# Properties

 Label 65.2.a.c Level $65$ Weight $2$ Character orbit 65.a Self dual yes Analytic conductor $0.519$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,2,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("65.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 65.a (trivial)

## 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: yes Analytic conductor: $$0.519027613138$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.73205 1.73205
−1.73205 2.73205 1.00000 −1.00000 −4.73205 2.00000 1.73205 4.46410 1.73205
1.2 1.73205 −0.732051 1.00000 −1.00000 −1.26795 2.00000 −1.73205 −2.46410 −1.73205
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 65.2.a.c 2
3.b odd 2 1 585.2.a.k 2
4.b odd 2 1 1040.2.a.h 2
5.b even 2 1 325.2.a.g 2
5.c odd 4 2 325.2.b.e 4
7.b odd 2 1 3185.2.a.k 2
8.b even 2 1 4160.2.a.y 2
8.d odd 2 1 4160.2.a.bj 2
11.b odd 2 1 7865.2.a.h 2
12.b even 2 1 9360.2.a.cm 2
13.b even 2 1 845.2.a.d 2
13.c even 3 2 845.2.e.e 4
13.d odd 4 2 845.2.c.e 4
13.e even 6 2 845.2.e.f 4
13.f odd 12 2 845.2.m.a 4
13.f odd 12 2 845.2.m.c 4
15.d odd 2 1 2925.2.a.z 2
15.e even 4 2 2925.2.c.v 4
20.d odd 2 1 5200.2.a.ca 2
39.d odd 2 1 7605.2.a.be 2
65.d even 2 1 4225.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.c 2 1.a even 1 1 trivial
325.2.a.g 2 5.b even 2 1
325.2.b.e 4 5.c odd 4 2
585.2.a.k 2 3.b odd 2 1
845.2.a.d 2 13.b even 2 1
845.2.c.e 4 13.d odd 4 2
845.2.e.e 4 13.c even 3 2
845.2.e.f 4 13.e even 6 2
845.2.m.a 4 13.f odd 12 2
845.2.m.c 4 13.f odd 12 2
1040.2.a.h 2 4.b odd 2 1
2925.2.a.z 2 15.d odd 2 1
2925.2.c.v 4 15.e even 4 2
3185.2.a.k 2 7.b odd 2 1
4160.2.a.y 2 8.b even 2 1
4160.2.a.bj 2 8.d odd 2 1
4225.2.a.w 2 65.d even 2 1
5200.2.a.ca 2 20.d odd 2 1
7605.2.a.be 2 39.d odd 2 1
7865.2.a.h 2 11.b odd 2 1
9360.2.a.cm 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2} - 2T - 2$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2} + 6T + 6$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 12$$
$19$ $$T^{2} + 2T - 26$$
$23$ $$T^{2} - 6T + 6$$
$29$ $$T^{2} + 12T + 24$$
$31$ $$T^{2} - 10T - 2$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} - 12$$
$43$ $$T^{2} - 10T - 2$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2} - 108$$
$59$ $$T^{2} + 6T - 138$$
$61$ $$T^{2} - 4T - 104$$
$67$ $$T^{2} + 8T - 92$$
$71$ $$T^{2} - 6T + 6$$
$73$ $$(T + 4)^{2}$$
$79$ $$T^{2} - 4T - 104$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 12T - 12$$
$97$ $$(T - 2)^{2}$$