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 $2$

Related objects

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: $$x^{6} + 277x^{4} + 19236x^{2} + 9216$$ x^6 + 277*x^4 + 19236*x^2 + 9216 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.

 $$f(q)$$ $$=$$ $$q - 27 \beta_1 q^{3} + (\beta_{3} - 37 \beta_1) q^{5} + ( - \beta_{4} + 23) q^{7} - 729 q^{9}+O(q^{10})$$ q - 27*b1 * q^3 + (b3 - 37*b1) * q^5 + (-b4 + 23) * q^7 - 729 * q^9 $$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})$$ q - 27*b1 * q^3 + (b3 - 37*b1) * q^5 + (-b4 + 23) * q^7 - 729 * q^9 + (8*b3 + 4*b2 - 1632*b1) * q^11 + (19*b3 - 3*b2 - 3642*b1) * q^13 + (27*b5 - 999) * q^15 + (20*b5 + 24*b4 - 4302) * q^17 + (68*b3 - 24*b2 - 14968*b1) * q^19 + (27*b2 - 621*b1) * q^21 + (76*b5 - 18*b4 + 2146) * q^23 + (-28*b5 + 60*b4 - 3099) * q^25 + 19683*b1 * q^27 + (-177*b3 - 186*b2 + 5963*b1) * q^29 + (268*b5 + 137*b4 + 36789) * q^31 + (216*b5 + 108*b4 - 44064) * q^33 + (512*b3 + 140*b2 - 22724*b1) * q^35 + (329*b3 - 171*b2 - 141616*b1) * q^37 + (513*b5 - 81*b4 - 98334) * q^39 + (-500*b5 - 480*b4 - 103282) * q^41 + (1692*b3 + 64*b2 + 37480*b1) * q^43 + (-729*b3 + 26973*b1) * q^45 + (1104*b5 - 734*b4 - 53726) * q^47 + (-1204*b5 + 444*b4 - 97463) * q^49 + (-540*b3 - 648*b2 + 116154*b1) * q^51 + (777*b3 + 810*b2 - 307091*b1) * q^53 + (3068*b5 + 1040*b4 - 786716) * q^55 + (1836*b5 - 648*b4 - 404136) * q^57 + (5448*b3 - 120*b2 - 335772*b1) * q^59 + (735*b3 + 1335*b2 - 1471404*b1) * q^61 + (729*b4 - 16767) * q^63 + (940*b5 + 720*b4 - 1586380) * q^65 + (-9992*b3 + 1672*b2 + 1061396*b1) * q^67 + (-2052*b3 + 486*b2 - 57942*b1) * q^69 + (-9404*b5 - 1438*b4 + 970974) * q^71 + (6060*b5 - 3108*b4 - 1842810) * q^73 + (756*b3 - 1620*b2 + 83673*b1) * q^75 + (8912*b3 + 588*b2 - 3114724*b1) * q^77 + (-10316*b5 + 1289*b4 + 78765) * q^79 + 531441 * q^81 + (-20912*b3 - 588*b2 + 1713736*b1) * q^83 + (-14738*b3 - 4560*b2 + 2281226*b1) * q^85 + (-4779*b5 - 5022*b4 + 161001) * q^87 + (-29320*b5 + 4848*b4 + 1598734) * q^89 + (6116*b3 + 7000*b2 + 1677300*b1) * q^91 + (-7236*b3 - 3699*b2 - 993303*b1) * q^93 + (-1188*b5 + 720*b4 - 5459004) * q^95 + (9200*b5 + 2568*b4 - 1583050) * q^97 + (-5832*b3 - 2916*b2 + 1189728*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 136 q^{7} - 4374 q^{9}+O(q^{10})$$ 6 * q + 136 * q^7 - 4374 * q^9 $$6 q + 136 q^{7} - 4374 q^{9} - 6048 q^{15} - 25804 q^{17} + 12688 q^{23} - 18418 q^{25} + 220472 q^{31} - 264600 q^{33} - 591192 q^{39} - 619652 q^{41} - 326032 q^{47} - 581482 q^{49} - 4724352 q^{55} - 2429784 q^{57} - 99144 q^{63} - 9518720 q^{65} + 5841776 q^{71} - 11075196 q^{73} + 495800 q^{79} + 3188646 q^{81} + 965520 q^{87} + 9660740 q^{89} - 32750208 q^{95} - 9511564 q^{97}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 277x^{4} + 19236x^{2} + 9216$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} + 181\nu^{3} + 5892\nu ) / 4032$$ (v^5 + 181*v^3 + 5892*v) / 4032 $$\beta_{2}$$ $$=$$ $$( -13\nu^{5} - 18481\nu^{3} - 1963572\nu ) / 4032$$ (-13*v^5 - 18481*v^3 - 1963572*v) / 4032 $$\beta_{3}$$ $$=$$ $$( -19\nu^{5} + 12689\nu^{3} + 2162100\nu ) / 4032$$ (-19*v^5 + 12689*v^3 + 2162100*v) / 4032 $$\beta_{4}$$ $$=$$ $$2\nu^{4} + 274\nu^{2} - 205$$ 2*v^4 + 274*v^2 - 205 $$\beta_{5}$$ $$=$$ $$( 2\nu^{4} + 306\nu^{2} + 2747 ) / 7$$ (2*v^4 + 306*v^2 + 2747) / 7
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} + 32\beta_1 ) / 96$$ (b3 + b2 + 32*b1) / 96 $$\nu^{2}$$ $$=$$ $$( 7\beta_{5} - \beta_{4} - 2952 ) / 32$$ (7*b5 - b4 - 2952) / 32 $$\nu^{3}$$ $$=$$ $$( -39\beta_{3} - 47\beta_{2} - 1352\beta_1 ) / 32$$ (-39*b3 - 47*b2 - 1352*b1) / 32 $$\nu^{4}$$ $$=$$ $$( -959\beta_{5} + 153\beta_{4} + 407704 ) / 32$$ (-959*b5 + 153*b4 + 407704) / 32 $$\nu^{5}$$ $$=$$ $$( 5095\beta_{3} + 6543\beta_{2} + 310888\beta_1 ) / 32$$ (5095*b3 + 6543*b2 + 310888*b1) / 32

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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 $$S_{8}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{6} + 243584T_{5}^{4} + 15873740800T_{5}^{2} + 127455068160000$$ T5^6 + 243584*T5^4 + 15873740800*T5^2 + 127455068160000 $$T_{7}^{3} - 68T_{7}^{2} - 1087632T_{7} + 353498688$$ T7^3 - 68*T7^2 - 1087632*T7 + 353498688

Hecke characteristic polynomials

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