Properties

Label 288.3.e.c
Level $288$
Weight $3$
Character orbit 288.e
Analytic conductor $7.847$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} +O(q^{10})\) \( q + \beta q^{5} + 24 q^{13} + 23 \beta q^{17} + 23 q^{25} -\beta q^{29} + 70 q^{37} + 49 \beta q^{41} -49 q^{49} -73 \beta q^{53} -22 q^{61} + 24 \beta q^{65} -96 q^{73} -46 q^{85} -119 \beta q^{89} -144 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 48q^{13} + 46q^{25} + 140q^{37} - 98q^{49} - 44q^{61} - 192q^{73} - 92q^{85} - 288q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.41421i
1.41421i
0 0 0 1.41421i 0 0 0 0 0
161.2 0 0 0 1.41421i 0 0 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.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 288.3.e.c 2
3.b odd 2 1 inner 288.3.e.c 2
4.b odd 2 1 CM 288.3.e.c 2
8.b even 2 1 576.3.e.d 2
8.d odd 2 1 576.3.e.d 2
12.b even 2 1 inner 288.3.e.c 2
16.e even 4 2 2304.3.h.d 4
16.f odd 4 2 2304.3.h.d 4
24.f even 2 1 576.3.e.d 2
24.h odd 2 1 576.3.e.d 2
48.i odd 4 2 2304.3.h.d 4
48.k even 4 2 2304.3.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.c 2 1.a even 1 1 trivial
288.3.e.c 2 3.b odd 2 1 inner
288.3.e.c 2 4.b odd 2 1 CM
288.3.e.c 2 12.b even 2 1 inner
576.3.e.d 2 8.b even 2 1
576.3.e.d 2 8.d odd 2 1
576.3.e.d 2 24.f even 2 1
576.3.e.d 2 24.h odd 2 1
2304.3.h.d 4 16.e even 4 2
2304.3.h.d 4 16.f odd 4 2
2304.3.h.d 4 48.i odd 4 2
2304.3.h.d 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( -24 + T )^{2} \)
$17$ \( 1058 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 2 + T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -70 + T )^{2} \)
$41$ \( 4802 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 10658 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( 22 + T )^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 96 + T )^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 28322 + T^{2} \)
$97$ \( ( 144 + T )^{2} \)
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