Properties

Label 968.1.f.b
Level $968$
Weight $1$
Character orbit 968.f
Self dual yes
Analytic conductor $0.483$
Analytic rank $0$
Dimension $2$
Projective image $D_{5}$
CM discriminant -8
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 968.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.483094932229\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
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 88)
Projective image: \(D_{5}\)
Projective field: Galois closure of 5.1.937024.1
Artin image: $D_5$
Artin field: Galois closure of 5.1.937024.1

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} + q^{8} + ( 1 - \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( -1 + \beta ) q^{3} + q^{4} + ( -1 + \beta ) q^{6} + q^{8} + ( 1 - \beta ) q^{9} + ( -1 + \beta ) q^{12} + q^{16} -\beta q^{17} + ( 1 - \beta ) q^{18} -\beta q^{19} + ( -1 + \beta ) q^{24} + q^{25} - q^{27} + q^{32} -\beta q^{34} + ( 1 - \beta ) q^{36} -\beta q^{38} + ( -1 + \beta ) q^{41} + ( -1 + \beta ) q^{43} + ( -1 + \beta ) q^{48} + q^{49} + q^{50} - q^{51} - q^{54} - q^{57} -\beta q^{59} + q^{64} -\beta q^{67} -\beta q^{68} + ( 1 - \beta ) q^{72} + ( -1 + \beta ) q^{73} + ( -1 + \beta ) q^{75} -\beta q^{76} + ( -1 + \beta ) q^{82} + ( -1 + \beta ) q^{83} + ( -1 + \beta ) q^{86} + ( -1 + \beta ) q^{89} + ( -1 + \beta ) q^{96} + ( -1 + \beta ) q^{97} + q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{6} + 2q^{8} + q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - q^{3} + 2q^{4} - q^{6} + 2q^{8} + q^{9} - q^{12} + 2q^{16} - q^{17} + q^{18} - q^{19} - q^{24} + 2q^{25} - 2q^{27} + 2q^{32} - q^{34} + q^{36} - q^{38} - q^{41} - q^{43} - q^{48} + 2q^{49} + 2q^{50} - 2q^{51} - 2q^{54} - 2q^{57} - q^{59} + 2q^{64} - q^{67} - q^{68} + q^{72} - q^{73} - q^{75} - q^{76} - q^{82} - q^{83} - q^{86} - q^{89} - q^{96} - q^{97} + 2q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(849\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
243.1
−0.618034
1.61803
1.00000 −1.61803 1.00000 0 −1.61803 0 1.00000 1.61803 0
243.2 1.00000 0.618034 1.00000 0 0.618034 0 1.00000 −0.618034 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.1.f.b 2
4.b odd 2 1 3872.1.f.a 2
8.b even 2 1 3872.1.f.a 2
8.d odd 2 1 CM 968.1.f.b 2
11.b odd 2 1 968.1.f.a 2
11.c even 5 2 88.1.l.a 4
11.c even 5 2 968.1.l.a 4
11.d odd 10 2 968.1.l.b 4
11.d odd 10 2 968.1.l.c 4
33.h odd 10 2 792.1.bu.a 4
44.c even 2 1 3872.1.f.b 2
44.g even 10 2 3872.1.t.a 4
44.g even 10 2 3872.1.t.c 4
44.h odd 10 2 352.1.t.a 4
44.h odd 10 2 3872.1.t.b 4
55.j even 10 2 2200.1.cl.a 4
55.k odd 20 4 2200.1.dd.a 8
88.b odd 2 1 3872.1.f.b 2
88.g even 2 1 968.1.f.a 2
88.k even 10 2 968.1.l.b 4
88.k even 10 2 968.1.l.c 4
88.l odd 10 2 88.1.l.a 4
88.l odd 10 2 968.1.l.a 4
88.o even 10 2 352.1.t.a 4
88.o even 10 2 3872.1.t.b 4
88.p odd 10 2 3872.1.t.a 4
88.p odd 10 2 3872.1.t.c 4
132.o even 10 2 3168.1.ck.a 4
176.v odd 20 4 2816.1.v.c 8
176.w even 20 4 2816.1.v.c 8
264.t odd 10 2 3168.1.ck.a 4
264.w even 10 2 792.1.bu.a 4
440.bh odd 10 2 2200.1.cl.a 4
440.bs even 20 4 2200.1.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.1.l.a 4 11.c even 5 2
88.1.l.a 4 88.l odd 10 2
352.1.t.a 4 44.h odd 10 2
352.1.t.a 4 88.o even 10 2
792.1.bu.a 4 33.h odd 10 2
792.1.bu.a 4 264.w even 10 2
968.1.f.a 2 11.b odd 2 1
968.1.f.a 2 88.g even 2 1
968.1.f.b 2 1.a even 1 1 trivial
968.1.f.b 2 8.d odd 2 1 CM
968.1.l.a 4 11.c even 5 2
968.1.l.a 4 88.l odd 10 2
968.1.l.b 4 11.d odd 10 2
968.1.l.b 4 88.k even 10 2
968.1.l.c 4 11.d odd 10 2
968.1.l.c 4 88.k even 10 2
2200.1.cl.a 4 55.j even 10 2
2200.1.cl.a 4 440.bh odd 10 2
2200.1.dd.a 8 55.k odd 20 4
2200.1.dd.a 8 440.bs even 20 4
2816.1.v.c 8 176.v odd 20 4
2816.1.v.c 8 176.w even 20 4
3168.1.ck.a 4 132.o even 10 2
3168.1.ck.a 4 264.t odd 10 2
3872.1.f.a 2 4.b odd 2 1
3872.1.f.a 2 8.b even 2 1
3872.1.f.b 2 44.c even 2 1
3872.1.f.b 2 88.b odd 2 1
3872.1.t.a 4 44.g even 10 2
3872.1.t.a 4 88.p odd 10 2
3872.1.t.b 4 44.h odd 10 2
3872.1.t.b 4 88.o even 10 2
3872.1.t.c 4 44.g even 10 2
3872.1.t.c 4 88.p odd 10 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{17}^{2} + T_{17} - 1 \) acting on \(S_{1}^{\mathrm{new}}(968, [\chi])\).

Hecke characteristic polynomials

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