Properties

Label 864.2.i.f
Level $864$
Weight $2$
Character orbit 864.i
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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.170772624.1
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + ( - \beta_{7} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + ( - \beta_{7} + \beta_1) q^{7} - \beta_{3} q^{11} + ( - \beta_{6} - 2 \beta_{4}) q^{13} + \beta_{5} q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} - \beta_{7} q^{23} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} - 4) q^{25} + ( - \beta_{6} - \beta_{5} + 2 \beta_{4} - 3) q^{29} + ( - \beta_{7} - 2 \beta_{2}) q^{31} + ( - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{35} + 4 q^{37} + \beta_{4} q^{41} - \beta_{3} q^{43} + (\beta_{7} - \beta_1) q^{47} + (3 \beta_{6} - 5 \beta_{4}) q^{49} + 4 q^{53} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{55} + ( - 2 \beta_{7} + \beta_{2}) q^{59} + (3 \beta_{6} + 3 \beta_{5} + 8 \beta_{4} - 5) q^{61} + ( - \beta_{6} - \beta_{5} - 8 \beta_{4} + 7) q^{65} + ( - 2 \beta_{7} + \beta_{2}) q^{67} + 2 \beta_1 q^{71} + (\beta_{5} + 8) q^{73} + ( - 3 \beta_{6} - 6 \beta_{4}) q^{77} + ( - \beta_{7} + 2 \beta_{3} + \beta_1) q^{79} + ( - \beta_{7} - 2 \beta_{3} + \beta_1) q^{83} - 8 \beta_{4} q^{85} + ( - 2 \beta_{5} - 8) q^{89} + (2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{91} + 4 \beta_{7} q^{95} + (9 \beta_{4} - 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 6 q^{13} - 4 q^{17} - 14 q^{25} - 10 q^{29} + 32 q^{37} + 4 q^{41} - 26 q^{49} + 32 q^{53} - 26 q^{61} + 30 q^{65} + 60 q^{73} - 18 q^{77} - 32 q^{85} - 56 q^{89} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 3\nu^{4} + 6\nu^{2} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} + 3\nu^{5} - 6\nu^{4} + 6\nu^{3} + 4\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 9\nu^{5} + 6\nu^{4} - 6\nu^{3} + 24\nu^{2} - 44\nu + 48 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 6\nu^{5} + 5\nu^{4} - 6\nu^{3} + 14\nu^{2} - 20\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 5\nu^{3} + 13\nu^{2} - 18\nu + 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} - 3\nu^{6} + 6\nu^{5} - 3\nu^{4} + 6\nu^{3} - 15\nu^{2} + 16\nu - 24 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 + 2 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - 3\beta_{6} - 3\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} + \beta_{5} + 10\beta_{4} + 3\beta_{3} + 3\beta_{2} + 3\beta _1 - 4 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + 3\beta_{6} + 3\beta_{5} + 3\beta_{4} - \beta_{3} - 2\beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - \beta_{6} - 2\beta_{5} - 17\beta_{4} + \beta_{2} + 32 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2\beta_{7} + 9\beta_{5} - 5\beta_{3} + 5\beta_{2} - \beta _1 + 24 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{7} - 13\beta_{6} + 13\beta_{5} - 5\beta_{4} + 5\beta_{3} + \beta _1 + 8 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-\beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.774115 + 1.18353i
1.41203 + 0.0786378i
−1.02187 + 0.977642i
0.335728 1.37379i
0.774115 1.18353i
1.41203 0.0786378i
−1.02187 0.977642i
0.335728 + 1.37379i
0 0 0 −1.68614 2.92048i 0 −2.35143 + 4.07279i 0 0 0
289.2 0 0 0 −1.68614 2.92048i 0 2.35143 4.07279i 0 0 0
289.3 0 0 0 1.18614 + 2.05446i 0 −1.10489 + 1.91373i 0 0 0
289.4 0 0 0 1.18614 + 2.05446i 0 1.10489 1.91373i 0 0 0
577.1 0 0 0 −1.68614 + 2.92048i 0 −2.35143 4.07279i 0 0 0
577.2 0 0 0 −1.68614 + 2.92048i 0 2.35143 + 4.07279i 0 0 0
577.3 0 0 0 1.18614 2.05446i 0 −1.10489 1.91373i 0 0 0
577.4 0 0 0 1.18614 2.05446i 0 1.10489 + 1.91373i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 577.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.i.f 8
3.b odd 2 1 288.2.i.f 8
4.b odd 2 1 inner 864.2.i.f 8
8.b even 2 1 1728.2.i.n 8
8.d odd 2 1 1728.2.i.n 8
9.c even 3 1 inner 864.2.i.f 8
9.c even 3 1 2592.2.a.x 4
9.d odd 6 1 288.2.i.f 8
9.d odd 6 1 2592.2.a.u 4
12.b even 2 1 288.2.i.f 8
24.f even 2 1 576.2.i.n 8
24.h odd 2 1 576.2.i.n 8
36.f odd 6 1 inner 864.2.i.f 8
36.f odd 6 1 2592.2.a.x 4
36.h even 6 1 288.2.i.f 8
36.h even 6 1 2592.2.a.u 4
72.j odd 6 1 576.2.i.n 8
72.j odd 6 1 5184.2.a.cf 4
72.l even 6 1 576.2.i.n 8
72.l even 6 1 5184.2.a.cf 4
72.n even 6 1 1728.2.i.n 8
72.n even 6 1 5184.2.a.cc 4
72.p odd 6 1 1728.2.i.n 8
72.p odd 6 1 5184.2.a.cc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.f 8 3.b odd 2 1
288.2.i.f 8 9.d odd 6 1
288.2.i.f 8 12.b even 2 1
288.2.i.f 8 36.h even 6 1
576.2.i.n 8 24.f even 2 1
576.2.i.n 8 24.h odd 2 1
576.2.i.n 8 72.j odd 6 1
576.2.i.n 8 72.l even 6 1
864.2.i.f 8 1.a even 1 1 trivial
864.2.i.f 8 4.b odd 2 1 inner
864.2.i.f 8 9.c even 3 1 inner
864.2.i.f 8 36.f odd 6 1 inner
1728.2.i.n 8 8.b even 2 1
1728.2.i.n 8 8.d odd 2 1
1728.2.i.n 8 72.n even 6 1
1728.2.i.n 8 72.p odd 6 1
2592.2.a.u 4 9.d odd 6 1
2592.2.a.u 4 36.h even 6 1
2592.2.a.x 4 9.c even 3 1
2592.2.a.x 4 36.f odd 6 1
5184.2.a.cc 4 72.n even 6 1
5184.2.a.cc 4 72.p odd 6 1
5184.2.a.cf 4 72.j odd 6 1
5184.2.a.cf 4 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} + 9T_{5}^{2} - 8T_{5} + 64 \) Copy content Toggle raw display
\( T_{7}^{8} + 27T_{7}^{6} + 621T_{7}^{4} + 2916T_{7}^{2} + 11664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T - 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 45 T^{2} + 432)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$29$ \( (T^{4} + 5 T^{3} + 27 T^{2} - 10 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 135 T^{6} + \cdots + 15116544 \) Copy content Toggle raw display
$37$ \( (T - 4)^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 36 T^{6} + 1269 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$47$ \( T^{8} + 27 T^{6} + 621 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$53$ \( (T - 4)^{8} \) Copy content Toggle raw display
$59$ \( T^{8} + 180 T^{6} + \cdots + 60886809 \) Copy content Toggle raw display
$61$ \( (T^{4} + 13 T^{3} + 201 T^{2} - 416 T + 1024)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 180 T^{6} + \cdots + 60886809 \) Copy content Toggle raw display
$71$ \( (T^{4} - 108 T^{2} + 1728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 48)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} + 135 T^{6} + \cdots + 15116544 \) Copy content Toggle raw display
$83$ \( T^{8} + 207 T^{6} + 41121 T^{4} + \cdots + 2985984 \) Copy content Toggle raw display
$89$ \( (T^{2} + 14 T + 16)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
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