Properties

Label 4608.2.k.g
Level $4608$
Weight $2$
Character orbit 4608.k
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - i ) q^{5} +O(q^{10})\) \( q + ( -1 - i ) q^{5} + ( -5 + 5 i ) q^{13} -8 q^{17} -3 i q^{25} + ( 3 - 3 i ) q^{29} + ( 7 + 7 i ) q^{37} -8 i q^{41} + 7 q^{49} + ( 9 + 9 i ) q^{53} + ( 11 - 11 i ) q^{61} + 10 q^{65} -6 i q^{73} + ( 8 + 8 i ) q^{85} + 10 i q^{89} -8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{5} - 10 q^{13} - 16 q^{17} + 6 q^{29} + 14 q^{37} + 14 q^{49} + 18 q^{53} + 22 q^{61} + 20 q^{65} + 16 q^{85} - 16 q^{97} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
1.00000i
1.00000i
0 0 0 −1.00000 + 1.00000i 0 0 0 0 0
3457.1 0 0 0 −1.00000 1.00000i 0 0 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 CM by \(\Q(\sqrt{-1}) \)
16.e even 4 1 inner
16.f odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.k.g 2
3.b odd 2 1 512.2.e.e yes 2
4.b odd 2 1 CM 4608.2.k.g 2
8.b even 2 1 4608.2.k.r 2
8.d odd 2 1 4608.2.k.r 2
12.b even 2 1 512.2.e.e yes 2
16.e even 4 1 inner 4608.2.k.g 2
16.e even 4 1 4608.2.k.r 2
16.f odd 4 1 inner 4608.2.k.g 2
16.f odd 4 1 4608.2.k.r 2
24.f even 2 1 512.2.e.d 2
24.h odd 2 1 512.2.e.d 2
32.g even 8 2 9216.2.a.p 2
32.h odd 8 2 9216.2.a.p 2
48.i odd 4 1 512.2.e.d 2
48.i odd 4 1 512.2.e.e yes 2
48.k even 4 1 512.2.e.d 2
48.k even 4 1 512.2.e.e yes 2
96.o even 8 2 1024.2.a.c 2
96.o even 8 2 1024.2.b.c 2
96.p odd 8 2 1024.2.a.c 2
96.p odd 8 2 1024.2.b.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.e.d 2 24.f even 2 1
512.2.e.d 2 24.h odd 2 1
512.2.e.d 2 48.i odd 4 1
512.2.e.d 2 48.k even 4 1
512.2.e.e yes 2 3.b odd 2 1
512.2.e.e yes 2 12.b even 2 1
512.2.e.e yes 2 48.i odd 4 1
512.2.e.e yes 2 48.k even 4 1
1024.2.a.c 2 96.o even 8 2
1024.2.a.c 2 96.p odd 8 2
1024.2.b.c 2 96.o even 8 2
1024.2.b.c 2 96.p odd 8 2
4608.2.k.g 2 1.a even 1 1 trivial
4608.2.k.g 2 4.b odd 2 1 CM
4608.2.k.g 2 16.e even 4 1 inner
4608.2.k.g 2 16.f odd 4 1 inner
4608.2.k.r 2 8.b even 2 1
4608.2.k.r 2 8.d odd 2 1
4608.2.k.r 2 16.e even 4 1
4608.2.k.r 2 16.f odd 4 1
9216.2.a.p 2 32.g even 8 2
9216.2.a.p 2 32.h odd 8 2

Hecke kernels

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

\( T_{5}^{2} + 2 T_{5} + 2 \)
\( T_{7} \)
\( T_{11} \)
\( T_{13}^{2} + 10 T_{13} + 50 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 + 2 T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( T^{2} \)
$13$ \( 50 + 10 T + T^{2} \)
$17$ \( ( 8 + T )^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( T^{2} \)
$37$ \( 98 - 14 T + T^{2} \)
$41$ \( 64 + T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 162 - 18 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 242 - 22 T + T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 36 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( T^{2} \)
$89$ \( 100 + T^{2} \)
$97$ \( ( 8 + T )^{2} \)
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