Properties

Label 1400.1.bs.c
Level $1400$
Weight $1$
Character orbit 1400.bs
Analytic conductor $0.699$
Analytic rank $0$
Dimension $8$
Projective image $D_{10}$
CM discriminant -56
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1400.bs (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
Defining polynomial: \(x^{8} - x^{6} + x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{10}\)
Projective field Galois closure of 10.0.10504375000000000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{20}^{8} q^{2} + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{3} -\zeta_{20}^{6} q^{4} -\zeta_{20}^{3} q^{5} + ( \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{6} - q^{7} -\zeta_{20}^{4} q^{8} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{9} +O(q^{10})\) \( q -\zeta_{20}^{8} q^{2} + ( \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{3} -\zeta_{20}^{6} q^{4} -\zeta_{20}^{3} q^{5} + ( \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{6} - q^{7} -\zeta_{20}^{4} q^{8} + ( -1 - \zeta_{20}^{2} - \zeta_{20}^{4} ) q^{9} -\zeta_{20} q^{10} + ( \zeta_{20} + \zeta_{20}^{3} ) q^{12} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{13} + \zeta_{20}^{8} q^{14} + ( 1 - \zeta_{20}^{8} ) q^{15} -\zeta_{20}^{2} q^{16} + ( -1 - \zeta_{20}^{2} + \zeta_{20}^{8} ) q^{18} + \zeta_{20}^{9} q^{20} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{21} + ( -\zeta_{20}^{2} + \zeta_{20}^{4} ) q^{23} + ( \zeta_{20} - \zeta_{20}^{9} ) q^{24} + \zeta_{20}^{6} q^{25} + ( -\zeta_{20} + \zeta_{20}^{9} ) q^{26} + ( \zeta_{20} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{27} + \zeta_{20}^{6} q^{28} + ( -\zeta_{20}^{6} - \zeta_{20}^{8} ) q^{30} - q^{32} + \zeta_{20}^{3} q^{35} + ( -1 + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{36} + ( 1 - \zeta_{20}^{6} - 2 \zeta_{20}^{8} ) q^{39} + \zeta_{20}^{7} q^{40} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} ) q^{42} + ( \zeta_{20}^{3} + \zeta_{20}^{5} + \zeta_{20}^{7} ) q^{45} + ( -1 + \zeta_{20}^{2} ) q^{46} + ( -\zeta_{20}^{7} - \zeta_{20}^{9} ) q^{48} + q^{49} + \zeta_{20}^{4} q^{50} + ( \zeta_{20}^{7} + \zeta_{20}^{9} ) q^{52} + ( -\zeta_{20}^{3} - \zeta_{20}^{5} - \zeta_{20}^{7} - \zeta_{20}^{9} ) q^{54} + \zeta_{20}^{4} q^{56} + ( -\zeta_{20}^{5} + \zeta_{20}^{9} ) q^{59} + ( -\zeta_{20}^{4} - \zeta_{20}^{6} ) q^{60} + ( \zeta_{20} - \zeta_{20}^{5} ) q^{61} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} ) q^{63} + \zeta_{20}^{8} q^{64} + ( \zeta_{20}^{4} + \zeta_{20}^{6} ) q^{65} + ( -\zeta_{20} - \zeta_{20}^{7} ) q^{69} + \zeta_{20} q^{70} + ( -\zeta_{20}^{4} - \zeta_{20}^{8} ) q^{71} + ( \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{72} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{75} + ( -\zeta_{20}^{4} - 2 \zeta_{20}^{6} - \zeta_{20}^{8} ) q^{78} + ( -1 + \zeta_{20}^{2} ) q^{79} + \zeta_{20}^{5} q^{80} + ( 1 + \zeta_{20}^{2} + \zeta_{20}^{4} + \zeta_{20}^{6} + \zeta_{20}^{8} ) q^{81} + ( \zeta_{20} + \zeta_{20}^{7} ) q^{83} + ( -\zeta_{20} - \zeta_{20}^{3} ) q^{84} + ( \zeta_{20} + \zeta_{20}^{3} + \zeta_{20}^{5} ) q^{90} + ( \zeta_{20} + \zeta_{20}^{3} ) q^{91} + ( 1 + \zeta_{20}^{8} ) q^{92} + ( -\zeta_{20}^{5} - \zeta_{20}^{7} ) q^{96} -\zeta_{20}^{8} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} + O(q^{10}) \) \( 8q + 2q^{2} - 2q^{4} - 8q^{7} + 2q^{8} - 8q^{9} - 2q^{14} + 10q^{15} - 2q^{16} - 12q^{18} - 4q^{23} + 2q^{25} + 2q^{28} - 8q^{32} - 8q^{36} + 10q^{39} - 6q^{46} + 8q^{49} - 2q^{50} - 2q^{56} + 8q^{63} - 2q^{64} + 4q^{71} - 2q^{72} - 6q^{79} + 8q^{81} + 6q^{92} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{20}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.587785 0.809017i
0.587785 + 0.809017i
0.951057 + 0.309017i
−0.951057 0.309017i
0.951057 0.309017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.309017 0.951057i −0.951057 + 0.690983i −0.809017 + 0.587785i −0.951057 + 0.309017i 0.951057 + 0.690983i −1.00000 0.809017 + 0.587785i 0.118034 0.363271i 0.587785 + 0.809017i
181.2 −0.309017 0.951057i 0.951057 0.690983i −0.809017 + 0.587785i 0.951057 0.309017i −0.951057 0.690983i −1.00000 0.809017 + 0.587785i 0.118034 0.363271i −0.587785 0.809017i
461.1 0.809017 0.587785i −0.587785 + 1.80902i 0.309017 0.951057i −0.587785 0.809017i 0.587785 + 1.80902i −1.00000 −0.309017 0.951057i −2.11803 1.53884i −0.951057 0.309017i
461.2 0.809017 0.587785i 0.587785 1.80902i 0.309017 0.951057i 0.587785 + 0.809017i −0.587785 1.80902i −1.00000 −0.309017 0.951057i −2.11803 1.53884i 0.951057 + 0.309017i
741.1 0.809017 + 0.587785i −0.587785 1.80902i 0.309017 + 0.951057i −0.587785 + 0.809017i 0.587785 1.80902i −1.00000 −0.309017 + 0.951057i −2.11803 + 1.53884i −0.951057 + 0.309017i
741.2 0.809017 + 0.587785i 0.587785 + 1.80902i 0.309017 + 0.951057i 0.587785 0.809017i −0.587785 + 1.80902i −1.00000 −0.309017 + 0.951057i −2.11803 + 1.53884i 0.951057 0.309017i
1021.1 −0.309017 + 0.951057i −0.951057 0.690983i −0.809017 0.587785i −0.951057 0.309017i 0.951057 0.690983i −1.00000 0.809017 0.587785i 0.118034 + 0.363271i 0.587785 0.809017i
1021.2 −0.309017 + 0.951057i 0.951057 + 0.690983i −0.809017 0.587785i 0.951057 + 0.309017i −0.951057 + 0.690983i −1.00000 0.809017 0.587785i 0.118034 + 0.363271i −0.587785 + 0.809017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1021.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
25.d even 5 1 inner
175.l odd 10 1 inner
200.t even 10 1 inner
1400.bs odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.bs.c 8
7.b odd 2 1 inner 1400.1.bs.c 8
8.b even 2 1 inner 1400.1.bs.c 8
25.d even 5 1 inner 1400.1.bs.c 8
56.h odd 2 1 CM 1400.1.bs.c 8
175.l odd 10 1 inner 1400.1.bs.c 8
200.t even 10 1 inner 1400.1.bs.c 8
1400.bs odd 10 1 inner 1400.1.bs.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.bs.c 8 1.a even 1 1 trivial
1400.1.bs.c 8 7.b odd 2 1 inner
1400.1.bs.c 8 8.b even 2 1 inner
1400.1.bs.c 8 25.d even 5 1 inner
1400.1.bs.c 8 56.h odd 2 1 CM
1400.1.bs.c 8 175.l odd 10 1 inner
1400.1.bs.c 8 200.t even 10 1 inner
1400.1.bs.c 8 1400.bs odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 5 T_{3}^{6} + 10 T_{3}^{4} + 25 \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \)
$3$ \( 25 + 10 T^{4} + 5 T^{6} + T^{8} \)
$5$ \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( T^{8} \)
$13$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 1 + 3 T + 4 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( 25 + 10 T^{4} + 5 T^{6} + T^{8} \)
$61$ \( 25 + 10 T^{4} + 5 T^{6} + T^{8} \)
$67$ \( T^{8} \)
$71$ \( ( 1 - 3 T + 4 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$73$ \( T^{8} \)
$79$ \( ( 1 + 2 T + 4 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$83$ \( 25 + 25 T^{2} + 10 T^{4} + T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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