Properties

Label 4608.2.c.c
Level $4608$
Weight $2$
Character orbit 4608.c
Analytic conductor $36.795$
Analytic rank $1$
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} + \beta q^{7} - 2 q^{13} - \beta q^{17} - 2 \beta q^{19} - 6 q^{23} - 3 q^{25} - 4 \beta q^{29} + 7 \beta q^{31} - 4 q^{35} + 2 q^{37} + 3 \beta q^{41} - 6 \beta q^{43} - 10 q^{47} + 5 q^{49} + 4 \beta q^{53} - 12 q^{59} - 10 q^{61} - 4 \beta q^{65} - 4 \beta q^{67} - 2 q^{71} - 10 q^{73} - 9 \beta q^{79} - 4 q^{83} + 4 q^{85} + 7 \beta q^{89} - 2 \beta q^{91} + 8 q^{95} + 12 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{13} - 12 q^{23} - 6 q^{25} - 8 q^{35} + 4 q^{37} - 20 q^{47} + 10 q^{49} - 24 q^{59} - 20 q^{61} - 4 q^{71} - 20 q^{73} - 8 q^{83} + 8 q^{85} + 16 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4607.1
1.41421i
1.41421i
0 0 0 2.82843i 0 1.41421i 0 0 0
4607.2 0 0 0 2.82843i 0 1.41421i 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
12.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.c.c 2
3.b odd 2 1 4608.2.c.d yes 2
4.b odd 2 1 4608.2.c.d yes 2
8.b even 2 1 4608.2.c.e yes 2
8.d odd 2 1 4608.2.c.f yes 2
12.b even 2 1 inner 4608.2.c.c 2
16.e even 4 2 4608.2.f.l 4
16.f odd 4 2 4608.2.f.j 4
24.f even 2 1 4608.2.c.e yes 2
24.h odd 2 1 4608.2.c.f yes 2
48.i odd 4 2 4608.2.f.j 4
48.k even 4 2 4608.2.f.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.c.c 2 1.a even 1 1 trivial
4608.2.c.c 2 12.b even 2 1 inner
4608.2.c.d yes 2 3.b odd 2 1
4608.2.c.d yes 2 4.b odd 2 1
4608.2.c.e yes 2 8.b even 2 1
4608.2.c.e yes 2 24.f even 2 1
4608.2.c.f yes 2 8.d odd 2 1
4608.2.c.f yes 2 24.h odd 2 1
4608.2.f.j 4 16.f odd 4 2
4608.2.f.j 4 48.i odd 4 2
4608.2.f.l 4 16.e even 4 2
4608.2.f.l 4 48.k even 4 2

Hecke kernels

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

\( T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{7}^{2} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{23} + 6 \) 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
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( T^{2} + 2 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 8 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 32 \) Copy content Toggle raw display
$31$ \( T^{2} + 98 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 18 \) Copy content Toggle raw display
$43$ \( T^{2} + 72 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 32 \) Copy content Toggle raw display
$59$ \( (T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 32 \) Copy content Toggle raw display
$71$ \( (T + 2)^{2} \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 162 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 98 \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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