Properties

Label 288.4.d.a
Level $288$
Weight $4$
Character orbit 288.d
Analytic conductor $16.993$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta q^{5} + 8 q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta q^{5} + 8 q^{7} + 3 \beta q^{11} - 10 \beta q^{13} + 14 q^{17} - 7 \beta q^{19} - 152 q^{23} + 13 q^{25} - 30 \beta q^{29} - 224 q^{31} - 16 \beta q^{35} - 46 \beta q^{37} + 70 q^{41} - 83 \beta q^{43} + 336 q^{47} - 279 q^{49} + 6 \beta q^{53} + 168 q^{55} - 101 \beta q^{59} - 18 \beta q^{61} - 560 q^{65} + 33 \beta q^{67} - 72 q^{71} - 294 q^{73} + 24 \beta q^{77} + 464 q^{79} + 103 \beta q^{83} - 28 \beta q^{85} - 266 q^{89} - 80 \beta q^{91} - 392 q^{95} + 994 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} + 26 q^{25} - 448 q^{31} + 140 q^{41} + 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} - 588 q^{73} + 928 q^{79} - 532 q^{89} - 784 q^{95} + 1988 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 10.5830i 0 8.00000 0 0 0
145.2 0 0 0 10.5830i 0 8.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.4.d.a 2
3.b odd 2 1 32.4.b.a 2
4.b odd 2 1 72.4.d.b 2
8.b even 2 1 inner 288.4.d.a 2
8.d odd 2 1 72.4.d.b 2
12.b even 2 1 8.4.b.a 2
15.d odd 2 1 800.4.d.a 2
15.e even 4 2 800.4.f.a 4
16.e even 4 2 2304.4.a.v 2
16.f odd 4 2 2304.4.a.bn 2
24.f even 2 1 8.4.b.a 2
24.h odd 2 1 32.4.b.a 2
48.i odd 4 2 256.4.a.j 2
48.k even 4 2 256.4.a.l 2
60.h even 2 1 200.4.d.a 2
60.l odd 4 2 200.4.f.a 4
120.i odd 2 1 800.4.d.a 2
120.m even 2 1 200.4.d.a 2
120.q odd 4 2 200.4.f.a 4
120.w even 4 2 800.4.f.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 12.b even 2 1
8.4.b.a 2 24.f even 2 1
32.4.b.a 2 3.b odd 2 1
32.4.b.a 2 24.h odd 2 1
72.4.d.b 2 4.b odd 2 1
72.4.d.b 2 8.d odd 2 1
200.4.d.a 2 60.h even 2 1
200.4.d.a 2 120.m even 2 1
200.4.f.a 4 60.l odd 4 2
200.4.f.a 4 120.q odd 4 2
256.4.a.j 2 48.i odd 4 2
256.4.a.l 2 48.k even 4 2
288.4.d.a 2 1.a even 1 1 trivial
288.4.d.a 2 8.b even 2 1 inner
800.4.d.a 2 15.d odd 2 1
800.4.d.a 2 120.i odd 2 1
800.4.f.a 4 15.e even 4 2
800.4.f.a 4 120.w even 4 2
2304.4.a.v 2 16.e even 4 2
2304.4.a.bn 2 16.f odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 112 \) acting on \(S_{4}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 112 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 252 \) Copy content Toggle raw display
$13$ \( T^{2} + 2800 \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 1372 \) Copy content Toggle raw display
$23$ \( (T + 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 25200 \) Copy content Toggle raw display
$31$ \( (T + 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 59248 \) Copy content Toggle raw display
$41$ \( (T - 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 192892 \) Copy content Toggle raw display
$47$ \( (T - 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 1008 \) Copy content Toggle raw display
$59$ \( T^{2} + 285628 \) Copy content Toggle raw display
$61$ \( T^{2} + 9072 \) Copy content Toggle raw display
$67$ \( T^{2} + 30492 \) Copy content Toggle raw display
$71$ \( (T + 72)^{2} \) Copy content Toggle raw display
$73$ \( (T + 294)^{2} \) Copy content Toggle raw display
$79$ \( (T - 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 297052 \) Copy content Toggle raw display
$89$ \( (T + 266)^{2} \) Copy content Toggle raw display
$97$ \( (T - 994)^{2} \) Copy content Toggle raw display
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