Properties

Label 864.2.c.b
Level 864
Weight 2
Character orbit 864.c
Analytic conductor 6.899
Analytic rank 0
Dimension 8
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} +O(q^{10})\) \( q + ( 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{5} + ( 1 - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{7} + ( 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{11} + ( 1 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{13} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( -1 + 2 \zeta_{24}^{4} ) q^{19} + ( 4 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{23} + ( -3 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{25} + ( 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{29} + ( -2 + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{31} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{35} + ( -3 + 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{37} + ( 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{41} + ( 2 - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{43} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{49} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{53} + 4 \zeta_{24}^{6} q^{55} + ( -8 \zeta_{24} - 8 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{59} + ( 1 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{61} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 10 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{65} + ( -1 + 2 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{67} + ( 8 \zeta_{24} + 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{71} + ( 3 - 8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{73} + ( -2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{77} + ( -3 + 6 \zeta_{24}^{4} - 10 \zeta_{24}^{6} ) q^{79} + ( 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{83} + ( 8 \zeta_{24}^{2} - 4 \zeta_{24}^{6} ) q^{85} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{89} + ( 5 - 10 \zeta_{24}^{4} - 8 \zeta_{24}^{6} ) q^{91} + ( -4 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{95} + 7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q + 8q^{13} - 24q^{25} - 24q^{37} + 8q^{61} + 24q^{73} + 56q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0 0 0 3.86370i 0 3.73205i 0 0 0
863.2 0 0 0 3.86370i 0 3.73205i 0 0 0
863.3 0 0 0 1.03528i 0 0.267949i 0 0 0
863.4 0 0 0 1.03528i 0 0.267949i 0 0 0
863.5 0 0 0 1.03528i 0 0.267949i 0 0 0
863.6 0 0 0 1.03528i 0 0.267949i 0 0 0
863.7 0 0 0 3.86370i 0 3.73205i 0 0 0
863.8 0 0 0 3.86370i 0 3.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.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.c.b 8
3.b odd 2 1 inner 864.2.c.b 8
4.b odd 2 1 inner 864.2.c.b 8
8.b even 2 1 1728.2.c.f 8
8.d odd 2 1 1728.2.c.f 8
9.c even 3 1 2592.2.s.c 8
9.c even 3 1 2592.2.s.g 8
9.d odd 6 1 2592.2.s.c 8
9.d odd 6 1 2592.2.s.g 8
12.b even 2 1 inner 864.2.c.b 8
24.f even 2 1 1728.2.c.f 8
24.h odd 2 1 1728.2.c.f 8
36.f odd 6 1 2592.2.s.c 8
36.f odd 6 1 2592.2.s.g 8
36.h even 6 1 2592.2.s.c 8
36.h even 6 1 2592.2.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.b 8 1.a even 1 1 trivial
864.2.c.b 8 3.b odd 2 1 inner
864.2.c.b 8 4.b odd 2 1 inner
864.2.c.b 8 12.b even 2 1 inner
1728.2.c.f 8 8.b even 2 1
1728.2.c.f 8 8.d odd 2 1
1728.2.c.f 8 24.f even 2 1
1728.2.c.f 8 24.h odd 2 1
2592.2.s.c 8 9.c even 3 1
2592.2.s.c 8 9.d odd 6 1
2592.2.s.c 8 36.f odd 6 1
2592.2.s.c 8 36.h even 6 1
2592.2.s.g 8 9.c even 3 1
2592.2.s.g 8 9.d odd 6 1
2592.2.s.g 8 36.f odd 6 1
2592.2.s.g 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 16 T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( ( 1 - 4 T^{2} + 6 T^{4} - 100 T^{6} + 625 T^{8} )^{2} \)
$7$ \( ( 1 - 14 T^{2} + 99 T^{4} - 686 T^{6} + 2401 T^{8} )^{2} \)
$11$ \( ( 1 + 28 T^{2} + 390 T^{4} + 3388 T^{6} + 14641 T^{8} )^{2} \)
$13$ \( ( 1 - 2 T + 15 T^{2} - 26 T^{3} + 169 T^{4} )^{4} \)
$17$ \( ( 1 - 20 T^{2} + 246 T^{4} - 5780 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 35 T^{2} + 361 T^{4} )^{4} \)
$23$ \( ( 1 - 20 T^{2} + 726 T^{4} - 10580 T^{6} + 279841 T^{8} )^{2} \)
$29$ \( ( 1 - 52 T^{2} + 1590 T^{4} - 43732 T^{6} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 68 T^{2} + 2310 T^{4} - 65348 T^{6} + 923521 T^{8} )^{2} \)
$37$ \( ( 1 + 6 T + 71 T^{2} + 222 T^{3} + 1369 T^{4} )^{4} \)
$41$ \( ( 1 - 100 T^{2} + 5094 T^{4} - 168100 T^{6} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 116 T^{2} + 6294 T^{4} - 214484 T^{6} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 + 76 T^{2} + 4662 T^{4} + 167884 T^{6} + 4879681 T^{8} )^{2} \)
$53$ \( ( 1 - 20 T^{2} - 1194 T^{4} - 56180 T^{6} + 7890481 T^{8} )^{2} \)
$59$ \( ( 1 + 28 T^{2} + 4806 T^{4} + 97468 T^{6} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 2 T + 15 T^{2} - 122 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( ( 1 - 134 T^{2} + 12699 T^{4} - 601526 T^{6} + 20151121 T^{8} )^{2} \)
$71$ \( ( 1 + 14 T^{2} + 5041 T^{4} )^{4} \)
$73$ \( ( 1 - 6 T + 107 T^{2} - 438 T^{3} + 5329 T^{4} )^{4} \)
$79$ \( ( 1 - 62 T^{2} + 2643 T^{4} - 386942 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 + 268 T^{2} + 30966 T^{4} + 1846252 T^{6} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 308 T^{2} + 39126 T^{4} - 2439668 T^{6} + 62742241 T^{8} )^{2} \)
$97$ \( ( 1 - 7 T + 97 T^{2} )^{8} \)
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