Properties

Label 2304.4.a.bq
Level $2304$
Weight $4$
Character orbit 2304.a
Self dual yes
Analytic conductor $135.940$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 384)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 4) q^{5} + (\beta + 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 4) q^{5} + (\beta + 8) q^{7} + (4 \beta - 4) q^{11} + ( - 2 \beta + 36) q^{13} + ( - 4 \beta + 18) q^{17} + ( - 4 \beta - 68) q^{19} + (2 \beta - 128) q^{23} + (8 \beta + 99) q^{25} + ( - 9 \beta - 76) q^{29} + (13 \beta - 40) q^{31} + (12 \beta + 240) q^{35} + (4 \beta + 68) q^{37} + (20 \beta + 218) q^{41} + (4 \beta + 356) q^{43} + (14 \beta + 112) q^{47} + (16 \beta - 71) q^{49} + (15 \beta - 172) q^{53} + (12 \beta + 816) q^{55} - 324 q^{59} + 324 q^{61} + (28 \beta - 272) q^{65} + (48 \beta - 228) q^{67} + ( - 2 \beta - 1024) q^{71} + (48 \beta + 330) q^{73} + (28 \beta + 800) q^{77} + ( - 59 \beta + 248) q^{79} + ( - 28 \beta + 388) q^{83} + (2 \beta - 760) q^{85} + ( - 8 \beta - 266) q^{89} + (20 \beta - 128) q^{91} + ( - 84 \beta - 1104) q^{95} + (88 \beta - 610) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{5} + 16 q^{7} - 8 q^{11} + 72 q^{13} + 36 q^{17} - 136 q^{19} - 256 q^{23} + 198 q^{25} - 152 q^{29} - 80 q^{31} + 480 q^{35} + 136 q^{37} + 436 q^{41} + 712 q^{43} + 224 q^{47} - 142 q^{49} - 344 q^{53} + 1632 q^{55} - 648 q^{59} + 648 q^{61} - 544 q^{65} - 456 q^{67} - 2048 q^{71} + 660 q^{73} + 1600 q^{77} + 496 q^{79} + 776 q^{83} - 1520 q^{85} - 532 q^{89} - 256 q^{91} - 2208 q^{95} - 1220 q^{97}+O(q^{100}) \) 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
−1.30278
2.30278
0 0 0 −10.4222 0 −6.42221 0 0 0
1.2 0 0 0 18.4222 0 22.4222 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 2304.4.a.bq 2
3.b odd 2 1 768.4.a.k 2
4.b odd 2 1 2304.4.a.bp 2
8.b even 2 1 2304.4.a.t 2
8.d odd 2 1 2304.4.a.s 2
12.b even 2 1 768.4.a.e 2
16.e even 4 2 1152.4.d.i 4
16.f odd 4 2 1152.4.d.o 4
24.f even 2 1 768.4.a.p 2
24.h odd 2 1 768.4.a.j 2
48.i odd 4 2 384.4.d.c 4
48.k even 4 2 384.4.d.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.4.d.c 4 48.i odd 4 2
384.4.d.e yes 4 48.k even 4 2
768.4.a.e 2 12.b even 2 1
768.4.a.j 2 24.h odd 2 1
768.4.a.k 2 3.b odd 2 1
768.4.a.p 2 24.f even 2 1
1152.4.d.i 4 16.e even 4 2
1152.4.d.o 4 16.f odd 4 2
2304.4.a.s 2 8.d odd 2 1
2304.4.a.t 2 8.b even 2 1
2304.4.a.bp 2 4.b odd 2 1
2304.4.a.bq 2 1.a even 1 1 trivial

Hecke kernels

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

\( T_{5}^{2} - 8T_{5} - 192 \) Copy content Toggle raw display
\( T_{7}^{2} - 16T_{7} - 144 \) Copy content Toggle raw display
\( T_{11}^{2} + 8T_{11} - 3312 \) Copy content Toggle raw display
\( T_{13}^{2} - 72T_{13} + 464 \) Copy content Toggle raw display
\( T_{17}^{2} - 36T_{17} - 3004 \) Copy content Toggle raw display
\( T_{19}^{2} + 136T_{19} + 1296 \) 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
$5$ \( T^{2} - 8T - 192 \) Copy content Toggle raw display
$7$ \( T^{2} - 16T - 144 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T - 3312 \) Copy content Toggle raw display
$13$ \( T^{2} - 72T + 464 \) Copy content Toggle raw display
$17$ \( T^{2} - 36T - 3004 \) Copy content Toggle raw display
$19$ \( T^{2} + 136T + 1296 \) Copy content Toggle raw display
$23$ \( T^{2} + 256T + 15552 \) Copy content Toggle raw display
$29$ \( T^{2} + 152T - 11072 \) Copy content Toggle raw display
$31$ \( T^{2} + 80T - 33552 \) Copy content Toggle raw display
$37$ \( T^{2} - 136T + 1296 \) Copy content Toggle raw display
$41$ \( T^{2} - 436T - 35676 \) Copy content Toggle raw display
$43$ \( T^{2} - 712T + 123408 \) Copy content Toggle raw display
$47$ \( T^{2} - 224T - 28224 \) Copy content Toggle raw display
$53$ \( T^{2} + 344T - 17216 \) Copy content Toggle raw display
$59$ \( (T + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T - 324)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 456T - 427248 \) Copy content Toggle raw display
$71$ \( T^{2} + 2048 T + 1047744 \) Copy content Toggle raw display
$73$ \( T^{2} - 660T - 370332 \) Copy content Toggle raw display
$79$ \( T^{2} - 496T - 662544 \) Copy content Toggle raw display
$83$ \( T^{2} - 776T - 12528 \) Copy content Toggle raw display
$89$ \( T^{2} + 532T + 57444 \) Copy content Toggle raw display
$97$ \( T^{2} + 1220 T - 1238652 \) Copy content Toggle raw display
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