# Properties

 Label 65.2.e.a Level $65$ Weight $2$ Character orbit 65.e Analytic conductor $0.519$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,2,Mod(16,65)]

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([0, 2]))

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

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

chi := DirichletCharacter("65.16");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.e (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: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 1 ) / 2$$ (v^3 + 1) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_1$$ b3 + b2 + b1 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 1$$ 2*b2 - 1

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
−0.809017 1.40126i 1.11803 + 1.93649i −0.309017 + 0.535233i 1.00000 1.80902 3.13331i 0.118034 0.204441i −2.23607 −1.00000 + 1.73205i −0.809017 1.40126i
16.2 0.309017 + 0.535233i −1.11803 1.93649i 0.809017 1.40126i 1.00000 0.690983 1.19682i −2.11803 + 3.66854i 2.23607 −1.00000 + 1.73205i 0.309017 + 0.535233i
61.1 −0.809017 + 1.40126i 1.11803 1.93649i −0.309017 0.535233i 1.00000 1.80902 + 3.13331i 0.118034 + 0.204441i −2.23607 −1.00000 1.73205i −0.809017 + 1.40126i
61.2 0.309017 0.535233i −1.11803 + 1.93649i 0.809017 + 1.40126i 1.00000 0.690983 + 1.19682i −2.11803 3.66854i 2.23607 −1.00000 1.73205i 0.309017 0.535233i
 $$n$$: e.g. 2-40 or 990-1000 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.e.a 4
3.b odd 2 1 585.2.j.e 4
4.b odd 2 1 1040.2.q.n 4
5.b even 2 1 325.2.e.b 4
5.c odd 4 2 325.2.o.a 8
13.b even 2 1 845.2.e.g 4
13.c even 3 1 inner 65.2.e.a 4
13.c even 3 1 845.2.a.e 2
13.d odd 4 2 845.2.m.e 8
13.e even 6 1 845.2.a.b 2
13.e even 6 1 845.2.e.g 4
13.f odd 12 2 845.2.c.c 4
13.f odd 12 2 845.2.m.e 8
39.h odd 6 1 7605.2.a.bf 2
39.i odd 6 1 585.2.j.e 4
39.i odd 6 1 7605.2.a.ba 2
52.j odd 6 1 1040.2.q.n 4
65.l even 6 1 4225.2.a.y 2
65.n even 6 1 325.2.e.b 4
65.n even 6 1 4225.2.a.u 2
65.q odd 12 2 325.2.o.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.e.a 4 1.a even 1 1 trivial
65.2.e.a 4 13.c even 3 1 inner
325.2.e.b 4 5.b even 2 1
325.2.e.b 4 65.n even 6 1
325.2.o.a 8 5.c odd 4 2
325.2.o.a 8 65.q odd 12 2
585.2.j.e 4 3.b odd 2 1
585.2.j.e 4 39.i odd 6 1
845.2.a.b 2 13.e even 6 1
845.2.a.e 2 13.c even 3 1
845.2.c.c 4 13.f odd 12 2
845.2.e.g 4 13.b even 2 1
845.2.e.g 4 13.e even 6 1
845.2.m.e 8 13.d odd 4 2
845.2.m.e 8 13.f odd 12 2
1040.2.q.n 4 4.b odd 2 1
1040.2.q.n 4 52.j odd 6 1
4225.2.a.u 2 65.n even 6 1
4225.2.a.y 2 65.l even 6 1
7605.2.a.ba 2 39.i odd 6 1
7605.2.a.bf 2 39.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + T^{3} + 2 T^{2} + \cdots + 1$$
$3$ $$T^{4} + 5T^{2} + 25$$
$5$ $$(T - 1)^{4}$$
$7$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$11$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$13$ $$(T^{2} + 2 T + 13)^{2}$$
$17$ $$T^{4} + 2 T^{3} + \cdots + 361$$
$19$ $$T^{4} + 4 T^{3} + \cdots + 1$$
$23$ $$T^{4} + 12 T^{3} + \cdots + 961$$
$29$ $$T^{4} - 6 T^{3} + \cdots + 121$$
$31$ $$T^{4}$$
$37$ $$(T^{2} + 3 T + 9)^{2}$$
$41$ $$T^{4} + 6 T^{3} + \cdots + 5041$$
$43$ $$T^{4} - 8 T^{3} + \cdots + 121$$
$47$ $$(T^{2} - 8 T - 64)^{2}$$
$53$ $$(T - 6)^{4}$$
$59$ $$T^{4} + 12 T^{3} + \cdots + 81$$
$61$ $$T^{4} + 2 T^{3} + \cdots + 32041$$
$67$ $$T^{4} + 8 T^{3} + \cdots + 841$$
$71$ $$T^{4} - 8 T^{3} + \cdots + 121$$
$73$ $$(T + 6)^{4}$$
$79$ $$T^{4}$$
$83$ $$(T^{2} - 80)^{2}$$
$89$ $$(T^{2} - 9 T + 81)^{2}$$
$97$ $$T^{4} + 2 T^{3} + \cdots + 361$$