Properties

Label 1400.1.cf.a
Level $1400$
Weight $1$
Character orbit 1400.cf
Analytic conductor $0.699$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -40
Inner twists $16$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1400.cf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
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 6.0.26891200.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -\zeta_{24}^{11} q^{2} -\zeta_{24}^{10} q^{4} -\zeta_{24}^{3} q^{7} -\zeta_{24}^{9} q^{8} -\zeta_{24}^{2} q^{9} +O(q^{10})\) \( q -\zeta_{24}^{11} q^{2} -\zeta_{24}^{10} q^{4} -\zeta_{24}^{3} q^{7} -\zeta_{24}^{9} q^{8} -\zeta_{24}^{2} q^{9} -\zeta_{24}^{4} q^{11} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{13} -\zeta_{24}^{2} q^{14} -\zeta_{24}^{8} q^{16} -\zeta_{24} q^{18} + ( -\zeta_{24}^{6} - \zeta_{24}^{10} ) q^{19} -\zeta_{24}^{3} q^{22} + \zeta_{24}^{5} q^{23} + ( 1 + \zeta_{24}^{4} ) q^{26} -\zeta_{24} q^{28} -\zeta_{24}^{7} q^{32} - q^{36} + \zeta_{24}^{11} q^{37} + ( -\zeta_{24}^{5} - \zeta_{24}^{9} ) q^{38} + ( \zeta_{24}^{4} + \zeta_{24}^{8} ) q^{41} -\zeta_{24}^{2} q^{44} + \zeta_{24}^{4} q^{46} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{47} + \zeta_{24}^{6} q^{49} + ( \zeta_{24}^{3} - \zeta_{24}^{11} ) q^{52} -\zeta_{24} q^{53} - q^{56} + \zeta_{24}^{5} q^{63} -\zeta_{24}^{6} q^{64} + \zeta_{24}^{11} q^{72} + \zeta_{24}^{10} q^{74} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{76} + \zeta_{24}^{7} q^{77} + \zeta_{24}^{4} q^{81} + ( \zeta_{24}^{3} + \zeta_{24}^{7} ) q^{82} -\zeta_{24} q^{88} + ( -\zeta_{24}^{4} - \zeta_{24}^{8} ) q^{91} + \zeta_{24}^{3} q^{92} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{94} + \zeta_{24}^{5} q^{98} + \zeta_{24}^{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 4q^{11} + 4q^{16} + 12q^{26} - 8q^{36} + 4q^{46} - 8q^{56} + 4q^{81} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{24}^{8}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
243.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i −0.866025 0.500000i 0
243.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i −0.866025 0.500000i 0
507.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i −0.866025 + 0.500000i 0
507.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i −0.866025 + 0.500000i 0
843.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 −0.707107 0.707107i 0.707107 0.707107i 0.866025 0.500000i 0
843.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0.707107 + 0.707107i −0.707107 + 0.707107i 0.866025 0.500000i 0
1307.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.866025 + 0.500000i 0
1307.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0.707107 0.707107i −0.707107 0.707107i 0.866025 + 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1307.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
8.d odd 2 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner
40.k even 4 2 inner
56.m even 6 1 inner
280.ba even 6 1 inner
280.bp odd 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.1.cf.a 8
5.b even 2 1 inner 1400.1.cf.a 8
5.c odd 4 2 inner 1400.1.cf.a 8
7.d odd 6 1 inner 1400.1.cf.a 8
8.d odd 2 1 inner 1400.1.cf.a 8
35.i odd 6 1 inner 1400.1.cf.a 8
35.k even 12 2 inner 1400.1.cf.a 8
40.e odd 2 1 CM 1400.1.cf.a 8
40.k even 4 2 inner 1400.1.cf.a 8
56.m even 6 1 inner 1400.1.cf.a 8
280.ba even 6 1 inner 1400.1.cf.a 8
280.bp odd 12 2 inner 1400.1.cf.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.1.cf.a 8 1.a even 1 1 trivial
1400.1.cf.a 8 5.b even 2 1 inner
1400.1.cf.a 8 5.c odd 4 2 inner
1400.1.cf.a 8 7.d odd 6 1 inner
1400.1.cf.a 8 8.d odd 2 1 inner
1400.1.cf.a 8 35.i odd 6 1 inner
1400.1.cf.a 8 35.k even 12 2 inner
1400.1.cf.a 8 40.e odd 2 1 CM
1400.1.cf.a 8 40.k even 4 2 inner
1400.1.cf.a 8 56.m even 6 1 inner
1400.1.cf.a 8 280.ba even 6 1 inner
1400.1.cf.a 8 280.bp odd 12 2 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1400, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( 1 + T^{4} )^{2} \)
$11$ \( ( 1 + T + T^{2} )^{4} \)
$13$ \( ( 9 + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 9 + 3 T^{2} + T^{4} )^{2} \)
$23$ \( 1 - T^{4} + T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( 1 - T^{4} + T^{8} \)
$41$ \( ( 3 + T^{2} )^{4} \)
$43$ \( T^{8} \)
$47$ \( 81 - 9 T^{4} + T^{8} \)
$53$ \( 1 - T^{4} + T^{8} \)
$59$ \( T^{8} \)
$61$ \( T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( T^{8} \)
show more
show less