Properties

Label 1400.1.m.d
Level $1400$
Weight $1$
Character orbit 1400.m
Analytic conductor $0.699$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -7
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.m (of order \(2\), degree \(1\), 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 6.0.109760000.1

$q$-expansion

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

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

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0 −0.500000 0.866025i 0 0 −1.00000 −1.00000 −1.00000 0
1301.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 −1.00000 −1.00000 −1.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
8.b even 2 1 inner
56.h odd 2 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\):

\( T_{3} \)
\( T_{11}^{2} + 3 \)
\( T_{23} + 1 \)
\( T_{113} - 1 \)

Hecke characteristic polynomials

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