# Properties

 Label 384.10.d.d Level $384$ Weight $10$ Character orbit 384.d Analytic conductor $197.774$ Analytic rank $0$ Dimension $10$ 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456$$ x^10 + 17386*x^8 + 91191193*x^6 + 169741365808*x^4 + 89987894131456*x^2 + 6183051813523456 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{52}\cdot 3^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 81 \beta_{5} q^{3} + (\beta_{6} + 112 \beta_{5}) q^{5} + ( - \beta_{2} + \beta_1 + 558) q^{7} - 6561 q^{9}+O(q^{10})$$ q + 81*b5 * q^3 + (b6 + 112*b5) * q^5 + (-b2 + b1 + 558) * q^7 - 6561 * q^9 $$q + 81 \beta_{5} q^{3} + (\beta_{6} + 112 \beta_{5}) q^{5} + ( - \beta_{2} + \beta_1 + 558) q^{7} - 6561 q^{9} + (\beta_{8} - 4 \beta_{7} - 11 \beta_{6} - 4503 \beta_{5}) q^{11} + ( - 4 \beta_{8} - \beta_{7} - 24 \beta_{6} + 5806 \beta_{5}) q^{13} + ( - 81 \beta_{2} - 9072) q^{15} + ( - \beta_{4} + 3 \beta_{3} - 93 \beta_{2} - 5 \beta_1 + 19568) q^{17} + (\beta_{9} + 15 \beta_{8} + 11 \beta_{7} + 31 \beta_{6} - 50574 \beta_{5}) q^{19} + (81 \beta_{7} - 81 \beta_{6} + 45198 \beta_{5}) q^{21} + ( - 4 \beta_{4} - 40 \beta_{3} + 82 \beta_{2} - 34 \beta_1 - 13536) q^{23} + (5 \beta_{4} + 50 \beta_{3} - 826 \beta_{2} + 21 \beta_1 - 131696) q^{25} - 531441 \beta_{5} q^{27} + ( - 4 \beta_{9} - 20 \beta_{8} + 342 \beta_{7} - 15 \beta_{6} + 7484 \beta_{5}) q^{29} + (16 \beta_{4} - 44 \beta_{3} + 1249 \beta_{2} - \beta_1 - 1488762) q^{31} + ( - 81 \beta_{3} + 891 \beta_{2} + 324 \beta_1 + 364743) q^{33} + ( - 14 \beta_{9} + 149 \beta_{8} + 330 \beta_{7} + \cdots - 2115873 \beta_{5}) q^{35}+ \cdots + ( - 6561 \beta_{8} + 26244 \beta_{7} + 72171 \beta_{6} + 29544183 \beta_{5}) q^{99}+O(q^{100})$$ q + 81*b5 * q^3 + (b6 + 112*b5) * q^5 + (-b2 + b1 + 558) * q^7 - 6561 * q^9 + (b8 - 4*b7 - 11*b6 - 4503*b5) * q^11 + (-4*b8 - b7 - 24*b6 + 5806*b5) * q^13 + (-81*b2 - 9072) * q^15 + (-b4 + 3*b3 - 93*b2 - 5*b1 + 19568) * q^17 + (b9 + 15*b8 + 11*b7 + 31*b6 - 50574*b5) * q^19 + (81*b7 - 81*b6 + 45198*b5) * q^21 + (-4*b4 - 40*b3 + 82*b2 - 34*b1 - 13536) * q^23 + (5*b4 + 50*b3 - 826*b2 + 21*b1 - 131696) * q^25 - 531441*b5 * q^27 + (-4*b9 - 20*b8 + 342*b7 - 15*b6 + 7484*b5) * q^29 + (16*b4 - 44*b3 + 1249*b2 - b1 - 1488762) * q^31 + (-81*b3 + 891*b2 + 324*b1 + 364743) * q^33 + (-14*b9 + 149*b8 + 330*b7 - 1119*b6 - 2115873*b5) * q^35 + (20*b9 - 28*b8 + 1579*b7 + 2766*b6 + 661650*b5) * q^37 + (324*b3 + 1944*b2 + 81*b1 - 470286) * q^39 + (-35*b4 - 89*b3 - 2593*b2 + 1113*b1 + 3247430) * q^41 + (-5*b9 - 809*b8 - 1215*b7 - 13905*b6 - 2189112*b5) * q^43 + (-6561*b6 - 734832*b5) * q^45 + (-44*b4 + 128*b3 + 21874*b2 + 914*b1 - 2868888) * q^47 + (35*b4 + 660*b3 + 6808*b2 + 6907*b1 + 8089412) * q^49 + (81*b9 + 243*b8 - 405*b7 - 7533*b6 + 1585008*b5) * q^51 + (-124*b9 - 532*b8 + 3658*b7 - 8041*b6 - 9668996*b5) * q^53 + (-24*b4 + 76*b3 + 25064*b2 - 680*b1 + 22332228) * q^55 + (81*b4 - 1215*b3 - 2511*b2 - 891*b1 + 4096494) * q^57 + (70*b9 - 320*b8 + 6122*b7 - 60392*b6 + 12713286*b5) * q^59 + (60*b9 + 4564*b8 - 2687*b7 - 20578*b6 - 37885314*b5) * q^61 + (6561*b2 - 6561*b1 - 3661038) * q^63 + (-91*b4 - 2021*b3 - 55869*b2 + 6545*b1 + 51149432) * q^65 + (-238*b9 + 3852*b8 - 8242*b7 - 35324*b6 + 6904614*b5) * q^67 + (324*b9 - 3240*b8 - 2754*b7 + 6642*b6 - 1096416*b5) * q^69 + (320*b4 - 4264*b3 + 103310*b2 + 6430*b1 + 72283356) * q^71 + (-125*b4 + 3300*b3 + 89128*b2 - 37093*b1 + 64257969) * q^73 + (-405*b9 + 4050*b8 + 1701*b7 - 66906*b6 - 10667376*b5) * q^75 + (140*b9 - 7316*b8 - 16728*b7 + 43208*b6 - 142319880*b5) * q^77 + (-24*b4 - 784*b3 + 47669*b2 + 17479*b1 - 11482422) * q^79 + 43046721 * q^81 + (322*b9 - 14289*b8 + 25490*b7 + 171139*b6 + 52304469*b5) * q^83 + (-780*b9 + 3040*b8 - 18164*b7 + 211342*b6 - 188208100*b5) * q^85 + (-324*b4 + 1620*b3 + 1215*b2 - 27702*b1 - 606204) * q^87 + (490*b4 + 5338*b3 + 38458*b2 - 92238*b1 + 170962186) * q^89 + (1225*b9 + 5379*b8 - 71757*b7 - 200253*b6 - 70608198*b5) * q^91 + (-1296*b9 - 3564*b8 - 81*b7 + 101169*b6 - 120589722*b5) * q^93 + (-364*b4 + 4336*b3 - 15620*b2 - 4396*b1 - 65714316) * q^95 + (-168*b4 + 15794*b3 + 6906*b2 + 43280*b1 + 391933488) * q^97 + (-6561*b8 + 26244*b7 + 72171*b6 + 29544183*b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 5576 q^{7} - 65610 q^{9}+O(q^{10})$$ 10 * q + 5576 * q^7 - 65610 * q^9 $$10 q + 5576 q^{7} - 65610 q^{9} - 90720 q^{15} + 195692 q^{17} - 135152 q^{23} - 1317134 q^{25} - 14887496 q^{31} + 3646296 q^{33} - 4703832 q^{39} + 32469956 q^{41} - 28692880 q^{47} + 80865242 q^{49} + 223324800 q^{55} + 40971096 q^{57} - 36584136 q^{63} + 511472000 q^{65} + 722817008 q^{71} + 642721212 q^{73} - 114892616 q^{79} + 430467210 q^{81} - 5955120 q^{87} + 1709981116 q^{89} - 657134976 q^{95} + 3919129836 q^{97}+O(q^{100})$$ 10 * q + 5576 * q^7 - 65610 * q^9 - 90720 * q^15 + 195692 * q^17 - 135152 * q^23 - 1317134 * q^25 - 14887496 * q^31 + 3646296 * q^33 - 4703832 * q^39 + 32469956 * q^41 - 28692880 * q^47 + 80865242 * q^49 + 223324800 * q^55 + 40971096 * q^57 - 36584136 * q^63 + 511472000 * q^65 + 722817008 * q^71 + 642721212 * q^73 - 114892616 * q^79 + 430467210 * q^81 - 5955120 * q^87 + 1709981116 * q^89 - 657134976 * q^95 + 3919129836 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 17386x^{8} + 91191193x^{6} + 169741365808x^{4} + 89987894131456x^{2} + 6183051813523456$$ :

 $$\beta_{1}$$ $$=$$ $$( - 303938737 \nu^{8} - 3215107672890 \nu^{6} + \cdots - 30\!\cdots\!72 ) / 11\!\cdots\!80$$ (-303938737*v^8 - 3215107672890*v^6 + 3241382041046999*v^4 + 39372542332398244320*v^2 - 30675130093745474697472) / 11335124540145070080 $$\beta_{2}$$ $$=$$ $$( 308496745 \nu^{8} + 5087199131946 \nu^{6} + \cdots + 16\!\cdots\!24 ) / 56\!\cdots\!40$$ (308496745*v^8 + 5087199131946*v^6 + 24531294476891041*v^4 + 42032665754616741984*v^2 + 16907197358261548705024) / 5667562270072535040 $$\beta_{3}$$ $$=$$ $$( 6881235923 \nu^{8} + 103669410123246 \nu^{6} + \cdots - 71\!\cdots\!48 ) / 56\!\cdots\!40$$ (6881235923*v^8 + 103669410123246*v^6 + 393773216593716155*v^4 + 319260286087400864544*v^2 - 71151925683932355797248) / 5667562270072535040 $$\beta_{4}$$ $$=$$ $$( - 69206986817 \nu^{8} + \cdots - 17\!\cdots\!60 ) / 37\!\cdots\!60$$ (-69206986817*v^8 - 1193682359534042*v^6 - 5994128580728845113*v^4 - 9190495077802831525408*v^2 - 1765644554467835537455360) / 3778374846715023360 $$\beta_{5}$$ $$=$$ $$( - 2120694601 \nu^{9} - 35965121354090 \nu^{7} + \cdots - 10\!\cdots\!56 \nu ) / 71\!\cdots\!40$$ (-2120694601*v^9 - 35965121354090*v^7 - 178036000609108353*v^5 - 287847352205197197280*v^3 - 101665712491052507469056*v) / 714188994814511671664640 $$\beta_{6}$$ $$=$$ $$( - 105159413737129 \nu^{9} + \cdots - 88\!\cdots\!76 \nu ) / 27\!\cdots\!60$$ (-105159413737129*v^9 - 1815721164824411178*v^7 - 9436226590353207197665*v^5 - 17499861895010179752898272*v^3 - 8847993327115736763349355776*v) / 27853370797765955194920960 $$\beta_{7}$$ $$=$$ $$( 692616151606351 \nu^{9} + \cdots + 24\!\cdots\!76 \nu ) / 55\!\cdots\!20$$ (692616151606351*v^9 + 11463452932003452870*v^7 + 54364405704052039582423*v^5 + 82045525327992466472874720*v^3 + 24786830874753611941577234176*v) / 55706741595531910389841920 $$\beta_{8}$$ $$=$$ $$( 125349951870869 \nu^{9} + \cdots + 82\!\cdots\!00 \nu ) / 34\!\cdots\!20$$ (125349951870869*v^9 + 2263141498772603874*v^7 + 12514731070482291137261*v^5 + 23546572660112177628434976*v^3 + 8276034472133012217961049600*v) / 3481671349720744399365120 $$\beta_{9}$$ $$=$$ $$( 18\!\cdots\!27 \nu^{9} + \cdots - 60\!\cdots\!72 \nu ) / 18\!\cdots\!40$$ (1859331957063327*v^9 + 23350209913126431334*v^7 + 29299602867070349292455*v^5 - 277339257500774865601557664*v^3 - 606287949148599693069277782272*v) / 18568913865177303463280640
 $$\nu$$ $$=$$ $$( -\beta_{9} + 53\beta_{7} + 148\beta_{6} + 21\beta_{5} ) / 9216$$ (-b9 + 53*b7 + 148*b6 + 21*b5) / 9216 $$\nu^{2}$$ $$=$$ $$( -\beta_{4} - 120\beta_{3} + 1652\beta_{2} - 1397\beta _1 - 10682541 ) / 3072$$ (-b4 - 120*b3 + 1652*b2 - 1397*b1 - 10682541) / 3072 $$\nu^{3}$$ $$=$$ $$( 4733\beta_{9} - 35136\beta_{8} - 467425\beta_{7} - 947588\beta_{6} - 1018767489\beta_{5} ) / 9216$$ (4733*b9 - 35136*b8 - 467425*b7 - 947588*b6 - 1018767489*b5) / 9216 $$\nu^{4}$$ $$=$$ $$( 45961\beta_{4} + 1008216\beta_{3} + 378892\beta_{2} + 15025565\beta _1 + 73666993461 ) / 3072$$ (45961*b4 + 1008216*b3 + 378892*b2 + 15025565*b1 + 73666993461) / 3072 $$\nu^{5}$$ $$=$$ $$( - 33885109 \beta_{9} + 371750688 \beta_{8} + 4421316473 \beta_{7} + 6168369604 \beta_{6} + 14034606037113 \beta_{5} ) / 9216$$ (-33885109*b9 + 371750688*b8 + 4421316473*b7 + 6168369604*b6 + 14034606037113*b5) / 9216 $$\nu^{6}$$ $$=$$ $$( - 218709739 \beta_{4} - 3328147432 \beta_{3} - 23500742980 \beta_{2} - 49005318759 \beta _1 - 206497978666895 ) / 1024$$ (-218709739*b4 - 3328147432*b3 - 23500742980*b2 - 49005318759*b1 - 206497978666895) / 1024 $$\nu^{7}$$ $$=$$ $$( 282754258541 \beta_{9} - 3486009452544 \beta_{8} - 42756986004625 \beta_{7} - 46818662199044 \beta_{6} - 15\!\cdots\!01 \beta_{5} ) / 9216$$ (282754258541*b9 - 3486009452544*b8 - 42756986004625*b7 - 46818662199044*b6 - 152234040158236401*b5) / 9216 $$\nu^{8}$$ $$=$$ $$( 7301243400377 \beta_{4} + 100824151254552 \beta_{3} + 963825255608204 \beta_{2} + \cdots + 56\!\cdots\!05 ) / 3072$$ (7301243400377*b4 + 100824151254552*b3 + 963825255608204*b2 + 1419861073468045*b1 + 5644853946928703205) / 3072 $$\nu^{9}$$ $$=$$ $$( - 25\!\cdots\!97 \beta_{9} + \cdots + 15\!\cdots\!05 \beta_{5} ) / 9216$$ (-2545032999188197*b9 + 32679647423748576*b8 + 414847783145681513*b7 + 397681582244268100*b6 + 1538701528012967889705*b5) / 9216

## 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
 8.96538i − 98.3652i 49.7838i − 26.9187i 66.5347i − 66.5347i 26.9187i − 49.7838i 98.3652i − 8.96538i
0 81.0000i 0 2526.54i 0 −4839.94 0 −6561.00 0
193.2 0 81.0000i 0 553.862i 0 12491.9 0 −6561.00 0
193.3 0 81.0000i 0 309.546i 0 −5272.51 0 −6561.00 0
193.4 0 81.0000i 0 332.682i 0 −3969.81 0 −6561.00 0
193.5 0 81.0000i 0 1878.17i 0 4378.40 0 −6561.00 0
193.6 0 81.0000i 0 1878.17i 0 4378.40 0 −6561.00 0
193.7 0 81.0000i 0 332.682i 0 −3969.81 0 −6561.00 0
193.8 0 81.0000i 0 309.546i 0 −5272.51 0 −6561.00 0
193.9 0 81.0000i 0 553.862i 0 12491.9 0 −6561.00 0
193.10 0 81.0000i 0 2526.54i 0 −4839.94 0 −6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.10 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.d yes 10
4.b odd 2 1 384.10.d.c 10
8.b even 2 1 inner 384.10.d.d yes 10
8.d odd 2 1 384.10.d.c 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.10.d.c 10 4.b odd 2 1
384.10.d.c 10 8.d odd 2 1
384.10.d.d yes 10 1.a even 1 1 trivial
384.10.d.d yes 10 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 $$S_{10}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{10} + 10424192 T_{5}^{8} + 27678481125376 T_{5}^{6} + \cdots + 73\!\cdots\!00$$ T5^10 + 10424192*T5^8 + 27678481125376*T5^6 + 12293551671041064960*T5^4 + 1697427125229861627494400*T5^2 + 73254457040173081338839040000 $$T_{7}^{5} - 2788T_{7}^{4} - 117213856T_{7}^{3} - 236229687680T_{7}^{2} + 1882375258450176T_{7} + 5540757921577073664$$ T7^5 - 2788*T7^4 - 117213856*T7^3 - 236229687680*T7^2 + 1882375258450176*T7 + 5540757921577073664

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$(T^{2} + 6561)^{5}$$
$5$ $$T^{10} + 10424192 T^{8} + \cdots + 73\!\cdots\!00$$
$7$ $$(T^{5} - 2788 T^{4} + \cdots + 55\!\cdots\!64)^{2}$$
$11$ $$T^{10} + 7487555664 T^{8} + \cdots + 11\!\cdots\!64$$
$13$ $$T^{10} + 61996270160 T^{8} + \cdots + 13\!\cdots\!00$$
$17$ $$(T^{5} - 97846 T^{4} + \cdots - 76\!\cdots\!48)^{2}$$
$19$ $$T^{10} + 1457745135696 T^{8} + \cdots + 71\!\cdots\!16$$
$23$ $$(T^{5} + 67576 T^{4} + \cdots + 14\!\cdots\!12)^{2}$$
$29$ $$T^{10} + 36404919905216 T^{8} + \cdots + 30\!\cdots\!00$$
$31$ $$(T^{5} + 7443748 T^{4} + \cdots + 10\!\cdots\!32)^{2}$$
$37$ $$T^{10} + 897755940902736 T^{8} + \cdots + 42\!\cdots\!36$$
$41$ $$(T^{5} - 16234978 T^{4} + \cdots - 81\!\cdots\!88)^{2}$$
$43$ $$T^{10} + \cdots + 45\!\cdots\!24$$
$47$ $$(T^{5} + 14346440 T^{4} + \cdots + 90\!\cdots\!60)^{2}$$
$53$ $$T^{10} + \cdots + 83\!\cdots\!00$$
$59$ $$T^{10} + \cdots + 42\!\cdots\!04$$
$61$ $$T^{10} + \cdots + 13\!\cdots\!00$$
$67$ $$T^{10} + \cdots + 14\!\cdots\!96$$
$71$ $$(T^{5} - 361408504 T^{4} + \cdots - 48\!\cdots\!76)^{2}$$
$73$ $$(T^{5} - 321360606 T^{4} + \cdots - 55\!\cdots\!56)^{2}$$
$79$ $$(T^{5} + 57446308 T^{4} + \cdots + 13\!\cdots\!52)^{2}$$
$83$ $$T^{10} + \cdots + 26\!\cdots\!24$$
$89$ $$(T^{5} - 854990558 T^{4} + \cdots - 65\!\cdots\!48)^{2}$$
$97$ $$(T^{5} - 1959564918 T^{4} + \cdots + 35\!\cdots\!00)^{2}$$