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.

\(f(q)\) \(=\) \( q + ( 2 + 8 \beta ) q^{5} + ( -4 - 8 \beta ) q^{7} +O(q^{10})\) \( q + ( 2 + 8 \beta ) q^{5} + ( -4 - 8 \beta ) q^{7} + ( 46 + 4 \beta ) q^{11} + ( -50 + 24 \beta ) q^{13} + ( -46 - 48 \beta ) q^{17} + ( -2 - 28 \beta ) q^{19} + ( -4 - 72 \beta ) q^{23} + ( 71 + 32 \beta ) q^{25} + ( 42 + 40 \beta ) q^{29} + ( -192 - 64 \beta ) q^{31} + ( -200 - 48 \beta ) q^{35} + ( 86 + 56 \beta ) q^{37} + ( 150 - 32 \beta ) q^{41} + ( -150 - 20 \beta ) q^{43} + ( 8 - 176 \beta ) q^{47} + ( -135 + 64 \beta ) q^{49} + ( -6 - 56 \beta ) q^{53} + ( 188 + 376 \beta ) q^{55} + ( -322 + 132 \beta ) q^{59} + ( -146 + 280 \beta ) q^{61} + ( 476 - 352 \beta ) q^{65} + ( 86 - 332 \beta ) q^{67} + ( -204 + 168 \beta ) q^{71} + ( 206 - 208 \beta ) q^{73} + ( -280 - 384 \beta ) q^{77} + ( -200 - 144 \beta ) q^{79} + ( 474 - 52 \beta ) q^{83} + ( -1244 - 464 \beta ) q^{85} + ( -286 + 464 \beta ) q^{89} + ( -376 + 304 \beta ) q^{91} + ( -676 - 72 \beta ) q^{95} + ( 1102 + 368 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{5} - 8q^{7} + O(q^{10}) \) \( 2q + 4q^{5} - 8q^{7} + 92q^{11} - 100q^{13} - 92q^{17} - 4q^{19} - 8q^{23} + 142q^{25} + 84q^{29} - 384q^{31} - 400q^{35} + 172q^{37} + 300q^{41} - 300q^{43} + 16q^{47} - 270q^{49} - 12q^{53} + 376q^{55} - 644q^{59} - 292q^{61} + 952q^{65} + 172q^{67} - 408q^{71} + 412q^{73} - 560q^{77} - 400q^{79} + 948q^{83} - 2488q^{85} - 572q^{89} - 752q^{91} - 1352q^{95} + 2204q^{97} + O(q^{100}) \)

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

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( 1 - 4 T + 62 T^{2} - 500 T^{3} + 15625 T^{4} \)
$7$ \( 1 + 8 T + 510 T^{2} + 2744 T^{3} + 117649 T^{4} \)
$11$ \( 1 - 92 T + 4730 T^{2} - 122452 T^{3} + 1771561 T^{4} \)
$13$ \( 1 + 100 T + 5166 T^{2} + 219700 T^{3} + 4826809 T^{4} \)
$17$ \( 1 + 92 T + 5030 T^{2} + 451996 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 4 T + 11370 T^{2} + 27436 T^{3} + 47045881 T^{4} \)
$23$ \( 1 + 8 T + 8798 T^{2} + 97336 T^{3} + 148035889 T^{4} \)
$29$ \( 1 - 84 T + 45742 T^{2} - 2048676 T^{3} + 594823321 T^{4} \)
$31$ \( 1 + 384 T + 84158 T^{2} + 11439744 T^{3} + 887503681 T^{4} \)
$37$ \( 1 - 172 T + 99294 T^{2} - 8712316 T^{3} + 2565726409 T^{4} \)
$41$ \( 1 - 300 T + 157270 T^{2} - 20676300 T^{3} + 4750104241 T^{4} \)
$43$ \( 1 + 300 T + 180314 T^{2} + 23852100 T^{3} + 6321363049 T^{4} \)
$47$ \( 1 - 16 T + 114782 T^{2} - 1661168 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 + 12 T + 288382 T^{2} + 1786524 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 + 644 T + 462170 T^{2} + 132264076 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 292 T + 240078 T^{2} + 66278452 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 - 172 T + 278250 T^{2} - 51731236 T^{3} + 90458382169 T^{4} \)
$71$ \( 1 + 408 T + 672766 T^{2} + 146027688 T^{3} + 128100283921 T^{4} \)
$73$ \( 1 - 412 T + 690678 T^{2} - 160275004 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 + 400 T + 963870 T^{2} + 197215600 T^{3} + 243087455521 T^{4} \)
$83$ \( 1 - 948 T + 1360138 T^{2} - 542054076 T^{3} + 326940373369 T^{4} \)
$89$ \( 1 + 572 T + 845846 T^{2} + 403242268 T^{3} + 496981290961 T^{4} \)
$97$ \( 1 - 2204 T + 2633478 T^{2} - 2011531292 T^{3} + 832972004929 T^{4} \)
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