# Properties

 Label 384.6.d.j Level $384$ Weight $6$ Character orbit 384.d Analytic conductor $61.587$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$61.5873868082$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956$$ x^12 + 338*x^10 + 43555*x^8 + 2692222*x^6 + 81680965*x^4 + 1098257588*x^2 + 4742525956 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{57}\cdot 3^{8}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_{4} q^{7} - 81 q^{9}+O(q^{10})$$ q + b6 * q^3 + b8 * q^5 + b4 * q^7 - 81 * q^9 $$q + \beta_{6} q^{3} + \beta_{8} q^{5} + \beta_{4} q^{7} - 81 q^{9} + ( - \beta_{11} + \beta_{6}) q^{11} + ( - 5 \beta_{8} - \beta_{7}) q^{13} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{15} + (\beta_{2} - 407) q^{17} + ( - 3 \beta_{11} + \beta_{10} + 14 \beta_{6}) q^{19} + ( - \beta_{9} + 7 \beta_{8} + 4 \beta_{7}) q^{21} + ( - 2 \beta_{4} + 4 \beta_{3} + 6 \beta_1) q^{23} + (\beta_{5} + 2 \beta_{2} - 1554) q^{25} - 81 \beta_{6} q^{27} + (5 \beta_{9} - 4 \beta_{8} - 5 \beta_{7}) q^{29} + ( - 13 \beta_{4} - 4 \beta_{3} + 25 \beta_1) q^{31} + ( - \beta_{5} + \beta_{2} - 60) q^{33} + (11 \beta_{11} - 6 \beta_{10} + 303 \beta_{6}) q^{35} + ( - 2 \beta_{9} - \beta_{8} + 29 \beta_{7}) q^{37} + (14 \beta_{4} + 5 \beta_{3} + 4 \beta_1) q^{39} + ( - 2 \beta_{5} - 3 \beta_{2} - 2093) q^{41} + (21 \beta_{11} + 3 \beta_{10} - 452 \beta_{6}) q^{43} - 81 \beta_{8} q^{45} + (30 \beta_{4} - 24 \beta_{3} + 52 \beta_1) q^{47} + (5 \beta_{5} - 8 \beta_{2} + 11296) q^{49} + (9 \beta_{11} + 9 \beta_{10} - 412 \beta_{6}) q^{51} + (9 \beta_{9} - 94 \beta_{8} + 27 \beta_{7}) q^{53} + (68 \beta_{4} + 28 \beta_{3} + 23 \beta_1) q^{55} + ( - 4 \beta_{5} - 5 \beta_{2} - 1095) q^{57} + ( - 32 \beta_{11} - 18 \beta_{10} + 1214 \beta_{6}) q^{59} + (20 \beta_{9} + 35 \beta_{8} + 57 \beta_{7}) q^{61} - 81 \beta_{4} q^{63} + ( - 6 \beta_{5} - 7 \beta_{2} + 21185) q^{65} + ( - 60 \beta_{11} + 10 \beta_{10} - 386 \beta_{6}) q^{67} + (16 \beta_{9} + 212 \beta_{8} + 26 \beta_{7}) q^{69} + ( - 190 \beta_{4} + 20 \beta_{3} - 78 \beta_1) q^{71} + (5 \beta_{5} - 8 \beta_{2} - 15587) q^{73} + ( - 54 \beta_{11} + 27 \beta_{10} - 1548 \beta_{6}) q^{75} + ( - 5 \beta_{9} - 245 \beta_{8} + 347 \beta_{7}) q^{77} + ( - 173 \beta_{4} - 44 \beta_{3} - 40 \beta_1) q^{79} + 6561 q^{81} + ( - 7 \beta_{11} - 30 \beta_{10} + 5621 \beta_{6}) q^{83} + (92 \beta_{9} - 1326 \beta_{8} + 116 \beta_{7}) q^{85} + (239 \beta_{4} - 31 \beta_{3} - 104 \beta_1) q^{87} + (8 \beta_{5} + 22 \beta_{2} - 47952) q^{89} + (57 \beta_{11} + 25 \beta_{10} - 4874 \beta_{6}) q^{91} + (30 \beta_{9} - 534 \beta_{8} + 69 \beta_{7}) q^{93} + (636 \beta_{4} + 132 \beta_{3} - 412 \beta_1) q^{95} + ( - 14 \beta_{5} + 62 \beta_{2} + 7838) q^{97} + (81 \beta_{11} - 81 \beta_{6}) q^{99}+O(q^{100})$$ q + b6 * q^3 + b8 * q^5 + b4 * q^7 - 81 * q^9 + (-b11 + b6) * q^11 + (-5*b8 - b7) * q^13 + (-b4 - b3 + b1) * q^15 + (b2 - 407) * q^17 + (-3*b11 + b10 + 14*b6) * q^19 + (-b9 + 7*b8 + 4*b7) * q^21 + (-2*b4 + 4*b3 + 6*b1) * q^23 + (b5 + 2*b2 - 1554) * q^25 - 81*b6 * q^27 + (5*b9 - 4*b8 - 5*b7) * q^29 + (-13*b4 - 4*b3 + 25*b1) * q^31 + (-b5 + b2 - 60) * q^33 + (11*b11 - 6*b10 + 303*b6) * q^35 + (-2*b9 - b8 + 29*b7) * q^37 + (14*b4 + 5*b3 + 4*b1) * q^39 + (-2*b5 - 3*b2 - 2093) * q^41 + (21*b11 + 3*b10 - 452*b6) * q^43 - 81*b8 * q^45 + (30*b4 - 24*b3 + 52*b1) * q^47 + (5*b5 - 8*b2 + 11296) * q^49 + (9*b11 + 9*b10 - 412*b6) * q^51 + (9*b9 - 94*b8 + 27*b7) * q^53 + (68*b4 + 28*b3 + 23*b1) * q^55 + (-4*b5 - 5*b2 - 1095) * q^57 + (-32*b11 - 18*b10 + 1214*b6) * q^59 + (20*b9 + 35*b8 + 57*b7) * q^61 - 81*b4 * q^63 + (-6*b5 - 7*b2 + 21185) * q^65 + (-60*b11 + 10*b10 - 386*b6) * q^67 + (16*b9 + 212*b8 + 26*b7) * q^69 + (-190*b4 + 20*b3 - 78*b1) * q^71 + (5*b5 - 8*b2 - 15587) * q^73 + (-54*b11 + 27*b10 - 1548*b6) * q^75 + (-5*b9 - 245*b8 + 347*b7) * q^77 + (-173*b4 - 44*b3 - 40*b1) * q^79 + 6561 * q^81 + (-7*b11 - 30*b10 + 5621*b6) * q^83 + (92*b9 - 1326*b8 + 116*b7) * q^85 + (239*b4 - 31*b3 - 104*b1) * q^87 + (8*b5 + 22*b2 - 47952) * q^89 + (57*b11 + 25*b10 - 4874*b6) * q^91 + (30*b9 - 534*b8 + 69*b7) * q^93 + (636*b4 + 132*b3 - 412*b1) * q^95 + (-14*b5 + 62*b2 + 7838) * q^97 + (81*b11 - 81*b6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 972 q^{9}+O(q^{10})$$ 12 * q - 972 * q^9 $$12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100})$$ 12 * q - 972 * q^9 - 4888 * q^17 - 18660 * q^25 - 720 * q^33 - 25096 * q^41 + 135564 * q^49 - 13104 * q^57 + 254272 * q^65 - 187032 * q^73 + 78732 * q^81 - 575544 * q^89 + 93864 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956$$ :

 $$\beta_{1}$$ $$=$$ $$( - 501856 \nu^{10} - 128371584 \nu^{8} - 9748135264 \nu^{6} - 138948034496 \nu^{4} + 6898201181664 \nu^{2} + \cdots + 100166634492736 ) / 228958809975$$ (-501856*v^10 - 128371584*v^8 - 9748135264*v^6 - 138948034496*v^4 + 6898201181664*v^2 + 100166634492736) / 228958809975 $$\beta_{2}$$ $$=$$ $$( - 978928 \nu^{10} - 55789152 \nu^{8} + 25346631296 \nu^{6} + 2989391325712 \nu^{4} + 103137790028736 \nu^{2} + \cdots + 899529825156571 ) / 125454890799$$ (-978928*v^10 - 55789152*v^8 + 25346631296*v^6 + 2989391325712*v^4 + 103137790028736*v^2 + 899529825156571) / 125454890799 $$\beta_{3}$$ $$=$$ $$( - 78708328 \nu^{10} - 20962386192 \nu^{8} - 1968575165032 \nu^{6} - 81171924503648 \nu^{4} + \cdots + 67215280640368 ) / 9409116809925$$ (-78708328*v^10 - 20962386192*v^8 - 1968575165032*v^6 - 81171924503648*v^4 - 1302445633736568*v^2 + 67215280640368) / 9409116809925 $$\beta_{4}$$ $$=$$ $$( - 8166241501 \nu^{10} - 2267974529364 \nu^{8} - 221787567202369 \nu^{6} + \cdots - 78\!\cdots\!94 ) / 395182906016850$$ (-8166241501*v^10 - 2267974529364*v^8 - 221787567202369*v^6 - 9280390651172366*v^4 - 157344992788946331*v^2 - 787485894434470994) / 395182906016850 $$\beta_{5}$$ $$=$$ $$( - 144829696 \nu^{10} - 34237610880 \nu^{8} - 2552278683136 \nu^{6} - 66770917381760 \nu^{4} - 538239146767872 \nu^{2} + \cdots - 26\!\cdots\!23 ) / 878184235593$$ (-144829696*v^10 - 34237610880*v^8 - 2552278683136*v^6 - 66770917381760*v^4 - 538239146767872*v^2 - 2613312602025923) / 878184235593 $$\beta_{6}$$ $$=$$ $$( 1551427 \nu^{11} + 374874240 \nu^{9} + 28047726409 \nu^{7} + 593227138388 \nu^{5} - 6428215193541 \nu^{3} + \cdots - 158055405918862 \nu ) / 25847648495928$$ (1551427*v^11 + 374874240*v^9 + 28047726409*v^7 + 593227138388*v^5 - 6428215193541*v^3 - 158055405918862*v) / 25847648495928 $$\beta_{7}$$ $$=$$ $$( 153683077859 \nu^{11} + 32357600561226 \nu^{9} + \cdots - 19\!\cdots\!54 \nu ) / 21\!\cdots\!50$$ (153683077859*v^11 + 32357600561226*v^9 + 1761152077469771*v^7 - 364455339395606*v^5 - 1364391707528118021*v^3 - 19305945416065315154*v) / 215989412744098350 $$\beta_{8}$$ $$=$$ $$( - 163026402997 \nu^{11} - 46810444439958 \nu^{9} + \cdots - 27\!\cdots\!18 \nu ) / 21\!\cdots\!50$$ (-163026402997*v^11 - 46810444439958*v^9 - 4862882862154093*v^7 - 226533816159932102*v^5 - 4575692678135776557*v^3 - 27750977157401138018*v) / 215989412744098350 $$\beta_{9}$$ $$=$$ $$( 126051978287 \nu^{11} + 34430061527418 \nu^{9} + \cdots - 70\!\cdots\!22 \nu ) / 10\!\cdots\!75$$ (126051978287*v^11 + 34430061527418*v^9 + 3032857817915303*v^7 + 75065074857055042*v^5 - 1945981054525522353*v^3 - 70453376018654091722*v) / 107994706372049175 $$\beta_{10}$$ $$=$$ $$( - 707802624817 \nu^{11} - 149633466506112 \nu^{9} + \cdots - 22\!\cdots\!82 \nu ) / 31\!\cdots\!24$$ (-707802624817*v^11 - 149633466506112*v^9 - 8988582027908371*v^7 - 174277841389849436*v^5 - 5585905248785114457*v^3 - 221548216506741642182*v) / 311024754351501624 $$\beta_{11}$$ $$=$$ $$( - 1650016024369 \nu^{11} - 481063311351552 \nu^{9} + \cdots - 27\!\cdots\!14 \nu ) / 31\!\cdots\!24$$ (-1650016024369*v^11 - 481063311351552*v^9 - 50193439542168979*v^7 - 2286863903934417884*v^5 - 43723863734518075737*v^3 - 270665448200217718214*v) / 311024754351501624
 $$\nu$$ $$=$$ $$( -9\beta_{10} + 9\beta_{9} + 9\beta_{8} - 27\beta_{7} - 83\beta_{6} ) / 2304$$ (-9*b10 + 9*b9 + 9*b8 - 27*b7 - 83*b6) / 2304 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} - 64\beta_{4} + 152\beta_{3} - 8\beta_{2} + 127\beta _1 - 129795 ) / 2304$$ (-b5 - 64*b4 + 152*b3 - 8*b2 + 127*b1 - 129795) / 2304 $$\nu^{3}$$ $$=$$ $$( 216\beta_{11} + 1368\beta_{10} - 2199\beta_{9} - 2175\beta_{8} + 1389\beta_{7} + 69904\beta_{6} ) / 4608$$ (216*b11 + 1368*b10 - 2199*b9 - 2175*b8 + 1389*b7 + 69904*b6) / 4608 $$\nu^{4}$$ $$=$$ $$( -251\beta_{5} + 10368\beta_{4} - 17928\beta_{3} + 2528\beta_{2} - 19440\beta _1 + 10420215 ) / 2304$$ (-251*b5 + 10368*b4 - 17928*b3 + 2528*b2 - 19440*b1 + 10420215) / 2304 $$\nu^{5}$$ $$=$$ $$( - 77112 \beta_{11} - 132066 \beta_{10} + 232027 \beta_{9} + 527603 \beta_{8} + 79127 \beta_{7} - 10638206 \beta_{6} ) / 4608$$ (-77112*b11 - 132066*b10 + 232027*b9 + 527603*b8 + 79127*b7 - 10638206*b6) / 4608 $$\nu^{6}$$ $$=$$ $$( 53528\beta_{5} - 1370048\beta_{4} + 1952464\beta_{3} - 399704\beta_{2} + 2860439\beta _1 - 970240704 ) / 2304$$ (53528*b5 - 1370048*b4 + 1952464*b3 - 399704*b2 + 2860439*b1 - 970240704) / 2304 $$\nu^{7}$$ $$=$$ $$( 6980904 \beta_{11} + 6871167 \beta_{10} - 12541321 \beta_{9} - 46005689 \beta_{8} - 11632181 \beta_{7} + 680775997 \beta_{6} ) / 2304$$ (6980904*b11 + 6871167*b10 - 12541321*b9 - 46005689*b8 - 11632181*b7 + 680775997*b6) / 2304 $$\nu^{8}$$ $$=$$ $$( - 2511861 \beta_{5} + 56030336 \beta_{4} - 71582536 \beta_{3} + 17976768 \beta_{2} - 130050800 \beta _1 + 32688443289 ) / 768$$ (-2511861*b5 + 56030336*b4 - 71582536*b3 + 17976768*b2 - 130050800*b1 + 32688443289) / 768 $$\nu^{9}$$ $$=$$ $$( - 2051442504 \beta_{11} - 1486228572 \beta_{10} + 2790312729 \beta_{9} + 13379824785 \beta_{8} + 3595812861 \beta_{7} - 166302107596 \beta_{6} ) / 4608$$ (-2051442504*b11 - 1486228572*b10 + 2790312729*b9 + 13379824785*b8 + 3595812861*b7 - 166302107596*b6) / 4608 $$\nu^{10}$$ $$=$$ $$( 943566077 \beta_{5} - 20134847104 \beta_{4} + 24059081432 \beta_{3} - 6840998480 \beta_{2} + 50313706612 \beta _1 - 10447619634033 ) / 2304$$ (943566077*b5 - 20134847104*b4 + 24059081432*b3 - 6840998480*b2 + 50313706612*b1 - 10447619634033) / 2304 $$\nu^{11}$$ $$=$$ $$( 273663797400 \beta_{11} + 165010354002 \beta_{10} - 316766850363 \beta_{9} - 1778470066323 \beta_{8} - 478276862391 \beta_{7} + 19986140244334 \beta_{6} ) / 4608$$ (273663797400*b11 + 165010354002*b10 - 316766850363*b9 - 1778470066323*b8 - 478276862391*b7 + 19986140244334*b6) / 4608

## 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
 − 4.65977i 2.80773i − 6.60695i 8.37346i 10.8525i − 8.76697i 8.76697i − 10.8525i − 8.37346i 6.60695i − 2.80773i 4.65977i
0 9.00000i 0 107.066i 0 150.154 0 −81.0000 0
193.2 0 9.00000i 0 46.1613i 0 −59.9278 0 −81.0000 0
193.3 0 9.00000i 0 21.1198i 0 −241.194 0 −81.0000 0
193.4 0 9.00000i 0 21.1198i 0 241.194 0 −81.0000 0
193.5 0 9.00000i 0 46.1613i 0 59.9278 0 −81.0000 0
193.6 0 9.00000i 0 107.066i 0 −150.154 0 −81.0000 0
193.7 0 9.00000i 0 107.066i 0 −150.154 0 −81.0000 0
193.8 0 9.00000i 0 46.1613i 0 59.9278 0 −81.0000 0
193.9 0 9.00000i 0 21.1198i 0 241.194 0 −81.0000 0
193.10 0 9.00000i 0 21.1198i 0 −241.194 0 −81.0000 0
193.11 0 9.00000i 0 46.1613i 0 −59.9278 0 −81.0000 0
193.12 0 9.00000i 0 107.066i 0 150.154 0 −81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 193.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 384.6.d.j 12
4.b odd 2 1 inner 384.6.d.j 12
8.b even 2 1 inner 384.6.d.j 12
8.d odd 2 1 inner 384.6.d.j 12
16.e even 4 1 768.6.a.be 6
16.e even 4 1 768.6.a.bf 6
16.f odd 4 1 768.6.a.be 6
16.f odd 4 1 768.6.a.bf 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.d.j 12 1.a even 1 1 trivial
384.6.d.j 12 4.b odd 2 1 inner
384.6.d.j 12 8.b even 2 1 inner
384.6.d.j 12 8.d odd 2 1 inner
768.6.a.be 6 16.e even 4 1
768.6.a.be 6 16.f odd 4 1
768.6.a.bf 6 16.e even 4 1
768.6.a.bf 6 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{6} + 14040T_{5}^{4} + 30489792T_{5}^{2} + 10895241728$$ T5^6 + 14040*T5^4 + 30489792*T5^2 + 10895241728 $$T_{7}^{6} - 84312T_{7}^{4} + 1601504448T_{7}^{2} - 4710436295168$$ T7^6 - 84312*T7^4 + 1601504448*T7^2 - 4710436295168

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{2} + 81)^{6}$$
$5$ $$(T^{6} + 14040 T^{4} + \cdots + 10895241728)^{2}$$
$7$ $$(T^{6} - 84312 T^{4} + \cdots - 4710436295168)^{2}$$
$11$ $$(T^{6} + 621360 T^{4} + \cdots + 32\!\cdots\!16)^{2}$$
$13$ $$(T^{6} + 478560 T^{4} + \cdots + 31\!\cdots\!32)^{2}$$
$17$ $$(T^{3} + 1222 T^{2} + \cdots - 2972835064)^{4}$$
$19$ $$(T^{6} + 10626864 T^{4} + \cdots + 83\!\cdots\!24)^{2}$$
$23$ $$(T^{6} - 16145760 T^{4} + \cdots - 30\!\cdots\!28)^{2}$$
$29$ $$(T^{6} + 77332440 T^{4} + \cdots + 14\!\cdots\!88)^{2}$$
$31$ $$(T^{6} - 94022424 T^{4} + \cdots - 33\!\cdots\!12)^{2}$$
$37$ $$(T^{6} + 183716352 T^{4} + \cdots + 21\!\cdots\!00)^{2}$$
$41$ $$(T^{3} + 6274 T^{2} + \cdots - 560226749800)^{4}$$
$43$ $$(T^{6} + 398631600 T^{4} + \cdots + 55\!\cdots\!04)^{2}$$
$47$ $$(T^{6} - 982944096 T^{4} + \cdots - 13\!\cdots\!52)^{2}$$
$53$ $$(T^{6} + 485370072 T^{4} + \cdots + 22\!\cdots\!48)^{2}$$
$59$ $$(T^{6} + 3145023792 T^{4} + \cdots + 10\!\cdots\!76)^{2}$$
$61$ $$(T^{6} + 1528967040 T^{4} + \cdots + 22\!\cdots\!48)^{2}$$
$67$ $$(T^{6} + 2669402928 T^{4} + \cdots + 20\!\cdots\!76)^{2}$$
$71$ $$(T^{6} - 4329842016 T^{4} + \cdots - 14\!\cdots\!48)^{2}$$
$73$ $$(T^{3} + 46758 T^{2} + \cdots - 12280634576248)^{4}$$
$79$ $$(T^{6} - 4272026904 T^{4} + \cdots - 20\!\cdots\!92)^{2}$$
$83$ $$(T^{6} + 13191609648 T^{4} + \cdots + 13\!\cdots\!64)^{2}$$
$89$ $$(T^{3} + 143886 T^{2} + \cdots + 28858879919016)^{4}$$
$97$ $$(T^{3} - 23466 T^{2} + \cdots - 592920131000504)^{4}$$