Properties

Label 4608.2.d.c
Level $4608$
Weight $2$
Character orbit 4608.d
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
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.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: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{7} +O(q^{10})\) \( q + ( -\zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{5} + ( -2 - \zeta_{8} + \zeta_{8}^{3} ) q^{7} + 2 \zeta_{8}^{2} q^{11} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{13} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{17} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{19} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + ( -1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{25} + ( \zeta_{8} - 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{29} + ( 6 - \zeta_{8} + \zeta_{8}^{3} ) q^{31} -2 \zeta_{8}^{2} q^{35} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{37} + ( -6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{43} + ( -4 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{47} + ( -1 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{49} + ( 7 \zeta_{8} + 2 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{53} + ( -4 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{55} -4 \zeta_{8}^{2} q^{59} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{61} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{65} -8 \zeta_{8}^{2} q^{67} + ( 12 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{71} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{73} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{77} + ( 10 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + ( -8 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{83} + ( -10 \zeta_{8} + 12 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{85} + 2 q^{89} + ( -4 \zeta_{8} - 4 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{91} + ( -8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{95} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} + O(q^{10}) \) \( 4 q - 8 q^{7} + 8 q^{17} + 16 q^{23} - 4 q^{25} + 24 q^{31} - 24 q^{41} - 16 q^{47} - 4 q^{49} - 16 q^{55} + 16 q^{65} + 48 q^{71} - 16 q^{73} + 40 q^{79} + 8 q^{89} - 32 q^{95} + 8 q^{97} + O(q^{100}) \)

Character values

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

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

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 3.41421i 0 −0.585786 0 0 0
2305.2 0 0 0 0.585786i 0 −3.41421 0 0 0
2305.3 0 0 0 0.585786i 0 −3.41421 0 0 0
2305.4 0 0 0 3.41421i 0 −0.585786 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
8.b even 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.c 4
3.b odd 2 1 1536.2.d.a 4
4.b odd 2 1 4608.2.d.o 4
8.b even 2 1 inner 4608.2.d.c 4
8.d odd 2 1 4608.2.d.o 4
12.b even 2 1 1536.2.d.f 4
16.e even 4 1 4608.2.a.e 2
16.e even 4 1 4608.2.a.r 2
16.f odd 4 1 4608.2.a.a 2
16.f odd 4 1 4608.2.a.n 2
24.f even 2 1 1536.2.d.f 4
24.h odd 2 1 1536.2.d.a 4
48.i odd 4 1 1536.2.a.b 2
48.i odd 4 1 1536.2.a.l yes 2
48.k even 4 1 1536.2.a.e yes 2
48.k even 4 1 1536.2.a.g yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.b 2 48.i odd 4 1
1536.2.a.e yes 2 48.k even 4 1
1536.2.a.g yes 2 48.k even 4 1
1536.2.a.l yes 2 48.i odd 4 1
1536.2.d.a 4 3.b odd 2 1
1536.2.d.a 4 24.h odd 2 1
1536.2.d.f 4 12.b even 2 1
1536.2.d.f 4 24.f even 2 1
4608.2.a.a 2 16.f odd 4 1
4608.2.a.e 2 16.e even 4 1
4608.2.a.n 2 16.f odd 4 1
4608.2.a.r 2 16.e even 4 1
4608.2.d.c 4 1.a even 1 1 trivial
4608.2.d.c 4 8.b even 2 1 inner
4608.2.d.o 4 4.b odd 2 1
4608.2.d.o 4 8.d odd 2 1

Hecke kernels

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

\( T_{5}^{4} + 12 T_{5}^{2} + 4 \)
\( T_{7}^{2} + 4 T_{7} + 2 \)
\( T_{17}^{2} - 4 T_{17} - 28 \)
\( T_{23}^{2} - 8 T_{23} + 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 + 12 T^{2} + T^{4} \)
$7$ \( ( 2 + 4 T + T^{2} )^{2} \)
$11$ \( ( 4 + T^{2} )^{2} \)
$13$ \( ( 8 + T^{2} )^{2} \)
$17$ \( ( -28 - 4 T + T^{2} )^{2} \)
$19$ \( ( 32 + T^{2} )^{2} \)
$23$ \( ( 8 - 8 T + T^{2} )^{2} \)
$29$ \( 4 + 12 T^{2} + T^{4} \)
$31$ \( ( 34 - 12 T + T^{2} )^{2} \)
$37$ \( 256 + 96 T^{2} + T^{4} \)
$41$ \( ( 4 + 12 T + T^{2} )^{2} \)
$43$ \( 256 + 96 T^{2} + T^{4} \)
$47$ \( ( -56 + 8 T + T^{2} )^{2} \)
$53$ \( 8836 + 204 T^{2} + T^{4} \)
$59$ \( ( 16 + T^{2} )^{2} \)
$61$ \( 256 + 96 T^{2} + T^{4} \)
$67$ \( ( 64 + T^{2} )^{2} \)
$71$ \( ( 136 - 24 T + T^{2} )^{2} \)
$73$ \( ( -16 + 8 T + T^{2} )^{2} \)
$79$ \( ( 82 - 20 T + T^{2} )^{2} \)
$83$ \( 15376 + 264 T^{2} + T^{4} \)
$89$ \( ( -2 + T )^{4} \)
$97$ \( ( -124 - 4 T + T^{2} )^{2} \)
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