Properties

Label 1536.2.d.d
Level $1536$
Weight $2$
Character orbit 1536.d
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q -\zeta_{8}^{2} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} +O(q^{10})\) \( q -\zeta_{8}^{2} q^{3} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{5} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{7} - q^{9} + 2 \zeta_{8}^{2} q^{11} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} + 2 q^{17} -4 \zeta_{8}^{2} q^{19} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{21} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + 3 q^{25} + \zeta_{8}^{2} q^{27} + ( 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{29} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{31} + 2 q^{33} -2 \zeta_{8}^{2} q^{35} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{37} -6 q^{41} + 8 \zeta_{8}^{2} q^{43} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{45} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{47} -5 q^{49} -2 \zeta_{8}^{2} q^{51} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{53} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{55} -4 q^{57} + 12 \zeta_{8}^{2} q^{59} + ( 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{61} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{63} -8 \zeta_{8}^{2} q^{67} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{69} + ( 10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{71} + 8 q^{73} -3 \zeta_{8}^{2} q^{75} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{77} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + q^{81} -6 \zeta_{8}^{2} q^{83} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{85} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{87} -2 q^{89} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{95} -14 q^{97} -2 \zeta_{8}^{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} + 8q^{17} + 12q^{25} + 8q^{33} - 24q^{41} - 20q^{49} - 16q^{57} + 32q^{73} + 4q^{81} - 8q^{89} - 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
769.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 1.00000i 0 1.41421i 0 1.41421 0 −1.00000 0
769.2 0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 0
769.3 0 1.00000i 0 1.41421i 0 −1.41421 0 −1.00000 0
769.4 0 1.00000i 0 1.41421i 0 1.41421 0 −1.00000 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 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1536.2.d.d 4
3.b odd 2 1 4608.2.d.g 4
4.b odd 2 1 inner 1536.2.d.d 4
8.b even 2 1 inner 1536.2.d.d 4
8.d odd 2 1 inner 1536.2.d.d 4
12.b even 2 1 4608.2.d.g 4
16.e even 4 1 1536.2.a.c 2
16.e even 4 1 1536.2.a.j yes 2
16.f odd 4 1 1536.2.a.c 2
16.f odd 4 1 1536.2.a.j yes 2
24.f even 2 1 4608.2.d.g 4
24.h odd 2 1 4608.2.d.g 4
48.i odd 4 1 4608.2.a.h 2
48.i odd 4 1 4608.2.a.j 2
48.k even 4 1 4608.2.a.h 2
48.k even 4 1 4608.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.c 2 16.e even 4 1
1536.2.a.c 2 16.f odd 4 1
1536.2.a.j yes 2 16.e even 4 1
1536.2.a.j yes 2 16.f odd 4 1
1536.2.d.d 4 1.a even 1 1 trivial
1536.2.d.d 4 4.b odd 2 1 inner
1536.2.d.d 4 8.b even 2 1 inner
1536.2.d.d 4 8.d odd 2 1 inner
4608.2.a.h 2 48.i odd 4 1
4608.2.a.h 2 48.k even 4 1
4608.2.a.j 2 48.i odd 4 1
4608.2.a.j 2 48.k even 4 1
4608.2.d.g 4 3.b odd 2 1
4608.2.d.g 4 12.b even 2 1
4608.2.d.g 4 24.f even 2 1
4608.2.d.g 4 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1536, [\chi])\):

\( T_{5}^{2} + 2 \)
\( T_{7}^{2} - 2 \)
\( T_{23}^{2} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 2 + T^{2} )^{2} \)
$7$ \( ( -2 + T^{2} )^{2} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( -2 + T )^{4} \)
$19$ \( ( 16 + T^{2} )^{2} \)
$23$ \( ( -8 + T^{2} )^{2} \)
$29$ \( ( 98 + T^{2} )^{2} \)
$31$ \( ( -50 + T^{2} )^{2} \)
$37$ \( ( 72 + T^{2} )^{2} \)
$41$ \( ( 6 + T )^{4} \)
$43$ \( ( 64 + T^{2} )^{2} \)
$47$ \( ( -8 + T^{2} )^{2} \)
$53$ \( ( 2 + T^{2} )^{2} \)
$59$ \( ( 144 + T^{2} )^{2} \)
$61$ \( ( 200 + T^{2} )^{2} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( -200 + T^{2} )^{2} \)
$73$ \( ( -8 + T )^{4} \)
$79$ \( ( -18 + T^{2} )^{2} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( 2 + T )^{4} \)
$97$ \( ( 14 + T )^{4} \)
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