# Properties

 Label 384.6.f.f Level $384$ Weight $6$ Character orbit 384.f Analytic conductor $61.587$ Analytic rank $0$ Dimension $20$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$61.5873868082$$ Analytic rank: $$0$$ Dimension: $$20$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816$$ x^20 + 5192*x^16 + 8441320*x^12 + 4098006217*x^8 + 8949568544*x^4 + 8386816 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{87}\cdot 3^{14}$$ 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 $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + \beta_{5} q^{7} + (\beta_{8} - 2) q^{9}+O(q^{10})$$ q - b2 * q^3 + (b4 - b2) * q^5 + b5 * q^7 + (b8 - 2) * q^9 $$q - \beta_{2} q^{3} + (\beta_{4} - \beta_{2}) q^{5} + \beta_{5} q^{7} + (\beta_{8} - 2) q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{11} + (\beta_{6} - \beta_{3} - 5 \beta_1) q^{13} + ( - \beta_{17} - \beta_{15} + 178) q^{15} + (\beta_{15} + \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{5}) q^{17} + ( - \beta_{16} - \beta_{14} + 2 \beta_{7} - 3 \beta_{4} + 19 \beta_{2}) q^{19} + (\beta_{18} - \beta_{16} - \beta_{14} - \beta_{7} + \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} - 8 \beta_1) q^{21} + ( - \beta_{17} + \beta_{15} - \beta_{13} - \beta_{9} + 5 \beta_{8} - 305) q^{23} + ( - 3 \beta_{17} - 2 \beta_{15} - \beta_{13} + 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + \cdots + 760) q^{25}+ \cdots + ( - 9 \beta_{19} - 73 \beta_{18} + 42 \beta_{16} + 83 \beta_{14} - 9 \beta_{7} + \cdots - 272 \beta_1) q^{99}+O(q^{100})$$ q - b2 * q^3 + (b4 - b2) * q^5 + b5 * q^7 + (b8 - 2) * q^9 + (-b3 + 2*b2 + b1) * q^11 + (b6 - b3 - 5*b1) * q^13 + (-b17 - b15 + 178) * q^15 + (b15 + b11 - b10 + b9 + b8 + 2*b5) * q^17 + (-b16 - b14 + 2*b7 - 3*b4 + 19*b2) * q^19 + (b18 - b16 - b14 - b7 + b6 + 2*b4 + 3*b3 - 3*b2 - 8*b1) * q^21 + (-b17 + b15 - b13 - b9 + 5*b8 - 305) * q^23 + (-3*b17 - 2*b15 - b13 + 2*b11 + 2*b9 + 2*b8 + 760) * q^25 + (b19 + b18 - b14 - 4*b6 + 25*b4 + 3*b3 - 2*b2 + 21*b1) * q^27 + (-b19 - 2*b18 - b16 + 3*b14 - 4*b7 - 9*b4 - 27*b2 - b1) * q^29 + (-5*b17 + b15 - 6*b12 + 3*b11 - b10 + b9 - 14*b8 - 5*b5 - 5) * q^31 + (-2*b17 - b15 + 3*b13 - 3*b12 + 4*b11 - 3*b10 + 3*b9 + b8 - 30*b5 + 379) * q^33 + (2*b19 + 4*b18 - 2*b16 + 2*b14 - 5*b7 + 8*b6 + b3 + 19*b2 - 38*b1) * q^35 + (b19 + 2*b18 - b16 + b14 - 16*b7 - b6 - 7*b3 + 156*b2 + 15*b1) * q^37 + (-8*b17 + 2*b15 + 3*b13 + 6*b12 + 4*b11 - 11*b10 + 3*b9 + b8 - 3*b5 + 137) * q^39 + (b17 - 3*b15 - b12 + 2*b11 - 4*b10 - 3*b9 + 58*b5 + 1) * q^41 + (4*b19 - 6*b18 + 3*b16 + b14 + 14*b7 - 119*b4 + 215*b2 - 3*b1) * q^43 + (-b19 + 7*b18 + 6*b16 - 6*b14 - 45*b7 - 2*b6 + 16*b4 + 6*b3 - 189*b2 + 4*b1) * q^45 + (-8*b17 + 5*b13 - 4*b12 + 5*b11 - 4*b10 + 29*b8 - 1302) * q^47 + (13*b17 + 10*b15 - 3*b13 + 10*b12 + 4*b11 + 10*b10 - 10*b9 - 12*b8 - 4048) * q^49 + (b19 - 5*b18 - 7*b16 + 6*b14 + 14*b7 + 12*b6 + 200*b4 - 3*b3 - 64*b2 - 94*b1) * q^51 + (6*b19 - 14*b18 + 2*b16 + 4*b14 - 26*b7 - 7*b4 - 287*b2 - 7*b1) * q^53 + (-7*b17 - 5*b15 - 38*b12 - 7*b11 + 27*b10 - 5*b9 - 26*b8 - 4*b5 - 7) * q^55 + (11*b17 + 9*b15 - 6*b13 + 33*b12 + 10*b11 - 13*b10 - 15*b9 - 17*b8 - 102*b5 - 4462) * q^57 + (-3*b7 - 32*b6 + 8*b3 + 75*b2 + 236*b1) * q^59 + (-7*b19 - 14*b18 + 7*b16 - 7*b14 - 80*b7 - 15*b6 + 7*b3 + 764*b2 + 73*b1) * q^61 + (4*b17 + 8*b15 - 15*b13 + 60*b12 + 7*b11 - 22*b10 - 6*b9 - 3*b8 + 15*b5 + 740) * q^63 + (-3*b17 - 9*b15 + 47*b12 + 24*b11 - 64*b10 - 9*b9 - 18*b8 + 110*b5 - 3) * q^65 + (-4*b19 + 22*b18 - 8*b16 - 22*b14 - 23*b7 - 376*b4 + 221*b2 + 11*b1) * q^67 + (8*b19 + 15*b18 - 9*b16 - 19*b14 - 147*b7 - 14*b6 + 33*b4 - 30*b3 + 312*b2 + 152*b1) * q^69 + (-9*b17 - 15*b15 - 10*b13 - 4*b12 - 9*b11 - 4*b10 + 15*b9 - 28*b8 + 3597) * q^71 + (4*b17 - 6*b15 + 21*b13 + 15*b12 + 47*b11 + 15*b10 + 6*b9 - 9*b8 - 3060) * q^73 + (-4*b19 + 24*b18 + 9*b16 - 31*b14 - 30*b7 + 16*b6 + 591*b4 - 39*b3 - 950*b2 - 211*b1) * q^75 + (-10*b19 + 2*b18 + 18*b16 + 36*b14 - 298*b7 - 102*b4 - 2480*b2 + b1) * q^77 + (31*b17 - 3*b15 - 48*b12 + 11*b11 + 6*b10 - 3*b9 + 90*b8 + 55*b5 + 31) * q^79 + (-13*b17 - 26*b15 - 12*b13 - 33*b12 + 65*b11 + 4*b10 + 6*b9 + 7*b8 + 12*b5 + 4472) * q^81 + (-10*b19 - 20*b18 + 10*b16 - 10*b14 + 34*b7 - 8*b6 - 7*b3 - 236*b2 + 268*b1) * q^83 + (16*b19 + 32*b18 - 16*b16 + 16*b14 - 352*b7 + 12*b6 + 84*b3 + 2912*b2 - 166*b1) * q^85 + (61*b17 - 15*b15 + 15*b13 + 66*b12 - 79*b11 + 19*b10 - 30*b9 + 11*b8 + 12*b5 + 7318) * q^87 + (-11*b17 + 32*b15 - 65*b12 + 121*b11 - 133*b10 + 32*b9 - b8 - 192*b5 - 11) * q^89 + (-20*b19 - 42*b18 - 3*b16 + 79*b14 - 19*b7 - 793*b4 + 652*b2 - 21*b1) * q^91 + (-23*b19 - 30*b18 - 9*b16 + 25*b14 - 384*b7 + 29*b6 - 147*b4 - 69*b3 - b2 - 233*b1) * q^93 + (63*b17 + b15 + 15*b13 + 40*b12 + 32*b11 + 40*b10 - b9 - 171*b8 - 19281) * q^95 + (-71*b17 - 36*b15 - 2*b13 + 37*b12 + 143*b11 + 37*b10 + 36*b9 + 103*b8 + 4625) * q^97 + (-9*b19 - 73*b18 + 42*b16 + 83*b14 - 9*b7 - 60*b6 + 221*b4 + 36*b3 - 452*b2 - 272*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 44 q^{9}+O(q^{10})$$ 20 * q - 44 * q^9 $$20 q - 44 q^{9} + 3568 q^{15} - 6112 q^{23} + 15228 q^{25} + 7592 q^{33} + 2800 q^{39} - 26112 q^{47} - 81044 q^{49} - 89296 q^{57} + 14816 q^{63} + 72224 q^{71} - 61256 q^{73} + 89588 q^{81} + 145648 q^{87} - 385504 q^{95} + 92808 q^{97}+O(q^{100})$$ 20 * q - 44 * q^9 + 3568 * q^15 - 6112 * q^23 + 15228 * q^25 + 7592 * q^33 + 2800 * q^39 - 26112 * q^47 - 81044 * q^49 - 89296 * q^57 + 14816 * q^63 + 72224 * q^71 - 61256 * q^73 + 89588 * q^81 + 145648 * q^87 - 385504 * q^95 + 92808 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816$$ :

 $$\beta_{1}$$ $$=$$ $$( - 1423883557 \nu^{18} - 7392863505464 \nu^{14} + \cdots - 13\!\cdots\!04 \nu^{2} ) / 12\!\cdots\!76$$ (-1423883557*v^18 - 7392863505464*v^14 - 12019769956470792*v^10 - 5835598220855211277*v^6 - 13007670695743153904*v^2) / 12441066209394876 $$\beta_{2}$$ $$=$$ $$( 98\!\cdots\!31 \nu^{19} + \cdots - 27\!\cdots\!52 \nu ) / 47\!\cdots\!24$$ (9897196433216131*v^19 - 1203647099846634*v^18 - 270188667554884*v^17 + 51386288160515109160*v^15 - 6249170883296925744*v^14 - 1403041960364985568*v^13 + 83545626263893633949496*v^11 - 10159676700829644475152*v^10 - 2281675296972463730592*v^9 + 40559125447481492306705755*v^7 - 4931652765669594474052026*v^6 - 1108326297000619810066276*v^5 + 88739499905504798275850032*v^3 - 10401372150024040249126560*v^2 - 2716987941781090225778752*v) / 47706811072134616986624 $$\beta_{3}$$ $$=$$ $$( 98\!\cdots\!31 \nu^{19} + \cdots - 27\!\cdots\!52 \nu ) / 23\!\cdots\!12$$ (9897196433216131*v^19 - 15915451814451280*v^18 - 270188667554884*v^17 + 51386288160515109160*v^15 - 82635465662585436544*v^14 - 1403041960364985568*v^13 + 83545626263893633949496*v^11 - 134361141842356595212416*v^10 - 2281675296972463730592*v^9 + 40559125447481492306705755*v^7 - 65245314725445836837899984*v^6 - 1108326297000619810066276*v^5 + 88739499905504798275850032*v^3 - 154061690339172308880983296*v^2 - 2716987941781090225778752*v) / 23853405536067308493312 $$\beta_{4}$$ $$=$$ $$( 30\!\cdots\!39 \nu^{19} + \cdots - 82\!\cdots\!20 \nu ) / 47\!\cdots\!24$$ (30254684389925939*v^19 - 1203647099846634*v^18 - 906246566233060*v^17 + 157082346303556144808*v^15 - 6249170883296925744*v^14 - 4706572393824709600*v^13 + 255389535531461665097400*v^11 - 10159676700829644475152*v^10 - 7653610766154145624224*v^9 + 123983907344652219970982027*v^7 - 4931652765669594474052026*v^6 - 3715616947437929477434372*v^5 + 270772046754320367251943728*v^3 - 10401372150024040249126560*v^2 - 8290688820128458882285120*v) / 47706811072134616986624 $$\beta_{5}$$ $$=$$ $$( - 15\!\cdots\!19 \nu^{19} - 190718048886868 \nu^{17} + \cdots - 43\!\cdots\!08 \nu ) / 23\!\cdots\!12$$ (-15521005659543719*v^19 - 190718048886868*v^17 - 80585432342910822920*v^15 - 988950337554660448*v^13 - 131019781055597514644952*v^11 - 1606888902993709344288*v^9 - 63608566098397355161726367*v^7 - 782126464676733006567412*v^5 - 140607991742328594445600880*v^3 - 4304180885439770158835008*v) / 23853405536067308493312 $$\beta_{6}$$ $$=$$ $$( 98\!\cdots\!31 \nu^{19} + \cdots - 27\!\cdots\!52 \nu ) / 23\!\cdots\!12$$ (9897196433216131*v^19 - 81380041758115848*v^18 - 270188667554884*v^17 + 51386288160515109160*v^15 - 422526616378001747904*v^14 - 1403041960364985568*v^13 + 83545626263893633949496*v^11 - 686960499971028754596672*v^10 - 2281675296972463730592*v^9 + 40559125447481492306705755*v^7 - 333501582428941867677393480*v^6 - 1108326297000619810066276*v^5 + 88739499905504798275850032*v^3 - 729274085647687680344570496*v^2 - 2716987941781090225778752*v) / 23853405536067308493312 $$\beta_{7}$$ $$=$$ $$( 98\!\cdots\!31 \nu^{19} + \cdots - 27\!\cdots\!52 \nu ) / 53\!\cdots\!36$$ (9897196433216131*v^19 + 1203647099846634*v^18 - 270188667554884*v^17 + 51386288160515109160*v^15 + 6249170883296925744*v^14 - 1403041960364985568*v^13 + 83545626263893633949496*v^11 + 10159676700829644475152*v^10 - 2281675296972463730592*v^9 + 40559125447481492306705755*v^7 + 4931652765669594474052026*v^6 - 1108326297000619810066276*v^5 + 88739499905504798275850032*v^3 + 10401372150024040249126560*v^2 - 2716987941781090225778752*v) / 5300756785792735220736 $$\beta_{8}$$ $$=$$ $$( 33\!\cdots\!59 \nu^{19} + \cdots + 18\!\cdots\!76 ) / 11\!\cdots\!56$$ (33223918422578959*v^19 + 1202751124775740*v^17 + 223781016937600*v^16 + 172498254772655186440*v^15 + 6243439556284290592*v^13 + 1161602805298361344*v^12 + 280452052933059953170968*v^11 + 10148010199633940645472*v^9 + 1884766712989508957184*v^8 + 136149161041151999900356423*v^7 + 4923588798997837228342300*v^5 + 910152561096506581944448*v^4 + 296071104113699520804345904*v^3 + 9066103485342945936749248*v + 1860793531876828969182976) / 11926702768033654246656 $$\beta_{9}$$ $$=$$ $$( - 91\!\cdots\!43 \nu^{19} + \cdots - 31\!\cdots\!08 ) / 23\!\cdots\!12$$ (-91641045763933943*v^19 - 2511168626661972*v^17 - 3736490033643376*v^16 - 475800727002404846984*v^15 - 13041469266095195232*v^13 - 19425038909702844544*v^12 - 773573429107681606303320*v^11 - 21209864066730927961632*v^9 - 31622670536389435561344*v^8 - 375548675864490608437113455*v^7 - 10301136505329465094095348*v^5 - 15374925115278095367161584*v^4 - 821606296663709482006857584*v^3 - 25155625463833593714310464*v - 31803526100205582191742208) / 23853405536067308493312 $$\beta_{10}$$ $$=$$ $$( - 98\!\cdots\!31 \nu^{19} - 270188667554884 \nu^{17} + 97754774539128 \nu^{16} + \cdots + 36\!\cdots\!60 ) / 14\!\cdots\!32$$ (-9897196433216131*v^19 - 270188667554884*v^17 + 97754774539128*v^16 - 51386288160515109160*v^15 - 1403041960364985568*v^13 + 503938742689487424*v^12 - 83545626263893633949496*v^11 - 2281675296972463730592*v^9 + 814022369194997354688*v^8 - 40559125447481492306705755*v^7 - 1108326297000619810066276*v^5 + 393049272788636932316472*v^4 - 88739499905504798275850032*v^3 - 2716987941781090225778752*v + 366413161745523324812160) / 1490837846004206780832 $$\beta_{11}$$ $$=$$ $$( 39\!\cdots\!97 \nu^{19} + \cdots + 21\!\cdots\!60 ) / 59\!\cdots\!28$$ (39870333278003297*v^19 + 1128594952003740*v^17 + 586528647234768*v^16 + 207006893553065845304*v^15 + 5860891097824818720*v^13 + 3023632456136924544*v^12 + 336558833425464917422440*v^11 + 9530993625508232138592*v^9 + 4884134215169984128128*v^8 + 163389767291029840752255401*v^7 + 4628624216220514263882876*v^5 + 2358295636731821593898832*v^4 + 357234773140922179315596944*v^3 + 10937814264516955005589440*v + 2198478970473139948872960) / 5963351384016827123328 $$\beta_{12}$$ $$=$$ $$( - 20\!\cdots\!35 \nu^{19} - 588217616893972 \nu^{17} + 97754774539128 \nu^{16} + \cdots + 36\!\cdots\!60 ) / 29\!\cdots\!64$$ (-20075940411571035*v^19 - 588217616893972*v^17 + 97754774539128*v^16 - 104234317232035626984*v^15 - 3054807177094847584*v^13 + 503938742689487424*v^12 - 169467580897677649523448*v^11 - 4967643031563304677408*v^9 + 814022369194997354688*v^8 - 82271516396066856138843891*v^7 - 2411971622219274643750324*v^5 + 393049272788636932316472*v^4 - 179755773329912582763896880*v^3 - 5503838380954774554031936*v + 366413161745523324812160) / 2981675692008413561664 $$\beta_{13}$$ $$=$$ $$( - 11\!\cdots\!53 \nu^{19} + \cdots + 87\!\cdots\!68 ) / 11\!\cdots\!56$$ (-112964584978585553*v^19 - 3459941028783220*v^17 - 2635180834563872*v^16 - 586512041878786877048*v^15 - 17965221751933928032*v^13 - 13730458003181659904*v^12 - 953569719783989788015848*v^11 - 29209997450650404922656*v^9 - 22362617543657660638464*v^8 - 462928695623211681404867225*v^7 - 14180837231438865756108052*v^5 - 10807361690101268464429088*v^4 - 1010540650395543879435539792*v^3 - 30941732014376855947928128*v + 8714690436337487277223168) / 11926702768033654246656 $$\beta_{14}$$ $$=$$ $$( 27\!\cdots\!59 \nu^{19} + \cdots - 74\!\cdots\!80 \nu ) / 23\!\cdots\!12$$ (273550794865247259*v^19 + 59579012815953128*v^18 - 10846138039111044*v^17 + 1420271807755214781288*v^15 + 309333103444795518656*v^14 - 56310455335340334816*v^13 + 2309109339829125767683320*v^11 + 502920040235337013546560*v^10 - 91536837939140688333216*v^9 + 1120979433070204767753750963*v^7 + 244147734875505197747625512*v^6 - 44409126668716174518168228*v^5 + 2431866912316621349007707568*v^3 + 530889482417632408282388096*v^2 - 74470744928237306644054080*v) / 23853405536067308493312 $$\beta_{15}$$ $$=$$ $$( - 29\!\cdots\!97 \nu^{19} + \cdots + 25\!\cdots\!76 ) / 23\!\cdots\!12$$ (-290984556299407697*v^19 - 9727675375316412*v^17 + 2506889803455152*v^16 - 1510790255638335965624*v^15 - 50502106603800938784*v^13 + 13070323357590222464*v^12 - 2456285746706041325329128*v^11 - 82097925264534571834464*v^9 + 21340958156850438809472*v^8 - 1192443642111402607839251993*v^7 - 39842669299316488464149148*v^5 + 10410225810775986744740912*v^4 - 2598032921345906606832933008*v^3 - 79552246375891269334805952*v + 25150633742487737654878976) / 23853405536067308493312 $$\beta_{16}$$ $$=$$ $$( 30\!\cdots\!73 \nu^{19} + \cdots - 82\!\cdots\!32 \nu ) / 23\!\cdots\!12$$ (300558176704002773*v^19 - 58375365716106494*v^18 - 9923184924394668*v^17 + 1560496465427573823128*v^15 - 303083932561498592912*v^14 - 51509984089179913632*v^13 + 2537099929191500548051080*v^11 - 492760363534507369071408*v^10 - 83725970002924566543840*v^9 + 1231677186694805707252665533*v^7 - 239216082109835603273573486*v^6 - 40628767844893762328782092*v^5 + 2684302543828071019313399504*v^3 - 520488110267608368033261536*v^2 - 82193345142267700663895232*v) / 23853405536067308493312 $$\beta_{17}$$ $$=$$ $$( 17\!\cdots\!71 \nu^{19} + \cdots - 55\!\cdots\!84 ) / 11\!\cdots\!56$$ (179412421823743471*v^19 + 5865443278334700*v^17 - 671343050812800*v^16 + 931508551424097249928*v^15 + 30452100864502509216*v^13 - 3484808415895084032*v^12 + 1514473825650109694357784*v^11 + 49506017849918286213600*v^9 - 5654300138968526871552*v^8 + 735227017705515681205580071*v^7 + 24028014829434540212792652*v^5 - 2730457683289519745833344*v^4 + 1602682858622942921044231600*v^3 + 49073938985062747821426624*v - 5594307298398520561795584) / 11926702768033654246656 $$\beta_{18}$$ $$=$$ $$( 45\!\cdots\!51 \nu^{19} + \cdots - 12\!\cdots\!92 \nu ) / 23\!\cdots\!12$$ (456538923650616251*v^19 + 62147674431019154*v^18 - 13897333871966884*v^17 + 2370349513639581202280*v^15 + 322669487284786540784*v^14 - 72142863285237778912*v^13 + 3853788689117486925291000*v^11 + 524602541523557121597264*v^10 - 117273416069688391038624*v^9 + 1870896183540064819836931667*v^7 + 254673718889186965643641250*v^6 - 56925458111697687383869636*v^5 + 4084614298650472921803221168*v^3 + 553760736126154797480507680*v^2 - 125066394667656821064457792*v) / 23853405536067308493312 $$\beta_{19}$$ $$=$$ $$( - 43\!\cdots\!63 \nu^{19} + \cdots + 11\!\cdots\!64 \nu ) / 11\!\cdots\!56$$ (-433138036298022363*v^19 + 57773542166183177*v^18 + 14088621761770188*v^17 - 2248850896642886572200*v^15 + 299959347119850130040*v^14 + 73140056947953003936*v^13 - 3656247768172405901157624*v^11 + 487680525184092546833832*v^10 + 118897174740823988202720*v^9 - 1774988181280282857780768627*v^7 + 236750255727000806036547473*v^6 + 57707311226608273668496428*v^5 - 3869656982989243288374525168*v^3 + 515287424192596347908698256*v^2 + 118488106618860533828758464*v) / 11926702768033654246656
 $$\nu$$ $$=$$ $$( - 4 \beta_{19} - 2 \beta_{18} - 4 \beta_{17} - 8 \beta_{16} - 12 \beta_{15} + 2 \beta_{14} - 6 \beta_{12} - 8 \beta_{11} + 9 \beta_{10} - 12 \beta_{9} - 24 \beta_{8} - 2 \beta_{7} - 24 \beta_{5} - 72 \beta_{4} + 58 \beta_{2} - \beta _1 - 4 ) / 4608$$ (-4*b19 - 2*b18 - 4*b17 - 8*b16 - 12*b15 + 2*b14 - 6*b12 - 8*b11 + 9*b10 - 12*b9 - 24*b8 - 2*b7 - 24*b5 - 72*b4 + 58*b2 - b1 - 4) / 4608 $$\nu^{2}$$ $$=$$ $$( - 6 \beta_{19} - 12 \beta_{18} + 6 \beta_{16} - 6 \beta_{14} + 16 \beta_{7} + 24 \beta_{6} + 72 \beta_{3} - 312 \beta_{2} - 1821 \beta_1 ) / 2304$$ (-6*b19 - 12*b18 + 6*b16 - 6*b14 + 16*b7 + 24*b6 + 72*b3 - 312*b2 - 1821*b1) / 2304 $$\nu^{3}$$ $$=$$ $$( 88 \beta_{19} + 2 \beta_{18} - 76 \beta_{17} + 188 \beta_{16} - 276 \beta_{15} + 10 \beta_{14} - 46 \beta_{12} - 184 \beta_{11} + 161 \beta_{10} - 276 \beta_{9} - 504 \beta_{8} + 90 \beta_{7} - 168 \beta_{5} + 864 \beta_{4} - 202 \beta_{2} + \cdots - 76 ) / 2304$$ (88*b19 + 2*b18 - 76*b17 + 188*b16 - 276*b15 + 10*b14 - 46*b12 - 184*b11 + 161*b10 - 276*b9 - 504*b8 + 90*b7 - 168*b5 + 864*b4 - 202*b2 + b1 - 76) / 2304 $$\nu^{4}$$ $$=$$ $$( - 178 \beta_{17} + 54 \beta_{15} + 132 \beta_{13} - 139 \beta_{12} + 32 \beta_{11} - 139 \beta_{10} - 54 \beta_{9} + 828 \beta_{8} - 398614 ) / 384$$ (-178*b17 + 54*b15 + 132*b13 - 139*b12 + 32*b11 - 139*b10 - 54*b9 + 828*b8 - 398614) / 384 $$\nu^{5}$$ $$=$$ $$( 8032 \beta_{19} - 1222 \beta_{18} + 7060 \beta_{17} + 17036 \beta_{16} + 25068 \beta_{15} + 2194 \beta_{14} - 1674 \beta_{12} + 14864 \beta_{11} - 8925 \beta_{10} + 25068 \beta_{9} + 46248 \beta_{8} + \cdots + 7060 ) / 4608$$ (8032*b19 - 1222*b18 + 7060*b17 + 17036*b16 + 25068*b15 + 2194*b14 - 1674*b12 + 14864*b11 - 8925*b10 + 25068*b9 + 46248*b8 + 16970*b7 + 4344*b5 + 35928*b4 + 100238*b2 - 611*b1 + 7060) / 4608 $$\nu^{6}$$ $$=$$ $$( 4581 \beta_{19} + 9162 \beta_{18} - 4581 \beta_{16} + 4581 \beta_{14} - 20632 \beta_{7} - 4416 \beta_{6} - 34272 \beta_{3} + 244740 \beta_{2} + 836934 \beta_1 ) / 576$$ (4581*b19 + 9162*b18 - 4581*b16 + 4581*b14 - 20632*b7 - 4416*b6 - 34272*b3 + 244740*b2 + 836934*b1) / 576 $$\nu^{7}$$ $$=$$ $$( - 368708 \beta_{19} + 92174 \beta_{18} + 326852 \beta_{17} - 779272 \beta_{16} + 1147980 \beta_{15} - 134030 \beta_{14} - 267850 \beta_{12} + 625448 \beta_{11} - 230257 \beta_{10} + \cdots + 326852 ) / 4608$$ (-368708*b19 + 92174*b18 + 326852*b17 - 779272*b16 + 1147980*b15 - 134030*b14 - 267850*b12 + 625448*b11 - 230257*b10 + 1147980*b9 + 2128536*b8 - 1034322*b7 - 83688*b5 - 257256*b4 - 8213590*b2 + 46087*b1 + 326852) / 4608 $$\nu^{8}$$ $$=$$ $$( 571154 \beta_{17} - 40194 \beta_{15} - 250743 \beta_{13} + 363389 \beta_{12} - 95119 \beta_{11} + 363389 \beta_{10} + 40194 \beta_{9} - 2084787 \beta_{8} + 773453720 ) / 384$$ (571154*b17 - 40194*b15 - 250743*b13 + 363389*b12 - 95119*b11 + 363389*b10 + 40194*b9 - 2084787*b8 + 773453720) / 384 $$\nu^{9}$$ $$=$$ $$( - 8488928 \beta_{19} + 2552678 \beta_{18} - 7507100 \beta_{17} - 17959684 \beta_{16} - 26448612 \beta_{15} - 3534506 \beta_{14} + 9354810 \beta_{12} - 13653640 \beta_{11} + \cdots - 7507100 ) / 2304$$ (-8488928*b19 + 2552678*b18 - 7507100*b17 - 17959684*b16 - 26448612*b15 - 3534506*b14 + 9354810*b12 - 13653640*b11 + 2578749*b10 - 26448612*b9 - 48969912*b8 - 27283954*b7 + 5407224*b5 + 17329536*b4 - 242765566*b2 + 1276339*b1 - 7507100) / 2304 $$\nu^{10}$$ $$=$$ $$( - 44634324 \beta_{19} - 89268648 \beta_{18} + 44634324 \beta_{16} - 44634324 \beta_{14} + 242551096 \beta_{7} + 13739208 \beta_{6} + 267048576 \beta_{3} + \cdots - 6769261395 \beta_1 ) / 2304$$ (-44634324*b19 - 89268648*b18 + 44634324*b16 - 44634324*b14 + 242551096*b7 + 13739208*b6 + 267048576*b3 - 2565998136*b2 - 6769261395*b1) / 2304 $$\nu^{11}$$ $$=$$ $$( 783400208 \beta_{19} - 251215430 \beta_{18} - 686063060 \beta_{17} + 1664137564 \beta_{16} - 2447537772 \beta_{15} + 348552578 \beta_{14} + 1084329706 \beta_{12} + \cdots - 686063060 ) / 4608$$ (783400208*b19 - 251215430*b18 - 686063060*b17 + 1664137564*b16 - 2447537772*b15 + 348552578*b14 + 1084329706*b12 - 1225436144*b11 + 72220477*b10 - 2447537772*b9 - 4505726952*b8 + 2692084170*b7 + 637597320*b5 - 3213250344*b4 + 25567450894*b2 - 125607715*b1 - 686063060) / 4608 $$\nu^{12}$$ $$=$$ $$( - 115686442 \beta_{17} + 3085668 \beta_{15} + 41452467 \beta_{13} - 71439844 \beta_{12} + 14259221 \beta_{11} - 71439844 \beta_{10} - 3085668 \beta_{9} + \cdots - 132898972992 ) / 32$$ (-115686442*b17 + 3085668*b15 + 41452467*b13 - 71439844*b12 + 14259221*b11 - 71439844*b10 - 3085668*b9 + 397768797*b8 - 132898972992) / 32 $$\nu^{13}$$ $$=$$ $$( 36200015860 \beta_{19} - 11607349246 \beta_{18} + 31239087268 \beta_{17} + 77360960312 \beta_{16} + 113560976172 \beta_{15} + 16568277838 \beta_{14} + \cdots + 31239087268 ) / 4608$$ (36200015860*b19 - 11607349246*b18 + 31239087268*b17 + 77360960312*b16 + 113560976172*b15 + 16568277838*b14 - 58121085882*b12 + 56058301448*b11 + 1753578855*b10 + 113560976172*b9 + 207278237976*b8 + 128057071970*b7 - 30380619432*b5 - 207089885016*b4 + 1273910659718*b2 - 5803674623*b1 + 31239087268) / 4608 $$\nu^{14}$$ $$=$$ $$( 101426598726 \beta_{19} + 202853197452 \beta_{18} - 101426598726 \beta_{16} + 101426598726 \beta_{14} - 639299912272 \beta_{7} + \cdots + 14259000539613 \beta_1 ) / 2304$$ (101426598726*b19 + 202853197452*b18 - 101426598726*b16 + 101426598726*b14 - 639299912272*b7 - 17816958744*b6 - 536312988936*b3 + 6456252710904*b2 + 14259000539613*b1) / 2304 $$\nu^{15}$$ $$=$$ $$( - 837238734952 \beta_{19} + 257591632462 \beta_{18} + 709507663276 \beta_{17} - 1802208541580 \beta_{16} + 2639447276532 \beta_{15} + \cdots + 709507663276 ) / 2304$$ (-837238734952*b19 + 257591632462*b18 + 709507663276*b17 - 1802208541580*b16 + 2639447276532*b15 - 385322704138*b14 - 1496780203490*b12 + 1298825572504*b11 + 119875381255*b10 + 2639447276532*b9 + 4767970266360*b8 - 2980631172042*b7 - 641717733144*b5 + 5906506126752*b4 - 30797988083654*b2 + 128795816231*b1 + 709507663276) / 2304 $$\nu^{16}$$ $$=$$ $$( 3116136283570 \beta_{17} - 73302787350 \beta_{15} - 1007072470572 \beta_{13} + 1953561401179 \beta_{12} - 216085951784 \beta_{11} + \cdots + 33\!\cdots\!18 ) / 384$$ (3116136283570*b17 - 73302787350*b15 - 1007072470572*b13 + 1953561401179*b12 - 216085951784*b11 + 1953561401179*b10 + 73302787350*b9 - 10575389683332*b8 + 3381941274608518) / 384 $$\nu^{17}$$ $$=$$ $$( - 77513966969728 \beta_{19} + 22194276830278 \beta_{18} - 64348134431764 \beta_{17} - 168193766477420 \beta_{16} - 245707733447148 \beta_{15} + \cdots - 64348134431764 ) / 4608$$ (-77513966969728*b19 + 22194276830278*b18 - 64348134431764*b17 - 168193766477420*b16 - 245707733447148*b15 - 35360109368242*b14 + 150744471746634*b12 - 121384448906576*b11 - 16149429237027*b10 - 245707733447148*b9 - 438752136742440*b8 - 273779681545418*b7 + 48857945778696*b5 + 636039724887720*b4 - 2926972036272398*b2 + 11097138415139*b1 - 64348134431764) / 4608 $$\nu^{18}$$ $$=$$ $$( - 56218106417985 \beta_{19} - 112436212835970 \beta_{18} + 56218106417985 \beta_{16} - 56218106417985 \beta_{14} + 402462815488456 \beta_{7} + \cdots - 76\!\cdots\!78 \beta_1 ) / 576$$ (-56218106417985*b19 - 112436212835970*b18 + 56218106417985*b16 - 56218106417985*b14 + 402462815488456*b7 + 12175161211104*b6 + 272859427594800*b3 - 3967362091335972*b2 - 7648494797519178*b1) / 576 $$\nu^{19}$$ $$=$$ $$( 35\!\cdots\!80 \beta_{19} - 932007071860718 \beta_{18} + \cdots - 29\!\cdots\!76 ) / 4608$$ (3590347721946980*b19 - 932007071860718*b18 - 2914330894771076*b17 + 7856712271069864*b16 - 11447059993016844*b15 + 1608023899036622*b14 + 7487865920732842*b12 - 5702502399756392*b11 - 913709357240255*b10 - 11447059993016844*b9 - 20190052677330072*b8 + 12463177451885682*b7 + 1627394785343592*b5 - 33170312933067672*b4 + 137646234066775222*b2 - 466003535930359*b1 - 2914330894771076) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 3.85040 + 3.85040i 3.85040 − 3.85040i −0.123732 + 0.123732i −0.123732 − 0.123732i −0.860472 − 0.860472i −0.860472 + 0.860472i −4.77687 − 4.77687i −4.77687 + 4.77687i −4.85801 + 4.85801i −4.85801 − 4.85801i 4.85801 − 4.85801i 4.85801 + 4.85801i 4.77687 + 4.77687i 4.77687 − 4.77687i 0.860472 + 0.860472i 0.860472 − 0.860472i 0.123732 − 0.123732i 0.123732 + 0.123732i −3.85040 − 3.85040i −3.85040 + 3.85040i
0 −15.4669 1.94284i 0 −101.098 0 200.335i 0 235.451 + 60.0993i 0
191.2 0 −15.4669 + 1.94284i 0 −101.098 0 200.335i 0 235.451 60.0993i 0
191.3 0 −14.0881 6.67280i 0 28.8990 0 44.6454i 0 153.948 + 188.014i 0
191.4 0 −14.0881 + 6.67280i 0 28.8990 0 44.6454i 0 153.948 188.014i 0
191.5 0 −10.7340 11.3040i 0 −4.25229 0 187.352i 0 −12.5625 + 242.675i 0
191.6 0 −10.7340 + 11.3040i 0 −4.25229 0 187.352i 0 −12.5625 242.675i 0
191.7 0 −6.91183 13.9723i 0 56.7417 0 130.269i 0 −147.453 + 193.149i 0
191.8 0 −6.91183 + 13.9723i 0 56.7417 0 130.269i 0 −147.453 193.149i 0
191.9 0 −1.14396 15.5464i 0 −71.6821 0 100.489i 0 −240.383 + 35.5689i 0
191.10 0 −1.14396 + 15.5464i 0 −71.6821 0 100.489i 0 −240.383 35.5689i 0
191.11 0 1.14396 15.5464i 0 71.6821 0 100.489i 0 −240.383 35.5689i 0
191.12 0 1.14396 + 15.5464i 0 71.6821 0 100.489i 0 −240.383 + 35.5689i 0
191.13 0 6.91183 13.9723i 0 −56.7417 0 130.269i 0 −147.453 193.149i 0
191.14 0 6.91183 + 13.9723i 0 −56.7417 0 130.269i 0 −147.453 + 193.149i 0
191.15 0 10.7340 11.3040i 0 4.25229 0 187.352i 0 −12.5625 242.675i 0
191.16 0 10.7340 + 11.3040i 0 4.25229 0 187.352i 0 −12.5625 + 242.675i 0
191.17 0 14.0881 6.67280i 0 −28.8990 0 44.6454i 0 153.948 188.014i 0
191.18 0 14.0881 + 6.67280i 0 −28.8990 0 44.6454i 0 153.948 + 188.014i 0
191.19 0 15.4669 1.94284i 0 101.098 0 200.335i 0 235.451 60.0993i 0
191.20 0 15.4669 + 1.94284i 0 101.098 0 200.335i 0 235.451 + 60.0993i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.20 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
12.b even 2 1 inner
24.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 384.6.f.f yes 20
3.b odd 2 1 384.6.f.e 20
4.b odd 2 1 384.6.f.e 20
8.b even 2 1 inner 384.6.f.f yes 20
8.d odd 2 1 384.6.f.e 20
12.b even 2 1 inner 384.6.f.f yes 20
24.f even 2 1 inner 384.6.f.f yes 20
24.h odd 2 1 384.6.f.e 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
384.6.f.e 20 3.b odd 2 1
384.6.f.e 20 4.b odd 2 1
384.6.f.e 20 8.d odd 2 1
384.6.f.e 20 24.h odd 2 1
384.6.f.f yes 20 1.a even 1 1 trivial
384.6.f.f yes 20 8.b even 2 1 inner
384.6.f.f yes 20 12.b even 2 1 inner
384.6.f.f yes 20 24.f 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 $$S_{6}^{\mathrm{new}}(384, [\chi])$$:

 $$T_{5}^{10} - 19432T_{5}^{8} + 117835648T_{5}^{6} - 256371264512T_{5}^{4} + 145811351932928T_{5}^{2} - 2553423501099008$$ T5^10 - 19432*T5^8 + 117835648*T5^6 - 256371264512*T5^4 + 145811351932928*T5^2 - 2553423501099008 $$T_{23}^{5} + 1528T_{23}^{4} - 15431552T_{23}^{3} - 8844092416T_{23}^{2} + 65002850471936T_{23} - 26581880445239296$$ T23^5 + 1528*T23^4 - 15431552*T23^3 - 8844092416*T23^2 + 65002850471936*T23 - 26581880445239296

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} + 22 T^{18} + \cdots + 71\!\cdots\!49$$
$5$ $$(T^{10} - 19432 T^{8} + \cdots - 25\!\cdots\!08)^{2}$$
$7$ $$(T^{10} + 104296 T^{8} + \cdots + 48\!\cdots\!72)^{2}$$
$11$ $$(T^{10} + 910868 T^{8} + \cdots + 74\!\cdots\!64)^{2}$$
$13$ $$(T^{10} + 1989584 T^{8} + \cdots + 53\!\cdots\!00)^{2}$$
$17$ $$(T^{10} + 8276896 T^{8} + \cdots + 81\!\cdots\!52)^{2}$$
$19$ $$(T^{10} - 12896104 T^{8} + \cdots - 26\!\cdots\!00)^{2}$$
$23$ $$(T^{5} + 1528 T^{4} + \cdots - 26\!\cdots\!96)^{4}$$
$29$ $$(T^{10} - 97006312 T^{8} + \cdots - 50\!\cdots\!92)^{2}$$
$31$ $$(T^{10} + 161397672 T^{8} + \cdots + 25\!\cdots\!08)^{2}$$
$37$ $$(T^{10} + 209940944 T^{8} + \cdots + 83\!\cdots\!76)^{2}$$
$41$ $$(T^{10} + 475505664 T^{8} + \cdots + 14\!\cdots\!00)^{2}$$
$43$ $$(T^{10} - 1117853800 T^{8} + \cdots - 80\!\cdots\!12)^{2}$$
$47$ $$(T^{5} + 6528 T^{4} + \cdots - 10\!\cdots\!80)^{4}$$
$53$ $$(T^{10} - 2394345768 T^{8} + \cdots - 49\!\cdots\!52)^{2}$$
$59$ $$(T^{10} + 2001626292 T^{8} + \cdots + 39\!\cdots\!36)^{2}$$
$61$ $$(T^{10} + 6490406864 T^{8} + \cdots + 41\!\cdots\!04)^{2}$$
$67$ $$(T^{10} - 6341733160 T^{8} + \cdots - 12\!\cdots\!00)^{2}$$
$71$ $$(T^{5} - 18056 T^{4} + \cdots - 96\!\cdots\!12)^{4}$$
$73$ $$(T^{5} + 15314 T^{4} + \cdots - 74\!\cdots\!40)^{4}$$
$79$ $$(T^{10} + 9549472360 T^{8} + \cdots + 37\!\cdots\!72)^{2}$$
$83$ $$(T^{10} + 9819894324 T^{8} + \cdots + 38\!\cdots\!76)^{2}$$
$89$ $$(T^{10} + 41200982944 T^{8} + \cdots + 22\!\cdots\!92)^{2}$$
$97$ $$(T^{5} - 23202 T^{4} + \cdots - 60\!\cdots\!04)^{4}$$