Properties

Label 4608.2.k.t
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: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

Character values

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

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

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.t yes 2
3.b odd 2 1 4608.2.k.d 2
4.b odd 2 1 4608.2.k.m yes 2
8.b even 2 1 4608.2.k.e yes 2
8.d odd 2 1 4608.2.k.l yes 2
12.b even 2 1 4608.2.k.k yes 2
16.e even 4 1 4608.2.k.e yes 2
16.e even 4 1 inner 4608.2.k.t yes 2
16.f odd 4 1 4608.2.k.l yes 2
16.f odd 4 1 4608.2.k.m yes 2
24.f even 2 1 4608.2.k.n yes 2
24.h odd 2 1 4608.2.k.u yes 2
32.g even 8 2 9216.2.a.c 2
32.h odd 8 2 9216.2.a.v 2
48.i odd 4 1 4608.2.k.d 2
48.i odd 4 1 4608.2.k.u yes 2
48.k even 4 1 4608.2.k.k yes 2
48.k even 4 1 4608.2.k.n yes 2
96.o even 8 2 9216.2.a.t 2
96.p odd 8 2 9216.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.k.d 2 3.b odd 2 1
4608.2.k.d 2 48.i odd 4 1
4608.2.k.e yes 2 8.b even 2 1
4608.2.k.e yes 2 16.e even 4 1
4608.2.k.k yes 2 12.b even 2 1
4608.2.k.k yes 2 48.k even 4 1
4608.2.k.l yes 2 8.d odd 2 1
4608.2.k.l yes 2 16.f odd 4 1
4608.2.k.m yes 2 4.b odd 2 1
4608.2.k.m yes 2 16.f odd 4 1
4608.2.k.n yes 2 24.f even 2 1
4608.2.k.n yes 2 48.k even 4 1
4608.2.k.t yes 2 1.a even 1 1 trivial
4608.2.k.t yes 2 16.e even 4 1 inner
4608.2.k.u yes 2 24.h odd 2 1
4608.2.k.u yes 2 48.i odd 4 1
9216.2.a.a 2 96.p odd 8 2
9216.2.a.c 2 32.g even 8 2
9216.2.a.t 2 96.o even 8 2
9216.2.a.v 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 \(S_{2}^{\mathrm{new}}(4608, [\chi])\):

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

Hecke characteristic polynomials

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