Properties

Label 1156.1.d.a
Level $1156$
Weight $1$
Character orbit 1156.d
Analytic conductor $0.577$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(0.576919154604\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 68)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.19652.1
Artin image: $SD_{16}$
Artin field: Galois closure of 8.2.26261675072.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(579\) \(581\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1155.1
1.41421i
1.41421i
1.00000 0 1.00000 1.41421i 0 0 1.00000 −1.00000 1.41421i
1155.2 1.00000 0 1.00000 1.41421i 0 0 1.00000 −1.00000 1.41421i
\(n\): 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 inner
68.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1156.1.d.a 2
4.b odd 2 1 CM 1156.1.d.a 2
17.b even 2 1 inner 1156.1.d.a 2
17.c even 4 2 1156.1.c.b 2
17.d even 8 2 68.1.f.a 2
17.d even 8 2 1156.1.f.b 2
17.e odd 16 8 1156.1.g.b 8
51.g odd 8 2 612.1.l.a 2
68.d odd 2 1 inner 1156.1.d.a 2
68.f odd 4 2 1156.1.c.b 2
68.g odd 8 2 68.1.f.a 2
68.g odd 8 2 1156.1.f.b 2
68.i even 16 8 1156.1.g.b 8
85.k odd 8 2 1700.1.n.a 2
85.m even 8 2 1700.1.p.a 2
85.n odd 8 2 1700.1.n.b 2
119.l odd 8 2 3332.1.m.b 2
119.q even 24 4 3332.1.bc.c 4
119.r odd 24 4 3332.1.bc.b 4
136.o even 8 2 1088.1.p.a 2
136.p odd 8 2 1088.1.p.a 2
204.p even 8 2 612.1.l.a 2
340.w even 8 2 1700.1.n.b 2
340.z even 8 2 1700.1.n.a 2
340.ba odd 8 2 1700.1.p.a 2
476.w even 8 2 3332.1.m.b 2
476.bg odd 24 4 3332.1.bc.c 4
476.bj even 24 4 3332.1.bc.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 17.d even 8 2
68.1.f.a 2 68.g odd 8 2
612.1.l.a 2 51.g odd 8 2
612.1.l.a 2 204.p even 8 2
1088.1.p.a 2 136.o even 8 2
1088.1.p.a 2 136.p odd 8 2
1156.1.c.b 2 17.c even 4 2
1156.1.c.b 2 68.f odd 4 2
1156.1.d.a 2 1.a even 1 1 trivial
1156.1.d.a 2 4.b odd 2 1 CM
1156.1.d.a 2 17.b even 2 1 inner
1156.1.d.a 2 68.d odd 2 1 inner
1156.1.f.b 2 17.d even 8 2
1156.1.f.b 2 68.g odd 8 2
1156.1.g.b 8 17.e odd 16 8
1156.1.g.b 8 68.i even 16 8
1700.1.n.a 2 85.k odd 8 2
1700.1.n.a 2 340.z even 8 2
1700.1.n.b 2 85.n odd 8 2
1700.1.n.b 2 340.w even 8 2
1700.1.p.a 2 85.m even 8 2
1700.1.p.a 2 340.ba odd 8 2
3332.1.m.b 2 119.l odd 8 2
3332.1.m.b 2 476.w even 8 2
3332.1.bc.b 4 119.r odd 24 4
3332.1.bc.b 4 476.bj even 24 4
3332.1.bc.c 4 119.q even 24 4
3332.1.bc.c 4 476.bg odd 24 4

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1156, [\chi])\).

Hecke characteristic polynomials

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