# Properties

 Label 65.2.a.a Level $65$ Weight $2$ Character orbit 65.a Self dual yes Analytic conductor $0.519$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - 2 * q^3 - q^4 - q^5 + 2 * q^6 - 4 * q^7 + 3 * q^8 + q^9 $$q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - q^{13} + 4 q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} - 6 q^{19} + q^{20} + 8 q^{21} - 2 q^{22} - 6 q^{23} - 6 q^{24} + q^{25} + q^{26} + 4 q^{27} + 4 q^{28} + 2 q^{29} - 2 q^{30} - 10 q^{31} - 5 q^{32} - 4 q^{33} - 2 q^{34} + 4 q^{35} - q^{36} - 2 q^{37} + 6 q^{38} + 2 q^{39} - 3 q^{40} - 6 q^{41} - 8 q^{42} + 10 q^{43} - 2 q^{44} - q^{45} + 6 q^{46} + 4 q^{47} + 2 q^{48} + 9 q^{49} - q^{50} - 4 q^{51} + q^{52} + 2 q^{53} - 4 q^{54} - 2 q^{55} - 12 q^{56} + 12 q^{57} - 2 q^{58} + 6 q^{59} - 2 q^{60} + 2 q^{61} + 10 q^{62} - 4 q^{63} + 7 q^{64} + q^{65} + 4 q^{66} - 4 q^{67} - 2 q^{68} + 12 q^{69} - 4 q^{70} + 6 q^{71} + 3 q^{72} - 6 q^{73} + 2 q^{74} - 2 q^{75} + 6 q^{76} - 8 q^{77} - 2 q^{78} - 12 q^{79} + q^{80} - 11 q^{81} + 6 q^{82} - 16 q^{83} - 8 q^{84} - 2 q^{85} - 10 q^{86} - 4 q^{87} + 6 q^{88} + 2 q^{89} + q^{90} + 4 q^{91} + 6 q^{92} + 20 q^{93} - 4 q^{94} + 6 q^{95} + 10 q^{96} - 2 q^{97} - 9 q^{98} + 2 q^{99}+O(q^{100})$$ q - q^2 - 2 * q^3 - q^4 - q^5 + 2 * q^6 - 4 * q^7 + 3 * q^8 + q^9 + q^10 + 2 * q^11 + 2 * q^12 - q^13 + 4 * q^14 + 2 * q^15 - q^16 + 2 * q^17 - q^18 - 6 * q^19 + q^20 + 8 * q^21 - 2 * q^22 - 6 * q^23 - 6 * q^24 + q^25 + q^26 + 4 * q^27 + 4 * q^28 + 2 * q^29 - 2 * q^30 - 10 * q^31 - 5 * q^32 - 4 * q^33 - 2 * q^34 + 4 * q^35 - q^36 - 2 * q^37 + 6 * q^38 + 2 * q^39 - 3 * q^40 - 6 * q^41 - 8 * q^42 + 10 * q^43 - 2 * q^44 - q^45 + 6 * q^46 + 4 * q^47 + 2 * q^48 + 9 * q^49 - q^50 - 4 * q^51 + q^52 + 2 * q^53 - 4 * q^54 - 2 * q^55 - 12 * q^56 + 12 * q^57 - 2 * q^58 + 6 * q^59 - 2 * q^60 + 2 * q^61 + 10 * q^62 - 4 * q^63 + 7 * q^64 + q^65 + 4 * q^66 - 4 * q^67 - 2 * q^68 + 12 * q^69 - 4 * q^70 + 6 * q^71 + 3 * q^72 - 6 * q^73 + 2 * q^74 - 2 * q^75 + 6 * q^76 - 8 * q^77 - 2 * q^78 - 12 * q^79 + q^80 - 11 * q^81 + 6 * q^82 - 16 * q^83 - 8 * q^84 - 2 * q^85 - 10 * q^86 - 4 * q^87 + 6 * q^88 + 2 * q^89 + q^90 + 4 * q^91 + 6 * q^92 + 20 * q^93 - 4 * q^94 + 6 * q^95 + 10 * q^96 - 2 * q^97 - 9 * q^98 + 2 * 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
 0
−1.00000 −2.00000 −1.00000 −1.00000 2.00000 −4.00000 3.00000 1.00000 1.00000
 $$n$$: 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.a 1
3.b odd 2 1 585.2.a.h 1
4.b odd 2 1 1040.2.a.f 1
5.b even 2 1 325.2.a.d 1
5.c odd 4 2 325.2.b.b 2
7.b odd 2 1 3185.2.a.e 1
8.b even 2 1 4160.2.a.q 1
8.d odd 2 1 4160.2.a.f 1
11.b odd 2 1 7865.2.a.c 1
12.b even 2 1 9360.2.a.ca 1
13.b even 2 1 845.2.a.a 1
13.c even 3 2 845.2.e.b 2
13.d odd 4 2 845.2.c.a 2
13.e even 6 2 845.2.e.a 2
13.f odd 12 4 845.2.m.b 4
15.d odd 2 1 2925.2.a.f 1
15.e even 4 2 2925.2.c.h 2
20.d odd 2 1 5200.2.a.d 1
39.d odd 2 1 7605.2.a.f 1
65.d even 2 1 4225.2.a.g 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(65))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T + 2$$
$5$ $$T + 1$$
$7$ $$T + 4$$
$11$ $$T - 2$$
$13$ $$T + 1$$
$17$ $$T - 2$$
$19$ $$T + 6$$
$23$ $$T + 6$$
$29$ $$T - 2$$
$31$ $$T + 10$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T - 10$$
$47$ $$T - 4$$
$53$ $$T - 2$$
$59$ $$T - 6$$
$61$ $$T - 2$$
$67$ $$T + 4$$
$71$ $$T - 6$$
$73$ $$T + 6$$
$79$ $$T + 12$$
$83$ $$T + 16$$
$89$ $$T - 2$$
$97$ $$T + 2$$