Properties

Label 1156.1.g.b
Level $1156$
Weight $1$
Character orbit 1156.g
Analytic conductor $0.577$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -4
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 1156 = 2^{2} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1156.g (of order \(8\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
Defining polynomial: \(x^{8} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{16}^{6} q^{2} -\zeta_{16}^{4} q^{4} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{5} + \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{16}^{6} q^{2} -\zeta_{16}^{4} q^{4} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{5} + \zeta_{16}^{2} q^{8} + \zeta_{16}^{2} q^{9} + ( \zeta_{16} + \zeta_{16}^{5} ) q^{10} - q^{16} - q^{18} + ( -\zeta_{16}^{3} + \zeta_{16}^{7} ) q^{20} -\zeta_{16}^{2} q^{25} + ( -\zeta_{16}^{3} - \zeta_{16}^{7} ) q^{29} -\zeta_{16}^{6} q^{32} -\zeta_{16}^{6} q^{36} + ( \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{37} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{40} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{41} + ( \zeta_{16} - \zeta_{16}^{5} ) q^{45} + \zeta_{16}^{6} q^{49} + q^{50} + ( \zeta_{16} + \zeta_{16}^{5} ) q^{58} + ( -\zeta_{16} - \zeta_{16}^{5} ) q^{61} + \zeta_{16}^{4} q^{64} + \zeta_{16}^{4} q^{72} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{73} + ( -\zeta_{16} + \zeta_{16}^{5} ) q^{74} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{80} + \zeta_{16}^{4} q^{81} + ( \zeta_{16}^{3} - \zeta_{16}^{7} ) q^{82} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{90} + ( \zeta_{16}^{3} + \zeta_{16}^{7} ) q^{97} -\zeta_{16}^{4} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 8q^{16} - 8q^{18} + 8q^{50} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times\).

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(\zeta_{16}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.923880 + 0.382683i
0.923880 0.382683i
−0.923880 0.382683i
0.923880 + 0.382683i
−0.382683 + 0.923880i
0.382683 0.923880i
−0.382683 0.923880i
0.382683 + 0.923880i
−0.707107 0.707107i 0 1.00000i −0.541196 1.30656i 0 0 0.707107 0.707107i 0.707107 0.707107i −0.541196 + 1.30656i
155.2 −0.707107 0.707107i 0 1.00000i 0.541196 + 1.30656i 0 0 0.707107 0.707107i 0.707107 0.707107i 0.541196 1.30656i
179.1 −0.707107 + 0.707107i 0 1.00000i −0.541196 + 1.30656i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i −0.541196 1.30656i
179.2 −0.707107 + 0.707107i 0 1.00000i 0.541196 1.30656i 0 0 0.707107 + 0.707107i 0.707107 + 0.707107i 0.541196 + 1.30656i
399.1 0.707107 0.707107i 0 1.00000i −1.30656 0.541196i 0 0 −0.707107 0.707107i −0.707107 0.707107i −1.30656 + 0.541196i
399.2 0.707107 0.707107i 0 1.00000i 1.30656 + 0.541196i 0 0 −0.707107 0.707107i −0.707107 0.707107i 1.30656 0.541196i
423.1 0.707107 + 0.707107i 0 1.00000i −1.30656 + 0.541196i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i −1.30656 0.541196i
423.2 0.707107 + 0.707107i 0 1.00000i 1.30656 0.541196i 0 0 −0.707107 + 0.707107i −0.707107 + 0.707107i 1.30656 + 0.541196i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 423.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
68.d odd 2 1 inner
68.f odd 4 2 inner
68.g odd 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.g.b 8
4.b odd 2 1 CM 1156.1.g.b 8
17.b even 2 1 inner 1156.1.g.b 8
17.c even 4 2 inner 1156.1.g.b 8
17.d even 8 4 inner 1156.1.g.b 8
17.e odd 16 2 68.1.f.a 2
17.e odd 16 2 1156.1.c.b 2
17.e odd 16 2 1156.1.d.a 2
17.e odd 16 2 1156.1.f.b 2
51.i even 16 2 612.1.l.a 2
68.d odd 2 1 inner 1156.1.g.b 8
68.f odd 4 2 inner 1156.1.g.b 8
68.g odd 8 4 inner 1156.1.g.b 8
68.i even 16 2 68.1.f.a 2
68.i even 16 2 1156.1.c.b 2
68.i even 16 2 1156.1.d.a 2
68.i even 16 2 1156.1.f.b 2
85.o even 16 1 1700.1.n.a 2
85.o even 16 1 1700.1.n.b 2
85.p odd 16 2 1700.1.p.a 2
85.r even 16 1 1700.1.n.a 2
85.r even 16 1 1700.1.n.b 2
119.p even 16 2 3332.1.m.b 2
119.s even 48 4 3332.1.bc.b 4
119.t odd 48 4 3332.1.bc.c 4
136.q odd 16 2 1088.1.p.a 2
136.s even 16 2 1088.1.p.a 2
204.t odd 16 2 612.1.l.a 2
340.bc odd 16 1 1700.1.n.a 2
340.bc odd 16 1 1700.1.n.b 2
340.bg even 16 2 1700.1.p.a 2
340.bj odd 16 1 1700.1.n.a 2
340.bj odd 16 1 1700.1.n.b 2
476.bf odd 16 2 3332.1.m.b 2
476.bk odd 48 4 3332.1.bc.b 4
476.bm even 48 4 3332.1.bc.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 17.e odd 16 2
68.1.f.a 2 68.i even 16 2
612.1.l.a 2 51.i even 16 2
612.1.l.a 2 204.t odd 16 2
1088.1.p.a 2 136.q odd 16 2
1088.1.p.a 2 136.s even 16 2
1156.1.c.b 2 17.e odd 16 2
1156.1.c.b 2 68.i even 16 2
1156.1.d.a 2 17.e odd 16 2
1156.1.d.a 2 68.i even 16 2
1156.1.f.b 2 17.e odd 16 2
1156.1.f.b 2 68.i even 16 2
1156.1.g.b 8 1.a even 1 1 trivial
1156.1.g.b 8 4.b odd 2 1 CM
1156.1.g.b 8 17.b even 2 1 inner
1156.1.g.b 8 17.c even 4 2 inner
1156.1.g.b 8 17.d even 8 4 inner
1156.1.g.b 8 68.d odd 2 1 inner
1156.1.g.b 8 68.f odd 4 2 inner
1156.1.g.b 8 68.g odd 8 4 inner
1700.1.n.a 2 85.o even 16 1
1700.1.n.a 2 85.r even 16 1
1700.1.n.a 2 340.bc odd 16 1
1700.1.n.a 2 340.bj odd 16 1
1700.1.n.b 2 85.o even 16 1
1700.1.n.b 2 85.r even 16 1
1700.1.n.b 2 340.bc odd 16 1
1700.1.n.b 2 340.bj odd 16 1
1700.1.p.a 2 85.p odd 16 2
1700.1.p.a 2 340.bg even 16 2
3332.1.m.b 2 119.p even 16 2
3332.1.m.b 2 476.bf odd 16 2
3332.1.bc.b 4 119.s even 48 4
3332.1.bc.b 4 476.bk odd 48 4
3332.1.bc.c 4 119.t odd 48 4
3332.1.bc.c 4 476.bm even 48 4

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16 \) acting on \(S_{1}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{4} )^{2} \)
$3$ \( T^{8} \)
$5$ \( 16 + T^{8} \)
$7$ \( T^{8} \)
$11$ \( T^{8} \)
$13$ \( T^{8} \)
$17$ \( T^{8} \)
$19$ \( T^{8} \)
$23$ \( T^{8} \)
$29$ \( 16 + T^{8} \)
$31$ \( T^{8} \)
$37$ \( 16 + T^{8} \)
$41$ \( 16 + T^{8} \)
$43$ \( T^{8} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( T^{8} \)
$61$ \( 16 + T^{8} \)
$67$ \( T^{8} \)
$71$ \( T^{8} \)
$73$ \( 16 + T^{8} \)
$79$ \( T^{8} \)
$83$ \( T^{8} \)
$89$ \( T^{8} \)
$97$ \( 16 + T^{8} \)
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