Properties

Label 384.8.a.s
Level $384$
Weight $8$
Character orbit 384.a
Self dual yes
Analytic conductor $119.956$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - 430 x^{2} - 2448 x + 12138\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 27 q^{3} + ( 48 + \beta_{2} ) q^{5} + ( 170 + \beta_{3} ) q^{7} + 729 q^{9} +O(q^{10})\) \( q + 27 q^{3} + ( 48 + \beta_{2} ) q^{5} + ( 170 + \beta_{3} ) q^{7} + 729 q^{9} + ( -1124 + \beta_{1} + 2 \beta_{2} + 8 \beta_{3} ) q^{11} + ( -3210 + 6 \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 1296 + 27 \beta_{2} ) q^{15} + ( -3738 + 13 \beta_{1} + 32 \beta_{2} - 30 \beta_{3} ) q^{17} + ( -5376 + 7 \beta_{1} + 56 \beta_{2} + 14 \beta_{3} ) q^{19} + ( 4590 + 27 \beta_{3} ) q^{21} + ( 13748 - 8 \beta_{1} - 180 \beta_{2} + 110 \beta_{3} ) q^{23} + ( 47683 - 56 \beta_{1} + 38 \beta_{2} + 98 \beta_{3} ) q^{25} + 19683 q^{27} + ( 60596 - 70 \beta_{1} + 357 \beta_{2} - 90 \beta_{3} ) q^{29} + ( 37858 - 2 \beta_{1} - 276 \beta_{2} - 59 \beta_{3} ) q^{31} + ( -30348 + 27 \beta_{1} + 54 \beta_{2} + 216 \beta_{3} ) q^{33} + ( 68496 - 49 \beta_{1} + 370 \beta_{2} + 252 \beta_{3} ) q^{35} + ( 28322 + 98 \beta_{1} + 317 \beta_{2} - 287 \beta_{3} ) q^{37} + ( -86670 + 162 \beta_{1} + 27 \beta_{2} - 27 \beta_{3} ) q^{39} + ( -59794 + 99 \beta_{1} + 1372 \beta_{2} - 278 \beta_{3} ) q^{41} + ( 373832 - 191 \beta_{1} + 1148 \beta_{2} - 458 \beta_{3} ) q^{43} + ( 34992 + 729 \beta_{2} ) q^{45} + ( 193092 - 124 \beta_{1} - 1272 \beta_{2} - 1146 \beta_{3} ) q^{47} + ( -394275 - 34 \beta_{1} + 986 \beta_{2} - 102 \beta_{3} ) q^{49} + ( -100926 + 351 \beta_{1} + 864 \beta_{2} - 810 \beta_{3} ) q^{51} + ( 597444 - 138 \beta_{1} - 471 \beta_{2} + 206 \beta_{3} ) q^{53} + ( 647520 - 602 \beta_{1} - 636 \beta_{2} + 2016 \beta_{3} ) q^{55} + ( -145152 + 189 \beta_{1} + 1512 \beta_{2} + 378 \beta_{3} ) q^{57} + ( 35308 + 784 \beta_{1} - 1292 \beta_{2} + 1876 \beta_{3} ) q^{59} + ( 307826 + 454 \beta_{1} + 2337 \beta_{2} + 1833 \beta_{3} ) q^{61} + ( 123930 + 729 \beta_{3} ) q^{63} + ( -260256 - 595 \beta_{1} - 9972 \beta_{2} - 1330 \beta_{3} ) q^{65} + ( -110348 - 616 \beta_{1} - 1452 \beta_{2} - 1996 \beta_{3} ) q^{67} + ( 371196 - 216 \beta_{1} - 4860 \beta_{2} + 2970 \beta_{3} ) q^{69} + ( 376876 + 1268 \beta_{1} - 68 \beta_{2} - 3570 \beta_{3} ) q^{71} + ( -379210 + 2 \beta_{1} + 4382 \beta_{2} - 3554 \beta_{3} ) q^{73} + ( 1287441 - 1512 \beta_{1} + 1026 \beta_{2} + 2646 \beta_{3} ) q^{75} + ( 3085112 + 42 \beta_{1} + 7592 \beta_{2} - 2936 \beta_{3} ) q^{77} + ( 2385234 + 2088 \beta_{1} + 12964 \beta_{2} - 67 \beta_{3} ) q^{79} + 531441 q^{81} + ( -1146900 - 751 \beta_{1} - 11242 \beta_{2} - 2508 \beta_{3} ) q^{83} + ( 1595712 - 1596 \beta_{1} - 24254 \beta_{2} - 6972 \beta_{3} ) q^{85} + ( 1636092 - 1890 \beta_{1} + 9639 \beta_{2} - 2430 \beta_{3} ) q^{87} + ( 40594 - 330 \beta_{1} - 5160 \beta_{2} + 8860 \beta_{3} ) q^{89} + ( -1170276 + 2457 \beta_{1} - 6832 \beta_{2} - 2998 \beta_{3} ) q^{91} + ( 1022166 - 54 \beta_{1} - 7452 \beta_{2} - 1593 \beta_{3} ) q^{93} + ( 7305312 - 4508 \beta_{1} - 10780 \beta_{2} + 7644 \beta_{3} ) q^{95} + ( 681690 + 2730 \beta_{1} - 17900 \beta_{2} + 7600 \beta_{3} ) q^{97} + ( -819396 + 729 \beta_{1} + 1458 \beta_{2} + 5832 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 108q^{3} + 192q^{5} + 680q^{7} + 2916q^{9} + O(q^{10}) \) \( 4q + 108q^{3} + 192q^{5} + 680q^{7} + 2916q^{9} - 4496q^{11} - 12840q^{13} + 5184q^{15} - 14952q^{17} - 21504q^{19} + 18360q^{21} + 54992q^{23} + 190732q^{25} + 78732q^{27} + 242384q^{29} + 151432q^{31} - 121392q^{33} + 273984q^{35} + 113288q^{37} - 346680q^{39} - 239176q^{41} + 1495328q^{43} + 139968q^{45} + 772368q^{47} - 1577100q^{49} - 403704q^{51} + 2389776q^{53} + 2590080q^{55} - 580608q^{57} + 141232q^{59} + 1231304q^{61} + 495720q^{63} - 1041024q^{65} - 441392q^{67} + 1484784q^{69} + 1507504q^{71} - 1516840q^{73} + 5149764q^{75} + 12340448q^{77} + 9540936q^{79} + 2125764q^{81} - 4587600q^{83} + 6382848q^{85} + 6544368q^{87} + 162376q^{89} - 4681104q^{91} + 4088664q^{93} + 29221248q^{95} + 2726760q^{97} - 3277584q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 430 x^{2} - 2448 x + 12138\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -96 \nu^{3} + 1360 \nu^{2} + 36928 \nu - 116144 \)\()/119\)
\(\beta_{2}\)\(=\)\((\)\( -200 \nu^{3} + 2516 \nu^{2} + 52816 \nu - 173740 \)\()/119\)
\(\beta_{3}\)\(=\)\((\)\( -8 \nu^{3} + 132 \nu^{2} + 1808 \nu - 13692 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{3} - 2 \beta_{2} + 7 \beta_{1}\)\()/768\)
\(\nu^{2}\)\(=\)\((\)\(38 \beta_{3} - 34 \beta_{2} + 17 \beta_{1} + 41280\)\()/192\)
\(\nu^{3}\)\(=\)\((\)\(173 \beta_{3} - 337 \beta_{2} + 338 \beta_{1} + 176256\)\()/96\)

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
−12.4323
22.6791
3.20119
−13.4480
0 27.0000 0 −432.433 0 113.601 0 729.000 0
1.2 0 27.0000 0 −76.3562 0 439.494 0 729.000 0
1.3 0 27.0000 0 170.323 0 −803.427 0 729.000 0
1.4 0 27.0000 0 530.466 0 930.332 0 729.000 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 384.8.a.s yes 4
4.b odd 2 1 384.8.a.o yes 4
8.b even 2 1 384.8.a.n 4
8.d odd 2 1 384.8.a.r yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.a.n 4 8.b even 2 1
384.8.a.o yes 4 4.b odd 2 1
384.8.a.r yes 4 8.d odd 2 1
384.8.a.s yes 4 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(384))\):

\( T_{5}^{4} - 192 T_{5}^{3} - 233184 T_{5}^{2} + 22830080 T_{5} + 2983276800 \)
\( T_{7}^{4} - 680 T_{7}^{3} - 627336 T_{7}^{2} + 407076832 T_{7} - 37318092656 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -27 + T )^{4} \)
$5$ \( 2983276800 + 22830080 T - 233184 T^{2} - 192 T^{3} + T^{4} \)
$7$ \( -37318092656 + 407076832 T - 627336 T^{2} - 680 T^{3} + T^{4} \)
$11$ \( 552050534693120 - 80873665280 T - 51757344 T^{2} + 4496 T^{3} + T^{4} \)
$13$ \( -2206787363930352 - 1704045801568 T - 118878120 T^{2} + 12840 T^{3} + T^{4} \)
$17$ \( 501535254519216912 - 12612841818976 T - 1527439272 T^{2} + 14952 T^{3} + T^{4} \)
$19$ \( 111267999147626496 - 4415480774656 T - 1128753024 T^{2} + 21504 T^{3} + T^{4} \)
$23$ \( -10957440785644181248 + 811363281794816 T - 12102667296 T^{2} - 54992 T^{3} + T^{4} \)
$29$ \( -3101636390659660800 + 8298669461030400 T - 34230604416 T^{2} - 242384 T^{3} + T^{4} \)
$31$ \( -76587946662315264624 + 2622223946440032 T - 16869354696 T^{2} - 151432 T^{3} + T^{4} \)
$37$ \( \)\(21\!\cdots\!80\)\( + 3518324458893280 T - 113704932456 T^{2} - 113288 T^{3} + T^{4} \)
$41$ \( \)\(51\!\cdots\!76\)\( + 67627712660395296 T - 452081286120 T^{2} + 239176 T^{3} + T^{4} \)
$43$ \( -\)\(38\!\cdots\!68\)\( + 266866222586916864 T + 281905350144 T^{2} - 1495328 T^{3} + T^{4} \)
$47$ \( \)\(37\!\cdots\!92\)\( + 1028659840610650880 T - 1611252798240 T^{2} - 772368 T^{3} + T^{4} \)
$53$ \( \)\(81\!\cdots\!76\)\( - 678316436078621184 T + 1983300053376 T^{2} - 2389776 T^{3} + T^{4} \)
$59$ \( -\)\(12\!\cdots\!84\)\( - 3421459186395628288 T - 5536960908960 T^{2} - 141232 T^{3} + T^{4} \)
$61$ \( \)\(70\!\cdots\!56\)\( + 3283458873252851936 T - 5254448671464 T^{2} - 1231304 T^{3} + T^{4} \)
$67$ \( \)\(86\!\cdots\!04\)\( - 1184110925351609600 T - 5894428427424 T^{2} + 441392 T^{3} + T^{4} \)
$71$ \( \)\(52\!\cdots\!72\)\( + 20540829503292883200 T - 18295169893920 T^{2} - 1507504 T^{3} + T^{4} \)
$73$ \( -\)\(15\!\cdots\!52\)\( - 25962509969892863072 T - 10236033852840 T^{2} + 1516840 T^{3} + T^{4} \)
$79$ \( -\)\(11\!\cdots\!48\)\( + \)\(47\!\cdots\!52\)\( T - 25872325295688 T^{2} - 9540936 T^{3} + T^{4} \)
$83$ \( -\)\(15\!\cdots\!92\)\( - \)\(10\!\cdots\!44\)\( T - 36667301759520 T^{2} + 4587600 T^{3} + T^{4} \)
$89$ \( -\)\(24\!\cdots\!04\)\( + \)\(23\!\cdots\!64\)\( T - 59295822522984 T^{2} - 162376 T^{3} + T^{4} \)
$97$ \( \)\(25\!\cdots\!00\)\( + 26618308655581340000 T - 128502414095400 T^{2} - 2726760 T^{3} + T^{4} \)
show more
show less