Properties

Label 2304.4.a.bn
Level $2304$
Weight $4$
Character orbit 2304.a
Self dual yes
Analytic conductor $135.940$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta q^{5} + 8 q^{7} + 3 \beta q^{11} - 10 \beta q^{13} + 14 q^{17} + 7 \beta q^{19} - 152 q^{23} - 13 q^{25} - 30 \beta q^{29} + 224 q^{31} + 16 \beta q^{35} + 46 \beta q^{37} - 70 q^{41} - 83 \beta q^{43} - 336 q^{47} - 279 q^{49} - 6 \beta q^{53} + 168 q^{55} - 101 \beta q^{59} - 18 \beta q^{61} - 560 q^{65} - 33 \beta q^{67} - 72 q^{71} + 294 q^{73} + 24 \beta q^{77} - 464 q^{79} - 103 \beta q^{83} + 28 \beta q^{85} + 266 q^{89} - 80 \beta q^{91} + 392 q^{95} + 994 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{7} + 28 q^{17} - 304 q^{23} - 26 q^{25} + 448 q^{31} - 140 q^{41} - 672 q^{47} - 558 q^{49} + 336 q^{55} - 1120 q^{65} - 144 q^{71} + 588 q^{73} - 928 q^{79} + 532 q^{89} + 784 q^{95} + 1988 q^{97}+O(q^{100}) \) 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
−2.64575
2.64575
0 0 0 −10.5830 0 8.00000 0 0 0
1.2 0 0 0 10.5830 0 8.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.4.a.bn 2
3.b odd 2 1 256.4.a.l 2
4.b odd 2 1 2304.4.a.v 2
8.b even 2 1 inner 2304.4.a.bn 2
8.d odd 2 1 2304.4.a.v 2
12.b even 2 1 256.4.a.j 2
16.e even 4 2 72.4.d.b 2
16.f odd 4 2 288.4.d.a 2
24.f even 2 1 256.4.a.j 2
24.h odd 2 1 256.4.a.l 2
48.i odd 4 2 8.4.b.a 2
48.k even 4 2 32.4.b.a 2
240.t even 4 2 800.4.d.a 2
240.z odd 4 2 800.4.f.a 4
240.bb even 4 2 200.4.f.a 4
240.bd odd 4 2 800.4.f.a 4
240.bf even 4 2 200.4.f.a 4
240.bm odd 4 2 200.4.d.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.b.a 2 48.i odd 4 2
32.4.b.a 2 48.k even 4 2
72.4.d.b 2 16.e even 4 2
200.4.d.a 2 240.bm odd 4 2
200.4.f.a 4 240.bb even 4 2
200.4.f.a 4 240.bf even 4 2
256.4.a.j 2 12.b even 2 1
256.4.a.j 2 24.f even 2 1
256.4.a.l 2 3.b odd 2 1
256.4.a.l 2 24.h odd 2 1
288.4.d.a 2 16.f odd 4 2
800.4.d.a 2 240.t even 4 2
800.4.f.a 4 240.z odd 4 2
800.4.f.a 4 240.bd odd 4 2
2304.4.a.v 2 4.b odd 2 1
2304.4.a.v 2 8.d odd 2 1
2304.4.a.bn 2 1.a even 1 1 trivial
2304.4.a.bn 2 8.b even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 112 \) Copy content Toggle raw display
\( T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 252 \) Copy content Toggle raw display
\( T_{13}^{2} - 2800 \) Copy content Toggle raw display
\( T_{17} - 14 \) Copy content Toggle raw display
\( T_{19}^{2} - 1372 \) 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
$5$ \( T^{2} - 112 \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 252 \) Copy content Toggle raw display
$13$ \( T^{2} - 2800 \) Copy content Toggle raw display
$17$ \( (T - 14)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 1372 \) Copy content Toggle raw display
$23$ \( (T + 152)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 25200 \) Copy content Toggle raw display
$31$ \( (T - 224)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 59248 \) Copy content Toggle raw display
$41$ \( (T + 70)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 192892 \) Copy content Toggle raw display
$47$ \( (T + 336)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 1008 \) Copy content Toggle raw display
$59$ \( T^{2} - 285628 \) Copy content Toggle raw display
$61$ \( T^{2} - 9072 \) Copy content Toggle raw display
$67$ \( T^{2} - 30492 \) Copy content Toggle raw display
$71$ \( (T + 72)^{2} \) Copy content Toggle raw display
$73$ \( (T - 294)^{2} \) Copy content Toggle raw display
$79$ \( (T + 464)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 297052 \) Copy content Toggle raw display
$89$ \( (T - 266)^{2} \) Copy content Toggle raw display
$97$ \( (T - 994)^{2} \) Copy content Toggle raw display
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