Properties

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

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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: \( x^{4} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} - 2) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{3} - 2) q^{7} - q^{9} + 2 \beta_1 q^{11} - 2 \beta_{2} q^{13} + ( - \beta_{3} + 2) q^{15} + (4 \beta_{3} - 2) q^{17} + 4 \beta_{2} q^{19} + (\beta_{2} + 2 \beta_1) q^{21} + (2 \beta_{3} - 4) q^{23} + (4 \beta_{3} - 1) q^{25} + \beta_1 q^{27} + (\beta_{2} - 2 \beta_1) q^{29} + ( - \beta_{3} + 6) q^{31} + 2 q^{33} - 2 \beta_1 q^{35} + ( - 4 \beta_{2} - 4 \beta_1) q^{37} - 2 \beta_{3} q^{39} + (4 \beta_{3} + 6) q^{41} + (4 \beta_{2} - 4 \beta_1) q^{43} + (\beta_{2} - 2 \beta_1) q^{45} + (6 \beta_{3} + 4) q^{47} + (4 \beta_{3} - 1) q^{49} + ( - 4 \beta_{2} + 2 \beta_1) q^{51} + (7 \beta_{2} + 2 \beta_1) q^{53} + (2 \beta_{3} - 4) q^{55} + 4 \beta_{3} q^{57} - 4 \beta_1 q^{59} + (4 \beta_{2} + 4 \beta_1) q^{61} + (\beta_{3} + 2) q^{63} + (4 \beta_{3} - 4) q^{65} + 8 \beta_1 q^{67} + ( - 2 \beta_{2} + 4 \beta_1) q^{69} + (2 \beta_{3} - 12) q^{71} + (4 \beta_{3} - 4) q^{73} + ( - 4 \beta_{2} + \beta_1) q^{75} + ( - 2 \beta_{2} - 4 \beta_1) q^{77} + ( - 3 \beta_{3} + 10) q^{79} + q^{81} + ( - 8 \beta_{2} + 2 \beta_1) q^{83} + (10 \beta_{2} - 12 \beta_1) q^{85} + (\beta_{3} - 2) q^{87} - 2 q^{89} + (4 \beta_{2} + 4 \beta_1) q^{91} + (\beta_{2} - 6 \beta_1) q^{93} + ( - 8 \beta_{3} + 8) q^{95} + (8 \beta_{3} + 2) q^{97} - 2 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 4 q^{9} + 8 q^{15} - 8 q^{17} - 16 q^{23} - 4 q^{25} + 24 q^{31} + 8 q^{33} + 24 q^{41} + 16 q^{47} - 4 q^{49} - 16 q^{55} + 8 q^{63} - 16 q^{65} - 48 q^{71} - 16 q^{73} + 40 q^{79} + 4 q^{81} - 8 q^{87} - 8 q^{89} + 32 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

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 0.585786i 0 −3.41421 0 −1.00000 0
769.2 0 1.00000i 0 3.41421i 0 −0.585786 0 −1.00000 0
769.3 0 1.00000i 0 3.41421i 0 −0.585786 0 −1.00000 0
769.4 0 1.00000i 0 0.585786i 0 −3.41421 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
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 1536.2.d.a 4
3.b odd 2 1 4608.2.d.c 4
4.b odd 2 1 1536.2.d.f 4
8.b even 2 1 inner 1536.2.d.a 4
8.d odd 2 1 1536.2.d.f 4
12.b even 2 1 4608.2.d.o 4
16.e even 4 1 1536.2.a.b 2
16.e even 4 1 1536.2.a.l yes 2
16.f odd 4 1 1536.2.a.e yes 2
16.f odd 4 1 1536.2.a.g yes 2
24.f even 2 1 4608.2.d.o 4
24.h odd 2 1 4608.2.d.c 4
48.i odd 4 1 4608.2.a.e 2
48.i odd 4 1 4608.2.a.r 2
48.k even 4 1 4608.2.a.a 2
48.k even 4 1 4608.2.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.b 2 16.e even 4 1
1536.2.a.e yes 2 16.f odd 4 1
1536.2.a.g yes 2 16.f odd 4 1
1536.2.a.l yes 2 16.e even 4 1
1536.2.d.a 4 1.a even 1 1 trivial
1536.2.d.a 4 8.b even 2 1 inner
1536.2.d.f 4 4.b odd 2 1
1536.2.d.f 4 8.d odd 2 1
4608.2.a.a 2 48.k even 4 1
4608.2.a.e 2 48.i odd 4 1
4608.2.a.n 2 48.k even 4 1
4608.2.a.r 2 48.i odd 4 1
4608.2.d.c 4 3.b odd 2 1
4608.2.d.c 4 24.h odd 2 1
4608.2.d.o 4 12.b even 2 1
4608.2.d.o 4 24.f even 2 1

Hecke kernels

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

\( T_{5}^{4} + 12T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 8T_{23} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 12T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} - 12 T + 34)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 204T^{2} + 8836 \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 96T^{2} + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 24 T + 136)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 82)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 264 T^{2} + 15376 \) Copy content Toggle raw display
$89$ \( (T + 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
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