# Properties

 Label 507.2.j.h Level $507$ Weight $2$ Character orbit 507.j Analytic conductor $4.048$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,2,Mod(316,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("507.316");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$507 = 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 507.j (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: $$4.04841538248$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: 12.0.17213603549184.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ x^12 - 5*x^10 + 19*x^8 - 28*x^6 + 31*x^4 - 6*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{4} + ( - \beta_{11} + \beta_{10} - \beta_{6} - \beta_{2}) q^{5} + (\beta_{8} - \beta_1) q^{6} + ( - \beta_{11} + 3 \beta_{10}) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} - \beta_{7} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b7 - 1) * q^3 + (b9 - b7 + b4 - b3 + 1) * q^4 + (-b11 + b10 - b6 - b2) * q^5 + (b8 - b1) * q^6 + (-b11 + 3*b10) * q^7 + (b11 - b8 + b2) * q^8 - b7 * q^9 $$q + \beta_1 q^{2} + (\beta_{7} - 1) q^{3} + (\beta_{9} - \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{4} + ( - \beta_{11} + \beta_{10} - \beta_{6} - \beta_{2}) q^{5} + (\beta_{8} - \beta_1) q^{6} + ( - \beta_{11} + 3 \beta_{10}) q^{7} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{8} - \beta_{7} q^{9} + (\beta_{7} - 2 \beta_{4} - 1) q^{10} + ( - 2 \beta_{6} + 4 \beta_{2} + 3 \beta_1) q^{11} + \beta_{3} q^{12} + (4 \beta_{5} - 1) q^{14} + (\beta_{6} + \beta_{2}) q^{15} + ( - 3 \beta_{9} - 2 \beta_{4}) q^{16} + (\beta_{9} - 3 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{17} - \beta_{8} q^{18} + ( - \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - 3 \beta_1) q^{19} + ( - \beta_{8} + \beta_1) q^{20} + (\beta_{11} - 3 \beta_{10} + 3 \beta_{6} + \beta_{2}) q^{21} + (3 \beta_{9} - \beta_{7} + 2 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + 3) q^{22} + (5 \beta_{9} - 3 \beta_{7} + 2 \beta_{4} + 3) q^{23} + ( - \beta_{2} + \beta_1) q^{24} + (3 \beta_{5} + 2 \beta_{3}) q^{25} + q^{27} + ( - 2 \beta_{6} + 2 \beta_{2} + 3 \beta_1) q^{28} + (2 \beta_{9} - 3 \beta_{7} - \beta_{4} + 3) q^{29} + ( - \beta_{7} + 2 \beta_{5} + 2 \beta_{4}) q^{30} + (2 \beta_{11} + 5 \beta_{10} + 5 \beta_{8} - 5 \beta_{6} + 2 \beta_{2}) q^{31} + ( - 4 \beta_{11} - \beta_{10} - 3 \beta_{8} + 3 \beta_1) q^{32} + (4 \beta_{11} + 2 \beta_{10} + 3 \beta_{8} - 3 \beta_1) q^{33} + (\beta_{11} - \beta_{8} + \beta_{2}) q^{34} + ( - 4 \beta_{9} - 5 \beta_{7} + 5 \beta_{5} + \beta_{4} + 4 \beta_{3} - 4) q^{35} + ( - \beta_{9} + \beta_{7} - \beta_{4} - 1) q^{36} + ( - 4 \beta_{6} - \beta_{2} - \beta_1) q^{37} + ( - 2 \beta_{5} + 3 \beta_{3} - 7) q^{38} + ( - 4 \beta_{5} - \beta_{3} + 4) q^{40} + \beta_1 q^{41} + ( - \beta_{7} + 4 \beta_{4} + 1) q^{42} + ( - 2 \beta_{9} + \beta_{7} - 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 2) q^{43} + ( - 3 \beta_{11} - 6 \beta_{10} + \beta_{8} + 6 \beta_{6} - 3 \beta_{2}) q^{44} + (\beta_{11} - \beta_{10}) q^{45} + (2 \beta_{11} + 3 \beta_{10} + 4 \beta_{8} - 4 \beta_1) q^{46} + (3 \beta_{11} + 7 \beta_{10} + 9 \beta_{8} - 7 \beta_{6} + 3 \beta_{2}) q^{47} + (3 \beta_{9} - \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 3) q^{48} + ( - 6 \beta_{9} - 4 \beta_{7} + \beta_{4} + 4) q^{49} + (3 \beta_{6} + \beta_{2} + \beta_1) q^{50} + (\beta_{3} + 2) q^{51} + (3 \beta_{5} + 2 \beta_{3} - 6) q^{53} + \beta_1 q^{54} + ( - 2 \beta_{9} - 3 \beta_{7} - 2 \beta_{4} + 3) q^{55} + (3 \beta_{9} + 3 \beta_{7} - 8 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} + 3) q^{56} + (\beta_{11} + 3 \beta_{10} - 3 \beta_{8} - 3 \beta_{6} + \beta_{2}) q^{57} + ( - \beta_{11} + 3 \beta_{10} - 2 \beta_{8} + 2 \beta_1) q^{58} + ( - 6 \beta_{11} - 8 \beta_{10} - 4 \beta_{8} + 4 \beta_1) q^{59} + \beta_{8} q^{60} + ( - 3 \beta_{9} + 7 \beta_{7} - 2 \beta_{5} - 5 \beta_{4} + 3 \beta_{3} - 3) q^{61} + (5 \beta_{9} + 3 \beta_{7} + 2 \beta_{4} - 3) q^{62} + ( - 3 \beta_{6} - \beta_{2}) q^{63} + (5 \beta_{5} + 3 \beta_{3} - 4) q^{64} + ( - 2 \beta_{5} + 3 \beta_{3} - 2) q^{66} + (4 \beta_{6} - 3 \beta_{2} - 4 \beta_1) q^{67} + ( - 3 \beta_{9} + 4 \beta_{7} - 2 \beta_{4} - 4) q^{68} + ( - 5 \beta_{9} + 3 \beta_{7} + 3 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 5) q^{69} + (\beta_{11} - 5 \beta_{10} - 8 \beta_{8} + 5 \beta_{6} + \beta_{2}) q^{70} + (2 \beta_{11} + \beta_{10} - 5 \beta_{8} + 5 \beta_1) q^{71} + ( - \beta_{11} + \beta_{8} - \beta_1) q^{72} + ( - \beta_{11} - 7 \beta_{10} - 2 \beta_{8} + 7 \beta_{6} - \beta_{2}) q^{73} + ( - \beta_{9} - 5 \beta_{5} - 6 \beta_{4} + \beta_{3} - 1) q^{74} + ( - 2 \beta_{9} + 2 \beta_{7} + \beta_{4} - 2) q^{75} + (4 \beta_{6} - 7 \beta_{2} - 6 \beta_1) q^{76} + ( - 2 \beta_{5} - 10 \beta_{3} + 9) q^{77} + (\beta_{5} + 5 \beta_{3} - 5) q^{79} + ( - 4 \beta_{6} - 3 \beta_{2} - \beta_1) q^{80} + (\beta_{7} - 1) q^{81} + (\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{82} + ( - 3 \beta_{11} - 6 \beta_{10} + \beta_{8} + 6 \beta_{6} - 3 \beta_{2}) q^{83} + (2 \beta_{11} + 2 \beta_{10} + 3 \beta_{8} - 3 \beta_1) q^{84} + (2 \beta_{11} - 2 \beta_{10} - \beta_{8} + \beta_1) q^{85} + ( - 4 \beta_{11} + 2 \beta_{10} - 5 \beta_{8} - 2 \beta_{6} - 4 \beta_{2}) q^{86} + ( - 2 \beta_{9} + 3 \beta_{7} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2) q^{87} + ( - 5 \beta_{9} + 6 \beta_{7} - 6 \beta_{4} - 6) q^{88} + (2 \beta_{6} + \beta_{2} - 2 \beta_1) q^{89} + ( - 2 \beta_{5} + 1) q^{90} + ( - 5 \beta_{5} - 6 \beta_{3} - 2) q^{92} + (5 \beta_{6} - 2 \beta_{2} - 5 \beta_1) q^{93} + (9 \beta_{9} + 6 \beta_{7} + 5 \beta_{4} - 6) q^{94} + (2 \beta_{9} - 2 \beta_{7} + 5 \beta_{5} + 7 \beta_{4} - 2 \beta_{3} + 2) q^{95} + (4 \beta_{11} + \beta_{10} + 3 \beta_{8} - \beta_{6} + 4 \beta_{2}) q^{96} + (10 \beta_{11} + 3 \beta_{10} + 4 \beta_{8} - 4 \beta_1) q^{97} + (\beta_{11} - 7 \beta_{10} - 9 \beta_{8} + 9 \beta_1) q^{98} + ( - 4 \beta_{11} - 2 \beta_{10} - 3 \beta_{8} + 2 \beta_{6} - 4 \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^2 + (b7 - 1) * q^3 + (b9 - b7 + b4 - b3 + 1) * q^4 + (-b11 + b10 - b6 - b2) * q^5 + (b8 - b1) * q^6 + (-b11 + 3*b10) * q^7 + (b11 - b8 + b2) * q^8 - b7 * q^9 + (b7 - 2*b4 - 1) * q^10 + (-2*b6 + 4*b2 + 3*b1) * q^11 + b3 * q^12 + (4*b5 - 1) * q^14 + (b6 + b2) * q^15 + (-3*b9 - 2*b4) * q^16 + (b9 - 3*b7 + b4 - b3 + 1) * q^17 - b8 * q^18 + (-b11 - 3*b10 + 3*b8 - 3*b1) * q^19 + (-b8 + b1) * q^20 + (b11 - 3*b10 + 3*b6 + b2) * q^21 + (3*b9 - b7 + 2*b5 + 5*b4 - 3*b3 + 3) * q^22 + (5*b9 - 3*b7 + 2*b4 + 3) * q^23 + (-b2 + b1) * q^24 + (3*b5 + 2*b3) * q^25 + q^27 + (-2*b6 + 2*b2 + 3*b1) * q^28 + (2*b9 - 3*b7 - b4 + 3) * q^29 + (-b7 + 2*b5 + 2*b4) * q^30 + (2*b11 + 5*b10 + 5*b8 - 5*b6 + 2*b2) * q^31 + (-4*b11 - b10 - 3*b8 + 3*b1) * q^32 + (4*b11 + 2*b10 + 3*b8 - 3*b1) * q^33 + (b11 - b8 + b2) * q^34 + (-4*b9 - 5*b7 + 5*b5 + b4 + 4*b3 - 4) * q^35 + (-b9 + b7 - b4 - 1) * q^36 + (-4*b6 - b2 - b1) * q^37 + (-2*b5 + 3*b3 - 7) * q^38 + (-4*b5 - b3 + 4) * q^40 + b1 * q^41 + (-b7 + 4*b4 + 1) * q^42 + (-2*b9 + b7 - 2*b5 - 4*b4 + 2*b3 - 2) * q^43 + (-3*b11 - 6*b10 + b8 + 6*b6 - 3*b2) * q^44 + (b11 - b10) * q^45 + (2*b11 + 3*b10 + 4*b8 - 4*b1) * q^46 + (3*b11 + 7*b10 + 9*b8 - 7*b6 + 3*b2) * q^47 + (3*b9 - b5 + 2*b4 - 3*b3 + 3) * q^48 + (-6*b9 - 4*b7 + b4 + 4) * q^49 + (3*b6 + b2 + b1) * q^50 + (b3 + 2) * q^51 + (3*b5 + 2*b3 - 6) * q^53 + b1 * q^54 + (-2*b9 - 3*b7 - 2*b4 + 3) * q^55 + (3*b9 + 3*b7 - 8*b5 - 5*b4 - 3*b3 + 3) * q^56 + (b11 + 3*b10 - 3*b8 - 3*b6 + b2) * q^57 + (-b11 + 3*b10 - 2*b8 + 2*b1) * q^58 + (-6*b11 - 8*b10 - 4*b8 + 4*b1) * q^59 + b8 * q^60 + (-3*b9 + 7*b7 - 2*b5 - 5*b4 + 3*b3 - 3) * q^61 + (5*b9 + 3*b7 + 2*b4 - 3) * q^62 + (-3*b6 - b2) * q^63 + (5*b5 + 3*b3 - 4) * q^64 + (-2*b5 + 3*b3 - 2) * q^66 + (4*b6 - 3*b2 - 4*b1) * q^67 + (-3*b9 + 4*b7 - 2*b4 - 4) * q^68 + (-5*b9 + 3*b7 + 3*b5 - 2*b4 + 5*b3 - 5) * q^69 + (b11 - 5*b10 - 8*b8 + 5*b6 + b2) * q^70 + (2*b11 + b10 - 5*b8 + 5*b1) * q^71 + (-b11 + b8 - b1) * q^72 + (-b11 - 7*b10 - 2*b8 + 7*b6 - b2) * q^73 + (-b9 - 5*b5 - 6*b4 + b3 - 1) * q^74 + (-2*b9 + 2*b7 + b4 - 2) * q^75 + (4*b6 - 7*b2 - 6*b1) * q^76 + (-2*b5 - 10*b3 + 9) * q^77 + (b5 + 5*b3 - 5) * q^79 + (-4*b6 - 3*b2 - b1) * q^80 + (b7 - 1) * q^81 + (b9 + b7 + b4 - b3 + 1) * q^82 + (-3*b11 - 6*b10 + b8 + 6*b6 - 3*b2) * q^83 + (2*b11 + 2*b10 + 3*b8 - 3*b1) * q^84 + (2*b11 - 2*b10 - b8 + b1) * q^85 + (-4*b11 + 2*b10 - 5*b8 - 2*b6 - 4*b2) * q^86 + (-2*b9 + 3*b7 + 3*b5 + b4 + 2*b3 - 2) * q^87 + (-5*b9 + 6*b7 - 6*b4 - 6) * q^88 + (2*b6 + b2 - 2*b1) * q^89 + (-2*b5 + 1) * q^90 + (-5*b5 - 6*b3 - 2) * q^92 + (5*b6 - 2*b2 - 5*b1) * q^93 + (9*b9 + 6*b7 + 5*b4 - 6) * q^94 + (2*b9 - 2*b7 + 5*b5 + 7*b4 - 2*b3 + 2) * q^95 + (4*b11 + b10 + 3*b8 - b6 + 4*b2) * q^96 + (10*b11 + 3*b10 + 4*b8 - 4*b1) * q^97 + (b11 - 7*b10 - 9*b8 + 9*b1) * q^98 + (-4*b11 - 2*b10 - 3*b8 + 2*b6 - 4*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 6 q^{3} - 2 q^{4} - 6 q^{9}+O(q^{10})$$ 12 * q - 6 * q^3 - 2 * q^4 - 6 * q^9 $$12 q - 6 q^{3} - 2 q^{4} - 6 q^{9} - 2 q^{10} + 4 q^{12} + 4 q^{14} + 10 q^{16} - 14 q^{17} + 10 q^{22} + 4 q^{23} + 20 q^{25} + 12 q^{27} + 16 q^{29} - 2 q^{30} - 36 q^{35} - 2 q^{36} - 80 q^{38} + 28 q^{40} - 2 q^{42} - 6 q^{43} + 10 q^{48} + 34 q^{49} + 28 q^{51} - 52 q^{53} + 26 q^{55} + 14 q^{56} + 26 q^{61} - 32 q^{62} - 16 q^{64} - 20 q^{66} - 14 q^{68} + 4 q^{69} - 14 q^{74} - 10 q^{75} + 60 q^{77} - 36 q^{79} - 6 q^{81} + 10 q^{82} + 16 q^{87} - 14 q^{88} + 4 q^{90} - 68 q^{92} - 64 q^{94} + 6 q^{95}+O(q^{100})$$ 12 * q - 6 * q^3 - 2 * q^4 - 6 * q^9 - 2 * q^10 + 4 * q^12 + 4 * q^14 + 10 * q^16 - 14 * q^17 + 10 * q^22 + 4 * q^23 + 20 * q^25 + 12 * q^27 + 16 * q^29 - 2 * q^30 - 36 * q^35 - 2 * q^36 - 80 * q^38 + 28 * q^40 - 2 * q^42 - 6 * q^43 + 10 * q^48 + 34 * q^49 + 28 * q^51 - 52 * q^53 + 26 * q^55 + 14 * q^56 + 26 * q^61 - 32 * q^62 - 16 * q^64 - 20 * q^66 - 14 * q^68 + 4 * q^69 - 14 * q^74 - 10 * q^75 + 60 * q^77 - 36 * q^79 - 6 * q^81 + 10 * q^82 + 16 * q^87 - 14 * q^88 + 4 * q^90 - 68 * q^92 - 64 * q^94 + 6 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559$$ (-25*v^11 + 95*v^9 - 361*v^7 + 155*v^5 - 30*v^3 - 1563*v) / 559 $$\beta_{3}$$ $$=$$ $$( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559$$ (25*v^10 - 95*v^8 + 361*v^6 - 155*v^4 + 30*v^2 + 1004) / 559 $$\beta_{4}$$ $$=$$ $$( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43$$ (3*v^10 - 20*v^8 + 76*v^6 - 139*v^4 + 124*v^2 - 24) / 43 $$\beta_{5}$$ $$=$$ $$( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559$$ (45*v^10 - 171*v^8 + 538*v^6 - 279*v^4 + 54*v^2 + 242) / 559 $$\beta_{6}$$ $$=$$ $$( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559$$ (70*v^11 - 266*v^9 + 899*v^7 - 434*v^5 + 84*v^3 + 1246*v) / 559 $$\beta_{7}$$ $$=$$ $$( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559$$ (114*v^10 - 545*v^8 + 2071*v^6 - 2831*v^4 + 3379*v^2 - 95) / 559 $$\beta_{8}$$ $$=$$ $$( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559$$ (114*v^11 - 545*v^9 + 2071*v^7 - 2831*v^5 + 3379*v^3 - 95*v) / 559 $$\beta_{9}$$ $$=$$ $$( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559$$ (-128*v^10 + 710*v^8 - 2698*v^6 + 4483*v^4 - 4402*v^2 + 852) / 559 $$\beta_{10}$$ $$=$$ $$( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559$$ (-242*v^11 + 1255*v^9 - 4769*v^7 + 7314*v^5 - 7781*v^3 + 1506*v) / 559 $$\beta_{11}$$ $$=$$ $$( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559$$ (-317*v^11 + 1540*v^9 - 5852*v^7 + 8338*v^5 - 9548*v^3 + 1848*v) / 559
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1$$ b9 + b7 + b4 - b3 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{11} + 3\beta_{8} + \beta_{2}$$ b11 + 3*b8 + b2 $$\nu^{4}$$ $$=$$ $$3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2$$ 3*b9 + 2*b7 + 4*b4 - 2 $$\nu^{5}$$ $$=$$ $$4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1$$ 4*b11 - b10 + 9*b8 - 9*b1 $$\nu^{6}$$ $$=$$ $$-5\beta_{5} + 9\beta_{3} - 14$$ -5*b5 + 9*b3 - 14 $$\nu^{7}$$ $$=$$ $$-5\beta_{6} - 14\beta_{2} - 28\beta_1$$ -5*b6 - 14*b2 - 28*b1 $$\nu^{8}$$ $$=$$ $$-28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28$$ -28*b9 - 14*b7 - 19*b5 - 47*b4 + 28*b3 - 28 $$\nu^{9}$$ $$=$$ $$-47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2}$$ -47*b11 + 19*b10 - 89*b8 - 19*b6 - 47*b2 $$\nu^{10}$$ $$=$$ $$-89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42$$ -89*b9 - 42*b7 - 155*b4 + 42 $$\nu^{11}$$ $$=$$ $$-155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1$$ -155*b11 + 66*b10 - 286*b8 + 286*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$$.

 $$n$$ $$170$$ $$340$$ $$\chi(n)$$ $$1$$ $$\beta_{7}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
316.1
 −1.56052 − 0.900969i −1.07992 − 0.623490i −0.385418 − 0.222521i 0.385418 + 0.222521i 1.07992 + 0.623490i 1.56052 + 0.900969i −1.56052 + 0.900969i −1.07992 + 0.623490i −0.385418 + 0.222521i 0.385418 − 0.222521i 1.07992 − 0.623490i 1.56052 − 0.900969i
−1.56052 0.900969i −0.500000 + 0.866025i 0.623490 + 1.07992i 1.44504i 1.56052 0.900969i −2.98349 + 1.72252i 1.35690i −0.500000 0.866025i 1.30194 2.25502i
316.2 −1.07992 0.623490i −0.500000 + 0.866025i −0.222521 0.385418i 2.80194i 1.07992 0.623490i 4.15860 2.40097i 3.04892i −0.500000 0.866025i −1.74698 + 3.02586i
316.3 −0.385418 0.222521i −0.500000 + 0.866025i −0.900969 1.56052i 0.246980i 0.385418 0.222521i −1.51816 + 0.876510i 1.69202i −0.500000 0.866025i −0.0549581 + 0.0951903i
316.4 0.385418 + 0.222521i −0.500000 + 0.866025i −0.900969 1.56052i 0.246980i −0.385418 + 0.222521i 1.51816 0.876510i 1.69202i −0.500000 0.866025i −0.0549581 + 0.0951903i
316.5 1.07992 + 0.623490i −0.500000 + 0.866025i −0.222521 0.385418i 2.80194i −1.07992 + 0.623490i −4.15860 + 2.40097i 3.04892i −0.500000 0.866025i −1.74698 + 3.02586i
316.6 1.56052 + 0.900969i −0.500000 + 0.866025i 0.623490 + 1.07992i 1.44504i −1.56052 + 0.900969i 2.98349 1.72252i 1.35690i −0.500000 0.866025i 1.30194 2.25502i
361.1 −1.56052 + 0.900969i −0.500000 0.866025i 0.623490 1.07992i 1.44504i 1.56052 + 0.900969i −2.98349 1.72252i 1.35690i −0.500000 + 0.866025i 1.30194 + 2.25502i
361.2 −1.07992 + 0.623490i −0.500000 0.866025i −0.222521 + 0.385418i 2.80194i 1.07992 + 0.623490i 4.15860 + 2.40097i 3.04892i −0.500000 + 0.866025i −1.74698 3.02586i
361.3 −0.385418 + 0.222521i −0.500000 0.866025i −0.900969 + 1.56052i 0.246980i 0.385418 + 0.222521i −1.51816 0.876510i 1.69202i −0.500000 + 0.866025i −0.0549581 0.0951903i
361.4 0.385418 0.222521i −0.500000 0.866025i −0.900969 + 1.56052i 0.246980i −0.385418 0.222521i 1.51816 + 0.876510i 1.69202i −0.500000 + 0.866025i −0.0549581 0.0951903i
361.5 1.07992 0.623490i −0.500000 0.866025i −0.222521 + 0.385418i 2.80194i −1.07992 0.623490i −4.15860 2.40097i 3.04892i −0.500000 + 0.866025i −1.74698 3.02586i
361.6 1.56052 0.900969i −0.500000 0.866025i 0.623490 1.07992i 1.44504i −1.56052 0.900969i 2.98349 + 1.72252i 1.35690i −0.500000 + 0.866025i 1.30194 + 2.25502i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 316.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.h 12
13.b even 2 1 inner 507.2.j.h 12
13.c even 3 1 507.2.b.g 6
13.c even 3 1 inner 507.2.j.h 12
13.d odd 4 1 507.2.e.j 6
13.d odd 4 1 507.2.e.k 6
13.e even 6 1 507.2.b.g 6
13.e even 6 1 inner 507.2.j.h 12
13.f odd 12 1 507.2.a.j 3
13.f odd 12 1 507.2.a.k yes 3
13.f odd 12 1 507.2.e.j 6
13.f odd 12 1 507.2.e.k 6
39.h odd 6 1 1521.2.b.m 6
39.i odd 6 1 1521.2.b.m 6
39.k even 12 1 1521.2.a.p 3
39.k even 12 1 1521.2.a.q 3
52.l even 12 1 8112.2.a.by 3
52.l even 12 1 8112.2.a.cf 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.f odd 12 1
507.2.a.k yes 3 13.f odd 12 1
507.2.b.g 6 13.c even 3 1
507.2.b.g 6 13.e even 6 1
507.2.e.j 6 13.d odd 4 1
507.2.e.j 6 13.f odd 12 1
507.2.e.k 6 13.d odd 4 1
507.2.e.k 6 13.f odd 12 1
507.2.j.h 12 1.a even 1 1 trivial
507.2.j.h 12 13.b even 2 1 inner
507.2.j.h 12 13.c even 3 1 inner
507.2.j.h 12 13.e even 6 1 inner
1521.2.a.p 3 39.k even 12 1
1521.2.a.q 3 39.k even 12 1
1521.2.b.m 6 39.h odd 6 1
1521.2.b.m 6 39.i odd 6 1
8112.2.a.by 3 52.l even 12 1
8112.2.a.cf 3 52.l even 12 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(507, [\chi])$$:

 $$T_{2}^{12} - 5T_{2}^{10} + 19T_{2}^{8} - 28T_{2}^{6} + 31T_{2}^{4} - 6T_{2}^{2} + 1$$ T2^12 - 5*T2^10 + 19*T2^8 - 28*T2^6 + 31*T2^4 - 6*T2^2 + 1 $$T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 1$$ T5^6 + 10*T5^4 + 17*T5^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 5 T^{10} + 19 T^{8} - 28 T^{6} + \cdots + 1$$
$3$ $$(T^{2} + T + 1)^{6}$$
$5$ $$(T^{6} + 10 T^{4} + 17 T^{2} + 1)^{2}$$
$7$ $$T^{12} - 38 T^{10} + 1063 T^{8} + \cdots + 707281$$
$11$ $$T^{12} - 61 T^{10} + 2735 T^{8} + \cdots + 3418801$$
$13$ $$T^{12}$$
$17$ $$(T^{6} + 7 T^{5} + 35 T^{4} + 84 T^{3} + \cdots + 49)^{2}$$
$19$ $$T^{12} - 101 T^{10} + \cdots + 163047361$$
$23$ $$(T^{6} - 2 T^{5} + 47 T^{4} + 252 T^{3} + \cdots + 6889)^{2}$$
$29$ $$(T^{6} - 8 T^{5} + 59 T^{4} - 126 T^{3} + \cdots + 1849)^{2}$$
$31$ $$(T^{6} + 110 T^{4} + 3681 T^{2} + \cdots + 38809)^{2}$$
$37$ $$T^{12} - 70 T^{10} + 3479 T^{8} + \cdots + 68574961$$
$41$ $$T^{12} - 5 T^{10} + 19 T^{8} - 28 T^{6} + \cdots + 1$$
$43$ $$(T^{6} + 3 T^{5} + 34 T^{4} - 133 T^{3} + \cdots + 841)^{2}$$
$47$ $$(T^{6} + 321 T^{4} + 30798 T^{2} + \cdots + 829921)^{2}$$
$53$ $$(T^{3} + 13 T^{2} + 40 T + 29)^{4}$$
$59$ $$T^{12} - 196 T^{10} + 36848 T^{8} + \cdots + 9834496$$
$61$ $$(T^{6} - 13 T^{5} + 157 T^{4} + \cdots + 49729)^{2}$$
$67$ $$T^{12} - 69 T^{10} + 3307 T^{8} + \cdots + 88529281$$
$71$ $$T^{12} - 194 T^{10} + \cdots + 45165175441$$
$73$ $$(T^{6} + 122 T^{4} + 4189 T^{2} + \cdots + 27889)^{2}$$
$79$ $$(T^{3} + 9 T^{2} - 22 T - 169)^{4}$$
$83$ $$(T^{6} + 146 T^{4} + 4401 T^{2} + \cdots + 1849)^{2}$$
$89$ $$T^{12} - 41 T^{10} + 1627 T^{8} + \cdots + 1$$
$97$ $$T^{12} - 363 T^{10} + \cdots + 1998607065841$$
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