Properties

Label 384.9.b.a
Level $384$
Weight $9$
Character orbit 384.b
Analytic conductor $156.433$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 384.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(156.433386263\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} + 200 x^{6} - 586 x^{5} + 10949 x^{4} - 20926 x^{3} + 78946 x^{2} - 68580 x + 222900\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} -\beta_{5} q^{5} -7 \beta_{4} q^{7} + 2187 q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} -\beta_{5} q^{5} -7 \beta_{4} q^{7} + 2187 q^{9} + ( -\beta_{2} - 61 \beta_{3} ) q^{11} + ( 22 \beta_{5} - 19 \beta_{6} ) q^{13} + ( -\beta_{1} + 27 \beta_{4} ) q^{15} + ( 11630 - \beta_{7} ) q^{17} + ( \beta_{2} - 667 \beta_{3} ) q^{19} + ( 63 \beta_{5} + 63 \beta_{6} ) q^{21} + ( -4 \beta_{1} - 1186 \beta_{4} ) q^{23} + ( 49489 - 2 \beta_{7} ) q^{25} -2187 \beta_{3} q^{27} + ( -1077 \beta_{5} - 490 \beta_{6} ) q^{29} + ( 44 \beta_{1} + 1183 \beta_{4} ) q^{31} + ( 133164 - 9 \beta_{7} ) q^{33} + ( 63 \beta_{2} - 6601 \beta_{3} ) q^{35} + ( 80 \beta_{5} - 1383 \beta_{6} ) q^{37} + ( 41 \beta_{1} + 3510 \beta_{4} ) q^{39} + ( 372322 + 47 \beta_{7} ) q^{41} + ( -139 \beta_{2} + 11049 \beta_{3} ) q^{43} -2187 \beta_{5} q^{45} + ( 256 \beta_{1} - 2874 \beta_{4} ) q^{47} + 2018065 q^{49} + ( -243 \beta_{2} - 11657 \beta_{3} ) q^{51} + ( 1889 \beta_{5} + 4802 \beta_{6} ) q^{53} + ( 44 \beta_{1} - 33428 \beta_{4} ) q^{55} + ( 1458972 + 9 \beta_{7} ) q^{57} + ( 760 \beta_{2} + 101684 \beta_{3} ) q^{59} + ( 13780 \beta_{5} - 8497 \beta_{6} ) q^{61} -15309 \beta_{4} q^{63} + ( 9628128 - 89 \beta_{7} ) q^{65} + ( -196 \beta_{2} - 79896 \beta_{3} ) q^{67} + ( 2898 \beta_{5} + 11646 \beta_{6} ) q^{69} + ( -252 \beta_{1} + 73282 \beta_{4} ) q^{71} + ( 23923582 - 176 \beta_{7} ) q^{73} + ( -486 \beta_{2} - 49543 \beta_{3} ) q^{75} + ( -49028 \beta_{5} + 10444 \beta_{6} ) q^{77} + ( 836 \beta_{1} - 76433 \beta_{4} ) q^{79} + 4782969 q^{81} + ( 1461 \beta_{2} - 227655 \beta_{3} ) q^{83} + ( 46098 \beta_{5} + 32240 \beta_{6} ) q^{85} + ( -587 \beta_{1} + 134919 \beta_{4} ) q^{87} + ( 52859054 - 278 \beta_{7} ) q^{89} + ( -2583 \beta_{2} - 859327 \beta_{3} ) q^{91} + ( 74889 \beta_{5} - 21339 \beta_{6} ) q^{93} + ( -772 \beta_{1} + 53084 \beta_{4} ) q^{95} + ( 3921070 + 1154 \beta_{7} ) q^{97} + ( -2187 \beta_{2} - 133407 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 17496 q^{9} + O(q^{10}) \) \( 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 200 x^{6} - 586 x^{5} + 10949 x^{4} - 20926 x^{3} + 78946 x^{2} - 68580 x + 222900\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -432 \nu^{6} + 1296 \nu^{5} - 76896 \nu^{4} + 151632 \nu^{3} - 3639600 \nu^{2} + 3564000 \nu - 12572496 \)\()/403\)
\(\beta_{2}\)\(=\)\((\)\( 422 \nu^{7} - 466693 \nu^{6} + 1475675 \nu^{5} - 68583127 \nu^{4} + 138435443 \nu^{3} - 536132632 \nu^{2} + 473860092 \nu + 89182264770 \)\()/5257910\)
\(\beta_{3}\)\(=\)\((\)\( -3798 \nu^{7} + 13293 \nu^{6} - 720243 \nu^{5} + 1767375 \nu^{4} - 35892171 \nu^{3} + 52077528 \nu^{2} - 94544604 \nu + 38651310 \)\()/5257910\)
\(\beta_{4}\)\(=\)\((\)\( -1288 \nu^{7} + 4508 \nu^{6} - 269884 \nu^{5} + 663440 \nu^{4} - 16358356 \nu^{3} + 23876348 \nu^{2} - 177074040 \nu + 84579636 \)\()/976469\)
\(\beta_{5}\)\(=\)\((\)\( 10064 \nu^{7} - 49762 \nu^{6} + 2067878 \nu^{5} - 8127346 \nu^{4} + 120250730 \nu^{3} - 342411844 \nu^{2} + 1396161240 \nu - 1195854900 \)\()/3307395\)
\(\beta_{6}\)\(=\)\((\)\( -10064 \nu^{7} - 81080 \nu^{6} - 1675352 \nu^{5} - 20265368 \nu^{4} - 64119512 \nu^{3} - 1244701616 \nu^{2} + 162952032 \nu - 4311284880 \)\()/3307395\)
\(\beta_{7}\)\(=\)\((\)\( 3072 \nu^{7} - 10752 \nu^{6} + 609024 \nu^{5} - 1495680 \nu^{4} + 31491840 \nu^{3} - 45747456 \nu^{2} + 82874880 \nu - 33862464 \)\()/2423\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{6} + 8 \beta_{5} + 12 \beta_{4} + 16 \beta_{3} + 432\)\()/864\)
\(\nu^{2}\)\(=\)\((\)\(36 \beta_{6} + 144 \beta_{5} + 144 \beta_{4} + 195 \beta_{3} + 27 \beta_{2} - 4 \beta_{1} - 497664\)\()/10368\)
\(\nu^{3}\)\(=\)\((\)\(-9 \beta_{7} + 652 \beta_{6} - 4784 \beta_{5} - 10992 \beta_{4} - 18621 \beta_{3} + 27 \beta_{2} - 4 \beta_{1} - 499392\)\()/6912\)
\(\nu^{4}\)\(=\)\((\)\(-27 \beta_{7} - 9312 \beta_{6} - 25728 \beta_{5} - 33120 \beta_{4} - 56340 \beta_{3} - 2484 \beta_{2} + 616 \beta_{1} + 42814656\)\()/10368\)
\(\nu^{5}\)\(=\)\((\)\(4275 \beta_{7} - 212796 \beta_{6} + 1199088 \beta_{5} + 2870928 \beta_{4} + 8207181 \beta_{3} - 12555 \beta_{2} + 3100 \beta_{1} + 216571968\)\()/20736\)
\(\nu^{6}\)\(=\)\((\)\(540 \beta_{7} + 106610 \beta_{6} + 286520 \beta_{5} + 365772 \beta_{4} + 1039705 \beta_{3} + 17505 \beta_{2} - 6923 \beta_{1} - 302102784\)\()/864\)
\(\nu^{7}\)\(=\)\((\)\(-535311 \beta_{7} + 23742708 \beta_{6} - 92469072 \beta_{5} - 236122704 \beta_{4} - 1026485487 \beta_{3} + 1514457 \beta_{2} - 592396 \beta_{1} - 26136385344\)\()/20736\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
1.36603 + 9.69292i
1.36603 + 2.01178i
1.36603 2.01178i
1.36603 9.69292i
−0.366025 + 2.01178i
−0.366025 + 9.69292i
−0.366025 9.69292i
−0.366025 2.01178i
0 −46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.2 0 −46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.3 0 −46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.4 0 −46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.5 0 46.7654 0 721.475i 0 1935.65i 0 2187.00 0
319.6 0 46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.7 0 46.7654 0 402.176i 0 1935.65i 0 2187.00 0
319.8 0 46.7654 0 721.475i 0 1935.65i 0 2187.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.9.b.a 8
4.b odd 2 1 inner 384.9.b.a 8
8.b even 2 1 inner 384.9.b.a 8
8.d odd 2 1 inner 384.9.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.9.b.a 8 1.a even 1 1 trivial
384.9.b.a 8 4.b odd 2 1 inner
384.9.b.a 8 8.b even 2 1 inner
384.9.b.a 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 682272 T_{5}^{2} + 84192825600 \) acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( -2187 + T^{2} )^{4} \)
$5$ \( ( 84192825600 + 682272 T^{2} + T^{4} )^{2} \)
$7$ \( ( 3746736 + T^{2} )^{4} \)
$11$ \( ( 84020850951158016 - 612159840 T^{2} + T^{4} )^{2} \)
$13$ \( ( 9971438023839744 + 1922702976 T^{2} + T^{4} )^{2} \)
$17$ \( ( -7909979324 - 23260 T + T^{2} )^{4} \)
$19$ \( ( 456063477443078400 - 2542536288 T^{2} + T^{4} )^{2} \)
$23$ \( ( \)\(93\!\cdots\!84\)\( + 237198222720 T^{2} + T^{4} )^{2} \)
$29$ \( ( \)\(27\!\cdots\!00\)\( + 1490396210208 T^{2} + T^{4} )^{2} \)
$31$ \( ( \)\(15\!\cdots\!00\)\( + 2886948519264 T^{2} + T^{4} )^{2} \)
$37$ \( ( \)\(57\!\cdots\!56\)\( + 7501371886080 T^{2} + T^{4} )^{2} \)
$41$ \( ( -17633303147132 - 744644 T + T^{2} )^{4} \)
$43$ \( ( \)\(30\!\cdots\!04\)\( - 12049696420704 T^{2} + T^{4} )^{2} \)
$47$ \( ( \)\(19\!\cdots\!84\)\( + 91745074569600 T^{2} + T^{4} )^{2} \)
$53$ \( ( \)\(14\!\cdots\!44\)\( + 88167062209824 T^{2} + T^{4} )^{2} \)
$59$ \( ( \)\(22\!\cdots\!24\)\( - 389367404122464 T^{2} + T^{4} )^{2} \)
$61$ \( ( \)\(66\!\cdots\!56\)\( + 463016972620800 T^{2} + T^{4} )^{2} \)
$67$ \( ( \)\(62\!\cdots\!36\)\( - 50799407453280 T^{2} + T^{4} )^{2} \)
$71$ \( ( \)\(13\!\cdots\!24\)\( + 908938263991680 T^{2} + T^{4} )^{2} \)
$73$ \( ( 323128538436100 - 47847164 T + T^{2} )^{4} \)
$79$ \( ( \)\(12\!\cdots\!00\)\( + 1858332724601184 T^{2} + T^{4} )^{2} \)
$83$ \( ( \)\(27\!\cdots\!00\)\( - 1499067578583648 T^{2} + T^{4} )^{2} \)
$89$ \( ( 2172311553439300 - 105718108 T + T^{2} )^{4} \)
$97$ \( ( -10698595013335484 - 7842140 T + T^{2} )^{4} \)
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