Properties

Label 1400.1.c.b
Level $1400$
Weight $1$
Character orbit 1400.c
Analytic conductor $0.699$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -7
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.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i 1.00000 0
349.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 1.00000 0
349.3 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 1.00000 0
349.4 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i 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 \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
8.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
56.h odd 2 1 inner
280.c 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.c.b 4
5.b even 2 1 inner 1400.1.c.b 4
5.c odd 4 1 1400.1.m.c 2
5.c odd 4 1 1400.1.m.d yes 2
7.b odd 2 1 CM 1400.1.c.b 4
8.b even 2 1 inner 1400.1.c.b 4
35.c odd 2 1 inner 1400.1.c.b 4
35.f even 4 1 1400.1.m.c 2
35.f even 4 1 1400.1.m.d yes 2
40.f even 2 1 inner 1400.1.c.b 4
40.i odd 4 1 1400.1.m.c 2
40.i odd 4 1 1400.1.m.d yes 2
56.h odd 2 1 inner 1400.1.c.b 4
280.c odd 2 1 inner 1400.1.c.b 4
280.s even 4 1 1400.1.m.c 2
280.s even 4 1 1400.1.m.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.c.b 4 1.a even 1 1 trivial
1400.1.c.b 4 5.b even 2 1 inner
1400.1.c.b 4 7.b odd 2 1 CM
1400.1.c.b 4 8.b even 2 1 inner
1400.1.c.b 4 35.c odd 2 1 inner
1400.1.c.b 4 40.f even 2 1 inner
1400.1.c.b 4 56.h odd 2 1 inner
1400.1.c.b 4 280.c odd 2 1 inner
1400.1.m.c 2 5.c odd 4 1
1400.1.m.c 2 35.f even 4 1
1400.1.m.c 2 40.i odd 4 1
1400.1.m.c 2 280.s even 4 1
1400.1.m.d yes 2 5.c odd 4 1
1400.1.m.d yes 2 35.f even 4 1
1400.1.m.d yes 2 40.i odd 4 1
1400.1.m.d yes 2 280.s even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

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