# Properties

 Label 165.4.a.a Level $165$ Weight $4$ Character orbit 165.a Self dual yes Analytic conductor $9.735$ Analytic rank $0$ 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] = [165,4,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("165.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$165 = 3 \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 165.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: $$9.73531515095$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 8 q^{4} - 5 q^{5} + 2 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 8 * q^4 - 5 * q^5 + 2 * q^7 + 9 * q^9 $$q - 3 q^{3} - 8 q^{4} - 5 q^{5} + 2 q^{7} + 9 q^{9} - 11 q^{11} + 24 q^{12} - 22 q^{13} + 15 q^{15} + 64 q^{16} + 72 q^{17} + 122 q^{19} + 40 q^{20} - 6 q^{21} + 72 q^{23} + 25 q^{25} - 27 q^{27} - 16 q^{28} + 96 q^{29} - 112 q^{31} + 33 q^{33} - 10 q^{35} - 72 q^{36} + 266 q^{37} + 66 q^{39} - 96 q^{41} - 382 q^{43} + 88 q^{44} - 45 q^{45} + 360 q^{47} - 192 q^{48} - 339 q^{49} - 216 q^{51} + 176 q^{52} + 318 q^{53} + 55 q^{55} - 366 q^{57} + 660 q^{59} - 120 q^{60} - 430 q^{61} + 18 q^{63} - 512 q^{64} + 110 q^{65} + 380 q^{67} - 576 q^{68} - 216 q^{69} + 168 q^{71} + 218 q^{73} - 75 q^{75} - 976 q^{76} - 22 q^{77} - 706 q^{79} - 320 q^{80} + 81 q^{81} + 1068 q^{83} + 48 q^{84} - 360 q^{85} - 288 q^{87} - 6 q^{89} - 44 q^{91} - 576 q^{92} + 336 q^{93} - 610 q^{95} + 686 q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 - 8 * q^4 - 5 * q^5 + 2 * q^7 + 9 * q^9 - 11 * q^11 + 24 * q^12 - 22 * q^13 + 15 * q^15 + 64 * q^16 + 72 * q^17 + 122 * q^19 + 40 * q^20 - 6 * q^21 + 72 * q^23 + 25 * q^25 - 27 * q^27 - 16 * q^28 + 96 * q^29 - 112 * q^31 + 33 * q^33 - 10 * q^35 - 72 * q^36 + 266 * q^37 + 66 * q^39 - 96 * q^41 - 382 * q^43 + 88 * q^44 - 45 * q^45 + 360 * q^47 - 192 * q^48 - 339 * q^49 - 216 * q^51 + 176 * q^52 + 318 * q^53 + 55 * q^55 - 366 * q^57 + 660 * q^59 - 120 * q^60 - 430 * q^61 + 18 * q^63 - 512 * q^64 + 110 * q^65 + 380 * q^67 - 576 * q^68 - 216 * q^69 + 168 * q^71 + 218 * q^73 - 75 * q^75 - 976 * q^76 - 22 * q^77 - 706 * q^79 - 320 * q^80 + 81 * q^81 + 1068 * q^83 + 48 * q^84 - 360 * q^85 - 288 * q^87 - 6 * q^89 - 44 * q^91 - 576 * q^92 + 336 * q^93 - 610 * q^95 + 686 * q^97 - 99 * 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
0 −3.00000 −8.00000 −5.00000 0 2.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$11$$ $$1$$

## 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 165.4.a.a 1
3.b odd 2 1 495.4.a.c 1
5.b even 2 1 825.4.a.e 1
5.c odd 4 2 825.4.c.g 2
11.b odd 2 1 1815.4.a.f 1
15.d odd 2 1 2475.4.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.a 1 1.a even 1 1 trivial
495.4.a.c 1 3.b odd 2 1
825.4.a.e 1 5.b even 2 1
825.4.c.g 2 5.c odd 4 2
1815.4.a.f 1 11.b odd 2 1
2475.4.a.f 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T - 2$$
$11$ $$T + 11$$
$13$ $$T + 22$$
$17$ $$T - 72$$
$19$ $$T - 122$$
$23$ $$T - 72$$
$29$ $$T - 96$$
$31$ $$T + 112$$
$37$ $$T - 266$$
$41$ $$T + 96$$
$43$ $$T + 382$$
$47$ $$T - 360$$
$53$ $$T - 318$$
$59$ $$T - 660$$
$61$ $$T + 430$$
$67$ $$T - 380$$
$71$ $$T - 168$$
$73$ $$T - 218$$
$79$ $$T + 706$$
$83$ $$T - 1068$$
$89$ $$T + 6$$
$97$ $$T - 686$$