Properties

Label 384.10.d.b
Level $384$
Weight $10$
Character orbit 384.d
Analytic conductor $197.774$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 81 \beta_{4} q^{3} + ( - \beta_{5} + 140 \beta_{4}) q^{5} + ( - \beta_{3} + 2 \beta_1 + 1704) q^{7} - 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 81 \beta_{4} q^{3} + ( - \beta_{5} + 140 \beta_{4}) q^{5} + ( - \beta_{3} + 2 \beta_1 + 1704) q^{7} - 6561 q^{9} + ( - 5 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - 9588 \beta_{4}) q^{11} + (4 \beta_{7} - 15 \beta_{6} - 5 \beta_{5} + 35820 \beta_{4}) q^{13} + ( - 81 \beta_1 + 11340) q^{15} + (60 \beta_{3} - 7 \beta_{2} + 99 \beta_1 + 1038) q^{17} + (61 \beta_{7} - 100 \beta_{6} + 33 \beta_{5} + 58860 \beta_{4}) q^{19} + ( - 81 \beta_{6} - 162 \beta_{5} - 138024 \beta_{4}) q^{21} + (110 \beta_{3} - 56 \beta_{2} + 24 \beta_1 + 576576) q^{23} + (204 \beta_{3} + 158 \beta_{2} - 426 \beta_1 - 594363) q^{25} + 531441 \beta_{4} q^{27} + ( - 652 \beta_{7} + 366 \beta_{6} - 167 \beta_{5} - 653500 \beta_{4}) q^{29} + ( - 95 \beta_{3} + 172 \beta_{2} - 1138 \beta_1 + 937416) q^{31} + (648 \beta_{3} + 405 \beta_{2} - 405 \beta_1 - 776628) q^{33} + ( - 665 \beta_{7} + 960 \beta_{6} - 4177 \beta_{5} - 4302960 \beta_{4}) q^{35} + (828 \beta_{7} + 513 \beta_{6} + 5173 \beta_{5} + 3851156 \beta_{4}) q^{37} + (1215 \beta_{3} - 324 \beta_{2} - 405 \beta_1 + 2901420) q^{39} + (1860 \beta_{3} - 235 \beta_{2} - 10817 \beta_1 - 5439862) q^{41} + (1509 \beta_{7} - 1564 \beta_{6} - 7247 \beta_{5} + 447468 \beta_{4}) q^{43} + (6561 \beta_{5} - 918540 \beta_{4}) q^{45} + ( - 3950 \beta_{3} - 672 \beta_{2} - 16820 \beta_1 - 6172752) q^{47} + ( - 348 \beta_{3} - 2492 \beta_{2} + 5288 \beta_1 - 9263351) q^{49} + ( - 567 \beta_{7} + 4860 \beta_{6} - 8019 \beta_{5} - 84078 \beta_{4}) q^{51} + (4020 \beta_{7} + 15570 \beta_{6} - 897 \beta_{5} - 2656004 \beta_{4}) q^{53} + ( - 3744 \beta_{3} + 5492 \beta_{2} - 44776 \beta_1 - 2378832) q^{55} + (8100 \beta_{3} - 4941 \beta_{2} + 2673 \beta_1 + 4767660) q^{57} + ( - 5328 \beta_{7} - 7944 \beta_{6} - 30648 \beta_{5} + 19616076 \beta_{4}) q^{59} + ( - 628 \beta_{7} - 9309 \beta_{6} - 7253 \beta_{5} - 5818932 \beta_{4}) q^{61} + (6561 \beta_{3} - 13122 \beta_1 - 11179944) q^{63} + ( - 24444 \beta_{3} + 4497 \beta_{2} + 30179 \beta_1 - 13783792) q^{65} + (7700 \beta_{7} - 5112 \beta_{6} - 117668 \beta_{5} + 53533068 \beta_{4}) q^{67} + ( - 4536 \beta_{7} + 8910 \beta_{6} - 1944 \beta_{5} - 46702656 \beta_{4}) q^{69} + ( - 1406 \beta_{3} - 24920 \beta_{2} - 38296 \beta_1 + 92718912) q^{71} + ( - 8796 \beta_{3} + 21044 \beta_{2} - 33352 \beta_1 - 188487930) q^{73} + (12798 \beta_{7} + 16524 \beta_{6} + 34506 \beta_{5} + 48143403 \beta_{4}) q^{75} + ( - 14476 \beta_{7} - 30900 \beta_{6} + \cdots - 180436192 \beta_{4}) q^{77}+ \cdots + (32805 \beta_{7} + 52488 \beta_{6} + 32805 \beta_{5} + 62906868 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4766\nu^{6} + 51851374\nu^{4} + 88099997114\nu^{2} - 43229725983072 ) / 34840779813 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 169786\nu^{6} + 1579828010\nu^{4} + 1426285970446\nu^{2} - 1771419698208576 ) / 243885458691 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 199366\nu^{6} + 2241395438\nu^{4} + 5124570485866\nu^{2} + 1439961094183104 ) / 243885458691 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28496\nu^{7} + 372522607\nu^{5} + 1264029013547\nu^{3} + 1137344409469551\nu ) / 4843958573246310 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 797877302 \nu^{7} - 9758638510954 \nu^{5} + \cdots - 22\!\cdots\!32 \nu ) / 75\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1012635742 \nu^{7} - 15512580206174 \nu^{5} + \cdots - 96\!\cdots\!32 \nu ) / 75\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 575969794 \nu^{7} + 7049643741998 \nu^{5} + \cdots + 18\!\cdots\!44 \nu ) / 10\!\cdots\!15 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 5\beta_{5} - 96\beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 48\beta_{3} + 13\beta_{2} - 353\beta _1 - 626976 ) / 192 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5251\beta_{7} - 2988\beta_{6} - 36371\beta_{5} - 24620160\beta_{4} ) / 192 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -307416\beta_{3} - 258409\beta_{2} + 3152165\beta _1 + 3849294240 ) / 192 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17617919\beta_{7} + 13863726\beta_{6} + 145754053\beta_{5} + 134259999792\beta_{4} ) / 96 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1228613316\beta_{3} + 1285518437\beta_{2} - 13182475621\beta _1 - 14273447764896 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -267619114645\beta_{7} - 229933464768\beta_{6} - 2397041072585\beta_{5} - 2381735984318112\beta_{4} ) / 192 \) 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
90.8862i
53.8781i
4.34998i
43.3581i
43.3581i
4.34998i
53.8781i
90.8862i
0 81.0000i 0 2178.24i 0 7658.16 0 −6561.00 0
193.2 0 81.0000i 0 473.533i 0 574.939 0 −6561.00 0
193.3 0 81.0000i 0 1428.10i 0 −6382.13 0 −6561.00 0
193.4 0 81.0000i 0 1783.68i 0 4965.03 0 −6561.00 0
193.5 0 81.0000i 0 1783.68i 0 4965.03 0 −6561.00 0
193.6 0 81.0000i 0 1428.10i 0 −6382.13 0 −6561.00 0
193.7 0 81.0000i 0 473.533i 0 574.939 0 −6561.00 0
193.8 0 81.0000i 0 2178.24i 0 7658.16 0 −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.8
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 384.10.d.b yes 8
4.b odd 2 1 384.10.d.a 8
8.b even 2 1 inner 384.10.d.b yes 8
8.d odd 2 1 384.10.d.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.d.a 8 4.b odd 2 1
384.10.d.a 8 8.d odd 2 1
384.10.d.b yes 8 1.a even 1 1 trivial
384.10.d.b yes 8 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 \(S_{10}^{\mathrm{new}}(384, [\chi])\):

\( T_{5}^{8} + 10189952T_{5}^{6} + 33495402213376T_{5}^{4} + 37796298115527475200T_{5}^{2} + 6903383511575235133440000 \) Copy content Toggle raw display
\( T_{7}^{4} - 6816T_{7}^{3} - 38951584T_{7}^{2} + 267125408256T_{7} - 139519152670464 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6561)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + 10189952 T^{6} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} - 6816 T^{3} + \cdots - 139519152670464)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 11978643008 T^{6} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{8} + 27798823744 T^{6} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4152 T^{3} + \cdots - 30\!\cdots\!80)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 1840089688640 T^{6} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2306304 T^{3} + \cdots - 24\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 126446858508928 T^{6} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{4} - 3749664 T^{3} + \cdots - 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 489154292072000 T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + 21759448 T^{3} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( (T^{4} + 24691008 T^{3} + \cdots - 77\!\cdots\!80)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} - 370875648 T^{3} + \cdots - 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 753951720 T^{3} + \cdots - 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 504186720 T^{3} + \cdots + 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 34\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + 668517224 T^{3} + \cdots + 15\!\cdots\!60)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 452408968 T^{3} + \cdots + 71\!\cdots\!20)^{2} \) Copy content Toggle raw display
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