# Properties

 Label 528.4.a.n Level $528$ Weight $4$ Character orbit 528.a Self dual yes Analytic conductor $31.153$ 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] = [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: $$2$$ Coefficient field: $$\Q(\sqrt{97})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 24$$ x^2 - x - 24 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 66) 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 $$\beta = \sqrt{97}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta + 5) q^{5} + ( - 3 \beta + 1) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b + 5) * q^5 + (-3*b + 1) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta + 5) q^{5} + ( - 3 \beta + 1) q^{7} + 9 q^{9} + 11 q^{11} + ( - 7 \beta + 7) q^{13} + (3 \beta - 15) q^{15} + (2 \beta - 40) q^{17} + ( - 10 \beta + 30) q^{19} + (9 \beta - 3) q^{21} + (5 \beta + 101) q^{23} + ( - 10 \beta - 3) q^{25} - 27 q^{27} + (6 \beta - 168) q^{29} + (16 \beta - 64) q^{31} - 33 q^{33} + ( - 16 \beta + 296) q^{35} + (32 \beta + 94) q^{37} + (21 \beta - 21) q^{39} + ( - 24 \beta - 66) q^{41} + (8 \beta - 240) q^{43} + ( - 9 \beta + 45) q^{45} + (3 \beta + 195) q^{47} + ( - 6 \beta + 531) q^{49} + ( - 6 \beta + 120) q^{51} + (35 \beta + 305) q^{53} + ( - 11 \beta + 55) q^{55} + (30 \beta - 90) q^{57} + (18 \beta - 186) q^{59} + (7 \beta + 525) q^{61} + ( - 27 \beta + 9) q^{63} + ( - 42 \beta + 714) q^{65} + ( - 16 \beta - 204) q^{67} + ( - 15 \beta - 303) q^{69} + ( - 15 \beta - 471) q^{71} + ( - 64 \beta + 370) q^{73} + (30 \beta + 9) q^{75} + ( - 33 \beta + 11) q^{77} + (27 \beta + 823) q^{79} + 81 q^{81} + ( - 52 \beta + 176) q^{83} + (50 \beta - 394) q^{85} + ( - 18 \beta + 504) q^{87} + (16 \beta + 1018) q^{89} + ( - 28 \beta + 2044) q^{91} + ( - 48 \beta + 192) q^{93} + ( - 80 \beta + 1120) q^{95} + (70 \beta + 168) q^{97} + 99 q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b + 5) * q^5 + (-3*b + 1) * q^7 + 9 * q^9 + 11 * q^11 + (-7*b + 7) * q^13 + (3*b - 15) * q^15 + (2*b - 40) * q^17 + (-10*b + 30) * q^19 + (9*b - 3) * q^21 + (5*b + 101) * q^23 + (-10*b - 3) * q^25 - 27 * q^27 + (6*b - 168) * q^29 + (16*b - 64) * q^31 - 33 * q^33 + (-16*b + 296) * q^35 + (32*b + 94) * q^37 + (21*b - 21) * q^39 + (-24*b - 66) * q^41 + (8*b - 240) * q^43 + (-9*b + 45) * q^45 + (3*b + 195) * q^47 + (-6*b + 531) * q^49 + (-6*b + 120) * q^51 + (35*b + 305) * q^53 + (-11*b + 55) * q^55 + (30*b - 90) * q^57 + (18*b - 186) * q^59 + (7*b + 525) * q^61 + (-27*b + 9) * q^63 + (-42*b + 714) * q^65 + (-16*b - 204) * q^67 + (-15*b - 303) * q^69 + (-15*b - 471) * q^71 + (-64*b + 370) * q^73 + (30*b + 9) * q^75 + (-33*b + 11) * q^77 + (27*b + 823) * q^79 + 81 * q^81 + (-52*b + 176) * q^83 + (50*b - 394) * q^85 + (-18*b + 504) * q^87 + (16*b + 1018) * q^89 + (-28*b + 2044) * q^91 + (-48*b + 192) * q^93 + (-80*b + 1120) * q^95 + (70*b + 168) * q^97 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 10 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 10 * q^5 + 2 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 10 q^{5} + 2 q^{7} + 18 q^{9} + 22 q^{11} + 14 q^{13} - 30 q^{15} - 80 q^{17} + 60 q^{19} - 6 q^{21} + 202 q^{23} - 6 q^{25} - 54 q^{27} - 336 q^{29} - 128 q^{31} - 66 q^{33} + 592 q^{35} + 188 q^{37} - 42 q^{39} - 132 q^{41} - 480 q^{43} + 90 q^{45} + 390 q^{47} + 1062 q^{49} + 240 q^{51} + 610 q^{53} + 110 q^{55} - 180 q^{57} - 372 q^{59} + 1050 q^{61} + 18 q^{63} + 1428 q^{65} - 408 q^{67} - 606 q^{69} - 942 q^{71} + 740 q^{73} + 18 q^{75} + 22 q^{77} + 1646 q^{79} + 162 q^{81} + 352 q^{83} - 788 q^{85} + 1008 q^{87} + 2036 q^{89} + 4088 q^{91} + 384 q^{93} + 2240 q^{95} + 336 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 10 * q^5 + 2 * q^7 + 18 * q^9 + 22 * q^11 + 14 * q^13 - 30 * q^15 - 80 * q^17 + 60 * q^19 - 6 * q^21 + 202 * q^23 - 6 * q^25 - 54 * q^27 - 336 * q^29 - 128 * q^31 - 66 * q^33 + 592 * q^35 + 188 * q^37 - 42 * q^39 - 132 * q^41 - 480 * q^43 + 90 * q^45 + 390 * q^47 + 1062 * q^49 + 240 * q^51 + 610 * q^53 + 110 * q^55 - 180 * q^57 - 372 * q^59 + 1050 * q^61 + 18 * q^63 + 1428 * q^65 - 408 * q^67 - 606 * q^69 - 942 * q^71 + 740 * q^73 + 18 * q^75 + 22 * q^77 + 1646 * q^79 + 162 * q^81 + 352 * q^83 - 788 * q^85 + 1008 * q^87 + 2036 * q^89 + 4088 * q^91 + 384 * q^93 + 2240 * q^95 + 336 * q^97 + 198 * q^99

## 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
 5.42443 −4.42443
0 −3.00000 0 −4.84886 0 −28.5466 0 9.00000 0
1.2 0 −3.00000 0 14.8489 0 30.5466 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.n 2
3.b odd 2 1 1584.4.a.ba 2
4.b odd 2 1 66.4.a.c 2
8.b even 2 1 2112.4.a.bi 2
8.d odd 2 1 2112.4.a.bb 2
12.b even 2 1 198.4.a.h 2
20.d odd 2 1 1650.4.a.s 2
20.e even 4 2 1650.4.c.u 4
44.c even 2 1 726.4.a.o 2
132.d odd 2 1 2178.4.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.4.a.c 2 4.b odd 2 1
198.4.a.h 2 12.b even 2 1
528.4.a.n 2 1.a even 1 1 trivial
726.4.a.o 2 44.c even 2 1
1584.4.a.ba 2 3.b odd 2 1
1650.4.a.s 2 20.d odd 2 1
1650.4.c.u 4 20.e even 4 2
2112.4.a.bb 2 8.d odd 2 1
2112.4.a.bi 2 8.b even 2 1
2178.4.a.bf 2 132.d odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 10T_{5} - 72$$ T5^2 - 10*T5 - 72 $$T_{7}^{2} - 2T_{7} - 872$$ T7^2 - 2*T7 - 872

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - 10T - 72$$
$7$ $$T^{2} - 2T - 872$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} - 14T - 4704$$
$17$ $$T^{2} + 80T + 1212$$
$19$ $$T^{2} - 60T - 8800$$
$23$ $$T^{2} - 202T + 7776$$
$29$ $$T^{2} + 336T + 24732$$
$31$ $$T^{2} + 128T - 20736$$
$37$ $$T^{2} - 188T - 90492$$
$41$ $$T^{2} + 132T - 51516$$
$43$ $$T^{2} + 480T + 51392$$
$47$ $$T^{2} - 390T + 37152$$
$53$ $$T^{2} - 610T - 25800$$
$59$ $$T^{2} + 372T + 3168$$
$61$ $$T^{2} - 1050 T + 270872$$
$67$ $$T^{2} + 408T + 16784$$
$71$ $$T^{2} + 942T + 200016$$
$73$ $$T^{2} - 740T - 260412$$
$79$ $$T^{2} - 1646 T + 606616$$
$83$ $$T^{2} - 352T - 231312$$
$89$ $$T^{2} - 2036 T + 1011492$$
$97$ $$T^{2} - 336T - 447076$$