# Properties

 Label 528.4.a.o 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: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \sqrt{33}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + (2 \beta + 8) q^{5} + ( - \beta - 1) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (2*b + 8) * q^5 + (-b - 1) * q^7 + 9 * q^9 $$q - 3 q^{3} + (2 \beta + 8) q^{5} + ( - \beta - 1) q^{7} + 9 q^{9} - 11 q^{11} + ( - 4 \beta - 38) q^{13} + ( - 6 \beta - 24) q^{15} + ( - 15 \beta - 13) q^{17} + ( - 9 \beta + 27) q^{19} + (3 \beta + 3) q^{21} - 112 q^{23} + (32 \beta + 71) q^{25} - 27 q^{27} + ( - 23 \beta + 111) q^{29} + (52 \beta + 20) q^{31} + 33 q^{33} + ( - 10 \beta - 74) q^{35} + ( - 22 \beta - 24) q^{37} + (12 \beta + 114) q^{39} + ( - \beta - 247) q^{41} + ( - 43 \beta + 33) q^{43} + (18 \beta + 72) q^{45} + ( - 24 \beta + 32) q^{47} + (2 \beta - 309) q^{49} + (45 \beta + 39) q^{51} + (64 \beta - 42) q^{53} + ( - 22 \beta - 88) q^{55} + (27 \beta - 81) q^{57} - 196 q^{59} + (26 \beta - 552) q^{61} + ( - 9 \beta - 9) q^{63} + ( - 108 \beta - 568) q^{65} + (76 \beta - 464) q^{67} + 336 q^{69} + (92 \beta - 228) q^{71} + (126 \beta - 296) q^{73} + ( - 96 \beta - 213) q^{75} + (11 \beta + 11) q^{77} + ( - 37 \beta + 115) q^{79} + 81 q^{81} + ( - 162 \beta - 174) q^{83} + ( - 146 \beta - 1094) q^{85} + (69 \beta - 333) q^{87} + ( - 4 \beta + 486) q^{89} + (42 \beta + 170) q^{91} + ( - 156 \beta - 60) q^{93} + ( - 18 \beta - 378) q^{95} + ( - 210 \beta - 592) q^{97} - 99 q^{99}+O(q^{100})$$ q - 3 * q^3 + (2*b + 8) * q^5 + (-b - 1) * q^7 + 9 * q^9 - 11 * q^11 + (-4*b - 38) * q^13 + (-6*b - 24) * q^15 + (-15*b - 13) * q^17 + (-9*b + 27) * q^19 + (3*b + 3) * q^21 - 112 * q^23 + (32*b + 71) * q^25 - 27 * q^27 + (-23*b + 111) * q^29 + (52*b + 20) * q^31 + 33 * q^33 + (-10*b - 74) * q^35 + (-22*b - 24) * q^37 + (12*b + 114) * q^39 + (-b - 247) * q^41 + (-43*b + 33) * q^43 + (18*b + 72) * q^45 + (-24*b + 32) * q^47 + (2*b - 309) * q^49 + (45*b + 39) * q^51 + (64*b - 42) * q^53 + (-22*b - 88) * q^55 + (27*b - 81) * q^57 - 196 * q^59 + (26*b - 552) * q^61 + (-9*b - 9) * q^63 + (-108*b - 568) * q^65 + (76*b - 464) * q^67 + 336 * q^69 + (92*b - 228) * q^71 + (126*b - 296) * q^73 + (-96*b - 213) * q^75 + (11*b + 11) * q^77 + (-37*b + 115) * q^79 + 81 * q^81 + (-162*b - 174) * q^83 + (-146*b - 1094) * q^85 + (69*b - 333) * q^87 + (-4*b + 486) * q^89 + (42*b + 170) * q^91 + (-156*b - 60) * q^93 + (-18*b - 378) * q^95 + (-210*b - 592) * q^97 - 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} + 16 q^{5} - 2 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 + 16 * q^5 - 2 * q^7 + 18 * q^9 $$2 q - 6 q^{3} + 16 q^{5} - 2 q^{7} + 18 q^{9} - 22 q^{11} - 76 q^{13} - 48 q^{15} - 26 q^{17} + 54 q^{19} + 6 q^{21} - 224 q^{23} + 142 q^{25} - 54 q^{27} + 222 q^{29} + 40 q^{31} + 66 q^{33} - 148 q^{35} - 48 q^{37} + 228 q^{39} - 494 q^{41} + 66 q^{43} + 144 q^{45} + 64 q^{47} - 618 q^{49} + 78 q^{51} - 84 q^{53} - 176 q^{55} - 162 q^{57} - 392 q^{59} - 1104 q^{61} - 18 q^{63} - 1136 q^{65} - 928 q^{67} + 672 q^{69} - 456 q^{71} - 592 q^{73} - 426 q^{75} + 22 q^{77} + 230 q^{79} + 162 q^{81} - 348 q^{83} - 2188 q^{85} - 666 q^{87} + 972 q^{89} + 340 q^{91} - 120 q^{93} - 756 q^{95} - 1184 q^{97} - 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 + 16 * q^5 - 2 * q^7 + 18 * q^9 - 22 * q^11 - 76 * q^13 - 48 * q^15 - 26 * q^17 + 54 * q^19 + 6 * q^21 - 224 * q^23 + 142 * q^25 - 54 * q^27 + 222 * q^29 + 40 * q^31 + 66 * q^33 - 148 * q^35 - 48 * q^37 + 228 * q^39 - 494 * q^41 + 66 * q^43 + 144 * q^45 + 64 * q^47 - 618 * q^49 + 78 * q^51 - 84 * q^53 - 176 * q^55 - 162 * q^57 - 392 * q^59 - 1104 * q^61 - 18 * q^63 - 1136 * q^65 - 928 * q^67 + 672 * q^69 - 456 * q^71 - 592 * q^73 - 426 * q^75 + 22 * q^77 + 230 * q^79 + 162 * q^81 - 348 * q^83 - 2188 * q^85 - 666 * q^87 + 972 * q^89 + 340 * q^91 - 120 * q^93 - 756 * q^95 - 1184 * 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
 −2.37228 3.37228
0 −3.00000 0 −3.48913 0 4.74456 0 9.00000 0
1.2 0 −3.00000 0 19.4891 0 −6.74456 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.o 2
3.b odd 2 1 1584.4.a.x 2
4.b odd 2 1 33.4.a.d 2
8.b even 2 1 2112.4.a.bh 2
8.d odd 2 1 2112.4.a.ba 2
12.b even 2 1 99.4.a.e 2
20.d odd 2 1 825.4.a.k 2
20.e even 4 2 825.4.c.i 4
28.d even 2 1 1617.4.a.j 2
44.c even 2 1 363.4.a.j 2
60.h even 2 1 2475.4.a.o 2
132.d odd 2 1 1089.4.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.d 2 4.b odd 2 1
99.4.a.e 2 12.b even 2 1
363.4.a.j 2 44.c even 2 1
528.4.a.o 2 1.a even 1 1 trivial
825.4.a.k 2 20.d odd 2 1
825.4.c.i 4 20.e even 4 2
1089.4.a.t 2 132.d odd 2 1
1584.4.a.x 2 3.b odd 2 1
1617.4.a.j 2 28.d even 2 1
2112.4.a.ba 2 8.d odd 2 1
2112.4.a.bh 2 8.b even 2 1
2475.4.a.o 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5}^{2} - 16T_{5} - 68$$ T5^2 - 16*T5 - 68 $$T_{7}^{2} + 2T_{7} - 32$$ T7^2 + 2*T7 - 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} - 16T - 68$$
$7$ $$T^{2} + 2T - 32$$
$11$ $$(T + 11)^{2}$$
$13$ $$T^{2} + 76T + 916$$
$17$ $$T^{2} + 26T - 7256$$
$19$ $$T^{2} - 54T - 1944$$
$23$ $$(T + 112)^{2}$$
$29$ $$T^{2} - 222T - 5136$$
$31$ $$T^{2} - 40T - 88832$$
$37$ $$T^{2} + 48T - 15396$$
$41$ $$T^{2} + 494T + 60976$$
$43$ $$T^{2} - 66T - 59928$$
$47$ $$T^{2} - 64T - 17984$$
$53$ $$T^{2} + 84T - 133404$$
$59$ $$(T + 196)^{2}$$
$61$ $$T^{2} + 1104 T + 282396$$
$67$ $$T^{2} + 928T + 24688$$
$71$ $$T^{2} + 456T - 227328$$
$73$ $$T^{2} + 592T - 436292$$
$79$ $$T^{2} - 230T - 31952$$
$83$ $$T^{2} + 348T - 835776$$
$89$ $$T^{2} - 972T + 235668$$
$97$ $$T^{2} + 1184 T - 1104836$$