Properties

Label 384.8.d.b
Level $384$
Weight $8$
Character orbit 384.d
Analytic conductor $119.956$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \( x^{6} + 277x^{4} + 19236x^{2} + 9216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 27 \beta_1 q^{3} + (\beta_{3} - 37 \beta_1) q^{5} + ( - \beta_{4} + 23) q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 27 \beta_1 q^{3} + (\beta_{3} - 37 \beta_1) q^{5} + ( - \beta_{4} + 23) q^{7} - 729 q^{9} + (8 \beta_{3} + 4 \beta_{2} - 1632 \beta_1) q^{11} + (19 \beta_{3} - 3 \beta_{2} - 3642 \beta_1) q^{13} + (27 \beta_{5} - 999) q^{15} + (20 \beta_{5} + 24 \beta_{4} - 4302) q^{17} + (68 \beta_{3} - 24 \beta_{2} - 14968 \beta_1) q^{19} + (27 \beta_{2} - 621 \beta_1) q^{21} + (76 \beta_{5} - 18 \beta_{4} + 2146) q^{23} + ( - 28 \beta_{5} + 60 \beta_{4} - 3099) q^{25} + 19683 \beta_1 q^{27} + ( - 177 \beta_{3} - 186 \beta_{2} + 5963 \beta_1) q^{29} + (268 \beta_{5} + 137 \beta_{4} + 36789) q^{31} + (216 \beta_{5} + 108 \beta_{4} - 44064) q^{33} + (512 \beta_{3} + 140 \beta_{2} - 22724 \beta_1) q^{35} + (329 \beta_{3} - 171 \beta_{2} - 141616 \beta_1) q^{37} + (513 \beta_{5} - 81 \beta_{4} - 98334) q^{39} + ( - 500 \beta_{5} - 480 \beta_{4} - 103282) q^{41} + (1692 \beta_{3} + 64 \beta_{2} + 37480 \beta_1) q^{43} + ( - 729 \beta_{3} + 26973 \beta_1) q^{45} + (1104 \beta_{5} - 734 \beta_{4} - 53726) q^{47} + ( - 1204 \beta_{5} + 444 \beta_{4} - 97463) q^{49} + ( - 540 \beta_{3} - 648 \beta_{2} + 116154 \beta_1) q^{51} + (777 \beta_{3} + 810 \beta_{2} - 307091 \beta_1) q^{53} + (3068 \beta_{5} + 1040 \beta_{4} - 786716) q^{55} + (1836 \beta_{5} - 648 \beta_{4} - 404136) q^{57} + (5448 \beta_{3} - 120 \beta_{2} - 335772 \beta_1) q^{59} + (735 \beta_{3} + 1335 \beta_{2} - 1471404 \beta_1) q^{61} + (729 \beta_{4} - 16767) q^{63} + (940 \beta_{5} + 720 \beta_{4} - 1586380) q^{65} + ( - 9992 \beta_{3} + 1672 \beta_{2} + 1061396 \beta_1) q^{67} + ( - 2052 \beta_{3} + 486 \beta_{2} - 57942 \beta_1) q^{69} + ( - 9404 \beta_{5} - 1438 \beta_{4} + 970974) q^{71} + (6060 \beta_{5} - 3108 \beta_{4} - 1842810) q^{73} + (756 \beta_{3} - 1620 \beta_{2} + 83673 \beta_1) q^{75} + (8912 \beta_{3} + 588 \beta_{2} - 3114724 \beta_1) q^{77} + ( - 10316 \beta_{5} + 1289 \beta_{4} + 78765) q^{79} + 531441 q^{81} + ( - 20912 \beta_{3} - 588 \beta_{2} + 1713736 \beta_1) q^{83} + ( - 14738 \beta_{3} - 4560 \beta_{2} + 2281226 \beta_1) q^{85} + ( - 4779 \beta_{5} - 5022 \beta_{4} + 161001) q^{87} + ( - 29320 \beta_{5} + 4848 \beta_{4} + 1598734) q^{89} + (6116 \beta_{3} + 7000 \beta_{2} + 1677300 \beta_1) q^{91} + ( - 7236 \beta_{3} - 3699 \beta_{2} - 993303 \beta_1) q^{93} + ( - 1188 \beta_{5} + 720 \beta_{4} - 5459004) q^{95} + (9200 \beta_{5} + 2568 \beta_{4} - 1583050) q^{97} + ( - 5832 \beta_{3} - 2916 \beta_{2} + 1189728 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 136 q^{7} - 4374 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 277x^{4} + 19236x^{2} + 9216 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 181\nu^{3} + 5892\nu ) / 4032 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -13\nu^{5} - 18481\nu^{3} - 1963572\nu ) / 4032 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -19\nu^{5} + 12689\nu^{3} + 2162100\nu ) / 4032 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\nu^{4} + 274\nu^{2} - 205 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{4} + 306\nu^{2} + 2747 ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 32\beta_1 ) / 96 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{5} - \beta_{4} - 2952 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -39\beta_{3} - 47\beta_{2} - 1352\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -959\beta_{5} + 153\beta_{4} + 407704 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5095\beta_{3} + 6543\beta_{2} + 310888\beta_1 ) / 32 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
11.6046i
11.9101i
0.694584i
0.694584i
11.9101i
11.6046i
0 27.0000i 0 349.927i 0 856.119 0 −729.000 0
193.2 0 27.0000i 0 96.4778i 0 −1147.84 0 −729.000 0
193.3 0 27.0000i 0 334.405i 0 359.725 0 −729.000 0
193.4 0 27.0000i 0 334.405i 0 359.725 0 −729.000 0
193.5 0 27.0000i 0 96.4778i 0 −1147.84 0 −729.000 0
193.6 0 27.0000i 0 349.927i 0 856.119 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.8.d.b yes 6
4.b odd 2 1 384.8.d.a 6
8.b even 2 1 inner 384.8.d.b yes 6
8.d odd 2 1 384.8.d.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.a 6 4.b odd 2 1
384.8.d.a 6 8.d odd 2 1
384.8.d.b yes 6 1.a even 1 1 trivial
384.8.d.b yes 6 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{6} + 243584T_{5}^{4} + 15873740800T_{5}^{2} + 127455068160000 \) Copy content Toggle raw display
\( T_{7}^{3} - 68T_{7}^{2} - 1087632T_{7} + 353498688 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 127455068160000 \) Copy content Toggle raw display
$7$ \( (T^{3} - 68 T^{2} - 1087632 T + 353498688)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 62402992 T^{4} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{6} + 138457776 T^{4} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{3} + 12902 T^{2} + \cdots - 3423209334392)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 2821164208 T^{4} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( (T^{3} - 6344 T^{2} + \cdots - 5673212960256)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 87314947392 T^{4} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$31$ \( (T^{3} - 110236 T^{2} + \cdots + 26\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 142490404528 T^{4} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{3} + 309826 T^{2} + \cdots - 36\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 712688054704 T^{4} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{3} + 163016 T^{2} + \cdots + 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 1939725513024 T^{4} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{6} + 7392682583856 T^{4} + \cdots + 72\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{6} + 10634490988848 T^{4} + \cdots + 49\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{6} + 31190766079536 T^{4} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{3} - 2920888 T^{2} + \cdots - 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 5537598 T^{2} + \cdots - 97\!\cdots\!56)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 247900 T^{2} + \cdots + 20\!\cdots\!56)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 115950455782576 T^{4} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{3} - 4830370 T^{2} + \cdots + 66\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 4755782 T^{2} + \cdots + 56\!\cdots\!96)^{2} \) Copy content Toggle raw display
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