Properties

Label 1152.4.a.s
Level $1152$
Weight $4$
Character orbit 1152.a
Self dual yes
Analytic conductor $67.970$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.9702003266\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 2 + 2 \beta ) q^{5} + ( -4 - 2 \beta ) q^{7} +O(q^{10})\) \( q + ( 2 + 2 \beta ) q^{5} + ( -4 - 2 \beta ) q^{7} + ( 46 + \beta ) q^{11} + ( -50 + 6 \beta ) q^{13} + ( -46 - 12 \beta ) q^{17} + ( -2 - 7 \beta ) q^{19} + ( -4 - 18 \beta ) q^{23} + ( 71 + 8 \beta ) q^{25} + ( 42 + 10 \beta ) q^{29} + ( -192 - 16 \beta ) q^{31} + ( -200 - 12 \beta ) q^{35} + ( 86 + 14 \beta ) q^{37} + ( 150 - 8 \beta ) q^{41} + ( -150 - 5 \beta ) q^{43} + ( 8 - 44 \beta ) q^{47} + ( -135 + 16 \beta ) q^{49} + ( -6 - 14 \beta ) q^{53} + ( 188 + 94 \beta ) q^{55} + ( -322 + 33 \beta ) q^{59} + ( -146 + 70 \beta ) q^{61} + ( 476 - 88 \beta ) q^{65} + ( 86 - 83 \beta ) q^{67} + ( -204 + 42 \beta ) q^{71} + ( 206 - 52 \beta ) q^{73} + ( -280 - 96 \beta ) q^{77} + ( -200 - 36 \beta ) q^{79} + ( 474 - 13 \beta ) q^{83} + ( -1244 - 116 \beta ) q^{85} + ( -286 + 116 \beta ) q^{89} + ( -376 + 76 \beta ) q^{91} + ( -676 - 18 \beta ) q^{95} + ( 1102 + 92 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 8q^{7} + O(q^{10}) \) \( 2q + 4q^{5} - 8q^{7} + 92q^{11} - 100q^{13} - 92q^{17} - 4q^{19} - 8q^{23} + 142q^{25} + 84q^{29} - 384q^{31} - 400q^{35} + 172q^{37} + 300q^{41} - 300q^{43} + 16q^{47} - 270q^{49} - 12q^{53} + 376q^{55} - 644q^{59} - 292q^{61} + 952q^{65} + 172q^{67} - 408q^{71} + 412q^{73} - 560q^{77} - 400q^{79} + 948q^{83} - 2488q^{85} - 572q^{89} - 752q^{91} - 1352q^{95} + 2204q^{97} + O(q^{100}) \)

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 −11.8564 0 9.85641 0 0 0
1.2 0 0 0 15.8564 0 −17.8564 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 1152.4.a.s 2
3.b odd 2 1 128.4.a.e 2
4.b odd 2 1 1152.4.a.t 2
8.b even 2 1 1152.4.a.q 2
8.d odd 2 1 1152.4.a.r 2
12.b even 2 1 128.4.a.g yes 2
24.f even 2 1 128.4.a.f yes 2
24.h odd 2 1 128.4.a.h yes 2
48.i odd 4 2 256.4.b.i 4
48.k even 4 2 256.4.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.4.a.e 2 3.b odd 2 1
128.4.a.f yes 2 24.f even 2 1
128.4.a.g yes 2 12.b even 2 1
128.4.a.h yes 2 24.h odd 2 1
256.4.b.h 4 48.k even 4 2
256.4.b.i 4 48.i odd 4 2
1152.4.a.q 2 8.b even 2 1
1152.4.a.r 2 8.d odd 2 1
1152.4.a.s 2 1.a even 1 1 trivial
1152.4.a.t 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1152))\):

\( T_{5}^{2} - 4 T_{5} - 188 \)
\( T_{7}^{2} + 8 T_{7} - 176 \)
\( T_{13}^{2} + 100 T_{13} + 772 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( -188 - 4 T + T^{2} \)
$7$ \( -176 + 8 T + T^{2} \)
$11$ \( 2068 - 92 T + T^{2} \)
$13$ \( 772 + 100 T + T^{2} \)
$17$ \( -4796 + 92 T + T^{2} \)
$19$ \( -2348 + 4 T + T^{2} \)
$23$ \( -15536 + 8 T + T^{2} \)
$29$ \( -3036 - 84 T + T^{2} \)
$31$ \( 24576 + 384 T + T^{2} \)
$37$ \( -2012 - 172 T + T^{2} \)
$41$ \( 19428 - 300 T + T^{2} \)
$43$ \( 21300 + 300 T + T^{2} \)
$47$ \( -92864 - 16 T + T^{2} \)
$53$ \( -9372 + 12 T + T^{2} \)
$59$ \( 51412 + 644 T + T^{2} \)
$61$ \( -213884 + 292 T + T^{2} \)
$67$ \( -323276 - 172 T + T^{2} \)
$71$ \( -43056 + 408 T + T^{2} \)
$73$ \( -87356 - 412 T + T^{2} \)
$79$ \( -22208 + 400 T + T^{2} \)
$83$ \( 216564 - 948 T + T^{2} \)
$89$ \( -564092 + 572 T + T^{2} \)
$97$ \( 808132 - 2204 T + T^{2} \)
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