Properties

Label 2304.2.a.t
Level $2304$
Weight $2$
Character orbit 2304.a
Self dual yes
Analytic conductor $18.398$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(18.3975326257\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{11} - 6 q^{17} - 3 \beta q^{19} - 5 q^{25} - 6 q^{41} + 3 \beta q^{43} - 7 q^{49} + 5 \beta q^{59} - 3 \beta q^{67} + 2 q^{73} - \beta q^{83} - 18 q^{89} - 10 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{17} - 10 q^{25} - 12 q^{41} - 14 q^{49} + 4 q^{73} - 36 q^{89} - 20 q^{97}+O(q^{100}) \) 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 0 0 0 0 0
1.2 0 0 0 0 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
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 2304.2.a.t 2
3.b odd 2 1 256.2.a.e 2
4.b odd 2 1 inner 2304.2.a.t 2
8.b even 2 1 inner 2304.2.a.t 2
8.d odd 2 1 CM 2304.2.a.t 2
12.b even 2 1 256.2.a.e 2
15.d odd 2 1 6400.2.a.by 2
16.e even 4 2 1152.2.d.c 2
16.f odd 4 2 1152.2.d.c 2
24.f even 2 1 256.2.a.e 2
24.h odd 2 1 256.2.a.e 2
48.i odd 4 2 128.2.b.a 2
48.k even 4 2 128.2.b.a 2
60.h even 2 1 6400.2.a.by 2
96.o even 8 2 1024.2.e.a 2
96.o even 8 2 1024.2.e.f 2
96.p odd 8 2 1024.2.e.a 2
96.p odd 8 2 1024.2.e.f 2
120.i odd 2 1 6400.2.a.by 2
120.m even 2 1 6400.2.a.by 2
240.t even 4 2 3200.2.d.c 2
240.z odd 4 2 3200.2.f.o 4
240.bb even 4 2 3200.2.f.o 4
240.bd odd 4 2 3200.2.f.o 4
240.bf even 4 2 3200.2.f.o 4
240.bm odd 4 2 3200.2.d.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.a 2 48.i odd 4 2
128.2.b.a 2 48.k even 4 2
256.2.a.e 2 3.b odd 2 1
256.2.a.e 2 12.b even 2 1
256.2.a.e 2 24.f even 2 1
256.2.a.e 2 24.h odd 2 1
1024.2.e.a 2 96.o even 8 2
1024.2.e.a 2 96.p odd 8 2
1024.2.e.f 2 96.o even 8 2
1024.2.e.f 2 96.p odd 8 2
1152.2.d.c 2 16.e even 4 2
1152.2.d.c 2 16.f odd 4 2
2304.2.a.t 2 1.a even 1 1 trivial
2304.2.a.t 2 4.b odd 2 1 inner
2304.2.a.t 2 8.b even 2 1 inner
2304.2.a.t 2 8.d odd 2 1 CM
3200.2.d.c 2 240.t even 4 2
3200.2.d.c 2 240.bm odd 4 2
3200.2.f.o 4 240.z odd 4 2
3200.2.f.o 4 240.bb even 4 2
3200.2.f.o 4 240.bd odd 4 2
3200.2.f.o 4 240.bf even 4 2
6400.2.a.by 2 15.d odd 2 1
6400.2.a.by 2 60.h even 2 1
6400.2.a.by 2 120.i odd 2 1
6400.2.a.by 2 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} - 8 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19}^{2} - 72 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 8 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 72 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 72 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 200 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 72 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 8 \) Copy content Toggle raw display
$89$ \( (T + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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