.707107 + 0.707107i 0 −0.363271 0.500000i
2278.5 0.437016 + 0.437016i 0 0.618034i −0.987688 0.156434i 0 −0.642040 + 0.642040i 0.707107 0.707107i 0 −0.363271 0.500000i
2278.6 0.437016 + 0.437016i 0 0.618034i −0.156434 0.987688i 0 1.26007 1.26007i 0.707107 0.707107i 0 0.363271 0.500000i
2278.7 1.14412 + 1.14412i 0 1.61803i −0.891007 + 0.453990i 0 −1.39680 + 1.39680i −0.707107 + 0.707107i 0 −1.53884 0.500000i
2278.8 1.14412 + 1.14412i 0 1.61803i 0.453990 0.891007i 0 −0.221232 + 0.221232i −0.707107 + 0.707107i 0 1.53884 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 847.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
159.d odd 2 1 CM by \(\Q(\sqrt{-159}) \)
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner
53.b even 2 1 inner
265.i odd 4 1 inner
795.m even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2385.1.q.b 16
3.b odd 2 1 inner 2385.1.q.b 16
5.c odd 4 1 inner 2385.1.q.b 16
15.e even 4 1 inner 2385.1.q.b 16
53.b even 2 1 inner 2385.1.q.b 16
159.d odd 2 1 CM 2385.1.q.b 16
265.i odd 4 1 inner 2385.1.q.b 16
795.m even 4 1 inner 2385.1.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2385.1.q.b 16 1.a even 1 1 trivial
2385.1.q.b 16 3.b odd 2 1 inner
2385.1.q.b 16 5.c odd 4 1 inner
2385.1.q.b 16 15.e even 4 1 inner
2385.1.q.b 16 53.b even 2 1 inner
2385.1.q.b 16 159.d odd 2 1 CM
2385.1.q.b 16 265.i odd 4 1 inner
2385.1.q.b 16 795.m even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 7T_{2}^{4} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2385, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 7 T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{8} + 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( (T^{8} + 8 T^{6} + 19 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} + 15 T^{4} + 25)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( (T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
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