# Properties

 Label 1452.4.a.b Level $1452$ Weight $4$ Character orbit 1452.a Self dual yes Analytic conductor $85.671$ 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] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, 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("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1452 = 2^{2} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1452.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: $$85.6707733283$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 132) 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 2 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 2 * q^7 + 9 * q^9 $$q - 3 q^{3} - 2 q^{7} + 9 q^{9} + 88 q^{13} + 66 q^{17} + 40 q^{19} + 6 q^{21} + 6 q^{23} - 125 q^{25} - 27 q^{27} + 54 q^{29} + 8 q^{31} - 106 q^{37} - 264 q^{39} - 354 q^{41} + 124 q^{43} + 546 q^{47} - 339 q^{49} - 198 q^{51} - 408 q^{53} - 120 q^{57} + 552 q^{59} - 404 q^{61} - 18 q^{63} - 4 q^{67} - 18 q^{69} + 126 q^{71} + 166 q^{73} + 375 q^{75} + 874 q^{79} + 81 q^{81} - 444 q^{83} - 162 q^{87} + 1002 q^{89} - 176 q^{91} - 24 q^{93} - 802 q^{97}+O(q^{100})$$ q - 3 * q^3 - 2 * q^7 + 9 * q^9 + 88 * q^13 + 66 * q^17 + 40 * q^19 + 6 * q^21 + 6 * q^23 - 125 * q^25 - 27 * q^27 + 54 * q^29 + 8 * q^31 - 106 * q^37 - 264 * q^39 - 354 * q^41 + 124 * q^43 + 546 * q^47 - 339 * q^49 - 198 * q^51 - 408 * q^53 - 120 * q^57 + 552 * q^59 - 404 * q^61 - 18 * q^63 - 4 * q^67 - 18 * q^69 + 126 * q^71 + 166 * q^73 + 375 * q^75 + 874 * q^79 + 81 * q^81 - 444 * q^83 - 162 * q^87 + 1002 * q^89 - 176 * q^91 - 24 * q^93 - 802 * q^97

## 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 0 0 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
$$2$$ $$-1$$
$$3$$ $$+1$$
$$11$$ $$-1$$

## 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 1452.4.a.b 1
11.b odd 2 1 132.4.a.b 1
33.d even 2 1 396.4.a.d 1
44.c even 2 1 528.4.a.i 1
88.b odd 2 1 2112.4.a.t 1
88.g even 2 1 2112.4.a.f 1
132.d odd 2 1 1584.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.a.b 1 11.b odd 2 1
396.4.a.d 1 33.d even 2 1
528.4.a.i 1 44.c even 2 1
1452.4.a.b 1 1.a even 1 1 trivial
1584.4.a.j 1 132.d odd 2 1
2112.4.a.f 1 88.g even 2 1
2112.4.a.t 1 88.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1452))$$:

 $$T_{5}$$ T5 $$T_{7} + 2$$ T7 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T$$
$13$ $$T - 88$$
$17$ $$T - 66$$
$19$ $$T - 40$$
$23$ $$T - 6$$
$29$ $$T - 54$$
$31$ $$T - 8$$
$37$ $$T + 106$$
$41$ $$T + 354$$
$43$ $$T - 124$$
$47$ $$T - 546$$
$53$ $$T + 408$$
$59$ $$T - 552$$
$61$ $$T + 404$$
$67$ $$T + 4$$
$71$ $$T - 126$$
$73$ $$T - 166$$
$79$ $$T - 874$$
$83$ $$T + 444$$
$89$ $$T - 1002$$
$97$ $$T + 802$$