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: \( x^{8} - 4x^{7} + 200x^{6} - 586x^{5} + 10949x^{4} - 20926x^{3} + 78946x^{2} - 68580x + 222900 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{5} q^{5} - 7 \beta_{4} q^{7} + 2187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{5} q^{5} - 7 \beta_{4} q^{7} + 2187 q^{9} + ( - 61 \beta_{3} - \beta_{2}) q^{11} + ( - 19 \beta_{6} + 22 \beta_{5}) q^{13} + (27 \beta_{4} - \beta_1) q^{15} + ( - \beta_{7} + 11630) q^{17} + ( - 667 \beta_{3} + \beta_{2}) q^{19} + (63 \beta_{6} + 63 \beta_{5}) q^{21} + ( - 1186 \beta_{4} - 4 \beta_1) q^{23} + ( - 2 \beta_{7} + 49489) q^{25} - 2187 \beta_{3} q^{27} + ( - 490 \beta_{6} - 1077 \beta_{5}) q^{29} + (1183 \beta_{4} + 44 \beta_1) q^{31} + ( - 9 \beta_{7} + 133164) q^{33} + ( - 6601 \beta_{3} + 63 \beta_{2}) q^{35} + ( - 1383 \beta_{6} + 80 \beta_{5}) q^{37} + (3510 \beta_{4} + 41 \beta_1) q^{39} + (47 \beta_{7} + 372322) q^{41} + (11049 \beta_{3} - 139 \beta_{2}) q^{43} - 2187 \beta_{5} q^{45} + ( - 2874 \beta_{4} + 256 \beta_1) q^{47} + 2018065 q^{49} + ( - 11657 \beta_{3} - 243 \beta_{2}) q^{51} + (4802 \beta_{6} + 1889 \beta_{5}) q^{53} + ( - 33428 \beta_{4} + 44 \beta_1) q^{55} + (9 \beta_{7} + 1458972) q^{57} + (101684 \beta_{3} + 760 \beta_{2}) q^{59} + ( - 8497 \beta_{6} + 13780 \beta_{5}) q^{61} - 15309 \beta_{4} q^{63} + ( - 89 \beta_{7} + 9628128) q^{65} + ( - 79896 \beta_{3} - 196 \beta_{2}) q^{67} + (11646 \beta_{6} + 2898 \beta_{5}) q^{69} + (73282 \beta_{4} - 252 \beta_1) q^{71} + ( - 176 \beta_{7} + 23923582) q^{73} + ( - 49543 \beta_{3} - 486 \beta_{2}) q^{75} + (10444 \beta_{6} - 49028 \beta_{5}) q^{77} + ( - 76433 \beta_{4} + 836 \beta_1) q^{79} + 4782969 q^{81} + ( - 227655 \beta_{3} + 1461 \beta_{2}) q^{83} + (32240 \beta_{6} + 46098 \beta_{5}) q^{85} + (134919 \beta_{4} - 587 \beta_1) q^{87} + ( - 278 \beta_{7} + 52859054) q^{89} + ( - 859327 \beta_{3} - 2583 \beta_{2}) q^{91} + ( - 21339 \beta_{6} + 74889 \beta_{5}) q^{93} + (53084 \beta_{4} - 772 \beta_1) q^{95} + (1154 \beta_{7} + 3921070) q^{97} + ( - 133407 \beta_{3} - 2187 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 17496 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 17496 q^{9} + 93040 q^{17} + 395912 q^{25} + 1065312 q^{33} + 2978576 q^{41} + 16144520 q^{49} + 11671776 q^{57} + 77025024 q^{65} + 191388656 q^{73} + 38263752 q^{81} + 422872432 q^{89} + 31368560 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -432\nu^{6} + 1296\nu^{5} - 76896\nu^{4} + 151632\nu^{3} - 3639600\nu^{2} + 3564000\nu - 12572496 ) / 403 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 422 \nu^{7} - 466693 \nu^{6} + 1475675 \nu^{5} - 68583127 \nu^{4} + 138435443 \nu^{3} - 536132632 \nu^{2} + 473860092 \nu + 89182264770 ) / 5257910 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3798 \nu^{7} + 13293 \nu^{6} - 720243 \nu^{5} + 1767375 \nu^{4} - 35892171 \nu^{3} + 52077528 \nu^{2} - 94544604 \nu + 38651310 ) / 5257910 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1288 \nu^{7} + 4508 \nu^{6} - 269884 \nu^{5} + 663440 \nu^{4} - 16358356 \nu^{3} + 23876348 \nu^{2} - 177074040 \nu + 84579636 ) / 976469 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10064 \nu^{7} - 49762 \nu^{6} + 2067878 \nu^{5} - 8127346 \nu^{4} + 120250730 \nu^{3} - 342411844 \nu^{2} + 1396161240 \nu - 1195854900 ) / 3307395 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10064 \nu^{7} - 81080 \nu^{6} - 1675352 \nu^{5} - 20265368 \nu^{4} - 64119512 \nu^{3} - 1244701616 \nu^{2} + 162952032 \nu - 4311284880 ) / 3307395 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3072 \nu^{7} - 10752 \nu^{6} + 609024 \nu^{5} - 1495680 \nu^{4} + 31491840 \nu^{3} - 45747456 \nu^{2} + 82874880 \nu - 33862464 ) / 2423 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + 8\beta_{5} + 12\beta_{4} + 16\beta_{3} + 432 ) / 864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 36\beta_{6} + 144\beta_{5} + 144\beta_{4} + 195\beta_{3} + 27\beta_{2} - 4\beta _1 - 497664 ) / 10368 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{7} + 652 \beta_{6} - 4784 \beta_{5} - 10992 \beta_{4} - 18621 \beta_{3} + 27 \beta_{2} - 4 \beta _1 - 499392 ) / 6912 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27 \beta_{7} - 9312 \beta_{6} - 25728 \beta_{5} - 33120 \beta_{4} - 56340 \beta_{3} - 2484 \beta_{2} + 616 \beta _1 + 42814656 ) / 10368 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4275 \beta_{7} - 212796 \beta_{6} + 1199088 \beta_{5} + 2870928 \beta_{4} + 8207181 \beta_{3} - 12555 \beta_{2} + 3100 \beta _1 + 216571968 ) / 20736 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 540 \beta_{7} + 106610 \beta_{6} + 286520 \beta_{5} + 365772 \beta_{4} + 1039705 \beta_{3} + 17505 \beta_{2} - 6923 \beta _1 - 302102784 ) / 864 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 535311 \beta_{7} + 23742708 \beta_{6} - 92469072 \beta_{5} - 236122704 \beta_{4} - 1026485487 \beta_{3} + 1514457 \beta_{2} - 592396 \beta _1 - 26136385344 ) / 20736 \) Copy content Toggle raw display

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 \( T_{5}^{4} + 682272T_{5}^{2} + 84192825600 \) acting on \(S_{9}^{\mathrm{new}}(384, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2187)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} + 682272 T^{2} + \cdots + 84192825600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 3746736)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 612159840 T^{2} + \cdots + 84\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1922702976 T^{2} + \cdots + 99\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 23260 T - 7909979324)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2542536288 T^{2} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 237198222720 T^{2} + \cdots + 93\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1490396210208 T^{2} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2886948519264 T^{2} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 7501371886080 T^{2} + \cdots + 57\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 744644 T - 17633303147132)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 12049696420704 T^{2} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 91745074569600 T^{2} + \cdots + 19\!\cdots\!84)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 88167062209824 T^{2} + \cdots + 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 389367404122464 T^{2} + \cdots + 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 463016972620800 T^{2} + \cdots + 66\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 50799407453280 T^{2} + \cdots + 62\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 908938263991680 T^{2} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 47847164 T + 323128538436100)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 105718108 T + 21\!\cdots\!00)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 7842140 T - 10\!\cdots\!84)^{4} \) Copy content Toggle raw display
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