# Properties

 Label 192.4.c.d Level $192$ Weight $4$ Character orbit 192.c Analytic conductor $11.328$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(191,192)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$192 = 2^{6} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 192.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$11.3283667211$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169$$ x^12 - 16*x^10 + 103*x^8 - 260*x^6 + 259*x^4 + 356*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{44}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 96) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9}+O(q^{10})$$ q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 $$q - \beta_{4} q^{3} + \beta_{8} q^{5} + \beta_1 q^{7} + (\beta_{5} - 2) q^{9} - \beta_{10} q^{11} + ( - \beta_{2} - 6) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_1) q^{15} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - 1) q^{17} + ( - \beta_{6} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{19} + ( - \beta_{9} + 3 \beta_{8} + \beta_{5} - \beta_{2} - 12) q^{21} + (2 \beta_{11} - 2 \beta_{6} + 4 \beta_{4} - \beta_{3}) q^{23} + (\beta_{8} - \beta_{7} + 4 \beta_{5} - 2 \beta_{2} - 12) q^{25} + (2 \beta_{11} + \beta_{10} + 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{27} + (9 \beta_{8} + 2 \beta_{7} + 4 \beta_{5} - 2) q^{29} + (2 \beta_{6} + 6 \beta_{4} + 3 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{9} + 11 \beta_{8} + \beta_{7} + 2 \beta_{5} - 2 \beta_{2} + 6) q^{33} + ( - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{6} + 13 \beta_{4} + 2 \beta_{3}) q^{35} + (2 \beta_{8} - 2 \beta_{7} + 8 \beta_{5} - \beta_{2}) q^{37} + ( - 3 \beta_{11} - 3 \beta_{10} - 5 \beta_{6} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{39} + ( - 3 \beta_{9} + 16 \beta_{8} - 3 \beta_{5}) q^{41} + (7 \beta_{6} + 21 \beta_{4} + 5 \beta_{3} + 10 \beta_1) q^{43} + ( - 3 \beta_{9} - 17 \beta_{8} + 2 \beta_{7} - \beta_{5} + 44) q^{45} + (4 \beta_{11} - 2 \beta_{10} + 10 \beta_{6} - 34 \beta_{4} - 2 \beta_{3}) q^{47} + ( - \beta_{8} + \beta_{7} - 4 \beta_{5} - 2 \beta_{2} - 44) q^{49} + (6 \beta_{11} - 3 \beta_{10} - 11 \beta_{6} + 3 \beta_{4} - 12 \beta_{3} + 12 \beta_1) q^{51} + (2 \beta_{9} - 15 \beta_{8} + 2 \beta_{5}) q^{53} + ( - 14 \beta_{6} - 42 \beta_{4} + 11 \beta_{3} + 6 \beta_1) q^{55} + (3 \beta_{9} - 10 \beta_{8} - 2 \beta_{7} + 2 \beta_{5} - 73) q^{57} + ( - 8 \beta_{11} + 4 \beta_{10} + 17 \beta_{6} - 43 \beta_{4} + 4 \beta_{3}) q^{59} + ( - 2 \beta_{8} + 2 \beta_{7} - 8 \beta_{5} + 3 \beta_{2} - 36) q^{61} + ( - 2 \beta_{11} - 4 \beta_{10} + 12 \beta_{6} + 10 \beta_{4} - 12 \beta_{3} + \cdots + 11 \beta_1) q^{63}+ \cdots + ( - 14 \beta_{11} + 14 \beta_{10} - 31 \beta_{6} - 17 \beta_{4} - 9 \beta_{3} + \cdots - 22 \beta_1) q^{99}+O(q^{100})$$ q - b4 * q^3 + b8 * q^5 + b1 * q^7 + (b5 - 2) * q^9 - b10 * q^11 + (-b2 - 6) * q^13 + (-b11 + b10 + b6 + b1) * q^15 + (-b9 - b8 + b7 + b5 - 1) * q^17 + (-b6 - 3*b4 + b3 - 2*b1) * q^19 + (-b9 + 3*b8 + b5 - b2 - 12) * q^21 + (2*b11 - 2*b6 + 4*b4 - b3) * q^23 + (b8 - b7 + 4*b5 - 2*b2 - 12) * q^25 + (2*b11 + b10 + 2*b6 - b4 - 3*b3 - 2*b1) * q^27 + (9*b8 + 2*b7 + 4*b5 - 2) * q^29 + (2*b6 + 6*b4 + 3*b3 - 5*b1) * q^31 + (b9 + 11*b8 + b7 + 2*b5 - 2*b2 + 6) * q^33 + (-4*b11 + 3*b10 - 3*b6 + 13*b4 + 2*b3) * q^35 + (2*b8 - 2*b7 + 8*b5 - b2) * q^37 + (-3*b11 - 3*b10 - 5*b6 + 4*b4 - 3*b3 - 6*b1) * q^39 + (-3*b9 + 16*b8 - 3*b5) * q^41 + (7*b6 + 21*b4 + 5*b3 + 10*b1) * q^43 + (-3*b9 - 17*b8 + 2*b7 - b5 + 44) * q^45 + (4*b11 - 2*b10 + 10*b6 - 34*b4 - 2*b3) * q^47 + (-b8 + b7 - 4*b5 - 2*b2 - 44) * q^49 + (6*b11 - 3*b10 - 11*b6 + 3*b4 - 12*b3 + 12*b1) * q^51 + (2*b9 - 15*b8 + 2*b5) * q^53 + (-14*b6 - 42*b4 + 11*b3 + 6*b1) * q^55 + (3*b9 - 10*b8 - 2*b7 + 2*b5 - 73) * q^57 + (-8*b11 + 4*b10 + 17*b6 - 43*b4 + 4*b3) * q^59 + (-2*b8 + 2*b7 - 8*b5 + 3*b2 - 36) * q^61 + (-2*b11 - 4*b10 + 12*b6 + 10*b4 - 12*b3 + 11*b1) * q^63 + (b9 - 38*b8 - 2*b7 - 3*b5 + 2) * q^65 + (-19*b6 - 57*b4 + 14*b3 - 8*b1) * q^67 + (b9 + 22*b8 - 4*b7 - 5*b5 + 4*b2 - 106) * q^69 + (-2*b11 + 6*b10 - 16*b6 + 50*b4 + b3) * q^71 + (-2*b8 + 2*b7 - 8*b5 + 10*b2 + 204) * q^73 + (-3*b10 + 23*b6 - 6*b4 - 15*b3 - 18*b1) * q^75 + (6*b9 + 22*b8 - 8*b7 - 10*b5 + 8) * q^77 + (34*b6 + 102*b4 + 13*b3 + 3*b1) * q^79 + (13*b8 - b7 - 6*b5 + 12*b2 + 246) * q^81 + (12*b11 - 17*b10 - 34*b6 + 90*b4 - 6*b3) * q^83 + (-8*b8 + 8*b7 - 32*b5 + 4*b2 + 264) * q^85 + (b11 + 17*b10 - 31*b6 - 6*b4 - 18*b3 - b1) * q^87 + (8*b9 - 21*b8 - 3*b7 + 2*b5 + 3) * q^89 + (28*b6 + 84*b4 + 23*b3 - 26*b1) * q^91 + (8*b9 - 27*b8 - 6*b7 - 8*b5 - b2 + 204) * q^93 + (-2*b11 + 8*b10 + 18*b6 - 52*b4 + b3) * q^95 + (b8 - b7 + 4*b5 + 12*b2 - 247) * q^97 + (-14*b11 + 14*b10 - 31*b6 - 17*b4 - 9*b3 - 22*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 20 q^{9}+O(q^{10})$$ 12 * q - 20 * q^9 $$12 q - 20 q^{9} - 72 q^{13} - 136 q^{21} - 132 q^{25} + 80 q^{33} + 24 q^{37} + 544 q^{45} - 540 q^{49} - 888 q^{57} - 456 q^{61} - 1312 q^{69} + 2424 q^{73} + 2924 q^{81} + 3072 q^{85} + 2360 q^{93} - 2952 q^{97}+O(q^{100})$$ 12 * q - 20 * q^9 - 72 * q^13 - 136 * q^21 - 132 * q^25 + 80 * q^33 + 24 * q^37 + 544 * q^45 - 540 * q^49 - 888 * q^57 - 456 * q^61 - 1312 * q^69 + 2424 * q^73 + 2924 * q^81 + 3072 * q^85 + 2360 * q^93 - 2952 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( -16388\nu^{10} + 259515\nu^{8} - 1685594\nu^{6} + 4399856\nu^{4} - 5980224\nu^{2} - 2762095 ) / 217953$$ (-16388*v^10 + 259515*v^8 - 1685594*v^6 + 4399856*v^4 - 5980224*v^2 - 2762095) / 217953 $$\beta_{2}$$ $$=$$ $$( 5904\nu^{10} - 97064\nu^{8} + 571840\nu^{6} - 896416\nu^{4} - 1055120\nu^{2} + 1560520 ) / 72651$$ (5904*v^10 - 97064*v^8 + 571840*v^6 - 896416*v^4 - 1055120*v^2 + 1560520) / 72651 $$\beta_{3}$$ $$=$$ $$( -544\nu^{10} + 8496\nu^{8} - 53344\nu^{6} + 135616\nu^{4} - 180960\nu^{2} - 83504 ) / 3573$$ (-544*v^10 + 8496*v^8 - 53344*v^6 + 135616*v^4 - 180960*v^2 - 83504) / 3573 $$\beta_{4}$$ $$=$$ $$( 2707 \nu^{11} + 10504 \nu^{10} - 41518 \nu^{9} - 183781 \nu^{8} + 254446 \nu^{7} + 1315990 \nu^{6} - 571012 \nu^{5} - 4058808 \nu^{4} + 500393 \nu^{3} + \cdots + 2381353 ) / 629642$$ (2707*v^11 + 10504*v^10 - 41518*v^9 - 183781*v^8 + 254446*v^7 + 1315990*v^6 - 571012*v^5 - 4058808*v^4 + 500393*v^3 + 5081284*v^2 + 95344*v + 2381353) / 629642 $$\beta_{5}$$ $$=$$ $$( - 2631 \nu^{11} - 7176 \nu^{10} + 57332 \nu^{9} + 97500 \nu^{8} - 464316 \nu^{7} - 531232 \nu^{6} + 1492582 \nu^{5} + 802880 \nu^{4} - 515419 \nu^{3} + \cdots - 7299877 ) / 314821$$ (-2631*v^11 - 7176*v^10 + 57332*v^9 + 97500*v^8 - 464316*v^7 - 531232*v^6 + 1492582*v^5 + 802880*v^4 - 515419*v^3 + 954824*v^2 - 2325140*v - 7299877) / 314821 $$\beta_{6}$$ $$=$$ $$( - 8121 \nu^{11} + 31512 \nu^{10} + 124554 \nu^{9} - 551343 \nu^{8} - 763338 \nu^{7} + 3947970 \nu^{6} + 1713036 \nu^{5} - 12176424 \nu^{4} - 1501179 \nu^{3} + \cdots + 7144059 ) / 629642$$ (-8121*v^11 + 31512*v^10 + 124554*v^9 - 551343*v^8 - 763338*v^7 + 3947970*v^6 + 1713036*v^5 - 12176424*v^4 - 1501179*v^3 + 15243852*v^2 - 286032*v + 7144059) / 629642 $$\beta_{7}$$ $$=$$ $$( 18023 \nu^{11} + 43056 \nu^{10} - 102208 \nu^{9} - 585000 \nu^{8} - 549788 \nu^{7} + 3187392 \nu^{6} + 5801042 \nu^{5} - 4817280 \nu^{4} + 2774507 \nu^{3} + \cdots + 44743725 ) / 944463$$ (18023*v^11 + 43056*v^10 - 102208*v^9 - 585000*v^8 - 549788*v^7 + 3187392*v^6 + 5801042*v^5 - 4817280*v^4 + 2774507*v^3 - 5728944*v^2 - 1691184*v + 44743725) / 944463 $$\beta_{8}$$ $$=$$ $$( 3815\nu^{11} - 60784\nu^{9} + 386308\nu^{7} - 931534\nu^{5} + 689195\nu^{3} + 2016192\nu ) / 72651$$ (3815*v^11 - 60784*v^9 + 386308*v^7 - 931534*v^5 + 689195*v^3 + 2016192*v) / 72651 $$\beta_{9}$$ $$=$$ $$( 168575 \nu^{11} + 21528 \nu^{10} - 2577340 \nu^{9} - 292500 \nu^{8} + 15775756 \nu^{7} + 1593696 \nu^{6} - 35075014 \nu^{5} - 2408640 \nu^{4} + \cdots + 21899631 ) / 944463$$ (168575*v^11 + 21528*v^10 - 2577340*v^9 - 292500*v^8 + 15775756*v^7 + 1593696*v^6 - 35075014*v^5 - 2408640*v^4 + 31664915*v^3 - 2864472*v^2 + 84598716*v + 21899631) / 944463 $$\beta_{10}$$ $$=$$ $$( - 525547 \nu^{11} + 8893158 \nu^{9} - 62629246 \nu^{7} + 194548900 \nu^{5} - 288501633 \nu^{3} - 73378376 \nu ) / 2833389$$ (-525547*v^11 + 8893158*v^9 - 62629246*v^7 + 194548900*v^5 - 288501633*v^3 - 73378376*v) / 2833389 $$\beta_{11}$$ $$=$$ $$( 1307467 \nu^{11} - 336856 \nu^{10} - 21563934 \nu^{9} + 5083299 \nu^{8} + 143863390 \nu^{7} - 30457882 \nu^{6} - 397870564 \nu^{5} + 71014216 \nu^{4} + \cdots - 44786495 ) / 5666778$$ (1307467*v^11 - 336856*v^10 - 21563934*v^9 + 5083299*v^8 + 143863390*v^7 - 30457882*v^6 - 397870564*v^5 + 71014216*v^4 + 510897681*v^3 - 97769724*v^2 + 122714864*v - 44786495) / 5666778
 $$\nu$$ $$=$$ $$( 3\beta_{9} + 6\beta_{8} + 32\beta_{6} + 3\beta_{5} - 96\beta_{4} ) / 384$$ (3*b9 + 6*b8 + 32*b6 + 3*b5 - 96*b4) / 384 $$\nu^{2}$$ $$=$$ $$( 4\beta_{8} - 4\beta_{7} - 8\beta_{6} + 16\beta_{5} - 24\beta_{4} + 3\beta_{3} - 24\beta _1 + 508 ) / 192$$ (4*b8 - 4*b7 - 8*b6 + 16*b5 - 24*b4 + 3*b3 - 24*b1 + 508) / 192 $$\nu^{3}$$ $$=$$ $$( 24 \beta_{11} + 12 \beta_{10} + 39 \beta_{9} - 126 \beta_{8} - 12 \beta_{7} + 124 \beta_{6} + 15 \beta_{5} - 396 \beta_{4} - 12 \beta_{3} + 12 ) / 384$$ (24*b11 + 12*b10 + 39*b9 - 126*b8 - 12*b7 + 124*b6 + 15*b5 - 396*b4 - 12*b3 + 12) / 384 $$\nu^{4}$$ $$=$$ $$( 7\beta_{8} - 7\beta_{7} - 24\beta_{6} + 28\beta_{5} - 72\beta_{4} + 42\beta_{3} + 9\beta_{2} - 120\beta _1 + 793 ) / 96$$ (7*b8 - 7*b7 - 24*b6 + 28*b5 - 72*b4 + 42*b3 + 9*b2 - 120*b1 + 793) / 96 $$\nu^{5}$$ $$=$$ $$( 90 \beta_{11} + 72 \beta_{10} + 372 \beta_{9} - 1216 \beta_{8} - 200 \beta_{7} + 232 \beta_{6} - 28 \beta_{5} - 786 \beta_{4} - 45 \beta_{3} + 200 ) / 384$$ (90*b11 + 72*b10 + 372*b9 - 1216*b8 - 200*b7 + 232*b6 - 28*b5 - 786*b4 - 45*b3 + 200) / 384 $$\nu^{6}$$ $$=$$ $$( - 4 \beta_{8} + 4 \beta_{7} - 80 \beta_{6} - 16 \beta_{5} - 240 \beta_{4} + 411 \beta_{3} + 18 \beta_{2} - 912 \beta _1 - 1084 ) / 96$$ (-4*b8 + 4*b7 - 80*b6 - 16*b5 - 240*b4 + 411*b3 + 18*b2 - 912*b1 - 1084) / 96 $$\nu^{7}$$ $$=$$ $$( - 450 \beta_{11} - 252 \beta_{10} + 2622 \beta_{9} - 8728 \beta_{8} - 1652 \beta_{7} - 2764 \beta_{6} - 682 \beta_{5} + 8742 \beta_{4} + 225 \beta_{3} + 1652 ) / 384$$ (-450*b11 - 252*b10 + 2622*b9 - 8728*b8 - 1652*b7 - 2764*b6 - 682*b5 + 8742*b4 + 225*b3 + 1652) / 384 $$\nu^{8}$$ $$=$$ $$( - 34 \beta_{8} + 34 \beta_{7} - 20 \beta_{6} - 136 \beta_{5} - 60 \beta_{4} + 223 \beta_{3} - 33 \beta_{2} - 452 \beta _1 - 4118 ) / 8$$ (-34*b8 + 34*b7 - 20*b6 - 136*b5 - 60*b4 + 223*b3 - 33*b2 - 452*b1 - 4118) / 8 $$\nu^{9}$$ $$=$$ $$( - 10680 \beta_{11} - 7176 \beta_{10} + 13431 \beta_{9} - 45018 \beta_{8} - 8904 \beta_{7} - 46024 \beta_{6} - 4377 \beta_{5} + 148752 \beta_{4} + 5340 \beta_{3} + 8904 ) / 384$$ (-10680*b11 - 7176*b10 + 13431*b9 - 45018*b8 - 8904*b7 - 46024*b6 - 4377*b5 + 148752*b4 + 5340*b3 + 8904) / 384 $$\nu^{10}$$ $$=$$ $$( - 9800 \beta_{8} + 9800 \beta_{7} - 1112 \beta_{6} - 39200 \beta_{5} - 3336 \beta_{4} + 21663 \beta_{3} - 11412 \beta_{2} - 42408 \beta _1 - 1134008 ) / 192$$ (-9800*b8 + 9800*b7 - 1112*b6 - 39200*b5 - 3336*b4 + 21663*b3 - 11412*b2 - 42408*b1 - 1134008) / 192 $$\nu^{11}$$ $$=$$ $$( - 106956 \beta_{11} - 73404 \beta_{10} + 30693 \beta_{9} - 103482 \beta_{8} - 21252 \beta_{7} - 436076 \beta_{6} - 11811 \beta_{5} + 1415184 \beta_{4} + 53478 \beta_{3} + 21252 ) / 384$$ (-106956*b11 - 73404*b10 + 30693*b9 - 103482*b8 - 21252*b7 - 436076*b6 - 11811*b5 + 1415184*b4 + 53478*b3 + 21252) / 384

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$$.

 $$n$$ $$65$$ $$127$$ $$133$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −2.59708 − 0.707107i −2.59708 + 0.707107i −1.64111 + 0.707107i −1.64111 − 0.707107i −0.248859 − 0.707107i −0.248859 + 0.707107i 0.248859 + 0.707107i 0.248859 − 0.707107i 1.64111 − 0.707107i 1.64111 + 0.707107i 2.59708 + 0.707107i 2.59708 − 0.707107i
0 −5.19416 0.143987i 0 7.96714i 0 25.6706i 0 26.9585 + 1.49578i 0
191.2 0 −5.19416 + 0.143987i 0 7.96714i 0 25.6706i 0 26.9585 1.49578i 0
191.3 0 −3.28223 4.02827i 0 2.00080i 0 14.5105i 0 −5.45398 + 26.4434i 0
191.4 0 −3.28223 + 4.02827i 0 2.00080i 0 14.5105i 0 −5.45398 26.4434i 0
191.5 0 −0.497717 5.17226i 0 18.4532i 0 17.1600i 0 −26.5046 + 5.14865i 0
191.6 0 −0.497717 + 5.17226i 0 18.4532i 0 17.1600i 0 −26.5046 5.14865i 0
191.7 0 0.497717 5.17226i 0 18.4532i 0 17.1600i 0 −26.5046 5.14865i 0
191.8 0 0.497717 + 5.17226i 0 18.4532i 0 17.1600i 0 −26.5046 + 5.14865i 0
191.9 0 3.28223 4.02827i 0 2.00080i 0 14.5105i 0 −5.45398 26.4434i 0
191.10 0 3.28223 + 4.02827i 0 2.00080i 0 14.5105i 0 −5.45398 + 26.4434i 0
191.11 0 5.19416 0.143987i 0 7.96714i 0 25.6706i 0 26.9585 1.49578i 0
191.12 0 5.19416 + 0.143987i 0 7.96714i 0 25.6706i 0 26.9585 + 1.49578i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 192.4.c.d 12
3.b odd 2 1 inner 192.4.c.d 12
4.b odd 2 1 inner 192.4.c.d 12
8.b even 2 1 96.4.c.a 12
8.d odd 2 1 96.4.c.a 12
12.b even 2 1 inner 192.4.c.d 12
16.e even 4 1 768.4.f.d 12
16.e even 4 1 768.4.f.i 12
16.f odd 4 1 768.4.f.d 12
16.f odd 4 1 768.4.f.i 12
24.f even 2 1 96.4.c.a 12
24.h odd 2 1 96.4.c.a 12
48.i odd 4 1 768.4.f.d 12
48.i odd 4 1 768.4.f.i 12
48.k even 4 1 768.4.f.d 12
48.k even 4 1 768.4.f.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.4.c.a 12 8.b even 2 1
96.4.c.a 12 8.d odd 2 1
96.4.c.a 12 24.f even 2 1
96.4.c.a 12 24.h odd 2 1
192.4.c.d 12 1.a even 1 1 trivial
192.4.c.d 12 3.b odd 2 1 inner
192.4.c.d 12 4.b odd 2 1 inner
192.4.c.d 12 12.b even 2 1 inner
768.4.f.d 12 16.e even 4 1
768.4.f.d 12 16.f odd 4 1
768.4.f.d 12 48.i odd 4 1
768.4.f.d 12 48.k even 4 1
768.4.f.i 12 16.e even 4 1
768.4.f.i 12 16.f odd 4 1
768.4.f.i 12 48.i odd 4 1
768.4.f.i 12 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 408T_{5}^{4} + 23232T_{5}^{2} + 86528$$ acting on $$S_{4}^{\mathrm{new}}(192, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 10 T^{10} + \cdots + 387420489$$
$5$ $$(T^{6} + 408 T^{4} + 23232 T^{2} + \cdots + 86528)^{2}$$
$7$ $$(T^{6} + 1164 T^{4} + 394800 T^{2} + \cdots + 40857664)^{2}$$
$11$ $$(T^{6} - 5400 T^{4} + \cdots - 3202560512)^{2}$$
$13$ $$(T^{3} + 18 T^{2} - 4500 T - 133928)^{4}$$
$17$ $$(T^{6} + 24576 T^{4} + \cdots + 391378894848)^{2}$$
$19$ $$(T^{6} + 8076 T^{4} + \cdots + 2518433856)^{2}$$
$23$ $$(T^{6} - 26976 T^{4} + \cdots - 124728147968)^{2}$$
$29$ $$(T^{6} + 118296 T^{4} + \cdots + 19385223967232)^{2}$$
$31$ $$(T^{6} + 58956 T^{4} + \cdots + 708600302656)^{2}$$
$37$ $$(T^{3} - 6 T^{2} - 96756 T + 3249144)^{4}$$
$41$ $$(T^{6} + 317280 T^{4} + \cdots + 359281246896128)^{2}$$
$43$ $$(T^{6} + 246252 T^{4} + \cdots + 24182238991936)^{2}$$
$47$ $$(T^{6} - 273792 T^{4} + \cdots - 165630483365888)^{2}$$
$53$ $$(T^{6} + 187224 T^{4} + \cdots + 152455972372992)^{2}$$
$59$ $$(T^{6} - 680472 T^{4} + \cdots - 61\!\cdots\!48)^{2}$$
$61$ $$(T^{3} + 114 T^{2} - 104724 T - 17941928)^{4}$$
$67$ $$(T^{6} + 1232556 T^{4} + \cdots + 53\!\cdots\!56)^{2}$$
$71$ $$(T^{6} - 589152 T^{4} + \cdots - 61\!\cdots\!68)^{2}$$
$73$ $$(T^{3} - 606 T^{2} - 319956 T + 145079064)^{4}$$
$79$ $$(T^{6} + 2122764 T^{4} + \cdots + 76\!\cdots\!44)^{2}$$
$83$ $$(T^{6} - 3104664 T^{4} + \cdots - 22\!\cdots\!92)^{2}$$
$89$ $$(T^{6} + 1258080 T^{4} + \cdots + 761571160915968)^{2}$$
$97$ $$(T^{3} + 738 T^{2} - 581844 T - 38714344)^{4}$$