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 $16$

Related objects

Downloads

Learn more

Newspace parameters

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

Self dual: no
Analytic conductor: \(0.349345508843\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{24}^{11} q^{2} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{3} -\zeta_{24}^{10} q^{4} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{6} + \zeta_{24}^{7} q^{7} -\zeta_{24}^{9} q^{8} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{9} +O(q^{10})\) \( q -\zeta_{24}^{11} q^{2} + ( \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{3} -\zeta_{24}^{10} q^{4} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{6} + \zeta_{24}^{7} q^{7} -\zeta_{24}^{9} q^{8} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} + \zeta_{24}^{10} ) q^{9} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{12} + \zeta_{24}^{6} q^{14} -\zeta_{24}^{8} q^{16} + ( -\zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{9} ) q^{18} + ( -1 - \zeta_{24}^{4} ) q^{21} -\zeta_{24}^{5} q^{23} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{24} + ( -\zeta_{24}^{7} - \zeta_{24}^{11} ) q^{27} + \zeta_{24}^{5} q^{28} + \zeta_{24}^{6} q^{29} -\zeta_{24}^{7} q^{32} + ( -1 - \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{36} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{41} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{42} -\zeta_{24}^{9} q^{43} -\zeta_{24}^{4} q^{46} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{48} -\zeta_{24}^{2} q^{49} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{54} + \zeta_{24}^{4} q^{56} + \zeta_{24}^{5} q^{58} + ( 1 + \zeta_{24}^{4} ) q^{61} + ( \zeta_{24} - \zeta_{24}^{5} - \zeta_{24}^{9} ) q^{63} -\zeta_{24}^{6} q^{64} + \zeta_{24}^{7} q^{67} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{69} + ( -\zeta_{24}^{3} + \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{72} + \zeta_{24}^{4} q^{81} + ( -\zeta_{24}^{3} - \zeta_{24}^{7} ) q^{82} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{83} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{84} -\zeta_{24}^{8} q^{86} + ( -\zeta_{24}^{3} + \zeta_{24}^{11} ) q^{87} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{89} -\zeta_{24}^{3} q^{92} + ( 1 + \zeta_{24}^{4} ) q^{96} -\zeta_{24} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 4q^{16} - 12q^{21} - 16q^{36} - 4q^{46} + 4q^{56} + 12q^{61} + 4q^{81} + 4q^{86} + 12q^{96} + O(q^{100}) \)

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.

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 607.2
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

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( 81 - 9 T^{4} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 1 - T^{4} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( 1 - T^{4} + T^{8} \)
$29$ \( ( 1 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( 3 + T^{2} )^{4} \)
$43$ \( ( 1 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( ( 3 - 3 T + T^{2} )^{4} \)
$67$ \( 1 - T^{4} + T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( ( 9 + T^{4} )^{2} \)
$89$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$97$ \( T^{8} \)
show more
show less