Properties

Label 4608.2.d.e
Level $4608$
Weight $2$
Character orbit 4608.d
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_1 q^{7} - \beta_{3} q^{11} + 2 \beta_{2} q^{13} - 4 q^{17} - 2 \beta_{3} q^{19} - 4 \beta_1 q^{23} - q^{25} - \beta_{2} q^{29} + \beta_1 q^{31} - \beta_{3} q^{35} - 2 \beta_{2} q^{37} + 4 q^{41} - 2 \beta_{3} q^{43} - 4 \beta_1 q^{47} - 5 q^{49} - 3 \beta_{2} q^{53} - 6 \beta_1 q^{55} + 4 \beta_{3} q^{59} + 2 \beta_{2} q^{61} - 12 q^{65} - 8 \beta_1 q^{71} + 4 q^{73} + 2 \beta_{2} q^{77} - 5 \beta_1 q^{79} + 3 \beta_{3} q^{83} - 4 \beta_{2} q^{85} + 16 q^{89} - 2 \beta_{3} q^{91} - 12 \beta_1 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{17} - 4 q^{25} + 16 q^{41} - 20 q^{49} - 48 q^{65} + 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2305.1
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0 0 0 2.44949i 0 −1.41421 0 0 0
2305.2 0 0 0 2.44949i 0 1.41421 0 0 0
2305.3 0 0 0 2.44949i 0 −1.41421 0 0 0
2305.4 0 0 0 2.44949i 0 1.41421 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
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 4608.2.d.e 4
3.b odd 2 1 4608.2.d.m 4
4.b odd 2 1 inner 4608.2.d.e 4
8.b even 2 1 inner 4608.2.d.e 4
8.d odd 2 1 inner 4608.2.d.e 4
12.b even 2 1 4608.2.d.m 4
16.e even 4 2 4608.2.a.v 4
16.f odd 4 2 4608.2.a.v 4
24.f even 2 1 4608.2.d.m 4
24.h odd 2 1 4608.2.d.m 4
48.i odd 4 2 4608.2.a.z yes 4
48.k even 4 2 4608.2.a.z yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.v 4 16.e even 4 2
4608.2.a.v 4 16.f odd 4 2
4608.2.a.z yes 4 48.i odd 4 2
4608.2.a.z yes 4 48.k even 4 2
4608.2.d.e 4 1.a even 1 1 trivial
4608.2.d.e 4 4.b odd 2 1 inner
4608.2.d.e 4 8.b even 2 1 inner
4608.2.d.e 4 8.d odd 2 1 inner
4608.2.d.m 4 3.b odd 2 1
4608.2.d.m 4 12.b even 2 1
4608.2.d.m 4 24.f even 2 1
4608.2.d.m 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 \(S_{2}^{\mathrm{new}}(4608, [\chi])\):

\( T_{5}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{17} + 4 \) Copy content Toggle raw display
\( T_{23}^{2} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$17$ \( (T + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$41$ \( (T - 4)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$89$ \( (T - 16)^{4} \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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