Properties

Label 1856.1.bh.a
Level $1856$
Weight $1$
Character orbit 1856.bh
Analytic conductor $0.926$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1856.bh (of order \(14\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.926264663447\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
Defining polynomial: \(x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{14}^{4} - \zeta_{14}^{6} ) q^{5} -\zeta_{14} q^{9} +O(q^{10})\) \( q + ( -\zeta_{14}^{4} - \zeta_{14}^{6} ) q^{5} -\zeta_{14} q^{9} + ( -\zeta_{14}^{2} + \zeta_{14}^{3} ) q^{13} + ( \zeta_{14}^{2} - \zeta_{14}^{5} ) q^{17} + ( -\zeta_{14} - \zeta_{14}^{3} - \zeta_{14}^{5} ) q^{25} -\zeta_{14}^{6} q^{29} + ( -\zeta_{14}^{4} + \zeta_{14}^{5} ) q^{37} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{41} + ( -1 + \zeta_{14}^{5} ) q^{45} -\zeta_{14} q^{49} + ( -1 + \zeta_{14}^{3} ) q^{53} + ( -\zeta_{14}^{2} - \zeta_{14}^{6} ) q^{61} + ( 1 - \zeta_{14} + \zeta_{14}^{2} + \zeta_{14}^{6} ) q^{65} + ( 1 + \zeta_{14}^{4} ) q^{73} + \zeta_{14}^{2} q^{81} + ( \zeta_{14} - \zeta_{14}^{2} - \zeta_{14}^{4} - \zeta_{14}^{6} ) q^{85} + ( \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{89} + ( 1 + \zeta_{14}^{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{5} - q^{9} + O(q^{10}) \) \( 6q + 2q^{5} - q^{9} + 2q^{13} - 2q^{17} - 3q^{25} + q^{29} + 2q^{37} - 2q^{41} - 5q^{45} - q^{49} - 5q^{53} + 2q^{61} + 3q^{65} + 5q^{73} - q^{81} + 4q^{85} - 2q^{89} + 5q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times\).

\(n\) \(321\) \(581\) \(639\)
\(\chi(n)\) \(-\zeta_{14}^{3}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
0 0 0 1.12349 1.40881i 0 0 0 −0.900969 0.433884i 0
575.1 0 0 0 1.12349 + 1.40881i 0 0 0 −0.900969 + 0.433884i 0
703.1 0 0 0 0.277479 + 1.21572i 0 0 0 0.623490 + 0.781831i 0
895.1 0 0 0 0.277479 1.21572i 0 0 0 0.623490 0.781831i 0
1151.1 0 0 0 −0.400969 + 0.193096i 0 0 0 −0.222521 + 0.974928i 0
1727.1 0 0 0 −0.400969 0.193096i 0 0 0 −0.222521 0.974928i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1727.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
29.d even 7 1 inner
116.j odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.1.bh.a 6
4.b odd 2 1 CM 1856.1.bh.a 6
8.b even 2 1 116.1.j.a 6
8.d odd 2 1 116.1.j.a 6
24.f even 2 1 1044.1.bb.a 6
24.h odd 2 1 1044.1.bb.a 6
29.d even 7 1 inner 1856.1.bh.a 6
40.e odd 2 1 2900.1.bj.a 6
40.f even 2 1 2900.1.bj.a 6
40.i odd 4 2 2900.1.bd.a 12
40.k even 4 2 2900.1.bd.a 12
116.j odd 14 1 inner 1856.1.bh.a 6
232.b odd 2 1 3364.1.j.d 6
232.g even 2 1 3364.1.j.d 6
232.k even 4 2 3364.1.h.e 12
232.l odd 4 2 3364.1.h.e 12
232.o even 14 1 3364.1.b.b 3
232.o even 14 2 3364.1.j.c 6
232.o even 14 1 3364.1.j.d 6
232.o even 14 2 3364.1.j.e 6
232.p odd 14 1 116.1.j.a 6
232.p odd 14 1 3364.1.b.c 3
232.p odd 14 2 3364.1.j.a 6
232.p odd 14 2 3364.1.j.b 6
232.s even 14 1 116.1.j.a 6
232.s even 14 1 3364.1.b.c 3
232.s even 14 2 3364.1.j.a 6
232.s even 14 2 3364.1.j.b 6
232.t odd 14 1 3364.1.b.b 3
232.t odd 14 2 3364.1.j.c 6
232.t odd 14 1 3364.1.j.d 6
232.t odd 14 2 3364.1.j.e 6
232.u odd 28 2 3364.1.d.a 6
232.u odd 28 4 3364.1.h.c 12
232.u odd 28 4 3364.1.h.d 12
232.u odd 28 2 3364.1.h.e 12
232.v even 28 2 3364.1.d.a 6
232.v even 28 4 3364.1.h.c 12
232.v even 28 4 3364.1.h.d 12
232.v even 28 2 3364.1.h.e 12
696.z odd 14 1 1044.1.bb.a 6
696.bf even 14 1 1044.1.bb.a 6
1160.bu even 14 1 2900.1.bj.a 6
1160.bw odd 14 1 2900.1.bj.a 6
1160.cp even 28 2 2900.1.bd.a 12
1160.cs odd 28 2 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 8.b even 2 1
116.1.j.a 6 8.d odd 2 1
116.1.j.a 6 232.p odd 14 1
116.1.j.a 6 232.s even 14 1
1044.1.bb.a 6 24.f even 2 1
1044.1.bb.a 6 24.h odd 2 1
1044.1.bb.a 6 696.z odd 14 1
1044.1.bb.a 6 696.bf even 14 1
1856.1.bh.a 6 1.a even 1 1 trivial
1856.1.bh.a 6 4.b odd 2 1 CM
1856.1.bh.a 6 29.d even 7 1 inner
1856.1.bh.a 6 116.j odd 14 1 inner
2900.1.bd.a 12 40.i odd 4 2
2900.1.bd.a 12 40.k even 4 2
2900.1.bd.a 12 1160.cp even 28 2
2900.1.bd.a 12 1160.cs odd 28 2
2900.1.bj.a 6 40.e odd 2 1
2900.1.bj.a 6 40.f even 2 1
2900.1.bj.a 6 1160.bu even 14 1
2900.1.bj.a 6 1160.bw odd 14 1
3364.1.b.b 3 232.o even 14 1
3364.1.b.b 3 232.t odd 14 1
3364.1.b.c 3 232.p odd 14 1
3364.1.b.c 3 232.s even 14 1
3364.1.d.a 6 232.u odd 28 2
3364.1.d.a 6 232.v even 28 2
3364.1.h.c 12 232.u odd 28 4
3364.1.h.c 12 232.v even 28 4
3364.1.h.d 12 232.u odd 28 4
3364.1.h.d 12 232.v even 28 4
3364.1.h.e 12 232.k even 4 2
3364.1.h.e 12 232.l odd 4 2
3364.1.h.e 12 232.u odd 28 2
3364.1.h.e 12 232.v even 28 2
3364.1.j.a 6 232.p odd 14 2
3364.1.j.a 6 232.s even 14 2
3364.1.j.b 6 232.p odd 14 2
3364.1.j.b 6 232.s even 14 2
3364.1.j.c 6 232.o even 14 2
3364.1.j.c 6 232.t odd 14 2
3364.1.j.d 6 232.b odd 2 1
3364.1.j.d 6 232.g even 2 1
3364.1.j.d 6 232.o even 14 1
3364.1.j.d 6 232.t odd 14 1
3364.1.j.e 6 232.o even 14 2
3364.1.j.e 6 232.t odd 14 2

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1856, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \)
$3$ \( T^{6} \)
$5$ \( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( T^{6} \)
$11$ \( T^{6} \)
$13$ \( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$17$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$19$ \( T^{6} \)
$23$ \( T^{6} \)
$29$ \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \)
$31$ \( T^{6} \)
$37$ \( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$41$ \( ( -1 - 2 T + T^{2} + T^{3} )^{2} \)
$43$ \( T^{6} \)
$47$ \( T^{6} \)
$53$ \( 1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6} \)
$59$ \( T^{6} \)
$61$ \( 1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6} \)
$67$ \( T^{6} \)
$71$ \( T^{6} \)
$73$ \( 1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6} \)
$79$ \( T^{6} \)
$83$ \( T^{6} \)
$89$ \( 1 - 3 T + 2 T^{2} + T^{3} + 4 T^{4} + 2 T^{5} + T^{6} \)
$97$ \( 1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6} \)
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