# Properties

 Label 165.4.a.b 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)

 $$f(q)$$ $$=$$ $$q + q^{2} + 3 q^{3} - 7 q^{4} - 5 q^{5} + 3 q^{6} + 36 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10})$$ q + q^2 + 3 * q^3 - 7 * q^4 - 5 * q^5 + 3 * q^6 + 36 * q^7 - 15 * q^8 + 9 * q^9 $$q + q^{2} + 3 q^{3} - 7 q^{4} - 5 q^{5} + 3 q^{6} + 36 q^{7} - 15 q^{8} + 9 q^{9} - 5 q^{10} + 11 q^{11} - 21 q^{12} + 2 q^{13} + 36 q^{14} - 15 q^{15} + 41 q^{16} + 66 q^{17} + 9 q^{18} + 140 q^{19} + 35 q^{20} + 108 q^{21} + 11 q^{22} - 68 q^{23} - 45 q^{24} + 25 q^{25} + 2 q^{26} + 27 q^{27} - 252 q^{28} + 150 q^{29} - 15 q^{30} - 128 q^{31} + 161 q^{32} + 33 q^{33} + 66 q^{34} - 180 q^{35} - 63 q^{36} - 314 q^{37} + 140 q^{38} + 6 q^{39} + 75 q^{40} - 118 q^{41} + 108 q^{42} + 172 q^{43} - 77 q^{44} - 45 q^{45} - 68 q^{46} - 324 q^{47} + 123 q^{48} + 953 q^{49} + 25 q^{50} + 198 q^{51} - 14 q^{52} + 82 q^{53} + 27 q^{54} - 55 q^{55} - 540 q^{56} + 420 q^{57} + 150 q^{58} - 740 q^{59} + 105 q^{60} + 122 q^{61} - 128 q^{62} + 324 q^{63} - 167 q^{64} - 10 q^{65} + 33 q^{66} - 124 q^{67} - 462 q^{68} - 204 q^{69} - 180 q^{70} - 988 q^{71} - 135 q^{72} + 2 q^{73} - 314 q^{74} + 75 q^{75} - 980 q^{76} + 396 q^{77} + 6 q^{78} + 1100 q^{79} - 205 q^{80} + 81 q^{81} - 118 q^{82} - 868 q^{83} - 756 q^{84} - 330 q^{85} + 172 q^{86} + 450 q^{87} - 165 q^{88} - 470 q^{89} - 45 q^{90} + 72 q^{91} + 476 q^{92} - 384 q^{93} - 324 q^{94} - 700 q^{95} + 483 q^{96} + 1186 q^{97} + 953 q^{98} + 99 q^{99}+O(q^{100})$$ q + q^2 + 3 * q^3 - 7 * q^4 - 5 * q^5 + 3 * q^6 + 36 * q^7 - 15 * q^8 + 9 * q^9 - 5 * q^10 + 11 * q^11 - 21 * q^12 + 2 * q^13 + 36 * q^14 - 15 * q^15 + 41 * q^16 + 66 * q^17 + 9 * q^18 + 140 * q^19 + 35 * q^20 + 108 * q^21 + 11 * q^22 - 68 * q^23 - 45 * q^24 + 25 * q^25 + 2 * q^26 + 27 * q^27 - 252 * q^28 + 150 * q^29 - 15 * q^30 - 128 * q^31 + 161 * q^32 + 33 * q^33 + 66 * q^34 - 180 * q^35 - 63 * q^36 - 314 * q^37 + 140 * q^38 + 6 * q^39 + 75 * q^40 - 118 * q^41 + 108 * q^42 + 172 * q^43 - 77 * q^44 - 45 * q^45 - 68 * q^46 - 324 * q^47 + 123 * q^48 + 953 * q^49 + 25 * q^50 + 198 * q^51 - 14 * q^52 + 82 * q^53 + 27 * q^54 - 55 * q^55 - 540 * q^56 + 420 * q^57 + 150 * q^58 - 740 * q^59 + 105 * q^60 + 122 * q^61 - 128 * q^62 + 324 * q^63 - 167 * q^64 - 10 * q^65 + 33 * q^66 - 124 * q^67 - 462 * q^68 - 204 * q^69 - 180 * q^70 - 988 * q^71 - 135 * q^72 + 2 * q^73 - 314 * q^74 + 75 * q^75 - 980 * q^76 + 396 * q^77 + 6 * q^78 + 1100 * q^79 - 205 * q^80 + 81 * q^81 - 118 * q^82 - 868 * q^83 - 756 * q^84 - 330 * q^85 + 172 * q^86 + 450 * q^87 - 165 * q^88 - 470 * q^89 - 45 * q^90 + 72 * q^91 + 476 * q^92 - 384 * q^93 - 324 * q^94 - 700 * q^95 + 483 * q^96 + 1186 * q^97 + 953 * q^98 + 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
1.00000 3.00000 −7.00000 −5.00000 3.00000 36.0000 −15.0000 9.00000 −5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(165))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T - 36$$
$11$ $$T - 11$$
$13$ $$T - 2$$
$17$ $$T - 66$$
$19$ $$T - 140$$
$23$ $$T + 68$$
$29$ $$T - 150$$
$31$ $$T + 128$$
$37$ $$T + 314$$
$41$ $$T + 118$$
$43$ $$T - 172$$
$47$ $$T + 324$$
$53$ $$T - 82$$
$59$ $$T + 740$$
$61$ $$T - 122$$
$67$ $$T + 124$$
$71$ $$T + 988$$
$73$ $$T - 2$$
$79$ $$T - 1100$$
$83$ $$T + 868$$
$89$ $$T + 470$$
$97$ $$T - 1186$$