# Properties

 Label 74.2.c.c Level $74$ Weight $2$ Character orbit 74.c Analytic conductor $0.591$ Analytic rank $0$ Dimension $6$ 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(47,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("74.47");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$74 = 2 \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 74.c (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: $$0.590892974957$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.4406832.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ x^6 - x^5 + 6*x^4 + 7*x^3 + 24*x^2 + 5*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 6x^{4} + 7x^{3} + 24x^{2} + 5x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -5\nu^{5} + 30\nu^{4} - 31\nu^{3} + 120\nu^{2} + 25\nu + 595 ) / 149$$ (-5*v^5 + 30*v^4 - 31*v^3 + 120*v^2 + 25*v + 595) / 149 $$\beta_{2}$$ $$=$$ $$( 7\nu^{5} - 42\nu^{4} + 103\nu^{3} - 168\nu^{2} - 35\nu - 386 ) / 149$$ (7*v^5 - 42*v^4 + 103*v^3 - 168*v^2 - 35*v - 386) / 149 $$\beta_{3}$$ $$=$$ $$( 30\nu^{5} - 31\nu^{4} + 186\nu^{3} + 174\nu^{2} + 744\nu + 6 ) / 149$$ (30*v^5 - 31*v^4 + 186*v^3 + 174*v^2 + 744*v + 6) / 149 $$\beta_{4}$$ $$=$$ $$( 88\nu^{5} - 81\nu^{4} + 486\nu^{3} + 719\nu^{2} + 1944\nu + 405 ) / 149$$ (88*v^5 - 81*v^4 + 486*v^3 + 719*v^2 + 1944*v + 405) / 149 $$\beta_{5}$$ $$=$$ $$( 125\nu^{5} - 154\nu^{4} + 775\nu^{3} + 725\nu^{2} + 2951\nu + 25 ) / 149$$ (125*v^5 - 154*v^4 + 775*v^3 + 725*v^2 + 2951*v + 25) / 149
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} ) / 2$$ (b5 - b4 - b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + \beta _1 - 4$$ b5 - 4*b3 + b1 - 4 $$\nu^{3}$$ $$=$$ $$( 5\beta_{2} + 7\beta _1 - 15 ) / 2$$ (5*b2 + 7*b1 - 15) / 2 $$\nu^{4}$$ $$=$$ $$-9\beta_{5} + 3\beta_{4} + 28\beta_{3} + 3\beta_{2}$$ -9*b5 + 3*b4 + 28*b3 + 3*b2 $$\nu^{5}$$ $$=$$ $$( -55\beta_{5} + 31\beta_{4} + 139\beta_{3} - 55\beta _1 + 139 ) / 2$$ (-55*b5 + 31*b4 + 139*b3 - 55*b1 + 139) / 2

## Character values

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

 $$n$$ $$39$$ $$\chi(n)$$ $$\beta_{3}$$

## 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
 −0.105378 − 0.182520i 1.43310 + 2.48220i −0.827721 − 1.43366i −0.105378 + 0.182520i 1.43310 − 2.48220i −0.827721 + 1.43366i
−0.500000 + 0.866025i −1.26704 2.19457i −0.500000 0.866025i −1.97779 3.42563i 2.53407 1.26704 + 2.19457i 1.00000 −1.71076 + 2.96312i 3.95558
47.2 −0.500000 + 0.866025i −0.258652 0.447998i −0.500000 0.866025i 2.10755 + 3.65038i 0.517304 0.258652 + 0.447998i 1.00000 1.36620 2.36632i −4.21509
47.3 −0.500000 + 0.866025i 1.52569 + 2.64257i −0.500000 0.866025i −0.629755 1.09077i −3.05137 −1.52569 2.64257i 1.00000 −3.15544 + 5.46539i 1.25951
63.1 −0.500000 0.866025i −1.26704 + 2.19457i −0.500000 + 0.866025i −1.97779 + 3.42563i 2.53407 1.26704 2.19457i 1.00000 −1.71076 2.96312i 3.95558
63.2 −0.500000 0.866025i −0.258652 + 0.447998i −0.500000 + 0.866025i 2.10755 3.65038i 0.517304 0.258652 0.447998i 1.00000 1.36620 + 2.36632i −4.21509
63.3 −0.500000 0.866025i 1.52569 2.64257i −0.500000 + 0.866025i −0.629755 + 1.09077i −3.05137 −1.52569 + 2.64257i 1.00000 −3.15544 5.46539i 1.25951
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.3 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 74.2.c.c 6
3.b odd 2 1 666.2.f.j 6
4.b odd 2 1 592.2.i.e 6
37.c even 3 1 inner 74.2.c.c 6
37.c even 3 1 2738.2.a.o 3
37.e even 6 1 2738.2.a.n 3
111.i odd 6 1 666.2.f.j 6
148.i odd 6 1 592.2.i.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.c.c 6 1.a even 1 1 trivial
74.2.c.c 6 37.c even 3 1 inner
592.2.i.e 6 4.b odd 2 1
592.2.i.e 6 148.i odd 6 1
666.2.f.j 6 3.b odd 2 1
666.2.f.j 6 111.i odd 6 1
2738.2.a.n 3 37.e even 6 1
2738.2.a.o 3 37.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 8T_{3}^{4} + 8T_{3}^{3} + 64T_{3}^{2} + 32T_{3} + 16$$ acting on $$S_{2}^{\mathrm{new}}(74, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + T + 1)^{3}$$
$3$ $$T^{6} + 8 T^{4} + 8 T^{3} + 64 T^{2} + \cdots + 16$$
$5$ $$T^{6} + T^{5} + 18 T^{4} + 25 T^{3} + \cdots + 441$$
$7$ $$T^{6} + 8 T^{4} - 8 T^{3} + 64 T^{2} + \cdots + 16$$
$11$ $$(T^{3} - 4 T^{2} - 16 T + 48)^{2}$$
$13$ $$(T^{2} + 2 T + 4)^{3}$$
$17$ $$T^{6} - 3 T^{5} + 38 T^{4} + \cdots + 3969$$
$19$ $$T^{6} + 8 T^{5} + 72 T^{4} + \cdots + 12544$$
$23$ $$(T^{3} - 8 T^{2} + 4 T + 12)^{2}$$
$29$ $$(T^{3} + 3 T^{2} - 5 T - 3)^{2}$$
$31$ $$(T^{3} - 4 T^{2} - 24 T - 16)^{2}$$
$37$ $$T^{6} + 11 T^{5} + 134 T^{4} + \cdots + 50653$$
$41$ $$T^{6} - 7 T^{5} + 62 T^{4} + 85 T^{3} + \cdots + 9$$
$43$ $$(T^{3} - 8 T^{2} - 20 T + 148)^{2}$$
$47$ $$(T^{3} + 16 T^{2} + 56 T + 48)^{2}$$
$53$ $$T^{6} + 22 T^{5} + 344 T^{4} + \cdots + 46656$$
$59$ $$T^{6} + 4 T^{5} + 176 T^{4} + \cdots + 451584$$
$61$ $$T^{6} + 9 T^{5} + 62 T^{4} + 157 T^{3} + \cdots + 49$$
$67$ $$T^{6} - 16 T^{5} + 192 T^{4} + \cdots + 256$$
$71$ $$T^{6} - 12 T^{5} + 104 T^{4} + \cdots + 1296$$
$73$ $$(T^{3} - 2 T^{2} - 20 T - 8)^{2}$$
$79$ $$T^{6} - 4 T^{5} + 88 T^{4} + \cdots + 20736$$
$83$ $$T^{6} + 4 T^{5} + 80 T^{4} + \cdots + 17424$$
$89$ $$T^{6} + 9 T^{5} + 230 T^{4} + \cdots + 301401$$
$97$ $$(T^{3} - 13 T^{2} + 27 T - 7)^{2}$$