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Properties

Label 1700.1.cf.a
Level $1700$
Weight $1$
Character orbit 1700.cf
Analytic conductor $0.848$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1700,1,Mod(19,1700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1700, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 36, 35]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1700.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1700.cf (of order \(40\), degree \(16\), 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: \(0.848410521476\)
Analytic rank: \(0\)
Dimension: \(16\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{17} q^{2} - \zeta_{40}^{14} q^{4} - \zeta_{40}^{11} q^{5} + \zeta_{40}^{11} q^{8} + \zeta_{40}^{13} q^{9} + \zeta_{40}^{8} q^{10} + ( - \zeta_{40}^{9} - \zeta_{40}^{7}) q^{13} - \zeta_{40}^{8} q^{16} + \cdots - \zeta_{40}^{12} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{10} + 4 q^{16} + 4 q^{26} + 4 q^{29} - 4 q^{41} + 4 q^{45} - 16 q^{53} - 16 q^{65} + 4 q^{68} - 4 q^{72} - 4 q^{73} - 4 q^{74} + 4 q^{82} - 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times\).

\(n\) \(477\) \(851\) \(1601\)
\(\chi(n)\) \(\zeta_{40}^{4}\) \(-1\) \(-\zeta_{40}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.987688 + 0.156434i
−0.891007 + 0.453990i
−0.987688 − 0.156434i
0.987688 − 0.156434i
−0.453990 − 0.891007i
−0.156434 − 0.987688i
0.453990 + 0.891007i
0.453990 − 0.891007i
0.891007 − 0.453990i
0.891007 + 0.453990i
0.987688 + 0.156434i
−0.891007 − 0.453990i
−0.156434 + 0.987688i
0.156434 + 0.987688i
−0.453990 + 0.891007i
0.156434 − 0.987688i
0.891007 + 0.453990i 0 0.587785 + 0.809017i −0.156434 − 0.987688i 0 0 0.156434 + 0.987688i 0.453990 + 0.891007i 0.309017 − 0.951057i
59.1 0.156434 + 0.987688i 0 −0.951057 + 0.309017i 0.453990 + 0.891007i 0 0 −0.453990 − 0.891007i −0.987688 − 0.156434i −0.809017 + 0.587785i
179.1 0.891007 − 0.453990i 0 0.587785 − 0.809017i −0.156434 + 0.987688i 0 0 0.156434 − 0.987688i 0.453990 − 0.891007i 0.309017 + 0.951057i
219.1 −0.891007 − 0.453990i 0 0.587785 + 0.809017i 0.156434 + 0.987688i 0 0 −0.156434 − 0.987688i −0.453990 − 0.891007i 0.309017 − 0.951057i
359.1 −0.987688 + 0.156434i 0 0.951057 − 0.309017i 0.891007 − 0.453990i 0 0 −0.891007 + 0.453990i 0.156434 − 0.987688i −0.809017 + 0.587785i
519.1 −0.453990 + 0.891007i 0 −0.587785 − 0.809017i −0.987688 + 0.156434i 0 0 0.987688 − 0.156434i −0.891007 + 0.453990i 0.309017 − 0.951057i
559.1 0.987688 − 0.156434i 0 0.951057 − 0.309017i −0.891007 + 0.453990i 0 0 0.891007 − 0.453990i −0.156434 + 0.987688i −0.809017 + 0.587785i
739.1 0.987688 + 0.156434i 0 0.951057 + 0.309017i −0.891007 − 0.453990i 0 0 0.891007 + 0.453990i −0.156434 − 0.987688i −0.809017 − 0.587785i
859.1 −0.156434 − 0.987688i 0 −0.951057 + 0.309017i −0.453990 − 0.891007i 0 0 0.453990 + 0.891007i 0.987688 + 0.156434i −0.809017 + 0.587785i
1039.1 −0.156434 + 0.987688i 0 −0.951057 − 0.309017i −0.453990 + 0.891007i 0 0 0.453990 − 0.891007i 0.987688 − 0.156434i −0.809017 − 0.587785i
1079.1 −0.891007 + 0.453990i 0 0.587785 − 0.809017i 0.156434 − 0.987688i 0 0 −0.156434 + 0.987688i −0.453990 + 0.891007i 0.309017 + 0.951057i
1239.1 0.156434 − 0.987688i 0 −0.951057 − 0.309017i 0.453990 − 0.891007i 0 0 −0.453990 + 0.891007i −0.987688 + 0.156434i −0.809017 − 0.587785i
1379.1 −0.453990 − 0.891007i 0 −0.587785 + 0.809017i −0.987688 − 0.156434i 0 0 0.987688 + 0.156434i −0.891007 − 0.453990i 0.309017 + 0.951057i
1419.1 0.453990 − 0.891007i 0 −0.587785 − 0.809017i 0.987688 − 0.156434i 0 0 −0.987688 + 0.156434i 0.891007 − 0.453990i 0.309017 − 0.951057i
1539.1 −0.987688 − 0.156434i 0 0.951057 + 0.309017i 0.891007 + 0.453990i 0 0 −0.891007 − 0.453990i 0.156434 + 0.987688i −0.809017 − 0.587785i
1579.1 0.453990 + 0.891007i 0 −0.587785 + 0.809017i 0.987688 + 0.156434i 0 0 −0.987688 − 0.156434i 0.891007 + 0.453990i 0.309017 + 0.951057i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
425.bd even 40 1 inner
1700.cf odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1700.1.cf.a ✓ 16
4.b odd 2 1 CM 1700.1.cf.a ✓ 16
17.d even 8 1 1700.1.cf.b yes 16
25.e even 10 1 1700.1.cf.b yes 16
68.g odd 8 1 1700.1.cf.b yes 16
100.h odd 10 1 1700.1.cf.b yes 16
425.bd even 40 1 inner 1700.1.cf.a ✓ 16
1700.cf odd 40 1 inner 1700.1.cf.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1700.1.cf.a ✓ 16 1.a even 1 1 trivial
1700.1.cf.a ✓ 16 4.b odd 2 1 CM
1700.1.cf.a ✓ 16 425.bd even 40 1 inner
1700.1.cf.a ✓ 16 1700.cf odd 40 1 inner
1700.1.cf.b yes 16 17.d even 8 1
1700.1.cf.b yes 16 25.e even 10 1
1700.1.cf.b yes 16 68.g odd 8 1
1700.1.cf.b yes 16 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{16} - 4 T_{29}^{15} + 10 T_{29}^{14} - 20 T_{29}^{13} + 34 T_{29}^{12} - 44 T_{29}^{11} + \cdots + 1 \) acting on \(S_{1}^{\mathrm{new}}(1700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - T^{12} + \cdots + 1 \) 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
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{8} + 8 T^{7} + 27 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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