# Properties

 Label 1452.4.a.q Level $1452$ Weight $4$ Character orbit 1452.a Self dual yes Analytic conductor $85.671$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# 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: $$4$$ Coefficient field: 4.4.20959101.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 90x^{2} + 91x + 2026$$ x^4 - 2*x^3 - 90*x^2 + 91*x + 2026 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (\beta_{3} + 3) q^{5} - \beta_1 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (b3 + 3) * q^5 - b1 * q^7 + 9 * q^9 $$q + 3 q^{3} + (\beta_{3} + 3) q^{5} - \beta_1 q^{7} + 9 q^{9} + \beta_{2} q^{13} + (3 \beta_{3} + 9) q^{15} + ( - \beta_{2} - \beta_1) q^{17} + (\beta_{2} + 3 \beta_1) q^{19} - 3 \beta_1 q^{21} + (11 \beta_{3} + 39) q^{23} + (6 \beta_{3} + 61) q^{25} + 27 q^{27} + (\beta_{2} - 5 \beta_1) q^{29} + ( - 18 \beta_{3} + 46) q^{31} + ( - 4 \beta_{2} - 10 \beta_1) q^{35} + ( - 6 \beta_{3} - 44) q^{37} + 3 \beta_{2} q^{39} + ( - 3 \beta_{2} + 9 \beta_1) q^{41} + (\beta_{2} - 11 \beta_1) q^{43} + (9 \beta_{3} + 27) q^{45} + (21 \beta_{3} + 213) q^{47} + (30 \beta_{3} + 395) q^{49} + ( - 3 \beta_{2} - 3 \beta_1) q^{51} + ( - 15 \beta_{3} + 231) q^{53} + (3 \beta_{2} + 9 \beta_1) q^{57} + (22 \beta_{3} - 90) q^{59} + (\beta_{2} - 10 \beta_1) q^{61} - 9 \beta_1 q^{63} + ( - 4 \beta_{2} + 32 \beta_1) q^{65} + (24 \beta_{3} - 44) q^{67} + (33 \beta_{3} + 117) q^{69} + ( - 13 \beta_{3} + 819) q^{71} + ( - 2 \beta_{2} + 2 \beta_1) q^{73} + (18 \beta_{3} + 183) q^{75} + (2 \beta_{2} + 27 \beta_1) q^{79} + 81 q^{81} + ( - 4 \beta_{2} + 38 \beta_1) q^{83} - 42 \beta_1 q^{85} + (3 \beta_{2} - 15 \beta_1) q^{87} + ( - 20 \beta_{3} - 786) q^{89} + ( - 132 \beta_{3} - 36) q^{91} + ( - 54 \beta_{3} + 138) q^{93} + (8 \beta_{2} + 62 \beta_1) q^{95} + ( - 30 \beta_{3} + 692) q^{97}+O(q^{100})$$ q + 3 * q^3 + (b3 + 3) * q^5 - b1 * q^7 + 9 * q^9 + b2 * q^13 + (3*b3 + 9) * q^15 + (-b2 - b1) * q^17 + (b2 + 3*b1) * q^19 - 3*b1 * q^21 + (11*b3 + 39) * q^23 + (6*b3 + 61) * q^25 + 27 * q^27 + (b2 - 5*b1) * q^29 + (-18*b3 + 46) * q^31 + (-4*b2 - 10*b1) * q^35 + (-6*b3 - 44) * q^37 + 3*b2 * q^39 + (-3*b2 + 9*b1) * q^41 + (b2 - 11*b1) * q^43 + (9*b3 + 27) * q^45 + (21*b3 + 213) * q^47 + (30*b3 + 395) * q^49 + (-3*b2 - 3*b1) * q^51 + (-15*b3 + 231) * q^53 + (3*b2 + 9*b1) * q^57 + (22*b3 - 90) * q^59 + (b2 - 10*b1) * q^61 - 9*b1 * q^63 + (-4*b2 + 32*b1) * q^65 + (24*b3 - 44) * q^67 + (33*b3 + 117) * q^69 + (-13*b3 + 819) * q^71 + (-2*b2 + 2*b1) * q^73 + (18*b3 + 183) * q^75 + (2*b2 + 27*b1) * q^79 + 81 * q^81 + (-4*b2 + 38*b1) * q^83 - 42*b1 * q^85 + (3*b2 - 15*b1) * q^87 + (-20*b3 - 786) * q^89 + (-132*b3 - 36) * q^91 + (-54*b3 + 138) * q^93 + (8*b2 + 62*b1) * q^95 + (-30*b3 + 692) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 $$4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100})$$ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 + 36 * q^15 + 156 * q^23 + 244 * q^25 + 108 * q^27 + 184 * q^31 - 176 * q^37 + 108 * q^45 + 852 * q^47 + 1580 * q^49 + 924 * q^53 - 360 * q^59 - 176 * q^67 + 468 * q^69 + 3276 * q^71 + 732 * q^75 + 324 * q^81 - 3144 * q^89 - 144 * q^91 + 552 * q^93 + 2768 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 90x^{2} + 91x + 2026$$ :

 $$\beta_{1}$$ $$=$$ $$( 4\nu^{3} - 6\nu^{2} - 174\nu + 88 ) / 7$$ (4*v^3 - 6*v^2 - 174*v + 88) / 7 $$\beta_{2}$$ $$=$$ $$( -4\nu^{3} + 6\nu^{2} + 258\nu - 130 ) / 7$$ (-4*v^3 + 6*v^2 + 258*v - 130) / 7 $$\beta_{3}$$ $$=$$ $$2\nu^{2} - 2\nu - 91$$ 2*v^2 - 2*v - 91
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 6 ) / 12$$ (b2 + b1 + 6) / 12 $$\nu^{2}$$ $$=$$ $$( 6\beta_{3} + \beta_{2} + \beta _1 + 552 ) / 12$$ (6*b3 + b2 + b1 + 552) / 12 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + 15\beta_{2} + 22\beta _1 + 275 ) / 4$$ (3*b3 + 15*b2 + 22*b1 + 275) / 4

## 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
 −5.75283 6.75283 7.73893 −6.73893
0 3.00000 0 −10.3041 0 −18.4086 0 9.00000 0
1.2 0 3.00000 0 −10.3041 0 18.4086 0 9.00000 0
1.3 0 3.00000 0 16.3041 0 −33.7213 0 9.00000 0
1.4 0 3.00000 0 16.3041 0 33.7213 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.4.a.q 4
11.b odd 2 1 inner 1452.4.a.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.4.a.q 4 1.a even 1 1 trivial
1452.4.a.q 4 11.b odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} - 6T_{5} - 168$$ T5^2 - 6*T5 - 168 $$T_{7}^{4} - 1476T_{7}^{2} + 385344$$ T7^4 - 1476*T7^2 + 385344

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 3)^{4}$$
$5$ $$(T^{2} - 6 T - 168)^{2}$$
$7$ $$T^{4} - 1476 T^{2} + 385344$$
$11$ $$T^{4}$$
$13$ $$T^{4} - 11556 T^{2} + 24662016$$
$17$ $$T^{4} - 13176 T^{2} + 42484176$$
$19$ $$T^{4} - 25272 T^{2} + 34777296$$
$23$ $$(T^{2} - 78 T - 19896)^{2}$$
$29$ $$T^{4} - 47736 T^{2} + 458655696$$
$31$ $$(T^{2} - 92 T - 55232)^{2}$$
$37$ $$(T^{2} + 88 T - 4436)^{2}$$
$41$ $$T^{4} + \cdots + 4127901264$$
$43$ $$T^{4} + \cdots + 8844511824$$
$47$ $$(T^{2} - 426 T - 32688)^{2}$$
$53$ $$(T^{2} - 462 T + 13536)^{2}$$
$59$ $$(T^{2} + 180 T - 77568)^{2}$$
$61$ $$T^{4} + \cdots + 6215213376$$
$67$ $$(T^{2} + 88 T - 100016)^{2}$$
$71$ $$(T^{2} - 1638 T + 640848)^{2}$$
$73$ $$T^{4} - 51552 T^{2} + 75527424$$
$79$ $$T^{4} + \cdots + 99443414016$$
$83$ $$T^{4} + \cdots + 1315989835776$$
$89$ $$(T^{2} + 1572 T + 546996)^{2}$$
$97$ $$(T^{2} - 1384 T + 319564)^{2}$$