Properties

Label 1344.1.bn.a
Level $1344$
Weight $1$
Character orbit 1344.bn
Analytic conductor $0.671$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1344.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.588.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{3} + \zeta_{6} q^{7} -\zeta_{6} q^{9} + q^{13} + \zeta_{6} q^{19} - q^{21} + \zeta_{6}^{2} q^{25} + q^{27} + \zeta_{6}^{2} q^{31} -\zeta_{6} q^{37} + \zeta_{6}^{2} q^{39} - q^{43} + \zeta_{6}^{2} q^{49} - q^{57} + 2 \zeta_{6} q^{61} -\zeta_{6}^{2} q^{63} -\zeta_{6}^{2} q^{67} -\zeta_{6}^{2} q^{73} -\zeta_{6} q^{75} -\zeta_{6} q^{79} + \zeta_{6}^{2} q^{81} + \zeta_{6} q^{91} -\zeta_{6} q^{93} + 2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{3} + q^{7} - q^{9} + O(q^{10}) \) \( 2q - q^{3} + q^{7} - q^{9} + 2q^{13} + q^{19} - 2q^{21} - q^{25} + 2q^{27} - q^{31} - q^{37} - q^{39} - 2q^{43} - q^{49} - 2q^{57} + 2q^{61} + q^{63} + q^{67} + q^{73} - q^{75} - q^{79} - q^{81} + q^{91} - q^{93} + 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 0.866025i 0 0 0 0.500000 0.866025i 0 −0.500000 + 0.866025i 0
641.1 0 −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 0 −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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.1.bn.a 2
3.b odd 2 1 CM 1344.1.bn.a 2
4.b odd 2 1 1344.1.bn.b 2
7.c even 3 1 inner 1344.1.bn.a 2
8.b even 2 1 336.1.bn.a 2
8.d odd 2 1 84.1.p.a 2
12.b even 2 1 1344.1.bn.b 2
21.h odd 6 1 inner 1344.1.bn.a 2
24.f even 2 1 84.1.p.a 2
24.h odd 2 1 336.1.bn.a 2
28.g odd 6 1 1344.1.bn.b 2
40.e odd 2 1 2100.1.bn.c 2
40.k even 4 2 2100.1.bh.a 4
56.e even 2 1 588.1.p.a 2
56.h odd 2 1 2352.1.bn.a 2
56.j odd 6 1 2352.1.d.b 1
56.j odd 6 1 2352.1.bn.a 2
56.k odd 6 1 84.1.p.a 2
56.k odd 6 1 588.1.c.b 1
56.m even 6 1 588.1.c.a 1
56.m even 6 1 588.1.p.a 2
56.p even 6 1 336.1.bn.a 2
56.p even 6 1 2352.1.d.a 1
72.l even 6 1 2268.1.m.a 2
72.l even 6 1 2268.1.bh.b 2
72.p odd 6 1 2268.1.m.a 2
72.p odd 6 1 2268.1.bh.b 2
84.n even 6 1 1344.1.bn.b 2
120.m even 2 1 2100.1.bn.c 2
120.q odd 4 2 2100.1.bh.a 4
168.e odd 2 1 588.1.p.a 2
168.i even 2 1 2352.1.bn.a 2
168.s odd 6 1 336.1.bn.a 2
168.s odd 6 1 2352.1.d.a 1
168.v even 6 1 84.1.p.a 2
168.v even 6 1 588.1.c.b 1
168.ba even 6 1 2352.1.d.b 1
168.ba even 6 1 2352.1.bn.a 2
168.be odd 6 1 588.1.c.a 1
168.be odd 6 1 588.1.p.a 2
280.bi odd 6 1 2100.1.bn.c 2
280.br even 12 2 2100.1.bh.a 4
504.ba odd 6 1 2268.1.bh.b 2
504.bt even 6 1 2268.1.m.a 2
504.ce odd 6 1 2268.1.m.a 2
504.cy even 6 1 2268.1.bh.b 2
840.cv even 6 1 2100.1.bn.c 2
840.dp odd 12 2 2100.1.bh.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.1.p.a 2 8.d odd 2 1
84.1.p.a 2 24.f even 2 1
84.1.p.a 2 56.k odd 6 1
84.1.p.a 2 168.v even 6 1
336.1.bn.a 2 8.b even 2 1
336.1.bn.a 2 24.h odd 2 1
336.1.bn.a 2 56.p even 6 1
336.1.bn.a 2 168.s odd 6 1
588.1.c.a 1 56.m even 6 1
588.1.c.a 1 168.be odd 6 1
588.1.c.b 1 56.k odd 6 1
588.1.c.b 1 168.v even 6 1
588.1.p.a 2 56.e even 2 1
588.1.p.a 2 56.m even 6 1
588.1.p.a 2 168.e odd 2 1
588.1.p.a 2 168.be odd 6 1
1344.1.bn.a 2 1.a even 1 1 trivial
1344.1.bn.a 2 3.b odd 2 1 CM
1344.1.bn.a 2 7.c even 3 1 inner
1344.1.bn.a 2 21.h odd 6 1 inner
1344.1.bn.b 2 4.b odd 2 1
1344.1.bn.b 2 12.b even 2 1
1344.1.bn.b 2 28.g odd 6 1
1344.1.bn.b 2 84.n even 6 1
2100.1.bh.a 4 40.k even 4 2
2100.1.bh.a 4 120.q odd 4 2
2100.1.bh.a 4 280.br even 12 2
2100.1.bh.a 4 840.dp odd 12 2
2100.1.bn.c 2 40.e odd 2 1
2100.1.bn.c 2 120.m even 2 1
2100.1.bn.c 2 280.bi odd 6 1
2100.1.bn.c 2 840.cv even 6 1
2268.1.m.a 2 72.l even 6 1
2268.1.m.a 2 72.p odd 6 1
2268.1.m.a 2 504.bt even 6 1
2268.1.m.a 2 504.ce odd 6 1
2268.1.bh.b 2 72.l even 6 1
2268.1.bh.b 2 72.p odd 6 1
2268.1.bh.b 2 504.ba odd 6 1
2268.1.bh.b 2 504.cy even 6 1
2352.1.d.a 1 56.p even 6 1
2352.1.d.a 1 168.s odd 6 1
2352.1.d.b 1 56.j odd 6 1
2352.1.d.b 1 168.ba even 6 1
2352.1.bn.a 2 56.h odd 2 1
2352.1.bn.a 2 56.j odd 6 1
2352.1.bn.a 2 168.i even 2 1
2352.1.bn.a 2 168.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{19}^{2} - T_{19} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1344, [\chi])\).

Hecke characteristic polynomials

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