Properties

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

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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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 + i ) q^{5} -4 i q^{7} +O(q^{10})\) \( q + ( 1 + i ) q^{5} -4 i q^{7} + ( 4 + 4 i ) q^{11} + ( 3 - 3 i ) q^{13} + 6 q^{17} + ( 4 - 4 i ) q^{19} + 8 i q^{23} -3 i q^{25} + ( 3 - 3 i ) q^{29} -4 q^{31} + ( 4 - 4 i ) q^{35} + ( 1 + i ) q^{37} + 2 i q^{41} + ( 4 + 4 i ) q^{43} -8 q^{47} -9 q^{49} + ( -7 - 7 i ) q^{53} + 8 i q^{55} + ( -3 + 3 i ) q^{61} + 6 q^{65} + ( -8 + 8 i ) q^{67} -10 i q^{73} + ( 16 - 16 i ) q^{77} + 12 q^{79} + ( -4 + 4 i ) q^{83} + ( 6 + 6 i ) q^{85} + 16 i q^{89} + ( -12 - 12 i ) q^{91} + 8 q^{95} + 8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{5} + 8 q^{11} + 6 q^{13} + 12 q^{17} + 8 q^{19} + 6 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{37} + 8 q^{43} - 16 q^{47} - 18 q^{49} - 14 q^{53} - 6 q^{61} + 12 q^{65} - 16 q^{67} + 32 q^{77} + 24 q^{79} - 8 q^{83} + 12 q^{85} - 24 q^{91} + 16 q^{95} + 16 q^{97} + O(q^{100}) \)

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 4.00000i 0 0 0
3457.1 0 0 0 1.00000 + 1.00000i 0 4.00000i 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
16.e even 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.u yes 2
3.b odd 2 1 4608.2.k.e yes 2
4.b odd 2 1 4608.2.k.n yes 2
8.b even 2 1 4608.2.k.d 2
8.d odd 2 1 4608.2.k.k yes 2
12.b even 2 1 4608.2.k.l yes 2
16.e even 4 1 4608.2.k.d 2
16.e even 4 1 inner 4608.2.k.u yes 2
16.f odd 4 1 4608.2.k.k yes 2
16.f odd 4 1 4608.2.k.n yes 2
24.f even 2 1 4608.2.k.m yes 2
24.h odd 2 1 4608.2.k.t yes 2
32.g even 8 2 9216.2.a.a 2
32.h odd 8 2 9216.2.a.t 2
48.i odd 4 1 4608.2.k.e yes 2
48.i odd 4 1 4608.2.k.t yes 2
48.k even 4 1 4608.2.k.l yes 2
48.k even 4 1 4608.2.k.m yes 2
96.o even 8 2 9216.2.a.v 2
96.p odd 8 2 9216.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.d 2 8.b even 2 1
4608.2.k.d 2 16.e even 4 1
4608.2.k.e yes 2 3.b odd 2 1
4608.2.k.e yes 2 48.i odd 4 1
4608.2.k.k yes 2 8.d odd 2 1
4608.2.k.k yes 2 16.f odd 4 1
4608.2.k.l yes 2 12.b even 2 1
4608.2.k.l yes 2 48.k even 4 1
4608.2.k.m yes 2 24.f even 2 1
4608.2.k.m yes 2 48.k even 4 1
4608.2.k.n yes 2 4.b odd 2 1
4608.2.k.n yes 2 16.f odd 4 1
4608.2.k.t yes 2 24.h odd 2 1
4608.2.k.t yes 2 48.i odd 4 1
4608.2.k.u yes 2 1.a even 1 1 trivial
4608.2.k.u yes 2 16.e even 4 1 inner
9216.2.a.a 2 32.g even 8 2
9216.2.a.c 2 96.p odd 8 2
9216.2.a.t 2 32.h odd 8 2
9216.2.a.v 2 96.o even 8 2

Hecke kernels

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

\( T_{5}^{2} - 2 T_{5} + 2 \)
\( T_{7}^{2} + 16 \)
\( T_{11}^{2} - 8 T_{11} + 32 \)
\( T_{13}^{2} - 6 T_{13} + 18 \)
\( T_{19}^{2} - 8 T_{19} + 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( 2 - 2 T + T^{2} \)
$7$ \( 16 + T^{2} \)
$11$ \( 32 - 8 T + T^{2} \)
$13$ \( 18 - 6 T + T^{2} \)
$17$ \( ( -6 + T )^{2} \)
$19$ \( 32 - 8 T + T^{2} \)
$23$ \( 64 + T^{2} \)
$29$ \( 18 - 6 T + T^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 2 - 2 T + T^{2} \)
$41$ \( 4 + T^{2} \)
$43$ \( 32 - 8 T + T^{2} \)
$47$ \( ( 8 + T )^{2} \)
$53$ \( 98 + 14 T + T^{2} \)
$59$ \( T^{2} \)
$61$ \( 18 + 6 T + T^{2} \)
$67$ \( 128 + 16 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 100 + T^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 32 + 8 T + T^{2} \)
$89$ \( 256 + T^{2} \)
$97$ \( ( -8 + T )^{2} \)
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