# Properties

 Label 528.4.a.m 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: $$\Q(\sqrt{185})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 46$$ x^2 - x - 46 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$\beta = \sqrt{185}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + ( - \beta - 3) q^{5} + (\beta - 11) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (-b - 3) * q^5 + (b - 11) * q^7 + 9 * q^9 $$q - 3 q^{3} + ( - \beta - 3) q^{5} + (\beta - 11) q^{7} + 9 q^{9} + 11 q^{11} + (\beta + 47) q^{13} + (3 \beta + 9) q^{15} + (6 \beta + 28) q^{17} + ( - 6 \beta - 38) q^{19} + ( - 3 \beta + 33) q^{21} + (5 \beta - 27) q^{23} + (6 \beta + 69) q^{25} - 27 q^{27} + (2 \beta + 52) q^{29} + ( - 8 \beta - 112) q^{31} - 33 q^{33} + (8 \beta - 152) q^{35} + ( - 16 \beta - 34) q^{37} + ( - 3 \beta - 141) q^{39} + ( - 4 \beta - 150) q^{41} + ( - 4 \beta - 228) q^{43} + ( - 9 \beta - 27) q^{45} + (27 \beta + 11) q^{47} + ( - 22 \beta - 37) q^{49} + ( - 18 \beta - 84) q^{51} + (35 \beta - 55) q^{53} + ( - 11 \beta - 33) q^{55} + (18 \beta + 114) q^{57} + ( - 22 \beta - 34) q^{59} + ( - \beta - 27) q^{61} + (9 \beta - 99) q^{63} + ( - 50 \beta - 326) q^{65} + (8 \beta - 780) q^{67} + ( - 15 \beta + 81) q^{69} + ( - 15 \beta + 137) q^{71} + (32 \beta + 274) q^{73} + ( - 18 \beta - 207) q^{75} + (11 \beta - 121) q^{77} + (23 \beta - 461) q^{79} + 81 q^{81} + ( - 44 \beta - 592) q^{83} + ( - 46 \beta - 1194) q^{85} + ( - 6 \beta - 156) q^{87} + (72 \beta - 478) q^{89} + (36 \beta - 332) q^{91} + (24 \beta + 336) q^{93} + (56 \beta + 1224) q^{95} + (70 \beta - 336) q^{97} + 99 q^{99}+O(q^{100})$$ q - 3 * q^3 + (-b - 3) * q^5 + (b - 11) * q^7 + 9 * q^9 + 11 * q^11 + (b + 47) * q^13 + (3*b + 9) * q^15 + (6*b + 28) * q^17 + (-6*b - 38) * q^19 + (-3*b + 33) * q^21 + (5*b - 27) * q^23 + (6*b + 69) * q^25 - 27 * q^27 + (2*b + 52) * q^29 + (-8*b - 112) * q^31 - 33 * q^33 + (8*b - 152) * q^35 + (-16*b - 34) * q^37 + (-3*b - 141) * q^39 + (-4*b - 150) * q^41 + (-4*b - 228) * q^43 + (-9*b - 27) * q^45 + (27*b + 11) * q^47 + (-22*b - 37) * q^49 + (-18*b - 84) * q^51 + (35*b - 55) * q^53 + (-11*b - 33) * q^55 + (18*b + 114) * q^57 + (-22*b - 34) * q^59 + (-b - 27) * q^61 + (9*b - 99) * q^63 + (-50*b - 326) * q^65 + (8*b - 780) * q^67 + (-15*b + 81) * q^69 + (-15*b + 137) * q^71 + (32*b + 274) * q^73 + (-18*b - 207) * q^75 + (11*b - 121) * q^77 + (23*b - 461) * q^79 + 81 * q^81 + (-44*b - 592) * q^83 + (-46*b - 1194) * q^85 + (-6*b - 156) * q^87 + (72*b - 478) * q^89 + (36*b - 332) * q^91 + (24*b + 336) * q^93 + (56*b + 1224) * q^95 + (70*b - 336) * q^97 + 99 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 6 q^{5} - 22 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 6 * q^5 - 22 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 6 q^{5} - 22 q^{7} + 18 q^{9} + 22 q^{11} + 94 q^{13} + 18 q^{15} + 56 q^{17} - 76 q^{19} + 66 q^{21} - 54 q^{23} + 138 q^{25} - 54 q^{27} + 104 q^{29} - 224 q^{31} - 66 q^{33} - 304 q^{35} - 68 q^{37} - 282 q^{39} - 300 q^{41} - 456 q^{43} - 54 q^{45} + 22 q^{47} - 74 q^{49} - 168 q^{51} - 110 q^{53} - 66 q^{55} + 228 q^{57} - 68 q^{59} - 54 q^{61} - 198 q^{63} - 652 q^{65} - 1560 q^{67} + 162 q^{69} + 274 q^{71} + 548 q^{73} - 414 q^{75} - 242 q^{77} - 922 q^{79} + 162 q^{81} - 1184 q^{83} - 2388 q^{85} - 312 q^{87} - 956 q^{89} - 664 q^{91} + 672 q^{93} + 2448 q^{95} - 672 q^{97} + 198 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 6 * q^5 - 22 * q^7 + 18 * q^9 + 22 * q^11 + 94 * q^13 + 18 * q^15 + 56 * q^17 - 76 * q^19 + 66 * q^21 - 54 * q^23 + 138 * q^25 - 54 * q^27 + 104 * q^29 - 224 * q^31 - 66 * q^33 - 304 * q^35 - 68 * q^37 - 282 * q^39 - 300 * q^41 - 456 * q^43 - 54 * q^45 + 22 * q^47 - 74 * q^49 - 168 * q^51 - 110 * q^53 - 66 * q^55 + 228 * q^57 - 68 * q^59 - 54 * q^61 - 198 * q^63 - 652 * q^65 - 1560 * q^67 + 162 * q^69 + 274 * q^71 + 548 * q^73 - 414 * q^75 - 242 * q^77 - 922 * q^79 + 162 * q^81 - 1184 * q^83 - 2388 * q^85 - 312 * q^87 - 956 * q^89 - 664 * q^91 + 672 * q^93 + 2448 * q^95 - 672 * 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
 7.30074 −6.30074
0 −3.00000 0 −16.6015 0 2.60147 0 9.00000 0
1.2 0 −3.00000 0 10.6015 0 −24.6015 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.m 2
3.b odd 2 1 1584.4.a.bd 2
4.b odd 2 1 264.4.a.g 2
8.b even 2 1 2112.4.a.bk 2
8.d odd 2 1 2112.4.a.bf 2
12.b even 2 1 792.4.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.g 2 4.b odd 2 1
528.4.a.m 2 1.a even 1 1 trivial
792.4.a.j 2 12.b even 2 1
1584.4.a.bd 2 3.b odd 2 1
2112.4.a.bf 2 8.d odd 2 1
2112.4.a.bk 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(528))$$:

 $$T_{5}^{2} + 6T_{5} - 176$$ T5^2 + 6*T5 - 176 $$T_{7}^{2} + 22T_{7} - 64$$ T7^2 + 22*T7 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 6T - 176$$
$7$ $$T^{2} + 22T - 64$$
$11$ $$(T - 11)^{2}$$
$13$ $$T^{2} - 94T + 2024$$
$17$ $$T^{2} - 56T - 5876$$
$19$ $$T^{2} + 76T - 5216$$
$23$ $$T^{2} + 54T - 3896$$
$29$ $$T^{2} - 104T + 1964$$
$31$ $$T^{2} + 224T + 704$$
$37$ $$T^{2} + 68T - 46204$$
$41$ $$T^{2} + 300T + 19540$$
$43$ $$T^{2} + 456T + 49024$$
$47$ $$T^{2} - 22T - 134744$$
$53$ $$T^{2} + 110T - 223600$$
$59$ $$T^{2} + 68T - 88384$$
$61$ $$T^{2} + 54T + 544$$
$67$ $$T^{2} + 1560 T + 596560$$
$71$ $$T^{2} - 274T - 22856$$
$73$ $$T^{2} - 548T - 114364$$
$79$ $$T^{2} + 922T + 114656$$
$83$ $$T^{2} + 1184T - 7696$$
$89$ $$T^{2} + 956T - 730556$$
$97$ $$T^{2} + 672T - 793604$$