Properties

Label 288.2.c.a
Level $288$
Weight $2$
Character orbit 288.c
Analytic conductor $2.300$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.29969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} +O(q^{10})\) \( q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{11} + 4 q^{13} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{17} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{29} + 4 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{35} -6 q^{37} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{41} -8 \zeta_{8}^{2} q^{43} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} -9 q^{49} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{53} -8 \zeta_{8}^{2} q^{55} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{59} -2 q^{61} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} -8 \zeta_{8}^{2} q^{67} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{71} + ( 16 \zeta_{8} + 16 \zeta_{8}^{3} ) q^{77} -4 \zeta_{8}^{2} q^{79} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{83} -6 q^{85} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{89} + 16 \zeta_{8}^{2} q^{91} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{13} + 12q^{25} - 24q^{37} - 36q^{49} - 8q^{61} - 24q^{85} - 32q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0 0 0 1.41421i 0 4.00000i 0 0 0
287.2 0 0 0 1.41421i 0 4.00000i 0 0 0
287.3 0 0 0 1.41421i 0 4.00000i 0 0 0
287.4 0 0 0 1.41421i 0 4.00000i 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
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 288.2.c.a 4
3.b odd 2 1 inner 288.2.c.a 4
4.b odd 2 1 inner 288.2.c.a 4
5.b even 2 1 7200.2.h.d 4
5.c odd 4 1 7200.2.o.a 4
5.c odd 4 1 7200.2.o.n 4
8.b even 2 1 576.2.c.c 4
8.d odd 2 1 576.2.c.c 4
9.c even 3 2 2592.2.s.d 8
9.d odd 6 2 2592.2.s.d 8
12.b even 2 1 inner 288.2.c.a 4
15.d odd 2 1 7200.2.h.d 4
15.e even 4 1 7200.2.o.a 4
15.e even 4 1 7200.2.o.n 4
16.e even 4 1 2304.2.f.c 4
16.e even 4 1 2304.2.f.e 4
16.f odd 4 1 2304.2.f.c 4
16.f odd 4 1 2304.2.f.e 4
20.d odd 2 1 7200.2.h.d 4
20.e even 4 1 7200.2.o.a 4
20.e even 4 1 7200.2.o.n 4
24.f even 2 1 576.2.c.c 4
24.h odd 2 1 576.2.c.c 4
36.f odd 6 2 2592.2.s.d 8
36.h even 6 2 2592.2.s.d 8
48.i odd 4 1 2304.2.f.c 4
48.i odd 4 1 2304.2.f.e 4
48.k even 4 1 2304.2.f.c 4
48.k even 4 1 2304.2.f.e 4
60.h even 2 1 7200.2.h.d 4
60.l odd 4 1 7200.2.o.a 4
60.l odd 4 1 7200.2.o.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.c.a 4 1.a even 1 1 trivial
288.2.c.a 4 3.b odd 2 1 inner
288.2.c.a 4 4.b odd 2 1 inner
288.2.c.a 4 12.b even 2 1 inner
576.2.c.c 4 8.b even 2 1
576.2.c.c 4 8.d odd 2 1
576.2.c.c 4 24.f even 2 1
576.2.c.c 4 24.h odd 2 1
2304.2.f.c 4 16.e even 4 1
2304.2.f.c 4 16.f odd 4 1
2304.2.f.c 4 48.i odd 4 1
2304.2.f.c 4 48.k even 4 1
2304.2.f.e 4 16.e even 4 1
2304.2.f.e 4 16.f odd 4 1
2304.2.f.e 4 48.i odd 4 1
2304.2.f.e 4 48.k even 4 1
2592.2.s.d 8 9.c even 3 2
2592.2.s.d 8 9.d odd 6 2
2592.2.s.d 8 36.f odd 6 2
2592.2.s.d 8 36.h even 6 2
7200.2.h.d 4 5.b even 2 1
7200.2.h.d 4 15.d odd 2 1
7200.2.h.d 4 20.d odd 2 1
7200.2.h.d 4 60.h even 2 1
7200.2.o.a 4 5.c odd 4 1
7200.2.o.a 4 15.e even 4 1
7200.2.o.a 4 20.e even 4 1
7200.2.o.a 4 60.l odd 4 1
7200.2.o.n 4 5.c odd 4 1
7200.2.o.n 4 15.e even 4 1
7200.2.o.n 4 20.e even 4 1
7200.2.o.n 4 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(288, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 2 + T^{2} )^{2} \)
$7$ \( ( 16 + T^{2} )^{2} \)
$11$ \( ( -32 + T^{2} )^{2} \)
$13$ \( ( -4 + T )^{4} \)
$17$ \( ( 18 + T^{2} )^{2} \)
$19$ \( T^{4} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( 2 + T^{2} )^{2} \)
$31$ \( ( 16 + T^{2} )^{2} \)
$37$ \( ( 6 + T )^{4} \)
$41$ \( ( 98 + T^{2} )^{2} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( ( -32 + T^{2} )^{2} \)
$53$ \( ( 18 + T^{2} )^{2} \)
$59$ \( ( -128 + T^{2} )^{2} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -32 + T^{2} )^{2} \)
$73$ \( T^{4} \)
$79$ \( ( 16 + T^{2} )^{2} \)
$83$ \( ( -32 + T^{2} )^{2} \)
$89$ \( ( 18 + T^{2} )^{2} \)
$97$ \( ( 8 + T )^{4} \)
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