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Properties

Label 700.1.bf.a
Level $700$
Weight $1$
Character orbit 700.bf
Analytic conductor $0.349$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -20
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,1,Mod(143,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.143");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 700.bf (of order \(12\), degree \(4\), 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: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.6722800.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{11} q^{2} + (\zeta_{24}^{9} + \zeta_{24}^{5}) q^{3} - \zeta_{24}^{10} q^{4} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{6} + \zeta_{24}^{7} q^{7} - \zeta_{24}^{9} q^{8} + (\zeta_{24}^{10} + \cdots - \zeta_{24}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{11} q^{2} + (\zeta_{24}^{9} + \zeta_{24}^{5}) q^{3} - \zeta_{24}^{10} q^{4} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{6} + \zeta_{24}^{7} q^{7} - \zeta_{24}^{9} q^{8} + (\zeta_{24}^{10} + \cdots - \zeta_{24}^{2}) q^{9} + \cdots - \zeta_{24} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{16} - 12 q^{21} - 16 q^{36} - 4 q^{46} + 4 q^{56} + 12 q^{61} + 4 q^{81} + 4 q^{86} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(-\zeta_{24}^{8}\) \(-1\) \(-\zeta_{24}^{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 − 0.965926i
−0.258819 + 0.965926i −1.67303 + 0.448288i −0.866025 − 0.500000i 0 − 1.73205i 0.965926 − 0.258819i 0.707107 − 0.707107i 1.73205 − 1.00000i 0
143.2 0.258819 − 0.965926i 1.67303 − 0.448288i −0.866025 − 0.500000i 0 − 1.73205i −0.965926 + 0.258819i −0.707107 + 0.707107i 1.73205 − 1.00000i 0
243.1 −0.965926 + 0.258819i 0.448288 − 1.67303i 0.866025 − 0.500000i 0 1.73205i 0.258819 − 0.965926i −0.707107 + 0.707107i −1.73205 − 1.00000i 0
243.2 0.965926 − 0.258819i −0.448288 + 1.67303i 0.866025 − 0.500000i 0 1.73205i −0.258819 + 0.965926i 0.707107 − 0.707107i −1.73205 − 1.00000i 0
507.1 −0.965926 − 0.258819i 0.448288 + 1.67303i 0.866025 + 0.500000i 0 − 1.73205i 0.258819 + 0.965926i −0.707107 − 0.707107i −1.73205 + 1.00000i 0
507.2 0.965926 + 0.258819i −0.448288 − 1.67303i 0.866025 + 0.500000i 0 − 1.73205i −0.258819 − 0.965926i 0.707107 + 0.707107i −1.73205 + 1.00000i 0
607.1 −0.258819 − 0.965926i −1.67303 − 0.448288i −0.866025 + 0.500000i 0 1.73205i 0.965926 + 0.258819i 0.707107 + 0.707107i 1.73205 + 1.00000i 0
607.2 0.258819 + 0.965926i 1.67303 + 0.448288i −0.866025 + 0.500000i 0 1.73205i −0.965926 − 0.258819i −0.707107 − 0.707107i 1.73205 + 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
20.e even 4 2 inner
28.f even 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner
140.s even 6 1 inner
140.x odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.1.bf.a ✓ 8
4.b odd 2 1 inner 700.1.bf.a ✓ 8
5.b even 2 1 inner 700.1.bf.a ✓ 8
5.c odd 4 2 inner 700.1.bf.a ✓ 8
7.d odd 6 1 inner 700.1.bf.a ✓ 8
20.d odd 2 1 CM 700.1.bf.a ✓ 8
20.e even 4 2 inner 700.1.bf.a ✓ 8
28.f even 6 1 inner 700.1.bf.a ✓ 8
35.i odd 6 1 inner 700.1.bf.a ✓ 8
35.k even 12 2 inner 700.1.bf.a ✓ 8
140.s even 6 1 inner 700.1.bf.a ✓ 8
140.x odd 12 2 inner 700.1.bf.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.1.bf.a ✓ 8 1.a even 1 1 trivial
700.1.bf.a ✓ 8 4.b odd 2 1 inner
700.1.bf.a ✓ 8 5.b even 2 1 inner
700.1.bf.a ✓ 8 5.c odd 4 2 inner
700.1.bf.a ✓ 8 7.d odd 6 1 inner
700.1.bf.a ✓ 8 20.d odd 2 1 CM
700.1.bf.a ✓ 8 20.e even 4 2 inner
700.1.bf.a ✓ 8 28.f even 6 1 inner
700.1.bf.a ✓ 8 35.i odd 6 1 inner
700.1.bf.a ✓ 8 35.k even 12 2 inner
700.1.bf.a ✓ 8 140.s even 6 1 inner
700.1.bf.a ✓ 8 140.x odd 12 2 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(700, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} - 9T^{4} + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 1)^{2} \) 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
$61$ \( (T^{2} - 3 T + 3)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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