# Properties

 Label 1452.4.a.m Level $1452$ Weight $4$ Character orbit 1452.a Self dual yes Analytic conductor $85.671$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + (11 \beta - 5) q^{5} + ( - 4 \beta + 13) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (11*b - 5) * q^5 + (-4*b + 13) * q^7 + 9 * q^9 $$q - 3 q^{3} + (11 \beta - 5) q^{5} + ( - 4 \beta + 13) q^{7} + 9 q^{9} + ( - 14 \beta + 51) q^{13} + ( - 33 \beta + 15) q^{15} + ( - 21 \beta + 16) q^{17} + (93 \beta - 85) q^{19} + (12 \beta - 39) q^{21} + 185 q^{23} + (11 \beta + 21) q^{25} - 27 q^{27} + (60 \beta - 118) q^{29} + ( - 209 \beta + 154) q^{31} + (119 \beta - 109) q^{35} + (154 \beta - 107) q^{37} + (42 \beta - 153) q^{39} + ( - 96 \beta + 235) q^{41} + (298 \beta + 5) q^{43} + (99 \beta - 45) q^{45} + (33 \beta - 388) q^{47} + ( - 88 \beta - 158) q^{49} + (63 \beta - 48) q^{51} + ( - 77 \beta + 312) q^{53} + ( - 279 \beta + 255) q^{57} + (11 \beta + 295) q^{59} + (357 \beta + 69) q^{61} + ( - 36 \beta + 117) q^{63} + (477 \beta - 409) q^{65} + (407 \beta - 575) q^{67} - 555 q^{69} + (99 \beta + 201) q^{71} + ( - 342 \beta - 456) q^{73} + ( - 33 \beta - 63) q^{75} + (336 \beta - 355) q^{79} + 81 q^{81} + ( - 266 \beta - 219) q^{83} + (50 \beta - 311) q^{85} + ( - 180 \beta + 354) q^{87} + ( - 220 \beta + 419) q^{89} + ( - 330 \beta + 719) q^{91} + (627 \beta - 462) q^{93} + ( - 377 \beta + 1448) q^{95} + (11 \beta + 151) q^{97}+O(q^{100})$$ q - 3 * q^3 + (11*b - 5) * q^5 + (-4*b + 13) * q^7 + 9 * q^9 + (-14*b + 51) * q^13 + (-33*b + 15) * q^15 + (-21*b + 16) * q^17 + (93*b - 85) * q^19 + (12*b - 39) * q^21 + 185 * q^23 + (11*b + 21) * q^25 - 27 * q^27 + (60*b - 118) * q^29 + (-209*b + 154) * q^31 + (119*b - 109) * q^35 + (154*b - 107) * q^37 + (42*b - 153) * q^39 + (-96*b + 235) * q^41 + (298*b + 5) * q^43 + (99*b - 45) * q^45 + (33*b - 388) * q^47 + (-88*b - 158) * q^49 + (63*b - 48) * q^51 + (-77*b + 312) * q^53 + (-279*b + 255) * q^57 + (11*b + 295) * q^59 + (357*b + 69) * q^61 + (-36*b + 117) * q^63 + (477*b - 409) * q^65 + (407*b - 575) * q^67 - 555 * q^69 + (99*b + 201) * q^71 + (-342*b - 456) * q^73 + (-33*b - 63) * q^75 + (336*b - 355) * q^79 + 81 * q^81 + (-266*b - 219) * q^83 + (50*b - 311) * q^85 + (-180*b + 354) * q^87 + (-220*b + 419) * q^89 + (-330*b + 719) * q^91 + (627*b - 462) * q^93 + (-377*b + 1448) * q^95 + (11*b + 151) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + q^{5} + 22 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + q^5 + 22 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + q^{5} + 22 q^{7} + 18 q^{9} + 88 q^{13} - 3 q^{15} + 11 q^{17} - 77 q^{19} - 66 q^{21} + 370 q^{23} + 53 q^{25} - 54 q^{27} - 176 q^{29} + 99 q^{31} - 99 q^{35} - 60 q^{37} - 264 q^{39} + 374 q^{41} + 308 q^{43} + 9 q^{45} - 743 q^{47} - 404 q^{49} - 33 q^{51} + 547 q^{53} + 231 q^{57} + 601 q^{59} + 495 q^{61} + 198 q^{63} - 341 q^{65} - 743 q^{67} - 1110 q^{69} + 501 q^{71} - 1254 q^{73} - 159 q^{75} - 374 q^{79} + 162 q^{81} - 704 q^{83} - 572 q^{85} + 528 q^{87} + 618 q^{89} + 1108 q^{91} - 297 q^{93} + 2519 q^{95} + 313 q^{97}+O(q^{100})$$ 2 * q - 6 * q^3 + q^5 + 22 * q^7 + 18 * q^9 + 88 * q^13 - 3 * q^15 + 11 * q^17 - 77 * q^19 - 66 * q^21 + 370 * q^23 + 53 * q^25 - 54 * q^27 - 176 * q^29 + 99 * q^31 - 99 * q^35 - 60 * q^37 - 264 * q^39 + 374 * q^41 + 308 * q^43 + 9 * q^45 - 743 * q^47 - 404 * q^49 - 33 * q^51 + 547 * q^53 + 231 * q^57 + 601 * q^59 + 495 * q^61 + 198 * q^63 - 341 * q^65 - 743 * q^67 - 1110 * q^69 + 501 * q^71 - 1254 * q^73 - 159 * q^75 - 374 * q^79 + 162 * q^81 - 704 * q^83 - 572 * q^85 + 528 * q^87 + 618 * q^89 + 1108 * q^91 - 297 * q^93 + 2519 * q^95 + 313 * 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
 −0.618034 1.61803
0 −3.00000 0 −11.7984 0 15.4721 0 9.00000 0
1.2 0 −3.00000 0 12.7984 0 6.52786 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.m 2
11.b odd 2 1 1452.4.a.l 2
11.d odd 10 2 132.4.i.a 4
33.f even 10 2 396.4.j.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.a 4 11.d odd 10 2
396.4.j.a 4 33.f even 10 2
1452.4.a.l 2 11.b odd 2 1
1452.4.a.m 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1452))$$:

 $$T_{5}^{2} - T_{5} - 151$$ T5^2 - T5 - 151 $$T_{7}^{2} - 22T_{7} + 101$$ T7^2 - 22*T7 + 101

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - T - 151$$
$7$ $$T^{2} - 22T + 101$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 88T + 1691$$
$17$ $$T^{2} - 11T - 521$$
$19$ $$T^{2} + 77T - 9329$$
$23$ $$(T - 185)^{2}$$
$29$ $$T^{2} + 176T + 3244$$
$31$ $$T^{2} - 99T - 52151$$
$37$ $$T^{2} + 60T - 28745$$
$41$ $$T^{2} - 374T + 23449$$
$43$ $$T^{2} - 308T - 87289$$
$47$ $$T^{2} + 743T + 136651$$
$53$ $$T^{2} - 547T + 67391$$
$59$ $$T^{2} - 601T + 90149$$
$61$ $$T^{2} - 495T - 98055$$
$67$ $$T^{2} + 743T - 69049$$
$71$ $$T^{2} - 501T + 50499$$
$73$ $$T^{2} + 1254 T + 246924$$
$79$ $$T^{2} + 374T - 106151$$
$83$ $$T^{2} + 704T + 35459$$
$89$ $$T^{2} - 618T + 34981$$
$97$ $$T^{2} - 313T + 24341$$