# Properties

 Label 528.4.a.p Level $528$ Weight $4$ Character orbit 528.a Self dual yes Analytic conductor $31.153$ Analytic rank $1$ 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: $$1$$ 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 33) 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta - 7) q^{5} + (2 \beta - 12) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b - 7) * q^5 + (2*b - 12) * q^7 + 9 * q^9 $$q + 3 q^{3} + ( - \beta - 7) q^{5} + (2 \beta - 12) q^{7} + 9 q^{9} + 11 q^{11} + (\beta + 15) q^{13} + ( - 3 \beta - 21) q^{15} + ( - 7 \beta + 53) q^{17} + (\beta - 25) q^{19} + (6 \beta - 36) q^{21} + (5 \beta - 67) q^{23} + (14 \beta + 21) q^{25} + 27 q^{27} + ( - 3 \beta - 99) q^{29} + ( - 4 \beta - 180) q^{31} + 33 q^{33} + ( - 2 \beta - 110) q^{35} + (26 \beta - 164) q^{37} + (3 \beta + 45) q^{39} + ( - 7 \beta - 391) q^{41} + (13 \beta - 193) q^{43} + ( - 9 \beta - 63) q^{45} + ( - 37 \beta - 133) q^{47} + ( - 48 \beta + 189) q^{49} + ( - 21 \beta + 159) q^{51} + ( - 27 \beta - 261) q^{53} + ( - 11 \beta - 77) q^{55} + (3 \beta - 75) q^{57} + (50 \beta + 86) q^{59} + (17 \beta - 389) q^{61} + (18 \beta - 108) q^{63} + ( - 22 \beta - 202) q^{65} + (48 \beta + 388) q^{67} + (15 \beta - 201) q^{69} + ( - 27 \beta - 315) q^{71} + ( - 14 \beta + 648) q^{73} + (42 \beta + 63) q^{75} + (22 \beta - 132) q^{77} + ( - 72 \beta - 326) q^{79} + 81 q^{81} + ( - 78 \beta + 162) q^{83} + ( - 4 \beta + 308) q^{85} + ( - 9 \beta - 297) q^{87} + (36 \beta - 378) q^{89} + (18 \beta + 14) q^{91} + ( - 12 \beta - 540) q^{93} + (18 \beta + 78) q^{95} + (96 \beta - 226) q^{97} + 99 q^{99}+O(q^{100})$$ q + 3 * q^3 + (-b - 7) * q^5 + (2*b - 12) * q^7 + 9 * q^9 + 11 * q^11 + (b + 15) * q^13 + (-3*b - 21) * q^15 + (-7*b + 53) * q^17 + (b - 25) * q^19 + (6*b - 36) * q^21 + (5*b - 67) * q^23 + (14*b + 21) * q^25 + 27 * q^27 + (-3*b - 99) * q^29 + (-4*b - 180) * q^31 + 33 * q^33 + (-2*b - 110) * q^35 + (26*b - 164) * q^37 + (3*b + 45) * q^39 + (-7*b - 391) * q^41 + (13*b - 193) * q^43 + (-9*b - 63) * q^45 + (-37*b - 133) * q^47 + (-48*b + 189) * q^49 + (-21*b + 159) * q^51 + (-27*b - 261) * q^53 + (-11*b - 77) * q^55 + (3*b - 75) * q^57 + (50*b + 86) * q^59 + (17*b - 389) * q^61 + (18*b - 108) * q^63 + (-22*b - 202) * q^65 + (48*b + 388) * q^67 + (15*b - 201) * q^69 + (-27*b - 315) * q^71 + (-14*b + 648) * q^73 + (42*b + 63) * q^75 + (22*b - 132) * q^77 + (-72*b - 326) * q^79 + 81 * q^81 + (-78*b + 162) * q^83 + (-4*b + 308) * q^85 + (-9*b - 297) * q^87 + (36*b - 378) * q^89 + (18*b + 14) * q^91 + (-12*b - 540) * q^93 + (18*b + 78) * q^95 + (96*b - 226) * q^97 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 - 14 * q^5 - 24 * q^7 + 18 * q^9 $$2 q + 6 q^{3} - 14 q^{5} - 24 q^{7} + 18 q^{9} + 22 q^{11} + 30 q^{13} - 42 q^{15} + 106 q^{17} - 50 q^{19} - 72 q^{21} - 134 q^{23} + 42 q^{25} + 54 q^{27} - 198 q^{29} - 360 q^{31} + 66 q^{33} - 220 q^{35} - 328 q^{37} + 90 q^{39} - 782 q^{41} - 386 q^{43} - 126 q^{45} - 266 q^{47} + 378 q^{49} + 318 q^{51} - 522 q^{53} - 154 q^{55} - 150 q^{57} + 172 q^{59} - 778 q^{61} - 216 q^{63} - 404 q^{65} + 776 q^{67} - 402 q^{69} - 630 q^{71} + 1296 q^{73} + 126 q^{75} - 264 q^{77} - 652 q^{79} + 162 q^{81} + 324 q^{83} + 616 q^{85} - 594 q^{87} - 756 q^{89} + 28 q^{91} - 1080 q^{93} + 156 q^{95} - 452 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 - 14 * q^5 - 24 * q^7 + 18 * q^9 + 22 * q^11 + 30 * q^13 - 42 * q^15 + 106 * q^17 - 50 * q^19 - 72 * q^21 - 134 * q^23 + 42 * q^25 + 54 * q^27 - 198 * q^29 - 360 * q^31 + 66 * q^33 - 220 * q^35 - 328 * q^37 + 90 * q^39 - 782 * q^41 - 386 * q^43 - 126 * q^45 - 266 * q^47 + 378 * q^49 + 318 * q^51 - 522 * q^53 - 154 * q^55 - 150 * q^57 + 172 * q^59 - 778 * q^61 - 216 * q^63 - 404 * q^65 + 776 * q^67 - 402 * q^69 - 630 * q^71 + 1296 * q^73 + 126 * q^75 - 264 * q^77 - 652 * q^79 + 162 * q^81 + 324 * q^83 + 616 * q^85 - 594 * q^87 - 756 * q^89 + 28 * q^91 - 1080 * q^93 + 156 * q^95 - 452 * 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 −16.8489 0 7.69772 0 9.00000 0
1.2 0 3.00000 0 2.84886 0 −31.6977 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.p 2
3.b odd 2 1 1584.4.a.bj 2
4.b odd 2 1 33.4.a.c 2
8.b even 2 1 2112.4.a.bg 2
8.d odd 2 1 2112.4.a.bn 2
12.b even 2 1 99.4.a.f 2
20.d odd 2 1 825.4.a.l 2
20.e even 4 2 825.4.c.h 4
28.d even 2 1 1617.4.a.k 2
44.c even 2 1 363.4.a.i 2
60.h even 2 1 2475.4.a.p 2
132.d odd 2 1 1089.4.a.u 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 4.b odd 2 1
99.4.a.f 2 12.b even 2 1
363.4.a.i 2 44.c even 2 1
528.4.a.p 2 1.a even 1 1 trivial
825.4.a.l 2 20.d odd 2 1
825.4.c.h 4 20.e even 4 2
1089.4.a.u 2 132.d odd 2 1
1584.4.a.bj 2 3.b odd 2 1
1617.4.a.k 2 28.d even 2 1
2112.4.a.bg 2 8.b even 2 1
2112.4.a.bn 2 8.d odd 2 1
2475.4.a.p 2 60.h even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 14T_{5} - 48$$ T5^2 + 14*T5 - 48 $$T_{7}^{2} + 24T_{7} - 244$$ T7^2 + 24*T7 - 244

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} + 14T - 48$$
$7$ $$T^{2} + 24T - 244$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} - 30T + 128$$
$17$ $$T^{2} - 106T - 1944$$
$19$ $$T^{2} + 50T + 528$$
$23$ $$T^{2} + 134T + 2064$$
$29$ $$T^{2} + 198T + 8928$$
$31$ $$T^{2} + 360T + 30848$$
$37$ $$T^{2} + 328T - 38676$$
$41$ $$T^{2} + 782T + 148128$$
$43$ $$T^{2} + 386T + 20856$$
$47$ $$T^{2} + 266T - 115104$$
$53$ $$T^{2} + 522T - 2592$$
$59$ $$T^{2} - 172T - 235104$$
$61$ $$T^{2} + 778T + 123288$$
$67$ $$T^{2} - 776T - 72944$$
$71$ $$T^{2} + 630T + 28512$$
$73$ $$T^{2} - 1296 T + 400892$$
$79$ $$T^{2} + 652T - 396572$$
$83$ $$T^{2} - 324T - 563904$$
$89$ $$T^{2} + 756T + 17172$$
$97$ $$T^{2} + 452T - 842876$$