Properties

Label 1400.1.y.b
Level $1400$
Weight $1$
Character orbit 1400.y
Analytic conductor $0.699$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -7
Inner twists $16$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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 6.0.78400000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{3} q^{7} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{6} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{7} q^{2} -\zeta_{24}^{2} q^{4} + \zeta_{24}^{3} q^{7} + \zeta_{24}^{9} q^{8} + \zeta_{24}^{6} q^{9} + q^{11} -\zeta_{24}^{10} q^{14} + \zeta_{24}^{4} q^{16} + \zeta_{24} q^{18} -\zeta_{24}^{7} q^{22} + \zeta_{24}^{9} q^{23} -\zeta_{24}^{5} q^{28} + ( \zeta_{24}^{2} - \zeta_{24}^{10} ) q^{29} -\zeta_{24}^{11} q^{32} -\zeta_{24}^{8} q^{36} -\zeta_{24}^{3} q^{37} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{43} -\zeta_{24}^{2} q^{44} + \zeta_{24}^{4} q^{46} + \zeta_{24}^{6} q^{49} -2 \zeta_{24}^{9} q^{53} - q^{56} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{58} + \zeta_{24}^{9} q^{63} -\zeta_{24}^{6} q^{64} + ( \zeta_{24}^{7} + \zeta_{24}^{11} ) q^{67} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{71} -\zeta_{24}^{3} q^{72} + \zeta_{24}^{10} q^{74} + \zeta_{24}^{3} q^{77} + ( -\zeta_{24}^{2} + \zeta_{24}^{10} ) q^{79} - q^{81} + ( -1 + \zeta_{24}^{8} ) q^{86} + \zeta_{24}^{9} q^{88} -\zeta_{24}^{11} q^{92} + \zeta_{24} q^{98} + \zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{11} + 4q^{16} + 4q^{36} + 4q^{46} - 8q^{56} - 8q^{81} - 12q^{86} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
307.2 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
307.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i 1.00000i 0
307.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 1.00000i 0
643.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
643.2 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
643.3 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 1.00000i 0
643.4 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 643.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner
40.e odd 2 1 inner
40.k even 4 2 inner
56.e even 2 1 inner
280.n even 2 1 inner
280.y odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.y.b 8
5.b even 2 1 inner 1400.1.y.b 8
5.c odd 4 2 inner 1400.1.y.b 8
7.b odd 2 1 CM 1400.1.y.b 8
8.d odd 2 1 inner 1400.1.y.b 8
35.c odd 2 1 inner 1400.1.y.b 8
35.f even 4 2 inner 1400.1.y.b 8
40.e odd 2 1 inner 1400.1.y.b 8
40.k even 4 2 inner 1400.1.y.b 8
56.e even 2 1 inner 1400.1.y.b 8
280.n even 2 1 inner 1400.1.y.b 8
280.y odd 4 2 inner 1400.1.y.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.y.b 8 1.a even 1 1 trivial
1400.1.y.b 8 5.b even 2 1 inner
1400.1.y.b 8 5.c odd 4 2 inner
1400.1.y.b 8 7.b odd 2 1 CM
1400.1.y.b 8 8.d odd 2 1 inner
1400.1.y.b 8 35.c odd 2 1 inner
1400.1.y.b 8 35.f even 4 2 inner
1400.1.y.b 8 40.e odd 2 1 inner
1400.1.y.b 8 40.k even 4 2 inner
1400.1.y.b 8 56.e even 2 1 inner
1400.1.y.b 8 280.n even 2 1 inner
1400.1.y.b 8 280.y odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + T^{4} )^{2} \)
$11$ \( ( -1 + T )^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( ( 1 + T^{4} )^{2} \)
$29$ \( ( -3 + T^{2} )^{4} \)
$31$ \( T^{8} \)
$37$ \( ( 1 + T^{4} )^{2} \)
$41$ \( T^{8} \)
$43$ \( ( 9 + T^{4} )^{2} \)
$47$ \( T^{8} \)
$53$ \( ( 16 + T^{4} )^{2} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( ( 9 + T^{4} )^{2} \)
$71$ \( ( 3 + T^{2} )^{4} \)
$73$ \( T^{8} \)
$79$ \( ( -3 + T^{2} )^{4} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
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