# Properties

 Label 441.2.f.g Level $441$ Weight $2$ Character orbit 441.f Analytic conductor $3.521$ 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] = [441,2,Mod(148,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

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

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

chi := DirichletCharacter("441.148");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.f (of order $$3$$, degree $$2$$, 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: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ x^12 - 7*x^10 + 37*x^8 - 78*x^6 + 123*x^4 - 36*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{6} - \beta_1) q^{2} + \beta_{10} q^{3} + (\beta_{7} - \beta_{6} - \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8}) q^{6} + (\beta_{4} + \beta_1 + 2) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + 1) q^{9}+O(q^{10})$$ q + (b6 - b1) * q^2 + b10 * q^3 + (b7 - b6 - b2 - 1) * q^4 + (-b5 + b3) * q^5 + (-b11 - b10 - b8) * q^6 + (b4 + b1 + 2) * q^8 + (b7 + b6 + b4 + 1) * q^9 $$q + (\beta_{6} - \beta_1) q^{2} + \beta_{10} q^{3} + (\beta_{7} - \beta_{6} - \beta_{2} - 1) q^{4} + ( - \beta_{5} + \beta_{3}) q^{5} + ( - \beta_{11} - \beta_{10} - \beta_{8}) q^{6} + (\beta_{4} + \beta_1 + 2) q^{8} + (\beta_{7} + \beta_{6} + \beta_{4} + 1) q^{9} + ( - \beta_{11} - 3 \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{5} - \beta_{3}) q^{10} + (\beta_{7} + \beta_{4} + \beta_{2}) q^{11} + (\beta_{8} + \beta_{5} - \beta_{3}) q^{12} + (\beta_{11} + \beta_{5} + \beta_{3}) q^{13} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 2) q^{15} + (\beta_{7} + 2 \beta_{6} + \beta_{4} - 2 \beta_1) q^{16} + (\beta_{11} + \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{5} + \beta_{3}) q^{17} + (2 \beta_{6} - \beta_{4} - \beta_{2} - 2 \beta_1 - 4) q^{18} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{5} - 2 \beta_{3}) q^{19} + (\beta_{11} + 4 \beta_{10} - \beta_{9} + 2 \beta_{8} - 4 \beta_{3}) q^{20} + ( - \beta_{2} - 1) q^{22} + (2 \beta_{7} - \beta_{6} - \beta_{2} - 1) q^{23} + (3 \beta_{10} + \beta_{8} - 2 \beta_{5} + \beta_{3}) q^{24} + (3 \beta_{6} + \beta_{2} - 3 \beta_1) q^{25} + ( - \beta_{11} - 2 \beta_{9} - \beta_{8} + \beta_{5}) q^{26} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{5} + 2 \beta_{3}) q^{27} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{4} + 4 \beta_{2} - \beta_1) q^{29} + (\beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{2} - 4 \beta_1 - 4) q^{30} + ( - 2 \beta_{10} - 2 \beta_{9} + \beta_{5} - \beta_{3}) q^{31} + (\beta_{6} - 3 \beta_{2} - 3) q^{32} + ( - \beta_{11} - \beta_{10} + \beta_{8} - \beta_{5} + 2 \beta_{3}) q^{33} + (2 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} - \beta_{8} + 4 \beta_{3}) q^{34} + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 4) q^{36} + ( - \beta_{4} + 2 \beta_1 - 2) q^{37} + (2 \beta_{11} + \beta_{9} + 2 \beta_{8} + \beta_{3}) q^{38} + (\beta_{7} - \beta_{6} + 2 \beta_{4} + 4 \beta_{2} - \beta_1 + 5) q^{39} + (\beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{5} + 5 \beta_{3}) q^{40} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{3}) q^{41} + ( - \beta_{7} - 2 \beta_{6} - \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{43} + (2 \beta_{4} + \beta_1 - 2) q^{44} - 3 \beta_{8} q^{45} + (\beta_{4} + 4 \beta_1 + 1) q^{46} + ( - \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - \beta_{8} - 5 \beta_{3}) q^{47} + ( - 3 \beta_{11} - 2 \beta_{10} - \beta_{8} - \beta_{5} + \beta_{3}) q^{48} + (3 \beta_{7} - 4 \beta_{6} - 9 \beta_{2} - 9) q^{50} + ( - 2 \beta_{7} - \beta_{6} - \beta_{4} + 4 \beta_{2} + 5 \beta_1 - 1) q^{51} + ( - \beta_{11} + \beta_{10} + \beta_{8} + \beta_{3}) q^{52} + ( - 2 \beta_{4} - 3 \beta_1 + 5) q^{53} + ( - \beta_{11} - 5 \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{5} - \beta_{3}) q^{54} + (\beta_{11} + \beta_{10} - 2 \beta_{8} + 2 \beta_{5} - 3 \beta_{3}) q^{55} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{4} + 5 \beta_{2} + \beta_1 + 1) q^{57} + (\beta_{7} - 7 \beta_{6} - \beta_{2} - 1) q^{58} + ( - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{5} - 3 \beta_{3}) q^{59} + (2 \beta_{7} - 5 \beta_{6} + \beta_{4} - 7 \beta_{2} + 7 \beta_1 + 4) q^{60} + (3 \beta_{11} + 2 \beta_{9} + 4 \beta_{8} + 3 \beta_{3}) q^{61} + (\beta_{11} + 7 \beta_{10} - \beta_{9} + 2 \beta_{8} - 2 \beta_{5} + \beta_{3}) q^{62} + (\beta_{4} - 2 \beta_1 - 3) q^{64} + ( - 3 \beta_{7} - 3 \beta_{4}) q^{65} - \beta_{3} q^{66} + ( - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{2} + 2) q^{67} + ( - 3 \beta_{10} - 3 \beta_{9} + 4 \beta_{5} - 4 \beta_{3}) q^{68} + (\beta_{9} + 2 \beta_{8} + \beta_{5}) q^{69} + (\beta_1 + 6) q^{71} + ( - \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 + 4) q^{72} + ( - \beta_{11} - \beta_{9} + \beta_{8} - \beta_{5} + 2 \beta_{3}) q^{73} + ( - 2 \beta_{7} - \beta_{6} - 2 \beta_{4} + 7 \beta_{2} + \beta_1) q^{74} + ( - 3 \beta_{11} - 4 \beta_{10} - 3 \beta_{8} + \beta_{3}) q^{75} + (2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{5} + 3 \beta_{3}) q^{76} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{4} - 5 \beta_{2} - 4 \beta_1 + 2) q^{78} + ( - 3 \beta_{7} + 3 \beta_{6} - 3 \beta_{4} - \beta_{2} - 3 \beta_1) q^{79} + ( - \beta_{11} - 4 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + 5 \beta_{5} - 4 \beta_{3}) q^{80} + ( - \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 - 5) q^{81} + (2 \beta_{11} + 7 \beta_{10} + \beta_{9} + 3 \beta_{8} - 3 \beta_{5} + \beta_{3}) q^{82} + ( - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} - 4 \beta_{8} - \beta_{3}) q^{83} + (6 \beta_{6} + 3 \beta_{2} + 3) q^{85} + ( - 2 \beta_{7} - \beta_{6} + 7 \beta_{2} + 7) q^{86} + (\beta_{11} - 5 \beta_{10} - 3 \beta_{8} + 2 \beta_{5} + 2 \beta_{3}) q^{87} + ( - \beta_{7} + \beta_{6} - \beta_{4} - \beta_{2} - \beta_1) q^{88} + (2 \beta_{11} + 6 \beta_{10} + \beta_{9} + 2 \beta_{8} - 2 \beta_{5}) q^{89} + ( - 3 \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 6 \beta_{5} - 6 \beta_{3}) q^{90} + (4 \beta_{6} + 9 \beta_{2} - 4 \beta_1) q^{92} + ( - 2 \beta_{7} - 4 \beta_{6} - \beta_{4} - 5 \beta_{2} - \beta_1 + 2) q^{93} + ( - \beta_{11} + 3 \beta_{10} + 3 \beta_{9} - 4 \beta_{5} + 2 \beta_{3}) q^{94} + ( - 3 \beta_{7} - 9 \beta_{2} - 9) q^{95} + (\beta_{9} - \beta_{5} - 2 \beta_{3}) q^{96} + (3 \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{3}) q^{97} + ( - 2 \beta_{7} + \beta_{6} + \beta_{4} - 3 \beta_{2} - 5) q^{99}+O(q^{100})$$ q + (b6 - b1) * q^2 + b10 * q^3 + (b7 - b6 - b2 - 1) * q^4 + (-b5 + b3) * q^5 + (-b11 - b10 - b8) * q^6 + (b4 + b1 + 2) * q^8 + (b7 + b6 + b4 + 1) * q^9 + (-b11 - 3*b10 - b9 - 2*b8 + 2*b5 - b3) * q^10 + (b7 + b4 + b2) * q^11 + (b8 + b5 - b3) * q^12 + (b11 + b5 + b3) * q^13 + (-2*b7 + 2*b6 - b4 + b2 - b1 + 2) * q^15 + (b7 + 2*b6 + b4 - 2*b1) * q^16 + (b11 + b10 + 2*b9 + 2*b8 - 2*b5 + b3) * q^17 + (2*b6 - b4 - b2 - 2*b1 - 4) * q^18 + (b11 + b9 - b8 + b5 - 2*b3) * q^19 + (b11 + 4*b10 - b9 + 2*b8 - 4*b3) * q^20 + (-b2 - 1) * q^22 + (2*b7 - b6 - b2 - 1) * q^23 + (3*b10 + b8 - 2*b5 + b3) * q^24 + (3*b6 + b2 - 3*b1) * q^25 + (-b11 - 2*b9 - b8 + b5) * q^26 + (-b11 + b10 + b9 + b8 - 2*b5 + 2*b3) * q^27 + (-2*b7 + b6 - 2*b4 + 4*b2 - b1) * q^29 + (b7 - b6 - b4 - 2*b2 - 4*b1 - 4) * q^30 + (-2*b10 - 2*b9 + b5 - b3) * q^31 + (b6 - 3*b2 - 3) * q^32 + (-b11 - b10 + b8 - b5 + 2*b3) * q^33 + (2*b11 - 5*b10 + 2*b9 - b8 + 4*b3) * q^34 + (2*b7 - 4*b6 + 2*b4 - 3*b2 + 3*b1 - 4) * q^36 + (-b4 + 2*b1 - 2) * q^37 + (2*b11 + b9 + 2*b8 + b3) * q^38 + (b7 - b6 + 2*b4 + 4*b2 - b1 + 5) * q^39 + (b11 + 2*b10 + 2*b9 - 3*b5 + 5*b3) * q^40 + (-b11 - 2*b10 - 2*b9 - 2*b3) * q^41 + (-b7 - 2*b6 - b4 + 2*b2 + 2*b1) * q^43 + (2*b4 + b1 - 2) * q^44 - 3*b8 * q^45 + (b4 + 4*b1 + 1) * q^46 + (-b11 + 3*b10 - 2*b9 - b8 - 5*b3) * q^47 + (-3*b11 - 2*b10 - b8 - b5 + b3) * q^48 + (3*b7 - 4*b6 - 9*b2 - 9) * q^50 + (-2*b7 - b6 - b4 + 4*b2 + 5*b1 - 1) * q^51 + (-b11 + b10 + b8 + b3) * q^52 + (-2*b4 - 3*b1 + 5) * q^53 + (-b11 - 5*b10 + b9 - 2*b8 + b5 - b3) * q^54 + (b11 + b10 - 2*b8 + 2*b5 - 3*b3) * q^55 + (2*b7 - 2*b6 - 2*b4 + 5*b2 + b1 + 1) * q^57 + (b7 - 7*b6 - b2 - 1) * q^58 + (-b11 + b10 + b9 + b5 - 3*b3) * q^59 + (2*b7 - 5*b6 + b4 - 7*b2 + 7*b1 + 4) * q^60 + (3*b11 + 2*b9 + 4*b8 + 3*b3) * q^61 + (b11 + 7*b10 - b9 + 2*b8 - 2*b5 + b3) * q^62 + (b4 - 2*b1 - 3) * q^64 + (-3*b7 - 3*b4) * q^65 - b3 * q^66 + (-3*b7 - 3*b6 + 2*b2 + 2) * q^67 + (-3*b10 - 3*b9 + 4*b5 - 4*b3) * q^68 + (b9 + 2*b8 + b5) * q^69 + (b1 + 6) * q^71 + (-b7 + 4*b6 + b4 - b2 + b1 + 4) * q^72 + (-b11 - b9 + b8 - b5 + 2*b3) * q^73 + (-2*b7 - b6 - 2*b4 + 7*b2 + b1) * q^74 + (-3*b11 - 4*b10 - 3*b8 + b3) * q^75 + (2*b11 + 2*b10 + 2*b9 + b5 + 3*b3) * q^76 + (b7 + 2*b6 + 2*b4 - 5*b2 - 4*b1 + 2) * q^78 + (-3*b7 + 3*b6 - 3*b4 - b2 - 3*b1) * q^79 + (-b11 - 4*b10 - 2*b9 - 5*b8 + 5*b5 - 4*b3) * q^80 + (-b7 + 4*b6 + b4 - b2 + b1 - 5) * q^81 + (2*b11 + 7*b10 + b9 + 3*b8 - 3*b5 + b3) * q^82 + (-2*b11 - 2*b10 - b9 - 4*b8 - b3) * q^83 + (6*b6 + 3*b2 + 3) * q^85 + (-2*b7 - b6 + 7*b2 + 7) * q^86 + (b11 - 5*b10 - 3*b8 + 2*b5 + 2*b3) * q^87 + (-b7 + b6 - b4 - b2 - b1) * q^88 + (2*b11 + 6*b10 + b9 + 2*b8 - 2*b5) * q^89 + (-3*b11 - 6*b10 - 3*b9 - 3*b8 + 6*b5 - 6*b3) * q^90 + (4*b6 + 9*b2 - 4*b1) * q^92 + (-2*b7 - 4*b6 - b4 - 5*b2 - b1 + 2) * q^93 + (-b11 + 3*b10 + 3*b9 - 4*b5 + 2*b3) * q^94 + (-3*b7 - 9*b2 - 9) * q^95 + (b9 - b5 - 2*b3) * q^96 + (3*b10 - b9 + b8 - 3*b3) * q^97 + (-2*b7 + b6 + b4 - 3*b2 - 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{2} - 6 q^{4} + 24 q^{8} + 12 q^{9}+O(q^{10})$$ 12 * q - 2 * q^2 - 6 * q^4 + 24 * q^8 + 12 * q^9 $$12 q - 2 q^{2} - 6 q^{4} + 24 q^{8} + 12 q^{9} - 8 q^{11} + 18 q^{15} - 6 q^{16} - 42 q^{18} - 6 q^{22} - 4 q^{23} - 12 q^{25} - 22 q^{29} - 48 q^{30} - 16 q^{32} - 30 q^{36} - 12 q^{37} + 24 q^{39} - 6 q^{43} - 28 q^{44} + 24 q^{46} - 56 q^{50} - 18 q^{51} + 56 q^{53} - 6 q^{57} - 18 q^{58} + 108 q^{60} - 48 q^{64} + 6 q^{65} + 76 q^{71} + 60 q^{72} - 36 q^{74} + 36 q^{78} + 6 q^{79} - 48 q^{81} + 30 q^{85} + 36 q^{86} + 6 q^{88} - 62 q^{92} + 42 q^{93} - 60 q^{95} - 48 q^{99}+O(q^{100})$$ 12 * q - 2 * q^2 - 6 * q^4 + 24 * q^8 + 12 * q^9 - 8 * q^11 + 18 * q^15 - 6 * q^16 - 42 * q^18 - 6 * q^22 - 4 * q^23 - 12 * q^25 - 22 * q^29 - 48 * q^30 - 16 * q^32 - 30 * q^36 - 12 * q^37 + 24 * q^39 - 6 * q^43 - 28 * q^44 + 24 * q^46 - 56 * q^50 - 18 * q^51 + 56 * q^53 - 6 * q^57 - 18 * q^58 + 108 * q^60 - 48 * q^64 + 6 * q^65 + 76 * q^71 + 60 * q^72 - 36 * q^74 + 36 * q^78 + 6 * q^79 - 48 * q^81 + 30 * q^85 + 36 * q^86 + 6 * q^88 - 62 * q^92 + 42 * q^93 - 60 * q^95 - 48 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 7x^{10} + 37x^{8} - 78x^{6} + 123x^{4} - 36x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( -49\nu^{10} + 259\nu^{8} - 1369\nu^{6} + 861\nu^{4} - 252\nu^{2} - 7266 ) / 4299$$ (-49*v^10 + 259*v^8 - 1369*v^6 + 861*v^4 - 252*v^2 - 7266) / 4299 $$\beta_{2}$$ $$=$$ $$( 148\nu^{10} - 987\nu^{8} + 5217\nu^{6} - 10175\nu^{4} + 17343\nu^{2} - 5076 ) / 4299$$ (148*v^10 - 987*v^8 + 5217*v^6 - 10175*v^4 + 17343*v^2 - 5076) / 4299 $$\beta_{3}$$ $$=$$ $$( -175\nu^{11} + 925\nu^{9} - 3661\nu^{7} - 1224\nu^{5} + 16296\nu^{3} - 43146\nu ) / 12897$$ (-175*v^11 + 925*v^9 - 3661*v^7 - 1224*v^5 + 16296*v^3 - 43146*v) / 12897 $$\beta_{4}$$ $$=$$ $$( 161\nu^{10} - 851\nu^{8} + 3884\nu^{6} - 2829\nu^{4} + 828\nu^{2} + 6678 ) / 4299$$ (161*v^10 - 851*v^8 + 3884*v^6 - 2829*v^4 + 828*v^2 + 6678) / 4299 $$\beta_{5}$$ $$=$$ $$( 269\nu^{11} - 2036\nu^{9} + 11990\nu^{7} - 31749\nu^{5} + 68325\nu^{3} - 32580\nu ) / 12897$$ (269*v^11 - 2036*v^9 + 11990*v^7 - 31749*v^5 + 68325*v^3 - 32580*v) / 12897 $$\beta_{6}$$ $$=$$ $$( -296\nu^{10} + 1974\nu^{8} - 10434\nu^{6} + 20350\nu^{4} - 30387\nu^{2} + 1554 ) / 4299$$ (-296*v^10 + 1974*v^8 - 10434*v^6 + 20350*v^4 - 30387*v^2 + 1554) / 4299 $$\beta_{7}$$ $$=$$ $$( -120\nu^{10} + 839\nu^{8} - 4230\nu^{6} + 8250\nu^{4} - 10034\nu^{2} + 630 ) / 1433$$ (-120*v^10 + 839*v^8 - 4230*v^6 + 8250*v^4 - 10034*v^2 + 630) / 1433 $$\beta_{8}$$ $$=$$ $$( -433\nu^{11} + 3517\nu^{9} - 19204\nu^{7} + 48756\nu^{5} - 71625\nu^{3} + 29142\nu ) / 12897$$ (-433*v^11 + 3517*v^9 - 19204*v^7 + 48756*v^5 - 71625*v^3 + 29142*v) / 12897 $$\beta_{9}$$ $$=$$ $$( 602\nu^{11} - 3182\nu^{9} + 16205\nu^{7} - 14877\nu^{5} + 20292\nu^{3} + 84969\nu ) / 12897$$ (602*v^11 - 3182*v^9 + 16205*v^7 - 14877*v^5 + 20292*v^3 + 84969*v) / 12897 $$\beta_{10}$$ $$=$$ $$( 1321\nu^{11} - 9439\nu^{9} + 50506\nu^{7} - 109806\nu^{5} + 175683\nu^{3} - 46701\nu ) / 12897$$ (1321*v^11 - 9439*v^9 + 50506*v^7 - 109806*v^5 + 175683*v^3 - 46701*v) / 12897 $$\beta_{11}$$ $$=$$ $$( -3580\nu^{11} + 23836\nu^{9} - 124762\nu^{7} + 240393\nu^{5} - 373386\nu^{3} + 23094\nu ) / 12897$$ (-3580*v^11 + 23836*v^9 - 124762*v^7 + 240393*v^5 - 373386*v^3 + 23094*v) / 12897
 $$\nu$$ $$=$$ $$( -\beta_{10} - \beta_{8} + 2\beta_{5} - 2\beta_{3} ) / 3$$ (-b10 - b8 + 2*b5 - 2*b3) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{6} + 2\beta_{2} + 2$$ b6 + 2*b2 + 2 $$\nu^{3}$$ $$=$$ $$( \beta_{11} + 3\beta_{10} - \beta_{9} + 4\beta_{8} + 4\beta_{5} - 5\beta_{3} ) / 3$$ (b11 + 3*b10 - b9 + 4*b8 + 4*b5 - 5*b3) / 3 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 5\beta_{6} - \beta_{4} + 7\beta_{2} - 5\beta_1$$ -b7 + 5*b6 - b4 + 7*b2 - 5*b1 $$\nu^{5}$$ $$=$$ $$( 11\beta_{11} + 42\beta_{10} + 4\beta_{9} + 32\beta_{8} - 16\beta_{5} + 2\beta_{3} ) / 3$$ (11*b11 + 42*b10 + 4*b9 + 32*b8 - 16*b5 + 2*b3) / 3 $$\nu^{6}$$ $$=$$ $$-7\beta_{4} - 23\beta _1 - 28$$ -7*b4 - 23*b1 - 28 $$\nu^{7}$$ $$=$$ $$( 37\beta_{11} + 141\beta_{10} + 53\beta_{9} + 67\beta_{8} - 134\beta_{5} + 118\beta_{3} ) / 3$$ (37*b11 + 141*b10 + 53*b9 + 67*b8 - 134*b5 + 118*b3) / 3 $$\nu^{8}$$ $$=$$ $$37\beta_{7} - 104\beta_{6} - 118\beta_{2} - 118$$ 37*b7 - 104*b6 - 118*b2 - 118 $$\nu^{9}$$ $$=$$ $$( -67\beta_{11} - 222\beta_{10} + 178\beta_{9} - 289\beta_{8} - 289\beta_{5} + 578\beta_{3} ) / 3$$ (-67*b11 - 222*b10 + 178*b9 - 289*b8 - 289*b5 + 578*b3) / 3 $$\nu^{10}$$ $$=$$ $$178\beta_{7} - 467\beta_{6} + 178\beta_{4} - 511\beta_{2} + 467\beta_1$$ 178*b7 - 467*b6 + 178*b4 - 511*b2 + 467*b1 $$\nu^{11}$$ $$=$$ $$( -1112\beta_{11} - 3891\beta_{10} - 289\beta_{9} - 2534\beta_{8} + 1267\beta_{5} + 379\beta_{3} ) / 3$$ (-1112*b11 - 3891*b10 - 289*b9 - 2534*b8 + 1267*b5 + 379*b3) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
148.1
 1.82904 − 1.05600i −1.82904 + 1.05600i −1.29589 + 0.748185i 1.29589 − 0.748185i 0.474636 − 0.274031i −0.474636 + 0.274031i 1.82904 + 1.05600i −1.82904 − 1.05600i −1.29589 − 0.748185i 1.29589 + 0.748185i 0.474636 + 0.274031i −0.474636 − 0.274031i
−1.23025 2.13086i −1.66238 + 0.486291i −2.02704 + 3.51094i −1.82904 + 3.16799i 3.08137 + 2.94405i 0 5.05408 2.52704 1.61680i 9.00071
148.2 −1.23025 2.13086i 1.66238 0.486291i −2.02704 + 3.51094i 1.82904 3.16799i −3.08137 2.94405i 0 5.05408 2.52704 1.61680i −9.00071
148.3 −0.119562 0.207087i −1.12441 + 1.31746i 0.971410 1.68253i 1.29589 2.24456i 0.407265 + 0.0753324i 0 −0.942820 −0.471410 2.96273i −0.619757
148.4 −0.119562 0.207087i 1.12441 1.31746i 0.971410 1.68253i −1.29589 + 2.24456i −0.407265 0.0753324i 0 −0.942820 −0.471410 2.96273i 0.619757
148.5 0.849814 + 1.47192i −1.40434 1.01381i −0.444368 + 0.769668i −0.474636 + 0.822093i 0.298820 2.92864i 0 1.88874 0.944368 + 2.84748i −1.61341
148.6 0.849814 + 1.47192i 1.40434 + 1.01381i −0.444368 + 0.769668i 0.474636 0.822093i −0.298820 + 2.92864i 0 1.88874 0.944368 + 2.84748i 1.61341
295.1 −1.23025 + 2.13086i −1.66238 0.486291i −2.02704 3.51094i −1.82904 3.16799i 3.08137 2.94405i 0 5.05408 2.52704 + 1.61680i 9.00071
295.2 −1.23025 + 2.13086i 1.66238 + 0.486291i −2.02704 3.51094i 1.82904 + 3.16799i −3.08137 + 2.94405i 0 5.05408 2.52704 + 1.61680i −9.00071
295.3 −0.119562 + 0.207087i −1.12441 1.31746i 0.971410 + 1.68253i 1.29589 + 2.24456i 0.407265 0.0753324i 0 −0.942820 −0.471410 + 2.96273i −0.619757
295.4 −0.119562 + 0.207087i 1.12441 + 1.31746i 0.971410 + 1.68253i −1.29589 2.24456i −0.407265 + 0.0753324i 0 −0.942820 −0.471410 + 2.96273i 0.619757
295.5 0.849814 1.47192i −1.40434 + 1.01381i −0.444368 0.769668i −0.474636 0.822093i 0.298820 + 2.92864i 0 1.88874 0.944368 2.84748i −1.61341
295.6 0.849814 1.47192i 1.40434 1.01381i −0.444368 0.769668i 0.474636 + 0.822093i −0.298820 2.92864i 0 1.88874 0.944368 2.84748i 1.61341
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 148.6 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.b odd 2 1 inner
9.c even 3 1 inner
63.l odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.f.g 12
3.b odd 2 1 1323.2.f.g 12
7.b odd 2 1 inner 441.2.f.g 12
7.c even 3 1 441.2.g.g 12
7.c even 3 1 441.2.h.g 12
7.d odd 6 1 441.2.g.g 12
7.d odd 6 1 441.2.h.g 12
9.c even 3 1 inner 441.2.f.g 12
9.c even 3 1 3969.2.a.be 6
9.d odd 6 1 1323.2.f.g 12
9.d odd 6 1 3969.2.a.bd 6
21.c even 2 1 1323.2.f.g 12
21.g even 6 1 1323.2.g.g 12
21.g even 6 1 1323.2.h.g 12
21.h odd 6 1 1323.2.g.g 12
21.h odd 6 1 1323.2.h.g 12
63.g even 3 1 441.2.h.g 12
63.h even 3 1 441.2.g.g 12
63.i even 6 1 1323.2.g.g 12
63.j odd 6 1 1323.2.g.g 12
63.k odd 6 1 441.2.h.g 12
63.l odd 6 1 inner 441.2.f.g 12
63.l odd 6 1 3969.2.a.be 6
63.n odd 6 1 1323.2.h.g 12
63.o even 6 1 1323.2.f.g 12
63.o even 6 1 3969.2.a.bd 6
63.s even 6 1 1323.2.h.g 12
63.t odd 6 1 441.2.g.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.f.g 12 1.a even 1 1 trivial
441.2.f.g 12 7.b odd 2 1 inner
441.2.f.g 12 9.c even 3 1 inner
441.2.f.g 12 63.l odd 6 1 inner
441.2.g.g 12 7.c even 3 1
441.2.g.g 12 7.d odd 6 1
441.2.g.g 12 63.h even 3 1
441.2.g.g 12 63.t odd 6 1
441.2.h.g 12 7.c even 3 1
441.2.h.g 12 7.d odd 6 1
441.2.h.g 12 63.g even 3 1
441.2.h.g 12 63.k odd 6 1
1323.2.f.g 12 3.b odd 2 1
1323.2.f.g 12 9.d odd 6 1
1323.2.f.g 12 21.c even 2 1
1323.2.f.g 12 63.o even 6 1
1323.2.g.g 12 21.g even 6 1
1323.2.g.g 12 21.h odd 6 1
1323.2.g.g 12 63.i even 6 1
1323.2.g.g 12 63.j odd 6 1
1323.2.h.g 12 21.g even 6 1
1323.2.h.g 12 21.h odd 6 1
1323.2.h.g 12 63.n odd 6 1
1323.2.h.g 12 63.s even 6 1
3969.2.a.bd 6 9.d odd 6 1
3969.2.a.bd 6 63.o even 6 1
3969.2.a.be 6 9.c even 3 1
3969.2.a.be 6 63.l odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{6} + T_{2}^{5} + 5T_{2}^{4} - 2T_{2}^{3} + 17T_{2}^{2} + 4T_{2} + 1$$ T2^6 + T2^5 + 5*T2^4 - 2*T2^3 + 17*T2^2 + 4*T2 + 1 $$T_{5}^{12} + 21T_{5}^{10} + 333T_{5}^{8} + 2106T_{5}^{6} + 9963T_{5}^{4} + 8748T_{5}^{2} + 6561$$ T5^12 + 21*T5^10 + 333*T5^8 + 2106*T5^6 + 9963*T5^4 + 8748*T5^2 + 6561

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + T^{5} + 5 T^{4} - 2 T^{3} + 17 T^{2} + \cdots + 1)^{2}$$
$3$ $$T^{12} - 6 T^{10} + 30 T^{8} - 99 T^{6} + \cdots + 729$$
$5$ $$T^{12} + 21 T^{10} + 333 T^{8} + \cdots + 6561$$
$7$ $$T^{12}$$
$11$ $$(T^{6} + 4 T^{5} + 17 T^{4} - 2 T^{3} + 5 T^{2} + \cdots + 1)^{2}$$
$13$ $$T^{12} + 39 T^{10} + 1170 T^{8} + \cdots + 6561$$
$17$ $$(T^{6} - 84 T^{4} + 1593 T^{2} + \cdots - 3969)^{2}$$
$19$ $$(T^{6} - 75 T^{4} + 1296 T^{2} + \cdots - 3969)^{2}$$
$23$ $$(T^{6} + 2 T^{5} + 29 T^{4} + 68 T^{3} + \cdots + 3481)^{2}$$
$29$ $$(T^{6} + 11 T^{5} + 107 T^{4} + 332 T^{3} + \cdots + 7921)^{2}$$
$31$ $$T^{12} + 129 T^{10} + \cdots + 6059221281$$
$37$ $$(T^{3} + 3 T^{2} - 24 T + 27)^{4}$$
$41$ $$T^{12} + 162 T^{10} + \cdots + 43046721$$
$43$ $$(T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729)^{2}$$
$47$ $$T^{12} + 183 T^{10} + \cdots + 37822859361$$
$53$ $$(T^{3} - 14 T^{2} + 11 T + 263)^{4}$$
$59$ $$T^{12} + 183 T^{10} + \cdots + 22430753361$$
$61$ $$T^{12} + 264 T^{10} + \cdots + 311374044081$$
$67$ $$(T^{6} + 111 T^{4} + 706 T^{3} + \cdots + 124609)^{2}$$
$71$ $$(T^{3} - 19 T^{2} + 116 T - 227)^{4}$$
$73$ $$(T^{6} - 75 T^{4} + 1296 T^{2} + \cdots - 3969)^{2}$$
$79$ $$(T^{6} - 3 T^{5} + 87 T^{4} + 20 T^{3} + \cdots + 11449)^{2}$$
$83$ $$T^{12} + 228 T^{10} + \cdots + 51769445841$$
$89$ $$(T^{6} - 246 T^{4} + 5751 T^{2} + \cdots - 3969)^{2}$$
$97$ $$T^{12} + 111 T^{10} + \cdots + 96059601$$