# Properties

 Label 1452.4.a.h Level $1452$ Weight $4$ Character orbit 1452.a Self dual yes Analytic conductor $85.671$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Learn more

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: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 4 q^{5} + 19 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 4 * q^5 + 19 * q^7 + 9 * q^9 $$q + 3 q^{3} + 4 q^{5} + 19 q^{7} + 9 q^{9} - 54 q^{13} + 12 q^{15} - 4 q^{17} - q^{19} + 57 q^{21} - 214 q^{23} - 109 q^{25} + 27 q^{27} - 102 q^{29} - 225 q^{31} + 76 q^{35} + 27 q^{37} - 162 q^{39} - 174 q^{41} - 76 q^{43} + 36 q^{45} + 234 q^{47} + 18 q^{49} - 12 q^{51} - 26 q^{53} - 3 q^{57} - 354 q^{59} - 403 q^{61} + 171 q^{63} - 216 q^{65} - 317 q^{67} - 642 q^{69} - 96 q^{71} - 117 q^{73} - 327 q^{75} + 1249 q^{79} + 81 q^{81} + 946 q^{83} - 16 q^{85} - 306 q^{87} - 916 q^{89} - 1026 q^{91} - 675 q^{93} - 4 q^{95} + 541 q^{97}+O(q^{100})$$ q + 3 * q^3 + 4 * q^5 + 19 * q^7 + 9 * q^9 - 54 * q^13 + 12 * q^15 - 4 * q^17 - q^19 + 57 * q^21 - 214 * q^23 - 109 * q^25 + 27 * q^27 - 102 * q^29 - 225 * q^31 + 76 * q^35 + 27 * q^37 - 162 * q^39 - 174 * q^41 - 76 * q^43 + 36 * q^45 + 234 * q^47 + 18 * q^49 - 12 * q^51 - 26 * q^53 - 3 * q^57 - 354 * q^59 - 403 * q^61 + 171 * q^63 - 216 * q^65 - 317 * q^67 - 642 * q^69 - 96 * q^71 - 117 * q^73 - 327 * q^75 + 1249 * q^79 + 81 * q^81 + 946 * q^83 - 16 * q^85 - 306 * q^87 - 916 * q^89 - 1026 * q^91 - 675 * q^93 - 4 * q^95 + 541 * 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 4.00000 0 19.0000 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.h yes 1
11.b odd 2 1 1452.4.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.g 1 11.b odd 2 1
1452.4.a.h yes 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1452))$$:

 $$T_{5} - 4$$ T5 - 4 $$T_{7} - 19$$ T7 - 19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 4$$
$7$ $$T - 19$$
$11$ $$T$$
$13$ $$T + 54$$
$17$ $$T + 4$$
$19$ $$T + 1$$
$23$ $$T + 214$$
$29$ $$T + 102$$
$31$ $$T + 225$$
$37$ $$T - 27$$
$41$ $$T + 174$$
$43$ $$T + 76$$
$47$ $$T - 234$$
$53$ $$T + 26$$
$59$ $$T + 354$$
$61$ $$T + 403$$
$67$ $$T + 317$$
$71$ $$T + 96$$
$73$ $$T + 117$$
$79$ $$T - 1249$$
$83$ $$T - 946$$
$89$ $$T + 916$$
$97$ $$T - 541$$
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