# Properties

 Label 1452.4.a.c 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 22 q^{5} + 20 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 22 * q^5 + 20 * q^7 + 9 * q^9 $$q - 3 q^{3} + 22 q^{5} + 20 q^{7} + 9 q^{9} - 22 q^{13} - 66 q^{15} - 110 q^{17} - 48 q^{19} - 60 q^{21} + 72 q^{23} + 359 q^{25} - 27 q^{27} + 142 q^{29} + 184 q^{31} + 440 q^{35} - 194 q^{37} + 66 q^{39} + 482 q^{41} + 80 q^{43} + 198 q^{45} + 392 q^{47} + 57 q^{49} + 330 q^{51} - 34 q^{53} + 144 q^{57} - 108 q^{59} - 382 q^{61} + 180 q^{63} - 484 q^{65} + 84 q^{67} - 216 q^{69} - 1040 q^{71} + 606 q^{73} - 1077 q^{75} + 1292 q^{79} + 81 q^{81} - 356 q^{83} - 2420 q^{85} - 426 q^{87} - 406 q^{89} - 440 q^{91} - 552 q^{93} - 1056 q^{95} + 1090 q^{97}+O(q^{100})$$ q - 3 * q^3 + 22 * q^5 + 20 * q^7 + 9 * q^9 - 22 * q^13 - 66 * q^15 - 110 * q^17 - 48 * q^19 - 60 * q^21 + 72 * q^23 + 359 * q^25 - 27 * q^27 + 142 * q^29 + 184 * q^31 + 440 * q^35 - 194 * q^37 + 66 * q^39 + 482 * q^41 + 80 * q^43 + 198 * q^45 + 392 * q^47 + 57 * q^49 + 330 * q^51 - 34 * q^53 + 144 * q^57 - 108 * q^59 - 382 * q^61 + 180 * q^63 - 484 * q^65 + 84 * q^67 - 216 * q^69 - 1040 * q^71 + 606 * q^73 - 1077 * q^75 + 1292 * q^79 + 81 * q^81 - 356 * q^83 - 2420 * q^85 - 426 * q^87 - 406 * q^89 - 440 * q^91 - 552 * q^93 - 1056 * q^95 + 1090 * 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 22.0000 0 20.0000 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.c 1
11.b odd 2 1 132.4.a.c 1
33.d even 2 1 396.4.a.a 1
44.c even 2 1 528.4.a.l 1
88.b odd 2 1 2112.4.a.n 1
88.g even 2 1 2112.4.a.a 1
132.d odd 2 1 1584.4.a.a 1

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

 $$T_{5} - 22$$ T5 - 22 $$T_{7} - 20$$ T7 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 22$$
$7$ $$T - 20$$
$11$ $$T$$
$13$ $$T + 22$$
$17$ $$T + 110$$
$19$ $$T + 48$$
$23$ $$T - 72$$
$29$ $$T - 142$$
$31$ $$T - 184$$
$37$ $$T + 194$$
$41$ $$T - 482$$
$43$ $$T - 80$$
$47$ $$T - 392$$
$53$ $$T + 34$$
$59$ $$T + 108$$
$61$ $$T + 382$$
$67$ $$T - 84$$
$71$ $$T + 1040$$
$73$ $$T - 606$$
$79$ $$T - 1292$$
$83$ $$T + 356$$
$89$ $$T + 406$$
$97$ $$T - 1090$$