Properties

Label 864.2.d.b
Level 864
Weight 2
Character orbit 864.d
Analytic conductor 6.899
Analytic rank 0
Dimension 4
CM discriminant -24
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 864.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( 1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( 1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + ( -2 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{11} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{25} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{29} + ( -5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{31} + ( 8 \zeta_{8} + 3 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{35} + ( 12 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{49} + ( 5 \zeta_{8} - 3 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{53} + ( 5 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{55} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{59} + ( -7 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{73} + ( -11 \zeta_{8} + 15 \zeta_{8}^{2} - 11 \zeta_{8}^{3} ) q^{77} + 10 q^{79} + ( -2 \zeta_{8} - 15 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{83} + ( -1 + 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{7} + O(q^{10}) \) \( 4q + 4q^{7} - 24q^{25} - 20q^{31} + 48q^{49} + 20q^{55} - 28q^{73} + 40q^{79} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
0.707107 + 0.707107i
−0.707107 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
0 0 0 4.41421i 0 −3.24264 0 0 0
433.2 0 0 0 1.58579i 0 5.24264 0 0 0
433.3 0 0 0 1.58579i 0 5.24264 0 0 0
433.4 0 0 0 4.41421i 0 −3.24264 0 0 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.d.b 4
3.b odd 2 1 inner 864.2.d.b 4
4.b odd 2 1 216.2.d.a 4
8.b even 2 1 inner 864.2.d.b 4
8.d odd 2 1 216.2.d.a 4
9.c even 3 2 2592.2.r.o 8
9.d odd 6 2 2592.2.r.o 8
12.b even 2 1 216.2.d.a 4
16.e even 4 1 6912.2.a.y 2
16.e even 4 1 6912.2.a.by 2
16.f odd 4 1 6912.2.a.z 2
16.f odd 4 1 6912.2.a.bz 2
24.f even 2 1 216.2.d.a 4
24.h odd 2 1 CM 864.2.d.b 4
36.f odd 6 2 648.2.n.p 8
36.h even 6 2 648.2.n.p 8
48.i odd 4 1 6912.2.a.y 2
48.i odd 4 1 6912.2.a.by 2
48.k even 4 1 6912.2.a.z 2
48.k even 4 1 6912.2.a.bz 2
72.j odd 6 2 2592.2.r.o 8
72.l even 6 2 648.2.n.p 8
72.n even 6 2 2592.2.r.o 8
72.p odd 6 2 648.2.n.p 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.a 4 4.b odd 2 1
216.2.d.a 4 8.d odd 2 1
216.2.d.a 4 12.b even 2 1
216.2.d.a 4 24.f even 2 1
648.2.n.p 8 36.f odd 6 2
648.2.n.p 8 36.h even 6 2
648.2.n.p 8 72.l even 6 2
648.2.n.p 8 72.p odd 6 2
864.2.d.b 4 1.a even 1 1 trivial
864.2.d.b 4 3.b odd 2 1 inner
864.2.d.b 4 8.b even 2 1 inner
864.2.d.b 4 24.h odd 2 1 CM
2592.2.r.o 8 9.c even 3 2
2592.2.r.o 8 9.d odd 6 2
2592.2.r.o 8 72.j odd 6 2
2592.2.r.o 8 72.n even 6 2
6912.2.a.y 2 16.e even 4 1
6912.2.a.y 2 48.i odd 4 1
6912.2.a.z 2 16.f odd 4 1
6912.2.a.z 2 48.k even 4 1
6912.2.a.by 2 16.e even 4 1
6912.2.a.by 2 48.i odd 4 1
6912.2.a.bz 2 16.f odd 4 1
6912.2.a.bz 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 22 T_{5}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - 2 T - 3 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \)
$11$ \( 1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{4} \)
$19$ \( ( 1 - 19 T^{2} )^{4} \)
$23$ \( ( 1 + 23 T^{2} )^{4} \)
$29$ \( ( 1 - 50 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4} )^{2} \)
$37$ \( ( 1 - 37 T^{2} )^{4} \)
$41$ \( ( 1 + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 43 T^{2} )^{4} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( 1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 + 10 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( ( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{4} \)
$83$ \( 1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2} \)
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