# Properties

 Label 528.2.bn.a Level $528$ Weight $2$ Character orbit 528.bn Analytic conductor $4.216$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,2,Mod(17,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 5, 9]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("528.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$528 = 2^{4} \cdot 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 528.bn (of order $$10$$, degree $$4$$, not minimal)

## 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: no Analytic conductor: $$4.21610122672$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: 8.0.185640625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81$$ x^8 - 3*x^7 + x^6 + x^5 + 4*x^4 + 3*x^3 + 9*x^2 - 81*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 66) Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{6} + \beta_{5} + \beta_1 - 2) q^{9}+O(q^{10})$$ q + b7 * q^3 + (-b3 - b2) * q^5 + (-b7 - b4 - b3 - b2) * q^7 + (b6 + b5 + b1 - 2) * q^9 $$q + \beta_{7} q^{3} + ( - \beta_{3} - \beta_{2}) q^{5} + ( - \beta_{7} - \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + (\beta_{6} + \beta_{5} + \beta_1 - 2) q^{9} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{11} + (2 \beta_{7} + 4 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 2) q^{13} + (2 \beta_{6} - \beta_{5} - 1) q^{15} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{17}+ \cdots + (2 \beta_{7} - 4 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + \cdots - 4) q^{99}+O(q^{100})$$ q + b7 * q^3 + (-b3 - b2) * q^5 + (-b7 - b4 - b3 - b2) * q^7 + (b6 + b5 + b1 - 2) * q^9 + (-b7 + b6 - 2*b4 - b3 + b1) * q^11 + (2*b7 + 4*b6 + 2*b5 + 2*b3 + 2*b1 + 2) * q^13 + (2*b6 - b5 - 1) * q^15 + (-2*b7 - 2*b6 - 2*b5 - 3*b4 - b3 - 2*b2 - 2) * q^17 + (2*b6 + 2*b5 + 2*b4 + 2*b3 + 3*b2 + b1 + 2) * q^19 + (2*b6 - b5 - b1 + 2) * q^21 + (2*b7 + 2*b6 + 2*b5 - 2*b4 - 2*b2 + 2*b1 + 2) * q^23 + (-2*b7 - 3*b6 - b5 + b4 - b3 + 2*b2 - b1 - 3) * q^25 + (-2*b7 + 2*b6 - b5 - 3*b2 - b1 + 2) * q^27 + (2*b7 + 6*b6 + b5 - 2*b4 + 2*b1 + 5) * q^29 + (2*b7 - 2*b5 + 2*b4 + b3 - 3*b2 - b1 - 1) * q^31 + (b7 + 6*b6 - 3*b4 + b3 - 3*b2 - b1 + 6) * q^33 + (-b7 + b5 - 3*b4 - 2*b2 + 3) * q^35 + (-2*b7 - 2*b6 - 2*b5 + 2*b4 - 2*b3 - 2*b2 - 4*b1) * q^37 + (-6*b2 - 2*b1) * q^39 + (-2*b7 - b6 + b5 - b4 - 3*b3 - b2 - b1) * q^41 + (2*b7 + 3*b6 + b5 + 2*b4 + 3*b2 + b1) * q^43 + (-b7 - 3*b4 + 2*b3) * q^45 + (4*b6 + 4*b4 + 2*b1 - 2) * q^47 + (5*b6 + 2*b5 - 2*b4 + b3 - b2 + 2*b1) * q^49 + (-3*b7 + 2*b6 - b5 - 6*b4 - 3*b3 - 3*b1 + 5) * q^51 + (2*b7 - 4*b6 + 2*b5 + 6*b4 + 3*b3 + 5*b2 + 3*b1 - 1) * q^53 + (2*b5 - 4*b4 + b3 - b2 + 4) * q^55 + (3*b7 - 3*b6 + 3*b4 + 2*b3 - 3*b2 + 3) * q^57 + (-b7 - 7*b6 - b5 + 3*b4 - b3 + 7*b2 - 2) * q^59 + (2*b7 + 2*b4 - 2*b3 - 2*b2 + 2*b1 - 2) * q^61 + (2*b7 - 2*b6 + b5 + 3*b3 + 3*b2 + b1 - 2) * q^63 - 4 * q^65 + (-3*b7 - b6 - 4*b5 - 3*b4 - 6*b3 - 5*b2 - 2*b1 - 1) * q^67 + (6*b6 - 2*b3 - 6*b2 - 2*b1) * q^69 + (-4*b6 + 2*b4 + 6*b2 - 8) * q^71 + (-b7 - 3*b6 - 3*b5 - b3 + b2 - 2) * q^73 + (b3 + 3*b2 + 2*b1 + 3) * q^75 + (b7 - 4*b6 + 2*b5 - b4 + b3 + b2 + 2*b1 - 2) * q^77 + (b7 - 4*b6 + b5 + 3*b4 + 3*b3 + 5*b2 + 3*b1 - 4) * q^79 + (-b7 - 7*b6 - 4*b5 + 3*b2 - 4*b1 - 1) * q^81 + (-b7 - 7*b6 - 4*b5 + 2*b4 - 2*b3 + b2 - 3*b1 - 1) * q^83 + (-2*b6 + 4*b5 - 2*b4 + 4*b3 + 6*b2 + 2*b1 + 4) * q^85 + (5*b7 + 8*b6 + 2*b5 - 3*b4 + 4*b3 - 6*b2 + 2) * q^87 + (-3*b6 + 3*b5 + 12*b4 + 9*b2 + 3*b1 - 6) * q^89 + (4*b2 - 4) * q^91 + (-5*b7 - 8*b6 - 2*b5 - 3*b4 - 3*b3 + 3*b2 - b1 - 11) * q^93 + (2*b7 - 4*b6 + 4*b4 - 2*b3 - 2*b2 - 4) * q^95 + (3*b7 - 3*b5 + 2*b3 - 2*b1 + 2) * q^97 + (2*b7 - 4*b6 + 2*b5 + 3*b4 + 3*b3 + 3*b2 - 2*b1 - 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 2 q^{3} - 18 q^{9}+O(q^{10})$$ 8 * q - 2 * q^3 - 18 * q^9 $$8 q - 2 q^{3} - 18 q^{9} + q^{11} - 9 q^{15} - 10 q^{17} + 15 q^{19} + 12 q^{21} - 6 q^{25} + 10 q^{27} + 23 q^{29} - 13 q^{31} + 17 q^{33} + 13 q^{35} + 6 q^{37} - 18 q^{39} + 2 q^{41} - 8 q^{45} - 10 q^{47} - 18 q^{49} + 30 q^{51} + 15 q^{53} + 14 q^{55} + 20 q^{57} + 25 q^{59} - 10 q^{61} - 16 q^{63} - 32 q^{65} + 2 q^{67} - 26 q^{69} - 40 q^{71} + 5 q^{73} + 34 q^{75} - 12 q^{77} - 10 q^{79} + 14 q^{81} + 21 q^{83} + 30 q^{85} - 42 q^{87} - 24 q^{91} - 53 q^{93} - 20 q^{95} + 9 q^{97} - 34 q^{99}+O(q^{100})$$ 8 * q - 2 * q^3 - 18 * q^9 + q^11 - 9 * q^15 - 10 * q^17 + 15 * q^19 + 12 * q^21 - 6 * q^25 + 10 * q^27 + 23 * q^29 - 13 * q^31 + 17 * q^33 + 13 * q^35 + 6 * q^37 - 18 * q^39 + 2 * q^41 - 8 * q^45 - 10 * q^47 - 18 * q^49 + 30 * q^51 + 15 * q^53 + 14 * q^55 + 20 * q^57 + 25 * q^59 - 10 * q^61 - 16 * q^63 - 32 * q^65 + 2 * q^67 - 26 * q^69 - 40 * q^71 + 5 * q^73 + 34 * q^75 - 12 * q^77 - 10 * q^79 + 14 * q^81 + 21 * q^83 + 30 * q^85 - 42 * q^87 - 24 * q^91 - 53 * q^93 - 20 * q^95 + 9 * q^97 - 34 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + x^{6} + x^{5} + 4x^{4} + 3x^{3} + 9x^{2} - 81x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{7} + \nu^{5} + 4\nu^{4} + 16\nu^{3} + 51\nu^{2} - 54\nu - 27 ) / 216$$ (v^7 + v^5 + 4*v^4 + 16*v^3 + 51*v^2 - 54*v - 27) / 216 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} - \nu^{5} - 4\nu^{4} - 16\nu^{3} + 21\nu^{2} - 18\nu + 27 ) / 72$$ (-v^7 - v^5 - 4*v^4 - 16*v^3 + 21*v^2 - 18*v + 27) / 72 $$\beta_{4}$$ $$=$$ $$( -11\nu^{7} + 6\nu^{6} + 7\nu^{5} + 16\nu^{4} + 28\nu^{3} - 69\nu^{2} - 216\nu + 351 ) / 216$$ (-11*v^7 + 6*v^6 + 7*v^5 + 16*v^4 + 28*v^3 - 69*v^2 - 216*v + 351) / 216 $$\beta_{5}$$ $$=$$ $$( \nu^{7} - 3\nu^{6} + \nu^{5} + \nu^{4} + 4\nu^{3} + 3\nu^{2} + 9\nu - 81 ) / 27$$ (v^7 - 3*v^6 + v^5 + v^4 + 4*v^3 + 3*v^2 + 9*v - 81) / 27 $$\beta_{6}$$ $$=$$ $$( 19\nu^{7} - 30\nu^{6} - 35\nu^{5} - 8\nu^{4} + 76\nu^{3} + 165\nu^{2} + 360\nu - 1107 ) / 216$$ (19*v^7 - 30*v^6 - 35*v^5 - 8*v^4 + 76*v^3 + 165*v^2 + 360*v - 1107) / 216 $$\beta_{7}$$ $$=$$ $$( 3\nu^{7} - 2\nu^{6} - 3\nu^{5} - 8\nu^{4} + 4\nu^{3} + 13\nu^{2} + 60\nu - 99 ) / 24$$ (3*v^7 - 2*v^6 - 3*v^5 - 8*v^4 + 4*v^3 + 13*v^2 + 60*v - 99) / 24
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 3\beta_{2} + \beta_1$$ b3 + 3*b2 + b1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3\beta_{4} - \beta_{3} + 3\beta_{2} + \beta_1$$ b7 + 3*b4 - b3 + 3*b2 + b1 $$\nu^{4}$$ $$=$$ $$-4\beta_{7} + 2\beta_{6} - \beta_{5} - 6\beta_{4} - 4\beta_{3} + 2$$ -4*b7 + 2*b6 - b5 - 6*b4 - 4*b3 + 2 $$\nu^{5}$$ $$=$$ $$2\beta_{7} - 6\beta_{6} + 6\beta_{5} + 12\beta_{2} + 6\beta _1 - 3$$ 2*b7 - 6*b6 + 6*b5 + 12*b2 + 6*b1 - 3 $$\nu^{6}$$ $$=$$ $$-2\beta_{6} - 8\beta_{5} - 6\beta_{4} - 8\beta_{3} + 12\beta_{2} + \beta _1 - 20$$ -2*b6 - 8*b5 - 6*b4 - 8*b3 + 12*b2 + b1 - 20 $$\nu^{7}$$ $$=$$ $$-2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 24\beta_{4} - 19\beta_{3} + 3\beta_{2} - 19\beta _1 + 22$$ -2*b7 - 2*b6 - 2*b5 - 24*b4 - 19*b3 + 3*b2 - 19*b1 + 22

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$$.

 $$n$$ $$133$$ $$145$$ $$353$$ $$463$$ $$\chi(n)$$ $$1$$ $$\beta_{4}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.245684 + 1.71454i 1.55470 − 0.763481i −1.52536 − 0.820539i 1.71634 + 0.232753i 1.71634 − 0.232753i −1.52536 + 0.820539i 1.55470 + 0.763481i −0.245684 − 1.71454i
0 −0.809017 1.53150i 0 2.01556 + 0.654895i 0 2.01556 + 2.77418i 0 −1.69098 + 2.47802i 0
17.2 0 −0.809017 + 1.53150i 0 −0.897526 0.291624i 0 −0.897526 1.23534i 0 −1.69098 2.47802i 0
161.1 0 0.309017 1.70426i 0 −1.56076 2.14820i 0 −1.56076 + 0.507121i 0 −2.80902 1.05329i 0
161.2 0 0.309017 + 1.70426i 0 0.442723 + 0.609356i 0 0.442723 0.143849i 0 −2.80902 + 1.05329i 0
305.1 0 0.309017 1.70426i 0 0.442723 0.609356i 0 0.442723 + 0.143849i 0 −2.80902 1.05329i 0
305.2 0 0.309017 + 1.70426i 0 −1.56076 + 2.14820i 0 −1.56076 0.507121i 0 −2.80902 + 1.05329i 0
497.1 0 −0.809017 1.53150i 0 −0.897526 + 0.291624i 0 −0.897526 + 1.23534i 0 −1.69098 + 2.47802i 0
497.2 0 −0.809017 + 1.53150i 0 2.01556 0.654895i 0 2.01556 2.77418i 0 −1.69098 2.47802i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 17.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.f even 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.bn.a 8
3.b odd 2 1 528.2.bn.b 8
4.b odd 2 1 66.2.h.b yes 8
11.d odd 10 1 528.2.bn.b 8
12.b even 2 1 66.2.h.a 8
33.f even 10 1 inner 528.2.bn.a 8
44.c even 2 1 726.2.h.d 8
44.g even 10 1 66.2.h.a 8
44.g even 10 1 726.2.b.e 8
44.g even 10 1 726.2.h.a 8
44.g even 10 1 726.2.h.c 8
44.h odd 10 1 726.2.b.c 8
44.h odd 10 1 726.2.h.f 8
44.h odd 10 1 726.2.h.h 8
44.h odd 10 1 726.2.h.j 8
132.d odd 2 1 726.2.h.j 8
132.n odd 10 1 66.2.h.b yes 8
132.n odd 10 1 726.2.b.c 8
132.n odd 10 1 726.2.h.f 8
132.n odd 10 1 726.2.h.h 8
132.o even 10 1 726.2.b.e 8
132.o even 10 1 726.2.h.a 8
132.o even 10 1 726.2.h.c 8
132.o even 10 1 726.2.h.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
66.2.h.a 8 12.b even 2 1
66.2.h.a 8 44.g even 10 1
66.2.h.b yes 8 4.b odd 2 1
66.2.h.b yes 8 132.n odd 10 1
528.2.bn.a 8 1.a even 1 1 trivial
528.2.bn.a 8 33.f even 10 1 inner
528.2.bn.b 8 3.b odd 2 1
528.2.bn.b 8 11.d odd 10 1
726.2.b.c 8 44.h odd 10 1
726.2.b.c 8 132.n odd 10 1
726.2.b.e 8 44.g even 10 1
726.2.b.e 8 132.o even 10 1
726.2.h.a 8 44.g even 10 1
726.2.h.a 8 132.o even 10 1
726.2.h.c 8 44.g even 10 1
726.2.h.c 8 132.o even 10 1
726.2.h.d 8 44.c even 2 1
726.2.h.d 8 132.o even 10 1
726.2.h.f 8 44.h odd 10 1
726.2.h.f 8 132.n odd 10 1
726.2.h.h 8 44.h odd 10 1
726.2.h.h 8 132.n odd 10 1
726.2.h.j 8 44.h odd 10 1
726.2.h.j 8 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 2T_{5}^{6} - 15T_{5}^{5} + 19T_{5}^{4} + 30T_{5}^{3} - 8T_{5}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(528, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + T^{3} + 5 T^{2} + 3 T + 9)^{2}$$
$5$ $$T^{8} - 2 T^{6} - 15 T^{5} + 19 T^{4} + \cdots + 16$$
$7$ $$T^{8} + 2 T^{6} + 25 T^{5} + 59 T^{4} + \cdots + 16$$
$11$ $$T^{8} - T^{7} + 15 T^{6} + 11 T^{5} + \cdots + 14641$$
$13$ $$T^{8} - 8 T^{6} - 120 T^{5} + \cdots + 4096$$
$17$ $$T^{8} + 10 T^{7} + 60 T^{6} + \cdots + 144400$$
$19$ $$T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 400$$
$23$ $$T^{8} + 88 T^{6} + 2544 T^{4} + \cdots + 30976$$
$29$ $$T^{8} - 23 T^{7} + 328 T^{6} + \cdots + 15376$$
$31$ $$T^{8} + 13 T^{7} + 168 T^{6} + \cdots + 1175056$$
$37$ $$T^{8} - 6 T^{7} + 32 T^{6} + \cdots + 30976$$
$41$ $$T^{8} - 2 T^{7} - 32 T^{6} + 91 T^{5} + \cdots + 16$$
$43$ $$T^{8} + 113 T^{6} + 4549 T^{4} + \cdots + 430336$$
$47$ $$T^{8} + 10 T^{7} - 8 T^{6} + \cdots + 1048576$$
$53$ $$T^{8} - 15 T^{7} + 100 T^{6} + \cdots + 2310400$$
$59$ $$T^{8} - 25 T^{7} + 182 T^{6} + \cdots + 844561$$
$61$ $$T^{8} + 10 T^{7} + 112 T^{6} + \cdots + 4096$$
$67$ $$(T^{4} - T^{3} - 149 T^{2} + 284 T + 3076)^{2}$$
$71$ $$(T^{4} + 20 T^{3} + 120 T^{2} + 120 T + 80)^{2}$$
$73$ $$T^{8} - 5 T^{7} - 50 T^{6} + \cdots + 24025$$
$79$ $$T^{8} + 10 T^{7} - 48 T^{6} + \cdots + 55696$$
$83$ $$T^{8} - 21 T^{7} + 252 T^{6} + \cdots + 737881$$
$89$ $$T^{8} + 513 T^{6} + \cdots + 12702096$$
$97$ $$T^{8} - 9 T^{7} + 282 T^{6} + \cdots + 2070721$$