Properties

Label 4608.2.d.n
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: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1536)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - 3 \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - 3 \beta_{3} q^{7} + 3 \beta_1 q^{11} + 4 \beta_{2} q^{13} + 6 q^{17} + 2 \beta_1 q^{19} - 2 \beta_{3} q^{23} + 3 q^{25} - \beta_{2} q^{29} + \beta_{3} q^{31} - 3 \beta_1 q^{35} + 6 \beta_{2} q^{37} - 2 q^{41} + 2 \beta_{3} q^{47} + 11 q^{49} - 7 \beta_{2} q^{53} - 6 \beta_{3} q^{55} + 2 \beta_1 q^{59} - 6 \beta_{2} q^{61} - 8 q^{65} + 4 \beta_1 q^{67} + 2 \beta_{3} q^{71} - 8 q^{73} - 18 \beta_{2} q^{77} - 9 \beta_{3} q^{79} - \beta_1 q^{83} + 6 \beta_{2} q^{85} + 2 q^{89} - 12 \beta_1 q^{91} - 4 \beta_{3} q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{17} + 12 q^{25} - 8 q^{41} + 44 q^{49} - 32 q^{65} - 32 q^{73} + 8 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) 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
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) 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 \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

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

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 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0 0 0 1.41421i 0 −4.24264 0 0 0
2305.2 0 0 0 1.41421i 0 4.24264 0 0 0
2305.3 0 0 0 1.41421i 0 −4.24264 0 0 0
2305.4 0 0 0 1.41421i 0 4.24264 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.n 4
3.b odd 2 1 1536.2.d.c 4
4.b odd 2 1 inner 4608.2.d.n 4
8.b even 2 1 inner 4608.2.d.n 4
8.d odd 2 1 inner 4608.2.d.n 4
12.b even 2 1 1536.2.d.c 4
16.e even 4 1 4608.2.a.g 2
16.e even 4 1 4608.2.a.l 2
16.f odd 4 1 4608.2.a.g 2
16.f odd 4 1 4608.2.a.l 2
24.f even 2 1 1536.2.d.c 4
24.h odd 2 1 1536.2.d.c 4
48.i odd 4 1 1536.2.a.d 2
48.i odd 4 1 1536.2.a.i yes 2
48.k even 4 1 1536.2.a.d 2
48.k even 4 1 1536.2.a.i yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.d 2 48.i odd 4 1
1536.2.a.d 2 48.k even 4 1
1536.2.a.i yes 2 48.i odd 4 1
1536.2.a.i yes 2 48.k even 4 1
1536.2.d.c 4 3.b odd 2 1
1536.2.d.c 4 12.b even 2 1
1536.2.d.c 4 24.f even 2 1
1536.2.d.c 4 24.h odd 2 1
4608.2.a.g 2 16.e even 4 1
4608.2.a.g 2 16.f odd 4 1
4608.2.a.l 2 16.e even 4 1
4608.2.a.l 2 16.f odd 4 1
4608.2.d.n 4 1.a even 1 1 trivial
4608.2.d.n 4 4.b odd 2 1 inner
4608.2.d.n 4 8.b even 2 1 inner
4608.2.d.n 4 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4608, [\chi])\):

\( T_{5}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 18 \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{23}^{2} - 8 \) 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
$5$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( (T + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 8)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T - 2)^{4} \) Copy content Toggle raw display
$97$ \( (T - 2)^{4} \) Copy content Toggle raw display
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