Properties

Label 384.8.d.f
Level $384$
Weight $8$
Character orbit 384.d
Analytic conductor $119.956$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 8 x^{15} - 1020 x^{14} + 7280 x^{13} + 388150 x^{12} - 2423904 x^{11} - 70542796 x^{10} + 375107560 x^{9} + 6894477923 x^{8} + \cdots + 694045717832241 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{108}\cdot 3^{28} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{3} - \beta_1 q^{5} + \beta_{10} q^{7} - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{3} - \beta_1 q^{5} + \beta_{10} q^{7} - 729 q^{9} + ( - \beta_{11} + 47 \beta_{8}) q^{11} + (\beta_{2} + 4 \beta_1) q^{13} + (\beta_{12} - \beta_{10}) q^{15} + (\beta_{5} + 4830) q^{17} + (\beta_{13} - 4 \beta_{9} - 260 \beta_{8}) q^{19} + (\beta_{4} + \beta_{2} - 11 \beta_1) q^{21} + ( - \beta_{15} - 2 \beta_{14} + \beta_{12} + 4 \beta_{10}) q^{23} + (\beta_{7} - 4 \beta_{5} - 2 \beta_{3} - 14779) q^{25} + 729 \beta_{8} q^{27} + ( - \beta_{6} + \beta_{4} - 6 \beta_{2} + 280 \beta_1) q^{29} + ( - 4 \beta_{15} + 3 \beta_{14} + 12 \beta_{12} + 39 \beta_{10}) q^{31} + (2 \beta_{7} + 2 \beta_{5} + 7 \beta_{3} + 34020) q^{33} + ( - 6 \beta_{13} + \beta_{11} + 115 \beta_{9} - 1427 \beta_{8}) q^{35} + ( - 3 \beta_{6} + 12 \beta_{4} + 7 \beta_{2} + 121 \beta_1) q^{37} + ( - 9 \beta_{14} - 9 \beta_{12} - 54 \beta_{10}) q^{39} + ( - 15 \beta_{7} + 20 \beta_{5} - 15 \beta_{3} - 58182) q^{41} + ( - 5 \beta_{13} + 30 \beta_{11} + 203 \beta_{9} + 2914 \beta_{8}) q^{43} + 729 \beta_1 q^{45} + (13 \beta_{15} - 4 \beta_{14} + 47 \beta_{12} + 56 \beta_{10}) q^{47} + (7 \beta_{7} - 28 \beta_{5} + 28 \beta_{3} + 72689) q^{49} + (9 \beta_{13} + 36 \beta_{11} + 54 \beta_{9} - 4842 \beta_{8}) q^{51} + (\beta_{6} + 41 \beta_{4} + 90 \beta_{2} + 744 \beta_1) q^{53} + (2 \beta_{15} + 45 \beta_{14} - 138 \beta_{12} + 132 \beta_{10}) q^{55} + (18 \beta_{7} - 63 \beta_{5} + 18 \beta_{3} - 189540) q^{57} + (30 \beta_{13} - 54 \beta_{11} + 515 \beta_{9} + 1518 \beta_{8}) q^{59} + ( - 9 \beta_{6} - 54 \beta_{4} + 73 \beta_{2} - 2777 \beta_1) q^{61} - 729 \beta_{10} q^{63} + ( - 51 \beta_{7} + 48 \beta_{5} - 87 \beta_{3} + 392688) q^{65} + (10 \beta_{13} - 270 \beta_{11} + 1067 \beta_{9} - 746 \beta_{8}) q^{67} + (9 \beta_{6} + 18 \beta_{4} - 144 \beta_{2} + 1413 \beta_1) q^{69} + (48 \beta_{15} - 6 \beta_{14} - 204 \beta_{12} + 762 \beta_{10}) q^{71} + (\beta_{7} + 86 \beta_{5} + 106 \beta_{3} - 170450) q^{73} + ( - 45 \beta_{13} - 261 \beta_{11} + 621 \beta_{9} + 14866 \beta_{8}) q^{75} + (27 \beta_{6} - 93 \beta_{4} - 492 \beta_{2} + 1525 \beta_1) q^{77} + ( - 22 \beta_{15} - 60 \beta_{14} + 282 \beta_{12} - 2657 \beta_{10}) q^{79} + 531441 q^{81} + ( - 30 \beta_{13} + 455 \beta_{11} + 1368 \beta_{9} + 15671 \beta_{8}) q^{83} + ( - 3 \beta_{6} - 114 \beta_{4} - 14 \beta_{2} - 15731 \beta_1) q^{85} + ( - 81 \beta_{15} + 63 \beta_{14} - 264 \beta_{12} + 57 \beta_{10}) q^{87} + ( - 21 \beta_{7} - 231 \beta_{5} + 141 \beta_{3} - 2693322) q^{89} + ( - 15 \beta_{13} + 1092 \beta_{11} + 1440 \beta_{9} - 63804 \beta_{8}) q^{91} + (36 \beta_{6} + 26 \beta_{4} + 269 \beta_{2} + 7535 \beta_1) q^{93} + (61 \beta_{15} + 80 \beta_{14} + 1427 \beta_{12} + 10070 \beta_{10}) q^{95} + ( - 190 \beta_{7} - 302 \beta_{5} - 100 \beta_{3} + \cdots + 2221070) q^{97}+ \cdots + (729 \beta_{11} - 34263 \beta_{8}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 11664 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 1020 x^{14} + 7280 x^{13} + 388150 x^{12} - 2423904 x^{11} - 70542796 x^{10} + 375107560 x^{9} + 6894477923 x^{8} + \cdots + 694045717832241 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 34\!\cdots\!03 \nu^{14} + \cdots + 43\!\cdots\!87 ) / 50\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!52 \nu^{14} + \cdots - 29\!\cdots\!93 ) / 71\!\cdots\!62 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!00 \nu^{14} + \cdots - 10\!\cdots\!70 ) / 11\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!82 \nu^{14} + \cdots + 70\!\cdots\!63 ) / 83\!\cdots\!39 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\!\cdots\!52 \nu^{14} + \cdots - 23\!\cdots\!50 ) / 11\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!97 \nu^{14} + \cdots - 47\!\cdots\!83 ) / 50\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!80 \nu^{14} + \cdots + 15\!\cdots\!68 ) / 11\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2164418201736 \nu^{15} + 16233136513020 \nu^{14} + \cdots - 17\!\cdots\!93 ) / 34\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!76 \nu^{15} + \cdots - 12\!\cdots\!36 ) / 80\!\cdots\!01 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43\!\cdots\!48 \nu^{15} + \cdots - 20\!\cdots\!17 ) / 99\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!32 \nu^{15} + \cdots + 41\!\cdots\!63 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 26\!\cdots\!52 \nu^{15} + \cdots - 93\!\cdots\!05 ) / 17\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!28 \nu^{15} + \cdots - 12\!\cdots\!72 ) / 77\!\cdots\!57 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 75\!\cdots\!56 \nu^{15} + \cdots - 25\!\cdots\!63 ) / 17\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 38\!\cdots\!32 \nu^{15} + \cdots - 11\!\cdots\!31 ) / 34\!\cdots\!42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 6\beta_{15} - 12\beta_{14} - 62\beta_{12} + 188\beta_{10} - 27\beta_{9} + 27648 ) / 55296 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12 \beta_{15} - 24 \beta_{14} - 124 \beta_{12} + 376 \beta_{10} - 54 \beta_{9} + 39 \beta_{7} + 12 \beta_{6} + 615 \beta_{5} - 40 \beta_{4} - 807 \beta_{3} + 56 \beta_{2} + 596 \beta _1 + 14542848 ) / 110592 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9888 \beta_{15} - 21928 \beta_{14} - 288 \beta_{13} - 46208 \beta_{12} + 8352 \beta_{11} + 130144 \beta_{10} - 43362 \beta_{9} + 1520928 \beta_{8} + 117 \beta_{7} + 36 \beta_{6} + 1845 \beta_{5} + \cdots + 43573248 ) / 221184 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4938 \beta_{15} - 10952 \beta_{14} - 144 \beta_{13} - 23042 \beta_{12} + 4176 \beta_{11} + 64884 \beta_{10} - 21654 \beta_{9} + 760464 \beta_{8} + 2637 \beta_{7} + 2606 \beta_{6} + \cdots + 1936935936 ) / 55296 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3453552 \beta_{15} - 8292808 \beta_{14} - 364800 \beta_{13} - 13398160 \beta_{12} + 5579520 \beta_{11} + 40659072 \beta_{10} - 19488042 \beta_{9} + 932847360 \beta_{8} + \cdots + 19296755712 ) / 221184 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2577822 \beta_{15} - 6192232 \beta_{14} - 273240 \beta_{13} - 9991046 \beta_{12} + 4174200 \beta_{11} + 30332188 \beta_{10} - 14561910 \beta_{9} + 697734360 \beta_{8} + \cdots + 642353144832 ) / 55296 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 304967922 \beta_{15} - 746679828 \beta_{14} - 51763824 \beta_{13} - 1133014618 \beta_{12} + 716269680 \beta_{11} + 3507185332 \beta_{10} - 2284093827 \beta_{9} + \cdots + 2231364049920 ) / 55296 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 603926685 \beta_{15} - 1478923888 \beta_{14} - 102890256 \beta_{13} - 2242743657 \beta_{12} + 1422804432 \beta_{11} + 6943671194 \beta_{10} + \cdots + 112235382065664 ) / 27648 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 143900998072 \beta_{15} - 353868327288 \beta_{14} - 32705604480 \beta_{13} - 529692128760 \beta_{12} + 443473625472 \beta_{11} + 1645677868496 \beta_{10} + \cdots + 13\!\cdots\!32 ) / 73728 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1061175725136 \beta_{15} - 2609731319784 \beta_{14} - 242209149840 \beta_{13} - 3905548300400 \beta_{12} + 3283426455120 \beta_{11} + \cdots + 15\!\cdots\!08 ) / 110592 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 152441390392872 \beta_{15} - 375229726968440 \beta_{14} - 43374027325728 \beta_{13} - 560059958667432 \beta_{12} + 585603407276448 \beta_{11} + \cdots + 17\!\cdots\!60 ) / 221184 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 111422746025514 \beta_{15} - 274269868316848 \beta_{14} - 31866140017008 \beta_{13} - 409341606733250 \beta_{12} + 430196563097136 \beta_{11} + \cdots + 13\!\cdots\!00 ) / 27648 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 26\!\cdots\!40 \beta_{15} + \cdots + 35\!\cdots\!08 ) / 110592 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 18\!\cdots\!36 \beta_{15} + \cdots + 19\!\cdots\!52 ) / 110592 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 18\!\cdots\!12 \beta_{15} + \cdots + 28\!\cdots\!88 ) / 221184 \) 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
9.25013 0.707107i
9.47155 + 0.707107i
19.4308 + 0.707107i
3.90187 0.707107i
−2.90187 0.707107i
−18.4308 + 0.707107i
−8.47155 + 0.707107i
−8.25013 0.707107i
−8.25013 + 0.707107i
−8.47155 0.707107i
−18.4308 0.707107i
−2.90187 + 0.707107i
3.90187 + 0.707107i
19.4308 0.707107i
9.47155 0.707107i
9.25013 + 0.707107i
0 27.0000i 0 484.521i 0 −826.966 0 −729.000 0
193.2 0 27.0000i 0 295.990i 0 1208.55 0 −729.000 0
193.3 0 27.0000i 0 183.972i 0 284.564 0 −729.000 0
193.4 0 27.0000i 0 124.096i 0 1165.97 0 −729.000 0
193.5 0 27.0000i 0 124.096i 0 −1165.97 0 −729.000 0
193.6 0 27.0000i 0 183.972i 0 −284.564 0 −729.000 0
193.7 0 27.0000i 0 295.990i 0 −1208.55 0 −729.000 0
193.8 0 27.0000i 0 484.521i 0 826.966 0 −729.000 0
193.9 0 27.0000i 0 484.521i 0 826.966 0 −729.000 0
193.10 0 27.0000i 0 295.990i 0 −1208.55 0 −729.000 0
193.11 0 27.0000i 0 183.972i 0 −284.564 0 −729.000 0
193.12 0 27.0000i 0 124.096i 0 −1165.97 0 −729.000 0
193.13 0 27.0000i 0 124.096i 0 1165.97 0 −729.000 0
193.14 0 27.0000i 0 183.972i 0 284.564 0 −729.000 0
193.15 0 27.0000i 0 295.990i 0 1208.55 0 −729.000 0
193.16 0 27.0000i 0 484.521i 0 −826.966 0 −729.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.16
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 384.8.d.f 16
4.b odd 2 1 inner 384.8.d.f 16
8.b even 2 1 inner 384.8.d.f 16
8.d odd 2 1 inner 384.8.d.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.8.d.f 16 1.a even 1 1 trivial
384.8.d.f 16 4.b odd 2 1 inner
384.8.d.f 16 8.b even 2 1 inner
384.8.d.f 16 8.d odd 2 1 inner

Hecke kernels

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

\( T_{5}^{8} + 371616T_{5}^{6} + 36963868032T_{5}^{4} + 1180874777241600T_{5}^{2} + 10720051644418560000 \) Copy content Toggle raw display
\( T_{7}^{8} - 3584928T_{7}^{6} + 4197967850880T_{7}^{4} - 1674896085793253376T_{7}^{2} + 109960973310053162225664 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{8} \) Copy content Toggle raw display
$5$ \( (T^{8} + 371616 T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 3584928 T^{6} + \cdots + 10\!\cdots\!64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 126799424 T^{6} + \cdots + 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 318245760 T^{6} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 19320 T^{3} + \cdots - 23\!\cdots\!36)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 5744950848 T^{6} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 17810534016 T^{6} + \cdots + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 96093574560 T^{6} + \cdots + 40\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 155053631136 T^{6} + \cdots + 23\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 815053715712 T^{6} + \cdots + 41\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 232728 T^{3} + \cdots + 31\!\cdots\!76)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 629422244928 T^{6} + \cdots + 59\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 1639414339200 T^{6} + \cdots + 32\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 6158991470496 T^{6} + \cdots + 29\!\cdots\!04)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 7502487677504 T^{6} + \cdots + 36\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 17143963828992 T^{6} + \cdots + 27\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 18631493656128 T^{6} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 27329684840064 T^{6} + \cdots + 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 681800 T^{3} + \cdots + 37\!\cdots\!64)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} - 64280992475808 T^{6} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 47708870931008 T^{6} + \cdots + 57\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 10773288 T^{3} + \cdots - 18\!\cdots\!84)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 8884280 T^{3} + \cdots - 86\!\cdots\!16)^{4} \) Copy content Toggle raw display
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