# 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 $2$

# Related objects

## 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: $$x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225$$ x^8 + 13062*x^6 + 45211107*x^4 + 45928424926*x^2 + 852972309225 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.

 $$f(q)$$ $$=$$ $$q - 81 \beta_{4} q^{3} + ( - \beta_{5} + 140 \beta_{4}) q^{5} + ( - \beta_{3} + 2 \beta_1 + 1704) q^{7} - 6561 q^{9}+O(q^{10})$$ q - 81*b4 * q^3 + (-b5 + 140*b4) * q^5 + (-b3 + 2*b1 + 1704) * q^7 - 6561 * q^9 $$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})$$ q - 81*b4 * q^3 + (-b5 + 140*b4) * q^5 + (-b3 + 2*b1 + 1704) * q^7 - 6561 * q^9 + (-5*b7 - 8*b6 - 5*b5 - 9588*b4) * q^11 + (4*b7 - 15*b6 - 5*b5 + 35820*b4) * q^13 + (-81*b1 + 11340) * q^15 + (60*b3 - 7*b2 + 99*b1 + 1038) * q^17 + (61*b7 - 100*b6 + 33*b5 + 58860*b4) * q^19 + (-81*b6 - 162*b5 - 138024*b4) * q^21 + (110*b3 - 56*b2 + 24*b1 + 576576) * q^23 + (204*b3 + 158*b2 - 426*b1 - 594363) * q^25 + 531441*b4 * q^27 + (-652*b7 + 366*b6 - 167*b5 - 653500*b4) * q^29 + (-95*b3 + 172*b2 - 1138*b1 + 937416) * q^31 + (648*b3 + 405*b2 - 405*b1 - 776628) * q^33 + (-665*b7 + 960*b6 - 4177*b5 - 4302960*b4) * q^35 + (828*b7 + 513*b6 + 5173*b5 + 3851156*b4) * q^37 + (1215*b3 - 324*b2 - 405*b1 + 2901420) * q^39 + (1860*b3 - 235*b2 - 10817*b1 - 5439862) * q^41 + (1509*b7 - 1564*b6 - 7247*b5 + 447468*b4) * q^43 + (6561*b5 - 918540*b4) * q^45 + (-3950*b3 - 672*b2 - 16820*b1 - 6172752) * q^47 + (-348*b3 - 2492*b2 + 5288*b1 - 9263351) * q^49 + (-567*b7 + 4860*b6 - 8019*b5 - 84078*b4) * q^51 + (4020*b7 + 15570*b6 - 897*b5 - 2656004*b4) * q^53 + (-3744*b3 + 5492*b2 - 44776*b1 - 2378832) * q^55 + (8100*b3 - 4941*b2 + 2673*b1 + 4767660) * q^57 + (-5328*b7 - 7944*b6 - 30648*b5 + 19616076*b4) * q^59 + (-628*b7 - 9309*b6 - 7253*b5 - 5818932*b4) * q^61 + (6561*b3 - 13122*b1 - 11179944) * q^63 + (-24444*b3 + 4497*b2 + 30179*b1 - 13783792) * q^65 + (7700*b7 - 5112*b6 - 117668*b5 + 53533068*b4) * q^67 + (-4536*b7 + 8910*b6 - 1944*b5 - 46702656*b4) * q^69 + (-1406*b3 - 24920*b2 - 38296*b1 + 92718912) * q^71 + (-8796*b3 + 21044*b2 - 33352*b1 - 188487930) * q^73 + (12798*b7 + 16524*b6 + 34506*b5 + 48143403*b4) * q^75 + (-14476*b7 - 30900*b6 - 70564*b5 - 180436192*b4) * q^77 + (76033*b3 + 2512*b2 - 120814*b1 - 126046680) * q^79 + 43046721 * q^81 + (-20443*b7 - 37712*b6 - 195603*b5 + 31850604*b4) * q^83 + (2624*b7 - 110928*b6 - 6894*b5 - 274245016*b4) * q^85 + (-29646*b3 + 52812*b2 - 13527*b1 - 52933500) * q^87 + (106728*b3 - 31186*b2 + 98298*b1 - 167129306) * q^89 + (-19271*b7 + 54172*b6 + 223165*b5 - 257374848*b4) * q^91 + (13932*b7 - 7695*b6 + 92178*b5 - 75930696*b4) * q^93 + (-218928*b3 + 6384*b2 + 207324*b1 + 67993776) * q^95 + (-135600*b3 - 2234*b2 + 91610*b1 - 113102242) * q^97 + (32805*b7 + 52488*b6 + 32805*b5 + 62906868*b4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10})$$ 8 * q + 13632 * q^7 - 52488 * q^9 $$8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225$$ :

 $$\beta_{1}$$ $$=$$ $$( 4766\nu^{6} + 51851374\nu^{4} + 88099997114\nu^{2} - 43229725983072 ) / 34840779813$$ (4766*v^6 + 51851374*v^4 + 88099997114*v^2 - 43229725983072) / 34840779813 $$\beta_{2}$$ $$=$$ $$( 169786\nu^{6} + 1579828010\nu^{4} + 1426285970446\nu^{2} - 1771419698208576 ) / 243885458691$$ (169786*v^6 + 1579828010*v^4 + 1426285970446*v^2 - 1771419698208576) / 243885458691 $$\beta_{3}$$ $$=$$ $$( 199366\nu^{6} + 2241395438\nu^{4} + 5124570485866\nu^{2} + 1439961094183104 ) / 243885458691$$ (199366*v^6 + 2241395438*v^4 + 5124570485866*v^2 + 1439961094183104) / 243885458691 $$\beta_{4}$$ $$=$$ $$( 28496\nu^{7} + 372522607\nu^{5} + 1264029013547\nu^{3} + 1137344409469551\nu ) / 4843958573246310$$ (28496*v^7 + 372522607*v^5 + 1264029013547*v^3 + 1137344409469551*v) / 4843958573246310 $$\beta_{5}$$ $$=$$ $$( - 797877302 \nu^{7} - 9758638510954 \nu^{5} + \cdots - 22\!\cdots\!32 \nu ) / 75\!\cdots\!05$$ (-797877302*v^7 - 9758638510954*v^5 - 29157793972297394*v^3 - 22780987632578534232*v) / 75081357885317805 $$\beta_{6}$$ $$=$$ $$( - 1012635742 \nu^{7} - 15512580206174 \nu^{5} + \cdots - 96\!\cdots\!32 \nu ) / 75\!\cdots\!05$$ (-1012635742*v^7 - 15512580206174*v^5 - 70850434762666054*v^3 - 96437550563928427332*v) / 75081357885317805 $$\beta_{7}$$ $$=$$ $$( 575969794 \nu^{7} + 7049643741998 \nu^{5} + \cdots + 18\!\cdots\!44 \nu ) / 10\!\cdots\!15$$ (575969794*v^7 + 7049643741998*v^5 + 21095692147663558*v^3 + 18573275336880625944*v) / 10725908269331115
 $$\nu$$ $$=$$ $$( \beta_{7} + 5\beta_{5} - 96\beta_{4} ) / 192$$ (b7 + 5*b5 - 96*b4) / 192 $$\nu^{2}$$ $$=$$ $$( 48\beta_{3} + 13\beta_{2} - 353\beta _1 - 626976 ) / 192$$ (48*b3 + 13*b2 - 353*b1 - 626976) / 192 $$\nu^{3}$$ $$=$$ $$( -5251\beta_{7} - 2988\beta_{6} - 36371\beta_{5} - 24620160\beta_{4} ) / 192$$ (-5251*b7 - 2988*b6 - 36371*b5 - 24620160*b4) / 192 $$\nu^{4}$$ $$=$$ $$( -307416\beta_{3} - 258409\beta_{2} + 3152165\beta _1 + 3849294240 ) / 192$$ (-307416*b3 - 258409*b2 + 3152165*b1 + 3849294240) / 192 $$\nu^{5}$$ $$=$$ $$( 17617919\beta_{7} + 13863726\beta_{6} + 145754053\beta_{5} + 134259999792\beta_{4} ) / 96$$ (17617919*b7 + 13863726*b6 + 145754053*b5 + 134259999792*b4) / 96 $$\nu^{6}$$ $$=$$ $$( 1228613316\beta_{3} + 1285518437\beta_{2} - 13182475621\beta _1 - 14273447764896 ) / 96$$ (1228613316*b3 + 1285518437*b2 - 13182475621*b1 - 14273447764896) / 96 $$\nu^{7}$$ $$=$$ $$( -267619114645\beta_{7} - 229933464768\beta_{6} - 2397041072585\beta_{5} - 2381735984318112\beta_{4} ) / 192$$ (-267619114645*b7 - 229933464768*b6 - 2397041072585*b5 - 2381735984318112*b4) / 192

## 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: 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 384.10.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 $$S_{10}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{8} + 10189952T_{5}^{6} + 33495402213376T_{5}^{4} + 37796298115527475200T_{5}^{2} + 6903383511575235133440000$$ T5^8 + 10189952*T5^6 + 33495402213376*T5^4 + 37796298115527475200*T5^2 + 6903383511575235133440000 $$T_{7}^{4} - 6816T_{7}^{3} - 38951584T_{7}^{2} + 267125408256T_{7} - 139519152670464$$ T7^4 - 6816*T7^3 - 38951584*T7^2 + 267125408256*T7 - 139519152670464

## Hecke characteristic polynomials

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