Properties

Label 864.2.i.c
Level $864$
Weight $2$
Character orbit 864.i
Analytic conductor $6.899$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + 2 \beta_{3} q^{17} + 4 q^{19} + (\beta_{2} - 3 \beta_1) q^{23} + ( - 4 \beta_1 + 4) q^{25} + (2 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 5) q^{29} + ( - \beta_{2} - 5 \beta_1) q^{31} + ( - \beta_{3} + 1) q^{35} + (2 \beta_{3} + 4) q^{37} + (2 \beta_{2} - 7 \beta_1) q^{41} + ( - 3 \beta_{3} + 3 \beta_{2} + 5 \beta_1 - 5) q^{43} + (3 \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{47} + 2 \beta_{2} q^{49} + (2 \beta_{3} - 4) q^{53} + (\beta_{3} + 1) q^{55} + ( - 3 \beta_{2} + 7 \beta_1) q^{59} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 3) q^{61} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{65} + (3 \beta_{2} - 5 \beta_1) q^{67} + (4 \beta_{3} + 2) q^{71} + 2 \beta_{3} q^{73} + 5 \beta_1 q^{77} + (\beta_{3} - \beta_{2} + 11 \beta_1 - 11) q^{79} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{83} - 2 \beta_{2} q^{85} + ( - 2 \beta_{3} + 8) q^{89} + ( - 3 \beta_{3} + 13) q^{91} - 4 \beta_1 q^{95} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} - 2 q^{7} - 2 q^{11} - 2 q^{13} + 16 q^{19} - 6 q^{23} + 8 q^{25} + 10 q^{29} - 10 q^{31} + 4 q^{35} + 16 q^{37} - 14 q^{41} - 10 q^{43} + 2 q^{47} - 16 q^{53} + 4 q^{55} + 14 q^{59} + 6 q^{61} - 2 q^{65} - 10 q^{67} + 8 q^{71} + 10 q^{77} - 22 q^{79} - 6 q^{83} + 32 q^{89} + 52 q^{91} - 8 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 −0.500000 0.866025i 0 −1.72474 + 2.98735i 0 0 0
289.2 0 0 0 −0.500000 0.866025i 0 0.724745 1.25529i 0 0 0
577.1 0 0 0 −0.500000 + 0.866025i 0 −1.72474 2.98735i 0 0 0
577.2 0 0 0 −0.500000 + 0.866025i 0 0.724745 + 1.25529i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.i.c 4
3.b odd 2 1 288.2.i.e yes 4
4.b odd 2 1 864.2.i.e 4
8.b even 2 1 1728.2.i.k 4
8.d odd 2 1 1728.2.i.m 4
9.c even 3 1 inner 864.2.i.c 4
9.c even 3 1 2592.2.a.s 2
9.d odd 6 1 288.2.i.e yes 4
9.d odd 6 1 2592.2.a.n 2
12.b even 2 1 288.2.i.c 4
24.f even 2 1 576.2.i.m 4
24.h odd 2 1 576.2.i.i 4
36.f odd 6 1 864.2.i.e 4
36.f odd 6 1 2592.2.a.o 2
36.h even 6 1 288.2.i.c 4
36.h even 6 1 2592.2.a.j 2
72.j odd 6 1 576.2.i.i 4
72.j odd 6 1 5184.2.a.by 2
72.l even 6 1 576.2.i.m 4
72.l even 6 1 5184.2.a.bu 2
72.n even 6 1 1728.2.i.k 4
72.n even 6 1 5184.2.a.bn 2
72.p odd 6 1 1728.2.i.m 4
72.p odd 6 1 5184.2.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.c 4 12.b even 2 1
288.2.i.c 4 36.h even 6 1
288.2.i.e yes 4 3.b odd 2 1
288.2.i.e yes 4 9.d odd 6 1
576.2.i.i 4 24.h odd 2 1
576.2.i.i 4 72.j odd 6 1
576.2.i.m 4 24.f even 2 1
576.2.i.m 4 72.l even 6 1
864.2.i.c 4 1.a even 1 1 trivial
864.2.i.c 4 9.c even 3 1 inner
864.2.i.e 4 4.b odd 2 1
864.2.i.e 4 36.f odd 6 1
1728.2.i.k 4 8.b even 2 1
1728.2.i.k 4 72.n even 6 1
1728.2.i.m 4 8.d odd 2 1
1728.2.i.m 4 72.p odd 6 1
2592.2.a.j 2 36.h even 6 1
2592.2.a.n 2 9.d odd 6 1
2592.2.a.o 2 36.f odd 6 1
2592.2.a.s 2 9.c even 3 1
5184.2.a.bj 2 72.p odd 6 1
5184.2.a.bn 2 72.n even 6 1
5184.2.a.bu 2 72.l even 6 1
5184.2.a.by 2 72.j odd 6 1

Hecke kernels

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

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 2T_{7}^{3} + 9T_{7}^{2} - 10T_{7} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + 9 T^{2} - 10 T + 25 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529 \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 33 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$29$ \( T^{4} - 10 T^{3} + 99 T^{2} - 10 T + 1 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + 81 T^{2} + \cdots + 361 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 14 T^{3} + 171 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + 129 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + 57 T^{2} + \cdots + 2809 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 14 T^{3} + 201 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} + 51 T^{2} + 90 T + 225 \) Copy content Toggle raw display
$67$ \( T^{4} + 10 T^{3} + 129 T^{2} + \cdots + 841 \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 92)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 22 T^{3} + 369 T^{2} + \cdots + 13225 \) Copy content Toggle raw display
$83$ \( T^{4} + 6 T^{3} + 33 T^{2} + 18 T + 9 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16 T + 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 2 T^{3} + 27 T^{2} - 46 T + 529 \) Copy content Toggle raw display
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