# Properties

 Label 224.2.q.a Level $224$ Weight $2$ Character orbit 224.q Analytic conductor $1.789$ 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] = [224,2,Mod(47,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("224.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$224 = 2^{5} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 224.q (of order $$6$$, degree $$2$$, 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: $$1.78864900528$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.144054149089536.2 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4$$ x^12 - 3*x^11 + x^9 + 48*x^8 - 189*x^7 + 431*x^6 - 654*x^5 + 624*x^4 - 340*x^3 + 96*x^2 - 12*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{8} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{5}) q^{7} + ( - \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{9}+O(q^{10})$$ q + b8 * q^3 + (-b7 - b4) * q^5 + (-b10 - b7 + b5) * q^7 + (-b9 + 2*b8 - b6 - b3 - b1 - 1) * q^9 $$q + \beta_{8} q^{3} + ( - \beta_{7} - \beta_{4}) q^{5} + ( - \beta_{10} - \beta_{7} + \beta_{5}) q^{7} + ( - \beta_{9} + 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 - 1) q^{9} + ( - \beta_{6} + \beta_{3} + 1) q^{11} + ( - \beta_{11} - \beta_{7} + 2 \beta_{5}) q^{13} + ( - \beta_{11} + \beta_{10} - \beta_{7} - \beta_{4}) q^{15} + ( - \beta_{8} + \beta_{3} + 2 \beta_1) q^{17} + ( - 2 \beta_{9} - \beta_{6} - \beta_{3} + \beta_1) q^{19} + (\beta_{11} - 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4}) q^{21} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{23} + 2 \beta_{3} q^{25} + (2 \beta_{6} - 2 \beta_{3} - \beta_1 - 1) q^{27} + ( - \beta_{11} + 2 \beta_{10} + \beta_{7}) q^{29} + (2 \beta_{11} + \beta_{10} - \beta_{5} + \beta_{4} + \beta_{2}) q^{31} + (2 \beta_{9} + \beta_{6}) q^{33} + (3 \beta_{9} - \beta_{8} + \beta_{6} + \beta_{3} - 2 \beta_1) q^{35} + (3 \beta_{11} + \beta_{7} - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{37} + ( - 2 \beta_{10} + \beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{2}) q^{39} + (3 \beta_{9} - 3 \beta_{8} + 2 \beta_{6} - 2 \beta_{3} - \beta_1 + 2) q^{41} + ( - 2 \beta_{11} + 2 \beta_{10} + \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{45} + ( - \beta_{11} + 2 \beta_{10} + 3 \beta_{7} - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{47} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{3} - 3 \beta_1 - 3) q^{49} + (2 \beta_{9} - 4 \beta_{8} + \beta_{6} + \beta_{3} + \beta_1 + 2) q^{51} + ( - 4 \beta_{10} - \beta_{5} - \beta_{4} + 4 \beta_{2}) q^{53} + (\beta_{10} + \beta_{4} - 2 \beta_{2}) q^{55} + ( - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_1 - 3) q^{57} + ( - \beta_{8} - 2 \beta_{6} + 2 \beta_{3} + 4 \beta_1 - 2) q^{59} + (\beta_{11} + 4 \beta_{10} + \beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{61} + (\beta_{11} - 2 \beta_{10} + 2 \beta_{7} - \beta_{5} - \beta_{4}) q^{63} + (\beta_{9} - 2 \beta_{8} - \beta_{6} - \beta_{3} - \beta_1 + 1) q^{65} + ( - 2 \beta_{9} + \beta_{8} + 4 \beta_{6} + 2 \beta_{3} - 6) q^{67} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{7} - 4 \beta_{5} - \beta_{4} - 4 \beta_{2}) q^{69} + (2 \beta_{11} + 2 \beta_{7} + 2 \beta_{4}) q^{71} + (\beta_{6} + 2 \beta_{3} + 4 \beta_1 + 1) q^{73} + (2 \beta_{9} + 2 \beta_{6} - 2) q^{75} + ( - 3 \beta_{11} + 2 \beta_{5}) q^{77} + ( - 2 \beta_{11} - 3 \beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{2}) q^{79} + (2 \beta_{9} - \beta_{8} + 3 \beta_{3} + 2) q^{81} + ( - 2 \beta_{9} + 2 \beta_{8} - 8 \beta_{6} - 4 \beta_{3} - 2 \beta_1 + 2) q^{83} + (\beta_{11} - 4 \beta_{10} - 3 \beta_{7} - \beta_{4}) q^{85} + ( - 2 \beta_{11} + 2 \beta_{10} - \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{87} + ( - \beta_{6} + 2) q^{89} + ( - 3 \beta_{9} + \beta_{8} - 2 \beta_{3} - 3 \beta_1 + 4) q^{91} + ( - 5 \beta_{11} - 3 \beta_{7} + 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{93} + ( - 5 \beta_{10} - 3 \beta_{7} + 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{2}) q^{95} + ( - 7 \beta_{9} + 7 \beta_{8} - 2 \beta_{6} - 2 \beta_{3} - \beta_1 - 6) q^{97} + (\beta_{9} + \beta_{8} + \beta_1 + 2) q^{99}+O(q^{100})$$ q + b8 * q^3 + (-b7 - b4) * q^5 + (-b10 - b7 + b5) * q^7 + (-b9 + 2*b8 - b6 - b3 - b1 - 1) * q^9 + (-b6 + b3 + 1) * q^11 + (-b11 - b7 + 2*b5) * q^13 + (-b11 + b10 - b7 - b4) * q^15 + (-b8 + b3 + 2*b1) * q^17 + (-2*b9 - b6 - b3 + b1) * q^19 + (b11 - 2*b10 + b7 - b5 + b4) * q^21 + (b7 - b4 - b2) * q^23 + 2*b3 * q^25 + (2*b6 - 2*b3 - b1 - 1) * q^27 + (-b11 + 2*b10 + b7) * q^29 + (2*b11 + b10 - b5 + b4 + b2) * q^31 + (2*b9 + b6) * q^33 + (3*b9 - b8 + b6 + b3 - 2*b1) * q^35 + (3*b11 + b7 - 3*b5 + 2*b4 - 2*b2) * q^37 + (-2*b10 + b7 - b5 + b4 + 2*b2) * q^39 + (3*b9 - 3*b8 + 2*b6 - 2*b3 - b1 + 2) * q^41 + (-2*b11 + 2*b10 + b7 + b5 - b4 + 2*b2) * q^45 + (-b11 + 2*b10 + 3*b7 - b5 + 2*b4 - b2) * q^47 + (-3*b9 + b8 - 2*b3 - 3*b1 - 3) * q^49 + (2*b9 - 4*b8 + b6 + b3 + b1 + 2) * q^51 + (-4*b10 - b5 - b4 + 4*b2) * q^53 + (b10 + b4 - 2*b2) * q^55 + (-2*b9 - 2*b8 + 2*b1 - 3) * q^57 + (-b8 - 2*b6 + 2*b3 + 4*b1 - 2) * q^59 + (b11 + 4*b10 + b7 + b5 + 2*b4 - 2*b2) * q^61 + (b11 - 2*b10 + 2*b7 - b5 - b4) * q^63 + (b9 - 2*b8 - b6 - b3 - b1 + 1) * q^65 + (-2*b9 + b8 + 4*b6 + 2*b3 - 6) * q^67 + (2*b11 + 2*b10 + 2*b7 - 4*b5 - b4 - 4*b2) * q^69 + (2*b11 + 2*b7 + 2*b4) * q^71 + (b6 + 2*b3 + 4*b1 + 1) * q^73 + (2*b9 + 2*b6 - 2) * q^75 + (-3*b11 + 2*b5) * q^77 + (-2*b11 - 3*b7 + 2*b5 + b4 - b2) * q^79 + (2*b9 - b8 + 3*b3 + 2) * q^81 + (-2*b9 + 2*b8 - 8*b6 - 4*b3 - 2*b1 + 2) * q^83 + (b11 - 4*b10 - 3*b7 - b4) * q^85 + (-2*b11 + 2*b10 - b7 + b5 - b4 + 2*b2) * q^87 + (-b6 + 2) * q^89 + (-3*b9 + b8 - 2*b3 - 3*b1 + 4) * q^91 + (-5*b11 - 3*b7 + 5*b5 - 2*b4 + 2*b2) * q^93 + (-5*b10 - 3*b7 + 2*b5 - 4*b4 + 5*b2) * q^95 + (-7*b9 + 7*b8 - 2*b6 - 2*b3 - b1 - 6) * q^97 + (b9 + b8 + b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 6 q^{3}+O(q^{10})$$ 12 * q + 6 * q^3 $$12 q + 6 q^{3} + 6 q^{11} - 6 q^{17} + 6 q^{19} - 6 q^{33} - 18 q^{35} - 12 q^{49} - 6 q^{51} - 36 q^{57} - 42 q^{59} - 12 q^{65} - 30 q^{67} + 18 q^{73} - 24 q^{75} + 6 q^{81} + 18 q^{89} + 72 q^{91} + 24 q^{99}+O(q^{100})$$ 12 * q + 6 * q^3 + 6 * q^11 - 6 * q^17 + 6 * q^19 - 6 * q^33 - 18 * q^35 - 12 * q^49 - 6 * q^51 - 36 * q^57 - 42 * q^59 - 12 * q^65 - 30 * q^67 + 18 * q^73 - 24 * q^75 + 6 * q^81 + 18 * q^89 + 72 * q^91 + 24 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3x^{11} + x^{9} + 48x^{8} - 189x^{7} + 431x^{6} - 654x^{5} + 624x^{4} - 340x^{3} + 96x^{2} - 12x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$( - 6789 \nu^{11} + 407977 \nu^{10} - 701704 \nu^{9} - 1014429 \nu^{8} - 865016 \nu^{7} + 19540967 \nu^{6} - 53823631 \nu^{5} + 103674982 \nu^{4} + \cdots - 9570816 ) / 5500348$$ (-6789*v^11 + 407977*v^10 - 701704*v^9 - 1014429*v^8 - 865016*v^7 + 19540967*v^6 - 53823631*v^5 + 103674982*v^4 - 118229724*v^3 + 58499210*v^2 - 2665028*v - 9570816) / 5500348 $$\beta_{2}$$ $$=$$ $$( - 76837 \nu^{11} - 171179 \nu^{10} + 509986 \nu^{9} + 1428477 \nu^{8} - 2520360 \nu^{7} - 4710591 \nu^{6} + 8507441 \nu^{5} - 20646084 \nu^{4} + \cdots - 7508164 ) / 5500348$$ (-76837*v^11 - 171179*v^10 + 509986*v^9 + 1428477*v^8 - 2520360*v^7 - 4710591*v^6 + 8507441*v^5 - 20646084*v^4 + 15569686*v^3 + 23756054*v^2 - 19439928*v - 7508164) / 5500348 $$\beta_{3}$$ $$=$$ $$( 54041 \nu^{11} - 302383 \nu^{10} + 439891 \nu^{9} + 184300 \nu^{8} + 2261772 \nu^{7} - 17461376 \nu^{6} + 49851815 \nu^{5} - 91525793 \nu^{4} + \cdots - 178398 ) / 2750174$$ (54041*v^11 - 302383*v^10 + 439891*v^9 + 184300*v^8 + 2261772*v^7 - 17461376*v^6 + 49851815*v^5 - 91525793*v^4 + 115305627*v^3 - 80402588*v^2 + 26100310*v - 178398) / 2750174 $$\beta_{4}$$ $$=$$ $$( - 127195 \nu^{11} + 82977 \nu^{10} + 563109 \nu^{9} + 592838 \nu^{8} - 5748844 \nu^{7} + 9850368 \nu^{6} - 14441625 \nu^{5} + 3585319 \nu^{4} + \cdots - 2572044 ) / 2750174$$ (-127195*v^11 + 82977*v^10 + 563109*v^9 + 592838*v^8 - 5748844*v^7 + 9850368*v^6 - 14441625*v^5 + 3585319*v^4 + 16374455*v^3 - 12968760*v^2 + 70472*v - 2572044) / 2750174 $$\beta_{5}$$ $$=$$ $$( 295642 \nu^{11} - 410251 \nu^{10} - 714509 \nu^{9} - 877952 \nu^{8} + 12614519 \nu^{7} - 34529048 \nu^{6} + 71188533 \nu^{5} - 75428613 \nu^{4} + \cdots + 15787424 ) / 5500348$$ (295642*v^11 - 410251*v^10 - 714509*v^9 - 877952*v^8 + 12614519*v^7 - 34529048*v^6 + 71188533*v^5 - 75428613*v^4 + 41121960*v^3 + 4078592*v^2 - 31162314*v + 15787424) / 5500348 $$\beta_{6}$$ $$=$$ $$( 2236 \nu^{11} - 4785 \nu^{10} - 6307 \nu^{9} + 1316 \nu^{8} + 113135 \nu^{7} - 323728 \nu^{6} + 580379 \nu^{5} - 648191 \nu^{4} + 212358 \nu^{3} + 208368 \nu^{2} + \cdots + 27316 ) / 41356$$ (2236*v^11 - 4785*v^10 - 6307*v^9 + 1316*v^8 + 113135*v^7 - 323728*v^6 + 580379*v^5 - 648191*v^4 + 212358*v^3 + 208368*v^2 - 142154*v + 27316) / 41356 $$\beta_{7}$$ $$=$$ $$( - 588552 \nu^{11} + 1471277 \nu^{10} + 959613 \nu^{9} - 594282 \nu^{8} - 29122539 \nu^{7} + 96716674 \nu^{6} - 193796667 \nu^{5} + 255549677 \nu^{4} + \cdots + 3770204 ) / 5500348$$ (-588552*v^11 + 1471277*v^10 + 959613*v^9 - 594282*v^8 - 29122539*v^7 + 96716674*v^6 - 193796667*v^5 + 255549677*v^4 - 178977012*v^3 + 33607448*v^2 + 7943010*v + 3770204) / 5500348 $$\beta_{8}$$ $$=$$ $$( 658600 \nu^{11} - 892121 \nu^{10} - 2171303 \nu^{9} - 1848624 \nu^{8} + 30777883 \nu^{7} - 72465116 \nu^{6} + 131465595 \nu^{5} - 131228611 \nu^{4} + \cdots - 5832856 ) / 5500348$$ (658600*v^11 - 892121*v^10 - 2171303*v^9 - 1848624*v^8 + 30777883*v^7 - 72465116*v^6 + 131465595*v^5 - 131228611*v^4 + 45177602*v^3 + 1135708*v^2 + 19832586*v - 5832856) / 5500348 $$\beta_{9}$$ $$=$$ $$( - 721311 \nu^{11} + 2509218 \nu^{10} - 349445 \nu^{9} - 2200105 \nu^{8} - 35847041 \nu^{7} + 152603387 \nu^{6} - 342514582 \nu^{5} + 520339833 \nu^{4} + \cdots - 2329388 ) / 5500348$$ (-721311*v^11 + 2509218*v^10 - 349445*v^9 - 2200105*v^8 - 35847041*v^7 + 152603387*v^6 - 342514582*v^5 + 520339833*v^4 - 478822416*v^3 + 211570982*v^2 - 29719866*v - 2329388) / 5500348 $$\beta_{10}$$ $$=$$ $$( 944653 \nu^{11} - 1864021 \nu^{10} - 2371448 \nu^{9} - 304551 \nu^{8} + 45642708 \nu^{7} - 132130403 \nu^{6} + 248926171 \nu^{5} - 285355566 \nu^{4} + \cdots + 938824 ) / 5500348$$ (944653*v^11 - 1864021*v^10 - 2371448*v^9 - 304551*v^8 + 45642708*v^7 - 132130403*v^6 + 248926171*v^5 - 285355566*v^4 + 136789536*v^3 + 36475726*v^2 - 33967380*v + 938824) / 5500348 $$\beta_{11}$$ $$=$$ $$( - 4510 \nu^{11} + 15163 \nu^{10} - 1205 \nu^{9} - 9766 \nu^{8} - 224935 \nu^{7} + 920210 \nu^{6} - 2085585 \nu^{5} + 3227207 \nu^{4} - 3042430 \nu^{3} + \cdots + 76880 ) / 26068$$ (-4510*v^11 + 15163*v^10 - 1205*v^9 - 9766*v^8 - 224935*v^7 + 920210*v^6 - 2085585*v^5 + 3227207*v^4 - 3042430*v^3 + 1622152*v^2 - 452518*v + 76880) / 26068
 $$\nu$$ $$=$$ $$( \beta_{8} + \beta_{7} + \beta_{2} - \beta_1 ) / 2$$ (b8 + b7 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{11} - 2\beta_{10} - 2\beta_{9} + 3\beta_{8} - 2\beta_{7} - 4\beta_{6} + \beta_{4} - 3\beta_{3} - 2\beta _1 + 1 ) / 2$$ (b11 - 2*b10 - 2*b9 + 3*b8 - 2*b7 - 4*b6 + b4 - 3*b3 - 2*b1 + 1) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{11} + 2\beta_{10} + 7\beta_{7} + 14\beta_{6} + 7\beta_{4} + 7\beta_{3} - \beta_{2} + 5\beta _1 - 3 ) / 2$$ (b11 + 2*b10 + 7*b7 + 14*b6 + 7*b4 + 7*b3 - b2 + 5*b1 - 3) / 2 $$\nu^{4}$$ $$=$$ $$( 4 \beta_{11} - 20 \beta_{10} - 8 \beta_{9} + 16 \beta_{8} - 12 \beta_{7} - 15 \beta_{6} + 3 \beta_{5} - 21 \beta_{4} - 14 \beta_{3} + 14 \beta_{2} - 22 \beta _1 - 25 ) / 2$$ (4*b11 - 20*b10 - 8*b9 + 16*b8 - 12*b7 - 15*b6 + 3*b5 - 21*b4 - 14*b3 + 14*b2 - 22*b1 - 25) / 2 $$\nu^{5}$$ $$=$$ $$( 14 \beta_{11} + 13 \beta_{10} - 37 \beta_{9} - 3 \beta_{8} + 8 \beta_{7} - 5 \beta_{6} - 13 \beta_{5} + 62 \beta_{4} - 13 \beta_{3} - 51 \beta_{2} + 55 \beta _1 + 56 ) / 2$$ (14*b11 + 13*b10 - 37*b9 - 3*b8 + 8*b7 - 5*b6 - 13*b5 + 62*b4 - 13*b3 - 51*b2 + 55*b1 + 56) / 2 $$\nu^{6}$$ $$=$$ $$( - 11 \beta_{11} + 24 \beta_{10} + 54 \beta_{9} - 9 \beta_{8} + 136 \beta_{7} + 217 \beta_{6} + 9 \beta_{5} - 68 \beta_{4} + 119 \beta_{3} + 78 \beta_{2} - 62 \beta _1 - 278 ) / 2$$ (-11*b11 + 24*b10 + 54*b9 - 9*b8 + 136*b7 + 217*b6 + 9*b5 - 68*b4 + 119*b3 + 78*b2 - 62*b1 - 278) / 2 $$\nu^{7}$$ $$=$$ $$( 114 \beta_{11} - 291 \beta_{10} - 326 \beta_{9} + 194 \beta_{8} - 542 \beta_{7} - 901 \beta_{6} - 55 \beta_{5} - 141 \beta_{4} - 580 \beta_{3} - 148 \beta_{2} - 165 \beta _1 + 247 ) / 2$$ (114*b11 - 291*b10 - 326*b9 + 194*b8 - 542*b7 - 901*b6 - 55*b5 - 141*b4 - 580*b3 - 148*b2 - 165*b1 + 247) / 2 $$\nu^{8}$$ $$=$$ $$( - 113 \beta_{11} + 1276 \beta_{10} + 404 \beta_{9} - 940 \beta_{8} + 1655 \beta_{7} + 2527 \beta_{6} - 195 \beta_{5} + 1312 \beta_{4} + 1588 \beta_{3} - 652 \beta_{2} + 1500 \beta _1 + 96 ) / 2$$ (-113*b11 + 1276*b10 + 404*b9 - 940*b8 + 1655*b7 + 2527*b6 - 195*b5 + 1312*b4 + 1588*b3 - 652*b2 + 1500*b1 + 96) / 2 $$\nu^{9}$$ $$=$$ $$( - 25 \beta_{11} - 2766 \beta_{10} + 72 \beta_{9} + 1773 \beta_{8} - 2716 \beta_{7} - 3969 \beta_{6} + 525 \beta_{5} - 5278 \beta_{4} - 2352 \beta_{3} + 2895 \beta_{2} - 5001 \beta _1 - 4406 ) / 2$$ (-25*b11 - 2766*b10 + 72*b9 + 1773*b8 - 2716*b7 - 3969*b6 + 525*b5 - 5278*b4 - 2352*b3 + 2895*b2 - 5001*b1 - 4406) / 2 $$\nu^{10}$$ $$=$$ $$( 1301 \beta_{11} + 4480 \beta_{10} - 4337 \beta_{9} - 3460 \beta_{8} - 1298 \beta_{7} - 4210 \beta_{6} - 2532 \beta_{5} + 12294 \beta_{4} - 2171 \beta_{3} - 10904 \beta_{2} + 12175 \beta _1 + 18488 ) / 2$$ (1301*b11 + 4480*b10 - 4337*b9 - 3460*b8 - 1298*b7 - 4210*b6 - 2532*b5 + 12294*b4 - 2171*b3 - 10904*b2 + 12175*b1 + 18488) / 2 $$\nu^{11}$$ $$=$$ $$( - 7243 \beta_{11} + 10057 \beta_{10} + 22468 \beta_{9} - 7768 \beta_{8} + 29731 \beta_{7} + 51343 \beta_{6} + 4507 \beta_{5} - 15000 \beta_{4} + 33291 \beta_{3} + 21451 \beta_{2} - 10568 \beta _1 - 47282 ) / 2$$ (-7243*b11 + 10057*b10 + 22468*b9 - 7768*b8 + 29731*b7 + 51343*b6 + 4507*b5 - 15000*b4 + 33291*b3 + 21451*b2 - 10568*b1 - 47282) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$129$$ $$197$$ $$\chi(n)$$ $$-1$$ $$\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 1.09935 + 0.468876i −0.0263223 + 0.217464i −2.37165 − 1.78079i 0.186445 + 1.54034i 2.00233 − 0.854000i 0.609850 − 0.457915i 1.09935 − 0.468876i −0.0263223 − 0.217464i −2.37165 + 1.78079i 0.186445 − 1.54034i 2.00233 + 0.854000i 0.609850 + 0.457915i
0 −1.18878 + 0.686340i 0 −0.345107 + 0.597743i 0 −2.63639 + 0.222310i 0 −0.557875 + 0.966267i 0
47.2 0 −1.18878 + 0.686340i 0 0.345107 0.597743i 0 2.63639 0.222310i 0 −0.557875 + 0.966267i 0
47.3 0 0.416472 0.240450i 0 −1.59713 + 2.76632i 0 0.694153 + 2.55307i 0 −1.38437 + 2.39779i 0
47.4 0 0.416472 0.240450i 0 1.59713 2.76632i 0 −0.694153 2.55307i 0 −1.38437 + 2.39779i 0
47.5 0 2.27230 1.31191i 0 −1.03926 + 1.80005i 0 1.25203 2.33076i 0 1.94224 3.36406i 0
47.6 0 2.27230 1.31191i 0 1.03926 1.80005i 0 −1.25203 + 2.33076i 0 1.94224 3.36406i 0
143.1 0 −1.18878 0.686340i 0 −0.345107 0.597743i 0 −2.63639 0.222310i 0 −0.557875 0.966267i 0
143.2 0 −1.18878 0.686340i 0 0.345107 + 0.597743i 0 2.63639 + 0.222310i 0 −0.557875 0.966267i 0
143.3 0 0.416472 + 0.240450i 0 −1.59713 2.76632i 0 0.694153 2.55307i 0 −1.38437 2.39779i 0
143.4 0 0.416472 + 0.240450i 0 1.59713 + 2.76632i 0 −0.694153 + 2.55307i 0 −1.38437 2.39779i 0
143.5 0 2.27230 + 1.31191i 0 −1.03926 1.80005i 0 1.25203 + 2.33076i 0 1.94224 + 3.36406i 0
143.6 0 2.27230 + 1.31191i 0 1.03926 + 1.80005i 0 −1.25203 2.33076i 0 1.94224 + 3.36406i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.2.q.a 12
3.b odd 2 1 2016.2.bs.a 12
4.b odd 2 1 56.2.m.a 12
7.b odd 2 1 1568.2.q.g 12
7.c even 3 1 1568.2.e.e 12
7.c even 3 1 1568.2.q.g 12
7.d odd 6 1 inner 224.2.q.a 12
7.d odd 6 1 1568.2.e.e 12
8.b even 2 1 56.2.m.a 12
8.d odd 2 1 inner 224.2.q.a 12
12.b even 2 1 504.2.bk.a 12
21.g even 6 1 2016.2.bs.a 12
24.f even 2 1 2016.2.bs.a 12
24.h odd 2 1 504.2.bk.a 12
28.d even 2 1 392.2.m.g 12
28.f even 6 1 56.2.m.a 12
28.f even 6 1 392.2.e.e 12
28.g odd 6 1 392.2.e.e 12
28.g odd 6 1 392.2.m.g 12
56.e even 2 1 1568.2.q.g 12
56.h odd 2 1 392.2.m.g 12
56.j odd 6 1 56.2.m.a 12
56.j odd 6 1 392.2.e.e 12
56.k odd 6 1 1568.2.e.e 12
56.k odd 6 1 1568.2.q.g 12
56.m even 6 1 inner 224.2.q.a 12
56.m even 6 1 1568.2.e.e 12
56.p even 6 1 392.2.e.e 12
56.p even 6 1 392.2.m.g 12
84.j odd 6 1 504.2.bk.a 12
168.ba even 6 1 504.2.bk.a 12
168.be odd 6 1 2016.2.bs.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.m.a 12 4.b odd 2 1
56.2.m.a 12 8.b even 2 1
56.2.m.a 12 28.f even 6 1
56.2.m.a 12 56.j odd 6 1
224.2.q.a 12 1.a even 1 1 trivial
224.2.q.a 12 7.d odd 6 1 inner
224.2.q.a 12 8.d odd 2 1 inner
224.2.q.a 12 56.m even 6 1 inner
392.2.e.e 12 28.f even 6 1
392.2.e.e 12 28.g odd 6 1
392.2.e.e 12 56.j odd 6 1
392.2.e.e 12 56.p even 6 1
392.2.m.g 12 28.d even 2 1
392.2.m.g 12 28.g odd 6 1
392.2.m.g 12 56.h odd 2 1
392.2.m.g 12 56.p even 6 1
504.2.bk.a 12 12.b even 2 1
504.2.bk.a 12 24.h odd 2 1
504.2.bk.a 12 84.j odd 6 1
504.2.bk.a 12 168.ba even 6 1
1568.2.e.e 12 7.c even 3 1
1568.2.e.e 12 7.d odd 6 1
1568.2.e.e 12 56.k odd 6 1
1568.2.e.e 12 56.m even 6 1
1568.2.q.g 12 7.b odd 2 1
1568.2.q.g 12 7.c even 3 1
1568.2.q.g 12 56.e even 2 1
1568.2.q.g 12 56.k odd 6 1
2016.2.bs.a 12 3.b odd 2 1
2016.2.bs.a 12 21.g even 6 1
2016.2.bs.a 12 24.f even 2 1
2016.2.bs.a 12 168.be odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(224, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$(T^{6} - 3 T^{5} + 9 T^{3} + 6 T^{2} - 9 T + 3)^{2}$$
$5$ $$T^{12} + 15 T^{10} + 174 T^{8} + \cdots + 441$$
$7$ $$T^{12} + 6 T^{10} - 33 T^{8} + \cdots + 117649$$
$11$ $$(T^{6} - 3 T^{5} + 12 T^{4} - 5 T^{3} + \cdots + 49)^{2}$$
$13$ $$(T^{6} - 36 T^{4} + 240 T^{2} - 336)^{2}$$
$17$ $$(T^{6} + 3 T^{5} - 12 T^{4} - 45 T^{3} + \cdots + 1083)^{2}$$
$19$ $$(T^{6} - 3 T^{5} - 18 T^{4} + 63 T^{3} + \cdots + 147)^{2}$$
$23$ $$T^{12} - 69 T^{10} + 3450 T^{8} + \cdots + 45252529$$
$29$ $$(T^{6} + 48 T^{4} + 576 T^{2} + 112)^{2}$$
$31$ $$T^{12} + 99 T^{10} + 8238 T^{8} + \cdots + 441$$
$37$ $$T^{12} - 117 T^{10} + \cdots + 3074369809$$
$41$ $$(T^{6} + 144 T^{4} + 5184 T^{2} + \cdots + 52272)^{2}$$
$43$ $$T^{12}$$
$47$ $$T^{12} + 123 T^{10} + 12990 T^{8} + \cdots + 6456681$$
$53$ $$T^{12} - 213 T^{10} + \cdots + 108651322129$$
$59$ $$(T^{6} + 21 T^{5} + 132 T^{4} + \cdots + 133563)^{2}$$
$61$ $$T^{12} + 207 T^{10} + \cdots + 42362284041$$
$67$ $$(T^{6} + 15 T^{5} + 198 T^{4} + \cdots + 52441)^{2}$$
$71$ $$(T^{2} + 28)^{6}$$
$73$ $$(T^{6} - 9 T^{5} - 36 T^{4} + 567 T^{3} + \cdots + 1323)^{2}$$
$79$ $$T^{12} - 213 T^{10} + \cdots + 31317319089$$
$83$ $$(T^{6} + 228 T^{4} + 816 T^{2} + 192)^{2}$$
$89$ $$(T^{2} - 3 T + 3)^{6}$$
$97$ $$(T^{6} + 360 T^{4} + 19248 T^{2} + \cdots + 134832)^{2}$$