Properties

Label 864.2.d.a
Level 864
Weight 2
Character orbit 864.d
Analytic conductor 6.899
Analytic rank 0
Dimension 4
CM no
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: \(\Q(i, \sqrt{7})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{5} - q^{7} +O(q^{10})\) \( q -\beta_{1} q^{5} - q^{7} + 3 \beta_{1} q^{11} -\beta_{3} q^{13} + \beta_{2} q^{17} -\beta_{3} q^{19} -\beta_{2} q^{23} + 4 q^{25} -6 \beta_{1} q^{29} + 7 q^{31} + \beta_{1} q^{35} -\beta_{3} q^{37} -\beta_{2} q^{41} -2 \beta_{3} q^{43} -6 q^{49} + 9 \beta_{1} q^{53} + 3 q^{55} -4 \beta_{1} q^{59} -\beta_{2} q^{65} + 3 \beta_{2} q^{71} + 3 q^{73} -3 \beta_{1} q^{77} -4 q^{79} -7 \beta_{1} q^{83} -\beta_{3} q^{85} + 2 \beta_{2} q^{89} + \beta_{3} q^{91} -\beta_{2} q^{95} + 7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{7} + 16q^{25} + 28q^{31} - 24q^{49} + 12q^{55} + 12q^{73} - 16q^{79} + 28q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 3 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( -\nu^{3} + 5 \nu \)
\(\beta_{3}\)\(=\)\( 4 \nu^{2} - 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 6\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{2} + 10 \beta_{1}\)\()/4\)

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
1.32288 + 0.500000i
−1.32288 + 0.500000i
−1.32288 0.500000i
1.32288 0.500000i
0 0 0 1.00000i 0 −1.00000 0 0 0
433.2 0 0 0 1.00000i 0 −1.00000 0 0 0
433.3 0 0 0 1.00000i 0 −1.00000 0 0 0
433.4 0 0 0 1.00000i 0 −1.00000 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
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 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.a 4
3.b odd 2 1 inner 864.2.d.a 4
4.b odd 2 1 216.2.d.b 4
8.b even 2 1 inner 864.2.d.a 4
8.d odd 2 1 216.2.d.b 4
9.c even 3 2 2592.2.r.p 8
9.d odd 6 2 2592.2.r.p 8
12.b even 2 1 216.2.d.b 4
16.e even 4 1 6912.2.a.bd 2
16.e even 4 1 6912.2.a.bv 2
16.f odd 4 1 6912.2.a.bc 2
16.f odd 4 1 6912.2.a.bu 2
24.f even 2 1 216.2.d.b 4
24.h odd 2 1 inner 864.2.d.a 4
36.f odd 6 2 648.2.n.n 8
36.h even 6 2 648.2.n.n 8
48.i odd 4 1 6912.2.a.bd 2
48.i odd 4 1 6912.2.a.bv 2
48.k even 4 1 6912.2.a.bc 2
48.k even 4 1 6912.2.a.bu 2
72.j odd 6 2 2592.2.r.p 8
72.l even 6 2 648.2.n.n 8
72.n even 6 2 2592.2.r.p 8
72.p odd 6 2 648.2.n.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 4.b odd 2 1
216.2.d.b 4 8.d odd 2 1
216.2.d.b 4 12.b even 2 1
216.2.d.b 4 24.f even 2 1
648.2.n.n 8 36.f odd 6 2
648.2.n.n 8 36.h even 6 2
648.2.n.n 8 72.l even 6 2
648.2.n.n 8 72.p odd 6 2
864.2.d.a 4 1.a even 1 1 trivial
864.2.d.a 4 3.b odd 2 1 inner
864.2.d.a 4 8.b even 2 1 inner
864.2.d.a 4 24.h odd 2 1 inner
2592.2.r.p 8 9.c even 3 2
2592.2.r.p 8 9.d odd 6 2
2592.2.r.p 8 72.j odd 6 2
2592.2.r.p 8 72.n even 6 2
6912.2.a.bc 2 16.f odd 4 1
6912.2.a.bc 2 48.k even 4 1
6912.2.a.bd 2 16.e even 4 1
6912.2.a.bd 2 48.i odd 4 1
6912.2.a.bu 2 16.f odd 4 1
6912.2.a.bu 2 48.k even 4 1
6912.2.a.bv 2 16.e even 4 1
6912.2.a.bv 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 9 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + T + 7 T^{2} )^{4} \)
$11$ \( ( 1 - 13 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 + 2 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 6 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 + 18 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 7 T + 31 T^{2} )^{4} \)
$37$ \( ( 1 - 46 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 54 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 26 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 + 47 T^{2} )^{4} \)
$53$ \( ( 1 - 25 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 102 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 61 T^{2} )^{4} \)
$67$ \( ( 1 - 67 T^{2} )^{4} \)
$71$ \( ( 1 - 110 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 3 T + 73 T^{2} )^{4} \)
$79$ \( ( 1 + 4 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 66 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{4} \)
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