# Properties

 Label 65.2.n.a Level $65$ Weight $2$ Character orbit 65.n Analytic conductor $0.519$ 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] = [65,2,Mod(9,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

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

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

chi := DirichletCharacter("65.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.n (of order $$6$$, 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: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1$$ x^12 - 8*x^10 + 54*x^8 - 78*x^6 + 92*x^4 - 10*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + ( - \beta_{11} + \beta_{4}) q^{3} + ( - \beta_{10} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{7} - 2 \beta_{6} - \beta_{5}) q^{9}+O(q^{10})$$ q - b4 * q^2 + (-b11 + b4) * q^3 + (-b10 - b6 - b2) * q^4 + (b8 - b7 + b6) * q^5 + (b10 + 2*b6 + b2) * q^6 + (-b3 - b1) * q^7 + (b11 - b9 - b8 + b7 - b6 - b4 + b3 - b1 - 1) * q^8 + (b7 - 2*b6 - b5) * q^9 $$q - \beta_{4} q^{2} + ( - \beta_{11} + \beta_{4}) q^{3} + ( - \beta_{10} - \beta_{6} - \beta_{2}) q^{4} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + (\beta_{10} + 2 \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{7} - 2 \beta_{6} - \beta_{5}) q^{9} + (\beta_{10} + \beta_{9} + 2 \beta_{6} + \beta_{5} + 2) q^{10} + (\beta_{9} - \beta_{8} + \beta_{5}) q^{11} + (\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{12} + ( - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{13} + (\beta_{2} - 4) q^{14} + (2 \beta_{11} - \beta_{10} - \beta_{6} - \beta_{4} - 1) q^{15} + ( - \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{6} - \beta_{5} - 3) q^{16} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{17} + ( - \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - 1) q^{18} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{19} + (\beta_{10} - \beta_{5} - \beta_{3} + \beta_{2} + 3 \beta_1) q^{20} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{2}) q^{21} + ( - \beta_{7} - \beta_{5} - \beta_1) q^{22} + (\beta_{11} - \beta_{4}) q^{23} + (2 \beta_{10} + \beta_{9} - \beta_{8} + 6 \beta_{6} + \beta_{5} + 6) q^{24} + ( - \beta_{11} + \beta_{9} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_1) q^{25} + ( - \beta_{10} + \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{2} + 2) q^{26} + ( - \beta_{11} - \beta_{4} - \beta_{3} - \beta_1) q^{27} + ( - 3 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} + 4 \beta_{4} + 1) q^{28} + (3 \beta_{6} + 3) q^{29} + ( - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_1) q^{30} + (2 \beta_{2} - 2) q^{31} + ( - \beta_{3} - 2 \beta_1) q^{32} + (3 \beta_{3} - \beta_1) q^{33} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{2} + 1) q^{34} + ( - \beta_{10} - 3 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} - \beta_{2} - \beta_1) q^{35} + ( - 2 \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - 1) q^{36} + ( - \beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + 1) q^{37} + (\beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - 3 \beta_{4} - 3 \beta_1 + 1) q^{38} + (2 \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{6} - \beta_{5} + 2 \beta_{2} - 4) q^{39} + ( - \beta_{11} - \beta_{9} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 6) q^{40} + (2 \beta_{10} + 3 \beta_{6} + 3) q^{41} + (2 \beta_{11} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - 5 \beta_{4} - 1) q^{42} + ( - 3 \beta_{3} + \beta_1) q^{43} + ( - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{2}) q^{44} + ( - \beta_{10} + 5 \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{45} + ( - \beta_{10} - 2 \beta_{6} - \beta_{2}) q^{46} + ( - 2 \beta_{11} - 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 6 \beta_{4} + \cdots - 2) q^{47}+ \cdots + (2 \beta_{2} + 8) q^{99}+O(q^{100})$$ q - b4 * q^2 + (-b11 + b4) * q^3 + (-b10 - b6 - b2) * q^4 + (b8 - b7 + b6) * q^5 + (b10 + 2*b6 + b2) * q^6 + (-b3 - b1) * q^7 + (b11 - b9 - b8 + b7 - b6 - b4 + b3 - b1 - 1) * q^8 + (b7 - 2*b6 - b5) * q^9 + (b10 + b9 + 2*b6 + b5 + 2) * q^10 + (b9 - b8 + b5) * q^11 + (b11 + b9 + b8 - b7 + b6 + 2*b4 + b3 + 2*b1 + 1) * q^12 + (-b11 - b9 - b8 - b6 - b5 + b4 - b3 + b1 - 1) * q^13 + (b2 - 4) * q^14 + (2*b11 - b10 - b6 - b4 - 1) * q^15 + (-b10 - b9 + b8 - 3*b6 - b5 - 3) * q^16 + (b7 + b5 + b3 + b1) * q^17 + (-b9 - b8 + b7 - b6 - 1) * q^18 + (-b7 - b6 + b5) * q^19 + (b10 - b5 - b3 + b2 + 3*b1) * q^20 + (-b9 + b8 - b7 + b6 - 2*b2) * q^21 + (-b7 - b5 - b1) * q^22 + (b11 - b4) * q^23 + (2*b10 + b9 - b8 + 6*b6 + b5 + 6) * q^24 + (-b11 + b9 - 2*b4 - b3 + b2 - 2*b1) * q^25 + (-b10 + b7 - 2*b6 - b5 - 2*b2 + 2) * q^26 + (-b11 - b4 - b3 - b1) * q^27 + (-3*b11 + b9 + b8 + b6 + b5 + 4*b4 + 1) * q^28 + (3*b6 + 3) * q^29 + (-b10 + b7 - b6 + b5 + b3 - b2 - 3*b1) * q^30 + (2*b2 - 2) * q^31 + (-b3 - 2*b1) * q^32 + (3*b3 - b1) * q^33 + (-b9 + b8 - b7 + b6 + b2 + 1) * q^34 + (-b10 - 3*b6 - 2*b5 + 2*b3 - b2 - b1) * q^35 + (-2*b10 + b9 - b8 - b6 + b5 - 1) * q^36 + (-b11 + b9 + b8 + b6 + b5 - b4 + 1) * q^37 + (b9 + b8 - b7 + b6 - 3*b4 - 3*b1 + 1) * q^38 + (2*b10 - b9 + b8 - 2*b6 - b5 + 2*b2 - 4) * q^39 + (-b11 - b9 + 3*b4 - b3 - 2*b2 + 3*b1 + 6) * q^40 + (2*b10 + 3*b6 + 3) * q^41 + (2*b11 - b9 - b8 - b6 - b5 - 5*b4 - 1) * q^42 + (-3*b3 + b1) * q^43 + (-b9 + b8 - b7 + b6 - b2) * q^44 + (-b10 + 5*b6 + b5 + b3 - b2 + 2*b1) * q^45 + (-b10 - 2*b6 - b2) * q^46 + (-2*b11 - 2*b9 - 2*b8 + 2*b7 - 2*b6 + 6*b4 - 2*b3 + 6*b1 - 2) * q^47 + (-b7 - b5 - 4*b3 + 5*b1) * q^48 + (b9 - b8 + b6 + b5 + 1) * q^49 + (-b11 - b10 + b9 - 5*b6 + b5 + 3*b4 - 5) * q^50 + (b9 - b8 + b7 - b6 + 2) * q^51 + (2*b11 - b9 - b8 - b6 - b5 - 6*b4 + 3*b3 - 4*b1 - 1) * q^52 + (4*b11 + b9 + b8 - b7 + b6 - 2*b4 + 4*b3 - 2*b1 + 1) * q^53 + (-b10 - 4*b6 - 4) * q^54 + (b11 + b10 - b9 - b8 - 5*b6 - b5 + 2*b4 - 5) * q^55 + (4*b10 - b7 + 5*b6 + b5 + 4*b2) * q^56 + (b11 + b4 + b3 + b1) * q^57 + 3*b1 * q^58 + (-2*b10 + b7 - b6 - b5 - 2*b2) * q^59 + (-3*b11 - 2*b9 - b4 - 3*b3 + 3*b2 - b1 - 10) * q^60 + (2*b7 - 3*b6 - 2*b5) * q^61 + (-2*b11 + 2*b9 + 2*b8 + 2*b6 + 2*b5 + 6*b4 + 2) * q^62 + (4*b11 + 2*b9 + 2*b8 + 2*b6 + 2*b5 - 2*b4 + 2) * q^63 + (2*b9 - 2*b8 + 2*b7 - 2*b6 - 1) * q^64 + (3*b11 + b10 - b9 + 5*b6 - b5 + b4 + 2*b3 + b2 - b1 + 4) * q^65 + b2 * q^66 + (b11 + 3*b4) * q^67 + (b11 + 2*b4) * q^68 + (-b7 + 5*b6 + b5) * q^69 + (b11 - b9 - 3*b8 + 3*b7 - 3*b6 - 3*b4 + b3 - b2 - 3*b1) * q^70 + (-b7 + 3*b6 + b5) * q^71 + (-b7 - b5 + 2*b3 - 6*b1) * q^72 + (2*b11 + b9 + b8 - b7 + b6 - 6*b4 + 2*b3 - 6*b1 + 1) * q^73 + (b10 - b7 + b5 + b2) * q^74 + (-3*b11 + 2*b10 - 2*b9 - 2*b5 - b4) * q^75 + (-b10 - b9 + b8 - 2*b6 - b5 - 2) * q^76 + (-3*b11 - 2*b9 - 2*b8 + 2*b7 - 2*b6 + 3*b4 - 3*b3 + 3*b1 - 2) * q^77 + (-2*b11 + 2*b9 + 2*b8 - b7 + 2*b6 + b5 + 6*b4 - 2*b3 + 3*b1 + 2) * q^78 + (-2*b2 - 8) * q^79 + (-b8 + 5*b6 - 5*b4 + 5) * q^80 + (2*b10 + 2*b9 - 2*b8 + 3*b6 + 2*b5 + 3) * q^81 + (-2*b7 - 2*b5 - 2*b3 + 7*b1) * q^82 + (-4*b11 + 2*b9 + 2*b8 - 2*b7 + 2*b6 + 2*b4 - 4*b3 + 2*b1 + 2) * q^83 + (-3*b10 + 3*b7 - 19*b6 - 3*b5 - 3*b2) * q^84 + (-2*b6 + 3*b5 - b3 + 3*b1) * q^85 - b2 * q^86 + (3*b3 - 3*b1) * q^87 + (b11 + 2*b9 + 2*b8 + 2*b6 + 2*b5 - 5*b4 + 2) * q^88 + (-4*b10 - b9 + b8 + 2*b6 - b5 + 2) * q^89 + (b11 - b9 + 2*b4 + b3 - b2 + 2*b1 + 5) * q^90 + (-2*b10 - 2*b9 + 2*b8 - 3*b7 + 3*b6 + b5 - 4) * q^91 + (-b11 - b9 - b8 + b7 - b6 - 2*b4 - b3 - 2*b1 - 1) * q^92 + (-2*b11 - 2*b9 - 2*b8 - 2*b6 - 2*b5 - 4*b4 - 2) * q^93 + (2*b10 - 2*b9 + 2*b8 + 10*b6 - 2*b5 + 10) * q^94 + (b10 - 3*b7 - 2*b6 - b5 - b3 + b2 - 2*b1) * q^95 + (-b9 + b8 - b7 + b6 - 3*b2 + 2) * q^96 + (2*b7 + 2*b5 + 3*b3 - 3*b1) * q^97 + (-b7 - b5) * q^98 + (2*b2 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 4 q^{4} - 6 q^{5} - 10 q^{6} + 6 q^{9}+O(q^{10})$$ 12 * q + 4 * q^4 - 6 * q^5 - 10 * q^6 + 6 * q^9 $$12 q + 4 q^{4} - 6 q^{5} - 10 q^{6} + 6 q^{9} + 7 q^{10} - 44 q^{14} - 4 q^{15} - 16 q^{16} + 12 q^{19} - q^{20} - 8 q^{21} + 32 q^{24} - 2 q^{25} + 24 q^{26} + 18 q^{29} + 4 q^{30} - 16 q^{31} + 16 q^{34} + 10 q^{35} - 2 q^{36} - 32 q^{39} + 70 q^{40} + 14 q^{41} - 4 q^{44} - 29 q^{45} + 10 q^{46} + 6 q^{49} - 31 q^{50} + 24 q^{51} - 22 q^{54} - 26 q^{55} - 16 q^{56} - 4 q^{59} - 96 q^{60} + 6 q^{61} - 12 q^{64} + 23 q^{65} + 4 q^{66} - 24 q^{69} + 20 q^{70} - 12 q^{71} + 8 q^{74} + 2 q^{75} - 10 q^{76} - 104 q^{79} + 33 q^{80} + 14 q^{81} + 90 q^{84} + 21 q^{85} - 4 q^{86} + 20 q^{89} + 62 q^{90} - 44 q^{91} + 56 q^{94} + 20 q^{95} + 12 q^{96} + 104 q^{99}+O(q^{100})$$ 12 * q + 4 * q^4 - 6 * q^5 - 10 * q^6 + 6 * q^9 + 7 * q^10 - 44 * q^14 - 4 * q^15 - 16 * q^16 + 12 * q^19 - q^20 - 8 * q^21 + 32 * q^24 - 2 * q^25 + 24 * q^26 + 18 * q^29 + 4 * q^30 - 16 * q^31 + 16 * q^34 + 10 * q^35 - 2 * q^36 - 32 * q^39 + 70 * q^40 + 14 * q^41 - 4 * q^44 - 29 * q^45 + 10 * q^46 + 6 * q^49 - 31 * q^50 + 24 * q^51 - 22 * q^54 - 26 * q^55 - 16 * q^56 - 4 * q^59 - 96 * q^60 + 6 * q^61 - 12 * q^64 + 23 * q^65 + 4 * q^66 - 24 * q^69 + 20 * q^70 - 12 * q^71 + 8 * q^74 + 2 * q^75 - 10 * q^76 - 104 * q^79 + 33 * q^80 + 14 * q^81 + 90 * q^84 + 21 * q^85 - 4 * q^86 + 20 * q^89 + 62 * q^90 - 44 * q^91 + 56 * q^94 + 20 * q^95 + 12 * q^96 + 104 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 8x^{10} + 54x^{8} - 78x^{6} + 92x^{4} - 10x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 16\nu^{10} - 108\nu^{8} + 729\nu^{6} - 184\nu^{4} + 20\nu^{2} + 3531 ) / 1222$$ (16*v^10 - 108*v^8 + 729*v^6 - 184*v^4 + 20*v^2 + 3531) / 1222 $$\beta_{3}$$ $$=$$ $$( 108\nu^{11} - 729\nu^{9} + 4768\nu^{7} - 1242\nu^{5} + 135\nu^{3} + 11156\nu ) / 1222$$ (108*v^11 - 729*v^9 + 4768*v^7 - 1242*v^5 + 135*v^3 + 11156*v) / 1222 $$\beta_{4}$$ $$=$$ $$( 135\nu^{11} - 1064\nu^{9} + 7182\nu^{7} - 9801\nu^{5} + 12236\nu^{3} - 1330\nu ) / 1222$$ (135*v^11 - 1064*v^9 + 7182*v^7 - 9801*v^5 + 12236*v^3 - 1330*v) / 1222 $$\beta_{5}$$ $$=$$ $$( - 92 \nu^{11} + 293 \nu^{10} + 621 \nu^{9} - 2436 \nu^{8} - 4039 \nu^{7} + 16443 \nu^{6} + 1058 \nu^{5} - 26893 \nu^{4} - 115 \nu^{3} + 28014 \nu^{2} - 5181 \nu - 3045 ) / 2444$$ (-92*v^11 + 293*v^10 + 621*v^9 - 2436*v^8 - 4039*v^7 + 16443*v^6 + 1058*v^5 - 26893*v^4 - 115*v^3 + 28014*v^2 - 5181*v - 3045) / 2444 $$\beta_{6}$$ $$=$$ $$( 135\nu^{10} - 1064\nu^{8} + 7182\nu^{6} - 9801\nu^{4} + 12236\nu^{2} - 1330 ) / 1222$$ (135*v^10 - 1064*v^8 + 7182*v^6 - 9801*v^4 + 12236*v^2 - 1330) / 1222 $$\beta_{7}$$ $$=$$ $$( - 92 \nu^{11} - 293 \nu^{10} + 621 \nu^{9} + 2436 \nu^{8} - 4039 \nu^{7} - 16443 \nu^{6} + 1058 \nu^{5} + 26893 \nu^{4} - 115 \nu^{3} - 28014 \nu^{2} - 5181 \nu + 3045 ) / 2444$$ (-92*v^11 - 293*v^10 + 621*v^9 + 2436*v^8 - 4039*v^7 - 16443*v^6 + 1058*v^5 + 26893*v^4 - 115*v^3 - 28014*v^2 - 5181*v + 3045) / 2444 $$\beta_{8}$$ $$=$$ $$( 563 \nu^{11} - 655 \nu^{10} - 4564 \nu^{9} + 5185 \nu^{8} + 30807 \nu^{7} - 34846 \nu^{6} - 46495 \nu^{5} + 47553 \nu^{4} + 52486 \nu^{3} - 52601 \nu^{2} - 5705 \nu + 524 ) / 2444$$ (563*v^11 - 655*v^10 - 4564*v^9 + 5185*v^8 + 30807*v^7 - 34846*v^6 - 46495*v^5 + 47553*v^4 + 52486*v^3 - 52601*v^2 - 5705*v + 524) / 2444 $$\beta_{9}$$ $$=$$ $$( 655 \nu^{11} + 92 \nu^{10} - 5185 \nu^{9} - 621 \nu^{8} + 34846 \nu^{7} + 4039 \nu^{6} - 47553 \nu^{5} - 1058 \nu^{4} + 52601 \nu^{3} + 115 \nu^{2} - 524 \nu + 2737 ) / 2444$$ (655*v^11 + 92*v^10 - 5185*v^9 - 621*v^8 + 34846*v^7 + 4039*v^6 - 47553*v^5 - 1058*v^4 + 52601*v^3 + 115*v^2 - 524*v + 2737) / 2444 $$\beta_{10}$$ $$=$$ $$( -405\nu^{10} + 3192\nu^{8} - 21546\nu^{6} + 29403\nu^{4} - 35486\nu^{2} + 324 ) / 1222$$ (-405*v^10 + 3192*v^8 - 21546*v^6 + 29403*v^4 - 35486*v^2 + 324) / 1222 $$\beta_{11}$$ $$=$$ $$\nu^{11} - 8\nu^{9} + 54\nu^{7} - 78\nu^{5} + 92\nu^{3} - 10\nu$$ v^11 - 8*v^9 + 54*v^7 - 78*v^5 + 92*v^3 - 10*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3\beta_{6} + 3$$ b10 + 3*b6 + 3 $$\nu^{3}$$ $$=$$ $$-\beta_{11} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + 5\beta_{4} - \beta_{3} + 5\beta _1 + 1$$ -b11 + b9 + b8 - b7 + b6 + 5*b4 - b3 + 5*b1 + 1 $$\nu^{4}$$ $$=$$ $$7\beta_{10} - \beta_{7} + 18\beta_{6} + \beta_{5} + 7\beta_{2}$$ 7*b10 - b7 + 18*b6 + b5 + 7*b2 $$\nu^{5}$$ $$=$$ $$-7\beta_{11} + 8\beta_{9} + 8\beta_{8} + 8\beta_{6} + 8\beta_{5} + 30\beta_{4} + 8$$ -7*b11 + 8*b9 + 8*b8 + 8*b6 + 8*b5 + 30*b4 + 8 $$\nu^{6}$$ $$=$$ $$-8\beta_{9} + 8\beta_{8} - 8\beta_{7} + 8\beta_{6} + 46\beta_{2} - 107$$ -8*b9 + 8*b8 - 8*b7 + 8*b6 + 46*b2 - 107 $$\nu^{7}$$ $$=$$ $$54\beta_{7} + 54\beta_{5} + 46\beta_{3} - 191\beta_1$$ 54*b7 + 54*b5 + 46*b3 - 191*b1 $$\nu^{8}$$ $$=$$ $$-299\beta_{10} - 54\beta_{9} + 54\beta_{8} - 689\beta_{6} - 54\beta_{5} - 689$$ -299*b10 - 54*b9 + 54*b8 - 689*b6 - 54*b5 - 689 $$\nu^{9}$$ $$=$$ $$299 \beta_{11} - 353 \beta_{9} - 353 \beta_{8} + 353 \beta_{7} - 353 \beta_{6} - 1233 \beta_{4} + 299 \beta_{3} - 1233 \beta _1 - 353$$ 299*b11 - 353*b9 - 353*b8 + 353*b7 - 353*b6 - 1233*b4 + 299*b3 - 1233*b1 - 353 $$\nu^{10}$$ $$=$$ $$-1939\beta_{10} + 353\beta_{7} - 4812\beta_{6} - 353\beta_{5} - 1939\beta_{2}$$ -1939*b10 + 353*b7 - 4812*b6 - 353*b5 - 1939*b2 $$\nu^{11}$$ $$=$$ $$1939\beta_{11} - 2292\beta_{9} - 2292\beta_{8} - 2292\beta_{6} - 2292\beta_{5} - 7984\beta_{4} - 2292$$ 1939*b11 - 2292*b9 - 2292*b8 - 2292*b6 - 2292*b5 - 7984*b4 - 2292

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/65\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$-1$$ $$\beta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 −2.20467 + 1.27287i −1.02826 + 0.593667i −0.286513 + 0.165418i 0.286513 − 0.165418i 1.02826 − 0.593667i 2.20467 − 1.27287i −2.20467 − 1.27287i −1.02826 − 0.593667i −0.286513 − 0.165418i 0.286513 + 0.165418i 1.02826 + 0.593667i 2.20467 + 1.27287i
−2.20467 1.27287i 1.86449 + 1.07646i 2.24039 + 3.88048i −0.817544 + 2.08125i −2.74039 4.74650i 2.54486 1.46928i 6.31544i 0.817544 + 1.41603i 4.45158 3.54786i
9.2 −1.02826 0.593667i 0.298874 + 0.172555i −0.295120 0.511162i 1.44045 1.71029i −0.204880 0.354863i 1.75765 1.01478i 3.07548i −1.44045 2.49493i −2.49650 + 0.903481i
9.3 −0.286513 0.165418i −2.33117 1.34590i −0.945274 1.63726i −2.12291 + 0.702335i 0.445274 + 0.771236i 2.90420 1.67674i 1.28714i 2.12291 + 3.67698i 0.724419 + 0.149939i
9.4 0.286513 + 0.165418i 2.33117 + 1.34590i −0.945274 1.63726i −2.12291 0.702335i 0.445274 + 0.771236i −2.90420 + 1.67674i 1.28714i 2.12291 + 3.67698i −0.492061 0.552395i
9.5 1.02826 + 0.593667i −0.298874 0.172555i −0.295120 0.511162i 1.44045 + 1.71029i −0.204880 0.354863i −1.75765 + 1.01478i 3.07548i −1.44045 2.49493i 0.465813 + 2.61378i
9.6 2.20467 + 1.27287i −1.86449 1.07646i 2.24039 + 3.88048i −0.817544 2.08125i −2.74039 4.74650i −2.54486 + 1.46928i 6.31544i 0.817544 + 1.41603i 0.846746 5.62912i
29.1 −2.20467 + 1.27287i 1.86449 1.07646i 2.24039 3.88048i −0.817544 2.08125i −2.74039 + 4.74650i 2.54486 + 1.46928i 6.31544i 0.817544 1.41603i 4.45158 + 3.54786i
29.2 −1.02826 + 0.593667i 0.298874 0.172555i −0.295120 + 0.511162i 1.44045 + 1.71029i −0.204880 + 0.354863i 1.75765 + 1.01478i 3.07548i −1.44045 + 2.49493i −2.49650 0.903481i
29.3 −0.286513 + 0.165418i −2.33117 + 1.34590i −0.945274 + 1.63726i −2.12291 0.702335i 0.445274 0.771236i 2.90420 + 1.67674i 1.28714i 2.12291 3.67698i 0.724419 0.149939i
29.4 0.286513 0.165418i 2.33117 1.34590i −0.945274 + 1.63726i −2.12291 + 0.702335i 0.445274 0.771236i −2.90420 1.67674i 1.28714i 2.12291 3.67698i −0.492061 + 0.552395i
29.5 1.02826 0.593667i −0.298874 + 0.172555i −0.295120 + 0.511162i 1.44045 1.71029i −0.204880 + 0.354863i −1.75765 1.01478i 3.07548i −1.44045 + 2.49493i 0.465813 2.61378i
29.6 2.20467 1.27287i −1.86449 + 1.07646i 2.24039 3.88048i −0.817544 + 2.08125i −2.74039 + 4.74650i −2.54486 1.46928i 6.31544i 0.817544 1.41603i 0.846746 + 5.62912i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.n.a 12
3.b odd 2 1 585.2.bs.a 12
4.b odd 2 1 1040.2.dh.a 12
5.b even 2 1 inner 65.2.n.a 12
5.c odd 4 2 325.2.e.e 12
13.b even 2 1 845.2.n.e 12
13.c even 3 1 inner 65.2.n.a 12
13.c even 3 1 845.2.b.d 6
13.d odd 4 2 845.2.l.f 24
13.e even 6 1 845.2.b.e 6
13.e even 6 1 845.2.n.e 12
13.f odd 12 2 845.2.d.d 12
13.f odd 12 2 845.2.l.f 24
15.d odd 2 1 585.2.bs.a 12
20.d odd 2 1 1040.2.dh.a 12
39.i odd 6 1 585.2.bs.a 12
52.j odd 6 1 1040.2.dh.a 12
65.d even 2 1 845.2.n.e 12
65.g odd 4 2 845.2.l.f 24
65.l even 6 1 845.2.b.e 6
65.l even 6 1 845.2.n.e 12
65.n even 6 1 inner 65.2.n.a 12
65.n even 6 1 845.2.b.d 6
65.q odd 12 2 325.2.e.e 12
65.q odd 12 2 4225.2.a.br 6
65.r odd 12 2 4225.2.a.bq 6
65.s odd 12 2 845.2.d.d 12
65.s odd 12 2 845.2.l.f 24
195.x odd 6 1 585.2.bs.a 12
260.v odd 6 1 1040.2.dh.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.n.a 12 1.a even 1 1 trivial
65.2.n.a 12 5.b even 2 1 inner
65.2.n.a 12 13.c even 3 1 inner
65.2.n.a 12 65.n even 6 1 inner
325.2.e.e 12 5.c odd 4 2
325.2.e.e 12 65.q odd 12 2
585.2.bs.a 12 3.b odd 2 1
585.2.bs.a 12 15.d odd 2 1
585.2.bs.a 12 39.i odd 6 1
585.2.bs.a 12 195.x odd 6 1
845.2.b.d 6 13.c even 3 1
845.2.b.d 6 65.n even 6 1
845.2.b.e 6 13.e even 6 1
845.2.b.e 6 65.l even 6 1
845.2.d.d 12 13.f odd 12 2
845.2.d.d 12 65.s odd 12 2
845.2.l.f 24 13.d odd 4 2
845.2.l.f 24 13.f odd 12 2
845.2.l.f 24 65.g odd 4 2
845.2.l.f 24 65.s odd 12 2
845.2.n.e 12 13.b even 2 1
845.2.n.e 12 13.e even 6 1
845.2.n.e 12 65.d even 2 1
845.2.n.e 12 65.l even 6 1
1040.2.dh.a 12 4.b odd 2 1
1040.2.dh.a 12 20.d odd 2 1
1040.2.dh.a 12 52.j odd 6 1
1040.2.dh.a 12 260.v odd 6 1
4225.2.a.bq 6 65.r odd 12 2
4225.2.a.br 6 65.q odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 8 T^{10} + 54 T^{8} - 78 T^{6} + \cdots + 1$$
$3$ $$T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16$$
$5$ $$(T^{6} + 3 T^{5} + 5 T^{4} + 10 T^{3} + \cdots + 125)^{2}$$
$7$ $$T^{12} - 24 T^{10} + 397 T^{8} + \cdots + 160000$$
$11$ $$(T^{6} + 13 T^{4} - 16 T^{3} + 169 T^{2} + \cdots + 64)^{2}$$
$13$ $$T^{12} - 15 T^{10} + 39 T^{8} + \cdots + 4826809$$
$17$ $$T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561$$
$19$ $$(T^{6} - 6 T^{5} + 37 T^{4} - 14 T^{3} + \cdots + 100)^{2}$$
$23$ $$T^{12} - 12 T^{10} + 109 T^{8} + \cdots + 16$$
$29$ $$(T^{2} - 3 T + 9)^{6}$$
$31$ $$(T^{3} + 4 T^{2} - 40 T + 40)^{4}$$
$37$ $$T^{12} - 35 T^{10} + 1062 T^{8} + \cdots + 28561$$
$41$ $$(T^{6} - 7 T^{5} + 78 T^{4} + 213 T^{3} + \cdots + 25)^{2}$$
$43$ $$T^{12} - 80 T^{10} + 6117 T^{8} + \cdots + 65536$$
$47$ $$(T^{6} + 236 T^{4} + 14640 T^{2} + \cdots + 270400)^{2}$$
$53$ $$(T^{6} + 171 T^{4} + 1040 T^{2} + \cdots + 400)^{2}$$
$59$ $$(T^{6} + 2 T^{5} + 59 T^{4} + 162 T^{3} + \cdots + 18496)^{2}$$
$61$ $$(T^{6} - 3 T^{5} + 58 T^{4} - 83 T^{3} + \cdots + 13225)^{2}$$
$67$ $$T^{12} - 100 T^{10} + \cdots + 406586896$$
$71$ $$(T^{6} + 6 T^{5} + 37 T^{4} + 46 T^{3} + \cdots + 676)^{2}$$
$73$ $$(T^{6} + 215 T^{4} + 13900 T^{2} + \cdots + 250000)^{2}$$
$79$ $$(T^{3} + 26 T^{2} + 180 T + 160)^{4}$$
$83$ $$(T^{6} + 276 T^{4} + 23600 T^{2} + \cdots + 640000)^{2}$$
$89$ $$(T^{6} - 10 T^{5} + 257 T^{4} + \cdots + 2515396)^{2}$$
$97$ $$T^{12} - 280 T^{10} + \cdots + 41740124416$$