Properties

Label 4608.2.a.n
Level $4608$
Weight $2$
Character orbit 4608.a
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta + 2) q^{5} + (\beta - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 2) q^{5} + (\beta - 2) q^{7} - 2 q^{11} + 2 \beta q^{13} + (4 \beta + 2) q^{17} + 4 \beta q^{19} + (2 \beta + 4) q^{23} + (4 \beta + 1) q^{25} + (\beta + 2) q^{29} + ( - \beta - 6) q^{31} - 2 q^{35} + ( - 4 \beta + 4) q^{37} + ( - 4 \beta + 6) q^{41} + ( - 4 \beta - 4) q^{43} + ( - 6 \beta + 4) q^{47} + ( - 4 \beta - 1) q^{49} + ( - 7 \beta + 2) q^{53} + ( - 2 \beta - 4) q^{55} + 4 q^{59} + ( - 4 \beta + 4) q^{61} + (4 \beta + 4) q^{65} - 8 q^{67} + (2 \beta + 12) q^{71} + (4 \beta + 4) q^{73} + ( - 2 \beta + 4) q^{77} + ( - 3 \beta - 10) q^{79} + (8 \beta + 2) q^{83} + (10 \beta + 12) q^{85} - 2 q^{89} + ( - 4 \beta + 4) q^{91} + (8 \beta + 8) q^{95} + ( - 8 \beta + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 4 q^{7} - 4 q^{11} + 4 q^{17} + 8 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 8 q^{37} + 12 q^{41} - 8 q^{43} + 8 q^{47} - 2 q^{49} + 4 q^{53} - 8 q^{55} + 8 q^{59} + 8 q^{61} + 8 q^{65} - 16 q^{67} + 24 q^{71} + 8 q^{73} + 8 q^{77} - 20 q^{79} + 4 q^{83} + 24 q^{85} - 4 q^{89} + 8 q^{91} + 16 q^{95} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 0 0 0.585786 0 −3.41421 0 0 0
1.2 0 0 0 3.41421 0 −0.585786 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\):

\( T_{5}^{2} - 4T_{5} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 28 \) Copy content Toggle raw display
\( T_{23}^{2} - 8T_{23} + 8 \) 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
$5$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 4T + 2 \) Copy content Toggle raw display
$11$ \( (T + 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 8 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( T^{2} - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 8T + 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$31$ \( T^{2} + 12T + 34 \) Copy content Toggle raw display
$37$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$41$ \( T^{2} - 12T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 94 \) Copy content Toggle raw display
$59$ \( (T - 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$67$ \( (T + 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 24T + 136 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$79$ \( T^{2} + 20T + 82 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
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