Properties

Label 1400.1.ba.a
Level $1400$
Weight $1$
Character orbit 1400.ba
Analytic conductor $0.699$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -40
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.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: no (minimal twist has level 280)
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.1960.1
Artin image $C_{12}\times S_3$
Artin field Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} -\zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{12}^{5} q^{2} -\zeta_{12}^{4} q^{4} -\zeta_{12}^{3} q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} -\zeta_{12}^{4} q^{11} -\zeta_{12}^{3} q^{13} + \zeta_{12}^{2} q^{14} -\zeta_{12}^{2} q^{16} -\zeta_{12} q^{18} -\zeta_{12}^{2} q^{19} + \zeta_{12}^{3} q^{22} + \zeta_{12}^{5} q^{23} + \zeta_{12}^{2} q^{26} -\zeta_{12} q^{28} + \zeta_{12} q^{32} + q^{36} -\zeta_{12}^{5} q^{37} + \zeta_{12} q^{38} - q^{41} -\zeta_{12}^{2} q^{44} -\zeta_{12}^{4} q^{46} -\zeta_{12}^{5} q^{47} - q^{49} -\zeta_{12} q^{52} + \zeta_{12} q^{53} + q^{56} -2 \zeta_{12}^{4} q^{59} -\zeta_{12}^{5} q^{63} - q^{64} + \zeta_{12}^{5} q^{72} + \zeta_{12}^{4} q^{74} - q^{76} -\zeta_{12} q^{77} + \zeta_{12}^{4} q^{81} -\zeta_{12}^{5} q^{82} + \zeta_{12} q^{88} + 2 \zeta_{12}^{2} q^{89} - q^{91} + \zeta_{12}^{3} q^{92} + \zeta_{12}^{4} q^{94} -\zeta_{12}^{5} q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 2q^{9} + 2q^{11} + 2q^{14} - 2q^{16} - 2q^{19} + 2q^{26} + 4q^{36} - 4q^{41} - 2q^{44} + 2q^{46} - 4q^{49} + 4q^{56} + 4q^{59} - 4q^{64} - 2q^{74} - 4q^{76} - 2q^{81} + 4q^{89} - 4q^{91} - 2q^{94} + 4q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
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 0.500000 0.866025i 0
51.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0 0 1.00000i 1.00000i 0.500000 0.866025i 0
851.1 −0.866025 + 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 0.500000 + 0.866025i 0
851.2 0.866025 0.500000i 0 0.500000 0.866025i 0 0 1.00000i 1.00000i 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
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
5.b even 2 1 inner
7.c even 3 1 inner
8.d odd 2 1 inner
35.j even 6 1 inner
56.k odd 6 1 inner
280.bi 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.ba.a 4
5.b even 2 1 inner 1400.1.ba.a 4
5.c odd 4 1 280.1.bi.a 2
5.c odd 4 1 280.1.bi.b yes 2
7.c even 3 1 inner 1400.1.ba.a 4
8.d odd 2 1 inner 1400.1.ba.a 4
15.e even 4 1 2520.1.ef.a 2
15.e even 4 1 2520.1.ef.b 2
20.e even 4 1 1120.1.by.a 2
20.e even 4 1 1120.1.by.b 2
35.f even 4 1 1960.1.bi.a 2
35.f even 4 1 1960.1.bi.b 2
35.j even 6 1 inner 1400.1.ba.a 4
35.k even 12 1 1960.1.i.b 1
35.k even 12 1 1960.1.i.c 1
35.k even 12 1 1960.1.bi.a 2
35.k even 12 1 1960.1.bi.b 2
35.l odd 12 1 280.1.bi.a 2
35.l odd 12 1 280.1.bi.b yes 2
35.l odd 12 1 1960.1.i.a 1
35.l odd 12 1 1960.1.i.d 1
40.e odd 2 1 CM 1400.1.ba.a 4
40.i odd 4 1 1120.1.by.a 2
40.i odd 4 1 1120.1.by.b 2
40.k even 4 1 280.1.bi.a 2
40.k even 4 1 280.1.bi.b yes 2
56.k odd 6 1 inner 1400.1.ba.a 4
105.x even 12 1 2520.1.ef.a 2
105.x even 12 1 2520.1.ef.b 2
120.q odd 4 1 2520.1.ef.a 2
120.q odd 4 1 2520.1.ef.b 2
140.w even 12 1 1120.1.by.a 2
140.w even 12 1 1120.1.by.b 2
280.y odd 4 1 1960.1.bi.a 2
280.y odd 4 1 1960.1.bi.b 2
280.bi odd 6 1 inner 1400.1.ba.a 4
280.bp odd 12 1 1960.1.i.b 1
280.bp odd 12 1 1960.1.i.c 1
280.bp odd 12 1 1960.1.bi.a 2
280.bp odd 12 1 1960.1.bi.b 2
280.br even 12 1 280.1.bi.a 2
280.br even 12 1 280.1.bi.b yes 2
280.br even 12 1 1960.1.i.a 1
280.br even 12 1 1960.1.i.d 1
280.bt odd 12 1 1120.1.by.a 2
280.bt odd 12 1 1120.1.by.b 2
840.dp odd 12 1 2520.1.ef.a 2
840.dp odd 12 1 2520.1.ef.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.1.bi.a 2 5.c odd 4 1
280.1.bi.a 2 35.l odd 12 1
280.1.bi.a 2 40.k even 4 1
280.1.bi.a 2 280.br even 12 1
280.1.bi.b yes 2 5.c odd 4 1
280.1.bi.b yes 2 35.l odd 12 1
280.1.bi.b yes 2 40.k even 4 1
280.1.bi.b yes 2 280.br even 12 1
1120.1.by.a 2 20.e even 4 1
1120.1.by.a 2 40.i odd 4 1
1120.1.by.a 2 140.w even 12 1
1120.1.by.a 2 280.bt odd 12 1
1120.1.by.b 2 20.e even 4 1
1120.1.by.b 2 40.i odd 4 1
1120.1.by.b 2 140.w even 12 1
1120.1.by.b 2 280.bt odd 12 1
1400.1.ba.a 4 1.a even 1 1 trivial
1400.1.ba.a 4 5.b even 2 1 inner
1400.1.ba.a 4 7.c even 3 1 inner
1400.1.ba.a 4 8.d odd 2 1 inner
1400.1.ba.a 4 35.j even 6 1 inner
1400.1.ba.a 4 40.e odd 2 1 CM
1400.1.ba.a 4 56.k odd 6 1 inner
1400.1.ba.a 4 280.bi odd 6 1 inner
1960.1.i.a 1 35.l odd 12 1
1960.1.i.a 1 280.br even 12 1
1960.1.i.b 1 35.k even 12 1
1960.1.i.b 1 280.bp odd 12 1
1960.1.i.c 1 35.k even 12 1
1960.1.i.c 1 280.bp odd 12 1
1960.1.i.d 1 35.l odd 12 1
1960.1.i.d 1 280.br even 12 1
1960.1.bi.a 2 35.f even 4 1
1960.1.bi.a 2 35.k even 12 1
1960.1.bi.a 2 280.y odd 4 1
1960.1.bi.a 2 280.bp odd 12 1
1960.1.bi.b 2 35.f even 4 1
1960.1.bi.b 2 35.k even 12 1
1960.1.bi.b 2 280.y odd 4 1
1960.1.bi.b 2 280.bp odd 12 1
2520.1.ef.a 2 15.e even 4 1
2520.1.ef.a 2 105.x even 12 1
2520.1.ef.a 2 120.q odd 4 1
2520.1.ef.a 2 840.dp odd 12 1
2520.1.ef.b 2 15.e even 4 1
2520.1.ef.b 2 105.x even 12 1
2520.1.ef.b 2 120.q odd 4 1
2520.1.ef.b 2 840.dp odd 12 1

Hecke kernels

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

Hecke characteristic polynomials

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