Properties

Label 1400.1.bf.b
Level $1400$
Weight $1$
Character orbit 1400.bf
Analytic conductor $0.699$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
RM discriminant 8
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.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{6}\)
Projective field Galois closure of 6.0.5378240000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{7} - q^{8} -\zeta_{6}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{6}^{2} q^{2} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{7} - q^{8} -\zeta_{6}^{2} q^{9} -\zeta_{6} q^{14} + \zeta_{6}^{2} q^{16} + ( -1 + \zeta_{6}^{2} ) q^{17} -\zeta_{6} q^{18} -\zeta_{6}^{2} q^{23} - q^{28} + ( 1 - \zeta_{6}^{2} ) q^{31} + \zeta_{6} q^{32} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{34} - q^{36} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{41} -\zeta_{6} q^{46} + ( -1 - \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + \zeta_{6}^{2} q^{56} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{62} -\zeta_{6} q^{63} + q^{64} + ( 1 + \zeta_{6} ) q^{68} + q^{71} + \zeta_{6}^{2} q^{72} + \zeta_{6}^{2} q^{79} -\zeta_{6} q^{81} + ( 1 + \zeta_{6} ) q^{82} + ( 1 + \zeta_{6} ) q^{89} - q^{92} + ( -1 + \zeta_{6}^{2} ) q^{94} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} + q^{7} - 2q^{8} + q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{4} + q^{7} - 2q^{8} + q^{9} - q^{14} - q^{16} - 3q^{17} - q^{18} + q^{23} - 2q^{28} + 3q^{31} + q^{32} - 2q^{36} - q^{46} - 3q^{47} - q^{49} - q^{56} - q^{63} + 2q^{64} + 3q^{68} + 2q^{71} - q^{72} - q^{79} - q^{81} + 3q^{82} + 3q^{89} - 2q^{92} - 3q^{94} - 2q^{98} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i 0
901.1 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0.500000 0.866025i −1.00000 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
7.d odd 6 1 inner
56.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.bf.b yes 2
5.b even 2 1 1400.1.bf.a 2
5.c odd 4 2 1400.1.bl.a 4
7.d odd 6 1 inner 1400.1.bf.b yes 2
8.b even 2 1 RM 1400.1.bf.b yes 2
35.i odd 6 1 1400.1.bf.a 2
35.k even 12 2 1400.1.bl.a 4
40.f even 2 1 1400.1.bf.a 2
40.i odd 4 2 1400.1.bl.a 4
56.j odd 6 1 inner 1400.1.bf.b yes 2
280.bk odd 6 1 1400.1.bf.a 2
280.bv even 12 2 1400.1.bl.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.bf.a 2 5.b even 2 1
1400.1.bf.a 2 35.i odd 6 1
1400.1.bf.a 2 40.f even 2 1
1400.1.bf.a 2 280.bk odd 6 1
1400.1.bf.b yes 2 1.a even 1 1 trivial
1400.1.bf.b yes 2 7.d odd 6 1 inner
1400.1.bf.b yes 2 8.b even 2 1 RM
1400.1.bf.b yes 2 56.j odd 6 1 inner
1400.1.bl.a 4 5.c odd 4 2
1400.1.bl.a 4 35.k even 12 2
1400.1.bl.a 4 40.i odd 4 2
1400.1.bl.a 4 280.bv even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + 3 T_{17} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

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