Properties

Label 1400.1.bs.a
Level $1400$
Weight $1$
Character orbit 1400.bs
Analytic conductor $0.699$
Analytic rank $0$
Dimension $4$
Projective image $D_{5}$
CM discriminant -56
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \(x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{5}\)
Projective field Galois closure of 5.1.1225000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{10}^{4} q^{2} + ( -1 + \zeta_{10} ) q^{3} -\zeta_{10}^{3} q^{4} -\zeta_{10}^{4} q^{5} + ( -1 - \zeta_{10}^{4} ) q^{6} + q^{7} + \zeta_{10}^{2} q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{10}^{4} q^{2} + ( -1 + \zeta_{10} ) q^{3} -\zeta_{10}^{3} q^{4} -\zeta_{10}^{4} q^{5} + ( -1 - \zeta_{10}^{4} ) q^{6} + q^{7} + \zeta_{10}^{2} q^{8} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{9} + \zeta_{10}^{3} q^{10} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{12} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{13} + \zeta_{10}^{4} q^{14} + ( 1 + \zeta_{10}^{4} ) q^{15} -\zeta_{10} q^{16} + ( 1 - \zeta_{10} + \zeta_{10}^{4} ) q^{18} -2 \zeta_{10}^{2} q^{19} -\zeta_{10}^{2} q^{20} + ( -1 + \zeta_{10} ) q^{21} + ( -\zeta_{10} + \zeta_{10}^{2} ) q^{23} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} -\zeta_{10}^{3} q^{25} + ( -\zeta_{10}^{2} + \zeta_{10}^{3} ) q^{26} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{27} -\zeta_{10}^{3} q^{28} + ( -\zeta_{10}^{3} + \zeta_{10}^{4} ) q^{30} + q^{32} -\zeta_{10}^{4} q^{35} + ( 1 - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{36} + 2 \zeta_{10} q^{38} + ( 1 - \zeta_{10}^{3} + 2 \zeta_{10}^{4} ) q^{39} + \zeta_{10} q^{40} + ( -1 - \zeta_{10}^{4} ) q^{42} + ( -1 + \zeta_{10} - \zeta_{10}^{4} ) q^{45} + ( 1 - \zeta_{10} ) q^{46} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{48} + q^{49} + \zeta_{10}^{2} q^{50} + ( \zeta_{10} - \zeta_{10}^{2} ) q^{52} + ( -1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{4} ) q^{54} + \zeta_{10}^{2} q^{56} + ( 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{57} + ( -1 - \zeta_{10}^{2} ) q^{59} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{60} + ( -1 + \zeta_{10}^{3} ) q^{61} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{63} + \zeta_{10}^{4} q^{64} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{65} + ( \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{69} + \zeta_{10}^{3} q^{70} + ( \zeta_{10}^{2} + \zeta_{10}^{4} ) q^{71} + ( \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{72} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{75} -2 q^{76} + ( \zeta_{10}^{2} - 2 \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{78} + ( 1 - \zeta_{10} ) q^{79} - q^{80} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} + \zeta_{10}^{4} ) q^{81} + ( \zeta_{10} + \zeta_{10}^{3} ) q^{83} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{84} + ( -1 + \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{90} + ( \zeta_{10}^{3} - \zeta_{10}^{4} ) q^{91} + ( 1 + \zeta_{10}^{4} ) q^{92} -2 \zeta_{10} q^{95} + ( -1 + \zeta_{10} ) q^{96} + \zeta_{10}^{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - q^{2} - 3q^{3} - q^{4} + q^{5} - 3q^{6} + 4q^{7} - q^{8} + 2q^{9} + O(q^{10}) \) \( 4q - q^{2} - 3q^{3} - q^{4} + q^{5} - 3q^{6} + 4q^{7} - q^{8} + 2q^{9} + q^{10} + 2q^{12} + 2q^{13} - q^{14} + 3q^{15} - q^{16} + 2q^{18} + 2q^{19} + q^{20} - 3q^{21} - 2q^{23} + 2q^{24} - q^{25} + 2q^{26} - q^{27} - q^{28} - 2q^{30} + 4q^{32} + q^{35} + 2q^{36} + 2q^{38} + q^{39} + q^{40} - 3q^{42} - 2q^{45} + 3q^{46} + 2q^{48} + 4q^{49} - q^{50} + 2q^{52} - q^{54} - q^{56} - 4q^{57} - 3q^{59} - 2q^{60} - 3q^{61} + 2q^{63} - q^{64} - 2q^{65} + 4q^{69} + q^{70} - 2q^{71} - 3q^{72} + 2q^{75} - 8q^{76} - 4q^{78} + 3q^{79} - 4q^{80} + 2q^{83} + 2q^{84} - 2q^{90} + 2q^{91} + 3q^{92} - 2q^{95} - 3q^{96} - q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{10}^{3}\)

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
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0.309017 + 0.951057i −1.30902 + 0.951057i −0.809017 + 0.587785i −0.309017 0.951057i −1.30902 0.951057i 1.00000 −0.809017 0.587785i 0.500000 1.53884i 0.809017 0.587785i
461.1 −0.809017 + 0.587785i −0.190983 + 0.587785i 0.309017 0.951057i 0.809017 0.587785i −0.190983 0.587785i 1.00000 0.309017 + 0.951057i 0.500000 + 0.363271i −0.309017 + 0.951057i
741.1 −0.809017 0.587785i −0.190983 0.587785i 0.309017 + 0.951057i 0.809017 + 0.587785i −0.190983 + 0.587785i 1.00000 0.309017 0.951057i 0.500000 0.363271i −0.309017 0.951057i
1021.1 0.309017 0.951057i −1.30902 0.951057i −0.809017 0.587785i −0.309017 + 0.951057i −1.30902 + 0.951057i 1.00000 −0.809017 + 0.587785i 0.500000 + 1.53884i 0.809017 + 0.587785i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
25.d even 5 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.a 4
7.b odd 2 1 1400.1.bs.b yes 4
8.b even 2 1 1400.1.bs.b yes 4
25.d even 5 1 inner 1400.1.bs.a 4
56.h odd 2 1 CM 1400.1.bs.a 4
175.l odd 10 1 1400.1.bs.b yes 4
200.t even 10 1 1400.1.bs.b yes 4
1400.bs odd 10 1 inner 1400.1.bs.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.bs.a 4 1.a even 1 1 trivial
1400.1.bs.a 4 25.d even 5 1 inner
1400.1.bs.a 4 56.h odd 2 1 CM
1400.1.bs.a 4 1400.bs odd 10 1 inner
1400.1.bs.b yes 4 7.b odd 2 1
1400.1.bs.b yes 4 8.b even 2 1
1400.1.bs.b yes 4 175.l odd 10 1
1400.1.bs.b yes 4 200.t even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 3 T_{3}^{3} + 4 T_{3}^{2} + 2 T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

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