# Properties

 Label 74.2.f.b Level $74$ Weight $2$ Character orbit 74.f Analytic conductor $0.591$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,2,Mod(7,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([16]))

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

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

chi := DirichletCharacter("74.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.f (of order $$9$$, degree $$6$$, 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: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{9})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625$$ x^12 - 24*x^10 + 264*x^8 - 1687*x^6 + 6600*x^4 - 15000*x^2 + 15625 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{7} + \beta_{4}) q^{2} + ( - \beta_{8} + \beta_1 - 1) q^{3} - \beta_{6} q^{4} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{6} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + (\beta_1 - 1) q^{8} + ( - \beta_{10} + 2 \beta_{7} - \beta_{2}) q^{9}+O(q^{10})$$ q + (-b7 + b4) * q^2 + (-b8 + b1 - 1) * q^3 - b6 * q^4 + (-b4 - b1) * q^5 + (-b4 - b3 + b2) * q^6 + (b11 - b9 - b7 + b6 - b5 + b4 + b3 - b1 + 1) * q^7 + (b1 - 1) * q^8 + (-b10 + 2*b7 - b2) * q^9 $$q + ( - \beta_{7} + \beta_{4}) q^{2} + ( - \beta_{8} + \beta_1 - 1) q^{3} - \beta_{6} q^{4} + ( - \beta_{4} - \beta_1) q^{5} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{6} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{7} + (\beta_1 - 1) q^{8} + ( - \beta_{10} + 2 \beta_{7} - \beta_{2}) q^{9} + (\beta_{7} + \beta_{5}) q^{10} + (\beta_{10} + \beta_{9} + \beta_{8} + \beta_{5} - \beta_1 + 1) q^{11} + (\beta_{9} + \beta_{5}) q^{12} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{11} + \beta_{10} + \beta_{8} - \beta_{6}) q^{14} + (\beta_{10} + \beta_{8} + \beta_{7} - \beta_{2} - \beta_1 + 1) q^{15} - \beta_{4} q^{16} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3}) q^{17} + (\beta_{11} - \beta_{9} + \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{2} - \beta_1 + 1) q^{18} + ( - \beta_{11} + 3 \beta_{6} - 4 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{19} + (\beta_{6} - \beta_{5} - \beta_1) q^{20} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{6} + 5 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 6 \beta_1 - 6) q^{21} + ( - \beta_{10} - \beta_{7} + \beta_{4} + \beta_{3}) q^{22} + (\beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 4 \beta_1) q^{23} - \beta_{10} q^{24} + (3 \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} + \beta_1 - 1) q^{25} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{26} + ( - \beta_{11} + \beta_{10} + \beta_{8} - 4 \beta_{7} - 5 \beta_{6} + 4 \beta_{4}) q^{27} + ( - \beta_{8} - \beta_{7} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{28} + (\beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - 2 \beta_1 + 2) q^{29} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 + 1) q^{30} + (\beta_{9} - \beta_{8} + 4 \beta_{7} - 4 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \cdots + 3) q^{31}+ \cdots + ( - \beta_{10} - \beta_{9} - 3 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} - 5 \beta_{4} + \cdots - 7) q^{99}+O(q^{100})$$ q + (-b7 + b4) * q^2 + (-b8 + b1 - 1) * q^3 - b6 * q^4 + (-b4 - b1) * q^5 + (-b4 - b3 + b2) * q^6 + (b11 - b9 - b7 + b6 - b5 + b4 + b3 - b1 + 1) * q^7 + (b1 - 1) * q^8 + (-b10 + 2*b7 - b2) * q^9 + (b7 + b5) * q^10 + (b10 + b9 + b8 + b5 - b1 + 1) * q^11 + (b9 + b5) * q^12 + (-b11 - b10 - b7 + b4 + b3 + b1 - 1) * q^13 + (-b11 + b10 + b8 - b6) * q^14 + (b10 + b8 + b7 - b2 - b1 + 1) * q^15 - b4 * q^16 + (b8 + b7 - b6 - b4 - b3) * q^17 + (b11 - b9 + b7 + 2*b6 - 2*b5 - b2 - b1 + 1) * q^18 + (-b11 + 3*b6 - 4*b5 - b4 - b3 + b2 + b1 - 1) * q^19 + (b6 - b5 - b1) * q^20 + (-b11 + b9 - b8 - b6 + 5*b5 - b4 - b3 + b2 + 6*b1 - 6) * q^21 + (-b10 - b7 + b4 + b3) * q^22 + (b11 - b9 - b8 - b7 + b6 - b5 + b4 + b3 - 4*b1) * q^23 - b10 * q^24 + (3*b7 + b6 - b5 - 3*b4 + b1 - 1) * q^25 + (-b11 - b8 + b7 - b6 - b2 + b1 - 1) * q^26 + (-b11 + b10 + b8 - 4*b7 - 5*b6 + 4*b4) * q^27 + (-b8 - b7 + b4 + b3 + b1 - 1) * q^28 + (b11 + b8 - b7 + b6 + b5 + b4 - 2*b1 + 2) * q^29 + (b11 - b9 - b7 + b6 - b5 + b4 + b3 - b1 + 1) * q^30 + (b9 - b8 + 4*b7 - 4*b6 + b5 - 2*b4 - 2*b3 + 2*b2 + b1 + 3) * q^31 + b5 * q^32 + (b11 + b10 - b9 + b6 - b5 - 5*b4 - b3 + b2 - b1 + 5) * q^33 + (b11 + b6 + b3 - b2) * q^34 + (2*b9 - b8 + b7 + 2*b5 - b2 + b1 - 1) * q^35 + (b11 + b10 - b9 + b8 + b6 - b5 - 2*b1 + 3) * q^36 + (b11 - 2*b10 - b9 - b8 - 4*b7 - b5 + 5*b4 + b3 - b2) * q^37 + (b9 - b8 + b5 + b1 + 3) * q^38 + (b9 + b8 - 5*b7 + 5*b5 + 4*b4 + b2 - 5*b1 + 1) * q^39 + (b7 + 1) * q^40 + (2*b11 - b10 - 2*b9 - 3*b7 - 2*b6 - 2*b5 + 2*b4 - 2*b3 + 2*b2 + b1 - 2) * q^41 + (-b10 + b9 - b8 + b5 - 5*b4 - b3 + b2 - 3*b1 - 1) * q^42 + (-b11 - b10 + b9 - b8 - b6 + b5 + 2*b4 + 2*b3 - 2*b2 + 2*b1 + 2) * q^43 + (-b11 + b7 - b6 - b2 + b1 - 1) * q^44 + (-b11 - b8 - b7 - 2*b6 - b4 + b2 + 2*b1 - 2) * q^45 + (-b11 + b10 + b8 + 4*b7 - b6 - b4 - b3 + b2) * q^46 + (b11 + b10 - 2*b9 - b8 + 5*b7 + b6 + 2*b5 - b4 - b3) * q^47 + (b7 - b2) * q^48 + (2*b10 - b9 + b8 - 2*b7 + 4*b6 - 5*b5 + 3*b4 + 2*b3 - b2 + 3*b1 - 3) * q^49 + (3*b6 - b4 + 1) * q^50 + (-5*b7 - 4*b5 + b4 + b3) * q^51 + (b11 - b9 - b8 + b6 - b5 - b3 + b2) * q^52 + (-b11 + b9 - b6 + b5 - 3*b4 + b1 - 1) * q^53 + (-b8 - b7 - 4*b6 + b4 + b3 + 5*b1 - 5) * q^54 + (-b11 + b8 - b6 - b4 - b3 + b2) * q^55 + (-b11 - b6 - b4 - b3 + b2 + b1 - 1) * q^56 + (-3*b11 - b10 + b7 + b6 - 4*b5 - b2 - b1 - 3) * q^57 + (b8 - b6 + b4 + b3 - b2 - 2*b1 + 1) * q^58 + (-b10 - b8 - b7 - 4*b4 + b2 + b1 - 1) * q^59 + (-b11 + b10 + b8 - b6) * q^60 + (-b11 + b10 + 2*b9 + 5*b7 + 3*b6 + 2*b5 + b3 - 2*b2 - 3*b1) * q^61 + (-b10 + 2*b9 - 4*b7 + 4*b6 - 2*b5 + 3*b4 - b3 + b2 + 4*b1 - 4) * q^62 + (3*b11 - 2*b10 - 2*b9 + b8 + 6*b7 + 7*b6 - 6*b5 - 2*b2 - 8*b1 + 8) * q^63 - b1 * q^64 + (b11 - b10 - b8 + b5 + b4 + b3 - b2 + b1) * q^65 + (b10 + b9 + b8 - 5*b7 + 5*b5 + 4*b4 + b2 - b1 + 1) * q^66 + (-b11 + b10 + 2*b9 + b8 + 5*b7 - b6 + 2*b5 - 2*b4 - 2*b3 + b2 + 4*b1 - 4) * q^67 + (-b9 + b8 - b4 - b1 + 1) * q^68 + (-b11 + 3*b10 + b9 + 3*b8 + 5*b7 - b6 + 5*b5 - b4 - b3 + 2*b1 - 2) * q^69 + (b11 - 2*b10 - b9 + b6 - b5 - b4 - b3 + b2 - b1 + 1) * q^70 + (2*b8 - 4*b4 + 2*b1 - 2) * q^71 + (b10 + b8 - 2*b7 + 2*b4 + b3 - b1 + 1) * q^72 + (2*b11 + 2*b10 - b9 + b8 + 2*b6 + 3*b5 - 2*b4 + 2*b3 - 2*b2 - 3*b1 + 6) * q^73 + (b10 - b9 + b8 + 2*b7 - 4*b6 - b5 - b4 - b3 - b2) * q^74 + (-b11 - b10 - b6 + 3*b4 + 3*b3 - 3*b2 + b1 - 1) * q^75 + (-b10 - 4*b7 + 3*b4 - b3 + b2) * q^76 + (b8 + 4*b7 - 4*b6 + 4*b5 - b1 + 5) * q^77 + (-b11 - b10 + b9 + 4*b7 - 5*b6 + b5 + b4 + b3 - b2 - 3*b1 - 1) * q^78 + (b11 + b10 - b9 + b8 - b7 + b6 + 3*b5 - 3*b4 + b3 - 6*b1 + 2) * q^79 + (-b7 + b6 - b5 + b4) * q^80 + (-b10 + 2*b9 - b8 - b5 - 6*b4 - 2*b3 + 2*b2 - 3*b1 - 1) * q^81 + (2*b10 + 2*b9 + 2*b8 + 2*b7 - 3*b6 + b5 - 4*b4 - b2 + 2*b1 - 2) * q^82 + (-b10 - b8 + 4*b7 + b4 + b3 - b2 + b1 - 1) * q^83 + (-b10 + b9 + 5*b7 + 5*b5 - b4 - b3) * q^84 + (-b10 - b9 - b8 - b7 + b6 - 2*b5 + 2*b2) * q^85 + (-b10 - 2*b9 - b8 - 3*b7 - 2*b5 + 3*b4 - b3 + b1 - 1) * q^86 + (b11 + 2*b10 - 2*b9 - 5*b7 + b6 - 2*b5 - b4 - b3 + 2*b2 + 3*b1 + 1) * q^87 + (b11 - b9 - b8 + b6 - b5) * q^88 + (-2*b11 - 3*b10 + 2*b9 - 2*b8 - 3*b7 - b6 + b5 + 5*b4 + b3 + 2*b2 + 4*b1 - 3) * q^89 + (-b11 + b9 - b8 - b6 + 2*b5 - b4 - b3 + b2 + 3*b1 - 3) * q^90 + (-b11 - 4*b7 - b6 - 4*b5 + b1 - 5) * q^91 + (b9 - b8 - b7 + 4*b6 - 3*b5 + b4 + b3 + b1 - 1) * q^92 + (3*b11 + b9 - 3*b8 + 3*b7 + 11*b6 - 7*b5 + 4*b4 + 4*b3 - 3*b2 + 4) * q^93 + (b11 + 2*b10 + b8 - b7 + 5*b6 - 4*b5 - b4 - b3 + 2*b2 - 6*b1 + 2) * q^94 + (2*b11 - b10 - 2*b9 - b7 + 2*b6 + 2*b5 + b4 + b3 + 2*b1 - 2) * q^95 + (b11 - b9 + b6 - b5 - b1 + 1) * q^96 + (b10 - b9 - b7 - 3*b6 - 2*b5 - 3*b3 - 8*b1) * q^97 + (-b11 + b10 - b9 - 2*b7 - 2*b6 - b5 - 3*b4 + b3 + b2 + b1 + 3) * q^98 + (-b10 - b9 - 3*b8 + 7*b7 + 4*b6 - 5*b5 - 5*b4 - b3 - 2*b2 + 7*b1 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9}+O(q^{10})$$ 12 * q - 3 * q^3 - 6 * q^5 + 6 * q^7 - 6 * q^8 - 3 * q^9 $$12 q - 3 q^{3} - 6 q^{5} + 6 q^{7} - 6 q^{8} - 3 q^{9} + 3 q^{11} - 3 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} - 3 q^{17} + 6 q^{18} - 3 q^{19} - 6 q^{20} - 33 q^{21} - 3 q^{22} - 21 q^{23} - 3 q^{24} - 6 q^{25} + 3 q^{27} - 3 q^{28} + 6 q^{29} + 6 q^{30} + 42 q^{31} + 57 q^{33} - 3 q^{34} - 9 q^{35} + 24 q^{36} - 3 q^{37} + 42 q^{38} - 24 q^{39} + 12 q^{40} - 21 q^{41} - 33 q^{42} + 36 q^{43} - 3 q^{44} - 6 q^{45} + 3 q^{46} + 9 q^{47} - 12 q^{49} + 12 q^{50} + 3 q^{52} - 6 q^{53} - 27 q^{54} - 3 q^{56} - 36 q^{57} - 3 q^{58} - 6 q^{59} + 3 q^{60} - 18 q^{61} - 33 q^{62} + 36 q^{63} - 6 q^{64} + 3 q^{65} + 3 q^{66} - 27 q^{67} + 6 q^{68} - 12 q^{69} - 18 q^{71} + 6 q^{72} + 54 q^{73} + 3 q^{74} - 6 q^{75} - 3 q^{76} + 51 q^{77} - 33 q^{78} - 12 q^{79} - 36 q^{81} - 18 q^{82} - 6 q^{83} - 6 q^{84} + 3 q^{85} + 39 q^{87} + 3 q^{88} - 15 q^{89} - 15 q^{90} - 51 q^{91} - 6 q^{92} + 45 q^{93} - 12 q^{94} - 15 q^{95} + 6 q^{96} - 42 q^{97} + 51 q^{98} - 33 q^{99}+O(q^{100})$$ 12 * q - 3 * q^3 - 6 * q^5 + 6 * q^7 - 6 * q^8 - 3 * q^9 + 3 * q^11 - 3 * q^12 - 6 * q^13 + 3 * q^14 + 6 * q^15 - 3 * q^17 + 6 * q^18 - 3 * q^19 - 6 * q^20 - 33 * q^21 - 3 * q^22 - 21 * q^23 - 3 * q^24 - 6 * q^25 + 3 * q^27 - 3 * q^28 + 6 * q^29 + 6 * q^30 + 42 * q^31 + 57 * q^33 - 3 * q^34 - 9 * q^35 + 24 * q^36 - 3 * q^37 + 42 * q^38 - 24 * q^39 + 12 * q^40 - 21 * q^41 - 33 * q^42 + 36 * q^43 - 3 * q^44 - 6 * q^45 + 3 * q^46 + 9 * q^47 - 12 * q^49 + 12 * q^50 + 3 * q^52 - 6 * q^53 - 27 * q^54 - 3 * q^56 - 36 * q^57 - 3 * q^58 - 6 * q^59 + 3 * q^60 - 18 * q^61 - 33 * q^62 + 36 * q^63 - 6 * q^64 + 3 * q^65 + 3 * q^66 - 27 * q^67 + 6 * q^68 - 12 * q^69 - 18 * q^71 + 6 * q^72 + 54 * q^73 + 3 * q^74 - 6 * q^75 - 3 * q^76 + 51 * q^77 - 33 * q^78 - 12 * q^79 - 36 * q^81 - 18 * q^82 - 6 * q^83 - 6 * q^84 + 3 * q^85 + 39 * q^87 + 3 * q^88 - 15 * q^89 - 15 * q^90 - 51 * q^91 - 6 * q^92 + 45 * q^93 - 12 * q^94 - 15 * q^95 + 6 * q^96 - 42 * q^97 + 51 * q^98 - 33 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625$$ :

 $$\beta_{1}$$ $$=$$ $$( -12\nu^{11} + 263\nu^{9} - 2568\nu^{7} + 13644\nu^{5} - 40150\nu^{3} + 52500\nu + 3125 ) / 6250$$ (-12*v^11 + 263*v^9 - 2568*v^7 + 13644*v^5 - 40150*v^3 + 52500*v + 3125) / 6250 $$\beta_{2}$$ $$=$$ $$( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} + 125\nu - 7500 ) / 250$$ (v^10 - 24*v^8 + 264*v^6 - 1562*v^4 + 5100*v^2 + 125*v - 7500) / 250 $$\beta_{3}$$ $$=$$ $$( \nu^{10} - 24\nu^{8} + 264\nu^{6} - 1562\nu^{4} + 5100\nu^{2} - 125\nu - 7500 ) / 250$$ (v^10 - 24*v^8 + 264*v^6 - 1562*v^4 + 5100*v^2 - 125*v - 7500) / 250 $$\beta_{4}$$ $$=$$ $$( \nu^{11} - 45 \nu^{10} - 24 \nu^{9} + 955 \nu^{8} + 264 \nu^{7} - 8880 \nu^{6} - 1687 \nu^{5} + 46040 \nu^{4} + 6600 \nu^{3} - 133000 \nu^{2} - 11875 \nu + 175000 ) / 6250$$ (v^11 - 45*v^10 - 24*v^9 + 955*v^8 + 264*v^7 - 8880*v^6 - 1687*v^5 + 46040*v^4 + 6600*v^3 - 133000*v^2 - 11875*v + 175000) / 6250 $$\beta_{5}$$ $$=$$ $$( \nu^{11} + 45 \nu^{10} - 24 \nu^{9} - 955 \nu^{8} + 264 \nu^{7} + 8880 \nu^{6} - 1687 \nu^{5} - 46040 \nu^{4} + 6600 \nu^{3} + 133000 \nu^{2} - 11875 \nu - 175000 ) / 6250$$ (v^11 + 45*v^10 - 24*v^9 - 955*v^8 + 264*v^7 + 8880*v^6 - 1687*v^5 - 46040*v^4 + 6600*v^3 + 133000*v^2 - 11875*v - 175000) / 6250 $$\beta_{6}$$ $$=$$ $$( 12 \nu^{11} + 40 \nu^{10} - 188 \nu^{9} - 835 \nu^{8} + 1393 \nu^{7} + 7560 \nu^{6} - 5719 \nu^{5} - 37605 \nu^{4} + 13000 \nu^{3} + 103125 \nu^{2} - 11875 \nu - 125000 ) / 6250$$ (12*v^11 + 40*v^10 - 188*v^9 - 835*v^8 + 1393*v^7 + 7560*v^6 - 5719*v^5 - 37605*v^4 + 13000*v^3 + 103125*v^2 - 11875*v - 125000) / 6250 $$\beta_{7}$$ $$=$$ $$( 12 \nu^{11} - 40 \nu^{10} - 188 \nu^{9} + 835 \nu^{8} + 1393 \nu^{7} - 7560 \nu^{6} - 5719 \nu^{5} + 37605 \nu^{4} + 13000 \nu^{3} - 103125 \nu^{2} - 11875 \nu + 125000 ) / 6250$$ (12*v^11 - 40*v^10 - 188*v^9 + 835*v^8 + 1393*v^7 - 7560*v^6 - 5719*v^5 + 37605*v^4 + 13000*v^3 - 103125*v^2 - 11875*v + 125000) / 6250 $$\beta_{8}$$ $$=$$ $$( \nu^{11} - 20 \nu^{10} - 24 \nu^{9} + 355 \nu^{8} + 264 \nu^{7} - 2905 \nu^{6} - 1687 \nu^{5} + 13240 \nu^{4} + 5975 \nu^{3} - 33000 \nu^{2} - 10000 \nu + 34375 ) / 1250$$ (v^11 - 20*v^10 - 24*v^9 + 355*v^8 + 264*v^7 - 2905*v^6 - 1687*v^5 + 13240*v^4 + 5975*v^3 - 33000*v^2 - 10000*v + 34375) / 1250 $$\beta_{9}$$ $$=$$ $$( - \nu^{11} - 20 \nu^{10} + 24 \nu^{9} + 355 \nu^{8} - 264 \nu^{7} - 2905 \nu^{6} + 1687 \nu^{5} + 13240 \nu^{4} - 5975 \nu^{3} - 33000 \nu^{2} + 10000 \nu + 34375 ) / 1250$$ (-v^11 - 20*v^10 + 24*v^9 + 355*v^8 - 264*v^7 - 2905*v^6 + 1687*v^5 + 13240*v^4 - 5975*v^3 - 33000*v^2 + 10000*v + 34375) / 1250 $$\beta_{10}$$ $$=$$ $$( 28 \nu^{11} + 100 \nu^{10} - 572 \nu^{9} - 1775 \nu^{8} + 4992 \nu^{7} + 14525 \nu^{6} - 23961 \nu^{5} - 66200 \nu^{4} + 62975 \nu^{3} + 168125 \nu^{2} - 72500 \nu - 184375 ) / 6250$$ (28*v^11 + 100*v^10 - 572*v^9 - 1775*v^8 + 4992*v^7 + 14525*v^6 - 23961*v^5 - 66200*v^4 + 62975*v^3 + 168125*v^2 - 72500*v - 184375) / 6250 $$\beta_{11}$$ $$=$$ $$( 28 \nu^{11} - 100 \nu^{10} - 572 \nu^{9} + 1775 \nu^{8} + 4992 \nu^{7} - 14525 \nu^{6} - 23961 \nu^{5} + 66200 \nu^{4} + 62975 \nu^{3} - 168125 \nu^{2} - 72500 \nu + 184375 ) / 6250$$ (28*v^11 - 100*v^10 - 572*v^9 + 1775*v^8 + 4992*v^7 - 14525*v^6 - 23961*v^5 + 66200*v^4 + 62975*v^3 - 168125*v^2 - 72500*v + 184375) / 6250
 $$\nu$$ $$=$$ $$-\beta_{3} + \beta_{2}$$ -b3 + b2 $$\nu^{2}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 4$$ -b11 + b10 + b9 + b8 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} + 5\beta_{5} + 5\beta_{4} - 3\beta_{3} + 3\beta_{2}$$ b9 - b8 + 5*b5 + 5*b4 - 3*b3 + 3*b2 $$\nu^{4}$$ $$=$$ $$- 7 \beta_{11} + 7 \beta_{10} + 7 \beta_{9} + 7 \beta_{8} - 5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5} + 5 \beta_{4} + \beta_{3} + \beta_{2} + 8$$ -7*b11 + 7*b10 + 7*b9 + 7*b8 - 5*b7 + 5*b6 - 5*b5 + 5*b4 + b3 + b2 + 8 $$\nu^{5}$$ $$=$$ $$-\beta_{11} - \beta_{10} + 11\beta_{9} - 11\beta_{8} + 35\beta_{5} + 35\beta_{4} - \beta_{3} + \beta_{2} - 8\beta _1 + 4$$ -b11 - b10 + 11*b9 - 11*b8 + 35*b5 + 35*b4 - b3 + b2 - 8*b1 + 4 $$\nu^{6}$$ $$=$$ $$- 26 \beta_{11} + 26 \beta_{10} + 25 \beta_{9} + 25 \beta_{8} - 50 \beta_{7} + 50 \beta_{6} - 55 \beta_{5} + 55 \beta_{4} + 15 \beta_{3} + 15 \beta_{2} - 21$$ -26*b11 + 26*b10 + 25*b9 + 25*b8 - 50*b7 + 50*b6 - 55*b5 + 55*b4 + 15*b3 + 15*b2 - 21 $$\nu^{7}$$ $$=$$ $$- 19 \beta_{11} - 19 \beta_{10} + 65 \beta_{9} - 65 \beta_{8} + 5 \beta_{7} + 5 \beta_{6} + 125 \beta_{5} + 125 \beta_{4} + 46 \beta_{3} - 46 \beta_{2} - 112 \beta _1 + 56$$ -19*b11 - 19*b10 + 65*b9 - 65*b8 + 5*b7 + 5*b6 + 125*b5 + 125*b4 + 46*b3 - 46*b2 - 112*b1 + 56 $$\nu^{8}$$ $$=$$ $$- 38 \beta_{11} + 38 \beta_{10} + 14 \beta_{9} + 14 \beta_{8} - 230 \beta_{7} + 230 \beta_{6} - 325 \beta_{5} + 325 \beta_{4} + 121 \beta_{3} + 121 \beta_{2} - 268$$ -38*b11 + 38*b10 + 14*b9 + 14*b8 - 230*b7 + 230*b6 - 325*b5 + 325*b4 + 121*b3 + 121*b2 - 268 $$\nu^{9}$$ $$=$$ $$- 192 \beta_{11} - 192 \beta_{10} + 218 \beta_{9} - 218 \beta_{8} + 120 \beta_{7} + 120 \beta_{6} + 70 \beta_{5} + 70 \beta_{4} + 282 \beta_{3} - 282 \beta_{2} - 826 \beta _1 + 413$$ -192*b11 - 192*b10 + 218*b9 - 218*b8 + 120*b7 + 120*b6 + 70*b5 + 70*b4 + 282*b3 - 282*b2 - 826*b1 + 413 $$\nu^{10}$$ $$=$$ $$118 \beta_{11} - 118 \beta_{10} - 430 \beta_{9} - 430 \beta_{8} - 130 \beta_{7} + 130 \beta_{6} - 1090 \beta_{5} + 1090 \beta_{4} + 631 \beta_{3} + 631 \beta_{2} - 1292$$ 118*b11 - 118*b10 - 430*b9 - 430*b8 - 130*b7 + 130*b6 - 1090*b5 + 1090*b4 + 631*b3 + 631*b2 - 1292 $$\nu^{11}$$ $$=$$ $$- 1279 \beta_{11} - 1279 \beta_{10} + 29 \beta_{9} - 29 \beta_{8} + 1560 \beta_{7} + 1560 \beta_{6} - 2150 \beta_{5} - 2150 \beta_{4} + 862 \beta_{3} - 862 \beta_{2} - 3752 \beta _1 + 1876$$ -1279*b11 - 1279*b10 + 29*b9 - 29*b8 + 1560*b7 + 1560*b6 - 2150*b5 - 2150*b4 + 862*b3 - 862*b2 - 3752*b1 + 1876

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/74\mathbb{Z}\right)^\times$$.

 $$n$$ $$39$$ $$\chi(n)$$ $$-\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 2.00752 + 0.984808i −2.00752 + 0.984808i −2.14169 + 0.642788i 2.14169 + 0.642788i −2.14169 − 0.642788i 2.14169 − 0.642788i −2.20976 + 0.342020i 2.20976 + 0.342020i 2.00752 − 0.984808i −2.00752 − 0.984808i −2.20976 − 0.342020i 2.20976 − 0.342020i
−0.939693 + 0.342020i −2.04963 0.746005i 0.766044 0.642788i −0.326352 1.85083i 2.18117 −0.711830 4.03699i −0.500000 + 0.866025i 1.34633 + 1.12971i 0.939693 + 1.62760i
7.2 −0.939693 + 0.342020i 1.72328 + 0.627223i 0.766044 0.642788i −0.326352 1.85083i −1.83388 0.598489 + 3.39420i −0.500000 + 0.866025i 0.278152 + 0.233397i 0.939693 + 1.62760i
9.1 0.173648 + 0.984808i −0.238878 + 1.35474i −0.939693 + 0.342020i 0.266044 + 0.223238i −1.37564 −0.365982 0.307095i −0.500000 0.866025i 1.04081 + 0.378824i −0.173648 + 0.300767i
9.2 0.173648 + 0.984808i 0.504922 2.86356i −0.939693 + 0.342020i 0.266044 + 0.223238i 2.90773 0.773586 + 0.649116i −0.500000 0.866025i −5.12593 1.86569i −0.173648 + 0.300767i
33.1 0.173648 0.984808i −0.238878 1.35474i −0.939693 0.342020i 0.266044 0.223238i −1.37564 −0.365982 + 0.307095i −0.500000 + 0.866025i 1.04081 0.378824i −0.173648 0.300767i
33.2 0.173648 0.984808i 0.504922 + 2.86356i −0.939693 0.342020i 0.266044 0.223238i 2.90773 0.773586 0.649116i −0.500000 + 0.866025i −5.12593 + 1.86569i −0.173648 0.300767i
49.1 0.766044 0.642788i −2.41262 2.02443i 0.173648 0.984808i −1.43969 + 0.524005i −3.14945 4.53424 1.65033i −0.500000 0.866025i 1.20148 + 6.81391i −0.766044 + 1.32683i
49.2 0.766044 0.642788i 0.972925 + 0.816381i 0.173648 0.984808i −1.43969 + 0.524005i 1.27006 −1.82850 + 0.665520i −0.500000 0.866025i −0.240839 1.36587i −0.766044 + 1.32683i
53.1 −0.939693 0.342020i −2.04963 + 0.746005i 0.766044 + 0.642788i −0.326352 + 1.85083i 2.18117 −0.711830 + 4.03699i −0.500000 0.866025i 1.34633 1.12971i 0.939693 1.62760i
53.2 −0.939693 0.342020i 1.72328 0.627223i 0.766044 + 0.642788i −0.326352 + 1.85083i −1.83388 0.598489 3.39420i −0.500000 0.866025i 0.278152 0.233397i 0.939693 1.62760i
71.1 0.766044 + 0.642788i −2.41262 + 2.02443i 0.173648 + 0.984808i −1.43969 0.524005i −3.14945 4.53424 + 1.65033i −0.500000 + 0.866025i 1.20148 6.81391i −0.766044 1.32683i
71.2 0.766044 + 0.642788i 0.972925 0.816381i 0.173648 + 0.984808i −1.43969 0.524005i 1.27006 −1.82850 0.665520i −0.500000 + 0.866025i −0.240839 + 1.36587i −0.766044 1.32683i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.f.b 12
3.b odd 2 1 666.2.x.g 12
4.b odd 2 1 592.2.bc.d 12
37.f even 9 1 inner 74.2.f.b 12
37.f even 9 1 2738.2.a.t 6
37.h even 18 1 2738.2.a.q 6
111.p odd 18 1 666.2.x.g 12
148.p odd 18 1 592.2.bc.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.f.b 12 1.a even 1 1 trivial
74.2.f.b 12 37.f even 9 1 inner
592.2.bc.d 12 4.b odd 2 1
592.2.bc.d 12 148.p odd 18 1
666.2.x.g 12 3.b odd 2 1
666.2.x.g 12 111.p odd 18 1
2738.2.a.q 6 37.h even 18 1
2738.2.a.t 6 37.f even 9 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 3 T_{3}^{11} + 6 T_{3}^{10} + 8 T_{3}^{9} + 24 T_{3}^{8} - 126 T_{3}^{7} - 151 T_{3}^{6} + 504 T_{3}^{5} + 384 T_{3}^{4} - 512 T_{3}^{3} + 1536 T_{3}^{2} - 3072 T_{3} + 4096$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} + T^{3} + 1)^{2}$$
$3$ $$T^{12} + 3 T^{11} + 6 T^{10} + 8 T^{9} + \cdots + 4096$$
$5$ $$(T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + 3 T^{2} + \cdots + 1)^{2}$$
$7$ $$T^{12} - 6 T^{11} + 24 T^{10} + \cdots + 4096$$
$11$ $$T^{12} - 3 T^{11} + 33 T^{10} + \cdots + 4096$$
$13$ $$T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 516961$$
$17$ $$T^{12} + 3 T^{11} + 18 T^{10} + \cdots + 26569$$
$19$ $$T^{12} + 3 T^{11} - 9 T^{10} + 228 T^{9} + \cdots + 4096$$
$23$ $$T^{12} + 21 T^{11} + 297 T^{10} + \cdots + 4096$$
$29$ $$T^{12} - 6 T^{11} + 60 T^{10} + \cdots + 1369$$
$31$ $$(T^{6} - 21 T^{5} + 24 T^{4} + \cdots + 130112)^{2}$$
$37$ $$T^{12} + 3 T^{11} + \cdots + 2565726409$$
$41$ $$T^{12} + 21 T^{11} + \cdots + 466905664$$
$43$ $$(T^{6} - 18 T^{5} + 51 T^{4} + 639 T^{3} + \cdots + 8704)^{2}$$
$47$ $$T^{12} - 9 T^{11} + \cdots + 6890328064$$
$53$ $$T^{12} + 6 T^{11} + 6 T^{10} + 109 T^{9} + \cdots + 289$$
$59$ $$T^{12} + 6 T^{11} + 12 T^{10} + \cdots + 1183744$$
$61$ $$T^{12} + 18 T^{11} + \cdots + 2801373184$$
$67$ $$T^{12} + 27 T^{11} + 444 T^{10} + \cdots + 262144$$
$71$ $$T^{12} + 18 T^{11} + 192 T^{10} + \cdots + 262144$$
$73$ $$(T^{6} - 27 T^{5} + 108 T^{4} + \cdots + 216289)^{2}$$
$79$ $$T^{12} + 12 T^{11} - 54 T^{10} + \cdots + 95883264$$
$83$ $$T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 1183744$$
$89$ $$T^{12} + 15 T^{11} + \cdots + 48144697561$$
$97$ $$T^{12} + 42 T^{11} + 1215 T^{10} + \cdots + 7529536$$