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Properties

Label 1156.1.g.a
Level $1156$
Weight $1$
Character orbit 1156.g
Analytic conductor $0.577$
Analytic rank $0$
Dimension $4$
Projective image $D_{2}$
CM/RM discs -4, -68, 17
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)
Artin image: $\OD_{32}$
Artin field: Galois closure of 16.8.732780301186512843008.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} -\zeta_{8} q^{9} +O(q^{10})\) \( q + \zeta_{8}^{3} q^{2} -\zeta_{8}^{2} q^{4} + \zeta_{8} q^{8} -\zeta_{8} q^{9} -2 \zeta_{8}^{2} q^{13} - q^{16} + q^{18} -\zeta_{8} q^{25} + 2 \zeta_{8} q^{26} -\zeta_{8}^{3} q^{32} + \zeta_{8}^{3} q^{36} -\zeta_{8}^{3} q^{49} + q^{50} -2 q^{52} -2 \zeta_{8}^{3} q^{53} + \zeta_{8}^{2} q^{64} -\zeta_{8}^{2} q^{72} + \zeta_{8}^{2} q^{81} + 2 \zeta_{8}^{2} q^{89} + \zeta_{8}^{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q - 4 q^{16} + 4 q^{18} + 4 q^{50} - 8 q^{52} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}\)

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
−0.707107 + 0.707107i
−0.707107 − 0.707107i
0.707107 + 0.707107i
0.707107 − 0.707107i
0.707107 + 0.707107i 0 1.00000i 0 0 0 −0.707107 + 0.707107i 0.707107 − 0.707107i 0
179.1 0.707107 − 0.707107i 0 − 1.00000i 0 0 0 −0.707107 − 0.707107i 0.707107 + 0.707107i 0
399.1 −0.707107 + 0.707107i 0 − 1.00000i 0 0 0 0.707107 + 0.707107i −0.707107 − 0.707107i 0
423.1 −0.707107 − 0.707107i 0 1.00000i 0 0 0 0.707107 − 0.707107i −0.707107 + 0.707107i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.b even 2 1 RM by \(\Q(\sqrt{17}) \)
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
17.c even 4 2 inner
17.d even 8 4 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.a 4
4.b odd 2 1 CM 1156.1.g.a 4
17.b even 2 1 RM 1156.1.g.a 4
17.c even 4 2 inner 1156.1.g.a 4
17.d even 8 4 inner 1156.1.g.a 4
17.e odd 16 2 68.1.d.a ✓ 1
17.e odd 16 2 1156.1.c.a 1
17.e odd 16 4 1156.1.f.a 2
51.i even 16 2 612.1.e.a 1
68.d odd 2 1 CM 1156.1.g.a 4
68.f odd 4 2 inner 1156.1.g.a 4
68.g odd 8 4 inner 1156.1.g.a 4
68.i even 16 2 68.1.d.a ✓ 1
68.i even 16 2 1156.1.c.a 1
68.i even 16 4 1156.1.f.a 2
85.o even 16 2 1700.1.d.b 2
85.p odd 16 2 1700.1.h.d 1
85.r even 16 2 1700.1.d.b 2
119.p even 16 2 3332.1.g.a 1
119.s even 48 4 3332.1.o.d 2
119.t odd 48 4 3332.1.o.c 2
136.q odd 16 2 1088.1.g.a 1
136.s even 16 2 1088.1.g.a 1
204.t odd 16 2 612.1.e.a 1
340.bc odd 16 2 1700.1.d.b 2
340.bg even 16 2 1700.1.h.d 1
340.bj odd 16 2 1700.1.d.b 2
476.bf odd 16 2 3332.1.g.a 1
476.bk odd 48 4 3332.1.o.d 2
476.bm even 48 4 3332.1.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.d.a ✓ 1 17.e odd 16 2
68.1.d.a ✓ 1 68.i even 16 2
612.1.e.a 1 51.i even 16 2
612.1.e.a 1 204.t odd 16 2
1088.1.g.a 1 136.q odd 16 2
1088.1.g.a 1 136.s even 16 2
1156.1.c.a 1 17.e odd 16 2
1156.1.c.a 1 68.i even 16 2
1156.1.f.a 2 17.e odd 16 4
1156.1.f.a 2 68.i even 16 4
1156.1.g.a 4 1.a even 1 1 trivial
1156.1.g.a 4 4.b odd 2 1 CM
1156.1.g.a 4 17.b even 2 1 RM
1156.1.g.a 4 17.c even 4 2 inner
1156.1.g.a 4 17.d even 8 4 inner
1156.1.g.a 4 68.d odd 2 1 CM
1156.1.g.a 4 68.f odd 4 2 inner
1156.1.g.a 4 68.g odd 8 4 inner
1700.1.d.b 2 85.o even 16 2
1700.1.d.b 2 85.r even 16 2
1700.1.d.b 2 340.bc odd 16 2
1700.1.d.b 2 340.bj odd 16 2
1700.1.h.d 1 85.p odd 16 2
1700.1.h.d 1 340.bg even 16 2
3332.1.g.a 1 119.p even 16 2
3332.1.g.a 1 476.bf odd 16 2
3332.1.o.c 2 119.t odd 48 4
3332.1.o.c 2 476.bm even 48 4
3332.1.o.d 2 119.s even 48 4
3332.1.o.d 2 476.bk odd 48 4

Hecke kernels

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

Hecke characteristic polynomials

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