Properties

Label 576.2.k.a
Level $576$
Weight $2$
Character orbit 576.k
Analytic conductor $4.599$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.k (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{5} + 2 i q^{7} +O(q^{10})\) \( q + ( 1 + i ) q^{5} + 2 i q^{7} + ( 1 + i ) q^{11} + ( -1 + i ) q^{13} + 2 q^{17} + ( -3 + 3 i ) q^{19} + 6 i q^{23} -3 i q^{25} + ( -3 + 3 i ) q^{29} + 8 q^{31} + ( -2 + 2 i ) q^{35} + ( 3 + 3 i ) q^{37} + ( -5 - 5 i ) q^{43} + 8 q^{47} + 3 q^{49} + ( 5 + 5 i ) q^{53} + 2 i q^{55} + ( -3 - 3 i ) q^{59} + ( -9 + 9 i ) q^{61} -2 q^{65} + ( 5 - 5 i ) q^{67} -10 i q^{71} -4 i q^{73} + ( -2 + 2 i ) q^{77} + ( -1 + i ) q^{83} + ( 2 + 2 i ) q^{85} -4 i q^{89} + ( -2 - 2 i ) q^{91} -6 q^{95} -2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 2q^{11} - 2q^{13} + 4q^{17} - 6q^{19} - 6q^{29} + 16q^{31} - 4q^{35} + 6q^{37} - 10q^{43} + 16q^{47} + 6q^{49} + 10q^{53} - 6q^{59} - 18q^{61} - 4q^{65} + 10q^{67} - 4q^{77} - 2q^{83} + 4q^{85} - 4q^{91} - 12q^{95} - 4q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(1\) \(i\)

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
1.00000i
1.00000i
0 0 0 1.00000 1.00000i 0 2.00000i 0 0 0
433.1 0 0 0 1.00000 + 1.00000i 0 2.00000i 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
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.2.k.a 2
3.b odd 2 1 64.2.e.a 2
4.b odd 2 1 144.2.k.a 2
8.b even 2 1 1152.2.k.a 2
8.d odd 2 1 1152.2.k.b 2
12.b even 2 1 16.2.e.a 2
15.d odd 2 1 1600.2.l.a 2
15.e even 4 1 1600.2.q.a 2
15.e even 4 1 1600.2.q.b 2
16.e even 4 1 inner 576.2.k.a 2
16.e even 4 1 1152.2.k.a 2
16.f odd 4 1 144.2.k.a 2
16.f odd 4 1 1152.2.k.b 2
24.f even 2 1 128.2.e.b 2
24.h odd 2 1 128.2.e.a 2
32.g even 8 2 9216.2.a.s 2
32.h odd 8 2 9216.2.a.d 2
48.i odd 4 1 64.2.e.a 2
48.i odd 4 1 128.2.e.a 2
48.k even 4 1 16.2.e.a 2
48.k even 4 1 128.2.e.b 2
60.h even 2 1 400.2.l.c 2
60.l odd 4 1 400.2.q.a 2
60.l odd 4 1 400.2.q.b 2
84.h odd 2 1 784.2.m.b 2
84.j odd 6 2 784.2.x.c 4
84.n even 6 2 784.2.x.f 4
96.o even 8 2 1024.2.a.b 2
96.o even 8 2 1024.2.b.e 2
96.p odd 8 2 1024.2.a.e 2
96.p odd 8 2 1024.2.b.b 2
240.t even 4 1 400.2.l.c 2
240.z odd 4 1 400.2.q.b 2
240.bb even 4 1 1600.2.q.a 2
240.bd odd 4 1 400.2.q.a 2
240.bf even 4 1 1600.2.q.b 2
240.bm odd 4 1 1600.2.l.a 2
336.v odd 4 1 784.2.m.b 2
336.br odd 12 2 784.2.x.c 4
336.bu even 12 2 784.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.2.e.a 2 12.b even 2 1
16.2.e.a 2 48.k even 4 1
64.2.e.a 2 3.b odd 2 1
64.2.e.a 2 48.i odd 4 1
128.2.e.a 2 24.h odd 2 1
128.2.e.a 2 48.i odd 4 1
128.2.e.b 2 24.f even 2 1
128.2.e.b 2 48.k even 4 1
144.2.k.a 2 4.b odd 2 1
144.2.k.a 2 16.f odd 4 1
400.2.l.c 2 60.h even 2 1
400.2.l.c 2 240.t even 4 1
400.2.q.a 2 60.l odd 4 1
400.2.q.a 2 240.bd odd 4 1
400.2.q.b 2 60.l odd 4 1
400.2.q.b 2 240.z odd 4 1
576.2.k.a 2 1.a even 1 1 trivial
576.2.k.a 2 16.e even 4 1 inner
784.2.m.b 2 84.h odd 2 1
784.2.m.b 2 336.v odd 4 1
784.2.x.c 4 84.j odd 6 2
784.2.x.c 4 336.br odd 12 2
784.2.x.f 4 84.n even 6 2
784.2.x.f 4 336.bu even 12 2
1024.2.a.b 2 96.o even 8 2
1024.2.a.e 2 96.p odd 8 2
1024.2.b.b 2 96.p odd 8 2
1024.2.b.e 2 96.o even 8 2
1152.2.k.a 2 8.b even 2 1
1152.2.k.a 2 16.e even 4 1
1152.2.k.b 2 8.d odd 2 1
1152.2.k.b 2 16.f odd 4 1
1600.2.l.a 2 15.d odd 2 1
1600.2.l.a 2 240.bm odd 4 1
1600.2.q.a 2 15.e even 4 1
1600.2.q.a 2 240.bb even 4 1
1600.2.q.b 2 15.e even 4 1
1600.2.q.b 2 240.bf even 4 1
9216.2.a.d 2 32.h odd 8 2
9216.2.a.s 2 32.g even 8 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 2 T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(576, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 - 2 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( 2 - 2 T + T^{2} \)
$13$ \( 2 + 2 T + T^{2} \)
$17$ \( ( -2 + T )^{2} \)
$19$ \( 18 + 6 T + T^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( 18 + 6 T + T^{2} \)
$31$ \( ( -8 + T )^{2} \)
$37$ \( 18 - 6 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 50 + 10 T + T^{2} \)
$47$ \( ( -8 + T )^{2} \)
$53$ \( 50 - 10 T + T^{2} \)
$59$ \( 18 + 6 T + T^{2} \)
$61$ \( 162 + 18 T + T^{2} \)
$67$ \( 50 - 10 T + T^{2} \)
$71$ \( 100 + T^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( 2 + 2 T + T^{2} \)
$89$ \( 16 + T^{2} \)
$97$ \( ( 2 + T )^{2} \)
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