Properties

Label 576.3.e.f
Level $576$
Weight $3$
Character orbit 576.e
Analytic conductor $15.695$
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: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.6948632272\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 4 q^{7} +O(q^{10})\) \( q + \beta q^{5} + 4 q^{7} + 4 \beta q^{11} -8 q^{13} -3 \beta q^{17} -16 q^{19} + 4 \beta q^{23} + 7 q^{25} -\beta q^{29} -44 q^{31} + 4 \beta q^{35} + 34 q^{37} + 11 \beta q^{41} -40 q^{43} + 20 \beta q^{47} -33 q^{49} -9 \beta q^{53} -72 q^{55} + 8 \beta q^{59} -50 q^{61} -8 \beta q^{65} + 8 q^{67} + 12 \beta q^{71} -16 q^{73} + 16 \beta q^{77} + 76 q^{79} + 28 \beta q^{83} + 54 q^{85} + 3 \beta q^{89} -32 q^{91} -16 \beta q^{95} + 176 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{7} + O(q^{10}) \) \( 2q + 8q^{7} - 16q^{13} - 32q^{19} + 14q^{25} - 88q^{31} + 68q^{37} - 80q^{43} - 66q^{49} - 144q^{55} - 100q^{61} + 16q^{67} - 32q^{73} + 152q^{79} + 108q^{85} - 64q^{91} + 352q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.41421i
1.41421i
0 0 0 4.24264i 0 4.00000 0 0 0
449.2 0 0 0 4.24264i 0 4.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 576.3.e.f 2
3.b odd 2 1 inner 576.3.e.f 2
4.b odd 2 1 576.3.e.c 2
8.b even 2 1 144.3.e.b 2
8.d odd 2 1 18.3.b.a 2
12.b even 2 1 576.3.e.c 2
16.e even 4 2 2304.3.h.c 4
16.f odd 4 2 2304.3.h.f 4
24.f even 2 1 18.3.b.a 2
24.h odd 2 1 144.3.e.b 2
40.e odd 2 1 450.3.d.f 2
40.f even 2 1 3600.3.l.d 2
40.i odd 4 2 3600.3.c.b 4
40.k even 4 2 450.3.b.b 4
48.i odd 4 2 2304.3.h.c 4
48.k even 4 2 2304.3.h.f 4
56.e even 2 1 882.3.b.a 2
56.k odd 6 2 882.3.s.b 4
56.m even 6 2 882.3.s.d 4
72.j odd 6 2 1296.3.q.f 4
72.l even 6 2 162.3.d.b 4
72.n even 6 2 1296.3.q.f 4
72.p odd 6 2 162.3.d.b 4
88.g even 2 1 2178.3.c.d 2
104.h odd 2 1 3042.3.c.e 2
104.m even 4 2 3042.3.d.a 4
120.i odd 2 1 3600.3.l.d 2
120.m even 2 1 450.3.d.f 2
120.q odd 4 2 450.3.b.b 4
120.w even 4 2 3600.3.c.b 4
168.e odd 2 1 882.3.b.a 2
168.v even 6 2 882.3.s.b 4
168.be odd 6 2 882.3.s.d 4
264.p odd 2 1 2178.3.c.d 2
312.h even 2 1 3042.3.c.e 2
312.w odd 4 2 3042.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 8.d odd 2 1
18.3.b.a 2 24.f even 2 1
144.3.e.b 2 8.b even 2 1
144.3.e.b 2 24.h odd 2 1
162.3.d.b 4 72.l even 6 2
162.3.d.b 4 72.p odd 6 2
450.3.b.b 4 40.k even 4 2
450.3.b.b 4 120.q odd 4 2
450.3.d.f 2 40.e odd 2 1
450.3.d.f 2 120.m even 2 1
576.3.e.c 2 4.b odd 2 1
576.3.e.c 2 12.b even 2 1
576.3.e.f 2 1.a even 1 1 trivial
576.3.e.f 2 3.b odd 2 1 inner
882.3.b.a 2 56.e even 2 1
882.3.b.a 2 168.e odd 2 1
882.3.s.b 4 56.k odd 6 2
882.3.s.b 4 168.v even 6 2
882.3.s.d 4 56.m even 6 2
882.3.s.d 4 168.be odd 6 2
1296.3.q.f 4 72.j odd 6 2
1296.3.q.f 4 72.n even 6 2
2178.3.c.d 2 88.g even 2 1
2178.3.c.d 2 264.p odd 2 1
2304.3.h.c 4 16.e even 4 2
2304.3.h.c 4 48.i odd 4 2
2304.3.h.f 4 16.f odd 4 2
2304.3.h.f 4 48.k even 4 2
3042.3.c.e 2 104.h odd 2 1
3042.3.c.e 2 312.h even 2 1
3042.3.d.a 4 104.m even 4 2
3042.3.d.a 4 312.w odd 4 2
3600.3.c.b 4 40.i odd 4 2
3600.3.c.b 4 120.w even 4 2
3600.3.l.d 2 40.f even 2 1
3600.3.l.d 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 18 \)
\( T_{7} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 18 + T^{2} \)
$7$ \( ( -4 + T )^{2} \)
$11$ \( 288 + T^{2} \)
$13$ \( ( 8 + T )^{2} \)
$17$ \( 162 + T^{2} \)
$19$ \( ( 16 + T )^{2} \)
$23$ \( 288 + T^{2} \)
$29$ \( 18 + T^{2} \)
$31$ \( ( 44 + T )^{2} \)
$37$ \( ( -34 + T )^{2} \)
$41$ \( 2178 + T^{2} \)
$43$ \( ( 40 + T )^{2} \)
$47$ \( 7200 + T^{2} \)
$53$ \( 1458 + T^{2} \)
$59$ \( 1152 + T^{2} \)
$61$ \( ( 50 + T )^{2} \)
$67$ \( ( -8 + T )^{2} \)
$71$ \( 2592 + T^{2} \)
$73$ \( ( 16 + T )^{2} \)
$79$ \( ( -76 + T )^{2} \)
$83$ \( 14112 + T^{2} \)
$89$ \( 162 + T^{2} \)
$97$ \( ( -176 + T )^{2} \)
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