# Properties

 Label 153.4.a.d Level $153$ Weight $4$ Character orbit 153.a Self dual yes Analytic conductor $9.027$ 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] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$153 = 3^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 153.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: $$9.02729223088$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 17) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} - 6 q^{5} - 28 q^{7} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 - 6 * q^5 - 28 * q^7 - 21 * q^8 $$q + 3 q^{2} + q^{4} - 6 q^{5} - 28 q^{7} - 21 q^{8} - 18 q^{10} + 24 q^{11} - 58 q^{13} - 84 q^{14} - 71 q^{16} - 17 q^{17} + 116 q^{19} - 6 q^{20} + 72 q^{22} + 60 q^{23} - 89 q^{25} - 174 q^{26} - 28 q^{28} - 30 q^{29} - 172 q^{31} - 45 q^{32} - 51 q^{34} + 168 q^{35} - 58 q^{37} + 348 q^{38} + 126 q^{40} + 342 q^{41} - 148 q^{43} + 24 q^{44} + 180 q^{46} - 288 q^{47} + 441 q^{49} - 267 q^{50} - 58 q^{52} - 318 q^{53} - 144 q^{55} + 588 q^{56} - 90 q^{58} - 252 q^{59} + 110 q^{61} - 516 q^{62} + 433 q^{64} + 348 q^{65} - 484 q^{67} - 17 q^{68} + 504 q^{70} + 708 q^{71} + 362 q^{73} - 174 q^{74} + 116 q^{76} - 672 q^{77} - 484 q^{79} + 426 q^{80} + 1026 q^{82} - 756 q^{83} + 102 q^{85} - 444 q^{86} - 504 q^{88} + 774 q^{89} + 1624 q^{91} + 60 q^{92} - 864 q^{94} - 696 q^{95} - 382 q^{97} + 1323 q^{98}+O(q^{100})$$ q + 3 * q^2 + q^4 - 6 * q^5 - 28 * q^7 - 21 * q^8 - 18 * q^10 + 24 * q^11 - 58 * q^13 - 84 * q^14 - 71 * q^16 - 17 * q^17 + 116 * q^19 - 6 * q^20 + 72 * q^22 + 60 * q^23 - 89 * q^25 - 174 * q^26 - 28 * q^28 - 30 * q^29 - 172 * q^31 - 45 * q^32 - 51 * q^34 + 168 * q^35 - 58 * q^37 + 348 * q^38 + 126 * q^40 + 342 * q^41 - 148 * q^43 + 24 * q^44 + 180 * q^46 - 288 * q^47 + 441 * q^49 - 267 * q^50 - 58 * q^52 - 318 * q^53 - 144 * q^55 + 588 * q^56 - 90 * q^58 - 252 * q^59 + 110 * q^61 - 516 * q^62 + 433 * q^64 + 348 * q^65 - 484 * q^67 - 17 * q^68 + 504 * q^70 + 708 * q^71 + 362 * q^73 - 174 * q^74 + 116 * q^76 - 672 * q^77 - 484 * q^79 + 426 * q^80 + 1026 * q^82 - 756 * q^83 + 102 * q^85 - 444 * q^86 - 504 * q^88 + 774 * q^89 + 1624 * q^91 + 60 * q^92 - 864 * q^94 - 696 * q^95 - 382 * q^97 + 1323 * q^98

## 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
3.00000 0 1.00000 −6.00000 0 −28.0000 −21.0000 0 −18.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$17$$ $$+1$$

## 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 153.4.a.d 1
3.b odd 2 1 17.4.a.a 1
4.b odd 2 1 2448.4.a.f 1
12.b even 2 1 272.4.a.d 1
15.d odd 2 1 425.4.a.d 1
15.e even 4 2 425.4.b.c 2
21.c even 2 1 833.4.a.a 1
24.f even 2 1 1088.4.a.a 1
24.h odd 2 1 1088.4.a.l 1
33.d even 2 1 2057.4.a.d 1
51.c odd 2 1 289.4.a.a 1
51.f odd 4 2 289.4.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 3.b odd 2 1
153.4.a.d 1 1.a even 1 1 trivial
272.4.a.d 1 12.b even 2 1
289.4.a.a 1 51.c odd 2 1
289.4.b.a 2 51.f odd 4 2
425.4.a.d 1 15.d odd 2 1
425.4.b.c 2 15.e even 4 2
833.4.a.a 1 21.c even 2 1
1088.4.a.a 1 24.f even 2 1
1088.4.a.l 1 24.h odd 2 1
2057.4.a.d 1 33.d even 2 1
2448.4.a.f 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(153))$$:

 $$T_{2} - 3$$ T2 - 3 $$T_{5} + 6$$ T5 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T + 6$$
$7$ $$T + 28$$
$11$ $$T - 24$$
$13$ $$T + 58$$
$17$ $$T + 17$$
$19$ $$T - 116$$
$23$ $$T - 60$$
$29$ $$T + 30$$
$31$ $$T + 172$$
$37$ $$T + 58$$
$41$ $$T - 342$$
$43$ $$T + 148$$
$47$ $$T + 288$$
$53$ $$T + 318$$
$59$ $$T + 252$$
$61$ $$T - 110$$
$67$ $$T + 484$$
$71$ $$T - 708$$
$73$ $$T - 362$$
$79$ $$T + 484$$
$83$ $$T + 756$$
$89$ $$T - 774$$
$97$ $$T + 382$$