# Properties

 Label 528.4.a.s Level $528$ Weight $4$ Character orbit 528.a Self dual yes Analytic conductor $31.153$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,4,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("528.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 528.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: $$31.1530084830$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.123209.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 66x - 136$$ x^3 - x^2 - 66*x - 136 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 264) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} - \beta_1 - 10) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b1 + 1) * q^5 + (-b2 - b1 - 10) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta_1 + 1) q^{5} + ( - \beta_{2} - \beta_1 - 10) q^{7} + 9 q^{9} - 11 q^{11} + ( - 2 \beta_{2} - \beta_1 + 15) q^{13} + (3 \beta_1 - 3) q^{15} + ( - \beta_{2} - 6 \beta_1 - 1) q^{17} + (\beta_{2} - 6 \beta_1 - 53) q^{19} + (3 \beta_{2} + 3 \beta_1 + 30) q^{21} + (6 \beta_{2} - 7 \beta_1 - 53) q^{23} + (4 \beta_{2} + 6 \beta_1 + 57) q^{25} - 27 q^{27} + ( - \beta_{2} + 2 \beta_1 + 75) q^{29} + ( - 12 \beta_{2} - 4 \beta_1 - 48) q^{31} + 33 q^{33} + ( - 6 \beta_{2} + 20 \beta_1 + 102) q^{35} + ( - 10 \beta_{2} + 4 \beta_1 + 160) q^{37} + (6 \beta_{2} + 3 \beta_1 - 45) q^{39} + (13 \beta_{2} + 8 \beta_1 + 115) q^{41} + (15 \beta_{2} + 16 \beta_1 - 25) q^{43} + ( - 9 \beta_1 + 9) q^{45} + (2 \beta_{2} - 9 \beta_1 + 85) q^{47} + (2 \beta_{2} + 10 \beta_1 + 229) q^{49} + (3 \beta_{2} + 18 \beta_1 + 3) q^{51} + (2 \beta_{2} - 13 \beta_1 + 95) q^{53} + (11 \beta_1 - 11) q^{55} + ( - 3 \beta_{2} + 18 \beta_1 + 159) q^{57} + ( - 4 \beta_{2} - 22 \beta_1 + 122) q^{59} + ( - 12 \beta_{2} - 47 \beta_1 + 215) q^{61} + ( - 9 \beta_{2} - 9 \beta_1 - 90) q^{63} + ( - 16 \beta_{2} - 2 \beta_1 + 58) q^{65} + ( - 12 \beta_{2} + 40 \beta_1 + 272) q^{67} + ( - 18 \beta_{2} + 21 \beta_1 + 159) q^{69} + ( - 14 \beta_{2} - 75 \beta_1 + 259) q^{71} + (26 \beta_{2} + 16 \beta_1 + 492) q^{73} + ( - 12 \beta_{2} - 18 \beta_1 - 171) q^{75} + (11 \beta_{2} + 11 \beta_1 + 110) q^{77} + (31 \beta_{2} - 23 \beta_1) q^{79} + 81 q^{81} + ( - 6 \beta_{2} + 16 \beta_1 + 774) q^{83} + (14 \beta_{2} + 46 \beta_1 + 1016) q^{85} + (3 \beta_{2} - 6 \beta_1 - 225) q^{87} + ( - 28 \beta_{2} - 24 \beta_1 - 602) q^{89} + ( - 26 \beta_{2} - 36 \beta_1 + 682) q^{91} + (36 \beta_{2} + 12 \beta_1 + 144) q^{93} + (34 \beta_{2} + 92 \beta_1 + 1102) q^{95} + ( - 18 \beta_{2} - 58 \beta_1 + 166) q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b1 + 1) * q^5 + (-b2 - b1 - 10) * q^7 + 9 * q^9 - 11 * q^11 + (-2*b2 - b1 + 15) * q^13 + (3*b1 - 3) * q^15 + (-b2 - 6*b1 - 1) * q^17 + (b2 - 6*b1 - 53) * q^19 + (3*b2 + 3*b1 + 30) * q^21 + (6*b2 - 7*b1 - 53) * q^23 + (4*b2 + 6*b1 + 57) * q^25 - 27 * q^27 + (-b2 + 2*b1 + 75) * q^29 + (-12*b2 - 4*b1 - 48) * q^31 + 33 * q^33 + (-6*b2 + 20*b1 + 102) * q^35 + (-10*b2 + 4*b1 + 160) * q^37 + (6*b2 + 3*b1 - 45) * q^39 + (13*b2 + 8*b1 + 115) * q^41 + (15*b2 + 16*b1 - 25) * q^43 + (-9*b1 + 9) * q^45 + (2*b2 - 9*b1 + 85) * q^47 + (2*b2 + 10*b1 + 229) * q^49 + (3*b2 + 18*b1 + 3) * q^51 + (2*b2 - 13*b1 + 95) * q^53 + (11*b1 - 11) * q^55 + (-3*b2 + 18*b1 + 159) * q^57 + (-4*b2 - 22*b1 + 122) * q^59 + (-12*b2 - 47*b1 + 215) * q^61 + (-9*b2 - 9*b1 - 90) * q^63 + (-16*b2 - 2*b1 + 58) * q^65 + (-12*b2 + 40*b1 + 272) * q^67 + (-18*b2 + 21*b1 + 159) * q^69 + (-14*b2 - 75*b1 + 259) * q^71 + (26*b2 + 16*b1 + 492) * q^73 + (-12*b2 - 18*b1 - 171) * q^75 + (11*b2 + 11*b1 + 110) * q^77 + (31*b2 - 23*b1) * q^79 + 81 * q^81 + (-6*b2 + 16*b1 + 774) * q^83 + (14*b2 + 46*b1 + 1016) * q^85 + (3*b2 - 6*b1 - 225) * q^87 + (-28*b2 - 24*b1 - 602) * q^89 + (-26*b2 - 36*b1 + 682) * q^91 + (36*b2 + 12*b1 + 144) * q^93 + (34*b2 + 92*b1 + 1102) * q^95 + (-18*b2 - 58*b1 + 166) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 9 q^{3} + 4 q^{5} - 28 q^{7} + 27 q^{9}+O(q^{10})$$ 3 * q - 9 * q^3 + 4 * q^5 - 28 * q^7 + 27 * q^9 $$3 q - 9 q^{3} + 4 q^{5} - 28 q^{7} + 27 q^{9} - 33 q^{11} + 48 q^{13} - 12 q^{15} + 4 q^{17} - 154 q^{19} + 84 q^{21} - 158 q^{23} + 161 q^{25} - 81 q^{27} + 224 q^{29} - 128 q^{31} + 99 q^{33} + 292 q^{35} + 486 q^{37} - 144 q^{39} + 324 q^{41} - 106 q^{43} + 36 q^{45} + 262 q^{47} + 675 q^{49} - 12 q^{51} + 296 q^{53} - 44 q^{55} + 462 q^{57} + 392 q^{59} + 704 q^{61} - 252 q^{63} + 192 q^{65} + 788 q^{67} + 474 q^{69} + 866 q^{71} + 1434 q^{73} - 483 q^{75} + 308 q^{77} - 8 q^{79} + 243 q^{81} + 2312 q^{83} + 2988 q^{85} - 672 q^{87} - 1754 q^{89} + 2108 q^{91} + 384 q^{93} + 3180 q^{95} + 574 q^{97} - 297 q^{99}+O(q^{100})$$ 3 * q - 9 * q^3 + 4 * q^5 - 28 * q^7 + 27 * q^9 - 33 * q^11 + 48 * q^13 - 12 * q^15 + 4 * q^17 - 154 * q^19 + 84 * q^21 - 158 * q^23 + 161 * q^25 - 81 * q^27 + 224 * q^29 - 128 * q^31 + 99 * q^33 + 292 * q^35 + 486 * q^37 - 144 * q^39 + 324 * q^41 - 106 * q^43 + 36 * q^45 + 262 * q^47 + 675 * q^49 - 12 * q^51 + 296 * q^53 - 44 * q^55 + 462 * q^57 + 392 * q^59 + 704 * q^61 - 252 * q^63 + 192 * q^65 + 788 * q^67 + 474 * q^69 + 866 * q^71 + 1434 * q^73 - 483 * q^75 + 308 * q^77 - 8 * q^79 + 243 * q^81 + 2312 * q^83 + 2988 * q^85 - 672 * q^87 - 1754 * q^89 + 2108 * q^91 + 384 * q^93 + 3180 * q^95 + 574 * q^97 - 297 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 66x - 136$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5\nu - 43$$ v^2 - 5*v - 43
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + 5\beta _1 + 91 ) / 2$$ (2*b2 + 5*b1 + 91) / 2

## 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
 9.47772 −2.33663 −6.14109
0 −3.00000 0 −16.9554 0 −27.3940 0 9.00000 0
1.2 0 −3.00000 0 6.67325 0 21.5303 0 9.00000 0
1.3 0 −3.00000 0 14.2822 0 −22.1363 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 528.4.a.s 3
3.b odd 2 1 1584.4.a.bn 3
4.b odd 2 1 264.4.a.i 3
8.b even 2 1 2112.4.a.bt 3
8.d odd 2 1 2112.4.a.bp 3
12.b even 2 1 792.4.a.n 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.i 3 4.b odd 2 1
528.4.a.s 3 1.a even 1 1 trivial
792.4.a.n 3 12.b even 2 1
1584.4.a.bn 3 3.b odd 2 1
2112.4.a.bp 3 8.d odd 2 1
2112.4.a.bt 3 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5}^{3} - 4T_{5}^{2} - 260T_{5} + 1616$$ T5^3 - 4*T5^2 - 260*T5 + 1616 $$T_{7}^{3} + 28T_{7}^{2} - 460T_{7} - 13056$$ T7^3 + 28*T7^2 - 460*T7 - 13056

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 3)^{3}$$
$5$ $$T^{3} - 4 T^{2} - 260 T + 1616$$
$7$ $$T^{3} + 28 T^{2} - 460 T - 13056$$
$11$ $$(T + 11)^{3}$$
$13$ $$T^{3} - 48 T^{2} - 1724 T - 2992$$
$17$ $$T^{3} - 4 T^{2} - 8996 T + 339408$$
$19$ $$T^{3} + 154 T^{2} - 3512 T - 376704$$
$23$ $$T^{3} + 158 T^{2} - 36800 T - 5902336$$
$29$ $$T^{3} - 224 T^{2} + 14604 T - 229664$$
$31$ $$T^{3} + 128 T^{2} - 83776 T - 9664512$$
$37$ $$T^{3} - 486 T^{2} + \cdots + 13846808$$
$41$ $$T^{3} - 324 T^{2} + \cdots + 22721952$$
$43$ $$T^{3} + 106 T^{2} + \cdots + 18380512$$
$47$ $$T^{3} - 262 T^{2} - 4864 T + 1674112$$
$53$ $$T^{3} - 296 T^{2} - 23500 T + 5201808$$
$59$ $$T^{3} - 392 T^{2} + \cdots + 29639424$$
$61$ $$T^{3} - 704 T^{2} + \cdots + 263227072$$
$67$ $$T^{3} - 788 T^{2} + \cdots + 199918912$$
$71$ $$T^{3} - 866 T^{2} + \cdots + 1016215296$$
$73$ $$T^{3} - 1434 T^{2} + \cdots + 194912296$$
$79$ $$T^{3} + 8 T^{2} - 915588 T - 315814432$$
$83$ $$T^{3} - 2312 T^{2} + \cdots - 365040384$$
$89$ $$T^{3} + 1754 T^{2} + \cdots - 261270536$$
$97$ $$T^{3} - 574 T^{2} + \cdots + 397963256$$