Properties

Label 2304.4.a.x
Level $2304$
Weight $4$
Character orbit 2304.a
Self dual yes
Analytic conductor $135.940$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.940400653\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} + 7 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} + 7 \beta q^{7} - 48 q^{11} - 12 \beta q^{13} - 54 q^{17} - 4 q^{19} - 50 \beta q^{23} - 113 q^{25} + 47 \beta q^{29} - 17 \beta q^{31} + 84 q^{35} + 94 \beta q^{37} - 294 q^{41} + 188 q^{43} + 146 \beta q^{47} + 245 q^{49} + 215 \beta q^{53} - 48 \beta q^{55} - 252 q^{59} + 26 \beta q^{61} - 144 q^{65} + 628 q^{67} + 2 \beta q^{71} + 1006 q^{73} - 336 \beta q^{77} + 387 \beta q^{79} + 720 q^{83} - 54 \beta q^{85} - 1482 q^{89} - 1008 q^{91} - 4 \beta q^{95} + 1822 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 96 q^{11} - 108 q^{17} - 8 q^{19} - 226 q^{25} + 168 q^{35} - 588 q^{41} + 376 q^{43} + 490 q^{49} - 504 q^{59} - 288 q^{65} + 1256 q^{67} + 2012 q^{73} + 1440 q^{83} - 2964 q^{89} - 2016 q^{91} + 3644 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 0 0 −3.46410 0 −24.2487 0 0 0
1.2 0 0 0 3.46410 0 24.2487 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.a.x 2
3.b odd 2 1 768.4.a.o 2
4.b odd 2 1 2304.4.a.bk 2
8.b even 2 1 2304.4.a.bk 2
8.d odd 2 1 inner 2304.4.a.x 2
12.b even 2 1 768.4.a.f 2
16.e even 4 2 576.4.d.c 4
16.f odd 4 2 576.4.d.c 4
24.f even 2 1 768.4.a.o 2
24.h odd 2 1 768.4.a.f 2
48.i odd 4 2 192.4.d.c 4
48.k even 4 2 192.4.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
192.4.d.c 4 48.i odd 4 2
192.4.d.c 4 48.k even 4 2
576.4.d.c 4 16.e even 4 2
576.4.d.c 4 16.f odd 4 2
768.4.a.f 2 12.b even 2 1
768.4.a.f 2 24.h odd 2 1
768.4.a.o 2 3.b odd 2 1
768.4.a.o 2 24.f even 2 1
2304.4.a.x 2 1.a even 1 1 trivial
2304.4.a.x 2 8.d odd 2 1 inner
2304.4.a.bk 2 4.b odd 2 1
2304.4.a.bk 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2304))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{7}^{2} - 588 \) Copy content Toggle raw display
\( T_{11} + 48 \) Copy content Toggle raw display
\( T_{13}^{2} - 1728 \) Copy content Toggle raw display
\( T_{17} + 54 \) Copy content Toggle raw display
\( T_{19} + 4 \) 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
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( T^{2} - 588 \) Copy content Toggle raw display
$11$ \( (T + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 1728 \) Copy content Toggle raw display
$17$ \( (T + 54)^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 30000 \) Copy content Toggle raw display
$29$ \( T^{2} - 26508 \) Copy content Toggle raw display
$31$ \( T^{2} - 3468 \) Copy content Toggle raw display
$37$ \( T^{2} - 106032 \) Copy content Toggle raw display
$41$ \( (T + 294)^{2} \) Copy content Toggle raw display
$43$ \( (T - 188)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 255792 \) Copy content Toggle raw display
$53$ \( T^{2} - 554700 \) Copy content Toggle raw display
$59$ \( (T + 252)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 8112 \) Copy content Toggle raw display
$67$ \( (T - 628)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 48 \) Copy content Toggle raw display
$73$ \( (T - 1006)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1797228 \) Copy content Toggle raw display
$83$ \( (T - 720)^{2} \) Copy content Toggle raw display
$89$ \( (T + 1482)^{2} \) Copy content Toggle raw display
$97$ \( (T - 1822)^{2} \) Copy content Toggle raw display
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