# Properties

 Label 65.2.e.b Level $65$ Weight $2$ Character orbit 65.e Analytic conductor $0.519$ Analytic rank $0$ Dimension $4$ CM no 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: $$\Q(\sqrt{-3}, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 4x^{2} + 3x + 9$$ x^4 - x^3 + 4*x^2 + 3*x + 9 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4x^{2} + 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12$$ (-v^3 + 4*v^2 - 4*v - 3) / 12 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 7 ) / 4$$ (v^3 + 7) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2} + \beta _1 - 1$$ b3 + 3*b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$4\beta_{3} - 7$$ 4*b3 - 7

## Character values

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

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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
 −0.651388 − 1.12824i 1.15139 + 1.99426i −0.651388 + 1.12824i 1.15139 − 1.99426i
−0.651388 1.12824i −0.500000 0.866025i 0.151388 0.262211i −1.00000 −0.651388 + 1.12824i 0.500000 0.866025i −3.00000 1.00000 1.73205i 0.651388 + 1.12824i
16.2 1.15139 + 1.99426i −0.500000 0.866025i −1.65139 + 2.86029i −1.00000 1.15139 1.99426i 0.500000 0.866025i −3.00000 1.00000 1.73205i −1.15139 1.99426i
61.1 −0.651388 + 1.12824i −0.500000 + 0.866025i 0.151388 + 0.262211i −1.00000 −0.651388 1.12824i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i 0.651388 1.12824i
61.2 1.15139 1.99426i −0.500000 + 0.866025i −1.65139 2.86029i −1.00000 1.15139 + 1.99426i 0.500000 + 0.866025i −3.00000 1.00000 + 1.73205i −1.15139 + 1.99426i
 $$n$$: 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.b 4
3.b odd 2 1 585.2.j.d 4
4.b odd 2 1 1040.2.q.o 4
5.b even 2 1 325.2.e.a 4
5.c odd 4 2 325.2.o.b 8
13.b even 2 1 845.2.e.d 4
13.c even 3 1 inner 65.2.e.b 4
13.c even 3 1 845.2.a.c 2
13.d odd 4 2 845.2.m.d 8
13.e even 6 1 845.2.a.f 2
13.e even 6 1 845.2.e.d 4
13.f odd 12 2 845.2.c.d 4
13.f odd 12 2 845.2.m.d 8
39.h odd 6 1 7605.2.a.bb 2
39.i odd 6 1 585.2.j.d 4
39.i odd 6 1 7605.2.a.bg 2
52.j odd 6 1 1040.2.q.o 4
65.l even 6 1 4225.2.a.t 2
65.n even 6 1 325.2.e.a 4
65.n even 6 1 4225.2.a.x 2
65.q odd 12 2 325.2.o.b 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - T_{2}^{3} + 4T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(65, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{3} + 4 T^{2} + \cdots + 9$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T + 1)^{4}$$
$7$ $$(T^{2} - T + 1)^{2}$$
$11$ $$T^{4} - 4 T^{3} + \cdots + 81$$
$13$ $$(T^{2} - 13)^{2}$$
$17$ $$T^{4} + 8 T^{3} + \cdots + 9$$
$19$ $$T^{4} - 4 T^{3} + \cdots + 81$$
$23$ $$(T^{2} - 3 T + 9)^{2}$$
$29$ $$T^{4} + 2 T^{3} + \cdots + 2601$$
$31$ $$(T + 4)^{4}$$
$37$ $$T^{4} + 13T^{2} + 169$$
$41$ $$(T^{2} + 3 T + 9)^{2}$$
$43$ $$T^{4} - 6 T^{3} + \cdots + 1849$$
$47$ $$(T^{2} - 4 T - 48)^{2}$$
$53$ $$(T^{2} - 8 T - 36)^{2}$$
$59$ $$T^{4} + 117 T^{2} + 13689$$
$61$ $$(T^{2} - T + 1)^{2}$$
$67$ $$(T^{2} - 7 T + 49)^{2}$$
$71$ $$T^{4} - 12 T^{3} + \cdots + 6561$$
$73$ $$(T^{2} + 16 T + 12)^{2}$$
$79$ $$(T^{2} + 4 T - 48)^{2}$$
$83$ $$(T^{2} + 4 T - 48)^{2}$$
$89$ $$T^{4} + 2 T^{3} + \cdots + 2601$$
$97$ $$T^{4} - 24 T^{3} + \cdots + 17161$$